Viscoplasticity - Wikipedia

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<span>Phenomenology</span> </div> </a> <button aria-controls="toc-Phenomenology-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Phenomenology subsection</span> </button> <ul id="toc-Phenomenology-sublist" class="vector-toc-list"> <li id="toc-Strain_hardening_test" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Strain_hardening_test"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Strain hardening test</span> </div> </a> <ul id="toc-Strain_hardening_test-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Creep_test" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Creep_test"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Creep test</span> </div> </a> <ul id="toc-Creep_test-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relaxation_test" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relaxation_test"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Relaxation test</span> </div> </a> <ul id="toc-Relaxation_test-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Rheological_models_of_viscoplasticity" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rheological_models_of_viscoplasticity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Rheological models of viscoplasticity</span> </div> </a> <button aria-controls="toc-Rheological_models_of_viscoplasticity-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Rheological models of viscoplasticity subsection</span> </button> <ul id="toc-Rheological_models_of_viscoplasticity-sublist" class="vector-toc-list"> <li id="toc-Perfectly_viscoplastic_solid_(Norton-Hoff_model)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Perfectly_viscoplastic_solid_(Norton-Hoff_model)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Perfectly viscoplastic solid (Norton-Hoff model)</span> </div> </a> <ul id="toc-Perfectly_viscoplastic_solid_(Norton-Hoff_model)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elastic_perfectly_viscoplastic_solid_(Bingham–Norton_model)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elastic_perfectly_viscoplastic_solid_(Bingham–Norton_model)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Elastic perfectly viscoplastic solid (Bingham–Norton model)</span> </div> </a> <ul id="toc-Elastic_perfectly_viscoplastic_solid_(Bingham–Norton_model)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elastoviscoplastic_hardening_solid" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elastoviscoplastic_hardening_solid"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Elastoviscoplastic hardening solid</span> </div> </a> <ul id="toc-Elastoviscoplastic_hardening_solid-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Strain-rate_dependent_plasticity_models" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Strain-rate_dependent_plasticity_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Strain-rate dependent plasticity models</span> </div> </a> <button aria-controls="toc-Strain-rate_dependent_plasticity_models-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon 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class="vector-toc-list"> </ul> </li> <li id="toc-Zerilli–Armstrong_flow_stress_model" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Zerilli–Armstrong_flow_stress_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.3</span> <span>Zerilli–Armstrong flow stress model</span> </div> </a> <ul id="toc-Zerilli–Armstrong_flow_stress_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mechanical_threshold_stress_flow_stress_model" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Mechanical_threshold_stress_flow_stress_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.4</span> <span>Mechanical threshold stress flow stress model</span> </div> </a> <ul id="toc-Mechanical_threshold_stress_flow_stress_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Preston–Tonks–Wallace_flow_stress_model" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Preston–Tonks–Wallace_flow_stress_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.5</span> <span>Preston–Tonks–Wallace flow stress model</span> </div> </a> <ul id="toc-Preston–Tonks–Wallace_flow_stress_model-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> 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searchaux" style="display:none">Theory in continuum mechanics</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Viscoplastic_elements.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Viscoplastic_elements.svg/240px-Viscoplastic_elements.svg.png" decoding="async" width="240" height="288" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Viscoplastic_elements.svg/360px-Viscoplastic_elements.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/Viscoplastic_elements.svg/480px-Viscoplastic_elements.svg.png 2x" data-file-width="509" data-file-height="611" /></a><figcaption>Figure 1. Elements used in one-dimensional models of viscoplastic materials.</figcaption></figure> <p><b>Viscoplasticity</b> is a theory in <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuum mechanics</a> that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the <a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">deformation</a> of the material depends on the rate at which <a href="/wiki/Structural_load" title="Structural load">loads</a> are applied.<sup id="cite_ref-Perzyna_1-0" class="reference"><a href="#cite_note-Perzyna-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The inelastic behavior that is the subject of viscoplasticity is <a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">plastic deformation</a> which means that the material undergoes unrecoverable deformations when a load level is reached. Rate-dependent plasticity is important for transient plasticity calculations. The main difference between rate-independent plastic and viscoplastic material models is that the latter exhibit not only permanent deformations after the application of loads but continue to undergo a <a href="/wiki/Creep_(deformation)" title="Creep (deformation)">creep</a> flow as a function of time under the influence of the applied load. </p><p>The elastic response of viscoplastic materials can be represented in one-dimension by <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hookean</a> <a href="/wiki/Spring_(device)" title="Spring (device)">spring</a> elements. Rate-dependence can be represented by nonlinear <a href="/wiki/Dashpot" title="Dashpot">dashpot</a> elements in a manner similar to <a href="/wiki/Viscoelasticity" title="Viscoelasticity">viscoelasticity</a>. Plasticity can be accounted for by adding sliding <a href="/wiki/Friction" title="Friction">frictional</a> elements as shown in Figure 1.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In the figure <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> is the <a href="/wiki/Young%27s_modulus" title="Young's modulus">modulus of elasticity</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> is the <a href="/wiki/Viscosity" title="Viscosity">viscosity</a> parameter and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> is a <a href="/wiki/Power-law" class="mw-redirect" title="Power-law">power-law</a> type parameter that represents non-linear dashpot <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\sigma (\mathrm {d} \varepsilon /\mathrm {d} t)=\sigma =\lambda (\mathrm {d} \varepsilon /\mathrm {d} t)^{1/N}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>σ</mi> <mo>=</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\sigma (\mathrm {d} \varepsilon /\mathrm {d} t)=\sigma =\lambda (\mathrm {d} \varepsilon /\mathrm {d} t)^{1/N}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94b715a7bee445a73d1ea15d07da8789c98212f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.8ex; height:3.343ex;" alt="{\displaystyle [\sigma (\mathrm {d} \varepsilon /\mathrm {d} t)=\sigma =\lambda (\mathrm {d} \varepsilon /\mathrm {d} t)^{1/N}]}"></span>. The sliding element can have a <a href="/wiki/Yield_(engineering)" title="Yield (engineering)">yield stress</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b22d96fceb022e70169a37383278199a26b3534a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.377ex; height:2.343ex;" alt="{\displaystyle \sigma _{y}}"></span>) that is <a href="/wiki/Strain_rate" title="Strain rate">strain rate</a> dependent, or even constant, as shown in Figure 1c. </p><p>Viscoplasticity is usually modeled in three-dimensions using <i>overstress models</i> of the Perzyna or Duvaut-Lions types.<sup id="cite_ref-Simo_3-0" class="reference"><a href="#cite_note-Simo-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In these models, the stress is allowed to increase beyond the rate-independent <a href="/wiki/Yield_surface" title="Yield surface">yield surface</a> upon application of a load and then allowed to relax back to the yield surface over time. The yield surface is usually assumed not to be rate-dependent in such models. An alternative approach is to add a <a href="/wiki/Strain_rate" title="Strain rate">strain rate</a> dependence to the yield stress and use the techniques of rate independent plasticity to calculate the response of a material.<sup id="cite_ref-Batra_4-0" class="reference"><a href="#cite_note-Batra-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>For <a href="/wiki/Metal" title="Metal">metals</a> and <a href="/wiki/Alloy" title="Alloy">alloys</a>, viscoplasticity is the <a href="/wiki/Macroscopic" class="mw-redirect" title="Macroscopic">macroscopic</a> behavior caused by a mechanism linked to the movement of <a href="/wiki/Dislocation" title="Dislocation">dislocations</a> in <a href="/wiki/Crystallite" title="Crystallite">grains</a>, with superposed effects of inter-crystalline gliding. The mechanism usually becomes dominant at temperatures greater than approximately one third of the absolute melting temperature. However, certain alloys exhibit viscoplasticity at room temperature (300 K). For <a href="/wiki/Polymer" title="Polymer">polymers</a>, <a href="/wiki/Wood" title="Wood">wood</a>, and <a href="/wiki/Bitumen" title="Bitumen">bitumen</a>, the theory of viscoplasticity is required to describe behavior beyond the limit of elasticity or <a href="/wiki/Viscoelasticity" title="Viscoelasticity">viscoelasticity</a>. </p><p>In general, viscoplasticity theories are useful in areas such as: </p> <ul><li>the calculation of permanent deformations,</li> <li>the prediction of the plastic collapse of structures,</li> <li>the investigation of stability,</li> <li>crash simulations,</li> <li>systems exposed to high temperatures such as turbines in engines, e.g. a power plant,</li> <li>dynamic problems and systems exposed to high strain rates.</li></ul> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Research on plasticity theories started in 1864 with the work of <a href="/wiki/Henri_Tresca" title="Henri Tresca">Henri Tresca</a>,<sup id="cite_ref-Tresca_5-0" class="reference"><a href="#cite_note-Tresca-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Saint_Venant" class="mw-redirect" title="Saint Venant">Saint Venant</a> (1870) and <a href="/wiki/Maurice_L%C3%A9vy" title="Maurice Lévy">Levy</a> (1871)<sup id="cite_ref-Levy_6-0" class="reference"><a href="#cite_note-Levy-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> on the <a href="/wiki/Yield_surface#Tresca_yield_surface" title="Yield surface">maximum shear criterion</a>.<sup id="cite_ref-Kojic_7-0" class="reference"><a href="#cite_note-Kojic-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> An improved plasticity model was presented in 1913 by <a href="/wiki/Von_Mises" title="Von Mises">Von Mises</a><sup id="cite_ref-Mises_8-0" class="reference"><a href="#cite_note-Mises-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> which is now referred to as the <a href="/wiki/Von_Mises_yield_criterion" title="Von Mises yield criterion">von Mises yield criterion</a>. In viscoplasticity, the development of a mathematical model heads back to 1910 with the representation of <a href="/wiki/Creep_(deformation)" title="Creep (deformation)">primary creep</a> by Andrade's law.<sup id="cite_ref-Betten_9-0" class="reference"><a href="#cite_note-Betten-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> In 1929, Norton<sup id="cite_ref-Norton_10-0" class="reference"><a href="#cite_note-Norton-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> developed a one-dimensional dashpot model which linked the rate of <a href="/wiki/Creep_(deformation)" title="Creep (deformation)">secondary creep</a> to the stress. In 1934, Odqvist<sup id="cite_ref-Odqvist_11-0" class="reference"><a href="#cite_note-Odqvist-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> generalized Norton's law to the multi-axial case. </p><p>Concepts such as the normality of plastic flow to the yield surface and flow rules for plasticity were introduced by <a href="/wiki/Prandtl" class="mw-redirect" title="Prandtl">Prandtl</a> (1924)<sup id="cite_ref-Prandtl_12-0" class="reference"><a href="#cite_note-Prandtl-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (August 2024)">full citation needed</span></a></i>]</sup> and Reuss (1930).<sup id="cite_ref-Reuss_13-0" class="reference"><a href="#cite_note-Reuss-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> In 1932, Hohenemser and <a href="/wiki/William_Prager" title="William Prager">Prager</a><sup id="cite_ref-Hohen_14-0" class="reference"><a href="#cite_note-Hohen-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> proposed the first model for slow viscoplastic flow. This model provided a relation between the <a href="/wiki/Deviatoric_stress" class="mw-redirect" title="Deviatoric stress">deviatoric stress</a> and the <a href="/wiki/Strain_rate" title="Strain rate">strain rate</a> for an incompressible <a href="/wiki/Bingham_plastic" title="Bingham plastic">Bingham solid</a><sup id="cite_ref-Bingham_15-0" class="reference"><a href="#cite_note-Bingham-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> However, the application of these theories did not begin before 1950, where limit theorems were discovered. </p><p>In 1960, the first <a href="/wiki/IUTAM" class="mw-redirect" title="IUTAM">IUTAM</a> Symposium "Creep in Structures" organized by Hoff<sup id="cite_ref-Hoff_16-0" class="reference"><a href="#cite_note-Hoff-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> provided a major development in viscoplasticity with the works of Hoff, Rabotnov, Perzyna, Hult, and Lemaitre for the <a href="/wiki/Work_hardening" title="Work hardening">isotropic hardening</a> laws, and those of Kratochvil, Malinini and Khadjinsky, Ponter and Leckie, and Chaboche for the <a href="/wiki/Work_hardening" title="Work hardening">kinematic hardening</a> laws. Perzyna, in 1963, introduced a viscosity coefficient that is temperature and time dependent.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The formulated models were supported by the <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> of <a href="/wiki/Irreversible_process" title="Irreversible process">irreversible processes</a> and the <a href="/wiki/Phenomenological_model" title="Phenomenological model">phenomenological</a> standpoint. The ideas presented in these works have been the basis for most subsequent research into rate-dependent plasticity. </p> <div class="mw-heading mw-heading2"><h2 id="Phenomenology">Phenomenology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=2" title="Edit section: Phenomenology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a qualitative analysis, several characteristic tests are performed to describe the phenomenology of viscoplastic materials. Some examples of these tests are <sup id="cite_ref-Betten_9-1" class="reference"><a href="#cite_note-Betten-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <ol><li>hardening tests at constant stress or strain rate,</li> <li>creep tests at constant force, and</li> <li>stress relaxation at constant elongation.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Strain_hardening_test">Strain hardening test</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=3" title="Edit section: Strain hardening test"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Visco79.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Visco79.svg/310px-Visco79.svg.png" decoding="async" width="310" height="272" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Visco79.svg/465px-Visco79.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/Visco79.svg/620px-Visco79.svg.png 2x" data-file-width="556" data-file-height="488" /></a><figcaption>Figure 2. Stress–strain response of a viscoplastic material at different strain rates. The dotted lines show the response if the strain-rate is held constant. The blue line shows the response when the strain rate is changed suddenly.</figcaption></figure> <p>One consequence of <a href="/wiki/Yield_(engineering)" title="Yield (engineering)">yielding</a> is that as plastic deformation proceeds, an increase in <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">stress</a> is required to produce additional <a href="/wiki/Finite_strain_theory" title="Finite strain theory">strain</a>. This phenomenon is known as <a href="/wiki/Work_hardening" title="Work hardening">Strain/Work hardening</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> For a viscoplastic material the hardening curves are not significantly different from those of rate-independent plastic material. Nevertheless, three essential differences can be observed. </p> <ol><li>At the same strain, the higher the rate of strain the higher the stress</li> <li>A change in the rate of strain during the test results in an immediate change in the stress–strain curve.</li> <li>The concept of a <a href="/wiki/Yield_(engineering)" title="Yield (engineering)">plastic yield limit</a> is no longer strictly applicable.</li></ol> <p>The hypothesis of partitioning the strains by decoupling the elastic and plastic parts is still applicable where the strains are small,<sup id="cite_ref-Simo_3-1" class="reference"><a href="#cite_note-Simo-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> i.e., </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\varepsilon }}={\boldsymbol {\varepsilon }}_{\mathrm {e} }+{\boldsymbol {\varepsilon }}_{\mathrm {vp} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}={\boldsymbol {\varepsilon }}_{\mathrm {e} }+{\boldsymbol {\varepsilon }}_{\mathrm {vp} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dddf4e270e5a3b75a5d6ff88d094eca21c83dcc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.605ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}={\boldsymbol {\varepsilon }}_{\mathrm {e} }+{\boldsymbol {\varepsilon }}_{\mathrm {vp} }}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\varepsilon }}_{\mathrm {e} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}_{\mathrm {e} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/763d15605c783f83ef7a4e8ef8ae4aa9a75b00a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.192ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}_{\mathrm {e} }}"></span> is the elastic strain and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\varepsilon }}_{\mathrm {vp} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}_{\mathrm {vp} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf25957bbc18ca81ebb29f1067f5f657e285998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.244ex; height:2.343ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}_{\mathrm {vp} }}"></span> is the viscoplastic strain. To obtain the stress–strain behavior shown in blue in the figure, the material is initially loaded at a strain rate of 0.1/s. The strain rate is then instantaneously raised to 100/s and held constant at that value for some time. At the end of that time period the strain rate is dropped instantaneously back to 0.1/s and the cycle is continued for increasing values of strain. There is clearly a lag between the strain-rate change and the stress response. This lag is modeled quite accurately by overstress models (such as the <a class="mw-selflink-fragment" href="#Perzyna_formulation">Perzyna model</a>) but not by models of rate-independent plasticity that have a rate-dependent yield stress. </p> <div class="mw-heading mw-heading3"><h3 id="Creep_test">Creep test</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=4" title="Edit section: Creep test"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Creep_test.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Creep_test.JPG/310px-Creep_test.JPG" decoding="async" width="310" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/8d/Creep_test.JPG 1.5x" data-file-width="389" data-file-height="258" /></a><figcaption>Figure 3a. Creep test</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:3StageCreep.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/3StageCreep.svg/310px-3StageCreep.svg.png" decoding="async" width="310" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/3StageCreep.svg/465px-3StageCreep.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/3StageCreep.svg/620px-3StageCreep.svg.png 2x" data-file-width="357" data-file-height="211" /></a><figcaption>Figure 3b. Strain as a function of time in a creep test</figcaption></figure> <p><a href="/wiki/Creep_(deformation)" title="Creep (deformation)">Creep</a> is the tendency of a solid material to slowly move or deform permanently under constant stresses. Creep tests measure the strain response due to a constant stress as shown in Figure 3. The classical creep curve represents the evolution of strain as a function of time in a material subjected to uniaxial stress at a constant temperature. The creep test, for instance, is performed by applying a constant force/stress and analyzing the strain response of the system. In general, as shown in Figure 3b this curve usually shows three phases or periods of behavior:<sup id="cite_ref-Betten_9-2" class="reference"><a href="#cite_note-Betten-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <ol><li>A <b>primary creep</b> stage, also known as transient creep, is the starting stage during which hardening of the material leads to a decrease in the rate of flow which is initially very high. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0\leq {\boldsymbol {\varepsilon }}\leq {\boldsymbol {\varepsilon }}_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mo>≤</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0\leq {\boldsymbol {\varepsilon }}\leq {\boldsymbol {\varepsilon }}_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10ddb2407dd6a11643cb87b7d96173db6712321c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.683ex; height:2.843ex;" alt="{\displaystyle (0\leq {\boldsymbol {\varepsilon }}\leq {\boldsymbol {\varepsilon }}_{1})}"></span>.</li> <li>The <b>secondary creep</b> stage, also known as the steady state, is where the strain rate is constant. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\boldsymbol {\varepsilon }}_{1}\leq {\boldsymbol {\varepsilon }}\leq {\boldsymbol {\varepsilon }}_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mo>≤</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\boldsymbol {\varepsilon }}_{1}\leq {\boldsymbol {\varepsilon }}\leq {\boldsymbol {\varepsilon }}_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78742bc508359b1c4dbf6f7433b562c7b98d8c95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.804ex; height:2.843ex;" alt="{\displaystyle ({\boldsymbol {\varepsilon }}_{1}\leq {\boldsymbol {\varepsilon }}\leq {\boldsymbol {\varepsilon }}_{2})}"></span>.</li> <li>A <b>tertiary creep</b> phase in which there is an increase in the strain rate up to the fracture strain. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\boldsymbol {\varepsilon }}_{2}\leq {\boldsymbol {\varepsilon }}\leq {\boldsymbol {\varepsilon }}_{R})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mo>≤</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\boldsymbol {\varepsilon }}_{2}\leq {\boldsymbol {\varepsilon }}\leq {\boldsymbol {\varepsilon }}_{R})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52feb6d7349439de29ff5e3ce94fc9e4d725377c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.229ex; height:2.843ex;" alt="{\displaystyle ({\boldsymbol {\varepsilon }}_{2}\leq {\boldsymbol {\varepsilon }}\leq {\boldsymbol {\varepsilon }}_{R})}"></span>.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Relaxation_test">Relaxation test</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=5" title="Edit section: Relaxation test"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Relaxation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Relaxation.svg/220px-Relaxation.svg.png" decoding="async" width="220" height="310" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Relaxation.svg/330px-Relaxation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/09/Relaxation.svg/440px-Relaxation.svg.png 2x" data-file-width="372" data-file-height="524" /></a><figcaption>Figure 4. a) Applied strain in a relaxation test and b) induced stress as functions of time over a short period for a viscoplastic material.</figcaption></figure> <p>As shown in Figure 4, the relaxation test<sup id="cite_ref-Francois_19-0" class="reference"><a href="#cite_note-Francois-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> is defined as the stress response due to a constant strain for a period of time. In viscoplastic materials, relaxation tests demonstrate the stress relaxation in uniaxial loading at a constant strain. In fact, these tests characterize the viscosity and can be used to determine the relation which exists between the stress and the rate of viscoplastic strain. The decomposition of strain rate is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}}{\mathrm {d} t}}={\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}_{\mathrm {e} }}{\mathrm {d} t}}+{\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}_{\mathrm {vp} }}{\mathrm {d} t}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}}{\mathrm {d} t}}={\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}_{\mathrm {e} }}{\mathrm {d} t}}+{\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}_{\mathrm {vp} }}{\mathrm {d} t}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a4841613604b8624f42b9417cb1d7224e2a7b81" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.218ex; height:7.176ex;" alt="{\displaystyle {\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}}{\mathrm {d} t}}={\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}_{\mathrm {e} }}{\mathrm {d} t}}+{\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}_{\mathrm {vp} }}{\mathrm {d} t}}~.}"></span> </p><p>The elastic part of the strain rate is given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}_{\mathrm {e} }}{\mathrm {d} t}}={\mathsf {E}}^{-1}~{\cfrac {\mathrm {d} {\boldsymbol {\sigma }}}{\mathrm {d} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}_{\mathrm {e} }}{\mathrm {d} t}}={\mathsf {E}}^{-1}~{\cfrac {\mathrm {d} {\boldsymbol {\sigma }}}{\mathrm {d} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb4f99a505734da5c652dea5f14ab8d0e0cdad42" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.443ex; height:7.176ex;" alt="{\displaystyle {\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}_{\mathrm {e} }}{\mathrm {d} t}}={\mathsf {E}}^{-1}~{\cfrac {\mathrm {d} {\boldsymbol {\sigma }}}{\mathrm {d} t}}}"></span> </p><p>For the flat region of the strain–time curve, the total strain rate is zero. Hence we have, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}_{\mathrm {vp} }}{\mathrm {d} t}}=-{\mathsf {E}}^{-1}~{\cfrac {\mathrm {d} {\boldsymbol {\sigma }}}{\mathrm {d} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}_{\mathrm {vp} }}{\mathrm {d} t}}=-{\mathsf {E}}^{-1}~{\cfrac {\mathrm {d} {\boldsymbol {\sigma }}}{\mathrm {d} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f68913efaac6ec8ab62aa419d9183e277508c233" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.304ex; height:7.176ex;" alt="{\displaystyle {\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}_{\mathrm {vp} }}{\mathrm {d} t}}=-{\mathsf {E}}^{-1}~{\cfrac {\mathrm {d} {\boldsymbol {\sigma }}}{\mathrm {d} t}}}"></span> </p><p>Therefore, the relaxation curve can be used to determine rate of viscoplastic strain and hence the viscosity of the dashpot in a one-dimensional viscoplastic material model. The residual value that is reached when the stress has plateaued at the end of a relaxation test corresponds to the upper limit of elasticity. For some materials such as rock salt such an upper limit of elasticity occurs at a very small value of stress and relaxation tests can be continued for more than a year without any observable plateau in the stress. </p><p>It is important to note that relaxation tests are extremely difficult to perform because maintaining the condition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}}{\mathrm {d} t}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}}{\mathrm {d} t}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcf6a30a2abb109880f4a29eb3b0df1f08f5050" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:7.619ex; height:7.176ex;" alt="{\displaystyle {\cfrac {\mathrm {d} {\boldsymbol {\varepsilon }}}{\mathrm {d} t}}=0}"></span> in a test requires considerable delicacy.<sup id="cite_ref-Cristescu_20-0" class="reference"><a href="#cite_note-Cristescu-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Rheological_models_of_viscoplasticity">Rheological models of viscoplasticity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=6" title="Edit section: Rheological models of viscoplasticity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One-dimensional constitutive models for viscoplasticity based on spring-dashpot-slider elements include<sup id="cite_ref-Simo_3-2" class="reference"><a href="#cite_note-Simo-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> the perfectly viscoplastic solid, the elastic perfectly viscoplastic solid, and the elastoviscoplastic hardening solid. The elements may be connected in <a href="/wiki/Series_circuits" class="mw-redirect" title="Series circuits">series</a> or in <a href="/wiki/Parallel_circuits" class="mw-redirect" title="Parallel circuits">parallel</a>. In models where the elements are connected in series the strain is additive while the stress is equal in each element. In parallel connections, the stress is additive while the strain is equal in each element. Many of these one-dimensional models can be generalized to three dimensions for the small strain regime. In the subsequent discussion, time rates strain and stress are written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\boldsymbol {\varepsilon }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\boldsymbol {\varepsilon }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b5512ac92557e02352f38280c37edf75feb050" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.419ex; height:2.176ex;" alt="{\displaystyle {\dot {\boldsymbol {\varepsilon }}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\boldsymbol {\sigma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">σ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\boldsymbol {\sigma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c126b5187517f81caf7772ac4d05aad833923146" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.594ex; height:2.176ex;" alt="{\displaystyle {\dot {\boldsymbol {\sigma }}}}"></span>, respectively. </p> <div class="mw-heading mw-heading3"><h3 id="Perfectly_viscoplastic_solid_(Norton-Hoff_model)"><span id="Perfectly_viscoplastic_solid_.28Norton-Hoff_model.29"></span>Perfectly viscoplastic solid (Norton-Hoff model)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=7" title="Edit section: Perfectly viscoplastic solid (Norton-Hoff model)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Visco89.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Visco89.svg/310px-Visco89.svg.png" decoding="async" width="310" height="139" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Visco89.svg/465px-Visco89.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/Visco89.svg/620px-Visco89.svg.png 2x" data-file-width="512" data-file-height="230" /></a><figcaption>Figure 5. Norton-Hoff model for perfectly viscoplastic solid</figcaption></figure> <p>In a perfectly viscoplastic solid, also called the Norton-Hoff model of viscoplasticity, the stress (as for viscous fluids) is a function of the rate of permanent strain. The effect of elasticity is neglected in the model, i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\varepsilon }}_{e}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}_{e}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be89a3d37b407ced5fd006430e9132e1b2a3e4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.489ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}_{e}=0}"></span> and hence there is no initial yield stress, i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{y}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/388e255b0a234069f5cda0deaab59031d1b0766f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.638ex; height:2.843ex;" alt="{\displaystyle \sigma _{y}=0}"></span>. The viscous dashpot has a response given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}=\eta ~{\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }\implies {\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }={\cfrac {\boldsymbol {\sigma }}{\eta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>=</mo> <mi>η</mi> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}=\eta ~{\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }\implies {\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }={\cfrac {\boldsymbol {\sigma }}{\eta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a064fa8d987e0126073675269be5b5e0cdfd9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.224ex; height:7.176ex;" alt="{\displaystyle {\boldsymbol {\sigma }}=\eta ~{\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }\implies {\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }={\cfrac {\boldsymbol {\sigma }}{\eta }}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span> is the viscosity of the dashpot. In the Norton-Hoff model the viscosity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span> is a nonlinear function of the applied stress and is given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =\lambda \left[{\cfrac {\lambda }{||{\boldsymbol {\sigma }}||}}\right]^{N-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <mi>λ</mi> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =\lambda \left[{\cfrac {\lambda }{||{\boldsymbol {\sigma }}||}}\right]^{N-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b220691b0cceba41c771b03519949ed642a7b8d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.144ex; height:8.009ex;" alt="{\displaystyle \eta =\lambda \left[{\cfrac {\lambda }{||{\boldsymbol {\sigma }}||}}\right]^{N-1}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> is a fitting parameter, λ is the kinematic viscosity of the material and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ||{\boldsymbol {\sigma }}||={\sqrt {{\boldsymbol {\sigma }}:{\boldsymbol {\sigma }}}}={\sqrt {\sigma _{ij}\sigma _{ij}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||{\boldsymbol {\sigma }}||={\sqrt {{\boldsymbol {\sigma }}:{\boldsymbol {\sigma }}}}={\sqrt {\sigma _{ij}\sigma _{ij}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d62ac04677fd4ea0bf490625f37be80b61f60b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:24.986ex; height:3.343ex;" alt="{\displaystyle ||{\boldsymbol {\sigma }}||={\sqrt {{\boldsymbol {\sigma }}:{\boldsymbol {\sigma }}}}={\sqrt {\sigma _{ij}\sigma _{ij}}}}"></span>. Then the viscoplastic strain rate is given by the relation </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }={\cfrac {\boldsymbol {\sigma }}{\lambda }}\left[{\cfrac {||{\boldsymbol {\sigma }}||}{\lambda }}\right]^{N-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }={\cfrac {\boldsymbol {\sigma }}{\lambda }}\left[{\cfrac {||{\boldsymbol {\sigma }}||}{\lambda }}\right]^{N-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1370b2175c227de2e1cdb90fbd43b87f037d0022" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.482ex; height:8.009ex;" alt="{\displaystyle {\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }={\cfrac {\boldsymbol {\sigma }}{\lambda }}\left[{\cfrac {||{\boldsymbol {\sigma }}||}{\lambda }}\right]^{N-1}}"></span> </p><p>In one-dimensional form, the Norton-Hoff model can be expressed as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =\lambda ~\left({\dot {\varepsilon }}_{\mathrm {vp} }\right)^{1/N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mi>λ</mi> <mtext> </mtext> <msup> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =\lambda ~\left({\dot {\varepsilon }}_{\mathrm {vp} }\right)^{1/N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62faae1691356009d4207052dc31a0794fcd7540" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.879ex; height:3.676ex;" alt="{\displaystyle \sigma =\lambda ~\left({\dot {\varepsilon }}_{\mathrm {vp} }\right)^{1/N}}"></span> </p><p>When <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=1.0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>1.0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=1.0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d14f8fc916734113fc5ed7e2953711caff552fa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.134ex; height:2.176ex;" alt="{\displaystyle N=1.0}"></span> the solid is <a href="/wiki/Viscoelastic" class="mw-redirect" title="Viscoelastic">viscoelastic</a>. </p><p>If we assume that plastic flow is <a href="/wiki/Isochoric_process" title="Isochoric process">isochoric</a> (volume preserving), then the above relation can be expressed in the more familiar form<sup id="cite_ref-Rappaz_21-0" class="reference"><a href="#cite_note-Rappaz-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {s}}=2K~\left({\sqrt {3}}{\dot {\varepsilon }}_{\mathrm {eq} }\right)^{m-1}~{\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">s</mi> </mrow> <mo>=</mo> <mn>2</mn> <mi>K</mi> <mtext> </mtext> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {s}}=2K~\left({\sqrt {3}}{\dot {\varepsilon }}_{\mathrm {eq} }\right)^{m-1}~{\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/102fa63ba2fa3b0e6cd50596c2f3daa047ba675e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.357ex; height:3.676ex;" alt="{\displaystyle {\boldsymbol {s}}=2K~\left({\sqrt {3}}{\dot {\varepsilon }}_{\mathrm {eq} }\right)^{m-1}~{\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adfccdf93cc6115ade3deffbbbf062d6849a1f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.234ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {s}}}"></span> is the <a href="/wiki/Deviatoric_stress" class="mw-redirect" title="Deviatoric stress">deviatoric stress</a> tensor, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\varepsilon }}_{\mathrm {eq} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\varepsilon }}_{\mathrm {eq} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49760aac6791343ca809719e4f7c802754615305" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.198ex; height:2.843ex;" alt="{\displaystyle {\dot {\varepsilon }}_{\mathrm {eq} }}"></span> is the <a href="/w/index.php?title=Equivalent_strain&action=edit&redlink=1" class="new" title="Equivalent strain (page does not exist)">von Mises equivalent strain</a> rate, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K,m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>,</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K,m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c34feca8786ed409ecf2ce21add401093c2e174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.14ex; height:2.509ex;" alt="{\displaystyle K,m}"></span> are material parameters. The equivalent strain rate is defined as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\bar {\epsilon }}}={\sqrt {{\frac {2}{3}}{\dot {\bar {\bar {\epsilon }}}}:{\dot {\bar {\bar {\epsilon }}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϵ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϵ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϵ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\bar {\epsilon }}}={\sqrt {{\frac {2}{3}}{\dot {\bar {\bar {\epsilon }}}}:{\dot {\bar {\bar {\epsilon }}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/069f9d99364e705758841207290cee1d6dbd49fb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.233ex; height:6.176ex;" alt="{\displaystyle {\dot {\bar {\epsilon }}}={\sqrt {{\frac {2}{3}}{\dot {\bar {\bar {\epsilon }}}}:{\dot {\bar {\bar {\epsilon }}}}}}}"></span> </p><p>These models can be applied in metals and alloys at temperatures higher than two thirds<sup id="cite_ref-Rappaz_21-1" class="reference"><a href="#cite_note-Rappaz-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> of their absolute melting point (in kelvins) and polymers/asphalt at elevated temperature. The responses for strain hardening, creep, and relaxation tests of such material are shown in Figure 6. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:PVS_VISCOUS2.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/PVS_VISCOUS2.JPG/400px-PVS_VISCOUS2.JPG" decoding="async" width="400" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/PVS_VISCOUS2.JPG/600px-PVS_VISCOUS2.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/PVS_VISCOUS2.JPG/800px-PVS_VISCOUS2.JPG 2x" data-file-width="924" data-file-height="389" /></a><figcaption>Figure 6: The response of perfectly viscoplastic solid to hardening, creep and relaxation tests</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Elastic_perfectly_viscoplastic_solid_(Bingham–Norton_model)"><span id="Elastic_perfectly_viscoplastic_solid_.28Bingham.E2.80.93Norton_model.29"></span>Elastic perfectly viscoplastic solid (Bingham–Norton model)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=8" title="Edit section: Elastic perfectly viscoplastic solid (Bingham–Norton model)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Visco98.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Visco98.jpg/310px-Visco98.jpg" decoding="async" width="310" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Visco98.jpg/465px-Visco98.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/a/a2/Visco98.jpg 2x" data-file-width="523" data-file-height="165" /></a><figcaption>Figure 7. The elastic perfectly viscoplastic material.</figcaption></figure> <p>Two types of elementary approaches can be used to build up an elastic-perfectly viscoplastic mode. In the first situation, the sliding friction element and the dashpot are arranged in parallel and then connected in series to the elastic spring as shown in Figure 7. This model is called the <b>Bingham–Maxwell model</b> (by analogy with the <a href="/wiki/Maxwell_material" title="Maxwell material">Maxwell model</a> and the <a href="/wiki/Bingham_plastic" title="Bingham plastic">Bingham model</a>) or the <b>Bingham–Norton model</b>.<sup id="cite_ref-Irgens_22-0" class="reference"><a href="#cite_note-Irgens-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> In the second situation, all three elements are arranged in parallel. Such a model is called a <b>Bingham–Kelvin model</b> by analogy with the <a href="/wiki/Kelvin_model" class="mw-redirect" title="Kelvin model">Kelvin model</a>. </p><p>For elastic-perfectly viscoplastic materials, the elastic strain is no longer considered negligible but the rate of plastic strain is only a function of the initial yield stress and there is no influence of hardening. The sliding element represents a constant yielding stress when the elastic limit is exceeded irrespective of the strain. The model can be expressed as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\boldsymbol {\sigma }}={\mathsf {E}}~{\boldsymbol {\varepsilon }}&&\mathrm {for} ~\|{\boldsymbol {\sigma }}\|<\sigma _{y}\\&{\dot {\boldsymbol {\varepsilon }}}={\dot {\boldsymbol {\varepsilon }}}_{\mathrm {e} }+{\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }={\mathsf {E}}^{-1}~{\dot {\boldsymbol {\sigma }}}+{\cfrac {\boldsymbol {\sigma }}{\eta }}\left[1-{\cfrac {\sigma _{y}}{\|{\boldsymbol {\sigma }}\|}}\right]&&\mathrm {for} ~\|{\boldsymbol {\sigma }}\|\geq \sigma _{y}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">σ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>≥</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\boldsymbol {\sigma }}={\mathsf {E}}~{\boldsymbol {\varepsilon }}&&\mathrm {for} ~\|{\boldsymbol {\sigma }}\|<\sigma _{y}\\&{\dot {\boldsymbol {\varepsilon }}}={\dot {\boldsymbol {\varepsilon }}}_{\mathrm {e} }+{\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }={\mathsf {E}}^{-1}~{\dot {\boldsymbol {\sigma }}}+{\cfrac {\boldsymbol {\sigma }}{\eta }}\left[1-{\cfrac {\sigma _{y}}{\|{\boldsymbol {\sigma }}\|}}\right]&&\mathrm {for} ~\|{\boldsymbol {\sigma }}\|\geq \sigma _{y}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d6fecf5508cd2d0e450dab5d4798bb9129c607d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:57.603ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}&{\boldsymbol {\sigma }}={\mathsf {E}}~{\boldsymbol {\varepsilon }}&&\mathrm {for} ~\|{\boldsymbol {\sigma }}\|<\sigma _{y}\\&{\dot {\boldsymbol {\varepsilon }}}={\dot {\boldsymbol {\varepsilon }}}_{\mathrm {e} }+{\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }={\mathsf {E}}^{-1}~{\dot {\boldsymbol {\sigma }}}+{\cfrac {\boldsymbol {\sigma }}{\eta }}\left[1-{\cfrac {\sigma _{y}}{\|{\boldsymbol {\sigma }}\|}}\right]&&\mathrm {for} ~\|{\boldsymbol {\sigma }}\|\geq \sigma _{y}\end{aligned}}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span> is the viscosity of the dashpot element. If the dashpot element has a response that is of the Norton form </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\boldsymbol {\sigma }}{\eta }}={\cfrac {\boldsymbol {\sigma }}{\lambda }}\left[{\cfrac {\|{\boldsymbol {\sigma }}\|}{\lambda }}\right]^{N-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\boldsymbol {\sigma }}{\eta }}={\cfrac {\boldsymbol {\sigma }}{\lambda }}\left[{\cfrac {\|{\boldsymbol {\sigma }}\|}{\lambda }}\right]^{N-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14fd540d4ecd33bf72b2fbeb72ec07f6f8e6d584" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.218ex; height:8.009ex;" alt="{\displaystyle {\cfrac {\boldsymbol {\sigma }}{\eta }}={\cfrac {\boldsymbol {\sigma }}{\lambda }}\left[{\cfrac {\|{\boldsymbol {\sigma }}\|}{\lambda }}\right]^{N-1}}"></span> </p><p>we get the Bingham–Norton model </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\boldsymbol {\varepsilon }}}={\mathsf {E}}^{-1}~{\dot {\boldsymbol {\sigma }}}+{\cfrac {\boldsymbol {\sigma }}{\lambda }}\left[{\cfrac {\|{\boldsymbol {\sigma }}\|}{\lambda }}\right]^{N-1}\left[1-{\cfrac {\sigma _{y}}{\|{\boldsymbol {\sigma }}\|}}\right]\quad \mathrm {for} ~\|{\boldsymbol {\sigma }}\|\geq \sigma _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">σ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>≥</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\boldsymbol {\varepsilon }}}={\mathsf {E}}^{-1}~{\dot {\boldsymbol {\sigma }}}+{\cfrac {\boldsymbol {\sigma }}{\lambda }}\left[{\cfrac {\|{\boldsymbol {\sigma }}\|}{\lambda }}\right]^{N-1}\left[1-{\cfrac {\sigma _{y}}{\|{\boldsymbol {\sigma }}\|}}\right]\quad \mathrm {for} ~\|{\boldsymbol {\sigma }}\|\geq \sigma _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf6270b11cbd6ca6cfc2a7c939e52397dc56677" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.422ex; height:8.009ex;" alt="{\displaystyle {\dot {\boldsymbol {\varepsilon }}}={\mathsf {E}}^{-1}~{\dot {\boldsymbol {\sigma }}}+{\cfrac {\boldsymbol {\sigma }}{\lambda }}\left[{\cfrac {\|{\boldsymbol {\sigma }}\|}{\lambda }}\right]^{N-1}\left[1-{\cfrac {\sigma _{y}}{\|{\boldsymbol {\sigma }}\|}}\right]\quad \mathrm {for} ~\|{\boldsymbol {\sigma }}\|\geq \sigma _{y}}"></span> </p><p>Other expressions for the strain rate can also be observed in the literature<sup id="cite_ref-Irgens_22-1" class="reference"><a href="#cite_note-Irgens-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> with the general form </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\boldsymbol {\varepsilon }}}={\mathsf {E}}^{-1}~{\dot {\boldsymbol {\sigma }}}+f({\boldsymbol {\sigma }},\sigma _{y})~{\boldsymbol {\sigma }}\quad \mathrm {for} ~\|{\boldsymbol {\sigma }}\|\geq \sigma _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">σ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>≥</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\boldsymbol {\varepsilon }}}={\mathsf {E}}^{-1}~{\dot {\boldsymbol {\sigma }}}+f({\boldsymbol {\sigma }},\sigma _{y})~{\boldsymbol {\sigma }}\quad \mathrm {for} ~\|{\boldsymbol {\sigma }}\|\geq \sigma _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b0a2338219b61ec7a746bb1afffec5111f612b7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.758ex; height:3.343ex;" alt="{\displaystyle {\dot {\boldsymbol {\varepsilon }}}={\mathsf {E}}^{-1}~{\dot {\boldsymbol {\sigma }}}+f({\boldsymbol {\sigma }},\sigma _{y})~{\boldsymbol {\sigma }}\quad \mathrm {for} ~\|{\boldsymbol {\sigma }}\|\geq \sigma _{y}}"></span> </p><p>The responses for strain hardening, creep, and relaxation tests of such material are shown in Figure 8. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:ELASTIC_PVISCO_solid.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/ELASTIC_PVISCO_solid.JPG/400px-ELASTIC_PVISCO_solid.JPG" decoding="async" width="400" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/ELASTIC_PVISCO_solid.JPG/600px-ELASTIC_PVISCO_solid.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/ELASTIC_PVISCO_solid.JPG/800px-ELASTIC_PVISCO_solid.JPG 2x" data-file-width="924" data-file-height="389" /></a><figcaption>Figure 8. The response of elastic perfectly viscoplastic solid to hardening, creep and relaxation tests.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Elastoviscoplastic_hardening_solid">Elastoviscoplastic hardening solid</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=9" title="Edit section: Elastoviscoplastic hardening solid"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An elastic-viscoplastic material with <a href="/wiki/Strain_hardening" class="mw-redirect" title="Strain hardening">strain hardening</a> is described by equations similar to those for an elastic-viscoplastic material with perfect plasticity. However, in this case the stress depends both on the plastic strain rate and on the plastic strain itself. For an elastoviscoplastic material the stress, after exceeding the yield stress, continues to increase beyond the initial yielding point. This implies that the yield stress in the sliding element increases with strain and the model may be expressed in generic terms as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\boldsymbol {\varepsilon }}={\boldsymbol {\varepsilon }}_{\mathrm {e} }={\mathsf {E}}^{-1}~{\boldsymbol {\sigma }}=~{\boldsymbol {\varepsilon }}&&\mathrm {for} ~||{\boldsymbol {\sigma }}||<\sigma _{y}\\&{\dot {\boldsymbol {\varepsilon }}}={\dot {\boldsymbol {\varepsilon }}}_{\mathrm {e} }+{\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }={\mathsf {E}}^{-1}~{\dot {\boldsymbol {\sigma }}}+f({\boldsymbol {\sigma }},\sigma _{y},{\boldsymbol {\varepsilon }}_{\mathrm {vp} })~{\boldsymbol {\sigma }}&&\mathrm {for} ~||{\boldsymbol {\sigma }}||\geq \sigma _{y}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">σ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\boldsymbol {\varepsilon }}={\boldsymbol {\varepsilon }}_{\mathrm {e} }={\mathsf {E}}^{-1}~{\boldsymbol {\sigma }}=~{\boldsymbol {\varepsilon }}&&\mathrm {for} ~||{\boldsymbol {\sigma }}||<\sigma _{y}\\&{\dot {\boldsymbol {\varepsilon }}}={\dot {\boldsymbol {\varepsilon }}}_{\mathrm {e} }+{\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }={\mathsf {E}}^{-1}~{\dot {\boldsymbol {\sigma }}}+f({\boldsymbol {\sigma }},\sigma _{y},{\boldsymbol {\varepsilon }}_{\mathrm {vp} })~{\boldsymbol {\sigma }}&&\mathrm {for} ~||{\boldsymbol {\sigma }}||\geq \sigma _{y}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed527c66f3d632933a817f7aa8ca02db13f8c4aa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:58.125ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}&{\boldsymbol {\varepsilon }}={\boldsymbol {\varepsilon }}_{\mathrm {e} }={\mathsf {E}}^{-1}~{\boldsymbol {\sigma }}=~{\boldsymbol {\varepsilon }}&&\mathrm {for} ~||{\boldsymbol {\sigma }}||<\sigma _{y}\\&{\dot {\boldsymbol {\varepsilon }}}={\dot {\boldsymbol {\varepsilon }}}_{\mathrm {e} }+{\dot {\boldsymbol {\varepsilon }}}_{\mathrm {vp} }={\mathsf {E}}^{-1}~{\dot {\boldsymbol {\sigma }}}+f({\boldsymbol {\sigma }},\sigma _{y},{\boldsymbol {\varepsilon }}_{\mathrm {vp} })~{\boldsymbol {\sigma }}&&\mathrm {for} ~||{\boldsymbol {\sigma }}||\geq \sigma _{y}\end{aligned}}}"></span> </p><p>This model is adopted when metals and alloys are at medium and higher temperatures and wood under high loads. The responses for strain hardening, creep, and relaxation tests of such a material are shown in Figure 9. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:PVS_VISCOUS3.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/PVS_VISCOUS3.JPG/400px-PVS_VISCOUS3.JPG" decoding="async" width="400" height="164" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/PVS_VISCOUS3.JPG/600px-PVS_VISCOUS3.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/PVS_VISCOUS3.JPG/800px-PVS_VISCOUS3.JPG 2x" data-file-width="951" data-file-height="390" /></a><figcaption>Figure 9. The response of elastoviscoplastic hardening solid to hardening, creep and relaxation tests.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Strain-rate_dependent_plasticity_models">Strain-rate dependent plasticity models</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=10" title="Edit section: Strain-rate dependent plasticity models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Classical phenomenological viscoplasticity models for <a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">small strains</a> are usually categorized into two types:<sup id="cite_ref-Simo_3-3" class="reference"><a href="#cite_note-Simo-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (August 2024)">full citation needed</span></a></i>]</sup> </p> <ul><li>the Perzyna formulation</li> <li>the Duvaut–Lions formulation</li></ul> <div class="mw-heading mw-heading3"><h3 id="Perzyna_formulation">Perzyna formulation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=11" title="Edit section: Perzyna formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the Perzyna formulation the plastic strain rate is assumed to be given by a constitutive relation of the form </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\varepsilon }}_{\mathrm {vp} }={\cfrac {\left\langle f({\boldsymbol {\sigma }},{\boldsymbol {q}})\right\rangle }{\tau }}{\cfrac {\partial f}{\partial {\boldsymbol {\sigma }}}}={\begin{cases}{\cfrac {f({\boldsymbol {\sigma }},{\boldsymbol {q}})}{\tau }}{\cfrac {\partial f}{\partial {\boldsymbol {\sigma }}}}&{\rm {if}}~f({\boldsymbol {\sigma }},{\boldsymbol {q}})>0\\0&{\rm {otherwise}}\\\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⟨</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">q</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">q</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">q</mi> </mrow> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">w</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\varepsilon }}_{\mathrm {vp} }={\cfrac {\left\langle f({\boldsymbol {\sigma }},{\boldsymbol {q}})\right\rangle }{\tau }}{\cfrac {\partial f}{\partial {\boldsymbol {\sigma }}}}={\begin{cases}{\cfrac {f({\boldsymbol {\sigma }},{\boldsymbol {q}})}{\tau }}{\cfrac {\partial f}{\partial {\boldsymbol {\sigma }}}}&{\rm {if}}~f({\boldsymbol {\sigma }},{\boldsymbol {q}})>0\\0&{\rm {otherwise}}\\\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8197c1c696f69e4aa06083500881ed52895647eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:53.024ex; height:9.509ex;" alt="{\displaystyle {\dot {\varepsilon }}_{\mathrm {vp} }={\cfrac {\left\langle f({\boldsymbol {\sigma }},{\boldsymbol {q}})\right\rangle }{\tau }}{\cfrac {\partial f}{\partial {\boldsymbol {\sigma }}}}={\begin{cases}{\cfrac {f({\boldsymbol {\sigma }},{\boldsymbol {q}})}{\tau }}{\cfrac {\partial f}{\partial {\boldsymbol {\sigma }}}}&{\rm {if}}~f({\boldsymbol {\sigma }},{\boldsymbol {q}})>0\\0&{\rm {otherwise}}\\\end{cases}}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(.,.)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mo>.</mo> <mo>,</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(.,.)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a15070f167ceeaab7da0cbf1f651bbe2b56c8be5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.19ex; height:2.843ex;" alt="{\displaystyle f(.,.)}"></span> is a <a href="/wiki/Yield_surface" title="Yield surface">yield function</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45fe1b9d8dcbc3103fc7805d69798bfe5ca5b16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.594ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\sigma }}}"></span> is the <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">Cauchy stress</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edf74db7c59a404f691ec204e3152a01ef488b83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {q}}}"></span> is a set of internal variables (such as the <a href="/wiki/Plastic_strain" class="mw-redirect" title="Plastic strain">plastic strain</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\varepsilon }}_{\mathrm {vp} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}_{\mathrm {vp} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf25957bbc18ca81ebb29f1067f5f657e285998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.244ex; height:2.343ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}_{\mathrm {vp} }}"></span>), <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> is a relaxation time. The notation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \dots \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mo>…</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \dots \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26212513a02b1ae540374fe070c0fd4f10a3fa00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.533ex; height:2.843ex;" alt="{\displaystyle \langle \dots \rangle }"></span> denotes the <a href="/wiki/Macaulay_brackets" title="Macaulay brackets">Macaulay brackets</a>. The flow rule used in various versions of the <i>Chaboche</i> model is a special case of Perzyna's flow rule<sup id="cite_ref-Lubliner1990_23-0" class="reference"><a href="#cite_note-Lubliner1990-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> and has the form </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\varepsilon }}_{\mathrm {vp} }=\left\langle {\frac {f}{f_{0}}}\right\rangle ^{n}sign({\boldsymbol {\sigma }}-{\boldsymbol {\chi }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>f</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>s</mi> <mi>i</mi> <mi>g</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">χ</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\varepsilon }}_{\mathrm {vp} }=\left\langle {\frac {f}{f_{0}}}\right\rangle ^{n}sign({\boldsymbol {\sigma }}-{\boldsymbol {\chi }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/253396bdcdc87dd394b8f83c490f73b89d9aefd4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.519ex; height:6.176ex;" alt="{\displaystyle {\dot {\varepsilon }}_{\mathrm {vp} }=\left\langle {\frac {f}{f_{0}}}\right\rangle ^{n}sign({\boldsymbol {\sigma }}-{\boldsymbol {\chi }})}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6423b30a4c5770c59b5ab92dcb4ce378755440ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{0}}"></span> is the quasistatic value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\chi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">χ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\chi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea5f1742b4bb7deb47d0b975d324618b20b95f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.669ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\chi }}}"></span> is a <i>backstress</i>. Several models for the backstress also go by the name <i>Chaboche model</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Duvaut–Lions_formulation"><span id="Duvaut.E2.80.93Lions_formulation"></span>Duvaut–Lions formulation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=12" title="Edit section: Duvaut–Lions formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Duvaut–Lions formulation is equivalent to the Perzyna formulation and may be expressed as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\varepsilon }}_{\mathrm {vp} }={\begin{cases}{\mathsf {C}}^{-1}:{\cfrac {{\boldsymbol {\sigma }}-{\mathcal {P}}{\boldsymbol {\sigma }}}{\tau }}&{\rm {{if}~f({\boldsymbol {\sigma }},{\boldsymbol {q}})>0}}\\0&{\rm {otherwise}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">f</mi> </mrow> <mtext> </mtext> <mi mathvariant="normal">f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">q</mi> </mrow> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">w</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\varepsilon }}_{\mathrm {vp} }={\begin{cases}{\mathsf {C}}^{-1}:{\cfrac {{\boldsymbol {\sigma }}-{\mathcal {P}}{\boldsymbol {\sigma }}}{\tau }}&{\rm {{if}~f({\boldsymbol {\sigma }},{\boldsymbol {q}})>0}}\\0&{\rm {otherwise}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e55f561fdece820ec60e3443505c5749ab5b14a5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:38.868ex; height:9.509ex;" alt="{\displaystyle {\dot {\varepsilon }}_{\mathrm {vp} }={\begin{cases}{\mathsf {C}}^{-1}:{\cfrac {{\boldsymbol {\sigma }}-{\mathcal {P}}{\boldsymbol {\sigma }}}{\tau }}&{\rm {{if}~f({\boldsymbol {\sigma }},{\boldsymbol {q}})>0}}\\0&{\rm {otherwise}}\end{cases}}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/699cb4847ecedddeeae2b69e2892ba9987f9b1d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle {\mathsf {C}}}"></span> is the elastic stiffness tensor, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}{\boldsymbol {\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}{\boldsymbol {\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c73c3a6819e2b2af055d617a640b672b4fa76de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.298ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}{\boldsymbol {\sigma }}}"></span> is the closest point projection of the stress state on to the boundary of the region that bounds all possible elastic stress states. The quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}{\boldsymbol {\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}{\boldsymbol {\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c73c3a6819e2b2af055d617a640b672b4fa76de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.298ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}{\boldsymbol {\sigma }}}"></span> is typically found from the rate-independent solution to a plasticity problem. </p> <div class="mw-heading mw-heading3"><h3 id="Flow_stress_models">Flow stress models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=13" title="Edit section: Flow stress models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f({\boldsymbol {\sigma }},{\boldsymbol {q}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f({\boldsymbol {\sigma }},{\boldsymbol {q}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac36b565d523556d5fd70de7a9bb48f9007a00ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.995ex; height:2.843ex;" alt="{\displaystyle f({\boldsymbol {\sigma }},{\boldsymbol {q}})}"></span> represents the evolution of the <a href="/wiki/Yield_surface" title="Yield surface">yield surface</a>. The yield function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is often expressed as an equation consisting of some invariant of stress and a model for the yield stress (or plastic flow stress). An example is <a href="/wiki/Von_Mises_yield_criterion" title="Von Mises yield criterion">von Mises</a> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f9986a8fbfd51097a5ff5e82d3252c9572b5835" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.344ex; height:2.509ex;" alt="{\displaystyle J_{2}}"></span> plasticity. In those situations the plastic strain rate is calculated in the same manner as in rate-independent plasticity. In other situations, the yield stress model provides a direct means of computing the plastic strain rate. </p><p>Numerous empirical and semi-empirical flow stress models are used the computational plasticity. The following temperature and strain-rate dependent models provide a sampling of the models in current use: </p> <ol><li>the Johnson–Cook model</li> <li>the Steinberg–Cochran–Guinan–Lund model.</li> <li>the Zerilli–Armstrong model.</li> <li>the Mechanical threshold stress model.</li> <li>the Preston–Tonks–Wallace model.</li></ol> <p>The Johnson–Cook (JC) model <sup id="cite_ref-Johnson83_24-0" class="reference"><a href="#cite_note-Johnson83-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> is purely empirical and is the most widely used of the five. However, this model exhibits an unrealistically small strain-rate dependence at high temperatures. The Steinberg–Cochran–Guinan–Lund (SCGL) model <sup id="cite_ref-Steinberg80_25-0" class="reference"><a href="#cite_note-Steinberg80-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Steinberg89_26-0" class="reference"><a href="#cite_note-Steinberg89-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> is semi-empirical. The model is purely empirical and strain-rate independent at high strain-rates. A dislocation-based extension based on <sup id="cite_ref-Hoge77_27-0" class="reference"><a href="#cite_note-Hoge77-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> is used at low strain-rates. The SCGL model is used extensively by the shock physics community. The Zerilli–Armstrong (ZA) model<sup id="cite_ref-Zerilli87_28-0" class="reference"><a href="#cite_note-Zerilli87-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> is a simple physically based model that has been used extensively. A more complex model that is based on ideas from dislocation dynamics is the Mechanical Threshold Stress (MTS) model.<sup id="cite_ref-Follans88_29-0" class="reference"><a href="#cite_note-Follans88-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> This model has been used to model the plastic deformation of copper, tantalum,<sup id="cite_ref-Chen96_30-0" class="reference"><a href="#cite_note-Chen96-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> alloys of steel,<sup id="cite_ref-Goto00_31-0" class="reference"><a href="#cite_note-Goto00-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Banerjee05b_32-0" class="reference"><a href="#cite_note-Banerjee05b-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> and aluminum alloys.<sup id="cite_ref-Puchi01_33-0" class="reference"><a href="#cite_note-Puchi01-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> However, the MTS model is limited to strain-rates less than around 10<sup>7</sup>/s. The Preston–Tonks–Wallace (PTW) model <sup id="cite_ref-Preston03_34-0" class="reference"><a href="#cite_note-Preston03-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> is also physically based and has a form similar to the MTS model. However, the PTW model has components that can model plastic deformation in the overdriven shock regime (strain-rates greater that 10<sup>7</sup>/s). Hence this model is valid for the largest range of strain-rates among the five flow stress models. </p> <div class="mw-heading mw-heading4"><h4 id="Johnson–Cook_flow_stress_model"><span id="Johnson.E2.80.93Cook_flow_stress_model"></span>Johnson–Cook flow stress model</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=14" title="Edit section: Johnson–Cook flow stress model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Johnson–Cook (JC) model <sup id="cite_ref-Johnson83_24-1" class="reference"><a href="#cite_note-Johnson83-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> is purely empirical and gives the following relation for the flow stress (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b22d96fceb022e70169a37383278199a26b3534a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.377ex; height:2.343ex;" alt="{\displaystyle \sigma _{y}}"></span>) </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{(1)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon _{\rm {p}}}},T)=\left[A+B(\varepsilon _{\rm {p}})^{n}\right]\left[1+C\ln({\dot {\varepsilon _{\rm {p}}}}^{*})\right]\left[1-(T^{*})^{m}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>(1)</mtext> </mrow> <mspace width="2em" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>C</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{(1)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon _{\rm {p}}}},T)=\left[A+B(\varepsilon _{\rm {p}})^{n}\right]\left[1+C\ln({\dot {\varepsilon _{\rm {p}}}}^{*})\right]\left[1-(T^{*})^{m}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e465f21e3f33e47fd9fb67b27f892a42cfc536fa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:62.775ex; height:3.009ex;" alt="{\displaystyle {\text{(1)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon _{\rm {p}}}},T)=\left[A+B(\varepsilon _{\rm {p}})^{n}\right]\left[1+C\ln({\dot {\varepsilon _{\rm {p}}}}^{*})\right]\left[1-(T^{*})^{m}\right]}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{\rm {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\rm {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49e885818c44b447ff969fd4d3e2ad7490a99a75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.23ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{\rm {p}}}"></span> is the <a href="/wiki/Infinitesimal_strain_theory#Equivalent_strain" title="Infinitesimal strain theory">equivalent plastic strain</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\varepsilon _{\rm {p}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\varepsilon _{\rm {p}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0927b98902d852820a2dd7011853a3d25389af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.23ex; height:2.843ex;" alt="{\displaystyle {\dot {\varepsilon _{\rm {p}}}}}"></span> is the plastic <a href="/wiki/Strain_rate" title="Strain rate">strain-rate</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C,n,m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C,n,m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84d9b8d8fe967deae3af67194f9b183d9cfd7b2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.844ex; height:2.509ex;" alt="{\displaystyle A,B,C,n,m}"></span> are material constants. </p><p>The normalized strain-rate and temperature in equation (1) are defined as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\varepsilon _{\rm {p}}}}^{*}:={\cfrac {\dot {\varepsilon _{\rm {p}}}}{\dot {\varepsilon _{\rm {p0}}}}}\qquad {\text{and}}\qquad T^{*}:={\cfrac {(T-T_{0})}{(T_{m}-T_{0})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mn>0</mn> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="2em" /> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>T</mi> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\varepsilon _{\rm {p}}}}^{*}:={\cfrac {\dot {\varepsilon _{\rm {p}}}}{\dot {\varepsilon _{\rm {p0}}}}}\qquad {\text{and}}\qquad T^{*}:={\cfrac {(T-T_{0})}{(T_{m}-T_{0})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8337733d16ccca6ce7c319a8ed15c9eab28265c6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.405ex; height:7.176ex;" alt="{\displaystyle {\dot {\varepsilon _{\rm {p}}}}^{*}:={\cfrac {\dot {\varepsilon _{\rm {p}}}}{\dot {\varepsilon _{\rm {p0}}}}}\qquad {\text{and}}\qquad T^{*}:={\cfrac {(T-T_{0})}{(T_{m}-T_{0})}}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\varepsilon _{\rm {p0}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mn>0</mn> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\varepsilon _{\rm {p0}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a64d2f719e75e07eb01285f33baef1e0349eadd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.052ex; height:2.843ex;" alt="{\displaystyle {\dot {\varepsilon _{\rm {p0}}}}}"></span> is the effective plastic strain-rate of the quasi-static test used to determine the yield and hardening parameters A,B and n. This is not as it is often thought just a parameter to make <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\varepsilon _{\rm {p}}}}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\varepsilon _{\rm {p}}}}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59fd0ce02ab8cff86bae893aa75210c1e732ef0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.284ex; height:3.009ex;" alt="{\displaystyle {\dot {\varepsilon _{\rm {p}}}}^{*}}"></span> non-dimensional.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}"></span> is a reference temperature, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6dcd584cf9192a7f162f3b62a63c49cf8b572f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.033ex; height:2.509ex;" alt="{\displaystyle T_{m}}"></span> is a reference <a href="/wiki/Melting_point" title="Melting point">melt temperature</a>. For conditions where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{*}<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{*}<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddf9c9e18b291478d39414b0c128294f7f18d14a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.035ex; height:2.343ex;" alt="{\displaystyle T^{*}<0}"></span>, we assume that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6100c5ebd48c6fd848709f2be624465203eb173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=1}"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Steinberg–Cochran–Guinan–Lund_flow_stress_model"><span id="Steinberg.E2.80.93Cochran.E2.80.93Guinan.E2.80.93Lund_flow_stress_model"></span>Steinberg–Cochran–Guinan–Lund flow stress model</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=15" title="Edit section: Steinberg–Cochran–Guinan–Lund flow stress model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Steinberg–Cochran–Guinan–Lund (SCGL) model is a semi-empirical model that was developed by Steinberg et al.<sup id="cite_ref-Steinberg80_25-1" class="reference"><a href="#cite_note-Steinberg80-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> for high strain-rate situations and extended to low strain-rates and bcc materials by Steinberg and Lund.<sup id="cite_ref-Steinberg89_26-1" class="reference"><a href="#cite_note-Steinberg89-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> The flow stress in this model is given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{(2)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon _{\rm {p}}}},T)=\left[\sigma _{a}f(\varepsilon _{\rm {p}})+\sigma _{t}({\dot {\varepsilon _{\rm {p}}}},T)\right]{\frac {\mu (p,T)}{\mu _{0}}};\quad \sigma _{a}f\leq \sigma _{\text{max}}~~{\text{and}}~~\sigma _{t}\leq \sigma _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>(2)</mtext> </mrow> <mspace width="2em" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>;</mo> <mspace width="1em" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mi>f</mi> <mo>≤</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{(2)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon _{\rm {p}}}},T)=\left[\sigma _{a}f(\varepsilon _{\rm {p}})+\sigma _{t}({\dot {\varepsilon _{\rm {p}}}},T)\right]{\frac {\mu (p,T)}{\mu _{0}}};\quad \sigma _{a}f\leq \sigma _{\text{max}}~~{\text{and}}~~\sigma _{t}\leq \sigma _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8dba4c5339724d639365222d2bb83e1334a7b82" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:80.572ex; height:6.176ex;" alt="{\displaystyle {\text{(2)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon _{\rm {p}}}},T)=\left[\sigma _{a}f(\varepsilon _{\rm {p}})+\sigma _{t}({\dot {\varepsilon _{\rm {p}}}},T)\right]{\frac {\mu (p,T)}{\mu _{0}}};\quad \sigma _{a}f\leq \sigma _{\text{max}}~~{\text{and}}~~\sigma _{t}\leq \sigma _{p}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ef75a8ddd0a1ba9772f635467a7aec9207f0c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.429ex; height:2.009ex;" alt="{\displaystyle \sigma _{a}}"></span> is the athermal component of the flow stress, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\varepsilon _{\rm {p}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\varepsilon _{\rm {p}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e270faf09d2aa47af8f3b5f689c2b8c523eb292f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.318ex; height:3.009ex;" alt="{\displaystyle f(\varepsilon _{\rm {p}})}"></span> is a function that represents strain hardening, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5dd6db0238ac32f34c6feb604748e253e842356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle \sigma _{t}}"></span> is the thermally activated component of the flow stress, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (p,T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (p,T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/877cdd1c0b9e23913e3d81c4c4a1cd9737c6b978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.051ex; height:2.843ex;" alt="{\displaystyle \mu (p,T)}"></span> is the pressure- and temperature-dependent shear modulus, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2fd9b8decb38a3cd158e7b6c0c6e2d987fefcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{0}}"></span> is the shear modulus at standard temperature and pressure. The saturation value of the athermal stress is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\text{max}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\text{max}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/148072fd2568d8481fde59970afa06d776fc815c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.618ex; height:2.009ex;" alt="{\displaystyle \sigma _{\text{max}}}"></span>. The saturation of the thermally activated stress is the <a href="/wiki/Peierls_stress" title="Peierls stress">Peierls stress</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd141c4b4b162275b62a7b903a1a065271fae3cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.387ex; height:2.343ex;" alt="{\displaystyle \sigma _{p}}"></span>). The shear modulus for this model is usually computed with the <a href="/wiki/Shear_modulus" title="Shear modulus">Steinberg–Cochran–Guinan shear modulus model</a>. </p><p>The strain hardening function (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>) has the form </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\varepsilon _{\rm {p}})=[1+\beta (\varepsilon _{\rm {p}}+\varepsilon _{\rm {p}}i)]^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>+</mo> <mi>β</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mi>i</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\varepsilon _{\rm {p}})=[1+\beta (\varepsilon _{\rm {p}}+\varepsilon _{\rm {p}}i)]^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3f13d292fe07503cf123b08412a9fbc3f334b9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.175ex; height:3.009ex;" alt="{\displaystyle f(\varepsilon _{\rm {p}})=[1+\beta (\varepsilon _{\rm {p}}+\varepsilon _{\rm {p}}i)]^{n}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe1162efdf4d398fabcbb240bb9bd527d611a4ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.761ex; height:2.509ex;" alt="{\displaystyle \beta ,n}"></span> are work hardening parameters, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{\rm {p}}i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\rm {p}}i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2474f2e9fd1f16c8323dbcaf08879f77d492b80b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.032ex; height:2.843ex;" alt="{\displaystyle \varepsilon _{\rm {p}}i}"></span> is the initial equivalent plastic strain. </p><p>The thermal component (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5dd6db0238ac32f34c6feb604748e253e842356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle \sigma _{t}}"></span>) is computed using a bisection algorithm from the following equation.<sup id="cite_ref-Steinberg89_26-2" class="reference"><a href="#cite_note-Steinberg89-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Hoge77_27-1" class="reference"><a href="#cite_note-Hoge77-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\varepsilon _{\rm {p}}}}=\left[{\frac {1}{C_{1}}}\exp \left[{\frac {2U_{k}}{k_{b}~T}}\left(1-{\frac {\sigma _{t}}{\sigma _{p}}}\right)^{2}\right]+{\frac {C_{2}}{\sigma _{t}}}\right]^{-1};\quad \sigma _{t}\leq \sigma _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mtext> </mtext> <mi>T</mi> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>;</mo> <mspace width="1em" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\varepsilon _{\rm {p}}}}=\left[{\frac {1}{C_{1}}}\exp \left[{\frac {2U_{k}}{k_{b}~T}}\left(1-{\frac {\sigma _{t}}{\sigma _{p}}}\right)^{2}\right]+{\frac {C_{2}}{\sigma _{t}}}\right]^{-1};\quad \sigma _{t}\leq \sigma _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ba016d1678a4edc7415428bb41260ff4a6f61a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.864ex; height:8.009ex;" alt="{\displaystyle {\dot {\varepsilon _{\rm {p}}}}=\left[{\frac {1}{C_{1}}}\exp \left[{\frac {2U_{k}}{k_{b}~T}}\left(1-{\frac {\sigma _{t}}{\sigma _{p}}}\right)^{2}\right]+{\frac {C_{2}}{\sigma _{t}}}\right]^{-1};\quad \sigma _{t}\leq \sigma _{p}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2U_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2U_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97b2bbbde94a83180558661b1d2ff0925990aa5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.839ex; height:2.509ex;" alt="{\displaystyle 2U_{k}}"></span> is the energy to form a <a href="/w/index.php?title=Kink-pair&action=edit&redlink=1" class="new" title="Kink-pair (page does not exist)">kink-pair</a> in a <a href="/w/index.php?title=Dislocation_segment&action=edit&redlink=1" class="new" title="Dislocation segment (page does not exist)">dislocation segment</a> of length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109c17027235d48b00201addbad2c8125389fdd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.675ex; height:2.509ex;" alt="{\displaystyle L_{d}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ad96a35bc2b042df24d22dfc172472bac6c62bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.149ex; height:2.509ex;" alt="{\displaystyle k_{b}}"></span> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd141c4b4b162275b62a7b903a1a065271fae3cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.387ex; height:2.343ex;" alt="{\displaystyle \sigma _{p}}"></span> is the <a href="/wiki/Peierls_stress" title="Peierls stress">Peierls stress</a>. The constants <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1},C_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{1},C_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b1a62be401c4c7a4aabc709187162795b01c896" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.466ex; height:2.509ex;" alt="{\displaystyle C_{1},C_{2}}"></span> are given by the relations </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}:={\frac {\rho _{d}L_{d}ab^{2}\nu }{2w^{2}}};\quad C_{2}:={\frac {D}{\rho _{d}b^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ν</mi> </mrow> <mrow> <mn>2</mn> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="1em" /> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mrow> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{1}:={\frac {\rho _{d}L_{d}ab^{2}\nu }{2w^{2}}};\quad C_{2}:={\frac {D}{\rho _{d}b^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b198887672e520db3f8a96735c149ae9f0de96ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.78ex; height:6.343ex;" alt="{\displaystyle C_{1}:={\frac {\rho _{d}L_{d}ab^{2}\nu }{2w^{2}}};\quad C_{2}:={\frac {D}{\rho _{d}b^{2}}}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab783435e470c8da42062d2910edfeb7008f1c72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.294ex; height:2.176ex;" alt="{\displaystyle \rho _{d}}"></span> is the <a href="/wiki/Dislocation" title="Dislocation">dislocation density</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109c17027235d48b00201addbad2c8125389fdd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.675ex; height:2.509ex;" alt="{\displaystyle L_{d}}"></span> is the length of a dislocation segment, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> is the distance between <a href="/w/index.php?title=Peierls_valleys&action=edit&redlink=1" class="new" title="Peierls valleys (page does not exist)">Peierls valleys</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> is the magnitude of the <a href="/wiki/Burgers_vector" title="Burgers vector">Burgers vector</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> is the <a href="/wiki/Debye_frequency" class="mw-redirect" title="Debye frequency">Debye frequency</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"></span> is the width of a <a href="/wiki/Pinning_points" title="Pinning points">kink loop</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> is the <a href="/wiki/Drag_coefficient" title="Drag coefficient">drag coefficient</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Zerilli–Armstrong_flow_stress_model"><span id="Zerilli.E2.80.93Armstrong_flow_stress_model"></span>Zerilli–Armstrong flow stress model</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=16" title="Edit section: Zerilli–Armstrong flow stress model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Zerilli–Armstrong (ZA) model <sup id="cite_ref-Zerilli87_28-1" class="reference"><a href="#cite_note-Zerilli87-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Zerilli93_36-0" class="reference"><a href="#cite_note-Zerilli93-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Zerilli04_37-0" class="reference"><a href="#cite_note-Zerilli04-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> is based on simplified dislocation mechanics. The general form of the equation for the flow stress is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{(3)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon _{\rm {p}}}},T)=\sigma _{a}+B\exp(-\beta T)+B_{0}{\sqrt {\varepsilon _{\rm {p}}}}\exp(-\alpha T)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>(3)</mtext> </mrow> <mspace width="2em" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>+</mo> <mi>B</mi> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>β</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </msqrt> </mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>α</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{(3)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon _{\rm {p}}}},T)=\sigma _{a}+B\exp(-\beta T)+B_{0}{\sqrt {\varepsilon _{\rm {p}}}}\exp(-\alpha T)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce67f04150c3ab1a7949f8d3c87bdf1ef86052d9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:62.356ex; height:3.343ex;" alt="{\displaystyle {\text{(3)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon _{\rm {p}}}},T)=\sigma _{a}+B\exp(-\beta T)+B_{0}{\sqrt {\varepsilon _{\rm {p}}}}\exp(-\alpha T)~.}"></span> In this model, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ef75a8ddd0a1ba9772f635467a7aec9207f0c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.429ex; height:2.009ex;" alt="{\displaystyle \sigma _{a}}"></span> is the athermal component of the flow stress given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{a}:=\sigma _{g}+{\frac {k_{h}}{\sqrt {l}}}+K\varepsilon _{\rm {p}}^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>:=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <msqrt> <mi>l</mi> </msqrt> </mfrac> </mrow> <mo>+</mo> <mi>K</mi> <msubsup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{a}:=\sigma _{g}+{\frac {k_{h}}{\sqrt {l}}}+K\varepsilon _{\rm {p}}^{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5ed598338c428ce8e14ea717940535425ff375f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.684ex; height:6.343ex;" alt="{\displaystyle \sigma _{a}:=\sigma _{g}+{\frac {k_{h}}{\sqrt {l}}}+K\varepsilon _{\rm {p}}^{n},}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/086f92de077f853afd7f5d22fb3d305cbf5e0ac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.349ex; height:2.343ex;" alt="{\displaystyle \sigma _{g}}"></span> is the contribution due to solutes and initial dislocation density, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{h}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{h}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e36b8b9441fed3f88a3e771633939a9202e4bf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle k_{h}}"></span> is the microstructural stress intensity, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> is the average grain diameter, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> is zero for fcc materials, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B,B_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B,B_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c5a6a6cfea606288ed06bcb6adcfe962bb51d93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.616ex; height:2.509ex;" alt="{\displaystyle B,B_{0}}"></span> are material constants. </p><p>In the thermally activated terms, the functional forms of the exponents <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> are </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\alpha _{0}-\alpha _{1}\ln({\dot {\varepsilon _{\rm {p}}}});\quad \beta =\beta _{0}-\beta _{1}\ln({\dot {\varepsilon _{\rm {p}}}});}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>;</mo> <mspace width="1em" /> <mi>β</mi> <mo>=</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\alpha _{0}-\alpha _{1}\ln({\dot {\varepsilon _{\rm {p}}}});\quad \beta =\beta _{0}-\beta _{1}\ln({\dot {\varepsilon _{\rm {p}}}});}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/415f2e1708fbafa31e7bf6d65a97213d3238aa51" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.255ex; height:3.009ex;" alt="{\displaystyle \alpha =\alpha _{0}-\alpha _{1}\ln({\dot {\varepsilon _{\rm {p}}}});\quad \beta =\beta _{0}-\beta _{1}\ln({\dot {\varepsilon _{\rm {p}}}});}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{0},\alpha _{1},\beta _{0},\beta _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{0},\alpha _{1},\beta _{0},\beta _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7379d06ae1141e6485396b430f47a0e8fa5c32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.926ex; height:2.509ex;" alt="{\displaystyle \alpha _{0},\alpha _{1},\beta _{0},\beta _{1}}"></span> are material parameters that depend on the type of material (fcc, bcc, hcp, alloys). The Zerilli–Armstrong model has been modified by <sup id="cite_ref-Abed05_38-0" class="reference"><a href="#cite_note-Abed05-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> for better performance at high temperatures. </p> <div class="mw-heading mw-heading4"><h4 id="Mechanical_threshold_stress_flow_stress_model">Mechanical threshold stress flow stress model</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=17" title="Edit section: Mechanical threshold stress flow stress model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Mechanical Threshold Stress (MTS) model<sup id="cite_ref-Follans88_29-1" class="reference"><a href="#cite_note-Follans88-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Goto00a_39-0" class="reference"><a href="#cite_note-Goto00a-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kocks01_40-0" class="reference"><a href="#cite_note-Kocks01-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup>) has the form </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{(4)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon }},T)=\sigma _{a}+(S_{i}\sigma _{i}+S_{e}\sigma _{e}){\frac {\mu (p,T)}{\mu _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>(4)</mtext> </mrow> <mspace width="2em" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{(4)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon }},T)=\sigma _{a}+(S_{i}\sigma _{i}+S_{e}\sigma _{e}){\frac {\mu (p,T)}{\mu _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b6a4dc20bf63ac84f64712924d086199afb3f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:49.098ex; height:6.176ex;" alt="{\displaystyle {\text{(4)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon }},T)=\sigma _{a}+(S_{i}\sigma _{i}+S_{e}\sigma _{e}){\frac {\mu (p,T)}{\mu _{0}}}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ef75a8ddd0a1ba9772f635467a7aec9207f0c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.429ex; height:2.009ex;" alt="{\displaystyle \sigma _{a}}"></span> is the athermal component of mechanical threshold stress, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab3208a7d0c634ef720e03ff5a9949e8310edc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.127ex; height:2.009ex;" alt="{\displaystyle \sigma _{i}}"></span> is the component of the flow stress due to intrinsic barriers to thermally activated dislocation motion and dislocation-dislocation interactions, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff8c275c9e5225e96ad468530d986ced509b797d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.326ex; height:2.009ex;" alt="{\displaystyle \sigma _{e}}"></span> is the component of the flow stress due to microstructural evolution with increasing deformation (strain hardening), (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i},S_{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{i},S_{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5105569f8eb33a05db386a06252b76f7bba66966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.682ex; height:2.509ex;" alt="{\displaystyle S_{i},S_{e}}"></span>) are temperature and strain-rate dependent scaling factors, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2fd9b8decb38a3cd158e7b6c0c6e2d987fefcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{0}}"></span> is the shear modulus at 0 K and ambient pressure. </p><p>The scaling factors take the <a href="/wiki/Arrhenius_equation" title="Arrhenius equation">Arrhenius</a> form </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}S_{i}&=\left[1-\left({\frac {k_{b}~T}{g_{0i}b^{3}\mu (p,T)}}\ln {\frac {\dot {\varepsilon _{\rm {0}}}}{\dot {\varepsilon }}}\right)^{1/q_{i}}\right]^{1/p_{i}}\\S_{e}&=\left[1-\left({\frac {k_{b}~T}{g_{0e}b^{3}\mu (p,T)}}\ln {\frac {\dot {\varepsilon _{\rm {0}}}}{\dot {\varepsilon }}}\right)^{1/q_{e}}\right]^{1/p_{e}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mtext> </mtext> <mi>T</mi> </mrow> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>i</mi> </mrow> </msub> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo>˙</mo> </mover> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mtext> </mtext> <mi>T</mi> </mrow> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>e</mi> </mrow> </msub> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo>˙</mo> </mover> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S_{i}&=\left[1-\left({\frac {k_{b}~T}{g_{0i}b^{3}\mu (p,T)}}\ln {\frac {\dot {\varepsilon _{\rm {0}}}}{\dot {\varepsilon }}}\right)^{1/q_{i}}\right]^{1/p_{i}}\\S_{e}&=\left[1-\left({\frac {k_{b}~T}{g_{0e}b^{3}\mu (p,T)}}\ln {\frac {\dot {\varepsilon _{\rm {0}}}}{\dot {\varepsilon }}}\right)^{1/q_{e}}\right]^{1/p_{e}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b566644deb9aecaa7ced39f6b36dcf2081bf8879" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:41.848ex; height:16.343ex;" alt="{\displaystyle {\begin{aligned}S_{i}&=\left[1-\left({\frac {k_{b}~T}{g_{0i}b^{3}\mu (p,T)}}\ln {\frac {\dot {\varepsilon _{\rm {0}}}}{\dot {\varepsilon }}}\right)^{1/q_{i}}\right]^{1/p_{i}}\\S_{e}&=\left[1-\left({\frac {k_{b}~T}{g_{0e}b^{3}\mu (p,T)}}\ln {\frac {\dot {\varepsilon _{\rm {0}}}}{\dot {\varepsilon }}}\right)^{1/q_{e}}\right]^{1/p_{e}}\end{aligned}}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ad96a35bc2b042df24d22dfc172472bac6c62bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.149ex; height:2.509ex;" alt="{\displaystyle k_{b}}"></span> is the Boltzmann constant, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> is the magnitude of the Burgers' vector, (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{0i},g_{0e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{0i},g_{0e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52c5ed91a8561347948e4cae371fba86d549a2b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.694ex; height:2.009ex;" alt="{\displaystyle g_{0i},g_{0e}}"></span>) are normalized activation energies, (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\varepsilon }},{\dot {\varepsilon _{\rm {0}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\varepsilon }},{\dot {\varepsilon _{\rm {0}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60bbe5baf8ad20df02bb4765bd7d7bd42068b6b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.528ex; height:2.509ex;" alt="{\displaystyle {\dot {\varepsilon }},{\dot {\varepsilon _{\rm {0}}}}}"></span>) are the strain-rate and reference strain-rate, and (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{i},p_{i},q_{e},p_{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{i},p_{i},q_{e},p_{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb5da70682fbaf6b913a1e6033e9a26ab0ef1163" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.111ex; height:2.009ex;" alt="{\displaystyle q_{i},p_{i},q_{e},p_{e}}"></span>) are constants. </p><p>The strain hardening component of the mechanical threshold stress (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff8c275c9e5225e96ad468530d986ced509b797d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.326ex; height:2.009ex;" alt="{\displaystyle \sigma _{e}}"></span>) is given by an empirical modified <a href="/w/index.php?title=Voce_law&action=edit&redlink=1" class="new" title="Voce law (page does not exist)">Voce law</a> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{(5)}}\qquad {\frac {d\sigma _{e}}{d\varepsilon _{\rm {p}}}}=\theta (\sigma _{e})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>(5)</mtext> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>θ</mi> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{(5)}}\qquad {\frac {d\sigma _{e}}{d\varepsilon _{\rm {p}}}}=\theta (\sigma _{e})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/655dce9b58a3f644f0817914a7a69f7697776e15" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.319ex; height:6.009ex;" alt="{\displaystyle {\text{(5)}}\qquad {\frac {d\sigma _{e}}{d\varepsilon _{\rm {p}}}}=\theta (\sigma _{e})}"></span> </p><p>where </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\theta (\sigma _{e})&=\theta _{0}[1-F(\sigma _{e})]+\theta _{IV}F(\sigma _{e})\\\theta _{0}&=a_{0}+a_{1}\ln {\dot {\varepsilon _{\rm {p}}}}+a_{2}{\sqrt {\dot {\varepsilon _{\rm {p}}}}}-a_{3}T\\F(\sigma _{e})&={\cfrac {\tanh \left(\alpha {\cfrac {\sigma _{e}}{\sigma _{es}}}\right)}{\tanh(\alpha )}}\\\ln({\cfrac {\sigma _{es}}{\sigma _{0es}}})&=\left({\frac {kT}{g_{0es}b^{3}\mu (p,T)}}\right)\ln \left({\cfrac {\dot {\varepsilon _{\rm {p}}}}{\dot {\varepsilon _{\rm {p}}}}}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>θ</mi> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>V</mi> </mrow> </msub> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </msqrt> </mrow> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>T</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>tanh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>s</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>tanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>s</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>e</mi> <mi>s</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>T</mi> </mrow> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>e</mi> <mi>s</mi> </mrow> </msub> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\theta (\sigma _{e})&=\theta _{0}[1-F(\sigma _{e})]+\theta _{IV}F(\sigma _{e})\\\theta _{0}&=a_{0}+a_{1}\ln {\dot {\varepsilon _{\rm {p}}}}+a_{2}{\sqrt {\dot {\varepsilon _{\rm {p}}}}}-a_{3}T\\F(\sigma _{e})&={\cfrac {\tanh \left(\alpha {\cfrac {\sigma _{e}}{\sigma _{es}}}\right)}{\tanh(\alpha )}}\\\ln({\cfrac {\sigma _{es}}{\sigma _{0es}}})&=\left({\frac {kT}{g_{0es}b^{3}\mu (p,T)}}\right)\ln \left({\cfrac {\dot {\varepsilon _{\rm {p}}}}{\dot {\varepsilon _{\rm {p}}}}}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b47d1a3a75941612979b468f4e8b7191e0de9f7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.338ex; width:41.144ex; height:25.843ex;" alt="{\displaystyle {\begin{aligned}\theta (\sigma _{e})&=\theta _{0}[1-F(\sigma _{e})]+\theta _{IV}F(\sigma _{e})\\\theta _{0}&=a_{0}+a_{1}\ln {\dot {\varepsilon _{\rm {p}}}}+a_{2}{\sqrt {\dot {\varepsilon _{\rm {p}}}}}-a_{3}T\\F(\sigma _{e})&={\cfrac {\tanh \left(\alpha {\cfrac {\sigma _{e}}{\sigma _{es}}}\right)}{\tanh(\alpha )}}\\\ln({\cfrac {\sigma _{es}}{\sigma _{0es}}})&=\left({\frac {kT}{g_{0es}b^{3}\mu (p,T)}}\right)\ln \left({\cfrac {\dot {\varepsilon _{\rm {p}}}}{\dot {\varepsilon _{\rm {p}}}}}\right)\end{aligned}}}"></span> </p><p>and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18b67de6bf25dba7a24e66967ff6319858798734" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.145ex; height:2.509ex;" alt="{\displaystyle \theta _{0}}"></span> is the hardening due to dislocation accumulation, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{IV}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{IV}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46481cf3d46511ea618777faf4f08dd536f0ab3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.415ex; height:2.509ex;" alt="{\displaystyle \theta _{IV}}"></span> is the contribution due to stage-IV hardening, (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0},a_{1},a_{2},a_{3},\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0},a_{1},a_{2},a_{3},\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f3a4eb089828a88552aabf5992d1636c07851dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.76ex; height:2.009ex;" alt="{\displaystyle a_{0},a_{1},a_{2},a_{3},\alpha }"></span>) are constants, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{es}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{es}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0a300ac746b57816efeb44c6ad78b9cbfa85fc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.097ex; height:2.009ex;" alt="{\displaystyle \sigma _{es}}"></span> is the stress at zero strain hardening rate, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{0es}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>e</mi> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{0es}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/579bd7654ac16431db6a7351fb01529a537a191c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.919ex; height:2.009ex;" alt="{\displaystyle \sigma _{0es}}"></span> is the saturation threshold stress for deformation at 0 K, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{0es}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>e</mi> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{0es}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b26edbefaaab46c3ca2a960d942650cd8d7971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.7ex; height:2.009ex;" alt="{\displaystyle g_{0es}}"></span> is a constant, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\varepsilon _{\rm {p}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\varepsilon _{\rm {p}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0927b98902d852820a2dd7011853a3d25389af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.23ex; height:2.843ex;" alt="{\displaystyle {\dot {\varepsilon _{\rm {p}}}}}"></span> is the maximum strain-rate. Note that the maximum strain-rate is usually limited to about <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{7}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8013a64c98fba31f457676460cc7752ddd4d491" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle 10^{7}}"></span>/s. </p> <div class="mw-heading mw-heading4"><h4 id="Preston–Tonks–Wallace_flow_stress_model"><span id="Preston.E2.80.93Tonks.E2.80.93Wallace_flow_stress_model"></span>Preston–Tonks–Wallace flow stress model</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=18" title="Edit section: Preston–Tonks–Wallace flow stress model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Preston–Tonks–Wallace (PTW) model<sup id="cite_ref-Preston03_34-1" class="reference"><a href="#cite_note-Preston03-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> attempts to provide a model for the flow stress for extreme strain-rates (up to 10<sup>11</sup>/s) and temperatures up to melt. A linear Voce hardening law is used in the model. The PTW flow stress is given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{(6)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon _{\rm {p}}}},T)={\begin{cases}2\left[\tau _{s}+\alpha \ln \left[1-\varphi \exp \left(-\beta -{\cfrac {\theta \varepsilon _{\rm {p}}}{\alpha \varphi }}\right)\right]\right]\mu (p,T)&{\text{thermal regime}}\\2\tau _{s}\mu (p,T)&{\text{shock regime}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>(6)</mtext> </mrow> <mspace width="2em" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>2</mn> <mrow> <mo>[</mo> <mrow> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>+</mo> <mi>α</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>φ</mi> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>β</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>φ</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>thermal regime</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>shock regime</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{(6)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon _{\rm {p}}}},T)={\begin{cases}2\left[\tau _{s}+\alpha \ln \left[1-\varphi \exp \left(-\beta -{\cfrac {\theta \varepsilon _{\rm {p}}}{\alpha \varphi }}\right)\right]\right]\mu (p,T)&{\text{thermal regime}}\\2\tau _{s}\mu (p,T)&{\text{shock regime}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f226906fc440c82bafbf328ce74c0df0fb054b9f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:89.756ex; height:10.509ex;" alt="{\displaystyle {\text{(6)}}\qquad \sigma _{y}(\varepsilon _{\rm {p}},{\dot {\varepsilon _{\rm {p}}}},T)={\begin{cases}2\left[\tau _{s}+\alpha \ln \left[1-\varphi \exp \left(-\beta -{\cfrac {\theta \varepsilon _{\rm {p}}}{\alpha \varphi }}\right)\right]\right]\mu (p,T)&{\text{thermal regime}}\\2\tau _{s}\mu (p,T)&{\text{shock regime}}\end{cases}}}"></span> </p><p>with </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha :={\frac {s_{0}-\tau _{y}}{d}};\quad \beta :={\frac {\tau _{s}-\tau _{y}}{\alpha }};\quad \varphi :=\exp(\beta )-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>;</mo> <mspace width="1em" /> <mi>β</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mi>α</mi> </mfrac> </mrow> <mo>;</mo> <mspace width="1em" /> <mi>φ</mi> <mo>:=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha :={\frac {s_{0}-\tau _{y}}{d}};\quad \beta :={\frac {\tau _{s}-\tau _{y}}{\alpha }};\quad \varphi :=\exp(\beta )-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49c1054ad3ab7dfe138da821c621b4918d2ac346" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:48.633ex; height:5.509ex;" alt="{\displaystyle \alpha :={\frac {s_{0}-\tau _{y}}{d}};\quad \beta :={\frac {\tau _{s}-\tau _{y}}{\alpha }};\quad \varphi :=\exp(\beta )-1}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c275a0c624748353b74062f56672cf02b250cab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.019ex; height:2.009ex;" alt="{\displaystyle \tau _{s}}"></span> is a normalized work-hardening saturation stress, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25c32f35eb134d23b3c45f1c878d59b0a112ede4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.145ex; height:2.009ex;" alt="{\displaystyle s_{0}}"></span> is the value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c275a0c624748353b74062f56672cf02b250cab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.019ex; height:2.009ex;" alt="{\displaystyle \tau _{s}}"></span> at 0K, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd30ece77dcd554d540234729f6212aeabeb6ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.065ex; height:2.343ex;" alt="{\displaystyle \tau _{y}}"></span> is a normalized yield stress, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> is the hardening constant in the Voce hardening law, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> is a dimensionless material parameter that modifies the Voce hardening law. </p><p>The saturation stress and the yield stress are given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\tau _{s}&=\max \left\{s_{0}-(s_{0}-s_{\infty }){\rm {{erf}\left[\kappa {\hat {T}}\ln \left({\cfrac {\gamma {\dot {\xi }}}{\dot {\varepsilon _{\rm {p}}}}}\right)\right],s_{0}\left({\cfrac {\dot {\varepsilon _{\rm {p}}}}{\gamma {\dot {\xi }}}}\right)^{s_{1}}}}\right\}\\\tau _{y}&=\max \left\{y_{0}-(y_{0}-y_{\infty }){\rm {{erf}\left[\kappa {\hat {T}}\ln \left({\cfrac {\gamma {\dot {\xi }}}{\dot {\varepsilon _{\rm {p}}}}}\right)\right],\min \left\{y_{1}\left({\cfrac {\dot {\varepsilon _{\rm {p}}}}{\gamma {\dot {\xi }}}}\right)^{y_{2}},s_{0}\left({\cfrac {\dot {\varepsilon _{\rm {p}}}}{\gamma {\dot {\xi }}}}\right)^{s_{1}}\right\}}}\right\}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">f</mi> </mrow> <mrow> <mo>[</mo> <mrow> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ξ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>,</mo> <msub> <mi mathvariant="normal">s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ξ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mrow> </mrow> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">f</mi> </mrow> <mrow> <mo>[</mo> <mrow> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ξ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>,</mo> <mo movablelimits="true" form="prefix">min</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi mathvariant="normal">y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ξ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mo>,</mo> <msub> <mi mathvariant="normal">s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ξ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mrow> <mo>}</mo> </mrow> </mrow> </mrow> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tau _{s}&=\max \left\{s_{0}-(s_{0}-s_{\infty }){\rm {{erf}\left[\kappa {\hat {T}}\ln \left({\cfrac {\gamma {\dot {\xi }}}{\dot {\varepsilon _{\rm {p}}}}}\right)\right],s_{0}\left({\cfrac {\dot {\varepsilon _{\rm {p}}}}{\gamma {\dot {\xi }}}}\right)^{s_{1}}}}\right\}\\\tau _{y}&=\max \left\{y_{0}-(y_{0}-y_{\infty }){\rm {{erf}\left[\kappa {\hat {T}}\ln \left({\cfrac {\gamma {\dot {\xi }}}{\dot {\varepsilon _{\rm {p}}}}}\right)\right],\min \left\{y_{1}\left({\cfrac {\dot {\varepsilon _{\rm {p}}}}{\gamma {\dot {\xi }}}}\right)^{y_{2}},s_{0}\left({\cfrac {\dot {\varepsilon _{\rm {p}}}}{\gamma {\dot {\xi }}}}\right)^{s_{1}}\right\}}}\right\}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45a0fe46dac78ab83be2929302d280c8c69d2eb9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:80.499ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\tau _{s}&=\max \left\{s_{0}-(s_{0}-s_{\infty }){\rm {{erf}\left[\kappa {\hat {T}}\ln \left({\cfrac {\gamma {\dot {\xi }}}{\dot {\varepsilon _{\rm {p}}}}}\right)\right],s_{0}\left({\cfrac {\dot {\varepsilon _{\rm {p}}}}{\gamma {\dot {\xi }}}}\right)^{s_{1}}}}\right\}\\\tau _{y}&=\max \left\{y_{0}-(y_{0}-y_{\infty }){\rm {{erf}\left[\kappa {\hat {T}}\ln \left({\cfrac {\gamma {\dot {\xi }}}{\dot {\varepsilon _{\rm {p}}}}}\right)\right],\min \left\{y_{1}\left({\cfrac {\dot {\varepsilon _{\rm {p}}}}{\gamma {\dot {\xi }}}}\right)^{y_{2}},s_{0}\left({\cfrac {\dot {\varepsilon _{\rm {p}}}}{\gamma {\dot {\xi }}}}\right)^{s_{1}}\right\}}}\right\}\end{aligned}}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d118938b0c46c9aa6aa9ea86aa4251443146b6af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.966ex; height:2.009ex;" alt="{\displaystyle s_{\infty }}"></span> is the value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c275a0c624748353b74062f56672cf02b250cab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.019ex; height:2.009ex;" alt="{\displaystyle \tau _{s}}"></span> close to the melt temperature, (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{0},y_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{0},y_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cdbacaefd97bfc51fdfbda4762db6df669f2185" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.242ex; height:2.009ex;" alt="{\displaystyle y_{0},y_{\infty }}"></span>) are the values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd30ece77dcd554d540234729f6212aeabeb6ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.065ex; height:2.343ex;" alt="{\displaystyle \tau _{y}}"></span> at 0 K and close to melt, respectively, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\kappa ,\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>κ</mi> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\kappa ,\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/325cb26b8294eee639f27303c9d257f1f797ce6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.445ex; height:2.843ex;" alt="{\displaystyle (\kappa ,\gamma )}"></span> are material constants, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {T}}=T/T_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {T}}=T/T_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f7f16de1a461e31f72d0bc326b33a21fd6cc8ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.704ex; height:3.343ex;" alt="{\displaystyle {\hat {T}}=T/T_{m}}"></span>, (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1},y_{1},y_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{1},y_{1},y_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e84ac22e47a3fe0721946bd59d75063419bcc4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.6ex; height:2.009ex;" alt="{\displaystyle s_{1},y_{1},y_{2}}"></span>) are material parameters for the high strain-rate regime, and </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\xi }}={\frac {1}{2}}\left({\cfrac {4\pi \rho }{3M}}\right)^{1/3}\left({\cfrac {\mu (p,T)}{\rho }}\right)^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ξ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>π</mi> <mi>ρ</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>M</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\xi }}={\frac {1}{2}}\left({\cfrac {4\pi \rho }{3M}}\right)^{1/3}\left({\cfrac {\mu (p,T)}{\rho }}\right)^{1/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34d488188da7c3e2a4a428cc405edc9f5ad47a41" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.703ex; height:8.009ex;" alt="{\displaystyle {\dot {\xi }}={\frac {1}{2}}\left({\cfrac {4\pi \rho }{3M}}\right)^{1/3}\left({\cfrac {\mu (p,T)}{\rho }}\right)^{1/2}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> is the density, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is the atomic mass. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Viscoelasticity" title="Viscoelasticity">Viscoelasticity</a></li> <li><a href="/wiki/Bingham_plastic" title="Bingham plastic">Bingham plastic</a></li> <li><a href="/wiki/Dashpot" title="Dashpot">Dashpot</a></li> <li><a href="/wiki/Creep_(deformation)" title="Creep (deformation)">Creep (deformation)</a></li> <li><a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">Plasticity (physics)</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li> <li><a href="/wiki/Quasi-solid" title="Quasi-solid">Quasi-solid</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Viscoplasticity&action=edit&section=20" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-Perzyna-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Perzyna_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFPerzyna1966" class="citation journal cs2">Perzyna, P. 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