Necking (engineering)

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decoding="async" width="220" height="66" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Necking_Example.png/330px-Necking_Example.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Necking_Example.png/440px-Necking_Example.png 2x" data-file-width="620" data-file-height="186" /></a><figcaption>The above image shows a test specimen, when of a certain kind of material, and experienced under a great enough load, experiences necking. The portion where necking occurs may be called the neck of the specimen. </figcaption></figure> <p>In <a href="/wiki/Engineering" title="Engineering">engineering</a> and <a href="/wiki/Materials_science" title="Materials science">materials science</a>, <b>necking</b> is a mode of tensile <a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">deformation</a> where relatively large amounts of <a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">strain</a> localize disproportionately in a small region of the material. The resulting prominent decrease in local cross-sectional area provides the basis for the name "neck". Because the local strains in the neck are large, necking is often closely associated with <a href="/wiki/Yield_(engineering)" title="Yield (engineering)">yielding</a>, a form of <a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">plastic</a> deformation associated with <a href="/wiki/Ductile" class="mw-redirect" title="Ductile">ductile</a> materials, often <a href="/wiki/Metal" title="Metal">metals</a> or <a href="/wiki/Polymer" title="Polymer">polymers</a>.<sup id="cite_ref-Kinloch_&_Young_1-0" class="reference"><a href="#cite_note-Kinloch_&_Young-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Once necking has begun, the neck becomes the exclusive location of yielding in the material, as the reduced area gives the neck the largest local <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">stress</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formation">Formation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Necking_(engineering)&action=edit&section=1" title="Edit section: Formation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Necking results from an <a href="/wiki/Instability" title="Instability">instability</a> during tensile deformation when the cross-sectional area of the sample decreases by a greater proportion than the material <a href="/wiki/Strain_hardening" class="mw-redirect" title="Strain hardening">strain hardens</a>. <a href="https://fr.wikipedia.org/wiki/Armand_Consid%C3%A8re" class="extiw" title="fr:Armand Considère">Armand Considère</a> published the basic criterion for necking in 1885, in the context of the stability of large scale structures such as bridges.<sup id="cite_ref-Considère_2-0" class="reference"><a href="#cite_note-Considère-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Three concepts provide the framework for understanding neck formation. </p> <ol><li>Before deformation, all real materials have heterogeneities such as flaws or local variations in dimensions or composition that cause local fluctuations in <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stresses</a> and <a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">strains</a>. To determine the location of the incipient neck, these fluctuations need only be <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> in magnitude.</li> <li>During plastic tensile deformation the material decreases in cross-sectional area due to the incompressibility of plastic flow. (Not due to the <a href="/wiki/Poisson%27s_ratio" title="Poisson's ratio">Poisson effect</a>, which is linked to elastic behaviour.)</li> <li>During plastic tensile deformation the material strain hardens. The amount of hardening varies with extent of deformation.</li></ol> <p>The latter two effects determine the stability while the first effect determines the neck's location. </p> <div class="mw-heading mw-heading3"><h3 id="The_Considère_treatment"><span id="The_Consid.C3.A8re_treatment"></span>The Considère treatment</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Necking_(engineering)&action=edit&section=2" title="Edit section: The Considère treatment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Instability (onset of necking) is expected to occur when an increase in the (local) strain produces no net increase in the load, <span class="texhtml mvar" style="font-style:italic;">F</span>. This will happen when </p> <dl><dd><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 \Delta F=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>F</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta F=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5ca34e404e4af194ca936d0e1a6c2e7918d5dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.938ex; height:2.176ex;" alt="{\displaystyle \Delta F=0}"></span></dd></dl> <p>This leads to </p> <dl><dd><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 {\begin{aligned}&F=A\sigma _{T}\\[4pt]\implies &dF=Ad\sigma _{T}+\sigma _{T}dA=0\\[4pt]\implies &{\frac {d\sigma _{T}}{\sigma _{T}}}=-{\frac {dA}{A}}={\frac {dL}{L}}=d\varepsilon _{T}\\\implies &\sigma _{T}={\frac {d\sigma _{T}}{d\varepsilon _{T}}}\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="0.7em 0.7em 0.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>F</mi> <mo>=</mo> <mi>A</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> </mtd> <mtd> <mi>d</mi> <mi>F</mi> <mo>=</mo> <mi>A</mi> <mi>d</mi> <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>T</mi> </mrow> </msub> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>A</mi> </mrow> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo>=</mo> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&F=A\sigma _{T}\\[4pt]\implies &dF=Ad\sigma _{T}+\sigma _{T}dA=0\\[4pt]\implies &{\frac {d\sigma _{T}}{\sigma _{T}}}=-{\frac {dA}{A}}={\frac {dL}{L}}=d\varepsilon _{T}\\\implies &\sigma _{T}={\frac {d\sigma _{T}}{d\varepsilon _{T}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5781ba557e2aea60523ad5c8f33762f32c59daf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:32.838ex; height:19.509ex;" alt="{\displaystyle {\begin{aligned}&F=A\sigma _{T}\\[4pt]\implies &dF=Ad\sigma _{T}+\sigma _{T}dA=0\\[4pt]\implies &{\frac {d\sigma _{T}}{\sigma _{T}}}=-{\frac {dA}{A}}={\frac {dL}{L}}=d\varepsilon _{T}\\\implies &\sigma _{T}={\frac {d\sigma _{T}}{d\varepsilon _{T}}}\end{aligned}}}"></span></dd></dl> <p>with the <span class="texhtml mvar" style="font-style:italic;">T</span> subscript being used to emphasize that these stresses and strains must be true values. Necking is thus predicted to start when the slope of the true stress / true strain curve falls to a value equal to the true stress at that point. </p> <div class="mw-heading mw-heading3"><h3 id="Application_to_metals">Application to metals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Necking_(engineering)&action=edit&section=3" title="Edit section: Application to metals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Necking commonly arises in both metals and polymers. However, while the phenomenon is caused by the same basic effect in both materials, they tend to have different types of (true) stress-strain curve, such that they should be considered separately in terms of necking behaviour. For metals, the (true) stress tends to rise monotonically with increasing strain, although the gradient (<a href="/wiki/Work_hardening" title="Work hardening">work hardening</a> rate) tends to fall off progressively. This is primarily due to a progressive fall in <a href="/wiki/Dislocation" title="Dislocation">dislocation</a> mobility, caused by interactions between them. With polymers, on the other hand, the curve can be more complex. For example, the gradient can in some cases rise sharply with increasing strain, due to the <a href="/wiki/Polymer_chains" class="mw-redirect" title="Polymer chains">polymer chains</a> becoming aligned as they reorganise during plastic deformation. This can lead to a stable neck. No effect of this type is possible in metals. </p><p>The figure shows a screenshot from an interactive simulation available on the <a href="/wiki/DoITPoMS" title="DoITPoMS">DoITPoMS</a> educational website. The construction is shown for a (true) stress-strain curve represented by a simple analytical expression (Ludwik-Hollomon). </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Considere_1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Considere_1.jpg/220px-Considere_1.jpg" decoding="async" width="220" height="248" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Considere_1.jpg/330px-Considere_1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Considere_1.jpg/440px-Considere_1.jpg 2x" data-file-width="3068" data-file-height="3460" /></a><figcaption>The Considère construction for prediction of the onset of necking, expressed as the gradient of the (true) stress-strain curve falling to the true stress, for a material conforming to the Ludwik-Hollomon relationship, with the parameter values shown.</figcaption></figure> <p>The condition can also be expressed in terms of the nominal strain: </p> <dl><dd><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 {\begin{aligned}{\frac {d\sigma _{T}}{d\varepsilon _{T}}}&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}{\frac {d\varepsilon _{N}}{d\varepsilon _{T}}}\\[4pt]&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}{\frac {dL/L_{0}}{dL/L}}\\[4pt]&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}{\frac {L}{L_{0}}}\\[4pt]&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}(1+\varepsilon _{\mathrm {N} })\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="0.7em 0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d\sigma _{T}}{d\varepsilon _{T}}}&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}{\frac {d\varepsilon _{N}}{d\varepsilon _{T}}}\\[4pt]&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}{\frac {dL/L_{0}}{dL/L}}\\[4pt]&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}{\frac {L}{L_{0}}}\\[4pt]&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}(1+\varepsilon _{\mathrm {N} })\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8570040d3cfc50e0452c31cb5a5663e5e74a0c09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.838ex; width:21.806ex; height:26.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {d\sigma _{T}}{d\varepsilon _{T}}}&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}{\frac {d\varepsilon _{N}}{d\varepsilon _{T}}}\\[4pt]&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}{\frac {dL/L_{0}}{dL/L}}\\[4pt]&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}{\frac {L}{L_{0}}}\\[4pt]&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}(1+\varepsilon _{\mathrm {N} })\end{aligned}}}"></span></dd></dl> <p>Therefore, at the instability point: </p> <pre><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}={\frac {d\sigma _{T}}{d\varepsilon _{N}}}(1+\varepsilon _{N})}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{T}={\frac {d\sigma _{T}}{d\varepsilon _{N}}}(1+\varepsilon _{N})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef073bd7c8bbacd3da8918d80979a5e7735d3d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.229ex; height:5.843ex;" alt="{\displaystyle \sigma _{T}={\frac {d\sigma _{T}}{d\varepsilon _{N}}}(1+\varepsilon _{N})}"></span> </pre> <p>It can therefore also be formulated in terms of a plot of true stress against nominal strain. On such a plot, necking will start where a line from the point <span class="texhtml"><i>ε<sub>N</sub></i> = –1</span> forms a tangent to the curve. This is shown in the next figure, which was obtained using the same Ludwik-Hollomon representation of the true stress – true strain relationship as that of the previous figure. </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Considere_2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Considere_2.jpg/220px-Considere_2.jpg" decoding="async" width="220" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Considere_2.jpg/330px-Considere_2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Considere_2.jpg/440px-Considere_2.jpg 2x" data-file-width="3351" data-file-height="3480" /></a><figcaption>The Considère construction for prediction of the onset of necking, expressed as the point where the gradient of the true stress – nominal strain curve extrapolates back to a nominal strain of -1 at zero stress, for a material conforming to the Ludwik-Hollomon relationship, with the parameter values shown.</figcaption></figure> <p>Importantly, the condition also corresponds to a peak (plateau) in the nominal stress – nominal strain plot. This can be seen on obtaining the gradient of such a plot by differentiating the expression for <span class="texhtml mvar" style="font-style:italic;">σ<sub>N</sub></span> with respect to <span class="texhtml mvar" style="font-style:italic;">ε<sub>N</sub></span>. </p> <dl><dd><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 {\begin{aligned}\sigma _{N}&={\frac {\sigma _{T}}{1+\varepsilon _{N}}}\\[4pt]\therefore {\frac {d\sigma _{N}}{d\varepsilon _{N}}}&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}{\frac {1}{1+\varepsilon _{N}}}-{\frac {\sigma _{T}}{(1+\varepsilon _{N})^{2}}}\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="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>∴</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma _{N}&={\frac {\sigma _{T}}{1+\varepsilon _{N}}}\\[4pt]\therefore {\frac {d\sigma _{N}}{d\varepsilon _{N}}}&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}{\frac {1}{1+\varepsilon _{N}}}-{\frac {\sigma _{T}}{(1+\varepsilon _{N})^{2}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81d861a442a8184cee57913a73baa0b2e03afa4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:36.875ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}\sigma _{N}&={\frac {\sigma _{T}}{1+\varepsilon _{N}}}\\[4pt]\therefore {\frac {d\sigma _{N}}{d\varepsilon _{N}}}&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}{\frac {1}{1+\varepsilon _{N}}}-{\frac {\sigma _{T}}{(1+\varepsilon _{N})^{2}}}\end{aligned}}}"></span></dd></dl> <p>Substituting for the true stress – nominal strain gradient (at the onset of necking): </p> <dl><dd><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 {\frac {d\sigma _{N}}{d\varepsilon _{N}}}={\frac {\sigma _{T}}{1+\varepsilon _{N}}}{\frac {1}{1+\varepsilon _{N}}}-{\frac {\sigma _{T}}{(1+\varepsilon _{N})^{2}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\sigma _{N}}{d\varepsilon _{N}}}={\frac {\sigma _{T}}{1+\varepsilon _{N}}}{\frac {1}{1+\varepsilon _{N}}}-{\frac {\sigma _{T}}{(1+\varepsilon _{N})^{2}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901b678846d6d217df2e6575388f4d78e8c41eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.976ex; height:6.176ex;" alt="{\displaystyle {\frac {d\sigma _{N}}{d\varepsilon _{N}}}={\frac {\sigma _{T}}{1+\varepsilon _{N}}}{\frac {1}{1+\varepsilon _{N}}}-{\frac {\sigma _{T}}{(1+\varepsilon _{N})^{2}}}=0}"></span></dd></dl> <p>This condition can also be seen in the two figures. Since many stress-strain curves are presented as nominal plots, and this is a simple condition that can be identified by visual inspection, it is in many ways the easiest criterion to use to establish the onset of necking. It also corresponds to the “strength” (<a href="/wiki/Ultimate_tensile_stress" class="mw-redirect" title="Ultimate tensile stress">ultimate tensile stress</a>), at least for metals that do neck (which covers the majority of “engineering” metals). On the other hand, the peak in a nominal stress-strain curve is commonly a fairly flat plateau, rather than a sharp maximum, so accurate assessment of the strain at the onset of necking may be difficult. Nevertheless, this strain is a meaningful indication of the “ductility” of the metal – more so than the commonly-used “nominal strain at fracture”, which depends on the aspect ratio of the gauge length of the tensile test-piece<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> – see the article on <a href="/wiki/Ductility" title="Ductility">ductility</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Application_to_polymers">Application to polymers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Necking_(engineering)&action=edit&section=4" title="Edit section: Application to polymers"><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:Stable_neck_MDPE.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Stable_neck_MDPE.jpg/110px-Stable_neck_MDPE.jpg" decoding="async" width="110" height="432" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Stable_neck_MDPE.jpg/165px-Stable_neck_MDPE.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Stable_neck_MDPE.jpg/220px-Stable_neck_MDPE.jpg 2x" data-file-width="504" data-file-height="1980" /></a><figcaption>A <a href="/wiki/Polyethylene" title="Polyethylene">polyethylene</a> sample that has necked under tension</figcaption></figure> <p>The tangent construction shown above is rarely used in interpreting the stress-strain curves of metals. However, it is popular for analysis of the tensile drawing of polymers.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> (since it allows study of the regime of stable necking). It may be noted that, for polymers, the strain is commonly expressed as a “draw ratio”, rather than a strain: in this case, extrapolation of the tangent is carried out to a draw ratio of zero, rather than a strain of -1. </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Consid%C3%A9re_graphical_criterion-labels.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Consid%C3%A9re_graphical_criterion-labels.svg/301px-Consid%C3%A9re_graphical_criterion-labels.svg.png" decoding="async" width="301" height="241" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Consid%C3%A9re_graphical_criterion-labels.svg/452px-Consid%C3%A9re_graphical_criterion-labels.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Consid%C3%A9re_graphical_criterion-labels.svg/602px-Consid%C3%A9re_graphical_criterion-labels.svg.png 2x" data-file-width="512" data-file-height="410" /></a><figcaption>Graphical construction indicating criteria for neck formation and neck stabilization.</figcaption></figure> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Considere_graphical_no-neck.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Considere_graphical_no-neck.svg/315px-Considere_graphical_no-neck.svg.png" decoding="async" width="315" height="252" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Considere_graphical_no-neck.svg/473px-Considere_graphical_no-neck.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Considere_graphical_no-neck.svg/630px-Considere_graphical_no-neck.svg.png 2x" data-file-width="512" data-file-height="410" /></a><figcaption>Graphical construction for a material that deforms homogeneously at all draw ratios.</figcaption></figure> <p>The plots relate (top) to a material that forms a stable neck and (bottom) a material that deforms homogeneously at all draw ratios. </p><p>As deformation proceeds, the geometric instability causes strain to continue concentrating in the neck until the material either ruptures or the necked material hardens enough, as indicated by the second tangent point in the top diagram, to cause other regions of the material to deform instead. The amount of strain in the stable neck is called the <i>natural draw ratio</i><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> because it is determined by the material's hardening characteristics, not the amount of drawing imposed on the material. Ductile polymers often exhibit stable necks because molecular orientation provides a mechanism for hardening that predominates at large strains.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </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=Necking_(engineering)&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Stress%E2%80%93strain_curve" title="Stress–strain curve">Stress–strain curve</a></li> <li><a href="/wiki/Trace_necking" class="mw-redirect" title="Trace necking">Trace necking</a></li> <li><a href="/wiki/Universal_testing_machine" title="Universal testing machine">Universal testing machine</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=Necking_(engineering)&action=edit&section=6" 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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Kinloch_&_Young-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kinloch_&_Young_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="CITEREFKinlochYoung1995" class="citation book cs1">Kinloch, AJ; Young, RJ (1995). <i>Fracture Behaviour of Polymers</i>. Chapman and Hall. p. 108. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789401715966" title="Special:BookSources/9789401715966"><bdi>9789401715966</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fracture+Behaviour+of+Polymers&rft.pages=108&rft.pub=Chapman+and+Hall&rft.date=1995&rft.isbn=9789401715966&rft.aulast=Kinloch&rft.aufirst=AJ&rft.au=Young%2C+RJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANecking+%28engineering%29" class="Z3988"></span></span> </li> <li id="cite_note-Considère-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Considère_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConsidère1885" class="citation journal cs1">Considère, Armand (1885). <i>Annales des Ponts et Chaussées</i>. <b>9</b>: 574.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+des+Ponts+et+Chauss%C3%A9es&rft.volume=9&rft.pages=574&rft.date=1885&rft.aulast=Consid%C3%A8re&rft.aufirst=Armand&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANecking+%28engineering%29" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Missing or empty <code class="cs1-code">|title=</code> (<a href="/wiki/Help:CS1_errors#citation_missing_title" title="Help:CS1 errors">help</a>)</span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaticKirbyJolles1988" class="citation journal cs1">Matic, P; Kirby, GC; Jolles, MI (1988). "The Relation of Tensile Specimen Size and Geometry Effects to Unique Constitutive Parameters for Ductile Materials". <i>Proceedings of the Royal Society of London A</i>. <b>417</b> (1853): 309–333. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1988RSPSA.417..309M">1988RSPSA.417..309M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1988.0063">10.1098/rspa.1988.0063</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:43033448">43033448</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+of+London+A&rft.atitle=The+Relation+of+Tensile+Specimen+Size+and+Geometry+Effects+to+Unique+Constitutive+Parameters+for+Ductile+Materials&rft.volume=417&rft.issue=1853&rft.pages=309-333&rft.date=1988&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A43033448%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frspa.1988.0063&rft_id=info%3Abibcode%2F1988RSPSA.417..309M&rft.aulast=Matic&rft.aufirst=P&rft.au=Kirby%2C+GC&rft.au=Jolles%2C+MI&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANecking+%28engineering%29" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcKinleyHassager1999" class="citation journal cs1">McKinley, GH; Hassager, O (1999). "The Considere Condition and Rapid Stretching of Linear and Branched Polymer Melts". <i>Journal of Rheology</i>. <b>43</b> (5): 1195–1212. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999JRheo..43.1195M">1999JRheo..43.1195M</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.498.6808">10.1.1.498.6808</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1122%2F1.551034">10.1122/1.551034</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Rheology&rft.atitle=The+Considere+Condition+and+Rapid+Stretching+of+Linear+and+Branched+Polymer+Melts&rft.volume=43&rft.issue=5&rft.pages=1195-1212&rft.date=1999&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.498.6808%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1122%2F1.551034&rft_id=info%3Abibcode%2F1999JRheo..43.1195M&rft.aulast=McKinley&rft.aufirst=GH&rft.au=Hassager%2C+O&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANecking+%28engineering%29" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCristMetaxas2004" class="citation journal cs1">Crist, B; Metaxas, C (2004). "Neck Propagation in Polyethylene". <i>Journal of Polymer Science Part B</i>. <b>42</b> (11): 2081–2091. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004JPoSB..42.2081C">2004JPoSB..42.2081C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fpolb.20087">10.1002/polb.20087</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Polymer+Science+Part+B&rft.atitle=Neck+Propagation+in+Polyethylene&rft.volume=42&rft.issue=11&rft.pages=2081-2091&rft.date=2004&rft_id=info%3Adoi%2F10.1002%2Fpolb.20087&rft_id=info%3Abibcode%2F2004JPoSB..42.2081C&rft.aulast=Crist&rft.aufirst=B&rft.au=Metaxas%2C+C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANecking+%28engineering%29" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSéguéla2007" class="citation journal cs1">Séguéla, R (2007). <a rel="nofollow" class="external text" href="https://hal.science/hal-04709308">"On the Natural Draw Ratio of Semi-Crystalline Polymers: Review of the Mechanical, Physical and Molecular Aspects"</a>. <i>Macromolecular Materials and Engineering</i>. <b>292</b> (3): 235–244. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fmame.200600389">10.1002/mame.200600389</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Macromolecular+Materials+and+Engineering&rft.atitle=On+the+Natural+Draw+Ratio+of+Semi-Crystalline+Polymers%3A+Review+of+the+Mechanical%2C+Physical+and+Molecular+Aspects&rft.volume=292&rft.issue=3&rft.pages=235-244&rft.date=2007&rft_id=info%3Adoi%2F10.1002%2Fmame.200600389&rft.aulast=S%C3%A9gu%C3%A9la&rft.aufirst=R&rft_id=https%3A%2F%2Fhal.science%2Fhal-04709308&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANecking+%28engineering%29" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaward2007" class="citation journal cs1">Haward, JN (2007). "Strain Hardening of High Density Polyethylene". <i>Journal of Polymer Science Part B</i>. <b>45</b> (9): 1090–1099. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007JPoSB..45.1090H">2007JPoSB..45.1090H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fpolb.21123">10.1002/polb.21123</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Polymer+Science+Part+B&rft.atitle=Strain+Hardening+of+High+Density+Polyethylene&rft.volume=45&rft.issue=9&rft.pages=1090-1099&rft.date=2007&rft_id=info%3Adoi%2F10.1002%2Fpolb.21123&rft_id=info%3Abibcode%2F2007JPoSB..45.1090H&rft.aulast=Haward&rft.aufirst=JN&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANecking+%28engineering%29" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Necking_(engineering)&oldid=1250980706">https://en.wikipedia.org/w/index.php?title=Necking_(engineering)&oldid=1250980706</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Materials_science" title="Category:Materials science">Materials science</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_errors:_missing_title" title="Category:CS1 errors: missing title">CS1 errors: missing title</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 13 October 2024, at 17:51<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Necking (engineering) - Wikipedia