= Necking
{disambiguate=engineering}
{wiki}
= Necking
{synonym}
<Necking> is localization of tensile <deformation> into a narrowing part of a specimen. A thinner cross-section carries greater local <stress> at the same axial <force>, encouraging further thinning. <Work hardening> opposes this feedback by increasing the <stress> required for additional <plastic deformation>.
For a uniformly extending specimen whose plastic flow approximately conserves <volume>, write $A=A_0e^{-\varepsilon}$ with logarithmic length change $\varepsilon=\log(L/L_0)$. Its load is $F=\sigma A$, where $\sigma$ is axial <Cauchy stress tensor>. Then
$$
\frac{dF}{d\varepsilon}=A\left(\frac{d\sigma}{d\varepsilon}-\sigma\right).
$$
The maximum-load condition is $d\sigma/d\varepsilon=\sigma$. This ideal estimate assumes uniform uniaxial deformation up to onset and neglects rate effects and damage. In <tensile testing>, recording both axial <force> and local specimen diameter distinguishes the start of localization from the later <fracture>; the maximum <engineering stress> need not be the maximum local <Cauchy stress tensor>.
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