= Tangent modulus
{title2=$E_{\rm tan}=d\sigma/d\varepsilon$}
{wiki}
The \b[tangent modulus] is the local slope of a specified uniaxial <stress-strain curve>:
$$
E_{\rm tan}(\varepsilon_0)=\left.\frac{d\sigma}{d\varepsilon}\right|_{\varepsilon=\varepsilon_0}.
$$
It converts a small additional <strain> near $\varepsilon_0$ into the corresponding first-order change of <stress>. For nonlinear <elastic deformation>, this slope can vary with the operating point.<Elastic Modulus in the Princeton Materials and Mechanics glossary>[Elastic Modulus entry, final definition and derivative formula]{ref} In uniaxial <linear elasticity>, $\sigma=E\varepsilon$, so the <tangent modulus> equals the constant <Young's modulus>.<Development of Constitutive Equations for Continuum, Beams and Plates by Tomasz Wierzbicki>[Lecture4, p4-1, equation4.1; p4-2, uniaxial-tension assumptions]{ref}
For an original nonlinear branch, stipulate
$$
\sigma(\varepsilon)=E_0\varepsilon+B\varepsilon^2,\qquad
E_0=2\,\mathrm{GPa},\quad B=100\,\mathrm{GPa}.
$$
Then $E_{\rm tan}=E_0+2B\varepsilon$. At $\varepsilon_0=0.02$, the stress is $80\,\mathrm{MPa}$ and the <tangent modulus> is $6\,\mathrm{GPa}$. The <secant modulus> from the stipulated unloaded origin is instead $\sigma/\varepsilon_0=4\,\mathrm{GPa}$. The two slopes answer different questions: the secant compares total stress and strain, while the tangent predicts a small further increment.
For $\Delta\varepsilon=0.001$, the tangent predicts $\Delta\sigma\simeq6\,\mathrm{MPa}$. The exact change in the chosen polynomial is $6.1\,\mathrm{MPa}$ because the omitted term is $B(\Delta\varepsilon)^2=0.1\,\mathrm{MPa}$. This is a <linear approximation> whose error decreases quadratically with the increment, not a claim that the nonlinear material obeys <Hooke's law> globally.
To estimate a <tangent modulus> experimentally, specify the stress and strain measures, temperature, loading history and loading rate. For <engineering stress> $\sigma=F/A_0$ and <engineering strain> $\varepsilon=\Delta L_g/L_g$ from a local <extensometer>,
$$
E_{\rm tan}=\frac{L_g}{A_0}\frac{dF}{d(\Delta L_g)}.
$$
Fit a local window about the chosen operating point and check how the estimate changes with that window. For an original $L_g=50\,\mathrm{mm}$, $A_0=10\,\mathrm{mm^2}$ and local slope $1200\,\mathrm{N\,mm^{-1}}$, this gives $6000\,\mathrm{N\,mm^{-2}}=6\,\mathrm{GPa}$. Differentiation amplifies measurement noise, and <crosshead displacement> can include <machine compliance>. Where <plastic deformation> or history-dependent response occurs, identify the loading branch: its local slope need not equal the unloading slope or an elastic material constant.
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