= Secant modulus
{title2=$E_{\rm sec}=\sigma/\varepsilon$}
The \b[secant modulus] at a specified point of a <stress-strain curve> is the slope of the line from the declared unloaded origin to that point:
$$
E_{\rm sec}(\varepsilon_*)=\frac{\sigma(\varepsilon_*)}{\varepsilon_*},\qquad \varepsilon_*\ne0.
$$
It depends on the chosen point when the curve is nonlinear. Mississippi State University's solid-mechanics laboratory material defines it geometrically in this way.<Secant Modulus in the Mississippi State University Virtual Laboratory for Solid Mechanics>[Strength and Stiffness Characteristics, Secant Modulus definition and diagram at pointA]{ref} Specify both the strain measure and the origin when reporting a <secant modulus>.
For an original branch $\sigma=E_0\varepsilon+B\varepsilon^2$, with $E_0=2\,\mathrm{GPa}$ and $B=100\,\mathrm{GPa}$,
$$
E_{\rm sec}=E_0+B\varepsilon,\qquad
E_{\rm tan}=E_0+2B\varepsilon.
$$
At $\varepsilon_*=0.02$, stress is $80\,\mathrm{MPa}$, the <secant modulus> is $4\,\mathrm{GPa}$ and the <tangent modulus> is $6\,\mathrm{GPa}$. The first slope relates accumulated stress and strain; the second predicts a small further increment. In <linear elasticity> through the unloaded origin they both equal <Young's modulus>.
The relation is especially clear for a smooth curve with $\sigma(0)=0$. The <fundamental theorem of calculus> gives
$$
E_{\rm sec}(\varepsilon_*)=
\frac1{\varepsilon_*}\int_0^{\varepsilon_*}E_{\rm tan}(u)\,du.
$$
Thus the <secant modulus> averages the local tangent slopes over that interval. A <chord modulus> instead joins two specified states, with slope $(\sigma_2-\sigma_1)/(\varepsilon_2-\varepsilon_1)$; its first state need not be the unloaded origin.
For a tensile measurement using initial cross-sectional area $A_0$ and local <gauge length> $L_g$, <engineering stress> and <engineering strain> give
$$
E_{\rm sec}=\frac{L_g}{A_0}\frac{F}{\Delta L_g},
$$
where extension is measured from the appropriate unloaded state. An original $F=800\,\mathrm N$, $A_0=10\,\mathrm{mm^2}$, $L_g=50\,\mathrm{mm}$ and $\Delta L_g=1\,\mathrm{mm}$ yields $E_{\rm sec}=4000\,\mathrm{N\,mm^{-2}}=4\,\mathrm{GPa}$. A local <extensometer> avoids attributing all <machine compliance> to specimen extension. Retain the loading history, temperature and target stress or strain: a secant from an arbitrary preload and an unloading slope are different measurements, particularly after <plastic deformation>.
Back to article page
