Secant modulus ($E_{\sec }=\sigma /\epsilon$)

The 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: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.^[ref] Specify both the strain measure and the origin when reporting a secant modulus.

For an original branch $\sigma =E_0\epsilon +B{\epsilon }^2$, with $E_0=2\mathrm{GPa}$ and $B=100\mathrm{GPa}$,At ${\epsilon }_{\ast }=0.02$, 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 givesThus the secant modulus averages the local tangent slopes over that interval. A chord modulus instead joins two specified states, with slope $({\sigma }_2-{\sigma }_1)/({\epsilon }_2-{\epsilon }_1)$; 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 givewhere extension is measured from the appropriate unloaded state. An original $F=800\mathrm{N}$, $A_0=10\mathrm{m}{\mathrm{m}}^2$, $L_g=50\mathrm{mm}$ and $\Delta L_g=1\mathrm{mm}$ yields $E_{\sec }=4000\mathrm{N}\mathrm{m}{\mathrm{m}}^{-2}=4\mathrm{GPa}$. 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.