Tangent stiffness ($k_{\tan }=dF/dx$)

The tangent stiffness at a point on a differentiable force-displacement curve is its local slope,It governs the first-order response to a small displacement change:The graphical interpretation is the slope of the tangent line. Structural stiffness measured from force-deflection behaviour includes geometry as well as material response.^[ref]

For the illustrative model $F(x)=ax+b{x}^3$, choose $a=100\mathrm{N}{\mathrm{m}}^{-1}$ and $b=1000\mathrm{N}{\mathrm{m}}^{-3}$. At $x_0=0.10\mathrm{m}$, the tangent stiffness is $130\mathrm{N}{\mathrm{m}}^{-1}$. An additional $\Delta x=0.001\mathrm{m}$ therefore predicts a force increase of $0.130\mathrm{N}$ to first order. Direct substitution gives $0.130301\mathrm{N}$; the difference is the higher-order contribution omitted by the tangent approximation. The secant stiffness from the origin is only $110\mathrm{N}{\mathrm{m}}^{-1}$ at this point.

In a displacement-controlled experiment, one can estimate the local slope with a central finite difference,Making $h$ smaller reduces the approximation error for a smooth exact curve, but measured differences can become dominated by measurement uncertainty. The force channel, displacement channel and loading branch must therefore be specified. Crosshead movement can include machine mechanical compliance, while an extensometer records a selected specimen extension.^[ref]

For a curve with hysteresis, loading and unloading can have different local slopes at the same displacement. At a sharp corner a unique derivative may not exist, and a finite-interval slope should be reported as such.