 Force-displacement curve

A force-displacement curve records how a measured force varies with a specified displacement during loading or unloading. The choice of displacement matters: it can describe specimen extension, motion of a machine crosshead or another reference. A curve can reveal resistance to motion, nonlinear response and differences between loading and unloading.

A concrete apparatus uses a moving crosshead, specimen fixtures and a load cell. The University of Illinois describes an Instron 4483 frame whose ball screws move the crosshead, whose load cell measures force, and whose encoder records crosshead position; an extensometer can separately measure specimen length change. The acquisition system converts those signals into engineering units and records them.^[ref]

Those displacement channels need not agree. Crosshead motion can include mechanical compliance of the machine, grip seating and deformation elsewhere in the load train. A specimen extensometer therefore answers a different geometrical question from the crosshead encoder.^[ref]

For a monotonic loading branch, the local slope $dF/dx$ is a tangent stiffness, while the ratio $F/x$ from a chosen zero point is a secant stiffness. These coincide for the linear model $F=kx$ but can differ for a nonlinear curve. The force here is the applied loading force in the positive displacement direction; a spring's restoring force has the opposite sign.

The work supplied along this one-dimensional branch isFor an own linear example, $k=200\mathrm{N}{\mathrm{m}}^{-1}$ and $x_1=0.05\mathrm{m}$ give final force $10\mathrm{N}$ and workThe nonlinear model $F=c{x}^3$, with $c=80000\mathrm{N}{\mathrm{m}}^{-3}$, reaches the same final force but givesIts final secant stiffness is $200\mathrm{N}{\mathrm{m}}^{-1}$, while its tangent stiffness is $3c{x}_1^2=600\mathrm{N}{\mathrm{m}}^{-1}$. A final force and displacement therefore do not determine either the full curve or its work integral.

When loading and unloading follow different branches, a single-valued relation between force and displacement may be insufficient. Hysteresis records that path dependence; relating a cycle's work to dissipation also requires an energy balance and the relevant state assumptions. Report the measured displacement channel, sign convention and loading path before treating a curve as a material property independent of its apparatus.