{title2=$F(x)$}

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.<Load Frames at the Mechanical Testing Instructional Laboratory>[Load Frames, apparatus and measurement-channel description]{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.<Extensometers in the Instron documentation>[Passage on crosshead displacement, machine compliance, grip seating and load-train deflection]{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 is
$$
W=\int_{x_0}^{x_1}F(x)\,dx.
$$
For an own linear example, $k=200\,\mathrm{N\,m^{-1}}$ and $x_1=0.05\,\mathrm m$ give final force $10\,\mathrm N$ and work
$$
W=\tfrac12kx_1^2=0.250\,\mathrm J.
$$
The nonlinear model $F=cx^3$, with $c=80000\,\mathrm{N\,m^{-3}}$, reaches the same final force but gives
$$
W=\frac{cx_1^4}{4}=0.125\,\mathrm J.
$$
Its final secant stiffness is $200\,\mathrm{N\,m^{-1}}$, while its tangent stiffness is $3cx_1^2=600\,\mathrm{N\,m^{-1}}$. A final force and displacement therefore do not determine either the full curve or its work integral.

\Image[/force-displacement-curve.png]
{title=Illustrative force-displacement curves with equal endpoints and different work}
{description=Own linear and cubic models, not experimental data. The horizontal axis displays millimetres; work is calculated after converting displacement to metres. Shading marks the 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.


 Force-displacement curve (source code)