= Mechanical compliance
{title2=$C=x/F$}
\b[Mechanical compliance] describes how much <displacement> a mechanical system produces per applied <force>. For a one-dimensional linear response measured from an unloaded reference,
$$
x=CF,\qquad C=\frac{x}{F}=\frac1k,
$$
where $k$ is the <stiffness>. Translational <mechanical compliance> has units $\mathrm{m\,N^{-1}}$. A larger value means a larger displacement under the same force. For a nonlinear <force-displacement curve>, the ratio $x/F$ and the local derivative $dx/dF$ need not agree; the latter is reciprocal to <tangent stiffness> where the inverse relation exists.
For a uniform bar in axial <linear elasticity>,
$$
x=\frac{FL}{EA},\qquad C=\frac{L}{EA},
$$
where $L$ is its length, $A$ its cross-sectional area and $E$ its <Young's modulus>. Thus doubling the length doubles <mechanical compliance>, while doubling the area halves it, with material and other assumptions fixed.<Structural Stiffness Concept by TU Delft OpenCourseWare>[Axial relation delta=PL/(EA) and discussion of material and geometrical influences]{ref}
The same idea matters when measuring a specimen in a testing machine. The machine frame, <load cell> and grips also deform, so a crosshead's total movement can exceed the specimen's extension.<Universal Testing Machine Compliance by Instron>[p1, What is it?]{ref} In an illustrative slowly loaded series model, each part carries the same force and their displacements add:
$$
x_{\rm total}=F(C_{\rm specimen}+C_{\rm machine}).
$$
Choose $C_{\rm specimen}=0.020\,\mathrm{mm\,N^{-1}}$ and $C_{\rm machine}=0.005\,\mathrm{mm\,N^{-1}}$. At $100\,\mathrm N$, total movement is $2.5\,\mathrm{mm}$ but specimen extension is $2.0\,\mathrm{mm}$. Treating the total as specimen extension would give an apparent <stiffness> of $40\,\mathrm{N\,mm^{-1}}$ instead of $50\,\mathrm{N\,mm^{-1}}$.
One apparatus-based correction loads a very stiff dummy specimen through the same load path, then subtracts its measured movement at matching forces. Its usefulness depends on reproducing the load path and making the dummy's own deformation sufficiently small.<Universal Testing Machine Compliance by Instron>[pp2–3, dummy specimen and force-matched displacement subtraction]{ref} An <extensometer> instead measures extension directly over its chosen specimen length.
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