= Machine compliance
{title2=$C_m$}
<Machine compliance> is the extra load-path <displacement> per applied <force> attributed to the apparatus in a specified measurement model. Its units are $\mathrm{m\,N^{-1}}$, the reciprocal units of a <spring constant>. Here $C_m$ includes the frame, <load cell>, adapters and grips insofar as their deformation contributes to the recorded <crosshead displacement> $x$. It is a property of that configuration and readout, rather than a universal constant of the testing machine.
Assume a uniform specimen with initial length $L$, constant $A_0$ and <Young's modulus> $E$, small axial <elastic deformation>, quasistatic loading, no slip, and a linear, length-independent apparatus response over the chosen <force> window. Equilibrium makes the same <force> act through specimen and apparatus, while their extensions add. The two <springs> in series therefore give
$$
x=x_0+\frac{FL}{EA_0}+C_mF,\qquad
q(L)=\frac{dx}{dF}=\frac{L}{EA_0}+C_m,
\qquad k_{\mathrm{obs}}=\frac1{q(L)}.
$$
The offset $x_0$ describes the displacement origin after seating; the slope $q$ describes total compliance. Fitting an intercept avoids forcing an arbitrary zero through the origin, but cannot repair slip or a curved response.
If all of $x-x_0$ is incorrectly attributed to specimen extension, the inferred modulus is
$$
E_{\mathrm{app}}=\frac{L}{A_0q(L)}
=\frac{E}{1+EA_0C_m/L}.
$$
For positive $C_m$, this is too small. The fractional bias is most serious for a short, thick, stiff specimen because its own compliance $L/(EA_0)$ is small. A local <extensometer> instead has $d(\Delta L_g)/dF=L_g/(EA_0)$ in this ideal model, excluding deformation outside its contacts.
A shaped specimen requires more care. With varying initial cross-sectional <area> $A(s)$ and uniform $E$, its axial compliance is $\int ds/[EA(s)]$ in the one-dimensional approximation. Shoulder and end deformation must be included explicitly or remain identical between tests and be absorbed into an effective intercept. Changing only the <extensometer> <gauge length> on an unchanged specimen does not change its total compliance and cannot perform a <two-length machine compliance separation>.
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