= Contact slip displacement
{title2=$\delta_s$}
The <contact slip displacement> is signed relative tangential <displacement> at a specified contact. For small motions in a fixed local frame, let $\mathbf t$ be a tangential <unit vector> and let $\mathbf u_1,\mathbf u_2$ describe the two contacting surfaces. Then
$$
\boxed{\delta_s=(\mathbf u_1-\mathbf u_2)\cdot\mathbf t.}
$$
This <scalar product> extracts relative motion along the contact. The corresponding relative normal <displacement> is a separate quantity controlling opening or closure. This local distinction matters for <inter-fibre sliding>: an entire bundle can move without adjacent <material fibres> slipping against one another.
As an original local example, take $\mathbf t=(1,0)$, $\mathbf u_1=(0.30,0.02)\,\mathrm{mm}$ and $\mathbf u_2=(0.10,0.02)\,\mathrm{mm}$. The <contact slip displacement> is $0.20\,\mathrm{mm}$ and the relative normal <displacement> is 0. Adding the same translation to both surfaces changes neither result. Reversing the orientation of $\mathbf t$ reverses the sign of $\delta_s$ while preserving its magnitude. If the contact frame rotates substantially or different material points enter contact, relative tangential <velocity> must be evaluated in the current contact frame and integrated consistently; projecting one final global <displacement> difference onto a final <tangent vector> is not a general finite-motion formula.
Under the ideal dry-<friction> model, <static friction> can transmit tangential <force> while the local slip rate is 0. Sliding introduces <kinetic friction>; with constant <normal force> $N$, its magnitude is $\mu_kN$, where $\mu_k$ is the <coefficient of friction>.<University Physics Volume 1 by William Moebs and others>[§6.2, static-friction bound and kinetic-friction model, equations 6.1–6.2]{ref} The <accumulated contact slip> and the <frictional dissipation from contact slip> are therefore different from net signed <displacement>:
$$
\ell_s=\int_0^t|\dot\delta_s(\tau)|\,d\tau,\qquad
D=\mu_kN\ell_s.
$$
The second expression follows from integrating the opposing frictional <force> over each increment of relative sliding; it assumes constant $N$ and $\mu_k$. For a prescribed out-and-back slip from 0 to $1\,\mathrm{mm}$ and back to 0, the final $\delta_s$ is 0 but $\ell_s=2\,\mathrm{mm}$. With stipulated $N=0.20\,\mathrm N$ and $\mu_k=0.25$, the dissipated <energy> is $100\,\mathrm{\mu J}$. \b[Zero net slip does not imply zero frictional dissipation.]
A simple measurement marks both surfaces close to a maintained contact, images their relative positions while tangential loading is varied, and resolves motion along the local contact direction. <Measurement calibration> converts image distances to lengths. Marks far from the contact can also move because the strands stretch or the fixtures deform; a grip-travel readout alone does not isolate <contact slip displacement>. Record both the signed motion and <accumulated contact slip> when comparing repeated bending cycles of an <unbonded material-fibre bundle>.
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