= Frictional dissipation from contact slip
{title2=$D=\int\mu_kN|\dot\delta_s|\,dt$}
<Frictional dissipation from contact slip> is the irreversible loss of mechanical energy associated with relative sliding at a contact. In the ideal <kinetic friction> model, friction opposes the local slip velocity and has magnitude $\mu_kN$, with nonnegative coefficient $\mu_k$ and <normal force> $N$. Its dissipated energy is
$$
D=\int\mu_k(t)N(t)|\dot\delta_s(t)|\,dt\geq0.
$$
The <work> calculation and the distinction between signed and accumulated motion are developed in <contact slip displacement>. The kinetic-friction force law is an ideal physical model, rather than a universal description of every interface.<University Physics Volume 1 by William Moebs and others>[§6.2, kinetic friction and equation6.2]{ref}
An original two-stage example keeps the loading history explicit. Let $\mu_k=0.20$. Slide forward $0.40\,\mathrm{mm}$ while $N=0.50\,\mathrm N$, then backward $0.60\,\mathrm{mm}$ while $N=0.10\,\mathrm N$. The final signed slip is $-0.20\,\mathrm{mm}$, the <accumulated contact slip> is $1.00\,\mathrm{mm}$, and
$$
D=0.20\bigl(0.50\times0.40+0.10\times0.60\bigr)\,\mathrm{N\,mm}
=0.052\,\mathrm{N\,mm}=52\,\mathrm{\mu J}.
$$
The two stages contribute $40$ and $12\,\mathrm{\mu J}$. Using the unweighted mean of the two normal loads would instead give $60\,\mathrm{\mu J}$; the correct weighting follows sliding distance.
A physical measurement records relative motion close to the contact together with tangential force and normal loading. Integrating the opposing tangential force over the resolved sliding increments estimates the lost mechanical energy under the stated model. Force transmitted during ideal sticking, where local slip rate is zero, contributes no slip dissipation in that model, even if other parts of the apparatus deform. Grip travel and fixture <mechanical compliance> therefore must be distinguished from local sliding.
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