Friction in continuum mechanics How does one include friction forces in the context of continuum mechanics? I imagine that one could rewrite the relation
$$
F\leq N
$$
from the classical mechanics in terms the strain on one surface inducing tractions on the other surface, wheres the sliding friction could be described as material failure.
Rather than to engage in a guesswork, I am looking for a canonical answer.
 A: I am not sure, whether it is a canonical answer or not, but the closest analogy which comes to my mind is the viscous fluid.
Viscosity emerges as a dissipative part of fluid stress-energy tensor. On a high level is the property of different layers of water to resist to the external stress, and balance non-uniformity of the velocity distribution.
It was observed long ago by Netwon (and this dependence is natural as the leading term in the gradient expansion), that the viscous forces are proportional to the velocity gradient:
$$
\tau \simeq \frac{\partial u}{\partial x}
$$
Then, using the property that in uniformly rotating fluid there are no viscous phenomena, one obtains, that
$$\tau_{ij} \sim \partial_i u_j + \partial_j u_i$$
Which is further decomposed into the traceless part and the trace
$$
\tau_{ij} \sim \eta(\partial_i u_j + \partial_j u_i - \frac{2}{3} \partial_k u_k) 
+ \zeta \partial_k u_k
$$
From the mass and momentum conservation, then follows the general equation, describing the motion of viscous fluid is the famous Navier-Stokes equation:
$$
\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {u} \right)=-\nabla p+\nabla \cdot \left\{\eta \left(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right)+\zeta (\nabla \cdot \mathbf {u} )\mathbf {I} \right\}+\rho \mathbf {g} 
$$
Where $\eta$ is the shear, and $\zeta$ is the bulk viscosity.
