**Appendix A. Model Validation**

The proposed framework has been tested against a Hertz indentation problem for validation. The solution of the FEM simulation is compared with the analytical solution of the equivalent half-plane 2*D* contact problem, in terms of the integral of the interface normal and tangential tractions *P<sup>z</sup>* and *Qx*, respectively, given the same imposed displacements history. A parabolic profile has been used as a first order approximation of a circular rigid cylinder with unitary radius *R*<sup>i</sup> . The profile makes contact on the flat side of a linear elastic semi-disk with plane strain Young's modulus *E* ∗ = 814.7 Pa and radius *R*<sup>d</sup> = 5*R*<sup>i</sup> , which simulates a half-plane. The load history includes two far-field displacements, imposed to the rigid profile. First, a normal displacement is applied, starting from zero and linearly increasing to a maximum value <sup>∆</sup>*z*,0/*R*<sup>i</sup> = <sup>1</sup> × <sup>10</sup> −3 , reached at time *t*0, see the black line in Figure A1. The normal displacement is then held constant, and a harmonic tangential displacement is applied, which increases up to a maximum *f*∆*z*,0, being *f* = 0.2 the coefficient of friction, and then makes a complete cycle, see the red line in Figure A1. Such a maximum value of horizontal displacement is chosen to cause the incipient sliding of the cylinder, and this is indeed what happens if the response of the system in terms of frictional reaction forces is analysed.

**Figure A1.** Imposed displacements.
