*Appendix A.1. Evaluation of Normal Reaction Forces*

For plane contact problems, displacements can only be evaluated to within an arbitrary constant or, equivalently, in reference to a datum point. For the 2*D* Hertz problem, the boundary displacements normal to the interface can be evaluated as ([11], pp. 20–24):

$$w(\mathbf{x}) = \begin{cases} -\frac{2P\_z}{\pi E^\*} \left[ \left(\frac{\chi}{d}\right)^2 + c\_0 \right], \text{ if } \mathbf{x} \le a\_\prime \\\ -\frac{2P\_z}{\pi E^\*} \left[ \log \left| \psi(\mathbf{x}) \right| + \frac{1}{2\psi(\mathbf{x})^2} + \frac{1}{2} + c\_0 \right], \text{ if } \mathbf{x} \ge a\_\prime \end{cases} \tag{A1}$$

where *c*<sup>0</sup> is the arbitrary constant, and:

$$
\psi(\mathbf{x}) = \frac{\mathbf{x}}{a} + \sqrt{\left(\frac{\mathbf{x}}{a}\right)^2 - 1}. \tag{A2}
$$

An additional equilibrium equation relates the value of the load with the extension of the contact semi-strip *a*:

$$a = \sqrt{\frac{4P\_z R\_i}{\pi E^\*}}.\tag{A3}$$

If the datum is set in correspondence of the point of the boundary *x* = *R*d, the relation between the imposed displacement and the resultant vertical load has the form:

$$w(0) - w(R\_{\rm d}) = \Delta\_{\rm z} = \frac{2P\_{\rm z}}{\pi E^\*} \left[ \log \psi(R\_{\rm d}) + \frac{1}{2\psi(R\_{\rm d})^2} + \frac{1}{2} \right],\tag{A4}$$

where *w*(0) is evaluated in coincidence of the point of first contact, coincident with the centre of the semi-disk. As a final step, the inversion of Equation (A4) for a given value of ∆*<sup>z</sup>* gives the desired *Pz*. The comparison with numerical results is shown in Figure A2, where diamond markers representing the FEM prediction show a very good accordance with the corresponding solid black line, that represents the analytical results.

**Figure A2.** Resulting integrals of surface normal tractions.

#### *Appendix A.2. Evaluation of Tangential Reaction Forces*

Finding *Q<sup>x</sup>* for a given displacement still requires the evaluation of the applied displacement history with respect to a reference value, still set in correspondence of *x* = *R*d. Since a closed form solution is not available for the tangential tractions, an extended version of the Jäger-Ciavarella theorem that accounts for variable normal and tangential loads have been used for evaluating the analytical solution of the problem, according to the algorithm presented in [12]. If a load path is defined in terms of ∆*<sup>z</sup>* and ∆*x*, then, according to the theorem, the tangential problem can be reduced to the normal one, since an increment in tangential forces can be evaluated as the difference between the actual vertical force and the vertical force related to a smaller imposed vertical displacement, multiplied by the coefficient of friction:

$$Q\_{\mathbf{X}} = f\left[P\_{\mathbf{z}}(\Delta\_{\mathbf{z}}) - P\_{\mathbf{z}}(\Delta\_{\mathbf{z}}^{\*})\right]. \tag{A5}$$

The value of ∆ ∗ *z* is a function of ∆*x*. For a constant normal load and an increasing tangential load, it can be evaluated as:

$$
\Delta\_z^\* = \Delta\_z - \frac{\Delta\_x}{f}.\tag{A6}
$$

For general loading scenarios, the principle can be extended and the correct value of ∆ ∗ *z* evaluated in terms of an equivalent path that respects both the equilibrium and the friction law. Results are shown in Figure A3,

**Figure A3.** Resulting integrals of surface tangential tractions.

Where good accordance is found between the analytical solution given by the solid black line and the numerical prediction, depicted by the red diamond markers. In the same figure, the limit of *gross slip* for forward and backward sliding is shown as well by means of positive and negative valued horizontal black dash-dotted lines, respectively. These values represent the upper and lower threshold for the values of *Qx*, and this condition is approached in correspondence of the related maximal tangential imposed displacement, cfr. Figure A1.

As a final remark, the differences between the numerical and the analytical results, for both normal and tangential forces, are due to the effect of coupling between normal and tangential tractions, which is not taken into account by the analytical approach. Moreover, another source of the small difference lies in the treatment of the friction law: FEM exploits a regularised Coulomb friction law, while the analytical approach exploits the classical one, where the stick-slip transition is abrupt.

#### **References**


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