properties $E_1,\nu_1$ and $E_2,\nu_2$, respectively, where $E$ and $\nu$ are the Young’s modulus and Poisson’s ratio, respectively. Schematically this problem can be presented as a contact between two spheres[^1], see Fig. \[fig:setup\]. We assume that the contacting solids possess rough surfaces, which, in the close vicinity of contact zone, can be described by analytical functions $z_1(x,y)$ and $z_2(x,y)$. Note that this roughness in case of curved surfaces shall be superimposed on the macroscopic shapes, which can be assumed to be $z_{si}=(x^2+y^2)/(2R_i)$. We assume that we can neglect the tangential (in-contact-plane) displacement of surface points, i.e. the effective roughness is given by $z^*(x,y) = z_1(x,y)-z_2(x,y)+z_0$, where $z_0$ is a constant. Under frictionless elastic contact between rough surfaces with relatively small slopes, this problem can be mapped on a problem between a smooth rigid sphere with effective radius $R^* = R_1R_2/(R_1+R_2)$ and a rough elastic half-space with effective elastic modulus $E^*$ [@johnson1985b; @barber2003bounds]: $$E^* = \frac{E_1E_2}{(1-\nu_2^2)E_1 + (1-\nu_1^2)E_2}.$$ In case of smooth surfaces, we recall [@johnson1985b] below the analytical solutions for this problem under the action of a squeezing force $N$. The contact radius $a$, the approach between distance points $\delta$, and the pressure distribution $p(x,y)$ are given by the following expressions: 