**Appendix A. Deng and Kesari's Model**

Deng and Kesari's model (DK) [14] is a combination of classical adhesive theories for smooth elastic spheres and Nayak's theory of rough surfaces [43]. In their work, DK used both JKR [18] and Maugis–Dugdale [44] theories for estimating the energy loss in a loading– unloading cycle of a single spherical asperity. In the limit of high Tabor parameters (*µ* ≥ 3), the two theories predict the same behavior. For this reason, as our study is focused on very soft materials, for which high Tabor numbers are expected, here we discuss DK's model in the framework of JKR theory.

DK gives an empirical estimate of the energy loss *θ* for a loading–unloading cycle of a single asperity of radius *R*; in the JKR limit the value of *θ* is

$$
\theta = 2E^\* \mathcal{R}^3 \bar{\theta} \tag{A1}
$$

being ¯*<sup>θ</sup>* <sup>≈</sup> 0.5262[*π*∆*γ*/(*<sup>E</sup>* <sup>∗</sup>*R*)]5/3 .

For a rough surface the energy loss of each asperity depends on the value of its radius of curvature. Following the work in [43], the variation of curvature in the population of all asperities contained in any unit region is

$$p\_k(t) = \sqrt{\frac{3}{4\pi}} (t^2 - 2 + 2e^{-t^2/2}) e^{-\frac{8(\zeta\_1^2 - \zeta\_2^2)t^2}{16\zeta\_1}}\tag{A2}$$

where *t* = − √ 3/*m*4*k*m, being *<sup>k</sup>*<sup>m</sup> ∈ (0, <sup>∞</sup>) the surface's mean curvature at the apex of an asperity, and the constants *C*<sup>1</sup> = *α*/(2*α* − 3) and *C*<sup>2</sup> = *C*<sup>1</sup> √ 12/*α* are related to the Nayak parameter *α* = *m*0*m*4/*m*<sup>2</sup> 2 . The quantities *m*0, *m*2, and *m*<sup>4</sup> are the spectral moments of surface roughness PSD. In particular, they are related to the rms roughness amplitude *h*rms, gradient *h* ′ rms and curvature *h* ′′ rms by *h*rms = √ *m*0, *h* ′ rms = √ 2*m*2, and *h* ′′ rms = √ 8/3*m*4.

Substituting (A2) in (A1) and integrating on the range of variation of *t*, the mean energy loss of contacting asperities can be written as

$$
\langle \theta \rangle = 0.5262 \times 2\mathbb{E}^\* \int\_{-\infty}^0 dt [\pi \Lambda \gamma / (\mathbb{E}^\* \mathbb{R}(t))]^{5/3} p\_k(t) \mathcal{R}(t)^3 \tag{A3}
$$

being *R*(*t*) = (1/*R*tip − √ *m*4/3*t*) −1 .In DK's model, *R*tip is the radius of the spherical indenter that is in contact with a nominally flat surface. In our case, as contact occurs between nominally flat surfaces, 1/*R*tip → 0.

Finally, the total energy loss can be computed as

$$
\Theta = \eta \cdot A \cdot \langle \theta \rangle \tag{A4}
$$

where *η* = *m*4/(6*π* √ 3*m*2) is the asperity density in a nominal contact region of unit area.

In the original DK's model, the contact area *A* is computed applying JKR theory to the macroscopic spherical indenter of radius *R*tip and neglecting roughness contribution. In particular, *A* is calculated as *A*<sup>∆</sup> max − *A*∆in, where *A*<sup>∆</sup> max and *A*∆in are the values of the contact area at the maximum indentation and the macroscopic jump-in instability of the spherical tip, respectively.

In this work, as we are dealing with the contact between two nominally flat surfaces, *A*∆in is interpreted as the true contact area of the first few asperities jumping into contact, while *A*<sup>∆</sup> max is the true area of contact obtained at the end of the loading phase. Moreover, as in our calculations *A*∆in ≪ *A*<sup>∆</sup> max, we have assumed *A*<sup>∆</sup> max − *A*∆in ≈ *A*<sup>∆</sup> max.

#### **References**


*Article*
