*JKR Theory*

Guduru [17] considered the contact between a rigid sphere of radius *R* that indents and elastic halfspace (Young modulus *E*, Poisson ratio *ν*) with an axisymmetric waviness of wavelength *λ* and amplitude *A* (see Figure 1).

In the system of reference shown in Figure 1, the axisymmetric waviness has the form

$$y(r) = -A\left(1 - \cos\left(\frac{2\pi r}{\lambda}\right)\right) \tag{1}$$

where *r* is the radial coordinate. Using the Hertzian approximation for the spherical profile [29] one gets the gap function

$$f\left(r\right) = \frac{r^2}{2R} + A\left(1 - \cos\left(\frac{2\pi r}{\lambda}\right)\right) \tag{2}$$

Guduru [17] solved the adhesive contact problem under the assumptions of compact contact area (i.e., there are no axisymmetric grooves within the contact patch) and in the limit of short range adhesion [18], which requires soft bodies into contact with large surface energy. Ensuring the contact area is compact requires that the gap function is strictly monotonically increasing

$$\frac{df(r)}{dr} > 0, \qquad r > 0 \tag{3}$$

which for the gap function (2) implies

$$\frac{\lambda^2}{AR} > 8.5761\tag{4}$$

**Figure 1.** The geometry of the axisymmetric contact problem. A rigid sphere of radius *R* indents an elastic halfspace with an axisymmetric waviness of wavelength *λ* and amplitude *A*. The sphere is approximated by a Hertzian profile.

Nevertheless, condition (4) is too restrictive. Indeed, Guduru [17] analysis holds at detachment if one requires that the normal load is increased from 0 to a value such that the contact radius *a* gets larger than a critical radius *r<sup>c</sup>* = 2*πAR*/*λ* for which the gap function is strictly monotone and any partial contact within the contact patch has coalesced. To this end it is evident from Johnson [30] analysis (strictly speaking that was a 2D problem) that a simply connected contact area would be achieved also when condition (4) is violated provided that the so-called "Johnson parameter" *αKL J* = 2*λw<sup>c</sup> π*2*A*2*E*∗ is sufficiently high to ensure spontaneous snap into full contact. By using three different solution approaches Guduru [17] obtained that the JKR adhesive solution for the geometry in Figure 1 can be written in dimensionless form as

$$\begin{cases} \overline{W}\_1 = 4\beta \left[ \frac{2\overline{\pi}^3}{3} + a \left( \frac{4\pi^2 \overline{\pi}^3}{3} + \frac{\pi \overline{a}}{2} H\_1 \left( 2\pi \overline{a} \right) - \pi^2 \overline{a}^2 H\_2 \left( 2\pi \overline{a} \right) \right) \right] \\ \overline{W} = \overline{W}\_1 - 4\sqrt{\beta \overline{a}^3} \\ \overline{\Delta} = \overline{a}^2 + a\pi^2 \overline{a} H\_0 \left( 2\pi \overline{a} \right) - \sqrt{\frac{\pi}{\beta}} \end{cases} \tag{5}$$

where the following dimensionless parameters have been defined

$$\alpha = \frac{AR}{\lambda^2}, \qquad \beta = \frac{\lambda^3 E^\*}{2\pi w\_c R^2}, \qquad \overline{W} = \frac{W}{\pi w\_c R'}, \qquad \overline{\Delta} = \frac{\Delta R}{\lambda^2}, \qquad \overline{a} = \frac{a}{\lambda'} \tag{6}$$

and *H<sup>n</sup>* (·) is the Struve function of order *n*, *E* ∗ = *E*/ 1 − *ν* 2 is the composite elastic modulus, *w<sup>c</sup>* is the surface energy per unit area, *W* is the external load, *W*<sup>1</sup> is the normal load in the adhesiveless problem and ∆ is the remote approach (>0 when the punch approaches the halfspace, see also [22]). Inspection of Equation (5) reveals that the Guduru problem, in the JKR regime, depends on two dimensionless parameters {*α*, *β*}: *α* represents the degree of waviness of the surface, with large (small)

*α* implying high (low) amplitude waviness, while *β* can be interpreted as the ratio between the elastic and the surface energy, with large (small) *β* implying a stiff (compliant) material [22]. For *α* = 0 the classical sphere-flat JKR solution is retrieved

$$\overline{\mathcal{W}} = 4\beta \frac{2\overline{a}^3}{3} - 4\sqrt{\beta \overline{a}^3} \tag{7}$$

$$
\overline{\Delta} = \overline{a}^2 - \sqrt{\frac{\overline{a}}{\beta}} \tag{8}
$$

### **3. Numerical Solution**
