The Contact Mechanics of Coated Elastic Solids: Effect of Coating Thickness and Stiffness

Abstract

We investigate the adhesive and adhesiveless contact between a rigid regular wavy profile and an elastic substrate coated with an elastic layer. The contact stiffness is strongly influenced by the elasticity and thickness of the coating layer. Specifically, coatings more compliant than the substrate entail larger contact area, and vice versa. Under adhesive conditions, thin soft coatings may lead to complete contact, regardless of the applied load. Moreover, enhanced adhesive strength (toughness) can be achieved at the pull-off by adopting stiff (compliant) coatings. We compare our exact model with the Bec/Tonck model, which is one of the most adopted approximate model for biological tissues contact mechanics. The results show that predictions of the proposed model are in better qualitative agreement with experimental results taken from the literature.

Appendix A: Derivation of the periodic Green’s functions

In this section, the periodic Green’s function for the layered geometry investigated in the present paper is derived according to the procedure given in Ref. [26].

First let us recall that for a general 1D + 1D elastic contact under linearity and translational invariance (i.e., in-plane homogeneity) conditions, the relation between the interfacial normal stresses $\sigma \left( x\right)$ and the surface normal displacements $u\left( x\right)$ can be written as a convolution product, i.e.,

$$\begin{aligned} u\left( x\right) =\int {\rm{d}}x^{\prime }G\left( x-x^{\prime }\right) \sigma \left( x^{\prime }\right) \end{aligned}$$

(13)

where $G\left( x\right)$ is the Green’s function. Taking the space Fourier transform of Eq. (13) one obtains

$$\begin{aligned} u\left( q\right) =M_{zz}\left( q\right) \sigma \left( q\right) \end{aligned}$$

(14)

The specific form of $M_{zz}\left( q\right)$ depends on the system geometry, on the material properties, and on how the system is constrained. For layered solids, according to Ref. [24], $M_{zz}\left( q\right)$ can be written as

$$\begin{aligned} M_{zz}\left( q\right) =-\frac{2\left( 1-\nu _{\text{l}}^{2}\right) }{E_{\text{l}}} \frac{1}{q}S\left( q\right) \end{aligned}$$

(15)

where $S\left( q\right)$, given by Eqs. (4–5), is a corrective factor taking into account the system geometry and boundary conditions. Notice in the case of homogenous elastic half-space $S\left( q\right) \rightarrow 1$.

Let us now consider the case of a layered half-space under a distribution of periodic forces with spatial periodicity $\lambda$. Such a distribution of loads can be represented by the surface stress field

$$\begin{aligned} \delta _{\lambda }\left( x\right) =\sum _{k=-\infty }^{+\infty }\delta \left( x-\frac{2\pi }{q_{0}}k\right) \end{aligned}$$

(16)

where $\delta \left( x\right)$ is the Dirac’s delta function, k is the wave number, and $q_{0}=2\pi /\lambda$ is the fundamental frequency. Substituting in Eq. (13)

$$\begin{aligned} G_{\lambda }\left( x\right) =\int dx^{\prime }G\left( x-x^{\prime }\right) \delta _{\lambda }\left( x^{\prime }\right) =\sum _{k=-\infty }^{+\infty }G\left( x-\frac{2\pi }{q_{0}}k\right) \end{aligned}$$

(17)

Taking the Fourier transform of Eq. (17) gives

$$\begin{aligned} M_{\lambda }\left( q\right) =\sum _{k=-\infty }^{+\infty }\int dxe^{-iqx} G\left( x-\frac{2\pi }{q_{0}}k\right) =\sum _{r=-\infty }^{+\infty } M_{zz}\left( q\right) \delta \left( \frac{q}{q_{0}}-r\right) \end{aligned}$$

(18)

Moving back to the space domain we finally have

$$\begin{aligned} G_{\lambda }\left( x\right) =\frac{1}{\left( 2\pi \right) ^{2}}\int dqM_{\lambda }\left( q\right) e^{iqx}=\left( \frac{q_{0}}{2\pi }\right) ^{2}\sum _{r=-\infty }^{+\infty }M_{zz}\left( rq_{0}\right) e^{irq_{0}x} \end{aligned}$$

(19)

Eq. (19) shows that the periodic Green’s function $G_{\lambda }\left( x\right)$ is a Fourier series with coefficients $\left[ q_{0}/\left( 2\pi \right) \right] ^{2}M_{zz}\left( rq_{0}\right)$, and can be very easily calculated with the fast Fourier transform numerical technique.