An extended JKR model for adhesion of a rigid sphere on a supported compressible elastic thin layer

Abstract

This study aimed to develop an explicit JKR-type model for adhesive contact of a rigid sphere (of radius R) on a compressible elastic thin layer (of thickness h, Young’s modulus E, and Poisson ratio $$\nu $$) bonded or sliding on a rigid substrate. Based on a simple Kerr model for a compressible elastic layer, an explicit expression for strain energy of the elastic layer is derived in terms of the two JKR-type variables $$\left( {\delta ,\;a} \right) $$, where a is the radius of contact zone and $$\delta $$ is the indentation depth of the rigid sphere. Thus the equilibrium values of $$\left( {\delta ,\;a} \right) $$ can be determined as the stationary point of the potential energy. The explicit model is justified by detailed comparison of the predicted results (for Poisson ratio $$\nu \le 0.45$$) with known data reported in recent literature. For example, the validity and accuracy of the present model are demonstrated for moderately soft elastic thin layers under the condition $${2\left( {1-v^{2}} \right) WR^{2}} \big /{\left( {Eh^{3}} \right) }\ge 100$$ (where W is adhesion energy per unit contact area, for instance, for typical materials with $$R=500\ \upmu \mathrm{m}$$ $$E=100\ \mathrm{KPa}$$ and $$W=100\ \mathrm{mJ}/\mathrm{m}^{{2}}$$, the condition requests $$\;h<20\ \upmu \mathrm{m}$$). As compared to existing methods that request more substantial numerical calculations, the present model achieves an explicit expression for strain energy of a compressible elastic layer and could offer a simpler analytical method for adhesion mechanics of a rigid sphere on a compressible thin elastic layer bonded or sliding on a rigid substrate.