Wavelet method applied to specific adhesion of elastic solids via molecular bonds

We propose a wavelet method to analyze the stochastic-elastic problem of specific adhesion between two elastic solids via ligand-receptor bond clusters, which is governed by a nonlinear integro-differential equation with a singular Cauchy kernel to describe the mean-field coupling between deformation of elastic materials and stochastic behavior of the molecular bonds. To solve this problem, Galerkin method based on a wavelet approximation scheme is adopted, and special treatment which transforms the singular Cauchy kernel into a smooth one has been proposed to avoid the cumbersome calculation of singular integrals. Numerical results demonstrate that the method is fully capable of solving the specific adhesion problems with complex nonlinear and singular equations. Based on the proposed method, investigations are performed to reveal the relation between steady-state pulling force and mean surface separation under different stress concentration indexes, which is crucial for assembling the overall constitutive relations for multicellular tumor spheroids and polymer-matrix microcomposites.