Analytical solutions for bonded elastically compressible layers

Abstract

Compression of elastic layers bonded between parallel plates often find applications in the mechanical characterization of soft materials or the transfer-printing of nanomembranes with polymeric stamps. In addition, annular rubbery gaskets and sealers are often under uniaxial compression during service. Analysis of elastic layers under compression has been focused on nearly incompressible materials such as rubbers, and empirical assumptions of displacements were adopted for simplicity. For compressible materials, solutions obtained by the method of averaged equilibrium are sufficient for effective compression modulus but inaccurate for the displacement or stress fields whereas solutions obtained by the method of series expansion are considerably complicated. In this paper, we report full field, closed-form solutions for bonded elastic layers (disks, annuli, annuli with rigid shafts, infinitely long strips) in compression using separation of variables without any pre-assumed deformation profile. These solutions can satisfy the exact forms of the equilibrium equations and all essential boundary conditions as well as the weak form of the natural boundary conditions. Therefore the predicted stress, displacement, and effective modulus have found excellent agreement with finite element modeling (FEM) results over a wide range of Poisson’s ratio and aspect ratio. The analytical and FEM results of the stress, displacement, and effective modulus are highly sensitive to Poisson’s ratio, especially near 0.5. Therefore we also propose a viable means to simultaneously measure the intrinsic Young’s modulus and Poisson’s ratio of elastically compressible layers without camera settings. When Poisson’s ratio approaches 0.5, our solutions can degenerate to classical solutions for incompressible elastic layers.