Nonlinear Analysis of Wrinkles in Film-Substrate Systems by Finite Element Method and Asymptotic Numerical Method

The purpose of the present paper is to present an efficient numerical model for the study of a nonlinear solid mechanics problem: the wrinkling in film/substrate systems, either planar or spherical. This numerical model will be based both on the Finite Element Method (FEM) and the Asymptotic Numerical Method (ANM). The mathematical and numerical aspects of ANM are given. ANM is a robust continuation method for solving nonlinear problems depending on a loading parameter. This technique has already been applied successfully to various fields in solid and fluid mechanics. We will present, with both theoretical and algorithm aspects, how ANM is used for solving nonlinear mechanical and nonlinear thermo-mechanical problems involving elastic behavior and geometrical nonlinearities. Then we propose the technical implementation of ANM for FEM in the framework of FreeFEM++. FEM software development platform, called FreeFEM++, because of its natural way to deal with variational formulations, is well adapted to implement ANM for instability problems in solid mechanics. In order to illustrate the great efficiency of FreeFEM++, we will show how to write elegant FreeFEM++ scripts to compute the different steps of ANM continuation for solid elastic structures considering simple geometries subjected to conservative loading. For validation purpose, we will show first the case of simple geometries, like a cantilever subjected to an applied force. The case of planar or spherical film/substrate systems will be studied in great details using the new developed numerical model. More precisely, we will show that this numerical model allows for the accurate prediction of equilibrium path in buckling analysis.