Mathematical Aspects of ANM/FEM Numerical Model, Applied to Nonlinear Elastic, and Thermo Elastic Analysis of Wrinkles in Film/Substrate Systems, and a New Implementation in the FreeFEM++ Language

The main purposes of the present paper are to present the mathematical and algorithmic aspects of the ANM/FEM numerical model and to show how it is applied to analyze elastic and thermo-elastic nonlinear solid mechanical problems. ANM is a robust continuation method based on a perturbation technique for solving nonlinear problems dependent on a loading parameter. Historically, this technique has been successfully applied to problems in various fields of solid and fluid mechanics. This paper shows how ANM is used to solve nonlinear elastic and nonlinear thermo-elastic problems involving elastic behavior and geometrical nonlinearities. The implementation of ANM for FEM in the FreeFEM++ language is then presented. The FEM software development platform, called FreeFEM++, is structured to work with variational formulations and, therefore, is well adapted to implement ANM for instability problems in solid mechanics. In order to illustrate the great efficiency of FreeFEM++, scripts will be presented for computing the different steps of ANM continuation for solid elastic structures, considering simple geometries subjected to conservative loading. For the purpose of validation, the problem of a cantilever subjected to an applied force is presented. Next, the new numerical model is applied to study wrinkles appearing in a planar film/substrate system that is subjected to compressive surface forces at the lateral faces of the film. Finally, the model is applied to a spherical film/substrate system subjected to thermo-elastic shrinkage. In both cases, the ANM/FEM prediction method, together with a Newton–Riks correction (if needed), identifies the equilibrium paths efficiently, especially after the post-buckling regime.