A symmetry-based study of global bifurcation-diagrams for heterogeneous solids and structures

Abstract

Many natural and artificial heterogeneous solids and structures are susceptible to surface instabilities such as wrinkling and folding among others. These instabilities can be found in multilayered systems employed in applications ranging from microelectronics to solar sails. Therefore there is interest in developing techniques to understand the mechanical behavior of these types of systems. A key feature in these systems is the presence of bifurcation of solutions from a parent state, but their mathematical treatment is cumbersome due to numerically ill-conditioned operators, the number of degrees of freedom involved, and the complex nature of the branches in equilibrium. Theoretical studies have confirmed the need for assessing their post-bifurcated range. However, most of the current methods for studying this range rely on asymptotic analyses, which are only locally valid, or imperfection-based numerical studies, which introduce biases and become tedious or fail in presence of multiple bifurcations. Symmetry-based methods have the potential to overcome these issues and have been successful in tracing branches efficiently once the symmetries of the branches are known. However, most of the computational tools based on these methods have not been optimized for branch-switching. In this work a symmetry-based computational framework is developed for generating bifurcation diagrams of connected branches in a perfect or unperturbed structure. The framework exploits the structure of the group representations of the problem and their building blocks, the irreducible representations, to facilitate branch-switching. The performance of this tool was contrasted with known results for the simplest model of a thin layer on a substrate: a periodic homogeneous beam on a nonlinear foundation under axial load. Finally, the developed framework was employed to assess the influence of the local mechanical and geometric properties on the post-bifurcated behavior of a periodic heterogeneous, inextensible and linearly elastic Euler-Bernoulli beam on a nonlinear elastic foundation under axial compression. Global bifurcation diagrams for a specific window of load and deformation were determined, including all the bifurcating branches allowed by the equivariant branching lemma for the particular symmetry of the problem.