Applications of Algebraic Topology in Elasticity

In this book chapter we discuss some applications of algebraic topology in elasticity. This includes the necessary and sufficient compatibility equations of nonlinear elasticity for non-simply-connected bodies when the ambient space is Euclidean. Algebraic topology is the natural tool to understand the topological obstructions to compatibility for both the deformation gradient F and the right Cauchy-Green strain C. We will investigate the relevance of homology, cohomology, and homotopy groups in elasticity. We will also use the relative homology groups in order to derive the compatibility equations in the presence of boundary conditions. The differential complex of nonlinear elasticity written in terms of the deformation gradient and the first Piola-Kirchhoff stress is also discussed.