Compatibility Equations of Nonlinear Elasticity for Non-Simply-Connected Bodies
Compatibility equations of elasticity are almost 150 years old. Interestingly they do not seem to have been rigorously studied for non-simply-connected bodies to this date. In this paper we derive necessary and sufficient compatibility equations of nonlinear elasticity for arbitrary non-simply-connected bodies when the ambient space is Euclidean. For a non-simply-connected body, a measure of strain may not be compatible even if the standard compatibility equations ("bulk" compatibility equations) are satisfied. It turns out that there may be topological obstructions to compatibility and this paper aims to understand them for both deformation gradient F and the right Cauchy-Green strain C. We show that the necessary and sufficient conditions for compatibility of deformation gradient F are vanishing of its exterior derivative and all its periods, i.e. its integral over generators of the first homology group of the material manifold. We will show that not every non-null-homotopic path requires supplementary compatibility equations for F and linearized strain e. We then find both necessary and sufficient compatibility conditions for the right Cauchy-Green strain tensor C for arbitrary non-simply-connected bodies when the material and ambient space manifolds have the same dimensions. We discuss the well-known necessary compatibility equations in the linearized setting and the Cesaro-Volterra path integral. We then obtain the sufficient conditions of compatibility for the linearized strain when the body is not simply-connected.