Linear and Nonlinear Elastostatics Complexes

The linear elastostatics complex can be used to find stable numerical schemes. In this paper, we show that the linear elastostatics complex on flat spaces is equivalent to the Calabi complex, which is a well-known complex in differential geometry. This enables us to obtain a coordinate-free expression for the linear compatibility equations on curved spaces with constant sectional curvatures and also enables us to introduce stress functions for the second Piola-Kirchhoff stress tensor of nonlinear elastostatics. We derive the nonlinear compatibility equations in terms of the Green deformation tensor C for motions of bodies and surfaces in curved ambient spaces with constant sectional curvatures. We write various complexes for nonlinear elastostatics. In particular, by considering the nonlinear compatibility problem in terms of the deformation gradient F and the first Piola-Kirchhoff stress tensor, we obtain a complex for nonlinear elastostatics that is isomorphic to the R3-valued de Rham complex. Therefore, we are able to formulate nonlinear elastostatics in terms of differential forms. This allows one to reformulate some important problems of nonlinear elasticity as some standard problems in differential geometry. The geometric approach presented in this work is crucial for understanding the connections between linear and nonlinear elastostatics and the Hodge Laplacian, which can enable one to convert numerical schemes of the Hodge Laplacian to those fo