Development of Geometric Formulation of Elasticity

Recently, new techniques have been presented that discretize continuous elasticity variables as cochains over a primal mesh, representing the solid, and an appropriately defined dual one. Discrete strain and stress can be then thought of as a vector-valued 1-form (or vector-valued 1-cochain on the primal mesh) and covector-valued 2-form (or vector-valued 2-cochain on the dual mesh) respectively. The governing equations can be formulated by requiring energy balance and invariance under time-dependent rigid translations and rotations of the ambient space. To obtain those, we project the discrete stress into normal and tangential components and formulate the boundary value problem with a system of two matrix equations. This allow for treating both classical and coupled-stress (micro-polar) elasticity. The link between discrete strains and stresses is provided by a material discrete Hodge star operator, which we define to include geometric and physical factors, such as lengths, areas, and moduli of elasticity and rigidity. The performance of the proposed formulation is demonstrated by a simple example.