Geometric modelling of elastic and elastic-plastic solids by separation of deformation energy and Prandtl operators

A geometric method for analysis of elastic and elastic-plastic solids is proposed. It involves operators on naturally discrete domains representing a material’s microstructure, rather than the classical discretisation of domains for solving continuum boundary value problems. Discrete microstructures are considered as general cell complexes, which are circumcentre-dual to simplicial cell complexes. The proposed method uses the separation of the total deformation energy into volumetric and distortional parts as a base for introducing elastoplastic material behaviour. Volumetric parts are obtained directly from volume changes of dual cells, and the distortional parts are calculated from the distance changes between primal and dual nodes. First, it is demonstrated that the method can accurately reproduce the elastic behaviour of solids with Poisson’s ratios in the thermodynamically admissible range from -0.99 to 0.49. Further verification of the method is demonstrated by excellent agreement between analytical results and simulations of the elastic deformation of a beam subjected to a vertical displacement. Second, the Prandtl operator approach is used to simulate the behaviour of the solid during cyclic loading, considering its elastoplastic material properties. Results from simulations of cyclic behaviour during alternating and variable load histories are compared to expected macroscopic behaviour as further support to the applicability of the method to elastic-plastic problems.