A discrete region model of isotropic elasticity

Abstract

A model of isotropic elastic materials is introduced that uses the positions of points within the material as the independent parameters. The energy of the material is defined as a function of these point positions. The gradient of this energy function provides the force at each point. From the point positions and the forces both stress and strain can be calculated as dependent parameters. This model is shown to reproduce the results of classical stress—strain, spring, and rubber elasticity models by changing the energy function and the equations defining stress and strain from the point positions. The discrete region model does this without large numbers of nodes, or being limited to modeling only deviatoric stress. The discrete region model also forms the basis of new elasticity models which are linear for large displacements and simultaneously rotationally invariant under rigid-body rotations. Finally a procedure is described for fitting the energy function for discrete region models to experimental data.