Deterministic calculation of elasto-plastic stress-strain behavior from arbitrary deformation fields

A system of partial differential equations is derived to compute the full-field stress from an observed strain field when the plastic component of the material constitutive equation is unknown. These equations generalize previously proposed equations that are valid in the case where the elastic strain is negligible. The system of equations assume that the constitutive relations are isotropic, but otherwise make few assumptions and can be directly applied (without modification) to cases of finite deformation, non-linear elasticity and plasticity, compressible materials, rate dependent materials, and a variety of different yield surface shapes and hardening laws. Unlike the prior linear hyperbolic partial differential equations, this system of equations is non-linear and time dependent. The newly proposed equations can be used to solve for the stress field (and hence constitutive equation), for arbitrary geometries and loading conditions when the full-field strain is known. This generalization resolves a significant challenge in computing the stress during elasto-plastic deformations: some regions of the material will be elastically unloading while others are undergoing significant plastic deformation. This problem significantly limited applications of the previously proposed equations, but is fully resolved here. A two-dimensional case study of necking in a uniaxial tensile test specimen is investigated to illustrate the method. The governing equations are numerically solved using an algorithm based on on the finite volume method developed in an accompanying work. This is validated against the solution to the forward problem and shown to give accurate results (within numerical error of the true solution). This case study thus demonstrates that the developed approach has unique capabilities, including the first theoretically exact solution to this well studied problem.