Cem Tasan | Searcharxiv

arXiv2023

Partial differential equations to determine elasto-plastic stress-strain behavior from measured kinematic fields

A system of partial differential equations (PDEs) is derived to compute the full-field stress from an observed kinematic field when the flow rule governing the plastic deformation is unknown. These equations generalize previously proposed equations that assume pure plastic behavior without elasticity. A method to numerically solve these equations is also presented. In addition to force balance, the equations are derived from the elastic-plastic decomposition of the deformation gradient, the assumption of isotropy, and the assumption that the function mapping the elastic strain to stress is known. The system of equations can be directly applied to complex geometries, finite deformation, non-linear elasticity and plasticity, compressible materials, rate dependent materials, and a variety of hardening laws. This system of PDEs is non-linear and time dependent. Furthermore, it overcomes an important prior limitation: it can be directly applied to cases where some regions of a body are elastically deforming while others are elasto-plastically deforming. A two-dimensional case study of necking in a uniaxial tensile specimen is investigated to illustrate and validate the method. The governing equations are numerically solved using strain fields output from a finite element simulation and validated against this same simulation showing accurate results.

cond-mat.mtrl-sci↗

arXiv2020

Full-field stress computation from measured deformation fields: a hyperbolic formulation

Recent developments in imaging techniques and correlation algorithms enable measurement of strain fields on a deforming material at high spatial and temporal resolution. In such cases, the computation of the stress field from the known deformation field becomes an interesting possibility. This is known as an inverse problem. Current approaches to this problem, such as the finite element update method, are generally over-determined and must rely on statistical approaches to minimize error. This provides approximate solutions in some cases, however, implementation difficulties, computational requirements, and accuracy are still significant challenges. Here, we show how the inverse problem can be formulated deterministically and solved exactly in two or three dimensions for large classes of materials including isotropic elastic solids, Newtonian fluids, non-Newtonian fluids, granular materials and isotropic plastic solids. This solution is based on a single assumption of the alignment of the principal directions of stress and strain or strain rate. No further assumptions regarding incompressibility, pressure independence, yield surface shape or the hardening law are necessary. This assumption leads to a closed, first order, linear system of hyperbolic partial differential equations with variable coefficients and can be solved to give the solution for any geometry and loading condition. We provide a numerical proof-of-principle study of the plastic deformation of a two-dimensional bar with spatially varying yield stress and strain hardening coefficient. The results are validated indicating the solution is exact up to numerical error. No model calibration or material parameters are required. Interestingly, this solution procedure lends itself to a simple physical interpretation of stress propagation through the material.

cond-mat.mtrl-sci↗