Preprint

Inserted: 10 apr 2026
Last Updated: 10 apr 2026

Year: 2026
Links: HAL preprint

Abstract:

We describe a novel formalism for elasto-plastic deformations based on the strain incompatibility, to model an elastic solid filled with dislocations, hence subject to plastic de- formations, at the macroscopic scale and under a time-discrete quasi-static framework. The main kinematical descriptor is the strain tensor E, that we define from a mesoscopic analysis. The associated state equations, as derived from a virtual work approach, are second-gradient in E, because they explicitly incorporate the incompatibility inc E, and are linear. Due to the introduction of an internal variable θ and a dissipation potential, related to the motion of dislo- cations, the overall problem is variational and nonlinear. It is numerically solved by alternating minimization with respect to (E, θ). Numerical simulations are performed to assess loading- unloading problems on some simple two-dimensional geometries. Elaborating on previous works by the authors, the present contribution is an in-depth description of an alternative approach to elasto-plastic deformations, grounded on a firm mathematical basis, and consolidated by numeri- cal results. By nature, this approach is multi-scale, therefore opening the way to a comprehensive study of strain incompatibility, dislocations and plasticity as a complex interaction of multiscale phenomena, of which hardening is only an archetypal example.

Keywords: Elasticity, plasticity, strain incompatibility, dislocations