Calculus of Variations and Geometric Measure Theory

S. Almi - D. Reggiani - F. Solombrino

Geometric rigidity for incompatible fields in the multi-well case and an application to strain-gradient plasticity

created by almi1 on 10 Oct 2023
modified by reggiani on 02 Nov 2023



Inserted: 10 oct 2023
Last Updated: 2 nov 2023

Year: 2023

ArXiv: 2311.00438 PDF


We derive a quantitative rigidity estimate for a multiwell problem in nonlinear elasticity with dislocations. Precisely, we show that the $L^{1^{*}}$-distance of a possibly incompatible strain field $\beta$ from a single well is controlled in terms of the $L^{1^{*}}$-distance from a finite set of wells, of ${\rm curl}\beta$, and of ${\rm div}\beta$. As a consequence, we derive a strain-gradient plasticity model as $\Gamma$-limit of a nonlinear finite dislocation model, containing a singular perturbation term accounting for the divergence of the strain field. This can also be seen as a generalization of the result of Alicandro et al. (2018) to the case of incompatible vector fields.