Speaker
Description
We present some recent developments in the derivation of a model for three-dimensional crystal plasticity via a Gamma convergence approach. It has long been understood that, in crystalline materials such as metals, a major role in the mechanisms involved in plastic deformations is played by a particular kind of defects of the underlying crystal lattice called dislocations. In the two-dimensional setting in [1], the authors studied a linear quadratic elastic energy in the presence of a diverging number of point dislocations, deriving a strain gradient plastic energy in the sense of Gamma convergence assuming well separation of the defects. The nonlinear case later studied in [2].To tackle the three-dimensional case, in [3] we studied an intermediate problem consisting in a (rescaled) version of a line tension energy that allows to understand the behaviour of the self-energy of the dislocations as the total number of defects diverge. The two-dimensional counterpart of this result was originally obtained in [4]. As a byproduct of this analysis, we obtain a density result for one rectifiable fields without boundary in the class of divergence-free measures. In this seminar, we present the results contained in [3] and some recent developments obtained in collaboration with Roberta Marziani and Adriana Garroni for the nonlinear three-dimensional elastic energy in the quadratic case.
[1] Garroni, A., Leoni, G., and Ponsiglione, M. "Gradient theory for plasticity via homogenization of discrete dislocations." Journal of the European Mathematical Society 012.5 (2010): 1231-1266
[2] Müller, S., Scardia, L., and Zeppieri, C. I. "Geometric Rigidity for Incompatible Fields, and an Application to Strain-Gradient Plasticity". Indiana University Mathematics Journal, 63(5) (2014): 1365–1396
[3] M. Fortuna, A. Garroni. "Homogenization of line tension energies". Nonlinear Analysis, 250 (2025): 113656
[4] Conti, S., Adriana G. and Müller S. “Homogenization of vector-valued partition problems and dislocation cell structures in the plane.” Bollettino dell'Unione Matematica Italiana, 10 (2017): 3-17.
