Regularity, Cavitation and Harmonic Dipoles in Nonlinear Elasticity

Explore the intricacies of Nonlinear Elasticity in this 37-minute lecture by Carlos Mora Corral, presented at the Workshop on "Between Regularity and Defects: Variational and Geometrical Methods in Materials Science" at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into the representation of deformations using Sobolev maps and the importance of positive gradient determinants for orientation preservation. Examine the relationship between integrability exponents of the gradient and cofactor gradient, and their impact on map regularity and local invertibility. Investigate the conditions under which cavitation can occur and how it is detected using the distributional determinant. Learn about the unique case where cavitation is the only possible singularity and its implications for map regularity. Discover the phenomenon of harmonic dipoles, where two cavities form, one reversing orientation and creating a hole filled by material from elsewhere. Understand how the distributional determinant and singular part of the inverse are used to describe this complex singularity. This lecture, based on joint work with Marco Barchiesi, Duvan Henao, and Rémy Rodiac, offers a comprehensive exploration of regularity, cavitation, and harmonic dipoles in Nonlinear Elasticity.