A fractional model of hyperelasticity.

Elastic materials are those that deform under the action of an applied force and recover their original configuration when the load

stops acting. When the elastic potential energy can be modeled as a variational principle we call then hyperplastic materials, and it is
the natural way to model large deformation in materials under the action of very big loads. In this case, deformations are
minimizers of the variational principle given by the potential energy. In this talk,  we first recall the classical existence theory
in hyperelasticity, in which the central requirement is polyconvexity of the integrand in the variational principle. Main ingredient
for obtaining the existence result is the weak continuity of the deformation gradient, which is itself a very remarkable compensated
compactness result. Then, we propose a fractional model for hyperelasticity by replacing the gradient of the deformation by its Riesz
fractional gradient. Functional space for this model will be a Bessel potential space, which is a fractional space in between the
Lebesgue and Sobolev spaces. We show well-posedness  of this new model by proving a nonlocal Piola identity, that yields to the weak
continuity of the Riesz fractional gradient. One remarkable and fortunate feature of this new model is that it allows for
singularities, such a fracture or cavitation, to happen in the optimal deformations. This was forbidden in the classical models on
Sobolev spaces. 

The talk will be intended for a broad mathematical audience.

This is a joint work with J. Cueto (UCLM) and C. Mora-Corral (UAM). (Nonlocal hyperelasticity and polyconvexity in fractional spaces,
arXiv:1812.05848, 2019.)