Carlos Mora-Corral

Existence for Nonlocal Variational Problems in Peridynamics

We present an existence theory based on minimization of the nonlocal energies appearing in peridynamics, which is a nonlocal continuum model in solid mechanics that avoids the use of deformation gradients. We employ the direct method of the calculus of variations in order to find minimizers of the energy of a deformation. Lower semicontinuity is proved under a weaker condition than convexity, whereas coercivity is proved via a nonlocal Poincare inequality. We cover Dirichlet, Neumann, and mixed boundary conditions. The existence theory is set in the Lebesgue

Lusin's condition and the distributional determinant for deformations with finite energy

L^p

Sobolev homeomorphisms with gradients of low rank via laminates

spaces and in the fractional Sobolev

Continuum limits of atomistic energies allowing smooth and sharp interfaces in 1D elasticity

W^{s,p}

Quasistatic Evolution of Cavities in Nonlinear Elasticity

spaces, for

Lower semicontinuity and relaxation via young measures for nonlocal variational problems and applications to peridynamics

0 < s < 1

Relaxation of nonlinear elastic energies involving the deformed configuration and applications to nematic elastomers

and

Quasistatic elastoplasticity via Peridynamics: existence and localization

1 < p < \infty

Invertibility and Weak Continuity of the Determinant for the Modelling of Cavitation and Fracture in Nonlinear Elasticity

.

Fracture Surfaces and the Regularity of Inverses for BV Deformations

Abstract. Based on a previous work by the authors on the modelling of cavitation and fracture in nonlinear elasticity, we give an alternative proof of a recent result by Csörnyei, Hencl and Malý on the regularity of the inverse of homeomorphisms in the Sobolev space . With this aim, we show that the notion of fracture surface introduced by the authors in their model corresponds precisely to the original notion of cavity surface in the cavitation models of Müller and Spector (1995) and Conti and De Lellis (2003). We also find that the surface energy introduced in the model for cavitation and fracture is related to Lusins condition (N) on the non-creation of matter. A fundamental question underlying this paper is whether necessarily implies that the deformation opens no cavities. We show that this is not true unless Müller and Spectors condition INV for the non-interpenetration of matter is satisfied. Having thus provided an additional justification of its importance, we prove the stability of this condition with respect to weak convergence in the critical space . Combining this with the work by Conti and De Lellis, we obtain an existence theory for cavitation in this critical case.