arXiv subjects

Carlos Mora-Corral

Publications and source records attributed to Carlos Mora-Corral.

22 records · Page 2Linked to original sources

arXiv2017

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

We start from a variational model for nematic elastomers that involves two energies: mechanical and nematic. The first one consists of a nonlinear elastic energy which is influenced by the orientation of the molecules of the nematic elastomer. The nematic energy is an Oseen--Frank energy in the deformed configuration. The constraint of the positivity of the determinant of the deformation gradient is imposed. The functionals are not assumed to have the usual polyconvexity or quasiconvexity assumptions to be lower semicontinuous. We instead compute its relaxation, that is, the lower semicontinuous envelope, which turns out to be the quasiconvexification of the mechanical term plus the tangential quasiconvexification of the nematic term. The main assumptions are that the quasiconvexification of the mechanical term is polyconvex and that the deformation is in the Sobolev space $W^{1,p}$ (with $p>n-1$ and $n$ the dimension of the space) and does not present cavitation.

math.AP↗

arXiv2016

Sobolev homeomorphisms with gradients of low rank via laminates

Let $Ω\subset \mathbb{R}^{n}$ be a bounded open set. Given $2\leq m\leq n$, we construct a convex function $ϕ:Ω\to \mathbb{R}$ whose gradient $f= \nabla ϕ$ is a Hölder continuous homeomorphism, $f$ is the identity on $\partial Ω$, the derivative $D f$ has rank $m-1$ a.e.\ in $Ω$ and $D f$ is in the weak $L^{m}$ space $L^{m,w}$. The proof is based on convex integration and staircase laminates.

math.CA↗

arXiv2008

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

In this paper we present two atomistic models for the energy of a one-dimensional elastic crystal. We assume that the macroscopic displacement equals the microscopic one. The energy of the first model is given by a two-body interaction potential, and we assume that the atoms follow a continuous and piecewise smooth macroscopic (continuum) deformation. We calculate the first terms of the Taylor expansion (with respect to the parameter representing the interatomic distance) of the atomistic energy, and obtain that the coefficients of that Taylor expansion represent, respectively, an elastic energy, a sharp-interface energy, and a smooth-interface energy. The second atomistic model is a variant of the first one, and its Taylor expansion predicts, in addition, a new term that accounts for the repulsion force between two sharp interfaces.

math-ph↗

arXiv2008

Approximation of Holder continuous homeomorphisms by piecewise affine homeomorphisms

This paper is concerned with the problem of approximating a homeomorphism by piecewise affine homeomorphisms. The main result is as follows: every homeomorphism from a planar domain with a polygonal boundary to R^2 that is globally Holder continuous of exponent α, and whose inverse is also globally Holder continuous of exponent αcan be approximated in the Holder norm of exponent βby piecewise affine homeomorphisms, for some βthat only depends on α. The proof is constructive. We adapt the proof of simplicial approximation in the supremum norm, and measure the side lengths and angles of the triangulation over which the approximating homeomorphism is piecewise affine. The approximation in the supremum norm, and a control on the minimum angle and on the ratio between the maximum and minimum side lengths of the triangulation suffice to obtain approximation in the Holder norm.

math.CA↗