Reduced models for linearly elastic thin films allowing for fracture, debonding or delamination

This work is devoted so show the appearance of different cracking modes in linearly elastic thin film systems by means of an asymptotic analysis as the thickness tends to zero. By superposing two thin plates, and upon suitable scaling law assumptions on the elasticity and fracture parameters, it is proven that either debonding or transverse cracks can emerge in the limit. A model coupling debonding, transverse cracks and delamination is also discussed

Energy estimates and cavity interaction for a critical-exponent cavitation model

We consider the minimization of \int_{\Omega_{\ep}} |D\vec u|^p \dd\vec x in a perforated domain \Omega_{\ep}:= \Omega \setminus \bigcup_{i=1}^M B_{\ep}(\vec a_i) of $\R^n$, among maps \vec u \in W^{1,p}(\Omega_{\ep}, \R^n) that are incompressible ($\det D\vec u\equiv 1$), invertible, and satisfy a Dirichlet boundary condition $\vec u= \vec g$ on $\partial \Omega$. If the volume enclosed by $\vec g (\partial \Omega)$ is greater than $|\Omega|$, any such deformation $\vec u$ is forced to map the small holes B_{\ep}(\vec a_i) onto macroscopically visible cavities (which do not disappear as \ep\to 0). We restrict our attention to the critical exponent $p=n$, where the energy required for cavitation is of the order of \sum_{i=1}^M v_i |\log \ep| and the model is suited, therefore, for an asymptotic analysis ($v_1,..., v_M$ denote the volumes of the cavities). In the spirit of the analysis of vortices in Ginzburg-Landau theory, we obtain estimates for the "renormalized" energy \frac{1}{n}\int_{\Omega_{\ep}} |\frac{D\vec u}{\sqrt{n-1}}|^p \dd\vec x - \sum_i v_i |\log \ep|, showing its dependence on the size and the shape of the cavities, on the initial distance between the cavitation points $\vec a_1,..., \vec a_M$, and on the distance from these points to the outer boundary $\partial \Omega$. Based on those estimates we conclude, for the case of two cavities, that either the cavities prefer to be spherical in shape and well separated, or to be very close to each other and appear as a single equivalent round cavity. This is in agreement with existing numerical simulations, and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence

Symmetry of uniaxial global Landau-de Gennes minimizers in the theory of nematic liquid crystals

We study uniaxial energy-minimizers within the Landau-de Gennes theory for nematic liquid crystals on a three-dimensional spherical droplet subject to homeotropic boundary conditions. We work in the low-temperature regime and show that uniaxial energy-minimizers necessarily have the structure of the well-studied radial-hedgehog solution in the low-temperature limit. An immediate consequence of this result is that Landau-de Gennes energy minimizers cannot be purely uniaxial for sufficiently low temperatures

Global invertibility of Sobolev maps

We define a class of Sobolev W 1,p (Ω , Rn ) functions, with p > n − 1, such that its trace on ∂Ω is also Sobolev, and do not present cavitation in the interior or on the boundary. We show that if a function in this class has positive Jacobian and coincides on the boundary with an injective map, then the function is itself injective. We then prove the existence of minimizers within this class for the type of functionals that appear in nonlinear elasticityC. Mora-Corral has been supported by the Spanish Ministry of Economy and Competitivity (Projects MTM2014-57769-C3-1-P, MTM2017-85934-C3-2-P and the “Ramón y Cajal” programme RYC-2010-06125) and the ERC Starting grant no. 307179. D. Henao has been supported by the FONDECYT project 1150038 of the Chilean Ministry of Education and the Millennium Nucleus Center for Analysis of PDE NC130017 of the Chilean Ministry of Econom

Γ-convergence Approximation of Fracture and Cavitation in Nonlinear Elasticity

The final publication is available at Springer via http://dx.doi.org/10.1007/s00205-014-0820-3Our starting point is a variational model in nonlinear elasticity that allows for cavitation and fracture that was introduced by Henao and Mora-Corral (Arch Rational Mech Anal 197:619–655, 2010). The total energy to minimize is the sum of the elastic energy plus the energy produced by crack and surface formation. It is a free discontinuity problem, since the crack set and the set of new surface are unknowns of the problem. The expression of the functional involves a volume integral and two surface integrals, and this fact makes the problem numerically intractable. In this paper we propose an approximation (in the sense of Γ-convergence) by functionals involving only volume integrals, which makes a numerical approximation by finite elements feasible. This approximation has some similarities to the Modica–Mortola approximation of the perimeter and the Ambrosio–Tortorelli approximation of the Mumford–Shah functional, but with the added difficulties typical of nonlinear elasticity, in which the deformation is assumed to be one-to-one and orientation-preservingD. Henao gratefully acknowledges the Chilean Ministry of Education’s support through the FONDE-CYT Iniciación project no. 11110011. C. Mora-Corral has been supported by Project MTM2011-28198 of the Spanish Ministry of Economy and Competitivity, the ERC Starting grant no. 307179, the “Ramón y Cajal” programme and the European Social Fund. X. Xu acknowledges the funding by NSFC 1100126

Investigación: "Salazon, prensado y secado de Tilapia del Nilo" (Sarotheradon Ni laticus)

Se pretendío presentar la importancia de la investigación y concretar los objetivos de la misma a través de 126 referencias bibliográficas se presenta una revisión de los siguientes aspectos pertinentes a la investigación