Some Numerical Simulations Based on Dacorogna Example Functions in Favor of Morrey Conjecture

Morrey Conjecture deals with two properties of functions which are known as quasi-convexity and rank-one convexity. It is well established that every function satisfying the quasi-convexity property also satisfies rank-one convexity. Morrey (1952) conjectured that the reversed implication will not always hold. In 1992, Vladimir Sverak found a counterexample to prove that Morrey Conjecture is true in three dimensional case. The planar case remains, however, open and interesting because of its connections to complex analysis, harmonic analysis, geometric function theory, probability, martingales, differential inclusions and planar non-linear elasticity. Checking analytically these notions is a very difficult task as the quasi-convexity criterion is of non-local type, especially for vector-valued functions. That's why we perform some numerical simulations based on a gradient descent algorithm using Dacorogna and Marcellini example functions. Our numerical results indicate that Morrey Conjecture holds true

A New Model for Void Coalescence by Internal Necking

A micromechanical model for predicting the strain increment required to bring a damaged material element from the onset of void coalescence up to final fracture is developed based on simple kinematics arguments. This strain increment controls the unloading slope and the energy dissipated during the final step of material failure. Proper prediction of the final drop of the load carrying capacity is an important ingredient of any ductile fracture model, especially at high stress triaxiality. The model has been motivated and verified by comparison to a large set of finite element void cell calculations.

Numerical Simulations of Void Linkage in Model Materials using a Nonlocal Ductile Damage Approximation

Experiments on the growth and linkage of 10 μm diameter holes laser drilled in high precision patterns into Al-plates were modelled with finite elements. The simulations used geometries identical to those of the experiments and incorporated ductile damage by element removal under the control of a ductile damage indicator based on the micromechanical studies of Rice and Tracey. A regularization of the problem was achieved through an integral-type nonlocal model based on the smoothing of the rate of a damage indicator D over a characteristic length L. The simulation does not predict the experimentally observed damage acceleration either in the case where no damage is included or when only a local damage model is used. However, the full three-dimensional simulations based on the nonlocal damage methodology do predict both the failure path and the failure strain at void linkage for almost all configurations studied. For the cases considered the critical parameter controlling the local deformations at void linkage was found to be the ratio between hole diameter and hole spacing

A generalized constitutive elasticity law for GLPD micromorphic materials, with application to the problem of a spherical shell subjected to axisymmetric loading conditions

In this work we propose to replace the GLPD hypo-elasticity law by a more rigorous generalized Hooke's law based on classical material symmetry characterization assumptions. This law introduces in addition to the two well-known Lame's moduli, five constitutive constants. An analytical solution is derived for the problem of a spherical shell subjected to axisymmetric loading conditions to illustrate the potential of the proposed generalized Hooke's law

New applications of a generalized Hooke’s law for second gradient materials

We provide analytical solutions to the problems of a circular bending of a beam in plane strain and the torsion of a non-circular cross-section beam, the beams obeying a second-gradient elasticity law proposed by the author, following a previous suggestion of Dell’Isola et al. (2009). The motivation was to find benchmark analytical solutions that can serve to grasp the physical foundations of second gradient elasticity laws for heterogeneous materials. The analytical solution of the circular beam problem presents the additional advantage to establish some nice properties on the unknown second gradient elastic moduli introduced by Enakoutsa (2014) model and the classical elasticity constants for both incompressible and compressible heterogeneous elastic materials. A framework to find the elastic moduli of the new model is also proposed

Modèles non locaux en rupture ductile des métaux

In the first part, we assess the practical efficacity of two proposals of modification of the Gurson model to circumvent the problem of unlimited strain and damage localization in this model. The assessment of the model is based on two criteria, absence of mesh size effect in finish elements computations and agreement of experimental and numerical results for some typical ductile fracture tests. The first proposal consisted of adopting some nonlocal evolution equation for the porosity involving some convolution integral. The second proposal is an extension of Gurson's condition of homogeneous boundary strain rate, to the case of conditions of inhomogeneous boundary strain rate. In the second part, one define a model for porous ductile material containing two "populations" of cavities, extending that of Perrin et al. (2000) to the case where continuous nucleation of secondary small voids is taking into account.On évalue dans la première partie, l'efficacité pratique de deux solutions au problème de la concentration infinie de la déformation et de l'endommagement dans le modèle de Gurson, sous l'angle de leur capacité à affranchir les résultats numériques qu'elles prédisent vis-à-vis de la taille de maille et à reproduire de manière satisfaisante les résultats d'expériences de rupture ductile. La première solution consiste à adopter dans les équations du modèle de Gurson une équation d'évolution non locale de la porosité, sous la forme d'une intégrale de convolution. La seconde est une extension de la technique d'homogénéisation de Gurson en condition de taux de déformation homogène au bord, au cas des conditions de taux de déformation inhomogène au bord. Dans la seconde partie, on définit un modèle pour un matériau ductile poreux à deux populations de cavités, étendant celui de Perrin et al. (2000) au cas de la prise en compte de la germination continue des petites cavités de la seconde populatio