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An asymptotically optimal model for isotropic heterogeneous linearly elastic plates

Ferdinando Auricchio, Carlo Lovadina, Alexandre L. Madureira (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we derive and analyze a Reissner-Mindlin-like model for isotropic heterogeneous linearly elastic plates. The modeling procedure is based on a Hellinger-Reissner principle, which we modify to derive consistent models. Due to the material heterogeneity, the classical polynomial profiles for the plate shear stress are replaced by more sophisticated choices, that are asymptotically correct. In the homogeneous case we recover a Reissner-Mindlin model with 5/6 as shear correction...

An asymptotically optimal model for isotropic heterogeneous linearly elastic plates

Ferdinando Auricchio, Carlo Lovadina, Alexandre L. Madureira (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, we derive and analyze a Reissner-Mindlin-like model for isotropic heterogeneous linearly elastic plates. The modeling procedure is based on a Hellinger-Reissner principle, which we modify to derive consistent models. Due to the material heterogeneity, the classical polynomial profiles for the plate shear stress are replaced by more sophisticated choices, that are asymptotically correct. In the homogeneous case we recover a Reissner-Mindlin model with 5 / 6 as shear correction factor....

Anisotropic geometric functionals and gradient flows

Giovanni Bellettini, Luca Mugnai (2009)

Banach Center Publications

We survey some recent results on the gradient flow of an anisotropic surface energy, the integrand of which is one-homogeneous in the normal vector. We discuss the reasons for assuming convexity of the anisotropy, and we review some known results in the smooth, mixed and crystalline case. In particular, we recall the notion of calibrability and the related facet-breaking phenomenon. Minimal barriers as weak solutions to the gradient flow in case of nonsmooth anisotropies are proposed. Furthermore,...

Convergence of the finite element method applied to an anisotropic phase-field model

Erik Burman, Daniel Kessler, Jacques Rappaz (2004)

Annales mathématiques Blaise Pascal

We formulate a finite element method for the computation of solutions to an anisotropic phase-field model for a binary alloy. Convergence is proved in the H 1 -norm. The convergence result holds for anisotropy below a certain threshold value. We present some numerical experiments verifying the theoretical results. For anisotropy below the threshold value we observe optimal order convergence, whereas in the case where the anisotropy is strong the numerical solution to the phase-field equation does not...

Finite elements methods for solving viscoelastic thin plates

Helena Růžičková, Alexander Ženíšek (1984)

Aplikace matematiky

The present paper deals with numerical solution of a viscoelastic plate. The discrete problem is defined by C 1 -elements and a linear multistep method. The effect of numerical integration is studied as well. The rate of cnvergence is established. Some examples are given in the conclusion.

Frictional contact of an anisotropic piezoelectric plate

Isabel N. Figueiredo, Georg Stadler (2009)

ESAIM: Control, Optimisation and Calculus of Variations

The purpose of this paper is to derive and study a new asymptotic model for the equilibrium state of a thin anisotropic piezoelectric plate in frictional contact with a rigid obstacle. In the asymptotic process, the thickness of the piezoelectric plate is driven to zero and the convergence of the unknowns is studied. This leads to two-dimensional Kirchhoff-Love plate equations, in which mechanical displacement and electric potential are partly decoupled. Based on this model numerical examples are presented...

Gradient theory for plasticity via homogenization of discrete dislocations

Adriana Garroni, Giovanni Leoni, Marcello Ponsiglione (2010)

Journal of the European Mathematical Society

We deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations. We restrict our analysis to the case of a cylindrical symmetry for the crystal under study, so that the mathematical formulation will involve a two-dimensional variational problem. The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so-called core region. We show that the Γ -limit of this energy (suitably rescaled),...

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