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Generalized Newton methods for the 2D-Signorini contact problem with friction in function space

Karl Kunisch, Georg Stadler (2005)

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

The 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented lagrangians allows to apply an infinite-dimensional...

Generalized Newton methods for the 2D-Signorini contact problem with friction in function space

Karl Kunisch, Georg Stadler (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented Lagrangians allows to apply an...

Geometrically nonlinear shape-memory polycrystals made from a two-variant material

Robert V. Kohn, Barbara Niethammer (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Bhattacharya and Kohn have used small-strain (geometrically linear) elasticity to analyze the recoverable strains of shape-memory polycrystals. The adequacy of small-strain theory is open to question, however, since some shape-memory materials recover as much as 10 percent strain. This paper provides the first progress toward an analogous geometrically nonlinear theory. We consider a model problem, involving polycrystals made from a two-variant elastic material in two space dimensions. The linear theory...

Global existence of weak solutions to the Fried-Gurtin model of phase transitions

Zenon Kosowski (2007)

Applicationes Mathematicae

We prove the existence of global in time weak solutions to a three-dimensional system of equations arising in a simple version of the Fried-Gurtin model for the isothermal phase transition in solids. In this model the phase is characterized by an order parameter. The problem considered here has the form of a coupled system of three-dimensional elasticity and parabolic equations. The system is studied with the help of the Faedo-Galerkin method using energy estimates.

Global well-posedness and blow up for the nonlinear fractional beam equations

Shouquan Ma, Guixiang Xu (2010)

Applicationes Mathematicae

We establish the Strichartz estimates for the linear fractional beam equations in Besov spaces. Using these estimates, we obtain global well-posedness for the subcritical and critical defocusing fractional beam equations. Of course, we need to assume small initial data for the critical case. In addition, by the convexity method, we show that blow up occurs for the focusing fractional beam equations with negative energy.

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),...

Ground states in complex bodies

Paolo Maria Mariano, Giuseppe Modica (2009)

ESAIM: Control, Optimisation and Calculus of Variations

A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappings and cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. Sobolev maps and cartesian currents describe the inner substructure of the material elements. Balance equations for irregular minimizers are derived. A contribution to the debate about the role of the balance...

Ground states in complex bodies

Paolo Maria Mariano, Giuseppe Modica (2008)

ESAIM: Control, Optimisation and Calculus of Variations

A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappings and Cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. Sobolev maps and Cartesian currents describe the inner substructure of the material elements. Balance equations for irregular minimizers are derived. A contribution to the debate about the role of the balance...

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