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Analysis and numerical approximation of an elastic frictional contact problem with normal compliance

Weimin Han, Mircea Sofonea (1999)

Applicationes Mathematicae

We consider the problem of frictional contact between an elastic body and an obstacle. The elastic constitutive law is assumed to be nonlinear. The contact is modeled with normal compliance and the associated version of Coulomb's law of dry friction. We present two alternative yet equivalent weak formulations of the problem, and establish existence and uniqueness results for both formulations using arguments of elliptic variational inequalities and fixed point theory. Moreover, we show the continuous...

Analysis and Numerical Approximation of an Electro-elastic Frictional Contact Problem

El. Essoufi, El. Benkhira, R. Fakhar (2010)

Mathematical Modelling of Natural Phenomena

We consider the problem of frictional contact between an piezoelectric body and a conductive foundation. The electro-elastic constitutive law is assumed to be nonlinear and the contact is modelled with the Signorini condition, nonlocal Coulomb friction law and a regularized electrical conductivity condition. The existence of a unique weak solution of the model is established. The finite elements approximation for the problem is presented, and error...

Analysis of a contact adhesive problem with normal compliance and nonlocal friction

Arezki Touzaline (2012)

Annales Polonici Mathematici

The paper deals with the problem of a quasistatic frictional contact between a nonlinear elastic body and a deformable foundation. The contact is modelled by a normal compliance condition in such a way that the penetration is restricted with a unilateral constraint and associated to the nonlocal friction law with adhesion. The evolution of the bonding field is described by a first-order differential equation. We establish a variational formulation of the mechanical problem and prove an existence...

Analysis of a discrete model for the contact problem between a membrane and an elastic obstacle

Aldo Maceri, Franco Maceri (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In questo lavoro viene risolto il problema del contatto tra una membrana ed un suolo od ostacolo elastico con una approssimazione lineare a tratti della soluzione. Sono date alcune formulazioni equivalenti del problema discreto e se ne discutono le corrispondenti proprietà computazionali.

Analysis of a frictionless contact problem for elastic bodies

S. Drabla, M. Sofonea, B. Teniou (1998)

Annales Polonici Mathematici

This paper deals with a nonlinear problem modelling the contact between an elastic body and a rigid foundation. The elastic constitutive law is assumed to be nonlinear and the contact is modelled by the well-known Signorini conditions. Two weak formulations of the model are presented and existence and uniqueness results are established using classical arguments of elliptic variational inequalities. Some equivalence results are presented and a strong convergence result involving a penalized problem...

Analysis of a viscoelastic antiplane contact problem with slip-dependent friction

Thierry-Vincent Hoarau-Mantel, Andaluzia Matei (2002)

International Journal of Applied Mathematics and Computer Science

We study a mathematical problem modelling the antiplane shear deformation of a viscoelastic body in frictional contact with a rigid foundation. The contact is bilateral and is modelled with a slip-dependent friction law. We present the classical formulation for the antiplane problem and write the corresponding variational formulation. Then we establish the existence of a unique weak solution to the model, by using the Banach fixed-point theorem and classical results for elliptic variational inequalities....

Analytical results on a model for damaging in domains and interfaces

Elena Bonetti, Michel Frémond (2011)

ESAIM: Control, Optimisation and Calculus of Variations

This paper deals with a model describing damage processes in a (nonlinear) elastic body which is in contact with adhesion with a rigid support. On the basis of phase transitions theory, we detail the derivation of the model written in terms of a PDE system, combined with suitable initial and boundary conditions. Some internal constraints on the variables are introduced in the equations and on the boundary, to get physical consistency. We prove the existence of global in time solutions (to a suitable...

Analytical results on a model for damaging in domains and interfaces*

Elena Bonetti, Michel Frémond (2011)

ESAIM: Control, Optimisation and Calculus of Variations

This paper deals with a model describing damage processes in a (nonlinear) elastic body which is in contact with adhesion with a rigid support. On the basis of phase transitions theory, we detail the derivation of the model written in terms of a PDE system, combined with suitable initial and boundary conditions. Some internal constraints on the variables are introduced in the equations and on the boundary, to get physical consistency. We prove the existence of global in time solutions (to a suitable...

Approximation and numerical realization of 3D contact problems with given friction and a coefficient of friction depending on the solution

Jaroslav Haslinger, Tomáš Ligurský (2009)

Applications of Mathematics

The paper presents the analysis, approximation and numerical realization of 3D contact problems for an elastic body unilaterally supported by a rigid half space taking into account friction on the common surface. Friction obeys the simplest Tresca model (a slip bound is given a priori) but with a coefficient of friction ℱ which depends on a solution. It is shown that a solution exists for a large class of ℱ and is unique provided that ℱ is Lipschitz continuous with a sufficiently small modulus of...

Approximation and numerical solution of contact problems with friction

Jaroslav Haslinger, Miroslav Tvrdý (1983)

Aplikace matematiky

The present paper deals with numerical solution of the contact problem with given friction. By a suitable choice of multipliers the whole problem is transformed to that of finding a saddle-point of the Lagrangian function ℒ on a certain convex set K × Λ . The approximation of this saddle-point is defined, the convergence is proved and the rate of convergence established. For the numerical realization Uzawa’s algorithm is used. Some examples are given in the conclusion.

Conical differentiability for bone remodeling contact rod models

Isabel N. Figueiredo, Carlos F. Leal, Cecília S. Pinto (2005)

ESAIM: Control, Optimisation and Calculus of Variations

We prove the conical differentiability of the solution to a bone remodeling contact rod model, for given data (applied loads and rigid obstacle), with respect to small perturbations of the cross section of the rod. The proof is based on the special structure of the model, composed of a variational inequality coupled with an ordinary differential equation with respect to time. This structure enables the verification of the two following fundamental results: the polyhedricity of a modified displacement...

Conical differentiability for bone remodeling contact rod models

Isabel N. Figueiredo, Carlos F. Leal, Cecília S. Pinto (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We prove the conical differentiability of the solution to a bone remodeling contact rod model, for given data (applied loads and rigid obstacle), with respect to small perturbations of the cross section of the rod. The proof is based on the special structure of the model, composed of a variational inequality coupled with an ordinary differential equation with respect to time. This structure enables the verification of the two following fundamental results: the polyhedricity of a modified displacement constraint...

Contact between elastic bodies. I. Continuous problems

Jaroslav Haslinger, Ivan Hlaváček (1980)

Aplikace matematiky

Problems of a unilateral contact between bounded bodies without friction are considered within the range of two-dimensional linear elastostatics. Two classes of problems are distinguished: those with a bounded contact zone and with an enlargign contact zone. Both classes can be formulated in terms of displacements by means of a variational inequality. The proofs of existence of a solution are presented and the uniqueness discussed.

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