Éléments de Mathemátiques. By N. BourbakiXIII . Livre VI. Intégration. Pp. 237. 2500 fr. 1952. (Actualités scientifiques et industrielles, 1175; Hermann, Paris)

Numerical analysis and simulations of a dynamic frictionless contact problem with damage

In this article we examine an evolution problem, which describes the dynamic contact of a viscoelastic body and a foundation. The contact is modeled by a general normal damped response condition and a friction law, which are nonmonotone, possibly multivalued and have the subdifferential form. First we derive a formulation of the model in the form of a multidimensional hemivariational inequality. Then we establish a priori estimates and we prove the existence of weak solutions by using a surjectivity result for pseudomonotone operators. Finally, we deliver conditions under which the solution of the hemivariational inequality is unique.

Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction

In this paper we consider mathematical models describing dynamic viscoelastic contact problems with the Kelvin–Voigt constitutive law and subdifferential boundary conditions. We treat evolution hemivariational inequalities which are weak formulations of contact problems. We establish the existence of solutions to hemivariational inequalities with different types of non-monotone multivalued boundary relations. These results are consequences of an existence theorem for second order evolution inclusions. In a particular case we deliver sufficient conditions under which the solution to a hemivariational inequality is unique. Finally, applications to several unilateral and bilateral problems in contact mechanics are provided.

A Unified Approach to Dynamic Contact Problems in Viscoelasticity

In this paper, we study a class of generalized differential hemivariational inequalities of parabolic type involving the time fractional order derivative operator in Banach spaces. We use the Rothe method combined with surjectivity of multivalued pseudomonotone operators and properties of the Clarke generalized gradient to establish existence of solution to the abstract inequality. As an illustrative application, a frictional quasistatic contact problem for viscoelastic materials with adhesion is investigated, in which the friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals, and the constitutive relation is modeled by the fractional Kelvin–Voigt law.

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A class of fractional differential hemivariational inequalities with application to contact problem

Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion

A model for the dynamic, adhesive, frictionless contact between a viscoelastic body and a deformable foundation is described. The adhesion process is modeled by a bonding field on the contact surface. The contact is described by a modified normal compliance condition. The tangential shear due to the bonding field is included. The problem is formulated as a coupled system of a variational equality for the displacements and a differential equation for the bonding field. The existence of a unique weak solution for the problem is established, together with a partial regularity result. The existence proof proceeds by construction of an appropriate mapping which is shown to be a contraction on a Hilbert space.

Dynamic frictionless contact with adhesion

A frictionless contact problem for elastic–viscoplastic materials with normal compliance and damage

We consider a mathematical model which describes the frictional contact between a viscoelastic body and a reactive foundation. The process is assumed to be dynamic and the contact is modeled with a general normal damped response condition and a local friction law. We present a variational formulation of the problem and prove the existence and uniqueness of the weak solution, using results on evolution equations with monotone operators and a fixed point argument. We then introduce and study a fully discrete numerical approximation scheme of the variational problem, in terms of the velocity variable. The numerical scheme has a unique solution. We derive error estimates under additional regularity assumptions on the data and the solution.

A Dynamic Frictional Contact Problem with Normal Damped Response

We consider two quasistatic problems which describe the frictional contact between a deformable body and an obstacle, the so-called foundation. In the first problem the body is assumed to have a viscoelastic behavior, while in the other it is assumed to be elastic. The frictional contact is modeled by a general velocity dependent dissipation functional. We derive weak formulations for the models and prove existence and uniqueness results. The proofs are based on the theory of evolution variational inequalities and fixed-point arguments. We also prove that the solution of the viscoelastic problem converges to the solution of the corresponding elastic problem, as the viscosity tensor converges to zero. Finally, we describe a number of concrete contact and friction conditions to which our results apply.

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Oanh Chau

Papers

Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion

Dynamic frictionless contact with adhesion

A frictionless contact problem for elastic–viscoplastic materials with normal compliance and damage

A Dynamic Frictional Contact Problem with Normal Damped Response

Quasistatic Frictional Problems for Elastic and Viscoelastic Materials