Quasistatic Contact Problem Research Articles

State-Dependent Implicit Sweeping Process in the Framework of Quasistatic Evolution Quasi-Variational Inequalities

This paper deals with the existence and uniqueness of solutions for a class of state-dependent sweeping processes with constrained velocity in Hilbert spaces without using any compactness assumption, which is known to be an open problem. To overcome the difficulty, we introduce a new notion called hypomonotonicity-like of the normal cone to the moving set, which is satisfied by many important cases. Combining this latter notion with an adapted Moreau’s catching-up algorithm and a Cauchy technique, we obtain the strong convergence of approximate solutions to the unique solution, which is a fundamental property. Using standard tools from convex analysis, we show the equivalence between this implicit state-dependent sweeping processes and quasistatic evolution quasi-variational inequalities. As an application, we study the state-dependent quasistatic frictional contact problem involving viscoelastic materials with short memory in contact mechanics.

On convergence of solutions to history-dependent variational–hemivariational inequalities

The goal of the paper is to study the behavior of solutions to the history-dependent variational–hemivariational inequalities under perturbation of the data. First, we give an existence and uniqueness result which has been obtained by using previous works. Second, we deliver a continuous dependence result when all the data are subjected to perturbations. Finally, a quasistatic viscoelastic frictional contact problem is presented to illustrate our main results.

Numerical analysis of history-dependent variational–hemivariational inequalities with applications in contact mechanics

Abstract This paper is devoted to numerical analysis of history-dependent variational– hemivariational inequalities arising in contact problems for viscoelastic material. We introduce both temporally semi-discrete approximation and fully discrete approximation for the problem, where the temporal integration is approximated by a trapezoidal rule and the spatial variable is approximated by the finite element method. We analyze the discrete schemes and derive error bounds. The results are applied for the numerical solution of a quasistatic contact problem. For the linear finite element method, we prove that the error estimation for the numerical solution is of optimal order under appropriate solution regularity assumptions.

A quasistatic frictional contact problem for viscoelastic materials with long memory

Abstract The aim of this paper is to study a quasistatic frictional contact problem for viscoelastic materials with long-term memory. The contact boundary conditions are governed by Tresca’s law, involving a slip dependent coefficient of friction. We focus our attention on the weak solvability of the problem within the framework of variational inequalities. The existence of a solution is obtained under a smallness assumption on a normal stress prescribed on the contact surface and on the coefficient of friction. The proof is based on a time discretization method, compactness and lower semicontinuity arguments.

Variational–hemivariational approach to quasistatic viscoplastic contact problem with normal compliance, unilateral constraint, memory term, friction and damage

The aim of this paper is to study a mathematical model which describes the quasistatic frictional contact between a viscoplastic body and a foundation. The material’s behavior is modeled with a rate-type constitutive law with internal state variable. The contact is modeled with normal compliance, unilateral constraint and memory term, friction and damage. We present the classical formulation of the problem and we derive its variational–hemivariational formulation. Finally, we prove its unique weak solvability. The proof is based on abstract results for a class of history-dependent variational–hemivariational inequalities.

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

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.

Nonsteady frictional heating on a sliding contact

We consider quasi-static contact problem on frictional heating on a sliding contact of a rotating rigid cylinder and a half-plane. The cylinder is pressed towards the half-plane material. The problem is reduced to solution of a singular integral equation with respect to contact stresses. Solution of the singular equation is looked for in a class of functions limited on the edge, with two additional conditions to determine timedependent boundaries of the contact area. Temperature at the contact and inside the half-plane is determined in terms of contact stresses.

An Implicit Sweeping Process Approach to Quasistatic Evolution Variational Inequalities

In this paper, we study a new variant of Moreau's sweeping process with velocity constraint. Based on an adapted version of Moreau's catching-up algorithm, we show the well-posedness (in the sense existence and uniqueness) of this problem in a general framework. We show the equivalence between this implicit sweeping process and a quasistatic evolution variational inequality. It is well known that the variational formulations of many mechanical problems with unilateral contact and friction lead to an evolution variational inequality. As an application, we reformulate the quasistatic antiplane frictional contact problem for linear elastic materials with short memory as an implicit sweeping process with velocity constraint. The link between the implicit sweeping process and the quasistatic evolution variational inequality is possible thanks to some standard tools from convex analysis and is new in the literature.

Analysis of a quasistatic contact problem with wear and damage for thermo-viscoelastic materials

Analysis of a quasistatic contact problem with wear and damage for thermo-viscoelastic materials

A variational inequality and applications to quasistatic problems with Coulomb friction

The aim of this paper is to study an evolution variational inequality that generalizes some contact problems with Coulomb friction in small deformation elasticity. Using an incremental procedure, appropriate estimates and convergence properties of the discrete solutions, the existence of a continuous solution is proved. This abstract result is applied to quasistatic contact problems with a local Coulomb friction law for nonlinear Hencky and also for linearly elastic materials.

Bilateral Contact Problem with Adhesion between Two Bodies for Viscoelastic with Long-term Memory and Damage

We consider a quasistatic contact problem between two viscoelastic bodies with long-term memory and damage. The contact is bilateral and the tangential shear due to the bonding field is included. The adhesion of the contact surfaces is taken into account and modelled by a surface variable, the bonding field. We prove the existence of a unique weak solution to the problem. The proof is based on arguments of time-dependent variational inequalities, parabolic inequalities, differential equations and fixed point.

Weak solvability of a fractional viscoelastic frictionless contact problem

The goal of this paper is to study a quasistatic frictionless contact problem for a viscoelastic body in which the constitutive equation is modeled with the fractional Kelvin–Voigt law and the contact condition is described by the Clarke subdifferential of a nonconvex and nonsmooth functional. The variational formulation of this problem is provided in the form of a fractional hemivariational inequality. In order to solve this inequality, we apply the Rothe method and prove that the associated abstract Volterra inclusion has at least one solution.

Quasistatic normal-compliance contact problem of visco-elastic bodies with Coulomb friction implemented by QP and SGBEM

The quasistatic normal-compliance contact problem of isotropic homogeneous linear visco-elastic bodies with Coulomb friction at small strains in Kelvin–Voigt rheology is considered. The discretization is made by a semi-implicit formula in time and the Symmetric Galerkin Boundary Element Method (SGBEM) in space, assuming that the ratio of the viscosity and elasticity moduli is a given relaxation-time coefficient. The obtained recursive minimization problem, formulated only on the contact boundary, has a nonsmooth cost function. If the normal compliance responds linearly and the 2D problems are considered, then the cost function is piecewise-quadratic, which after a certain transformation gets the quadratic programming (QP) structure. However, it would lead to second-order cone programming in 3D problems. Finally, several computational tests are presented and analysed, with an additional discussion on numerical stability and convergence of the involved approximated Poincaré–Steklov operators.

Quasistatic bilateral contact problem with time delay for viscoelastic materials

In this paper we study the mathematical model which describes quasistatic frictional contact problems between a deformable body and a rigid foundation. In this model the contact is bilateral and the behaviour of the material is described by a viscoelastic constitutive law with time delay. The variational formulation of the mathematical model is given as a hemivariational inequality of elliptic type. Based on some recent results for the elliptic hemivariational inequality and Banach’s fixed point theorem we prove existence and uniqueness of the solution for the obtained hemivariational inequality.

A quasistatic frictional contact problem with damage involving viscoelastic materials with short memory

We consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is assumed to be quasistatic and the material behaviour is described by a viscoelastic constitutive law with damage. The friction and contact are modeled with subdifferential boundary conditions. We derive the variational formulation of the problem which is a coupled system of a hemivariational inequality for the velocity and a parabolic variational inequality for the damage field. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of time-dependent stationary inclusions and a fixed point theorem.

A quasistatic frictional contact problem with normal compliance and unilateral constraint

A mathematical model describing the quasistatic process of frictional contact between a nonlinearly elastic body and an elastic-rigid foundation is considered. The contact is modeled by normal compliance with unilateral constraint, and the friction by a slip-dependent version of Coulomb’s law. A weak formulation of the problem is derived and, under a smallness assumption on contact and friction functions, an existence result is proved by using incremental techniques, Kakutani’s fixed point theorem, and compactness, monotonicity, and lower semicontinuity arguments.

A quasistatic viscoplastic contact problem with normal compliance, unilateral constraint, memory term and friction

The goal of this paper is to deal with a mathematical model which describes the quasistatic frictional contact between a viscoplastic body and a foundation. The contact is modeled with normal compliance, unilateral constraint and memory term. We present the classical formulation of the problem together with the list of assumptions on the data. Then we derive the variational–hemivariational formulation of the model and we prove its unique weak solvability. The proof is based on a recent abstract result of a class of history-dependent variational–hemivariational inequalities.

A quasistatic viscoelastic frictional contact problem with multivalued normal compliance, unilateral constraint and material damage

The goal of this paper is to deal with a mathematical model which describes the quasistatic frictional contact between a viscoelastic body and a foundation. We are interested in the foundation which is made of a hard material covered by a thin layer composed of a rigid crust and a viscoplastic material with finite thickness. The mechanical damage, caused by excessive stress or strain, is emerged in the viscoelastic body during the contact process. We provide the variational formulation of the model which is a system consisting of a history-dependent variational–hemivariational inequality, a fully nonlinear integral equation and a parabolic variational inequality. Then, in order to solve this complex system, we deliver abstract results on a class of history-dependent variational–hemivariational inequalities. Finally, our abstract results are applied to derive the unique weak solvability of the frictional contact problem under consideration.

An efficient quasi-optimal space-time PGD application to frictional contact mechanics

The proper generalized decomposition (PGD) aims at finding the solution of a generic problems into a low rank approximation. On the contrary to the singular value decomposition (SVD), such a low rank approximation is generally not the optimal one leading to memory issues and loss of computational efficiency. Nonetheless, the computational cost of the SVD is generally prohibitive to be performed. In this paper, authors suggest an algorithm to address this issue. First, the algorithm is described and studied in details. It consists in a cheap iterative method compressing a low rank expansion. It will be shown that given a low rank approximation, the SVD of a provided low rank approximation can be reached at convergence. Behavior of the method is exhibited on a numerical application. Second, the algorithm is embedded into a general space-time PGD solver to compress the iterated separated form for the solution. An application to a quasi-static frictional contact problem is illustrated. Then, efficiency of such a compressing method will be demonstrated.

Quasistatic contact problem between thermo-electroelastic bodies with long-term memory and adhesion

We study of a quasistatic frictional contact problem between two thermo-electroelastic bodies with adhesion. The temperature of the materials caused by elastic deformations. The contact is modelled with a version of normal compliance condition and the associated Coulomb’s law of friction in which the adhesion of contact surfaces is taken into account. We establish a variational formulation for the model and we prove the existence of a unique weak solution to the problem. The proof is based on a classical existence anduniqueness result on parabolic equalities, differential equations and fixed point arguments.