Viscoelastic Contact Problem Research Articles

A class of evolutionary history-dependent variational-hemivariational inequalities with strong and weak solutions

AbstractIn this paper we examine the well-posedness of evolutionary variational-hemivariational inequalities involving a constraint set and history-dependent operators. The strong and weak formulations of such inequalities are studied. First, the existence and uniqueness of solutions to both formulations are proved, and results on the dependence of solution on functional parameters are delivered. Next, the well-posedness is established for a general form of history-dependent variational-hemivariational inequalities with constraints by using a fixed point theorem. Finally, the results are applied to a dynamic frictional contact problem in viscoelasticity in which the contact is described by Signorini-type unilateral boundary condition with a nonmonotone Clarke’s relation.

Modelling and analysis of a viscoelastic contact problem with unilateral constraints

Modelling and analysis of a viscoelastic contact problem with unilateral constraints

Virtual element method for solving a viscoelastic contact problem with long memory

In this paper, we use the virtual element method to solve a history-dependent hemivariational inequality arising in contact problems. The contact problem concerns the deformation of a viscoelastic body with long memory, subjected to a contact condition with non-monotone normal compliance and unilateral constraints. A fully discrete scheme based on the trapezoidal rule for the discretization of the time integration and the virtual element method for the spatial discretization are analyzed. We provide a unified priori error analysis for both internal and external approximations. For the linear virtual element method, we obtain the optimal order error estimate. Finally, three numerical examples are reported, providing numerical evidence of the theoretically predicted optimal convergence orders.

A doubly history‐dependent quasivariational inequality arising in viscoelastic frictional contact problems with wear

AbstractWe aim here to investigate a new mathematical model that describes the contact between a viscoelastic body, accounting for long memory and wear effects, and an obstacle referred to as the foundation. The contact model is governed by a normal compliance condition, coupled with Coulomb's law of dry friction for sliding, and wear effects. We derive the variational formulation of the model, which involves coupling of a quasi‐variational inequality with a nonlinear equation. By pursuing the abstract history‐dependent quasi‐variational inequalities and leveraging the fixed point theorem, we establish results concerning both existence and uniqueness.

Hyperbolic quasivariational inequalities with applications to dynamic viscoelastic frictional contact problem

In this paper, we consider a quasivariational inequalities of hyperbolic type, the existence, uniqueness and regularity of a solution to the quasivariational inequalities is proved by using Rothe method and fixed-point technique. We illustrate the abstract results by an application to dynamic contact frictional problem for viscoelastic materials where the friction bound function depends on the slip.

A numerical model to simulate the transient frictional viscoelastic sliding contact

Sliding motion has always been one of the major concerns when it comes to the analysis of viscoelastic contact problems. A new model simulating the transient sliding contact of smooth viscoelastic surfaces is developed in this paper. By taking the dry contact friction and the coupling between shear tractions and normal pressure into account, the effect of the early partial slip period, which is often neglected in the study of viscoelastic sliding contact problems, is investigated numerically. Compared with solutions based on the frictionless assumption, the steady-state pressure profile is found to be slightly different under the effect of the partial slip regime, including a lower peak pressure and the shift of the contacting region in the direction opposite to the sliding motion. Furthermore, the time required for the viscoelastic contact to reach its steady state is delayed owing to the partial slip period preceding the global sliding motion.

Duality arguments in the analysis of a viscoelastic contact problem

We consider a mathematical model which describes the quasistatic frictionless contact of a viscoelastic body with a rigid-plastic foundation. We describe the mechanical assumptions, list the hypotheses on the data and provide three different variational formulations of the model in which the unknowns are the displacement field, the stress field and the strain field, respectively. These formulations have a different structure. Nevertheless, we prove that they are pairwise dual of each other. Then, we deduce the unique weak solvability of the contact problem as well as the Lipschitz continuity of its weak solution with respect to the data. The proofs are based on recent results on history-dependent variational inequalities and inclusions. Finally, we present numerical simulations in the study of the contact problem, together with the corresponding mechanical interpretations.

A time-fractional of a viscoelastic frictionless contact problem with normal compliance

A time-fractional of a viscoelastic frictionless contact problem with normal compliance

The contact condition of viscoelastic contact problems

Three contact states in the viscoelastic contact deformation area between the rigid cylinder and the viscoelastic foundation are discussed based on various material mechanical properties between the rigid and the viscoelastic materials. On this basis, the relationship model between the node displacements and the node stresses in the boundary contact area is established, and the corresponding numerical analytical algorithms are used to simulate the viscoelastic contact deformation. Meanwhile, an example is given to simulate the contact deformation of the conveyor belt.

Inverse problems for constrained parabolic variational-hemivariational inequalities *

In this paper we study a novel class of inverse problems for parabolic variational–hemivariational inequalities with a unilateral constraint. A theorem on the well-posedness for weak solution is established. Based on a new continuous dependence result, we prove the nonemptiness and stability of the set of optimal solutions to the corresponding inverse problems. We illustrate the results by a quasistatic nonsmooth frictional viscoelastic contact problem with a unilateral constraint for which we derive results on the unique weak solvability, and existence and stability of solutions to the associated inverse problems.

Numerical analysis of a non-clamped dynamic thermoviscoelastic contact problem

In this work, we analyze a non-clamped dynamic viscoelastic contact problem involving thermal effect. The friction law is described by a non-monotone relation between the tangential stress and the tangential velocity. This leads to a system of second-order inclusion for displacement and a parabolic equation for temperature. We provide a fully discrete approximation of the problem and find optimal error estimates without any smallness assumption on the data. The theoretical result is illustrated by numerical simulations.

A new class of fully history-dependent variational-hemivariational inequalities with application to contact mechanics

In this paper, we consider the behaviour of solutions to a class of fully history-dependent variational-hemivariational inequalities with respect to the perturbation of the data. First, the existence and uniqueness of the solution to a class of fully history-dependent variational-hemivariational inequalities is obtained by using a fixed point theorem. Second, we obtain continuous dependence result of solutions with respect to all the data of variational-hemivariational inequalities. Meanwhile, the convergence results of the solutions for the special case of abstract variational inequalities are also given. Finally, to illustrate our main results, we consider a class of viscoelastic contact problem with a long memory. By using our abstract result, we get the continuous dependence of the solutions to frictional contact problem with respect to all the data.

A Review on Modelling of Viscoelastic Contact Problems

Approaches to solving viscoelastic problems have received extensive attention in recent decades as viscoelastic materials have been widely applied in various fields. An overview of relevant modelling approaches is provided in the paper. The review starts with a brief introduction of some basic terminologies and theories that are commonly used to describe the contact behaviour of viscoelastic materials. By building up the complexity of contact problems, including dry contact, lubricated contact, thermoviscoelastic contact and non-linear viscoelastic contact, tentative analytical solutions are first introduced as essential milestones. Afterwards, a series of numerical models for the various types of contact problems with and without surface roughness are presented and discussed. Examples, in which computational tools were employed to assist the analysis of viscoelastic components in different fields, are given as case studies to demonstrate that a comprehensive numerical framework is currently being developed to address complex viscoelastic contact problems that are prevalent in real life.

Analysis of quasistatic viscoelastic viscoplastic piezoelectric contact problem with friction and adhesion

"In this paper we study the process of bilateral contact with adhesion and friction between a piezoelectric body and an insulator obstacle, the socalled foundation. The material's behavior is assumed to be electro-viscoelastic- viscoplastic; the process is quasistatic, the contact is modeled by a general non-local friction law with adhesion. The adhesion process is modeled by a bonding field on the contact surface. We derive a variational formulation for the problem and then, under a smallness assumption on the coe cient of friction, we prove the existence of a unique weak solution to the model. The proofs are based on a general results on elliptic variational inequalities and fixed point arguments."

Viscoelastic steady-state rolling contacts: A generalized boundary element formulation for conformal and non-conformal geometries

A general Boundary Element methodology is developed to study two-dimensional steady-state viscoelastic circular contact problems. The methodology relies on an ad hoc defined steady-state viscoelastic Green’s function, which accounts for the circular domain, typical, for example, of a mechanical pin joint, and can be employed to manage any real viscoelastic material, characterized by a continuum spectrum of relaxation times. The contact problem can be then easily formulated in the form of a spatial convolution product between surface stresses and displacement, which paves the way to solve contact problems of a countless number of engineering relevant systems, where multiple viscoelastic contacts occur. These include, inter alia, viscoelastic rolling element bearings, and pin joints.

A new class of differential quasivariational inequalities with an application to a quasistatic viscoelastic frictional contact problem

The overarching goal of this paper is to introduce and investigate a new nonlinear system driven by a nonlinear differential equation, a history-dependent quasivariational inequality and a parabolic variational inequality in Banach spaces. Such a system can be used to model quasistatic frictional contact problems for viscoelastic materials with long memory, damage and wear. By using the Banach fixed point theorem, we prove an existence and uniqueness theorem of solution for such a system under some mild conditions. As a novel application, we obtain a unique solvability of a quasistatic viscoelastic frictional contact problem with long memory, damage and wear.

Analysis of a Frictionless Electro Viscoelastic Contact Problem with Signorini Conditions

This study considered a mathematical model to describe the process of a quasi-static contact between a piezoelectric body and an electrically conductive foundation. The behavior of the material was modeled with a nonlinear electro-viscoelastic constitutive law, the contact was frictionless, and the result was described with the Signorini condition. A variational formulation was derived for the problem, proving the existence and uniqueness of a weak solution of the model. The proof was based on arguments for nonlinear equations with maximal monotone operators.

A new class of history-dependent quasi variational–hemivariational inequalities with constraints

In this paper we consider an abstract class of time-dependent quasi variational–hemivariational inequalities which involves history-dependent operators and a set of unilateral constraints. First, we establish the existence and uniqueness of solution by using a recent result for elliptic variational–hemivariational inequalities in reflexive Banach spaces combined with a fixed-point principle for history-dependent operators. Then, we apply the abstract result to show the unique weak solvability to a quasistatic viscoelastic frictional contact problem. The contact law involves a unilateral Signorini-type condition for the normal velocity and the nonmonotone normal damped response condition while the friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip.

Well-posedness of constrained evolutionary differential variational–hemivariational inequalities with applications

A system of a first order history-dependent evolutionary variational– hemivariational inequality with unilateral constraints coupled with a nonlinear ordinary differential equation in a Banach space is studied. Based on a fixed point theorem for history dependent operators, results on the well-posedness of the system are proved. Existence, uniqueness, continuous dependence of the solution on the data, and the solution regularity are established. Two applications of dynamic problems from contact mechanics illustrate the abstract results. First application is a unilateral viscoplastic frictionless contact problem which leads to a hemivariational inequality for the velocity field, and the second one deals with a viscoelastic frictional contact problem which is described by a variational inequality.

Dynamic viscoelastic unilateral constrained contact problems with thermal effects

A new model that describes a dynamic frictional contact between a viscoelastic body and an obstacle is investigated in this paper. We consider a nonlinear viscoelastic constitutive law which involves a convex subdifferential inclusion term and thermal effects. The contact condition is modeled with unilateral constraint condition for a version of normal velocity. The boundary conditions that describe the contact, friction and heat flux are govern by the generalized Clarke multivalued subdifferential. We derive a coupled system of two nonlinear first order evolution inclusions problems, which consists of a parabolic variational-hemivariational inequality for the displacement and a hemivariational inequality of parabolic type for the temperature. Then, the unique weak solvability of the contact problem is obtained by virtue of a fixed point theorem and the surjectivity result of multivalued maps. Finally, we deliver a continuous dependence result on a coupled system when the data are subjected to perturbations.