Quasistatic Frictional Contact Problem Research Articles

A quasistatic electro-elastic contact problem with long memory and slip dependent coefficient of friction

In this paper we consider a mathematical model describing a quasistatic frictional contact problem between a deformable body and an obstacle, say a foundation. We assume that the behavior of the material is described by a linear electro-elastic constitutive law with long memory. The contact is modeled with a version of Coulomb's law of dry friction in which the normal stress is prescribed on the contact surface. Moreover, we consider a slip dependent coefficient of friction. We derive a variational formulation for the model, in the form of a coupled system for the displacements and the electric potential. Under a less assumption on the coefficient of friction, we prove the existence result of weak solutions of the model. We can show the uniqueness of solution by adding another condition (3.10). The proofs are based on arguments of time-dependent variational inequalities, differential equations and Banach fixed point theorem.

Analysis and Numerical Simulation of Time-Fractional Derivative Contact Problem with Friction in Thermo-Viscoelasticity

Abstract The objective of this study is to analyze a quasistatic frictional contact problem involving the interaction between a thermo-viscoelastic body and a thermally conductive foundation. The constitutive relation in our investigation is constructed using a fractional Kelvin–Voigt model to describe displacement behavior. Additionally, the heat conduction aspect is governed by a time-fractional derivative parameter that is associated with temperature. The contact is modeled using the Signorini condition, which is a version of Coulomb’s law for dry friction. We develop a variational formulation for the problem and establish the existence of its weak solution using a combination of techniques, including the theory of monotone operators, Caputo derivative, Galerkin method, and the Banach fixed point theorem. To demonstrate the effectiveness of our approach, we include several numerical simulations that showcase the performance of the method.

Optimal control and feedback control for nonlinear impulsive evolutionary equations

The aim of this paper is to study the existence, optimal control and feedback control for a class of nonlinear impulsive evolution equations involving weakly continuous operators. We first obtain the existence result of the evolution equation without impulse in any subinterval by Rothe method, and construct a solution of impulsive evolution equation by using this result. Moreover, we obtain the existence of optimal control pairs for the optimal control problem and feasible pairs for the feedback control system by establishing several sufficient assumptions. An application on a quasistatic frictional contact problem is provided.

Quasistatic frictional contact problem with damage for thermo-electro-elastic-viscoplastic bodies

Quasistatic frictional contact problem with damage for thermo-electro-elastic-viscoplastic bodies

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.

VARIATIONAL ANALYSIS OF AN ELASTIC-THERMO-VISCOPLASTIC CONTACT PROBLEM WITH NORMAL DAMPED RESPONSE

We consider a quasistatic frictional contact problem between an elastic-viscoplastic body and an obstacle. The contact is modelled with normal damped response and a local friction law. The material is elastic-viscoplastic with two internal variables which may describe a temperature parameter and the damage of the contacting surface. We provide a variational formulation of the problem and prove the existence of a unique weak solution to the model. The proof is based on arguments of evolution equations with monotone operators, a classical existence and uniqueness result on parabolic inequalities and fixed point.

Impulsive hemivariational inequality for a class of history-dependent quasistatic frictional contact problems

<p style='text-indent:20px;'>This paper deals with a class of history-dependent frictional contact problem with the surface traction affected by the impulsive differential equation. The weak formulation of the contact problem is a history-dependent hemivariational inequality with the impulsive differential equation. By virtue of the surjectivity of multivalued pseudomonotone operator theorem and the Rothe method, existence and uniqueness results on the abstract impulsive differential hemivariational inequalities is established. In addition, we consider the stability of the solution to impulsive differential hemivariational inequalities in relation to perturbation data. Finally, the existence and uniqueness of weak solution to the contact problem is proved by means of abstract results.</p>

On the existence of solutions to quasistatic frictional rational contact problems with limited interpenetration

The existence of solutions to quasistatic frictional contact problems with limited interpenetration with an ahead prescribed bound is proved here. If the depth of the interpenetration tends to zero, then there are some sequence of solutions of such problems and a solution of the corresponding Signorini contact problem such that it is the limit of the sequence.

Weak formulations of quasistatic frictional contact problems

We consider a general mathematical model which describes the quasistatic contact of a deformable body with an obstacle, the so-called foundation. The material’s behaviour is modeled with a visco-elastic-type constitutive law and the contact is described with a general interface law associated to a version of Coulomb’s law of dry friction. We list the assumptions on the data and provide relevant examples of constitutive laws and boundary conditions. Then, we derive two different variational formulations of the model in which the unknowns are the displacement and the strain field, respectively. We prove the equivalence of these formulations. Finally, we use recent arguments of sweeping process in order to obtain the existence of a unique weak solution to the contact model.

Optimal Control for Time-Dependent Variational–Hemivariational Inequalities

The present work is intended to investigate optimal control for time-dependent variational–hemivariational inequalities in which the constraint set depends on time. Based on the existence, uniqueness and boundedness of the solution to the inequality, we deliver two continuous dependence results with respect to the time, and then, an existence result for an optimal control problem is presented. Finally, a semipermeability problem and a quasistatic frictional contact problem are given to illustrate our main results.

Hertz-type frictional contact problem of a bidirectionally graded half-plane indented by a sliding rounded punch

The quasi-static frictional contact problem of a rigid rounded punch sliding over a bidirectionally graded half-plane is treated within the framework of plane-strain elasticity. The material heterogeneity in the half-plane is assumed to entail independent shear modulus variations along the depth and longitudinal directions in terms of exponential functions. In-plane field variables are converted through the Fourier transform method and subsequently reduced to a Cauchy singular integral equation (SIE) as a function of the unknown contact stress. The SIE is solved using a collocation approach on the roots of the Chebyshev polynomial of the first kind. A finite element approach (FEA) is also implemented for the same contact problem employing the homogeneous finite element method and the augmented Lagrange algorithm in the environment of a structural analysis software. Besides the verification results generated by the foregoing techniques, the effects of such factors as the friction coefficient and the degree of material heterogeneities are also addressed on the contact stress, the in-plane tensile stress, the normal contact force and on the centerline position of the punch. The bidirectional gradation is evidently shown superior to the conventional one-directional gradation, as far as their potentials are considered for the alleviation of the contact-induced failure risks.

Almost History-Dependent Variational-Hemivariational Inequality for Frictional Contact Problems

We study two quasi-static frictional contact problems on an infinite time interval for a viscoelastic material with long memory and a viscoplastic material, respectively, and with the constitutive ...

Analysis of Quasistatic Frictional Contact Problem with Subdifferential Form, Unilateral Condition and Long-Term Memory

We consider a quasistatic problem which models the contact between a deformable body and an obstacle called foundation. The material is assumed to have a viscoelastic behavior that we model with a constitutive law with long-term memory, thus at each moment of time, the stress tensor depends not only on the present strain tensor, but also on its whole history. In Contact Mechanics, history-dependent operators could arise both in the constitutive law of the material and in the frictional contact conditions. The mathematical analysis of contact models leads to the study of variational and hemivariational inequalities. For this reason a large number of contact problems lead to inequalities which involve history dependent operators, called history dependent inequalities. Such inequalities could be variational or hemivariational and variational hemivariational. In this paper we derive a weak formulation of the problem and, under appropriate regularity hypotheses, we stablish an existence and uniqueness result. The proof of the result is based on arguments of variational inequalities monotone operators and Banach fixed point theorem.

On the Signorini’s Contact Problem with Non-local Coulomb’s Friction in Thermo-Piezoelectricity

This paper deals with the quasi-static frictional contact problem between a thermo-piezoelectric body and a rigid foundation. The body’s material is modelled with a linear thermo–electro-elastic constitutive laws. The contact is described with a temperature depends Signorini conditions and a version of Coulomb’s friction law with slip dependent coefficient of friction. We derive a variational formulation for the model, it is a coupled system for the displacements, the electric potential and the temperature. We then prove, under some smallness assumptions of the friction coefficient, existence of weak solution to the problem. Finally, we study the continuous dependence of solutions on friction coefficient and on displacement and temperature initial data.

A new class of quasistatic frictional contact problems governed by a variational–hemivariational inequality

A quasistatic nonsmooth frictional contact problem for a viscoelastic material is studied. The contact is modeled by a multivalued normal damped response condition with the Clarke generalized gradient of a locally Lipschitz superpotential and the friction is described by a version of the Coulomb law of dry friction with the friction bound depending on the regularized normal stress. The weak formulation of the contact problem is a history-dependent variational–hemivariational inequality coupled with an operator equation for the stress field. A result on the unique weak solvability to this coupled system is proved through a recent contribution on history-dependent subdifferential inclusions and a fixed point approach.

Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics

In this paper, an abstract evolutionary hemivariational inequality with a history-dependent operator is studied. First, a result on its unique solvability and solution regularity is proved by applying the Rothe method. Next, we introduce a numerical scheme to solve the inequality and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modeled with a viscoelastic constitutive law, the contact is given in the form of multivalued normal compliance, and friction is described with a subgradient of a locally Lipschitz potential. Finally, for the contact problem, we provide the optimal error estimate.

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.

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.

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.