Subdifferential Boundary Conditions Research Articles

Weak solvability of antiplane frictional contact problems for elastic cylinders

We study a mathematical model which describes the antiplane shear deformations of a cylinder in frictional contact with a rigid foundation. The process is static, the material behavior is described with a linearly elastic constitutive law and friction is modeled with a general slip dependent subdifferential boundary condition. We derive a variational formulation of the model which is in a form of a hemivariational inequality for the displacement field. Then we prove the existence of a weak solution to the model and, under additional assumptions, its uniqueness. The proofs are based on abstract results for operator inclusions in Banach spaces. Finally, we present concrete examples of friction laws for which our results are valid.

Solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders

We study a mathematical model which describes the antiplane shear deformations of a cylinder in frictional contact with a rigid foundation. The process is dynamic, the material behavior is described with a linearly viscoelastic constitutive law and friction is modeled with a general subdifferential boundary condition. We derive a variational formulation of the model which is in a form of an evolutionary hemivariational inequality for the displacement field. Then we prove the existence of a weak solution to the model. The proof is based on an abstract result for second order evolutionary inclusions in Banach spaces. Also, we prove that, under additional assumptions, the weak solution to the model is unique. We complete our results with concrete examples of friction laws for which our results are valid.

INTEGRODIFFERENTIAL HEMIVARIATIONAL INEQUALITIES WITH APPLICATIONS TO VISCOELASTIC FRICTIONAL CONTACT

We consider a class of abstract second-order evolutionary inclusions involving a Volterra-type integral term, for which we prove an existence and uniqueness result. The proof is based on arguments of evolutionary inclusions with monotone operators and the Banach fixed point theorem. Next, we apply this result to prove the solvability of a class of second-order integrodifferential hemivariational inequalities and, under an additional assumption, its unique solvability. Then we consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is dynamic, the material behavior is described with a viscoelastic constitutive law involving a long memory term and the contact is modelled with subdifferential boundary conditions. We derive the variational formulation of the problem which is of the form of an integrodifferential hemivariational inequality for the displacement field. Then we use our abstract results to prove the existence of a unique weak solution to the frictional contact model.

Analysis of a dynamic Elastic-Viscoplastic contact problem with friction

We consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is dynamic, the material behavior is described with an elastic-viscoplastic constitutive law and the frictional contact is modeled with subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving an integral equation coupled with an evolutionary hemivariational inequality. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of abstract second order evolutionary inclusions with monotone operators and a fixed point theorem.

Homogenization of boundary hemivariational inequalities in linear elasticity

In this paper we consider a mathematical model describing static elastic contact problems with the Hooke constitutive law and subdifferential boundary conditions. We treat boundary hemivariational inequalities which are weak formulations of contact problems. We establish existence and uniqueness of solutions to hemivariational inequalities. Using the notion of H-convergence of elasticity tensors we investigate the limit behavior of the sequence of solutions to hemivariational inequalities.

Variational inequality for the rotating Navier–Stokes equations with subdifferential boundary conditions

A steady viscous incompressible fluid through a rotating channel satisfies the rotating Navier–Stokes equations. Subdifferential boundary conditions are imposed. A variational inequality is derived for this system. Furthermore, under some assumptions, the weak solution of the variational inequality exists and is unique.

Hemivariational inequality approach to the dynamic viscoelastic contact problem with nonmonotone normal compliance and slip-dependent friction

We examine a mathematical model which describes dynamic viscoelastic contact problems with nonmonotone normal compliance condition and the slip displacement dependent friction. First, we derive a weak formulation of the model in the form of a hemivariational inequality. Then we embed the hemivariational inequality into a class of second-order evolution inclusions for which we provide a result on the existence of a solution. We conclude with examples of the subdifferential boundary conditions for contact with normal compliance and the slip dependent friction.

Existence and Uniqueness to a Dynamic Bilateral Frictional Contact Problem in Viscoelasticity

In this paper we examine an evolution problem which describes the dynamic bilateral contact of a viscoelastic body and a foundation. The contact is modeled by a friction multivalued subdifferential boundary condition which incorporates the Coulomb law of friction, the SJK model and the orthotropic friction law. The main result concerns the existence and uniqueness of weak solutions to the hyperbolic variational inequality when the friction coefficient is sufficiently small. The proof is based on a surjectivity result for multivalued operators and a fixed point argument.

A Unified Approach to Dynamic Contact Problems in Viscoelasticity

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.

Hemivariational inequalities in thermoviscoelasticity

In this paper we present the existence and uniqueness of the weak solution for a dynamic thermoviscoelastic problem which describes frictional contact between a body and a foundation. We employ the Kelvin–Voigt viscoelastic law which includes the thermal effects and consider the general nonmonotone and multivalued subdifferential boundary conditions. The model consists of the system of the hemivariational inequality of hyperbolic type for the displacement and the parabolic hemivariational inequality for the temperature. The existence of solutions is obtained from a surjectivity result for operators of pseudomonotone type. The uniqueness holds for a large class of operators of subdifferential type satisfying a relaxed monotonicity condition.

A system of evolution hemivariational inequalities modeling thermoviscoelastic frictional contact

In this paper we prove the existence and uniqueness of the weak solution for a dynamic thermoviscoelastic problem which describes frictional contact between a body and a foundation. We employ the Kelvin–Voigt viscoelastic law, include the thermal effects and consider the general nonmonotone and multivalued subdifferential boundary conditions. The model consists of the system of the hemivariational inequality of hyperbolic type for the displacement and the parabolic hemivariational inequality for the temperature. The existence of solutions is proved by using a surjectivity result for operators of pseudomonotone type. The uniqueness is obtained for a large class of operators of subdifferential type satisfying a relaxed monotonicity condition.

Discontinuous Solutions of the Hamilton--Jacobi Equation for Exit Time Problems

In general, the value function associated with an exit time problem is a discontinuous function. We prove that the lower (upper) semicontinuous envelope of the value function is a supersolution (subsolution) of the Hamilton--Jacobi equation involving the proximal subdifferentials (superdifferentials) with subdifferential-type (superdifferential-type) mixed boundary condition. We also show that if the value function is upper semicontinuous, then it is the maximum subsolution of the Hamilton--Jacobi equation involving the proximal superdifferentials with the natural boundary condition, and if the value function is lower semicontinuous, then it is the minimum solution of the Hamilton--Jacobi equation involving the proximal subdifferentials with a natural boundary condition. Futhermore, if a compatibility condition is satisfied, then the value function is the unique lower semicontinuous solution of the Hamilton--Jacobi equation with a natural boundary condition and a subdifferential type boundary condition. Some conditions ensuring lower semicontinuity of the value functions are also given.

Weak formulation of stochastic parabolic systems with subdifferential boundary conditions

AbstractThis paper discusses the mathematical formulation of the stochastic parabolic system, which is described by the multivalued operator called subdifferential operator, that plays an important role in the modeling of the free boundary problem such as viscoelastic and obstacle problems. For various deterministic distributed systems containing a subdifferential operator, the existence of the strong or weak solution is already shown.With respect to the stochastic distributed system, on the other hand, the existence of a strong solution has not been shown except for the system containing a subdifferential operator of a special convex function. As to the existence of the weak solution, there is a report of the case where the subdifferential operator appears in the interior of the system, but there is no report on the general stochastic system containing a subdifferential operator in the boundary condition. Another point is that a strong constraint is imposed on the initial value and the input function in the proof for the existence of the strong solution.From those viewpoints, this paper aims at ameliorating such a constraint and formulates the system by the weak solution. First, the existence condition for the weak solution is shown. Then it is shown that there exists the maximal solution in the set of weak solutions and an estimate for the maximal solution is derived which has a stronger significance than in the past formulation.

On some nonmonotone subdifferential boundary conditions in elastostatics

In the framework of linear elasticity some boundary conditions derived from nonconvex non-differentiable superpotentials are considered by making use of the notion of the generalized Clarke's gradient, [4, 5]. The existence of solutions to the corresponding boundary value problems, termed hemivariational inequalities (cf. [8–11]), is established. The method employed consists in finding the conditions sufficient for the generalized Clarke's gradient of a certain class of Lipschitz functions to be pseudo-monotone in the sense of F. E. Browder and P. Hess, [2]. The obtained results are applied to the investigation of a such problem in which the destruction of the support is taken into account.

A variational-hemivariational inequality approach to the laminated plate theory under subdifferential boundary conditions

In this paper the delamination problem for laminated plates is studied. A nonmonotone multivalued law is introduced in order to describe the interlaminar bonding forces. This law is written as the generalized gradient in the sense of F. H. Clarke of an appropriately defined nonconvex superpotential. Moreover, monotone boundary conditions of the subdifferential type are assumed to hold. The problem is formulated as a variational-hemivariational inequality expressing the principle of virtual work in inequality form. By using compactness and monotonicity arguments, the existence and the approximation of the solution of this inequality are investigated.

Laminated Orthotropic Plates under Subdifferential Boundary Conditions. A Variational‐Hemivariational Inequality Approach

AbstractThis paper deals with an orthotropic laminated plate concerning the delamination problem. A nonmonotone multivalued law simulates the mechanical behaviour of the binding material. This law is written in terms of a generalized gradient (in the sense of F. H. Clarke) of an appropriately defined nonconvex superpotential. Moreover, monotone, e.g. frictional slipallowing boundary conditions of subdifferential type are assumed to hold, which are obtained from a convex superpotential through subdifferentiation. The problem is formulated as a variational‐hemivariational inequality which physically expresses the principle of virtual work in inequality form. The existence and the approximation of the solution of this inequality is discussed by an appropriate combination of compactness, monotonicity and average value arguments.