Existence of a class of variational inequalities modelling quasi‐static viscoelastic contact problems

Abstract

This work concerns the existence for a class of variational inequalities arising in quasi-static frictional contact problems for viscoelastic materials. We first consider dynamic contact problems described by evolutionary variational inequalities with a small parameter in the inertial term. Then we study the asymptotic behavior of their solutions when the parameter tends to zero. Based on a time-discretization technique and monotone operators theory, the inequalities are solved in the form of evolutionary inclusions in the framework of evolution triples. We show that the limit functions of dynamic problems are solutions to the quasi-static variational inequalities. Applications of the abstract result to frictional contact problems with multivalued constitutive law and normal damped response are given. Moreover, some comments on nonmonotone boundary problem are presented. K E Y W O R D S hemivariational inequality, nonlinear inclusion, quasi-static problem, variational inequality, viscoelastic contact A M S S U B J E C T C L A S S I F I C A T I O N S (( 1.1) andHere, is a Banach space of admissible displacements, and are operators related to the viscoelastic constitutive law, is a convex functional related to contact boundary conditions, and is a linear operator on . The parameter is positive, the function represents the given body forces and surface traction, and 0 , 1 represents the initial displacement and velocity, PENG respectively. Moreover, the derivative of displacement ′ is with respect to the time variable , the interval of interest is [0, ], and the symbol ⟨⋅, ⋅⟩ denotes the duality pairing between the Banach space and its dual * . Recently, the existence and uniqueness of the weak solutions for dynamic contact problems have been established in numerous publications (see e.g., [4][5][6][7][8]). The motion equation for the models of these problems is ′′ − div = , where denotes the displacement vector, is the stress tensor and the mass density is taken to be one. This equation represents the balance of the momentum in the system and governs the evolution of the state of the body. The variational formulations the equation are expressed by variational (resp. hemivariational) inequalities when the multivalued monotone (resp. nonmonotone) boundary conditions are considered.The present work is motivated by the quasi-static model of viscoelastic contact problems. This model comes from the situations in which the system configuration and the external forces and traction evolve slowly in time in such a way that the accelerations in the system are rather small and negligible so that the inertial term can be neglected. Thus the equation of motion is approximated by the equilibrium equation −div = . In this case, at each time instant the system is very close to equilibrium and the external forces are balanced by the internal stresses. The variational formulations of these problems can be described by variational inequalities of the form: find ∶ [0, ] → such that for almost all ∈ [0, ]

Optimal Control of a Quasistatic Frictional Contact Problem with History-Dependent Operators

“…The main novelty is that we prove a continuous dependence result for the solution map of a quasistatic problem. Compared with dynamic problems, quasistatic problems [5,10] are more difficult to derive a continuous dependence result and there is little relevant literature. Moreover, we consider control variables with regard to both boundary and initial conditions, and a cost functional that combines observations within the domain, on the boundary and at the terminal time.…”

Existence for a quasistatic variational-hemivariational inequality

“…Inequalities, including variational and hemivariational inequalities, were initially developed in monographs [5,9,18] to study contact problems. In the past several decades, there is considerable literature devoted to the mathematical theory of these inequalities and their applications to dynamic, static and quasistatic contact problems for elastic and viscoelastic materials; see [12,17,14,26,25,24,23,6,7,19,11,13,15,16,10] and the references therein. The study of inclusion (1), as is stated in [12], is motivated by a quasistatic model of contact problems for viscoelastic materials.…”

“…To the best of our knowledge, the aforementioned open problem to (1) hasn't been solved in the past years. Recently, we have considered an inclusion similar to (1) in [19], where A is maximal monotone, B is linear and bounded, and J is a convex functional of the velocity u (t). The existence of weak solutions is proved by the Rothe method and a vanishing acceleration technique.…”

“…The existence of weak solutions is proved by the Rothe method and a vanishing acceleration technique. Motivated by [12,19] and the literature in this area, we study the inclusion (1) in case that A is linear and B is a nonlinear subdifferential operator via the theory of monotone operators and Rothe method. The work extends the known existence result of a quasistatic hemivariational inequality in [12].…”

“…Theorem 3.2 could be developed in case that A and ∂ cl J in (1) are replaced by A(t) and ∂ cl J(t), respectively. We refer readers to [19,20] for the technique of dealing with time-dependent operators in Rothe method. .…”