Search

We consider a mathematical model which describes the frictional contact between a deformable body and an obstacle, say a foundation. The body is assumed to be linear elastic and 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. The novelty consists here in the fact that we consider a slip dependent coefficient of friction and a quasistatic process. We present two alternative yet equivalent formulations of the problem and establish existence and uniqueness results. The proofs are based on a new result obtained in [10] in the study of evolutionary variational inequalities.

A quasistatic process of contact with adhesion between an elastic beam and a foundation is considered. The contact is modeled with the Signorini condition when the foundation is rigid, and with normal compliance when it is deformable. The adhesion is modeled by introducing the bonding function β, the evolution of which is described by an ordinary differential equation. The existence and uniqueness of the weak solution for each of the problems is established. A fully-discrete scheme for numerical solutions of the problem with normal compliance is described.

We consider a class of evolutionary variational inequalities arising in quasistatic frictional contact problems for linear elastic materials. We indicate sufficient conditions in order to have the existence, the uniqueness and the Lipschitz continuous dependence of the solution with respect to the data, respectively. The existence of the solution is obtained using a time-discretization method, compactness and lower semicontinuity arguments. In the study of the discrete problems we use a recent result obtained by the authors (2000). Further, we apply the abstract results in the study of a number of mechanical problems modeling the frictional contact between a deformable body and a foundation. The material is assumed to have linear elastic behavior and the processes are quasistatic. The first problem concerns a model with normal compliance and a version of Coulomb's law of dry friction, for which we prove the existence of a weak solution. We then consider a problem of bilateral contact with Tresca's friction law and a problem involving a simplified version of Coulomb's friction law. For these two problems we prove the existence, the uniqueness and the Lipschitz continuous dependence of the weak solution with respect to the data.

We analyze a nonlinear abstract evolution problem describing a class of frictional contact processes between a viscoelastic body and a foundation. The problem is set as a time-dependent differential inclusion. The existence of a unique solution is established using the theory of elliptic variational inequalities and Banach's fixed point theorem. A dual formulation of the problem is also introduced and an equivalence result between the two problems is proved. Finally, the abstract results obtained are used to solve some frictional contact problems for viscoelastic materials.

We consider a stationary boundary value problem for the Bingham fluid with friction. This mechanical problem leads to an abstract class of variational inequalities for which we study the blocking property. In this way we can find the link between the yield limit, the friction coefficient and the external forces for which the Bingham fluid is blocked.

The purpose of this paper is to generalize a property of the variational inequality which describes the flow of the Bingham body through a cylinder (see Duvaut and Lions[1], p. 314). We introduce the blocking property and the blocking space for a class of variational inequalities. In the last part of the paper we present some examples, applications and mechanical interpretations of the concepts we have introduced.

We consider a mathematical model which describes the frictionless contact between a viscoelastic body and a deformable foundation. We model the material's behavior with a nonlinear Kelvin-Voigt constitutive law. The process is assumed to be quasistatic and the contact is modeled with a general normal damped response condition. We present the variational formulation of the problem including the existence of a unique weak solution to the model. We then study the numerical approach to the problem using a fully discrete finite-elements scheme with an explicit discretization in time. We state the existence of the unique solution for the scheme and derive an error estimate on the approximate solutions. Finally, we present some numerical results involving examples in one and two dimensions.

We consider a mathematical model which describes the contact between a viscoelastic body and a rigid-deformable foundation with memory effects. We derive a variational formulation of the model which is in the form of a history-dependent variational inequality for the displacement field. Then we prove the existence of a unique weak solution to the problem. We also study the continuous dependence of the solution with respect to the data and prove two convergence results, under different assumptions on the data. The proofs are based on arguments of lower semicontinuity, pseudomonotonicity, and compactness. Finally, we use our convergence results in the study of several optimization problems associated to the viscoelastic contact model.

In this paper, we consider a penalty based numerical method to solve a model contact problem with unilateral constraint that is described by a constrained stationary hemivariational inequality. The penalty technique is applied to approximately enforce the constraint condition, and a corresponding numerical method using finite elements is introduced. We show the convergence of the penalty based numerical solutions to the solution of the constrained hemivariational inequality as both the mesh-size and the penalty parameter approach zero.

We consider an abstract mixed variational problem governed by a nonlinear operator $A$ and a bifunctional $J$, in a real reflexive Banach space $X$. The operator $A$ is assumed to be continuous, Lipschitz continuous on each bounded subset of $X,$ and generalized monotone. First, we pay attention to the unique solvability of the problem. Next, we prove a continuous dependence result of the solution with respect to the data. Based on this result we prove the existence of at least one solution for an associated optimization problem. Finally, we apply our abstract results to the well-posedness and the optimization of an antiplane frictional contact model for nonlinearly elastic materials of Hencky-type., 20 paged

Contact phenomena arise in a variety of industrial process and engineering applications. For this reason, contact mechanics has attracted substantial attention from research communities. Mathematical problems from contact mechanics have been studied extensively for over half a century. Effort was initially focused on variational inequality formulations, and in the past ten years considerable effort has been devoted to contact problems in the form of hemivariational inequalities. This article surveys recent development in studies of hemivariational inequalities arising in contact mechanics. We focus on contact problems with elastic and viscoelastic materials, in the framework of linearized strain theory, with a particular emphasis on their numerical analysis. We begin by introducing three representative mathematical models which describe the contact between a deformable body in contact with a foundation, in static, history-dependent and dynamic cases. In weak formulations, the models we consider lead to various forms of hemivariational inequalities in which the unknown is either the displacement or the velocity field. Based on these examples, we introduce and study three abstract hemivariational inequalities for which we present existence and uniqueness results, together with convergence analysis and error estimates for numerical solutions. The results on the abstract hemivariational inequalities are general and can be applied to the study of a variety of problems in contact mechanics; in particular, they are applied to the three representative mathematical models. We present numerical simulation results giving numerical evidence on the theoretically predicted optimal convergence order; we also provide mechanical interpretations of simulation results.

We study a dynamic frictionless contact problem between a viscoelastic body and an obstacle, the so-called foundation. The contact is subjected to an adhesion effect, whose evolution is described by an ordinary differential equation. For the variational formulation of the contact problem, we present and prove an existence and uniqueness result. A fully discrete scheme is introduced to solve the problem. Under certain solution regularity assumptions, we derive an optimal order error estimate. Some numerical examples are included to show the performance of the method.

International audience; We prove an existence and uniqueness result for a class of subdifferential inclusions which involve a history-dependent operator. Then we specialize this result in the study of a class of history-dependent hemivariational inequalities. Problems of such kind arise in a large number of mathematical models which describe quasistatic processes of contact. To provide an example we consider an elastic beam in contact with a reactive obstacle. The contact is modeled with a new and nonstandard condition which involves both the subdifferential of a nonconvex and nonsmooth function and a Volterra-type integral term. We derive a variational formulation of the problem which is in the form of a history-dependent hemivariational inequality for the displacement field. Then, we use our abstract result to prove its unique weak solvability. Finally, we consider a numerical approximation of the model, solve effectively the approximate problems and provide numerical simulations.

We consider a mathematical model which describes the frictional contact between a viscoelastic body and a foundation. The contact is modelled with normal compliance associated to a rate-and-state version of Coulomb's law of dry friction. We start by presenting a description of the friction law, together with some examples used in geophysics. Then we state the classical formulation of the problem, list the assumptions on the data and derive a variational formulation of the model. It is in a form of a differential variational inequality in which the unknowns are the displacement field and the surface state variable. Next, we prove the unique weak solvability of the problem. The proof is based on arguments of history-dependent variational inequalities and nonlinear implicit integral equations in Banach spaces.

A preprint peer-reviewed and recommended by PCI Mathematical & Computational Biology (https://mcb.peercommunityin.org/); International audience; Antimicrobial efficacy is traditionally described by a single value, the minimal inhibitory concentration (MIC), which is the lowest concentration that prevents visible growth of the bacterial population. As a consequence, bacteria are classically qualitatively categorized as resistant if therapeutic concentrations are below MIC and susceptible otherwise. However, there is a continuity in the space of the bacterial resistance levels. Here, we introduce a model of within-host evolution of resistance under treatment that considers resistance as a continuous quantitative trait, describing the level of resistance of the bacterial population. The use of integro-differential equations allows to simultaneously track the dynamics of the bacterial population size and the evolution of its level of resistance. We analyze this model to characterize the conditions; in terms of (a) the efficiency of the drug measured by the antimicrobial activity relatively to the host immune response, and (b) the cost-benefit of resistance; that (i) prevents bacterial growth to make the patient healthy, and (ii) ensures the emergence of a bacterial population with a minimal level of resistance in case of treatment failure. We investigate how chemotherapy (i.e., drug treatment) impacts bacterial population structure at equilibrium, focusing on the level of evolved resistance by the bacterial population in presence of antimicrobial pressure. We show that this level is explained by the reproduction number R0. We also explore the impact of the initial bacterial population size and their average resistance level on the minimal duration of drug administration in preventing bacterial growth and the emergence of resistant bacterial population.

We consider a general boundary value problem involving operators of the form div(a(·, ∇u(·)) in which a is a Carathéodory function satisfying a p(·)-growth condition. We are interested on the weak solvability of the problem and, to this end, we start by introducing the Lebesgue and Sobolev spaces with variable exponent, together with their main properties. Then, we state and prove our main existence and uniqueness result, Theorem 3.1. The proof is based on a Weierstrass-type argument. We also consider two antiplane contact problems for nonhomogenous elastic materials of Hencky-type. The contact is frictional and it is modelled with a regularized version of Tresca’s friction law and a power-law friction, respectively. We prove that the problems cast in the abstract setting, then we use Theorem 3.1 to deduce their unique weak solvability.

We consider a new class of inclusions in Hilbert spaces for which we provide an existence and uniqueness result. The proof is based on arguments of monotonicity, convexity and fixed point. We use this result to establish the unique solvability of an associated class of Moreau’s sweeping processes. Next, we give two applications in Solid Mechanics. The first one concerns the study of a time-dependent constitutive law with unilateral constraints and memory term. The second one is related to a frictional contact problem for viscoelastic materials. For both problems we describe the physical setting, list the assumptions on the data and provide existence and uniqueness results.

We consider a variational–hemivariational inequality in a real Hilbert space, which depends on two parameters. We prove that the inequality is governed by a maximal monotone operator, then we deduce various existence, uniqueness and equivalence results. The proofs are based on the theory of maximal monotone operators, fixed point arguments and the properties of the subdifferential, both in the sense of Clarke and in the sense of convex analysis. These results lay the background in the study of various classes of inequalities. We use them to prove existence, uniqueness and continuous dependence results for the solution of elliptic and history-dependent variational–hemivariational inequalities. We also present some iterative methods in solving these inequalities, together with various convergence results.

We consider a rate-type constitutitve law given by an implicit nonlinear differential equation in the space of second order symmetric tensors on R d , in which the unknowns are the stress and the linearized strain fields. We list the assumptions on the constitutive functions then we state and prove its well-posedness with respect to two different Tykhonov triples. We use these well-posedness properties in order to deduce two convergence results. Finally, we provide the mechanical interpretation of these results as well as some concluding remarks.

International audience; Elastic wave propagation and scattering in a media containing a continuous density of defect is modeled with a geometrical approach. The material is supposed to be a Riemann–Cartan manifold with a connection enriched by a nonzero torsion. The study is followed until to reveal analytical solutions. The scattering of a defective domain shows explicitly some non-classical phenomena.