Abstract

We consider a mathematical model which describes the dynamic evolution of a viscoelastic body in frictional contact with an obstacle. The contact is modelled with a combination of a normal compliance and a normal damped response law associated with a slip rate-dependent version of Coulomb’s law of dry friction. We derive a variational formulation and an existence and uniqueness result of the weak solution of the problem is presented. Next, we introduce a fully discrete approximation of the variational problem based on a finite element method and on an implicit time integration scheme. We study this fully discrete approximation schemes and bound the errors of the approximate solutions. Under regularity assumptions imposed on the exact solution, optimal order error estimates are derived for the fully discrete solution. Finally, after recalling the solution of the frictional contact problem, some numerical simulations are provided in order to illustrate both the behavior of the solution related to the frictional contact conditions and the theoretical error estimate result.

1. Introduction

So far, dynamic contact problems abound in industry and everyday life. For this very reason, special care has been given to the modelling, mathematical analysis, and numerical solution of such problems for several decades, be it in the engineering or the mathematical literature. Such problems involve frictional contact phenomena that leads, because of their inherent complexity, to nonlinear, nonsmooth, and nonconvex mathematical problems.

Over the last decades, various contact boundary conditions have been used to model contact phenomena and their modelling is still under investigation; see, for instance, [1–6]. Among these laws, the normal compliance condition introduced in [7] remains one of the most popular contact models used in the literature; see [4, 8–11]. It represents a regularization of the so-called Signorini contact condition, expressed in terms of unilateral constraints for the displacement field, and described the contact with a deformable foundation. Such a law allows penetrations, while the Signorini condition does not. Note that these contact conditions, used in most of the related works on the subject, are formulated in terms of the displacement field. However, normal compliance conditions expressed in terms of the normal velocity seem to be more appropriate when the contact surfaces are lubricated, as mentioned in [5, 12]. Such kind of conditions is called normal damped response conditions. In this current paper, we model the contact with new and nonstandard conditions which involve both the normal compliance law and normal damped response law and, as such, describes the contact with a specific particular deformable foundation. Furthermore, one cannot hope to fully grasp the phenomenon of contact without taking into account the friction. In the literature, it is generally modelled by either the so-called Tresca or Coulomb friction laws. Normally, one would expect the friction coefficient to be constant; however, such a classical approach has shown its own limits for friction-induced phenomena such as stick-slip motion. For this very reason, several authors have introduced nonmonotone versions of friction laws [5, 13].

Due to all the motivations mentioned above, in this work, we consider the mathematical study of a dynamic frictional contact problem in which the contact is modelled with a combination of normal compliance law and normal damped response law, associated with a nonmonotone friction law involving a slip rate dependent friction.

The current study represents continuation of [14–17] and its aim is to provide the variational and numerical analysis of a new and nonstandard dynamic frictional contact problem supported by numerical simulation. With respect to these articles, this paper presents several traits of novelty that we describe in what follows. Here, we no longer have finite penetration in contrast to [14–16], where a combination of a normal compliance law and unilateral constraint was used, nor finite velocity as in [17], where a combination of a normal damped response law and unilateral constraint in velocity was considered. Note that we consider here a dynamic contact model, since the static case had been studied in [15, 16] for a material whose behavior was described by only an elastic constitutive law. We insist on the fact that, in the problem considered in this paper, the behavior of the foundation is described by a Kelvin-Voigt-like foundation modelled by a combination of normal compliance and normal damped response. Furthermore, such considerations lead to nonstandard Coulomb’s law of dry friction where the threshold depends on the tangential velocity, through the coefficient of friction, and both the normal displacement and normal velocity, because of the normal compliance and normal damped response, respectively. Another trait of novelty arises from the mathematical analysis with in particular the numerical analysis of such a problem by considering a dynamic process. Indeed, in this study, we derive optimal order error estimates in the fully discrete case under regularity assumptions imposed on the exact solution. Finally, we provide numerical simulations to validate the bound of the error estimate and to illustrate the mechanical behavior of the concrete physical body.

The rest of the paper is structured as follows. In Section 2 we introduce the notation we shall use as well as some preliminary material. In Section 3 we present the classical formulation of the frictional contact problem, we list the assumptions on the data, and we derive the variational formulation of the initial and approximate problem. Then, the existence and uniqueness result of the weak solution of the problem is presented. Section 4 is devoted to the presentation of the error estimate result in the fully discrete numerical case. In Section 5, the numerical strategy used to solve the frictional contact problem is presented, followed by numerical simulations on a two-dimensional example including numerical validation of the optimal error estimate established in Section 4.

2. Notation and Preliminaries

We present the notation and some preliminary material which will be of use later on. Everywhere in this paper, we use the notation for the set of positive integers and will represent the set of nonnegative real numbers; that is, . Let . Then, we denote by the space of second-order symmetric tensors on . The inner product and norm on and are defined by Here and below the indices , , , and run between and and, unless otherwise stated, the summation convention over repeated indices is used.

Let be a bounded domain with a Lipschitz continuous boundary and let be a measurable part of such that . We use the notation for a typical point in and we denote by the outward unit normal at . Also, an index that follows a comma represents the partial derivative with respect to the corresponding component of the spatial variable; for example, . We use standard notations for the Lebesgue and Sobolev spaces associated with and and, moreover, we consider the spacesHere and represent the deformation and divergence operators given by The spaces , , , and are real Hilbert spaces endowed with the inner products and the associated norms , , and , respectively. We denote by and the normal and the tangential component of on , respectively, given by , . For a regular function we denote by and the normal and the tangential components of the vector on , respectively, and we recall that and .

Next recall the definitions of classical (one-sided) directional derivative and its generalization in the sense of Clarke. Let be a Banach space and its dual. For a function , the directional derivative of at in the direction is defined by whenever this limit exists. The Clarke generalized directional derivative of a locally Lipschitz function at the point in the direction is defined by The Clarke subdifferential of at is a subset of given byA locally Lipschitz function is said to be regular (in the sense of Clarke) at if, for all , the directional derivative exists and . The function is regular (in the sense of Clarke) on if it is regular at every point .

Now let us consider some material useful for the numerical analysis of the problem. We will need the following Gronwall inequalities, proved in [18].

Lemma 1. Let be given. For a positive integer we define . Assume that and are two sequences of nonnegative numbers satisfying for a positive constant independent of or . Then there exists a positive constant , independent of or , such that

3. Mechanical Problem and Variational Formulation

We consider a viscoelastic body that occupies the bounded domain with , its boundary. We assume that is a Lipschitz continuous boundary divided into three measurable parts , , and such that the (. We use the notations and for the displacement field and the stress field, respectively; therefore, , , and represent the linearized strain tensor, the velocity field, and the acceleration field, respectively. We denote by the unit outward normal, defined almost everywhere on . Let be the time interval of interest with . The body is clamped on and, therefore, the displacement field vanishes there. A volume force of density acts in , and surface tractions of density act on . In this study, we consider the dynamic contact with friction between a viscoelastic body and a foundation on . The material’s behavior is modelled by a viscoelastic constitutive law of the formHere is the linearized strain tensor, is a nonlinear operator which describes the viscous properties of the material, and is a nonlinear operator which describes its elastic behavior. Various examples and mechanical interpretations in the study of viscoelastic materials of the form (10) can be found in [19] and the references therein. Such kinds of constitutive laws were used in the literature in order to model the behavior of real materials like rubbers, rocks, metals, pastes, and polymers. In addition, the foundation is made of a soft material, where the contact is modelled with normal compliance and normal damped response. The friction is based on nonmonotone Coulomb’s law of dry friction during the contact with the soft material. The boundary conditions on the contact surface are described in what follows.

When the body moves towards the obstacle, the contact is described by both a normal compliance condition and a normal damped response condition. In this case, the foundation is characterized by a Kelvin-Voigt like model. Therefore,Here represents the positive normal compliance function such that for and represents the positive normal damped response function such that for . A similar rheological model was used in [20] to describe the behavior of soft clay with application on rheological consolidation.

Furthermore, the contact is associated with Coulomb’s law of dry friction in which the coefficient of friction is assumed to depend on the slip rate . Therefore, when , the friction condition is described byDetails on the normal compliance conditions associated with Coulomb’s law of dry friction can be found in [4, 5, 18], for instance. To simplify the notation and when it is sensible, we omit the dependence in position of the variables.

With these preliminaries, the classical formulation of our dynamic frictional contact problem is the following.

Problem . Find a displacement field and a stress field such thatfor all and, moreover,

Here, (13) represents the viscoelastic constitutive law of the material. Equation (14) is the classic equilibrium equation which includes the acceleration term; denotes the density of the material and is assumed to be a constant for the sake of simplicity. Conditions (15) and (16) are the homogeneous displacement and traction boundary conditions, respectively, and (19) represents the initial conditions, with and being the initial displacement and velocity, respectively. Our interest lies in the contact conditions (17) and (18). In this case, the contact condition (17) allows nonlimited penetration. It could be assimilated to a Kelvin-Voigt law since the normal stress on the contact boundary is described by a combination of a normal compliance law and normal damped response law. The friction condition, described in (18), represents Coulomb’s law of dry friction where is the slip rate dependent coefficient of friction.

In order to introduce a weak formulation of the mechanical Problem , we consider a closed subspace of

Completeness of the space follows from the assumption , which allows the use of Korn’s inequality, for some constant , depending only on and ,On , we use the inner product given byand let be the associated norm; that is,

It follows from (21) and (23) that and are equivalent norms on and therefore is a real Hilbert space. Also, note that since we obtain that

The duality pairing between and is denoted by . Identifying with its dual, we have an evolution triple with dense, continuous, and compact embedding. By the Sobolev trace theorem and by (25), there exists a constant depending only on the domains , , and such thatBy (26) there exists a continuous trace operator and for the function we still denote by its trace . In what follows, we need the spaces , , and where the time derivative involved in the definition of is understood in the sense of vector valued distributions. Equipped with the norm , the space becomes a separable Hilbert space. We also have and the following inequality holds, where is a constant,It is well known that the embedding and embedding are continuous. Moreover, choosing we introduce the space equipped with norm. Note that and with both embeddings being compact, and hence where both embeddings are again compact.

We introduce now Ehrling’s lemma, proved in [21].

Lemma 2. Let , , and be three Banach spaces such that with the injection of into being continuous and the injection of into being compact. Then, for each , there exists some constant such that

Using this lemma, we get the following inequality:In the study of the contact problem, we assume the following properties on the data..The viscosity operator satisfies.The elasticity operator satisfies.The normal compliance function satisfies.The normal damped response function satisfies.The friction bound satisfies The assumption (35)(c) represents a one-side Lipschitz condition, which allows the function to decrease with a rate not faster than ..The force and the traction densities satisfy Also, we assume that the initial values satisfy the following.. and .

Using the Clarke subdifferential (7), the friction condition (18) can be expressed in another form. Indeed, define a function byThen, assuming , the condition (18) is equivalent to the following subdifferential inclusion: where denotes the Clarke subdifferential of at the point .

Properties of the function are summarized in the next lemma.

Lemma 3. If the assumptions hold, then the function defined by (37) is locally Lipschitz, and If furthermore the assumption holds, then we have

Proof. First, we have to proof that is locally Lipschitz. In order to do so, let and . For , we get With and Proposition 5.6.28(ii) in [22], we have the following characterization of the Clarke subdifferential : Then, the other properties follow straightforwardly.

We define the operators , , functional , and , defined by

If holds, the operator has the following properties:Under the assumptions , the operator satisfies the following properties:Assuming , the functional is locally Lipschitz, and we have the following inequalities on :