The present paper represents a continuation of Sofonea and Matei’s paper (Sofonea, M. and Matei, A. (2011) History-dependent quasivariational inequalities arising in contact mechanics.ย Eur. J. Appl. Math.ย 22, 471โ491). There a new class of variational inequalities involving history-dependent operators was considered, an abstract existence and uniqueness result was proved and it was completed with a regularity result. Moreover, these results were used in the analysis of various frictional and frictionless models of contact.
In this current paper we present a penalization method in the study of such inequalities. We start with an example which motivates our study; it concerns a mathematical model which describes the quasistatic contact between a viscoelastic body and a foundation; the material’s behaviour is modelled with a constitutive law with long memory, the contact is frictionless and is modelled with a multivalued normal compliance condition and unilateral constraint. Then we introduce the abstract variational inequalities together with their penalizations.
We prove the unique solvability of the penalized problems and the convergence of their solutions to the solution of the original problem, as the penalization parameter converges to zero. Finally, we turn back to our contact model, apply our abstract results in the study of this problem and provide their mechanical interpretation.
Authors
Mircea Sofonea
(Laboratoire de Mathรฉmatiques et Physique, Universitรฉ de Perpignan)
Flavius Patrulescu (Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)
M. Sofonea, F. Pฤtrulescu, Penalization of history-dependent variational inequalities, European J. Appl. Math., vol. 25, no. 2 (2014), pp. 155-176 DOI: 10.1017/S0956792513000363
Penalization of History-Dependent Variational Inequalities
M. Sofonea 1, F. Pฤtrulescu 2 1 Laboratoire de Mathรฉmatiques et Physique
Universitรฉ de Perpignan, 52 Avenue de Paul Alduy, 66860 Perpignan, France
2 Tiberiu Popoviciu Institute of Numerical Analysis
P.O. Box 68-1, 400110 Cluj-Napoca, Romania
Abstract
The present paper represents a continuation of [21]. There, a new class of variational inequalities involving history-dependent operators was considered, an abstract existence and uniqueness result was proved and it was completed with a regularity result. Moreover, these results were used in the analysis of various frictional and frictionless models of contact. In this current paper we present a penalization method in the study of such inequalities. We start with an example which motivates our study; it concerns a mathematical model which describes the quasistatic contact between a viscoelastic body and a foundation; the materialโs behaviour is modelled with a constitutive law with long memory, the contact is frictionless and is modelled with a multivalued normal compliance condition and unilateral constraint. Then, we introduce the abstract variational inequalities together with their penalizations. We prove the unique solvability of the penalized problems and the convergence of their solutions to the solution of the original problem, as the penalization parameter converges to zero. Finally, we turn back to our a contact model, apply our abstract results in the study of this problem and provide their mechanical interpretation.
The theory of variational inequalities plays an important role in the study of nonlinear boundary value problems arising in mechanics, physics and engineering science. At the heart of this theory is the intrinsic inclusion of free boundaries in an elegant mathematical formulation. General results on the analysis of the variational inequalities, including existence and uniqueness results, can be found in [1, 2, 11, 13, 17, 22, for instance. Details concerning the numerical analysis of variational inequalities, including solution algorithms and error estimates, can be found in [6, 10]. References in the study of mathematical and numerical analysis of variational inequalities arising in hardening plasticity include [7, 8).
Phenomena of contact between deformable bodies abound in industry and everyday life. For this reason, considerable progress has been achieved recently in modelling, mathematical analysis and numerical simulations of various contact processes and, as a result, a general mathematical theory of contact mechanics is currently emerging. It is concerned with the mathematical structures which underlie general contact problems with different constitutive laws, i.e. materials, varied geometries and different contact conditions. To this end, it uses various mathematical concepts which include both variational and hemivariational inequalities and multivalued inclusions. An early attempt to study frictional contact problems within the framework of variational inequalities was made in [4]. An excellent reference on analysis and numerical approximations of contact problems involving elastic materials with or without friction is 12]. The variational analysis of various contact problems can be found in the monographs [5, 9, 10, 12, 16, 17, 20. The state of the art in the field can be found in the proceedings [14, 18, 24] and in the special issue [19], as well.
Existence, uniqueness and regularity results in the study of a new class of variational inequalities were proved in [21. There, the first trait of novelty lies in the fact that, unlike the results obtained in literature, the variational inequalities considered were defined on an unbounded interval of time. The second novelty was related to their special structure, which involves two nondifferentiable convex functionals, one of them depending on the history of the solution. This class of variational inequalities represents a general framework in which a large number of quasistatic contact problems, associated with various constitutive laws and frictional contact conditions, can be cast, as exemplified in [22].
Our intention in this current paper is to present a penalization method in the study of the variational inequalities introduced in [21 and to apply it to a new model of contact. Penalization methods in the study of elliptic variational inequalities were used by many authors, mainly for numerical reasons. Details can be found in [6] and the references therein. The main ingredient of these methods arises from the fact that they remove the constraints by considering penalized problems defined on the whole space; these approximative problems have unique solutions which converge to
the solutions of the original problems, as the penalization parameter converges to zero.
The rest of the paper is structured as follows. In Section 2 we present a new mathematical model of contact which is of applied interest and which motivates the abstract study we present in this paper. In Section 3 we state the abstract problem and recall its unique solvability, obtained in [21]. Then we state the penalized problems and prove our main result, Theorem 3.2. The proof of this theorem is given in Section 4. Further, we illustrate the use of the abstract results in the study of the contact model introduced in Section 2. To this end, in Section 5 we list the assumptions on the data and derive the variational formulation. Then we state and prove Theorem 5.1 which concerns the unique weak solvability of the model. Next, in Section 6, we use our abstract penalization method. Our main result in this section is given by Theorem 6.1 which states the existence of a unique weak solution of the penalized contact problems and its convergence to the weak solution of the original contact model. Finally, in Section 7, we present some concluding remarks.
2 A viscoelastic contact problem
The physical setting is as follows. A viscoelastic body occupies a bounded domain with a Lipschitz continuous boundary which is divided into three measurable parts and , such that meas . The body is subject to the action of body forces of density . We also assume that it is fixed on and surface tractions of density act on . On , the body is in frictionless contact with a deformable obstacle, the so-called foundation. We assume that the contact process is quasistatic, we study it in the interval of time , and we denote by and the outward unit normal at and the space of second order symmetric tensors on , respectively. Then, the classical formulation of the contact problem we consider in the rest of this paper is the following.
Problem . Find a displacement field and a stress field : such that
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
for all , and there exists which satisfies
for all .
Here and below, in order to simplify the notation, we do not indicate explicitly the dependence of various functions on the spatial variable . Equation (2.1) represents the viscoelastic constitutive law with long memory in which is the elasticity operator, represents the relaxation tensor and denotes the linearized strain tensor. Equation (2.2) represents the equation of equilibrium in which Div denotes the divergence operator for tensor valued functions. Conditions (2.3) and (2.4) are the displacement boundary condition and the traction boundary condition, respectively. Condition (2.5) is the frictionless condition and it shows that the tangential stress on the contact surface, denoted by , vanishes. More details on the equations and conditions (2.1)-(2.5) can be found in 22.
We now describe the contact condition (2.6) in which our main interest lies and which represents the main novelty of the model. Here denotes the normal stress, is the normal displacement and may be interpreted as the penetration of the bodyโs surface asperities and those of the foundation. Moreover, is a Lipschitz continuous increasing function which vanishes for a negative argument, is a positive function and . This condition can be derived in the following way. Let be given. First, we assume that the penetration is limited by the bound and, therefore, the normal displacement satisfies the inequality
(2.7)
Next, we assume that the normal stress has an additive decomposition of the form
(2.8)
in which the function describes the deformability of the foundation and the functions describe the rigidity and the memory properties of the foundation, respectively. We assume that satisfies a normal compliance contact condition, that is
(2.9)
The part of the normal stress satisfies the Signorini condition in the form with a gap function, i.e.
(2.10)
Finally, the function satisfies the condition
We combine (2.8), (2.9) and write to see that
(2.12)
Then we substitute equality (2.12) in (2.10) and use (2.7), (2.11) to obtain the contact condition (2.6).
We now present additional details of the contact condition (2.6). The inequalities and equalities below in this section are valid at an arbitrary point . First, we recall that (2.6) describes a condition with unilateral constraint, since inequality (2.7) holds at each moment of time. Next, assume that at a given moment there is separation between the body and the foundation, i.e. . Then, since , (2.6) shows that , i.e. the reaction of the foundation vanishes. Note that the same behaviour of the normal stress is described both in the classical normal compliance condition and in the Signorini contact condition, when there is separation. Assume now that at the moment there is penetration which did not reach the bound , i.e. . Then (2.6) yields
(2.13)
This equality shows that, at the moment , the reaction of the foundation depends on the penetration and represents a normal compliance-type condition. Note that (2.6) also shows that if at the moment we have penetration which satisfies then . Indeed, if then (2.13) holds and this implies that . We conclude from above that if then there is no penetration and, therefore, represents a yield limit of the normal pressure, under which the penetration is not possible. This kind of behaviour characterizes a rigidelastic foundation.
In conclusion, condition (2.6) shows that when there is separation between the bodyโs surface and the foundation then the normal stress vanishes; the penetration arises only if the normal stress reaches the critical value ; when there is penetration the contact follows a normal compliance condition of the form (2.13) but up to the limit and then, when this limit is reached, the contact follows a Signorini-type unilateral condition with the gap . For this reason we refer to this condition as to a multivalued normal compliance contact condition with unilateral constraint. It can be interpreted physically as follows. The foundation is assumed to be made of a hard material covered by a thin layer of a soft material with thickness . The soft material has a rigid-elastic behaviour, i.e. is deformable, allows penetration, but only if the normal stress arrives to the yield value ; the contact with this layer is modelled
with normal compliance, as shown in equality (2.13). The hard material is perfectly rigid and, therefore, it does not allow penetration; the contact with this material is modelled with the Signorini contact condition.
Two questions arise in the study of the unilateral contact problem . The first one concerns its unique solvability; the second one concerns the approach of the solution by the solution of a contact model with normal compliance without unilateral constraint. The answers to the questions above are provided by the variational analysis of this contact problem, presented in Section 5 and 6. This analysis is carried out based on the abstract existence, uniqueness and convergence result that we present in the next section.
3 Abstract problem and main result
Everywhere below we use the notation for the set of positive integers and . For each normed space we use the notation for the space of continuous functions defined on with values in . For a subset we still use the symbol for the set of continuous functions defined on with values in . It is well known that, if is a Banach space, then can be organized in a canonical way as a Frรฉchet space, i.e. as a complete metric space in which the corresponding topology is induced by a countable family of seminorms. Details can be found in 3 and [15], for instance. Here we only need to recall that the convergence of a sequence to the element , in the space , can be described as follows:
Consider now a real Hilbert space with inner product and associated norm . Also, let be a subset of , let be two operators, and let be two functions. We assume in what follows that
(3.2)
and is strongly monotone and Lipschitz continuous operator, i.e.
Moreover, we assume that the operator satisfies the following condition:
Following the terminology in [21, 22] we refer to an operator which satisfies (3.4) as a history-dependent operator. Finally, we suppose that
(3.5)
(3.6)
With the data above, we consider the following problem.
Problem . Find a function such that, for all , the inequality below holds:
(3.7)
Following [21, 22] we refer to (3.7) as a history-dependent variational inequality. It represents the framework in which the variational formulation of a large number of contact problems can be cast, with the appropriate choice of spaces and operators. Details can be found in [9, 21, 22, 23] and the references therein. The solvability of Problem is provided by the following existence and uniqueness result, proved in [21.
Theorem 3.1 Let be a Hilbert space and assume that (3.2)-(3.6) hold. Then, Problem has a unique solution .
In order to formulate the penalized problems associated to Problem we consider an operator which satisfies the following conditions:
Note that conditions (3.8) (a) and (b) show that is a monotone Lipschtz continuous operator. Also, note that such an operator always exists. For example consider the operator defined by
where represents the projection operator onto . Then, using the properties of the projections, it is easy to see that the operator satisfies condition (3.8).
Next, for each we consider the following problem.
Problem . Find a function such that, for all , the inequality below holds:
(3.9)
Note that, in contrast to Problem , in Problem the constraint is removed and is replaced with an additional term which contains the penalization parameter . For this reason, we refer to Problem as a penalized problem associated to Problem .
We have the following existence, uniqueness and convergence result, which represents the main result of this section.
Theorem 3.2 Let be a Hilbert space and assume that (3.2)-(3.6), (3.8) hold. Then:
1.
For each Problem has a unique solution which satisfies .
2.
The solution of Problem converges to the solution of Problem , that is
(3.10)
for each .
Note that the convergence (3.10) above is understood in the following sense: for each and for every sequence converging to 0 as we have as .
4 Proof of Theorem 3.2
The proof of Theorem 3.2 will be carried out in several steps that we present in what follows. To this end, below in this section we assume that (3.2)-(3.6), (3.8) hold and
we denote by a positive constant which may depend on and , but is independent of , and whose value may change from line to line. The following lemma shows the unique solvability of the nonlinear inequality (3.9).
Lemma 4.1 For each there exists a unique function which satisfies the inequality (3.9) for all .
Proof. Let . Using (3.3) and (3.8) it is easy to show that the operator
is a strongly monotone Lipschitz continuous operator on . Lemma 4.1 is now a consequence of Theorem 3.1 used with .
Next, we consider the following intermediate problem.
Problem . Find a function such that, for all , the inequality below holds:
(4.1)
Note that inequality (3.9) is a history-dependent variational inequality, since the operator is applied to the unknown . In contrast, the variational inequality (4.1) is a time-dependent variational inequality, since here is a given function. The following lemma shows the unique solvability of the nonlinear inequality (4.1).
Lemma 4.2 For each there exists a unique function which satisfies the inequality (4.1), for all .
Proof. The proof is obtained by similar arguments to those used in the proof of Lemma 4.1.
Next we investigate the properties of the sequence for a fixed .
Lemma 4.3 For each there exists a subsequence of the sequence , again denoted , which converges weakly to , i.e.
(4.2)
Proof. Let and let . We use (4.1) to obtain
and, therefore,
(4.3)
We use (3.5) to see that there exist and , which do not depend on , such that
and, therefore,
(4.4)
Then, we combine (4.3), (3.3), (3.8) (c) and (4.4) to find that
(4.5)
We use now (4.5), the elementary inequality
and the triangle inequality
As a result we deduce that there exists which depends on but does not depend on such that
(4.6)
Inequality (4.6) shows that the sequence is bounded in . Therefore, it follows that there exists a subsequence of the sequence , again denoted and an element such that
(4.7)
Next, we investigate the properties of the element . First of all, we show that . To this end, we use (4.1) to deduce that
(4.8)
We now write
then we use the Lipschitz continuity of the operator and inequality (4.4) to obtain that
We combine now this inequality and (4.6) to see that there exists a positive constant which depends on and , but is independent on , such that
(4.9)
We take now in (4.9), then we pass to the upper limit as in the resulting inequality to obtain
Therefore, using assumption (3.8) (a), (b) the convergence (4.7) and standard arguments on pseudomonotone operators (see Proposition 1.23 in [22, for instance) we deduce that
(4.10)
On the other hand, the inequality (4.9) implies that
(4.11)
We combine the inequalities (4.10) and (4.11) to see that
and, taking in this inequality yields . We conclude that and, using assumption (3.8)(d) it follows that
(4.12)
Next, from inequality (4.1) and assumption (3.8) (c) we find that
(4.13)
We now take in (4.13) and obtain
then we pass to the upper limit as in this inequality and use the weak convergence (4.7) and the assumption (3.5). As a result we obtain
(4.14)
and, using again the argument on pseudomonotonicity employed in the proof of Lemma 4.3, it follows that
(4.15)
On the other hand, passing to the lower limit as in (4.13) and using (4.7) yields
(4.16)
We combine now the inequalities (4.15) and (4.16) to see that
(4.17)
Next, we take in (4.17) and in (3.7). Then, adding the resulting inequalities and using the strong monotonicity of the operator we obtain that
(4.18)
which concludes the proof.
The next step is provided by the following weak convergence result.
Lemma 4.4 For each the whole sequence converges weakly in to as .
Proof. Let . A carefully examination of the proof of Lemma 4.3 shows that any weak convergent subsequence of the sequence converges weakly to , where, recall, is the element of which solves the variational inequality (3.7) at the moment . This inequality has a unique solution and, moreover, estimate (4.6) shows that the sequence is bounded in . Lemma 4.4 is now a consequence of a standard compactness argument.
We proceed with the following strong convergence result.
Lemma 4.5 For each the sequence converges strongly in to , that is
(4.19)
Proof. Let and . We take in (4.15) to see that
then we combine this inequality with (4.14) to obtain that
Finally, we use (4.18) to find that
(4.20)
On the other hand, from the weak convergence of the sequence to , guaranteed by Lemma 4.4, it follows that
(4.21)
Next, from the strong monotonicity of the operator we have
(4.22)
The strong convergence (4.19) is now a consequence of (4.20)-(4.22).
The last step is provided by the following strong convergence result.
Lemma 4.6 For each the sequence converges strongly in to , that is
(4.23)
Proof. Let and be such that . Let also . We take in (4.1) and in (3.9). Then, adding the resulting inequalities we deduce that
Next, we use the monotony of the operator , (3.8)(a), to obtain that
Therefore, using (3.3) (a) yields
(4.24)
We now combine (4.24) and (3.4) to find that
It follows from here that
and, using a Gronwallโs argument, we obtain that
(4.25)
Note that for all and, therefore, (4.25) yields
(4.26)
On the other hand, by estimate (4.6), Lemma 4.5 and Lebesgueโs convergence theorem it follows that
(4.27)
We use now (4.26), (4.27) and (4.19) to obtain the convergence (4.23), which concludes the proof.
We end this section with the remark that the points 1) and 2) of Theorem 3.2 correspond to Lemmas 4.1 and 4.6, respectively. Therefore, we conclude from here that the proof of Theorem 3.2 is complete.
5 Existence and uniqueness
We turn now to the variational analysis of problem . To this end, we need further notation and preliminaries. First, we use the notation for a typical point in and we denote by the outward unit normal at . Here and below the indices run between 1 and and, unless stated otherwise, the summation convention over repeated indices is used. An index that follows a comma represents the partial derivative with respect to the corresponding component of the spatial variable, e.g. . Recall that the inner product and norm on and are defined by
We use standard notation for the Lebesgue and Sobolev spaces associated to and and, moreover, we consider the following spaces:
These are real Hilbert spaces endowed with the inner products
Here and below and Div are the deformation and the divergence operators, respectively, defined by
Completeness of the space ( ) follows from the assumption meas , which allows the use of Kornโs inequality.
For an element we still write for the trace of on the boundary and we denote by and the normal and tangential components of on , given by . Let be a measurable part of . Then, by the Sobolev trace theorem, there exists a positive constant which depends on and such that
(5.1)
Also, for a regular function we use the notation and for the normal and the tangential trace, i.e. and . Moreover, we recall that
the following Greenโs formula holds:
(5.2)
Finally, we denote by the space of fourth order tensor fields given by
and we recall that is a real Banach space with the norm
Moreover, a simple calculation shows that
(5.3)
Next, we list the assumptions on the data, derive the variational formulation of the problem and then we state and prove its unique weak solvability. To this end we assume that the elasticity operator and the relaxation tensor satisfy the following conditions.
The densities of body forces and surface tractions are such that
(5.6)
Finally, the normal compliance function and the surface yield function satisfy
(5.8)
In what follows we consider the set of admissible displacements defined by
(5.9)
Moreover, we define the operator and the functions , by equalities
(5.10)
(5.11)
(5.12)
Here and below, for we denote by its positive part, i.e. . Note that assumptions (5.6)-(5.8) imply that the integrals in (5.10)-(5.12) are welldefined.
Assume in what follows that ( ) are sufficiently regular functions which satisfy (2.1)-(2.6) and let and be given. First, we use Greenโs formula (5.2) and the equilibrium equation (2.2) to see that
We split the surface integral over and and, since a.e. on , on , we deduce that
Moreover, since
taking into account the frictionless condition (2.5) we obtain
(5.13)
We write now
then we use the contact conditions (2.6) and the definition (5.9) of the set to see that
(5.14)
We use (2.6), again, and the hypothesis (5.8) on function to deduce that
(5.15)
Then we add the inequalities (5.14) and (5.15) and integrate the result on to find that
(5.16)
Finally, we combine (5.13) and (5.16) and use the definitions (5.10)-(5.12) to deduce that
(5.17)
We now substitute the constitutive law (2.1) in (5.17) to obtain the following variational formulation of Problem .
Problem . Find a displacement field such that, for all , the inequality below holds:
(5.18)
In the study of the problem we have the following existence and uniqueness result.
Theorem 5.1 Assume that (5.4)-(5.8) hold. Then, Problem has a unique solution which satisfies .
Proof. To solve the variational inequality (5.18) we use Theorem 3.1 with and . To this end we consider the operators and defined by
(5.19)
(5.20)
It is easy to see that condition (3.2) holds. Next, we use (5.4), (5.7) and (5.1) to see that the operator satisfies conditions (3.3) with and . Let . Then, a simple calculation based on assumption (5.5) and inequality (5.3) shows that
(5.21)
This inequality shows that the operator , defined by (5.20), satisfies condition (3.4) with
Next, we use condition (5.8) to see that the functional defined by (5.11) is a seminorm on and, moreover, it satisfies
(5.22)
Inequality (5.22) shows that the seminorm is continuous on and, therefore, (3.5) holds. Finally, using assumption (5.6) and definition (5.12) we deduce that which shows that (3.6) holds, too.
It follows now from Theorem 3.1 that there exists a unique function which satisfies the inequality
(5.23)
for all . And, using (5.19) and (5.20) we deduce that there exists a unique function such that (5.18) holds for all , which concludes the proof.
Let be the function defined by (2.1). Then, it follows from (5.4) and (5.5) that . Moreover, it is easy to see that (5.17) holds for all and, using standard arguments, it results from here that
(5.24)
Therefore, using the regularity in (5.6) we deduce that which implies that . A couple of functions ( ) which satisfies (2.1), (5.18) for all is called a weak solution to the contact problem . We conclude that Theorem 5.1 provides the unique weak solvability of Problem . Moreover, the regularity of the weak solution is .
6 Penalization
In this section we show how the abstract result in Theorem 3.2 can be used in the study of the contact problem . To this end, for each we consider the following contact problem.
Problem . Find a displacement field and a stress field such that
(6.1)
(6.2)
(6.3)
(6.4)
(6.5)
for all , and there exists which satisfies
for all .
Here and below and represent the normal and the tangential components of the functions and , respectively. Note that the contact condition (6.6) can be obtained from the contact condition (2.6) in the limit when . For this reason, its mechanical interpretation is similar to that of condition (2.6) and could be summarised as follows: when there is separation between the bodyโs surface and the foundation then the normal stress vanishes; the penetration arises only if the normal stress reaches the critical value ; when there is penetration the contact follows a normal compliance condition of the form (2.13). For this reason we refer to this
condition as to a multivalued normal compliance contact condition. It models the case when the foundation is assumed to have a rigid-elastic behaviour. Arguments similar to those used in [9, 20] show that can be interpreted as a deformability coefficient of the hard layer of the foundation.
Using notation (5.10)-(5.12) by similar arguments as those used in the case of Problem we obtain the following variational formulation of Problem .
Problem . Find a displacement field such that, for all , the inequality below holds:
(6.7)
We have the following existence, uniqueness and convergence result, which states the unique solvability of Problem and describes the behaviour of its solution as .
Theorem 6.1 Assume that (5.4)-(5.8) hold. Then:
1.
For each Problem has a unique solution which satisfies .
2.
The solution of Problem converges to the solution of Problem , that is
(6.8)
for all .
Proof. We use Theorem 3.2 with and . To this end we define the operator by equality
(6.9)
We use (5.1) and (5.7) to show that is a monotone Lipschitz continuous operator with Lipschitz constant , i.e. it satisfies condition (3.8) (a) and (b).
Assume now that and . Then, using (5.9) and (5.7) it is easy to see that
and, therefore
which shows that (3.8) (c) holds, too.
Finally, assume that . Then, and, therefore,
(6.10)
We use (5.7) to obtain the inequality
Therefore, since the integrand in (6.10) is positive, we deduce from (6.10) that
This equality combined with assumption (5.7) (d) implies that a.e. on and, therefore, we deduce that . Conversely, if it follows that a.e. on and using assumption (5.7) (d) we deduce that a.e. on . From the definition (6.9) of the operator we deduce that for all , which implies that . It follows from above that satisfies the condition (3.8)(d).
We now turn back to (5.19) and (5.20). Thus, it is easy to see that is a solution to Problem iff
(6.11)
for all . Moreover, is a solution to Problem iff satisfies inequality (5.23) for all . Recall also that the operator satisfies condition (3.8). Theorem 6.1 is now a consequence of Theorem 3.2.
Note that the convergence result (6.8) can be easily extended to the weak solutions of the problems and . Indeed, let and be the functions defined by (6.1) and (2.1), respectively, and let be such that . Then, following the arguments presented in Section 5, it follows that and, moreover,
(6.12)
Therefore, using (2.1), (6.1) and (6.12) as well as the properties of the operators and we deduce that
(6.13)
Next, we take in (6.11), then we use the properties of the operators combined with those of the functional . As a result we obtain
We now use the property (5.21) of the operator and a Gronwall argument to see that
(6.14)
where represents a constant which depends on but is independent on . Then, we use the inequality (6.13), the convergence (6.8), the estimate (6.14) and Lebesqueโs theorem to deduce that
(6.15)
In addition to the mathematical interest in the convergence result (6.8), (6.15), it is important from the mechanical point of view, since it shows that the weak solution of the viscoelastic contact problem with multivalued normal compliance and unilateral constraint may be approached as closely as one wishes by the solution of the viscoelastic contact problem with multivalued normal compliance, with a sufficiently small deformability coefficient.
7 Conclusion
We presented a penalization method for a class of history-dependent variational inequalities in Hilbert spaces. It contains the existence and the uniqueness of the solution for the penalized problems as well as its convergence to the solution of the original problem. The proofs were based on arguments of compactness and monotonicity. The method can be applied in the study of a large class of nonlinear boundary value problems with unilateral constraints. To provide an example, we presented a new model of quasistatic frictionless contact with viscoelastic materials which, in the variational formulation, leads to a history-dependent variational inequality for the displacement field. We applied the abstract penalization method in the study of this contact problem and we presented the mechanical interpretation of the corresponding results. A numerical validation of the convergence result included in this method will be provided in a forthcoming paper.
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