In this paper, we consider two quasistatic contact problems. The materialโs behavior is modelled with an elastic constitutive law for the first problem and a viscoplastic constitutive law for the second problem. The novelty arises in the fact that the contact is frictionless and is modelled with a condition which involves normal compliance and memory term. Moreover, for the second problem we consider a condition with unilateral constraint. For each problem we derive a variational formulation of the model and prove its unique solvability. Also, we analyze the dependence of the solution with respect to the data.
Authors
Flavius Patrulescu (Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)
Ahmad Ramadan (Laboratoire de Mathรฉmatiques et Physique, Universitรฉ de Perpignan)
[1] C. Corduneanu, Problemes globaux dans la theorie des equations Integrales de Volterra, Ann. Math. Pure Appl., 67 (1965), 349-363.
[2] M. Barboteu, A. Matei, Sofonea, Analysis of quasistatic viscoplastic contact problems with normal compliance, Quart. J. of Mech. and Appl. Math., 65 (2012), 555-579.
[3] M. Barboteu, F. Patrulescu, A. Ramadan,M. Sofonea, History-dependent contact models for viscoplastic materials, IMA J. Appl. Math., 79, no. 6 (2014), 1180-1200.
[4] W. Han, M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics, 30, American Mathematical SocietyโInternational Press, Sommerville, MA (2002).
[5] J.J. Massera, J.J. Schaffer, Linear Differential Equations and Function Spaces, Academic Press, New York-London (1966).
[6] M. Sofonea, A. Matei. History-dependent quasivariational inequalities arising in Contact Mechanics, Eur. J. Appl. Math., 22 (2011), 471-491.
[7] M. Sofonea, A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, 398, Cambridge University Press, Cambridge (2012).
[8] M. Sofonea, F. Patrulescu, Analysis of a history-dependent frictionless contact problem, Math. and Mech. of Solids,18 (2013), 409-430.
Paper (preprint) in HTML form
CONVERGENCE RESULTS FOR CONTACT PROBLEMS WITH MEMORY TERM
FLAVIUS PฤTRULESCU 1,2 and AHMAD RAMADAN 3
In this paper we consider two quasistatic contact problems. The materialโs behavior is modelled with an elastic constitutive law for the first problem and a viscoplastic constitutive law for the second problem. The novelty arises in the fact that the contact is frictionless and is modelled with a condition which involves normal compliance and memory term. Moreover, for the second problem we consider a condition with unilateral constraint. For each problem we derive a variational formulation of the model and prove its unique solvability. Also, we analyze the dependence of the solution with respect to the data.
The first aim of this paper is to study a quasistatic frictionless contact problem for elastic materials, within the framework of the Mathematical Theory of Contact Mechanics. We model the behavior of the material with a constitutive law of the form
(1.1)
where denotes the displacement field, represents the stress field, is the linearized strain tensor and is a fourth order tensor which describes the elastic properties of the material. The contact is modelled with a condition which involves normal compliance, memory term and infinite penetration. We prove the unique solvability of this model by using new arguments on historydependent variational inequalities presented in [6]. Also, we state and prove the dependence of the solution with respect to the data.
The second aim is to study the continuous dependence of the solution of a quasistatic frictionless contact problem for rate-type viscoplastic materials. We model the behavior of the material with a constitutive law of the form
(1.2)
Here is a nonlinear constitutive function which describes the viscoplastic properties of the material. In (1.2) and everywhere in this paper the dot above a variable represents derivative with respect to the time variable . The second part represents a continuation of [3] where a contact problem for viscoplastic materials of the form (1.2) was considered. The process was assumed to be quasistatic and the contact was modelled by using the normal compliance condition, finite penetration and memory term. The unique solvability of the solution was obtained. Also, the convergence of the solution of the problem with infinite penetration to the solution of the problem with finite penetration as the stiffness coefficient converges to infinity was proved. In the present paper we analyse the dependence of the solution of the viscoplastic contact problem in 3 with respect to the data.
The rest of the paper is structured as follows. In Section 2 we provide the notation we shall use as well as some preliminary material. In Section 3 we present the classical formulation of the first problem, list the assumption on the data and derive the variational formulation. Then we state and prove the unique weak solvability of the problem, Theorem 3.1, and a convergence result, Theorem 3.2. In Section 4 we introduce the classical formulation of the second problem and resume the results on its unique weak solvability obtained in 3. Then we state and prove a convergence result, Theorem 4.3,
2 NOTATION AND PRELIMINARIES
Everywhere in this paper we use the notation for the set of positive integers and will represent the set of nonnegative real numbers, i.e. . We denote by the space of second order symmetric tensors on . The inner product and norm on and are defined by
Let be a bounded domain with a Lipschitz continuous boundary and let be a measurable part of such that meas . 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. . We consider the spaces
These are real Hilbert spaces endowed with the inner products
and the associated norms and , respectively. Here represents the deformation operator given by
Also, we define the space
(2.1)
which is a Hilbert space endowed with the inner product
and the associated norm . Here Div represents the divergence operator given by .
The assumption meas allows the use of Kornโs inequality which involves the completeness of the space ( ). For an element we still write for its trace 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
(2.2)
Also, for a regular stress function we use the notation and for the normal and the tangential components, i.e. and . For the convenience of the reader we recall the following Greenโs formula:
(2.3)
We denote by the space of fourth order tensor fields given by
and is a real Banach space with the norm . Moreover, a simple calculation shows that
(2.4)
For each Banach space we use the notation for the space of continuous functions defined on with values on . It is well known that can be organized in a canonical way as a Frรฉchet space. Details can be found in (1) and 5, for instance. Here we restrict ourseleves to recall that the convergence of a sequence to the element , in the space , can be described as follows:
Let be a real Hilbert space with inner product and associated norm . Let be a subset of and consider the operators , and function . We are interested in the problem of finding a function such that , for all , and inequality below holds
(2.6)
In the study of (2.6) we assume that
(2.7) is a nonempty, closed, convex subset of
and is a strongly monotone Lipschitz continuous operator, i.e.
(2.8)
The operator satisfies
(2.9)
for all , and, finally, we assume that
(2.10) .
The next results, proved in [7], will be used in the rest of this paper.
THEOREM 2.1 Let be an Hilbert space and assume that (2.7)-(2.10) hold. Then, the inequality (2.6) has a unique solution .
COROLLARY 2.2 Let be an Hilbert space and assume that (2.8)-(2.10) hold. Then there exists a unique function such that
(2.11)
To avoid any confusion, we note that here and below the notation and are short hand notation for and , for all .
We end this section with a short description of the physical setting of the two contact problems.
An elastic body in the first problem and a viscoplastic body in the second problem occupies a bounded domain with a Lipschitz continuous boundary , 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 process is quasistatic and is studied in the interval of time .
3 ANALYSIS OF AN ELASTIC CONTACT PROBLEM
The classical formulation of the first contact problem is the following.
Problem . Find a displacement field and a stress field such that, for each ,
Here and below, in order to simplify the notation, we do not indicate explicitly the dependence of various functions on the variables or . Equation (3.1) represents the elastic constitutive law of the material. Equation (3.2) is the equation of equilibrium, conditions (3.3), (3.4) represent the displacement and traction boundary conditions, respectively, and condition (3.5) shows that the contact follows a normal compliance condition with memory term. At the moment , the reaction of the foundation depends both on the current value of the penetration (represented by the term as well as on the history of the penetration (represented by the integral term). Finally, (3.6) is the frictionless condition.
We assume that the body forces and surface tractions have the regularity
(3.7)
Also, we assume that the normal compliance function verifies
and the surface memory function satisfies
Further details on the contact condition (3.5), normal compliance function and surface memory function can be found in 4 or 8 .
We turn now to the variational formulation of Problem . To this end, we assume in what follows that ( ) are sufficiently regular functions which satisfy (3.1)-(3.6). Let and be given. We use Greenโs formula (2.3), equation of equilibrium (3.2), we split the boundary integral over and and, since on and on , it follows that
(3.10)
Moreover, since on , condition (3.6) implies that . We use the contact condition (3.5) to see that
We combine the above relations to deduce that
(3.11)
We use now (3.1) and (3.11) to derive the following variational formulation of the frictionless contact problem (3.1)-(3.6).
Problem . Find a displacement field such that , for all , and the equality below holds
(3.12)
Next, we prove the unique weak solvability of the variational problem . To this end we assume that the elasticity tensor satisfies the following conditions.
We have the following existence and uniqueness result.
THEOREM 3.1 Assume that (3.13), (3.7)-(3.9) hold. Then, Problem has a unique solution which satisfies .
Proof. We start with by providing an equivalent form to Problem . To this end, we use the Riesz representation Theorem to define the operators and the function by equalities
(3.14)
(3.15)
(3.16)
Then, it is easy to see that Problem is equivalent to the problem of finding a function such that for all and the equality below holds
(3.17)
To solve the variational equation (3.17) we use Corollary 2.2 with . To this end, we consider the operator defined by
(3.18)
Then, for all the equality (3.17) can be written as
Using (3.13), (3.8) and the definition of the operator we deduce that the operator is strongly monotone and Lipschitz continuous, i.e. it verifies (2.8).
Let . Then, a simple calculation based on assumption (3.9) and inequality (2.2) shows that the following inequality holds:
(3.19)
This inequality implies that the operator given by (3.15) satisfies (2.9) with
(3.20)
Finally, using (3.7) and (3.16) we deduce that and, therefore, (2.10) holds. It follows now from Corollary 2.2 that there exists a unique function which satisfies the equation
And, using (3.15), (3.16) and (3.18) we deduce that there exists a unique solution to the equality (3.12) for all , which concludes the proof.
Let be the function defined by (3.1). Then, it follows from (3.13) that . Moreover, it is easy to see that (3.11) holds for all and, using standard arguments, it results from here that
(3.21)
Therefore, using the regularity in (3.7) we deduce that which implies that . A couple of functions ( ) which satisfies (3.1), (3.12) for all is called a weak solution to the contact problem . We conclude that Theorem 3.1 provides the unique weak solvability of Problem . Moreover, the regularity of the weak solution is .
We study now the dependence of the solution of Problem with respect to perturbations of the data. To this end, we assume in what follows that (3.13), (3.7)-(3.9) hold and we denote by the solution of Problem obtained in Theorem 3.1. For each let and be perturbations of and which satisfy conditions (3.7)-(3.9). We consider the following variational problem.
Problem . Find a displacement field such that , for all , and the equality below holds for all :
(3.22)
Note that, here and below, represents the normal component of the function .
It follows from Theorem 3.1 that, for each Problem has a unique solution . Consider now the following assumptions
We have the following convergence result.
THEOREM 3.2 Under assumptions (3.23)-(3.26) the solution of Problem converges to the solution of Problem , i.e.
(3.27) in as .
Proof. Let . We use the Riesz representation Theorem to define the operators and the function by equalities
,
,
.
It follows from the proof of Theorem 3.1 that is a solution of Problem iff solves equality (3.17), for all . In a similar way, is a solution of Problem iff , for all and the following equality:
(3.31)
holds for all .
Let and let . We take in 3.31 and in (3.17) and add the resulting equalities to obtain
(3.32)
Next, we use the definitions (3.28) and (3.14), the monotonicity of the function and assumption (3.26) to see that
Therefore, using the trace inequality (2.2), after some elementary calculus we find that
(3.33)
On the other hand the operator verifies (2.9), i.e.
(3.34)
Using trace inequality we obtain
(3.35)
From (3.34) and (3.35) we conclude that
where
(3.37)
We also note that
(3.38)
where
(3.39)
Finally, using assumption (3.13) it follows that
(3.40)
We combine (3.32), (3.33), (3.36), (3.38) and (3.40) to deduce that
(3.41)
Denote meas .
Then, (3.41) yields
(3.42)
and, using the Gronwall inequality we obtain that
(3.43)
We use assumption (3.23) to see that the sequence defined by (3.37) is bounded. Therefore, there exists which depends on and is independent of such that
(3.44)
We pas to the upper bound as in (3.43) and use (3.44) to obtain
(3.45)
We use now assumptions (3.23)-(3.25) and definitions (3.37), (3.39) to see that
(3.46)
We combine now (3.46) and (3.26) (b) with inequality (3.45) to obtain
(3.47)
The convergence (3.47) shows that (3.27) holds, which concludes the proof.
Note that the convergence result in Theorem 3.2 can be easily extended to the corresponding stress functions. Indeed, let be the function defined by (3.1) and, for all , denote by the function given by
(3.48)
for all . Then, it follows that and, moreover, (3.22) yields
(3.49) .
We combine now equalities (3.1), (3.21), (3.48) and (3.49), then we use the convergences (3.24) and (3.27) to see that
(3.50)
4 A CONVERGENCE RESULT FOR A VISCOPLASTIC CONTACT PROBLEM
The classical formulation of the second contact problem is the following.
Problem . Find a displacement field and a stress field such that for all
The difference between problems and consists in the fact that equation (4.1) represents the viscoplastic constitutive law of the material and condition (4.5) shows that the contact follows a normal compliance condition with memory term and unilateral constraint. Finally, (4.7) represents the initial conditions in which and denote the initial displacement and the initial stress field, respectively.
We assume that the elasticity tensor and the nonlinear constitutive function satisfy the following conditions:
(4.8)
Also, as in the case of the first problem we assume that the normal compliance function verifies (3.8), the surface memory function satisfies (3.9), the body
forces and the surface tractions have the regularity (3.7) and, finally, we assume that the initial data verify
(4.10) ,
where denotes the set of admissible displacements defined by
(4.11) a.e. .
The following existence and uniqueness result is proved in 2 .
LEMMA 4.1 Assume that (4.8), (4.9) and (4.10) hold. Then, for each function there exists a unique function such that
(4.12)
for all . Moreover, the operator satisfies the following property: for every there exists such that and
(4.13)
We use (3.15), the Rieszโs representation Theorem and the above lemma to define the operator by equality
(4.14)
The variational formulation of Problem , derived in [3], is the following.
Problem . Find a displacement field and a stress field , such that
(4.15)
(4.16)
hold, for all .
In the study of the problem we have the following existence and uniqueness result.
THEOREM 4.2 Assume that (3.7) -(3.9) and (4.8) -(4.10) hold. Then, Problem has a unique solution, which satisfies
(4.17)
The proof of Theorem 4.2 can be found in [3]. It is based on arguments of history-dependent variational inequalities developed in 6.
We study now the dependence of the solution of Problem with respect to perturbations of the data. To this end, we assume in what follows that (3.7)-(3.9), (3.13), (4.8)-(4.10) hold and we denote by ( ) the solution of Problem obtained in Theorem 4.2. For each let , and be perturbations of and , respectively, which satisfy conditions (3.8), (3.9), (3.7), (4.10). We define the operators and by
(4.18)
(4.19)
and we consider the following variational problem.
Problem . Find a displacement field and a stress field , such that
(4.20)
(4.21)
It follows from Theorem 4.2 that, for each Problem has a unique solution ( ) with the regularity . Consider now the assumptions (3.23)-(3.26) and
(4.22)
We have the following convergence result.
THEOREM 4.3 Under assumptions (3.23)-(3.26) and (4.22) the solution ( ) of Problem converges to the solution ( ) of Problem , i.e.
(4.23)
Proof. Let and let . We take in (4.21) and in (4.16) and add the resulting inequalities to obtain
(4.24)
We have
(4.25)
Using Lemma 4.1, (2.4) and (4.8) we have the following inequality
(4.26)
where
(4.27)
and is a positive constant which depends on and .
From (4.25), (3.36) and (4.26) we obtain
(4.28)
Next, we combine (4.24), (4.8) (c), (3.33), (3.38) and (4.28) to see that
(4.29)
Next, using (4.15), (4.20), (2.4), (4.8), (4.9) and (4.26) we deduce that
(4.30)
where is a positive generic constant and whose value may change from line to line.
Following, we use the notation for and we add now inequalities (4.30) and (4.29) to obtain
(4.31)
Then, we use Gronwall inequality to see that
(4.32)
We pass to the upper bound as in (4.32) and use (3.44) to obtain
Finally, (4.22) yields
(4.33) as .
We use now (3.26) (b), (3.46) and (4.33) in the above inequality to obtain
(4.34) as .
Since the convergence (4.34) holds for each we deduce from (2.5) that (4.23) holds, which concludes the proof.
In addition to the mathematical interest in the convergence results (3.27), (3.50), (4.23) it is of importance from mechanical point of view, since it states that the weak solution of the problems (3.1)-(3.6) and (4.1)-(4.7) depends continuously on the normal compliance function, the surface memory function, the densities of body forces and surface tractions. Moreover, for the second problem the weak solution depends continuously on the initial data too.
References
[1] C. Corduneanu, Problรจmes globaux dans la thรฉorie des รฉquations intรฉgrales de Volterra, Ann. Math. Pure Appl. 67 (1965), 349-363.
[2] M. Barboteu, A. Matei and M. Sofonea, Analysis of quasistatic viscoplastic contact problems with normal compliance, Quart. J. of Mech. and Appl. Math. 65 (2012), 555-579.
[3] M. Barboteu, F. Pฤtrulescu, A. Ramadan and M. Sofonea, Historydependent contact models for viscoplastic materials, The IMA J. of Appl. Math., IMA J. Appl. Math. 79 (2014), 6, 1180-1200.
[4] W. Han and M. Sofonea, Quasistatic Contact Problem in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics 30, American Mathematical Society-International Press, 2002.
[5] J. J. Massera and J. J. Schรคffer, Linear Differential Equations and Function Spaces, Academic Press, New York-London, 1966.
[6] M. Sofonea and A. Matei, History-dependent quasivariational inequalities arising in Contact Mechanics, European J. of Appl. Math. 22 (2011), 471-491.
[7] M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, Cambridge University Press 398, Cambridge, 2012.
[8] M. Sofonea and F. Pฤtrulescu, Analysis of a history-dependent frictionless contact problem, Math. and Mech. of Solids 18 (2013), 409-430.
Tiberiu Popoviciu Institute of Numerical Analysis Romanian Academy, Cluj-Napoca, Romania fpatrulescu@ictp.acad.ro 2 Faculty of Mathematics and Computer Science, Babeล-Bolyai University, Cluj-Napoca, Romania
3 Laboratoire de Mathรฉmatiques et Physique, Universitรฉ de Perpignan, Perpignan, France ahmad.ramadan@univ-perp.fr