A Quasistatic Contact Problem for Viscoelastic Materials with Slip-Dependent Friction and Time Delay
A mathematical model which describes an explicit time-dependent quasistatic frictional contact problem between a deformable body and a foundation is introduced and studied, in which the contact is bilateral, the friction is modeled with Tresca’s friction law with the friction bound depending on the total slip, and the behavior of the material is described with a viscoelastic constitutive law with time delay. The variational formulation of the mathematical model is given as a quasistatic integro-differential variational inequality system. Based on arguments of the time-dependent variational inequality and Banach's fixed point theorem, an existence and uniqueness of the solution for the quasistatic integro-differential variational inequality system is proved under some suitable conditions. Furthermore, the behavior of the solution with respect to perturbations of time-delay term is considered and a convergence result is also given.
The phenomena of contact between deformable bodies or between deformable and rigid bodies are abound in industry and daily life. Contact of braking pads with wheels and that of tires with roads are just a few simple examples . Because of the importance of contact processes in structural and mechanical systems, a considerable effort has been made in their modeling and numerical simulations (see [1–4] and the references therein). What is worth to be taken particularly is some engineering papers that discussed the developed mathematical modeling to a practically interesting problem [5, 6]. Owing to their inherent complexity, contact phenomena are modeled by nonlinear evolutionary problems that are difficult to analyze (see ). The first work concerned with the study of frictional contact problems within the framework of variational inequalities was made in . Comprehensive references for the analysis and numerical approximation of contact problems include [1, 8]. Mathematical, mechanical, and numerical state-of-the-art on contact mechanics can be found in the proceedings [9, 10] and in the special issue .
When a viscoelastic material undergoes a small deformation gradient with a relatively slow force applied, the process of the relative contact problem can be modeled by a quasistatic system. At a relatively short duration, the effect of temperature changes caused by energy dissipation on the deformation of the material is usually negligible. Rigorous mathematical treatment of quasistatic problems is a test of recent years. The reason lies in the considerable difficulties that the process of frictional contact presents in the modeling and analysis because of the complicated surface phenomena involved (see ). By employing Banach's fixed point theorem, Chau et al.  got some existence and uniqueness results for two quasistatic problems which describe the frictional contact between a deformable body and an obstacle. They also proved that the solution of the viscoelastic problem converges to the solution of the corresponding elastic problem. By using arguments for time-dependent elliptic variational inequalities and Banach's fixed point theorem, Rodriguez-Aros et al.  dealt with the existence of a unique solution to an evolutionary variational inequality with Volterra-type integral term. Delost and Fabre  presented a valid approximation method for a quasistatic abstract variational inequality with time-independent constraint and applied its results to the approximation of the quasistatic evolution of an elastic body in bilateral contact with a rigid foundation. Very recently, Vollebregt and Schuttelaars  studied the quasistatic analysis of a contact problem with slip-velocity-dependent friction. For more works concerned with the quasistatic contact problems, we refer to [1, 16] and the references therein.
Quasistatic contact problems for viscoelastic or other materials with explicit time-dependent operators were investigated in a large number of papers. Applying the theory of evolutionary hemivariational inequality, Migórski et al.  proved the existence and the regularity properties of the unique weak solution to a nonlinear explicit time-dependent elastic-viscoplastic frictional contact problem with multivalued subdifferential boundary conditions. In , Migórski et al. considered a class of quasistatic contact problems for explicit time-dependent viscoelastic materials with subdifferential frictional contact conditions. Based on the fixed point theorem for multivalued mappings and variational-hemivariational inequality theory, Costea and Matei  proved the existence of weak solution for the general and unified framework contact models. They also discussed the uniqueness, the boundedness, and the stability of the weak solution under some suitable conditions.
It is well known that time-delay phenomena are frequently encountered in various technical systems, such as electric, pneumatic and hydraulic networks, and chemical processes. For example, regarding polymer under the action of alternative stress, the stress will lag behind the strain, which is just a time delay phenomenon. It gives us a mechanism of the time-delay phenomena appeared in contact problems. Comincioli  proved the existence and uniqueness for a kind of variational inequality with time-delay. For general results of variational inequalities with time delay, we refer to [21–23]. However, to the best of our knowledge, there is no papers to study contact problems for viscoelastic materials with time-delay.
Motivated and inspired by the work mentioned above, in this paper, we introduce and study a mathematical model which describes an explicit time-dependent quasistatic frictional contact problem between a deformable body and a foundation, in which the contact is bilateral, the friction is modeled with Tresca's friction law with the friction bound depending on the total slip, and the behavior of the material is described with a viscoelastic constitutive law with time delay. We give the variational formulation of the mathematical model as a quasistatic integro-differential variational inequality system. By using the arguments of time-dependent variational inequality and Banach's fixed point theorem, we prove an existence and uniqueness of the solution for the quasistatic integro-differential variational inequality system under some suitable conditions. Furthermore, we consider the behavior of the solution with respect to perturbations of time-delay term and show a convergence result. The results presented in this paper generalize and improve some known results of [1, 24].
The paper is structured as follows. In Section 2, we list the necessary assumptions on the data and derive the variational formulation for the problem. In this part, an example which is assumed to the Kelvin-Voigt viscoelastic constitutive law with long memory is given, which represents a constitutive equation of the form (2.19). In Section 3, we prove the existence and uniqueness of the solution to the quasistatic integro-differential variational inequality system. In Section 4, we study the behavior of the solution with respect to perturbations of time-delay term and derive the convergence result.
Let be a -dimensional Euclidean space and the space of second order symmetric tensors on . Let be open, connected, and bounded with a Lipschitz boundary that is divided into three disjoint measurable parts , , and such that . Let be the Lebesgue space of -integrable functions and the Sobolev space of functions whose weak derivatives of orders less than or equal to are -integrable on . Let .
Since the boundary is Lipschitz continuous, the outward unit normal which is denoted by exists a.e. on . For , and let be the bounded time interval of interest. Let be the range of displacement . Since the body is clamped on , the displacement field vanishes there. Surface traction of density acts on and a body force of density is applied in . The contact is bilateral, that is, the normal displacement vanishes on at any time.
The canonical inner products and corresponding norms on and are defined as follows: Everywhere in the sequel the index and run between and and the summation convention over repeated indices is implied.
In the following we denote where and are Hilbert spaces with the canonical inner products. The associated norms on the spaces will be denoted by and , respectively.
Define It is easy to verify that is a real Hilbert space. Since is a closed subspace of the space and , the following Korn's inequality holds: where denotes a positive constant depending only on and . We define the inner product and the norm on by It follows that and are equivalent norms on . Thus, is a real Hilbert space and is also a real Hilbert space under the inner product of the space given by (2.5).
For every element , we also use the notation for the trace of on and we denote by and the normal and the tangential components of on given by We also denote by and the normal and the tangential traces of a function , and we recall that when is a regular function, that is, , then and the following Green's formula holds:
We model the friction with Tresca's friction law, where the friction bound is assumed to depend on the accumulated slip of the surface. In this model we try to incorporate changes in the contact surface structure resulted from sliding. Therefore, on with being the accumulated slip at the point on over the time period as It follows that on . When the strict inequality holds, the material point is in the stick zone: , while when the equality holds, , the material point is in the slip zone: for some .
Let be a constant satisfying and set . Let be the Borel -algebra of the interval and be a given finite signed measure defined on . Zhu  defined the time-delay operator as follows: for any , In order to make the above integral coherent, we always take the integrand to be a Borel correction of (by which we mean a Borel measurable function that is equal to almost everywhere).
Some special cases of the operator are as follows: (i) Let , , and where and . Then it is easy to see that which can be used to describe the countably many discrete delays.(ii) Let , be a -algebra of , a Lebesgue measure, a Lebesgue measurable function, and Then Moreover, letting , then
Remark 2.1. It is easy to see that, for any , as an element in is independent of the choices of Borel corrections for .
Remark 2.2. Since is a very general regular measure, (2.10) can be used in many cases such as finitely many and countably many discrete delays. At this stage, we note that (2.10) contains a very wide class of time-delay operators.
The following lemma is a fundamental result for operator .
Lemma 2.3 (see ). For , we have . Furthermore, for any and , the following inequality holds:
Now we consider the contact problem. For any , based on (2.10), we derive the time-delay operator of the form
Remark 2.4. Replacing with in (2.10) and letting and in Lemma 2.3, it is easy to know that
Under the previous assumptions, the classical formulation of the frictional contact problem with total slip dependent friction bound and the time-delay is as follows. For any , find a displacement field and a stress field such that
We present a short description of the equations and conditions in Problems (2.19)–(2.24). For more details and mechanical interpretation, we refer to [1, 16]. Here (2.19) represents the viscoelastic constitutive law in which , , and are given nonlinear operators, called the viscosity operator, elasticity operator, and time-delay operator, respectively. The prime represents the derivative with respect to the time variable, and therefore represents the velocity field. Note that the explicit dependence of the viscosity, elasticity, and time-delay operators , , and with respect to the time variable means that the model involve the situations when the properties of the material depend on the temperature, that is, its evolution in time is prescribed. Equality (2.20) represents the equilibrium equation where represents the divergence of stress. Conditions (2.21) and (2.22) are the displacement and traction boundary conditions, respectively. Equation (2.23) represents the frictional contact conditions and (2.24) is the initial condition in which the function denotes the initial displacement field.
In the study of mechanical problems (2.19)–(2.24), we assume that , , , and satisfy the following conditions.
H(): is an operator such that (i), for all , a.e. with ;(ii), for all , a.e. with ;(iii)for any is measurable on ;(iv)the mapping .
H(): is an operator such that (i), for all , a.e. with ;(ii)for any is measurable on ;(iii)the mapping .
H(): is an operator such that (i)there exists such that for all , , ;(ii)for any is measurable;(iii)the mapping ;(iv) exists such that .
H(): is an operator such that (i), for all , a.e. with ;(ii)for any is measurable on ;(iii), for all , a.e. with ;(iv)the mapping .
In the following, we provide an elementary example of the mechanical problem which hold the constitutive law equation (2.19).
Example 2.5. Let and be nonlinear operators which describe the viscous and the elastic properties of the material and satisfy the conditions H() and H(), respectively, while is the linear relaxation operator. The following example is assumed to be the Kelvin-Voigt viscoelastic constitutive law with long memory of the form which represents a constitutive equation of the form (2.19).
Contact problems involving viscoelastic materials with long memory have been studied in [25, 26]. For more detail on the long memory models, we refer to [27, 28].
The famous time-temperature superposition principle tells us that when materials are applied with the alternating stress, the reaction time is an inverse proportion to the effect of the frequency. Hence, the influence of increasing the time (or reducing the frequency) and elevating temperature to materials is equivalent.
The sinusoidally driven indentation test was shown to be effective for viability characterization of articular cartilage. Based on the viscoelastic correspondence principle, Argatov  described the mechanical response of the articular cartilage layer in the framework of viscoelastic model. Using the asymptotic modeling approach, Argatov analyzed and interpreted the results of the indentation test. Now, deriving from the and in , and noting the relationship between time and frequency, we write the viscoelastic constitutive law in the following form: where and are some parameters which rely on the characteristic relaxation time of strain under an applied step in stress, the equilibrium elastic modulus, and the glass elastic modulus.
It is easy to verify that and satisfy the assumption H() and H(), respectively.
Next, we denote by the element of given by
When we assume that the body force and surface traction satisfy and , we can get
Let be the functional defined as follows:
We notice that, by the assumption H(), the integral in (2.29) is well defined.
Lemma 2.6 (Gronwall's inequality). Assume that satisfy where is a constant. Then Moreover, if is nondecreasing, then
Proceeding in a standard way with these notations, we combine (2.8)–(2.24) to obtain the following variational formulation.
Problem 1. Find a displacement such that (2.24) holds and We first introduce the following problem.
Problem 2. Find a displacement such that (2.24) holds and
For solving the above problems, we derive some results for an elliptic variational inequality of the second kind: Given , find such that
Lemma 2.7 (see ). Let be a proper, convex, and lower semicontinuous functional. Then for any , there exists a unique element such that
Lemma 2.8. Let be a Hilbert space. Assume that H() holds and is a proper, convex, lower semicontinuous functional. Then for any , variational inequality (2.35) has a unique solution.
Proof. For any , let be a parameter to be chosen later. Since is again a proper, convex, and lower semicontinuous functional, we can define an operator by
where (see (2.5)). We will show that with a suitable choice of the operator is a contractive mapping on . To this end, let . Since is a nonexpansive mapping, it follows from (2.37) that
Using the assumption H() and (2.5), we obtain
If , then
we deduce that and
which shows that is a contractive mapping. Therefore, has a fixed point , that is,
It follows that
Since , we deduce from the above inequality that is a solution of variational inequality (2.35).
To show the uniqueness, we assume that there exist two solutions of variational inequality (2.35). Then for any and , we have Since is proper, we know that and . Taking in the first inequality and in the second one and adding the corresponding inequalities, we get Using (2.5) and H(), we obtain that , which completes the proof of Lemma 2.8.
Remark 2.9. Lemma 2.8 is a generalization of Theorem 4.1 of .
3. Main Results
In this section, we present an existence and uniqueness result concerned with the solution of Problem 1. Throughout this section, we assume that H(), H(), H(), H(), and (2.28) hold.
Theorem 3.1. Problem 2 has a unique solution .
The proof of Theorem 3.1 is based on fixed point arguments and is established in several steps. Let and be arbitrarily given. We consider the following auxiliary variational problem.
Problem 3. Find such that for any
Lemma 3.2. There exists a unique solution to Problem 3.
Proof. For each fixed , in terms of hypotheses (2.9), H(), H(), and (2.29), Problem 3 is an elliptic variational inequality on . It follows from Lemma 2.8 that Problem 3 is uniquely solvable. Let be the unique solution of Problem 3. Now we show that .
Suppose that . For simplicity we write , and with . Using (3.1) for , we have By adding two inequalities with in (3.2) and in (3.3), we get where It implies that By H(), we get Constituting a trace operator that , since is a linear continuous operator, it implies that there exists a constant such that
It follows from (2.9), (2.29), (3.8), and H() that By (3.6)-(3.7) and (3.9), we have which implies that . This completes the proof of Lemma 3.2.
In order to get the unique solution of Problem 2, we derive the following operator defined by
Lemma 3.3. For any , the operator has a unique fixed point .
Proof. Let and . We denote by the solution of Problem 3 with for . By an argument similar to that used in obtaining (3.6), we get where Using (2.29), (3.8), (2.9), and H(), we deduce that, for any , where . By using the similar method in obtaining (3.10), we have Since , we rewrite the above inequality as For , let where is a constant which will be chosen later. Clearly, defines a norm on the space and Thus, and so the operator is a contraction on the space endowed with the equivalent norm if we choose such that . Therefore, the operator has a unique fixed point , which completes the proof of Lemma 3.3.
In what follows, for any , we write By , (3.11), and (3.20), we have Taking in (3.1) and using (3.20) and (3.21), we deduce that, for any , Let be the function given by In addition, we define the operator by
Lemma 3.4. The operator has a unique fixed point .
Proof. For any , let , and with . Using (3.22) and arguments similar to those used in the proof of Lemma 3.3, we obtain where An application of the Gronwall inequality yields where Thus, where . For the operator defined in (3.24), by , (2.5), Remark 2.4, and H(), we obtain where . By using (3.30) and the similar proof of Lemma 3.3, we get the result of Lemma 3.4. This completes the proof.
Now we prove Theorem 3.1.
Proof of Theorem 3.1. Let be the fixed point of and let be the function defined by (3.24) for . For any and a.e. , it follows from and (3.22) that
Now inequality (2.34) follows from (3.24) and (3.30). Moreover, since (3.23) implies , we conclude that is a solution of Problem 2.
Let be two solutions to Problem 2 and let for . Then we have For a.e. , by the similar argument used in obtaining (3.6), we have where Using (2.29), (3.8), (2.9), and H(), we deduce that From the assumption H() and relations (3.30)–(3.35), we know that, for a.e. , where An application of the Gronwall inequality yields where Recalling the definition (3.32) of and , and letting , we obtain and so the Gronwall inequality implies that . By definition (3.32), we see that , which completes the proof of Theorem 3.1.
Theorem 3.5. Problem 1 has a unique solution .
Proof. Let and denote by the solution of the following problem:
From Theorem 3.1, we know that there exists a unique solution for problem (3.41).
Consider the operator defined by Now we show that the operator has a unique fixed point. In fact, for any , let and be the corresponding solutions to (3.41). Then it is easy to see that . For any and , by the similar argument used in obtaining (3.33), we have By H(), H(), (2.5), (3.30), (3.32), and (3.35), we get where An application of the Gronwall inequality yields where . From H(), (3.43), and (3.47), we have where . Iterating the last inequality times, we obtain which leads to Since , the previous inequality implies that, for large enough, a power of is a contraction. It follows that there exists a unique element such that . Moreover, since we deduce that is also a fixed point of the operator . By the uniqueness of the fixed point that , we know that is a fixed point of . The uniqueness of the fixed point of results straightforward from the uniqueness of the fixed point of . This implies that is the unique solution of Problem 1, which completes the proof of Theorem 3.5.
Remark 3.6. When and all the viscosity and elasticity operators and are explicitly time dependent, Theorem 3.5 reduces to Theorem 10.2 of . Furthermore, Theorem 3.5 is also a generalization of Theorem 2.1 of .
4. A Convergence Result
In this section, we study the dependence of the solution to Problem 1 with respect to perturbations of the operator . We assume that H(), H(), H(), and H() hold and, for any , let be a perturbation of the operator .
We consider the following problem.
Problem 4. Find such that
It follows from Theorem 3.5 that, for each , Problem 4 has a unique solution denoted by .
In order to get the convergence result, we need the following assumption:
Now we give the convergence result.
Theorem 4.1. Assume that ), ), ), ), and (2.28) hold. Then the solution of Problem 4 converges to the solution of Problem 1, that is,
Proof. For any and a.e. , let