Abstract
In this paper, we take into consideration the mathematical analysis of time-dependent quasistatic processes involving the contact between a solid body and an extremely rigid structure, referred to as a foundation. It is assumed that the constitutive law is fractional long-memory viscoelastic. The contact is considered to be bilateral and is modeled around Tresca’s law. We establish the existence of the generalized solution’s result. The proof is supported by the surjectivity of the multivalued maximum monotone operator, Rothe’s semidiscretization method, and arguments for evolutionary variational inequality.
1. Introduction
It is well known that the empirical models containing the fractional derivative fit the experimental data more precisely than the model containing the integer derivative. The early application of the fractional derivative in viscoelasticity dates back to the beginning of this century. Germent and Baglet proposed the fractional Kelvin–Voigt model, and at the same time, Koeller obtained the fractional Maxwell model [1, 2]. The theoretical and numerical investigation has been done for fractional differential equations and inequalities in finite-dimensional spaces. On this subject, we cite, for instance [3–6]. The application of the variational framework, more precisely the variational inequalities, proves to be very useful in the study of engineering and mechanical problems. It was initiated by Panagiotopoulos; see monographs [7, 8], and several results have emerged; see [9–13]. In this paper, we are interested in studying the contact problem for a fractional viscoelastic law with long-term memory. The corresponding constitutive law iswhere , and represent the fractional viscoelasticity, elasticity operator, and relaxation tensor, and stand for the stress and the displacement fields, respectively, and is related to Caputo fractional derivative of order and .
The long-memory viscoelastic constitutive law has been the subject of numerous papers; see [14–16]. The distinction of this study is the selection of contact with Tresca’s friction type, governed by a fractional constitutive law, and the use of Rothe’s approach and the maximum monotone operators’ surjectivity results to demonstrate the existence of a solution to a variational inequality. Actually, there are only a few contributions to the literature on the resolution of variational and hemivariational inequalities using Rothe’s approach; see, for example, [18–21].
This article is structured according to the following outlines: in Section 2, we state the equilibrium equations modeling the process of viscoelastic material coming into contact with a foundation. We give some of the notations and the assumptions on data. Then, we obtain the variational formulation of the problem and present the main results concerning the existence of a weak solution. In Section 3, we first prove the existence of the discrete generalized solution to the discrete problem, in the second, we establish the a priori estimates, and in the last, we pass to the limit to obtain the existence of solutions to the variational problem.
2. Setting of the Problem
Before setting up the physical model, we give some definitions and preliminary results.
Definition 1. For any positive integer and , the Caputo fractional derivative and fractional integral of order of a given function , are respectively defined aswhere is the standard Gamma function.
Definition 2. Let be a real Banach space. An operator is said to be(a)est hemicontinuous if the real function is continuous from [0,1] in , for all ,.(b)bounded, if maps bounded sets of into bounded sets of .(c)is monotone if, .(d) is maximal monotone if, is monotone, and for all ,, such that then .(e)The generalized gradient (subdifferential) of a convex function at is a subset of the dual space given by(f)A sequence is weakly convergent to , noted , if for all , we have where represents the duality between and .
Theorem 1 (see [22]). Let is convex, lower semicontinuous, and . Then, is maximal monotone.
Theorem 2 (see [22]). Let the following hypotheses hold : the operator is maximal monotone. : the operator is monotone hemicontinuous and bounded. : is -coercive, meaning that a point and a number exist, that way Hence, is surjective.
2.1. Physical Setting
Suppose that in the reference configuration, the domain which is considered bounded with a smooth boundary . Three open, disconnected sections make up this boundary: , , and such that . Let time interval of interest, where .
The body is submitted to the action of body forces of density . It also submitted to mechanical constraints on the boundary. The body is supposed squeezed into , thus the displacement field vanishes over . On , there are the density of traction forces denoted . At some point, the body comes into contact over .
Let denote by the space of the second order of symmetric tensors and by and , respectively, the scalar product and the Euclidean norm in (resp in ). We adopt the summation convention over repeated indices. All indices , take values from 1 to .
Let , we shall use the notationwith and are, respectively, operators of deformation and divergence defined by
The space , , , and are Hilbert spaces endowed with the inner products given by
We use the notation for the trace of on . The normal and tangential components of on are represented by the symbols and , respectively, and are determined by the formulas , , , and . The trace of the regular stress field applied to is noted .
The physical model for the process is as follows:
Problem 1. Find a displacement field The boundary and initial conditions
For the sake of simplifying the notations in equations (10)–(16) and below, we do not show explicitly the dependence of all functions on, respectively, space and time variables and . The fractional viscoelastic constituency with memory law is given by (10), for , where is linear viscosity function, is linear elasticity, and the linear relaxation function. The expression (11) is the equilibrium equation, we suppose the process is quasistatic. Here, the boundary conditions on displacement and traction are represented by (12) and (13). Conditions (14) and (15) are the bilateral contact and the friction conditions, respectively, it is modeled by Tresca’s law with an imposed friction bound (hence and the nonseparation condition ). The equation given in (16) is the initial condition on displacement. To be able to give a variational formulation of the above problem, we give the following additional notations. We setequipped with the inner product for , . In view of , there exist, respectively, and , called Korn’s and Poincaré constants, such that
Let denote by the Bochner–Lebesgue function space. Since the space is reflexive, it remains that and its dual space are reflexive too. To study the problem and present our main result, we make the following assumptions: the viscosity operator , the elasticity one , and the relaxation tensor satisfy the conditions
The initial data and the density forces satisfy
3. Variational Formulation of the Problem
In this section, we establish the existence of a weak solution to Problem 1. First, we multiply the equilibrium equation by the test function, taking into account the initial and boundary conditions, and we obtain Problem 2 in the form of a variational inequality. Then, we reformulate the problem in operational form (Problem 3), which allows us to pass from a heavy scripture relating to tensorial products to a more convenient one. In order to apply the classical results of functional inclusions, we rewrite the last problem in the form of a nonlinear inclusion, thus giving Problem 4. Note that Problem 2, 3, and 4 are equivalent.
To obtain the variational formulation of Problem 1, we assume that the pair of functions and are sufficiently smooth. We recall that Green’s formula holds:
From the (11), the boundary conditions (12)–(15) and equality (23), we obtain for all
Further,
Let define , for all , expression (25) leads to the following variational formulation:
Problem 2. Find , such that
Let define the operators , , and by the expressionswhere is defined below. Note that since is convex, we have
The following problem can be established in light of all those considerations.
Problem 3. Find , such thatx
Let now suppose that be a solution to Problem 3 and put , for where is the Caputo fractional derivative. Note that when , then have a sense.
So, we may rewrite our problem as a nonlinear inclusion, driven by a fractional integral operator, and it is given by the problem below.
Problem 4. Find , such that
Using data from Problems 3 and 4, we reformulate the assumptions. : is coercive, i.e., there exists a constant , such that : . there exists , such that, and . .
Let be fixed, , , and for all , be defined by , for . Consider the so-called Rothe problem, i.e., the discretized problem corresponding to Problem 4, given by as follows:
Problem 5. Find , such that , andwhere and for , and , are defined by
First, we shall show the existence of solution for a Problem 5.
Theorem 3. Suppose that conditions , , , and are verified. Hence, there exists, such that for all, there is at least one solution to Problem 5.
Proof. Given , we claim that there exist , such that (33) hold. From equality (34), we obtain elements . We have to show that there exists elements and such that inclusion (33) holds. Let denote byTo accomplish this, we will demonstrate that the multivalued operatorSuppositions , , and imply the linearity and boundedness of the operatorso, it is hemicontinuous, and satisfies the conditionThus, under the condition , the operatorFor the mapping , which it is, continuous and convex for all , then by Theorem 1, we obtain that is maximal monotone for all . The coercivity is so fulfilled if . Therefore, for all , where is given byWe may now use Theorem 2 to guarantee the surjectivity of the operator , for all . Thus, we conclude that there exists elements , satisfying the (33). This completes the proof of the theorem.
Note that is solution of (33), also means that there exists , such that
It is noteworthy that hereafter, we use to denote a generic positive constant independent of . The sequence of solutions to Rothe Problem 5 is estimated in the finding that follows:
Lemma 1. Let’s say that all four hypotheses , , , and , are fulfilled, then there existsindependent of , such that for all, the solutions of Problem 5 satisfywhere the constant here is independent of and .
Proof. We multiply equation (33) by taking and to getBy the expression (34) of and hypotheses and , we havewhereAccording to the definition of , we havewhere is Poincaré’s constant. This implies thatNow, we conclude that because of the coercivity of the operator and expressions (46), (47), and (50),Thus, givesIf we choose , we deduce that(note that we have ). Subsequently,From hypothesis , , for all ; . Next, thanks to the generalized discrete Gronwall inequality, see that forSo, the estimate (43) is verified.
Next, by the definition of the , which is written in terms of , we find the following estimate:Thus concludes the proof of Lemma 1.
It is now necessary to demonstrate our major result, that a solution to Problem 3 exists. We define the interpolate functions , byas , where is a solution to Problem 4.
Lemma 2. Assume that , , , and , hold. Let , where , and be a sequence such that , as , to make simple, let it denoted by . Then, we have
Proof. Let , then by applying estimate (43), we haveHence, it follows that is bounded in and consequently, we deduce that there exists , such thatFor any and , note that function for , is in , where , indeedHence, this leadsThus, we deduce thatfor all . Moreover, remark that we haveso, it entailsUsing estimate (43) again, one hasfor . Therefore, estimate (67), ensure that, for all Employing the latter and taking convergence (64) into account, we get
Lemma 3. Let and are solutions of equation (42). Then, we havewhere and is given by the convergence (59).
Proof. Let’s remind that , where ; furthermore, using the compactness of trace map , it follows from the weak convergence , in that , strongly in . Subsequently, we deduce thatFor all , we have firstand secondThis leads toThen, there exists a subsequence noted , such that weakly in ( is reflexive space). We claim that . Indeed, we have that the subdifferential is identical to , where . It is sufficient to prove then thatOtherwise, let us show that , with . Combining the strong convergencewith the weak convergence of a subsequence of to , we pass to the limit as , we haveSince , and is linear, it follows thatThe uniqueness of limit guarantees that . Now by definition of the subdifferential of , we haveThis entails is in . Thus, complete the proof of the lemma.
Theorem 4. Let , , , and be satisfied. Assume that , where . Then, there exists a solution and , satisfying Problem 3.
Proof. Consider now , , and , the Nemytskii operators that correspond to , , and , respectively, which are defined by , and . Since and weakly in , as , then we haveNote that , then . Now by hypothesis that , one hasFurthermore, we use the convergenceApplying the Lebesgue dominated convergence theorem, we deduce thatNow by hypothesis on , , for Using the boundedness of , and the Lipschitz continuity of , this impliesThat is to concludeSince , it is obvious that belongs to . That leads toIt is clear that strongly in , as .
The above convergences entail for all for and . By the strong convergence of to in and the fact that and belong to , we haveAll above convergences imply thatWe obtain the following inequality, for all Finally, taking into account the formulation (25), we deduce that and in is a solution of Problem 3. This completes the proof of Theorem 4.
4. Conclusion
In this paper, we have already considered the problem of the deformation of a body that comes into contact with a rigid foundation. The novelty here was that we took into account the combination of a fractional Kelvin–Voigt’s constitutive law and a frictional Tresca’s contact [17] law. Also, we have established the existence of weak solutions by going through a series of problems that allows us to find the first discrete solutions by means of Rothe’s method and some results on nonlinear inclusions. Then, by passing to the limit, we have found a weak solution to the problem, satisfying a certain regularity.
Data Availability
No data were used to support this study.
Disclosure
This work is performed as part of the employment of the authors.
Conflicts of Interest
No Conflicts of Interest are related to this work
Acknowledgments
L. Ait Kaki and N. Arar acknowledge the support from Directorate General for Scientific Research and Technological Development (DGRSDT), Algeria.