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 timedependent 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.


Introduction
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 1 .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 1 .The first work concerned with the study of frictional contact problems within the framework of variational inequalities was made in 7 .Comprehensive 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 timedependent 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.

Preliminaries
Let R d be a d-dimensional Euclidean space and S d the space of second order symmetric tensors on R d .Let Ω ⊂ R d be open, connected, and bounded with a Lipschitz boundary Γ that is divided into three disjoint measurable parts Γ 1 , Γ 2 , and Γ 3 such that meas Γ 1 > 0. Let L 2 Ω be the Lebesgue space of 2-integrable functions and W k,p Ω the Sobolev space of functions whose weak derivatives of orders less than or equal to k are p-integrable on Ω.Let H k Ω W k,2 Ω .Since the boundary is Lipschitz continuous, the outward unit normal which is denoted by ν exists a.e. on Γ.For T > 0, and let I ˙ 0, T be the bounded time interval of interest.Let u be the range of displacement u.Since the body is clamped on Γ 1 , the displacement field vanishes there.Surface traction of density f 2 acts on Γ 2 and a body force of density f 0 is applied in Ω.The contact is bilateral, that is, the normal displacement u ν vanishes on Γ 3 at any time.
The canonical inner products and corresponding norms on R d and S d are defined as follows: Everywhere in the sequel the index i and j run between 1 and d and the summation convention over repeated indices is implied.
In the following we denote where H and Q are Hilbert spaces with the canonical inner products.The associated norms on the spaces will be denoted by • H and • Q , respectively.Define It is easy to verify that H 1 , • H 1 is a real Hilbert space.Since V is a closed subspace of the space H 1 and meas Γ 1 > 0, the following Korn's inequality holds: where ι denotes a positive constant depending only on Ω and Γ 1 .We define the inner product It follows that • H 1 and • V are equivalent norms on V .Thus, V, • V is a real Hilbert space and V 1 is also a real Hilbert space under the inner product of the space V given by 2.5 .
For every element v ∈ H 1 , we also use the notation v for the trace of v on Γ and we denote by v ν and v τ the normal and the tangential components of v on Γ given by We also denote by σ ν and σ τ the normal and the tangential traces of a function σ ∈ Q, and we recall that when σ is a regular function, that is, σ ∈ C 1 Ω d×d s , then and the following Green's formula holds: We model the friction with Tresca's friction law, where the friction bound g 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, g g t, S t u on Γ 3 × I with S t u x being the accumulated slip at the point x on Γ 3 over the time period I as S t u t 0 uτ s ds, t ∈ I. 2.9 It follows that σ τ ≤ g S t u on Γ 3 .When the strict inequality holds, the material point is in the stick zone: uτ 0, while when the equality holds, σ τ g S t u , the material point is in the slip zone: σ τ −λ uτ for some λ > 0.
Let r be a constant satisfying 0 < r < T and set Q −r Ω × −r, 0 .Let B be the Borel σalgebra of the interval −r, 0 and μ • be a given finite signed measure defined on −r, 0 , B .Zhu 22 defined the time-delay operator G as follows: for any

2.10
In order to make the above integral coherent, we always take the integrand to be a Borel correction of h by which we mean a Borel measurable function that is equal to h almost everywhere .Some special cases of the operator G are as follows: i Let Ω 1 {ω 1 , ω 2 , . . ., ω n , . ..},L 1 2 Ω 1 , and The following lemma is a fundamental result for operator G.

2.16
Now we consider the contact problem.For any u ∈ u , based on 2.10 , we derive the time-delay operator G of the form 2.17 Remark 2.4.Replacing x with ε u in 2.10 and letting s 0 0 and g Gh in Lemma 2.3, it is easy to know that G ≤ μ −r, 0 .

2.18
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 u ∈ u , find a displacement field u : Ω × I → R d and a stress field σ : 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 A, B, and G 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 u represents the velocity field.Note that the explicit dependence of the viscosity, elasticity, and time-delay operators A, B, and G 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 Divσ σ ij,j 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 u 0 denotes the initial displacement field.
In the study of mechanical problems 2.19 -2.24 , we assume that A, B, g, and h satisfy the following conditions.
H A : In the following, we provide an elementary example of the mechanical problem which hold the constitutive law equation 2.19 .
Example 2.5.Let A and B be nonlinear operators which describe the viscous and the elastic properties of the material and satisfy the conditions H A and H B , respectively, while C 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 5 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 30 and 115 in 5 , and noting the relationship between time and frequency, we write the viscoelastic constitutive law in the following form: where a 1 , a 2 and b 1 , b 2 , b 3 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 A t, ε u t ˙ a 1 t 2 / a 2 t 2 ε u t and B t, ε u t ˙ b 1 b 2 t 2 / b 3 t 2 ε u t satisfy the assumption H A and H B , respectively.
Next, we denote by f t the element of V 1 given by

2.27
When we assume that the body force and surface traction satisfy f 0 ∈ C I; H and f 2 ∈ C I; L 2 Γ 2 , we can get

2.28
Let j : R be the functional defined as follows:

2.29
We notice that, by the assumption H g , the integral in 2.29 is well defined.
where c > 0 is a constant.Then g s e c t−s ds, t ∈ a, b .

2.31
Moreover, if g is nondecreasing, then f t ≤ g t e c t−a , t ∈ a, b .

2.32
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 u : I → V 1 such that 2.24 holds and

2.33
We first introduce the following problem.
Problem 2. Find a displacement u : I → V 1 such that 2.24 holds and

2.34
For solving the above problems, we derive some results for an elliptic variational inequality of the second kind: Given f ∈ X, find u ∈ V such that

2.35
Lemma 2.7 see 1 .Let j : V → R be a proper, convex, and lower semicontinuous functional.Then for any f ∈ V , there exists a unique element u : Proof.For any f ∈ V , let ρ > 0 be a parameter to be chosen later.Since ρj : V → R is again a proper, convex, and lower semicontinuous functional, we can define an operator T : 5 .We will show that with a suitable choice of ρ the operator T is a contractive mapping on I × V .To this end, let u, v ∈ V .Since Prox is a nonexpansive mapping, it follows from 2.37 that

2.38
Using the assumption H A and 2.5 , we obtain

2.40
Taking we deduce that α ∈ 0, 1 and which shows that T : I × V → V is a contractive mapping.Therefore, T has a fixed point u, that is,

2.44
It follows that

2.45
Since ρ > 0, we deduce from the above inequality that u is a solution of variational inequality 2.35 .
To show the uniqueness, we assume that there exist two solutions u 1 , u 2 ∈ V of variational inequality 2.35 .Then for any v ∈ V and a.e.t ∈ I, we have

2.46
Since j is proper, we know that j u 1 < ∞ and j u 2 < ∞.Taking v u 2 in the first inequality and v u 1 in the second one and adding the corresponding inequalities, we get 2.47 Using 2.5 and H A , we obtain that u 1 u 2 , which completes the proof of Lemma 2.8.

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 A , H B , H g , 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 η ∈ C I; Q and ξ ∈ C I; V 1 be arbitrarily given.We consider the following auxiliary variational problem.

3.3
By adding two inequalities with v w 2 in 3.2 and v w 1 in 3.3 , we get

3.5
It implies that

3.6
By H A , we get

3.7
Constituting a trace operator γ : V → L 2 Γ 3 that γv v| Γ 3 , since γ is a linear continuous operator, it implies that there exists a constant c > 0 such that

3.8
It follows from 2.9 , 2.29 , 3.8 , and H g that 3.9 By 3.6 -3.7 and 3.9 , we have which implies that w ηξ ∈ C I; V 1 .This completes the proof of Lemma 3.2.
In order to get the unique solution of Problem 2, we derive the following operator We denote by w i the solution of Problem 3 with ξ ξ i for i 1, 2. By an argument similar to that used in obtaining 3.6 , we get 12 where D t, ξ 1 , ξ 2 , w 1 , w 2 j t, S t ξ 1 ; w 2 − j t, S t ξ 1 ; w 1 j t, S t ξ 2 ; w 1 − j t, S t ξ 2 ; w 2 .3.13 Using 2.29 , 3.8 , 2.9 , and H g , we deduce that, for any t ∈ I, where c 1 cL 2 .By using the similar method in obtaining 3.10 , we have

3.15
Since w i w ηξ i Λ η ξ i , we rewrite the above inequality as where β > 0 is a constant which will be chosen later.Clearly, • β defines a norm on the space C I; V 1 and

3.18
Thus, and so the operator Λ η is a contraction on the space C I; V 1 endowed with the equivalent norm • β if we choose β such that Mβ > c 1 .Therefore, the operator Λ η has a unique fixed point ξ η ∈ C I; V 1 , which completes the proof of Lemma 3.
In what follows, for any η ∈ C I; Q , we write w η w ηξ η .

3.21
Taking ξ ξ η in 3.1 and using 3.20 and 3.21 , we deduce that, for any

3.22
Let w η : I → V 1 be the function given by u η t t 0 w η x ds u 0 , a.e.t ∈ I.

3.23
In addition, we define the operator Λ :
Proof.For any η 1 , η 2 ∈ C I; Q , let u i u η i , u i u η i and w i w η i with i 1, 2. 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

3.44
By H A , H h , 2.5 , 3.30 , 3.32 , and 3.35 , we get  Remark 3.6.When G 0 and all the viscosity and elasticity operators A and B are explicitly time dependent, Theorem 3.5 reduces to Theorem 10.2 of 1 .Furthermore, Theorem 3.5 is also a generalization of Theorem 2.1 of 24 .

A Convergence Result
In this section, we study the dependence of the solution to Problem 1 with respect to perturbations of the operator h.We assume that H A , H B , H g , and H h hold and, for any β > 0, let h β be a perturbation of the operator h.We consider the following problem.
Problem 4. Find u β : It follows from the Gronwall inequality that w β w and so the convergence result 4.4 is a consequence of 3.32 .This completes the proof.
Since μ 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.
There exists a unique solution w ηξ ∈ C I; V 1 to Problem 3.Proof.For each fixed t ∈ I, in terms of hypotheses 2.9 , H A , H g , and 2.29 , Problem 3 is an elliptic variational inequality on Q.It follows from Lemma 2.8 that Problem 3 is uniquely solvable.Let w ηξ t ∈ V 1 be the unique solution of Problem 3. Now we show thatw ηξ t ∈ C I, V 1 .Suppose that t 1 , t 2 ∈ I.For simplicity we write w ηξ t i w i , η t i η i and f t i f i with i 1, 2. Using 3.1 for t t 1 , t 2 , we have , S t ξ ; v − j t, S t ξ ; w ηξ t ≥ f t , v − w ηξ t V , a.e.t ∈ I. 3.1 Lemma 3.2.
L 3 |μ| −r, 0 c 4 .By using 3.30 and the similar proof of Lemma 3.3, we get the result ofLemma 3.4.This completes the proof.Proof of Theorem 3.1.Let η * ∈ C I; Q be the fixed point of Λ and let u η * ∈ C 1 I; Q be the function defined by 3.24 for η η * .For any v ∈ V 1 and a.e.t ∈ I, it follows from uη * w η * and 3.22 thatA t, ε uη * t , ε v − uη * t Q η * , ε v − uη * t Q j t, S t uη * ; v − j t, S t uη * ; uη * t ≥ f t , v − uη * t V .3.31Now inequality 2.34 follows from 3.24 and 3.30 .Moreover, since 3.23 implies u η * 0 u 0 , we conclude that u η * is a solution of Problem 2.Let u 1 , u 2 ∈ C I; V 1 be two solutions to Problem 2 and let w i ui for i 1, 2. Then we have For a.e.t ∈ I, by the similar argument used in obtaining 3.6 , we haveA t, ε w 1 t − A t, ε w 2 t , ε w 1 t − w 2 t Q ≤ Λη 1 t − Λη 2 t , ε w 1 t − w 2 t Q D t, w 1 , w 2 , − w 2 s V ds w 1 t − w 2 t V , a.e.t ∈ I.Recalling the definition 3.32 of u 1 and u 2 , and letting c 8 c 7 1 T , we obtainw 1 t − w 2 t V ≤ c 8ζ 1 and u 2 u ζ 2 be the corresponding solutions to 3.41 .Then it is easy to see that u 1 , u 2 ∈ C I; V 1 .For any u 1 ∈ u 1 and u 2 ∈ u 2 , by the similar argument used in obtaining 3.33 , we have t 0 w 1 s − w 2 s V ds, a.e.t ∈ I 3.40 and so the Gronwall inequality implies that w 1 w 2 .By definition 3.32 , we see that u 1 u 2 , which completes the proof of Theorem 3.1.Theorem 3.5.Problem 1 has a unique solution u ∈ C 1 I; V 1 .Proof.Let ζ ∈ C I; V 1 and denote by u ζ ∈ C I; V 1 the solution of the following problem:A t, ε uζ t , ε v − uζ t Q Gh t, ε u ζ , ε v − uζ t Q j t, S t uζ ; v − j t, S t uζ ; uζ t ≥ f t − ζ t , v − uζ t V , ∀v, u ζ ∈ V 1 ,a.e.t ∈ I, Now we show that the operator Υ has a unique fixed point.In fact, for any ζ 1 , ζ 2 ∈ C I; V 1 , let u 1 u a.e.t ∈ I. − 1 ! 1/2 0, the previous inequality implies that, for p large enough, a power Υ p of Υ is a contraction.It follows that there exists a unique elementζ * ∈ V 1 such that Υ p ζ * ζ * .Moreover, sinceΥ is also a fixed point of the operator Υ p .By the uniqueness of the fixed point that Υζ * ζ * , we know that ζ * is a fixed point of Υ.The uniqueness of the fixed point of Υ results straightforward from the uniqueness of the fixed point of Υ p .This implies that u ζ * is the unique solution of Problem 1, which completes the proof of Theorem 3.5.