Hyperbolic Hemivariational Inequalities for Dynamic Viscoelastic Contact Problems

The paper deals with second order nonlinear evolution inclusions and their applications. First, we study an evolution inclusion involving Volterra-type integral operator which is considered within the framework of an evolution triple of spaces. We provide a result on the unique solvability of the Cauchy problem for the inclusion. Next, we examine a dynamic frictional contact problem of viscoelasticity for materials with long memory and derive a weak formulation of the model in the form of a hemivariational inequality. Then, we embed the hemivariational inequality into a class of second order evolution inclusions involving Volterra-type integral operator and indicate how the result on evolution inclusion is applicable to the model of the contact problem. We conclude with examples of the subdifferential boundary conditions for different types of frictional contact.


Introduction
An important number of problems arising in Mechanics, Physics and Engineering Science lead to mathematical models expressed in terms of nonlinear inclusions and hemivariational inequalities. For this reason the mathematical literature dedicated to this field is extensive and the progress made in the last decades is impressive. It concerns both results on the existence, uniqueness, regularity and behavior of solutions for various classes of nonlinear inclusions as well as results on numerical approach of the solution for the corresponding problems.
The purpose of this paper is to use a recent result on unique solvability of the following second order evolution inclusion which is considered on a finite time interval in the framework of evolution triple of spaces (V , H, V * ) and show how the result on the evolution inclusion is applicable to the model of the contact problem. We provide conditions on a unique solvability of the inclusion which were studier earlier in [18,19]. Subsequently, we consider the class of evolution hemivariational inequalities of second order of the form where g 0 denotes the generalized directional derivative of Clarke of a possibly nonconvex function g, γ is a trace map and ·, · stands for the duality pairing between V * and V .
Our study includes the modeling of a mechanical problem and its variational analysis. We derive the hemivariational inequality for the displacement field from nonconvex superpotentials through the generalized Clarke subdifferential. The novelty of the model is to deal with nonlinear elasticity and viscosity operators and to consider the coupling between two kinds of nonmonotone possibly multivalued boundary conditions which depend on the normal (respectively, tangential) components of both the displacement and velocity. We recall that the notion of hemivariational inequality is based on the generalized gradient of Clarke [6] and has been introduced in the early 1980s by Panagiotopoulos [33,34]. We also note that the existence of solutions to the second order evolution inclusions as well as to the corresponding dynamic hemivariational inequalities has been studied, for instance, in Migórski [22][23][24][25], Migórski and Ochal [27,28] and Kulig [17,18].
Finally, in order to illustrate the cross fertilization between rigorous mathematical description and Nonlinear Analysis on one hand, and modeling and applications on the other hand, we provide several examples of contact and friction subdifferential boundary conditions.
The paper is structured as follows. In Sect. 2 we recall some preliminary material. In Sect. 3 we describe the dynamic viscoelastic contact problem and present its classical and weak formulation. Next, we recall a result on the existence and uniqueness of solutions to the Cauchy problem for the second order nonlinear evolution inclusion involving a Volterratype integral operator. We establish the link between a nonlinear evolution inclusion and the hemivariational inequality, and apply the aforementioned results to the viscoelastic contact problem with a memory term. The review of several examples of contact and friction subdifferential boundary conditions which illustrates the applicability of our results is provided in Sect. 4.

Preliminaries
In this section we recall the notation and basic definitions needed in the sequel.
Given a Banach space (X, · X ), we use the symbol w-X to denote the space X endowed with the weak topology. The class of linear and bounded operators from X to X * is denoted by L(X, X * ). If U ⊂ X, then we have U X = sup{ x X : x ∈ U }. The duality between X and its dual is denoted by ·, · X * ×X .
The linear space of second order symmetric tensors on R d will be denoted by S d . The inner product and the corresponding norm on S d is defined similarly to the inner product on R d , i.e., The summation convention over repeated indices is used here and below. Let ⊂ R d be a bounded domain with Lipschitz boundary . In what follows we will need the following Hilbert spaces with their inner products Here ε : H 1 ( ; R d ) → L 2 ( ; S d ) denotes the deformation operator, Div : H 1 → L 2 ( ; R d ) stands for the divergence operator, where ε(u) = ε ij (u) , ε ij (u) = 1 2 (u i,j + u j,i ), Div σ = {σ ij,j }.
The index following comma indicates a partial derivative.
Given v ∈ H 1 we denote its trace γ v on by v, where γ : H 1 ( ; R d ) → H 1/2 ( ; R d ) ⊂ L 2 ( ; R d ) is the trace map. Let n denote the outward unit normal vector to . Since is Lipschitz continuous, the normal vector is defined a.e. on . For v ∈ H 1/2 ( ; R d ) we denote its normal and tangential components by v N = v · n and v T = v − v N n.
Let V be a separable Banach space. We identify H with its dual and we consider a Gelfand triple V ⊂ H ⊂ V * where all embeddings are compact, dense and continuous (see [7,44]). We will need the following spaces V = L 2 (0, T ; V ) and W = {w ∈ V : w ∈ V * }. The time derivative involved in the definition of W is understood in the sense of vector valued distributions. It is well known that W is a separable Banach space equipped with the Let us recall some definitions needed in the next sections.
Measurable multifunction Let ( , ) be a measurable space, X be a separable Banach space and F : → 2 X . The multifunction F is said to be measurable if for every Generalized directional derivative Let X be a Banach space. The generalized directional derivative of Clarke of locally Lipschitz function h : X → R at x ∈ X in the direction v ∈ X, denoted by h 0 (x; v), is defined by (cf. [6]) Generalized gradient Let X be a Banach space. The generalized gradient of a function h : X → R at x ∈ X, denoted by ∂h(x), is a subset of a dual space X * given by ∂h( Regular function A locally Lipschitz function h is called regular (in the sense of Clarke) at x ∈ X if for all v ∈ X the one-sided directional derivative h (x; v) exists and satisfies Finally we state results needed in a sequel whose proofs can be found in Kulig [18].

Lemma 1
Let X and Y be Banach spaces and ϕ : X × Y → R be such that where ∂ϕ denotes the generalized gradient of ϕ(x, ·).
Then ϕ is continuous on X × Y .

Dynamic Viscoelastic Contact Problem with Memory Term
In this section we present a short description of the modeled process, give its weak formulation which is a hyperbolic hemivariational inequality and obtain results on existence and uniqueness of weak solutions.

Physical Setting of the Problem
The physical setting and the process are as follows. The set is occupied by a viscoelastic body in R d (d = 2, 3 in applications) which is referred to as the reference configuration. We assume that is a bounded domain with Lipschitz boundary which is divided into three mutually disjoint measurable parts D , N and C with m( D ) > 0.
We study the process of evolution of the mechanical state in time interval [0, T ], 0 < T < ∞. The system evolves in time as a result of applied volume forces and surface tractions. The description of this evolution is done by introducing a vector function u = u(x, t) = (u 1 (x, t), . . . , u d (x, t)) which describes the displacement at time t of a particle that has the position x = (x 1 , . . . , x d ) in the reference configuration. We denote by σ = σ (x, t) = (σ ij (x, t)) the stress tensor and by ε(u) = (ε ij (u)) the linearized (small) strain tensor whose components are given by (a compatibility condition) ε ij = ε ij (u) = 1 2 (u i,j + u j,i ), where i, j = 1, . . . , d. In cases where an index appears twice, we use the summation convention. We also put Q = × (0, T ).
Since the process is dynamic, we deal with the dynamic equation of motion representing momentum conservation (cf. [11,33]) and governing the evolution of the state of the body where Div denotes the divergence operator for tensor valued functions and f 0 is the density of applied volume forces such as gravity. We assume that the mass density is constant and set equal to one.
In the model the material is assumed to be viscoelastic and for its description we suppose a general constitutive law of the form Here A is a nonlinear operator describing the purely viscous properties of the material while B and C are the nonlinear elasticity and the linear relaxation operators, respectively. Note that the operators A and B may depend explicitly on the time variable and this is the case when the viscosity and elasticity properties of the material depend on the temperature field which plays the role of a parameter and whose evolution in time is prescribed. When C = 0 the constitutive law (1) reduces to a viscoelastic constitutive law (the so called Kelvin-Voigt law) with short memory and in the case when A = 0, it reduces to an elastic constitutive law with long memory. Next, we describe the boundary conditions. The body is supposed to be held fixed on the part D of the surface, so the displacement u = 0 on D × (0, T ). On the part N a prescribed surface force (traction) f 1 = f 1 (x, t) is applied, thus we have the condition σ (t)ν = f 1 on N × (0, T ). Here ν ∈ R d denotes the outward unit normal to and σ (t)ν represents the boundary stress vector. The body may come in contact over the part C of its surface. As it is met in the literature (cf. [10,11,41,42]) the conditions on the contact surface are naturally divided to conditions in the normal direction and those in the tangential direction, cf. Sect. 5.4 of [11] for the normal approach and the tangential process. In the model under consideration, the frictional contact on the part C is described by the subdifferential boundary conditions of the form where σ ν and σ τ , u ν and u τ , u ν and u τ denote the normal and the tangential components of the stress tensor, the displacement and the velocity, respectively. The functions j k , k = 1, . . . , 4 are prescribed and locally Lipschitz in their last variables. The component σ τ represents the friction force on the contact surface and ∂j k , k = 1, . . . , 4 denote the Clarke subdifferentials of the superpotentials j k , k = 1, . . . , 4 with respect to their last variables. Concrete examples of contact models which lead to aforementioned subdifferential boundary conditions will be provided in Sect. 4.
Finally, we prescribe the initial conditions for the displacement and the velocity, i.e., where u 0 and u 1 denote the initial displacement and the initial velocity, respectively. In what follows we skip occasionally the dependence of various functions on the spatial variable x ∈ ∪ . Collecting the equations and conditions described above, we obtain the following formulation of the mechanical problem: find a displacement field u : Q → R d and a stress field σ : Q → S d such that The problem above represents the classical formulation of the viscoelastic frictional contact problem. The conditions (6) and (7) introduce one of the main difficulties to the problem since the superpotentials are nonconvex and nonsmooth in general. This is the reason why the problem (2)-(8) has no classical solutions, i.e., solutions which posses all necessary classical derivatives and satisfy the relations in the usual sense at each point and at each time instant. In the following we formulate the above problem in a weak sense.

Weak Formulation of the Problem
In this section we give a weak formulation of the classical viscoelastic frictional contact problem (2)- (8). Due to the Clarke subdifferential boundary conditions (6) and (7) this formulation will be a hyperbolic hemivariational inequality. We introduce This is a closed subspace of H 1 and so it is a Hilbert space with the inner product and the corresponding norm given by The duality pairing between V * and V is denoted by ·, · . We admit the following hypotheses on the data of the problem (2)- (8).

H (A): The viscosity operator
Remark 3 It should be remarked that the growth condition H (A)(iii) excludes terms with power greater than one, but is satisfied within linearized viscoelasticity, and is satisfied by truncated operators, cf. [11,41].
Thus the hypothesis H (B) is more general than the ones considered in [23-27, 32, 35] where the elasticity operator is assumed to be linear (which corresponds to the Hooke law).

H (C):
The relaxation operator C : .
The functions j k for k = 1, 2 satisfy the following The functions j k for k = 3, 4 satisfy the corresponding conditions with the last variable being in R d .
Moreover, we need the following hypothesis.
The above hypotheses are realistic with respect to the physical data and the process modeling. We will see this in the specific examples of contact laws which are given in Sect. 4.
for a.e. t ∈ (0, T ). Assuming that the functions in the problem (2)-(8) are sufficiently regular, using the equation of motion (2) and the Green formula (cf. [44]), we obtain for a.e. t ∈ (0, T ). From the boundary conditions (4) and (5), we have On the other hand, the subdifferential boundary conditions (6) and (7) imply Using the constitutive law (3) and the above relations, we obtain the following weak formulation of the problem (2)-(8) which is called hemivariational inequality.

Evolution Inclusion for Hemivariational Inequality
In this section we state a result on the existence of solutions to second order evolution inclusions and apply it to an abstract hemivariational inequality. To this end, let where A, B : (0, T ) × V → V * are nonlinear operators, C(t) is a bounded linear operator for a.e. t ∈ (0, T ) and F : (0, T ) × V × V → 2 Z * is a multivalued mapping. Let us notice that the initial conditions in Problem P have sense in V and H since the embeddings {v ∈ V | v ∈ W} ⊂ C(0, T ; V ) and W ⊂ C(0, T ; H ) are continuous (see [7,44]). A solution to Problem P is understood as follows.
We will need the following hypotheses on the data.

Theorem 7 Under the hypotheses H (A), H (B), H (C), H (F ) 1 , (H 0 ), (H 1 ) and (H 2 ), Problem P admits a unique solution.
For the proof, we refer to Theorem 8 of [19]. We are now in a position to apply Theorem 7 to the hemivariational inequality problem we are dealing with. We define the following operators A, B, C : for u, v ∈ V , a.e. t ∈ (0, T ).

Proof By H (A)(iii) and Hölder's inequality, we have
Since the latter is separable, from the Pettis measurability theorem, it fol- is monotone for a.e. t ∈ (0, T ). From Proposition 26.12 of Zeidler [44], we know that the operator A(t, ·) is continuous for a.e. t ∈ (0, T ). Hence, in particular, it is hemicontinuous proves that H (A)(v) is satisfied and ends the proof of the lemma.

Lemma 9
Under the hypothesis H (B), the operator B : Proof The measurability of B(·, v) for all v ∈ V is shown analogously as in the proof of Lemma 8. Indeed, using H (B)(ii) and Hölder's inequality, we have Using (13), we easily obtain that and Hölder's inequality, we get follows. The proof of the lemma is thus complete.

Lemma 10 Under the hypothesis H (C)
, the operator C defined by (11) satisfies H (C).
Proof From the hypothesis H (C), we have Since c(x, t) = {c ij kl (x, t)} and c ij kl ∈ L ∞ (Q), using the Hölder inequality we readily obtain that C ∈ L 2 (0, T ; L(V , V * )).
We also observe that if H (f ) holds then (H 0 ) is satisfied as well. Now, in order to formulate Problem (HVI) in the form of evolution inclusion, we extend the pointwise relations (6) and (7) to relations involving multifunctions. To this end, we consider the function for In what follows, we will need the following hypotheses.
The next step is to study the integral functional corresponding to superpotentials which appear in the boundary conditions. Let us consider the functional G : for w, z, u, v ∈ L 2 ( C ; R d ), t ∈ (0, T ), where the integrand g is given by (14). We introduce the following hypothesis.

Lemma 12 Under the hypotheses H (g) and H (g) reg hold the functional G defined by
Proof First, from H (g)(ii) and Lemma 1, it follows that g(x, t, ·, ·, ·, ·) is continuous on (R d ) 4

t, w(x), z(x), u(x), v(x)) is integrable and from Fubini's theorem, we infer that G(·, w, z, u, v) is measurable and H (G)(i) holds.
Now, let w, z ∈ L 2 ( C ; R d ) and let g :
Next, by the Fatou lemma, we have Since the function g(x, t, ζ, ρ, ·, ·) is regular in the sense of Clarke. By exploiting the Fatou lemma and the above, we obtain for all w, z, u, v, u, v ∈ L 2 ( C ; R d ), a.e. t ∈ (0, T ). Hence G (u,v) (t, w, z, u, v; u, v) exists and which means that G(t, w, z, ·, ·) is regular for all w, z ∈ L 2 ( C ; R d ) and a.e. t ∈ (0, T ). The two above also imply that H (G)(iv) holds. When −g(x, t, ζ, ρ, ·, ·) is regular in the sense of Clarke, we proceed analogously as above and deduce the regularity of −G(t, w, z, ·, ·). From the property a.e. t ∈ (0, T ) (cf. Proposition 2.1.1 of [6]), we again get equality (20).
Finally, we suppose the hypotheses H (g)(iv). Let t ∈ (0, T ), w, z, u, v, u, v ∈ L 2 ( C ; R d ) and {w n }, {z n }, {u n }, {v n } be sequences in L 2 ( C ; R d ) such that w n → w, z n → z, u n → u and v n → v in L 2 ( C ; R d ). We may assume by passing to subsequences, if necessary, that w n ( with w 0 , z 0 , u 0 , v 0 ∈ L 2 ( C ; R d ). By the Fatou lemma and H (g)(iv), we obtain lim sup G 0 t, w n , z n , u n , v n , u, v w, z, u, v, u, v) for all u, v ∈ L 2 ( C ; R d ) and a.e. t ∈ (0, T ). This means that G 0 (t, ·, ·, ·, ·, u, v) is upper semicontinuous on L 2 ( C ; R d ) 4 for all u, v ∈ L 2 ( C ; R d ) and a.e. t ∈ (0, T ). This completes the proof that the functional G satisfies H (G)(v). The proof of the lemma is done. Now we are in a position to carry out the last step of the construction of the multifunction which will appear in the evolution inclusion. We introduce the following operators

t, w n (x), z n (x), u n (x), v n (x); u(x), v(x) d
We define the following multivalued mapping F : where ∂G denotes the Clarke subdifferential of the functional G = G(t, w, z, u, v) defined by (19) with respect to (u, v). Before we establish the properties of the multifunction F given by (22), we need the following auxiliary lemma (cf. [18]).
Proof The fact that the mapping F has nonempty and convex values follows from the nonemptiness and convexity of values of the Clarke subdifferential of G (cf. Proposition 2.1.2 of [6]). Because the values of the subdifferential ∂G(t, w, z, ·, ·) are weakly closed subsets of L 2 ( C ; R d ), using H (G) 1 , we can also easily check that the mapping F has closed values in Z * .
On the other hand, we can readily verify that SR * : L 2 ( C ; R d ) 2 → Z * is a linear continuous operator. These properties ensure the applicability of Lemma 13. So we have that (0, T ) t → SR * ∂G(t, w, z, u, v) is measurable. As a consequence the multifunction F (·, u, v) is measurable for all u, v ∈ V . Next we will prove the upper semicontinuity of F (t, ·, ·) for a.e. t ∈ (0, T ). According to Proposition 4.1.4 of [7], we show that for every weakly closed subset K of Z * , the set We can find ζ n ∈ F (t, u n , v n ) ∩ K for n ∈ N. By the definition of F , we have ζ n = ζ 1 n + ζ 2 n , (ζ 1 n , ζ 2 n ) = (γ * η 1 n , γ * η 2 n ) with (η 1 n , η 2 n ) ∈ L 2 ( C ; R d ) and η 1 n , η 2 n ∈ ∂G(t, γ u n , γ v n , γ u n , γ v n ) for a.e. t ∈ (0, T ).
Using the continuity of the trace operator, we have γ u n → γ u, γ v n → γ v in L 2 ( C ; R d ).
Since by H (G)(iii) the operator ∂G(t, ·, ·, ·, ·) is bounded (it maps bounded sets into bounded sets), from (23), it follows that the sequence {(η 1 n , η 2 n )} remains in a bounded subset of L 2 ( C ; R d ) 2 . Thus, by passing to a subsequence, if necessary, we may suppose that . Now, we will use the fact that the graph of ∂G(t, ·, ·, ·, ·) is closed in 2 )-topology for a.e. t ∈ (0, T ), which will be showed at the end of this proof. Hence and from (23), we obtain Furthermore, since {ζ n } also remains in a bounded subset of Z * , we may assume that ζ n → ζ weakly in Z * . Because ζ n ∈ K and K is weakly closed in Z * , it follows that ζ ∈ K. By the continuity and linearity of the operator γ * , we obtain γ * η 1 n → γ * η 1 , γ * η 2 n → γ * η 2 weakly in Z * .
To complete the proof, it is enough to show that the graph of ∂G(t, ·, ·, ·, ·) is closed in ∈ (0, T ). This is a simple consequence of H (G) (v). Indeed, let t ∈ (0, T ), {w n }, {z n }, {u n }, {v n } be sequences in 2 and (η 1 n , η 2 n ) ∈ ∂G(t, w n , z n , u n , v n ). The latter means that The hypothesis H (G)(v) implies w, z, u, v). The above finishes the proof that the graph is closed. This argument completes the proof of the lemma.
In order to prove that the multifunction F defined by (22) satisfies the hypothesis H (F ) 1 , we need additional conditions on the superpotentials j k for k = 1, . . . , 4.

Lemma 16 Assume that the hypotheses
Then the multifunction F : (0, T ) × V × V → 2 Z * defined by (22) with the functional G given by (19) and its integrand g defined by (14), satisfies the condition H (F ) 1 with m 2 = c e k 1 γ 2 and m 3 = c e k 2 γ 2 .
Proof It is clear that under the hypotheses, the condition H (j) reg holds. By Lemma 11 we know that the integrand g given by (14) satisfies H (g) and H (g) reg . Hence by Lemma 12, it follows that the functional G given by (19) satisfies H (G). Using Lemma 14, under H (G), we obtain that the multifunction F satisfies H (F ). Now, it is enough to prove that the multifunction F satisfies H (F ) 1 (iv). We suppose (24), the case when (25) holds can be treated analogously. We show that the following inequality holds: (24), it follows that g(x, t, ζ, ρ, ·, ·) is regular for all ζ , ρ ∈ R d , a.e. (x, t) ∈ C × (0, T ). Using this regularity, by Theorem 2.3.10 of [6] and Proposition 5.6.33 of [7], we have ∂g(x, t, ζ, ρ, ξ, η) ⊂ ∂ ξ g(x, t, ζ, ρ, ξ, η) × ∂ η g(x, t, ζ, ρ, ξ, η) where ∂g denotes the subdifferential of g with respect to (ξ, η), 2. For simplicity of notation we omit the dependence on (x, t). Then for k = 1, 2 we have χ k ∈ ∂j 1 (x, t, ξ k , η k , ξ kν )ν + (∂j 3 (x, t, ξ k , η k , ξ kτ )) τ and σ k ∈ ∂j 2 (x, t, ξ k , η k , η kν )ν + (∂j 4 (x, t, ξ k , η k , η kτ )) τ which means that for i = 1, 2. By the hypotheses H (j k ) 1 for k = 1, 3, we have whereas for k = 2, 4 we have Using the last four inequalities and the fact that (ζ τ , ρ) Hence the proof of the property (26) is complete. Next we will prove that the subdifferential ∂G of the functional G defined by (19) satisfies the condition for all w i , z i ∈ L 2 ( C ; R d ), a.e. t ∈ (0, T ) with k 1 , k 2 ≥ 0, where ∂G denotes the subdifferential of G(t, w, z, ·, ·). Similarly as in the proof of Lemma 12 and Theorem 2.7.5 of Clarke [6], we use the property that if (u, v) ∈ ∂G(t, w, z, u, v) for a.e. t ∈ (0, T ) then ∈ (0, T ). From the aforementioned property, we know that for a.e. (x, t) ∈ C × (0, T ). Exploiting the inequality (26), we have for a.e. x ∈ C . Integrating this inequality over C and applying the Hölder inequality, we obtain which means that (27) is satisfied. Finally we show that the multifunction F defined by (22) satisfies H (F ) 1 (iv). Let u i , v i ∈ V , t ∈ (0, T ) and z i ∈ F (t, u i , v i ) for i = 1, 2. By the definition of F , we have (27) and the continuity of the trace operator, we obtain where c e > 0 is the embedding constant of V into Z and γ is the norm of the trace operator. Thus the condition H (F ) 1 (iv) holds with m 2 = c e k 1 γ 2 and m 3 = c e k 2 γ 2 . The proof of the lemma is complete.
In order to formulate and prove the results on the existence and uniqueness of solutions to the hemivariational inequality in Problem (HVI), we need the following two lemmas.
for all v ∈ V , a.e. t ∈ (0, T ), where the operators A, B and C are defined by (9), (10) and (11), respectively. From H (j) reg , by Lemmas 11 and 12, we know that equalities (15) and (20) hold, which implies for all u, v ∈ V , a.e. t ∈ (0, T ). Therefore, from (37), we have for a.e. t ∈ (0, T ) which means that u is a solution Problem P. The case when j 2 = j 4 = 0 can be treated in an analogous way. This completes the proof of the lemma.
The following are the existence result for the hemivariational inequality in Problem (HVI) which are the direct conclusion from the lemmas above and Theorem 7.

Applications to Viscoelastic Mechanical Problems
The aim of this section is to explain, by providing several examples, formulations of multivalued boundary conditions of mechanics. We consider boundary conditions resulting from convex or nonconvex and nonsmooth potentials using the concept of a subdifferential. We restrict ourselves to one-dimensional examples, referring to Chap. 4.6 of [30] for two-and three-dimensional contact laws. We present specific examples of contact and friction laws which can be met in mechanics and which lead to the subdifferential boundary conditions of the form −σ ν (t) ∈ ∂j 1 x, t, u(t), u (t), u ν (t) + ∂j 2 x, t, u(t), u (t), u ν (t) , −σ τ (t) ∈ ∂j 3 x, t, u(t), u (t), u τ (t) + ∂j 4 x, t, u(t), u (t), u τ (t) on C × (0, T ).

Prescribed Normal Stress and Nonmonotone Friction Laws
Let us consider the following boundary conditions on C × (0, T ): Equation (40) states that the normal stress is prescribed on C × (0, T ) and is given by S = S(x, t) ≥ 0. Such a condition makes sense when the real contact area is close to the nominal one and the surfaces are conforming. Then S = S(x, t) is the contact pressure and it is given by the ratio of the total applied force to the nominal contact area. It is considered (see Chaps. 2.6 and 10.1 of Shillor et al. [41]) to be a good approximation when the load is light and the contact surface is transmitted by the asperity tips only. This law is of the form (38) with j 1 (x, t, ζ, ρ, r) = S(x, t)r and j 2 = 0, where S ∈ L ∞ ( C × (0, T )), S ≥ 0 is a given normal stress. It is clear that j 1 (x, t, ζ, ρ, ·) is convex (hence regular), and that H (j 1 ) and H (j 1 ) 1 hold.

Nonmonotone Friction Independent of Slip and Slip Rate
We consider the nonmonotone friction laws which are independent of the slip displacement and the slip rate. This is the case when the superpotential j 4 = j 4 (x, t, ζ, ρ, θ) is independent of (ζ, ρ) and nonconvex in θ . Then the friction law (41) takes the form −σ τ (t) ∈ ∂j 4 x, t, u τ (t) on C × (0, T ).
This law appears (cf. Sect. 7.2 of Panagiotopoulos [34]) in the tangential direction of the adhesive interface and describes the partial cracking and crushing of the adhesive bonding material. Several examples of zig-zag friction laws from Sect. 2.4 of Panagiotopoulos [34] can be formulated in the form (42). For instance, let j 4 : R → R be given by j 4 (r) = min{ϕ 1 (r), ϕ 2 (r)}, where ϕ 1 (r) = ar 2 , ϕ 2 (r) = a 2 (r 2 + 1), r ∈ R (for simplicity we also drop the (x, t)-dependence) and a > 0. Its subdifferential is as follows: By Theorem 2.5.1 of [6], we know that ∂j 4 (r) ⊂ co{ϕ 1 (r), ϕ 2 (r)} and that the subdifferential ∂j 4 has at most linear growth. Since j 4 is the minimum of the strictly differentiable functions, the function −j 4 is regular. From the above and Proposition 2.1.1 of [6], the hypothesis H (j 4 ) holds.

Contact with Nonmonotone Normal Damped Response
This contact condition is of the form (38) with j 1 = 0 and it models the situations with granular or wet surfaces in which the response of the foundation depends on the normal velocity of the body. For simplicity, we describe the case when The specific example of the nonmonotone normal damped response condition is given by the following nonconvex, regular and d.c. function It is clear that |∂j 2 (r)| ≤ 1 + |r| for r ∈ R; The function j 2 can be represented as the difference of convex functions, i.e. j 2 (r) = ϕ 1 (r) − ϕ 2 (r), r ∈ R, where Since ϕ 1 , ϕ 2 are convex functions, ∂ϕ 1 , ∂ϕ 2 have a sublinear growth with ∂ϕ 2 being a singleton, we deduce by Lemma 14 of [29] that j 2 is regular. From the above and the Proposition 2.1.1 of [6], it is obvious that H (j 2 ) holds. Next, we verify that η 1 ≤ η 2 − (r 1 − r 2 ) for all r 1 < r 2 and η i ∈ ∂j 2 (r i ), i = 1, 2 which implies relaxed monotonicity condition (∂j 2 (r 1 ) − ∂j 2 (r 2 ))(r 1 − r 2 ) ≥ −|r 1 − r 2 | 2 (cf. Remark 15), as well as H (j 2 ) 1 .

Viscous Contact with Tresca's Friction Law
We consider a model of damped response contact with time-dependent Tresca's friction law.

Other Nonmonotone Friction Contact Laws
In this part we comment on the boundary conditions expressed in the form −σ τ (t) ∈ ∂j 3 x, t, u(t), u (t), u τ (t) .
This relation describes the tangential contact law between reinforcement and concrete in a concrete structure. In literature, cf. Chap. 2.4 in Panagiotopoulos [34] (diagrams of Fig. 2.4.1), Chap. 1.4 in Naniewicz and Panagiotopoulos [30] (diagrams of Fig. 1.4.3), one can find a couple of examples of the superpotential j 3 which describes such type of contact. We give two examples of nonconvex functions which appear in (43). In the first example the superpotential j 3 : R → R and its subdifferential are of the form 22 3 if r ≥ 3, It is easy to check that the function j 3 satisfies H (j 3 ). Furthermore, j 3 can be represented as the difference of convex functions, j 3 (r) = ϕ 1 (r) − ϕ 2 (r), r ∈ R with ϕ 1 (r) = 0 ifr < 0, 2r 2 if r ≥ 0, Since ϕ 1 , ϕ 2 are convex functions and ∂ϕ 1 is a singleton, from Lemma 14 of [29] we deduce that the function −j 3 is regular. Moreover, its subdifferential ∂j 3 is Lipschitz which implies that the condition H (j 3 ) 1 is fulfilled.
In the second example, we consider the function j 3 : R → R such that Similarly to the previous case, j 3 satisfies H (j 3 ) 1 . It can be also represented as the difference of convex functions, j 3 (r) = ϕ 1 (r) − ϕ 2 (r), r ∈ R, where Again, from the fact that ϕ 1 and ϕ 2 are convex functions and ∂ϕ 1 is a singleton, we conclude that −j 3 is regular.
We end this section with indications on specific applications of research on contact problems. It is of importance to provide various applications of the theoretical results to contact problems arising in real world. The applications concern the following areas.
Construction and exploitation of machines The understanding of contact problems are extremely important in various branches of engineering such as structural foundations, bearings, metal forming processes, drilling problems, the simulation of car crashes, the car braking system, contact of train wheels with the rails, a shoe with the floor, machine tools, bearings, motors, turbines, cooling of electronic devices, joints in mechanical devices, ski lubricants, and many more, cf., e.g., Andrews et al. [4], Chau et al. [5], Kuttler and Shillor [20,21], Rochdi et al. [37] and Sofonea and Matei [43].
Biomechanics The applications concerns the medical field of arthoplasty where bonding between the bone implant and the tissue is of considerable importance. Artificial implants of knee and hip prostheses (both cemented and cement-less) demonstrate that the adhesion is important at the bone-implant interface. These applications are related to contact modeling and design of biomechanical parts like human joints, implants or teeth, cf. Panagiotopoulos [34], Rojek and Telega [38], Rojek et al. [39], Shillor et al. [41] and Sofonea et al. [42].

Medicine and biology
Results are applicable to nonmonotone semipermeable membranes and walls (biological and artificial), cf. Duvaut and Lions [10]. In particular, contact problems for piezoelectric materials will continue to play a decisive role in the field of ultrasonic transducers for imaging applications, e.g., medical imaging (sonogram), nondestructive testing and high power applications (medical treatment, sonochemistry and industrial processing), cf. Shillor et al. [41], Sofonea et al. [42].