Abstract
The paper deals with second order nonlinear evolution inclusions and their applications. First, we study an evolution inclusion involving Volterratype 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 Volterratype 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.
Keywords
Evolution inclusion Pseudomonotone operator Volterratype operator Multifunction Hyperbolic Contact problem Hemivariational inequality Viscoelasticity Clarke subdifferentialMathematics Subject Classification
35L90 35R70 45P05 47H04 47H05 74H20 74H251 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.
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 [22–25], 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.
2 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 wX to denote the space X endowed with the weak topology. The class of linear and bounded operators from X to X ^{∗} is denoted by \(\mathcal{L}(X,X^{*})\). If U⊂X, then we have ∥U∥_{ X }=sup{∥x∥_{ X }:x∈U}. The duality between X and its dual is denoted by \(\langle\cdot,\cdot\rangle_{X^{*} \times X}\).
Given v∈H _{1} we denote its trace γv on Γ by v, where γ:H ^{1}(Γ;ℝ^{ d })→H ^{1/2}(Γ;ℝ^{ d })⊂L ^{2}(Γ;ℝ^{ d }) is the trace map. Let n denote the outward unit normal vector to Γ. Since Γ is Lipschitz continuous, the normal vector is defined a.e. on Γ. For v∈H ^{1/2}(Γ;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 \(\mathcal{V} = L^{2}(0,T;V)\) and \(\mathcal{W} = \{ w \in\mathcal{V} : w' \in\mathcal{V}^{*} \}\). The time derivative involved in the definition of \(\mathcal{W}\) is understood in the sense of vector valued distributions. It is well known that \(\mathcal{W}\) is a separable Banach space equipped with the norm \(\ v \_{\mathcal{W}} = \v\_{\mathcal{V}} + \v\_{\mathcal{V}^{*}}\) and \(\mathcal{W} \subset\mathcal{V} \subset L^{2}(0,T;H) \subset {\mathcal{V}^{*}}\).
Let us recall some definitions needed in the next sections.
Measurable multifunction
Let (Ω,Σ) be a measurable space, \(\mathcal{X}\) be a separable Banach space and \(F \colon\Omega\to{2}^{\mathcal{X}}\). The multifunction F is said to be measurable if for every U⊂X open, we have F ^{−}(U)={ω∈Ω:F(ω)∩U≠∅}∈Σ.
Generalized directional derivative
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 \(\partial{h}(x) =\{ \zeta\in X^{*} \mid h^{0}(x; v) \ge\langle\zeta, v \rangle_{X^{*}\times X}\ \mbox{for all} \ v \in X \}\).
Regular function
A locally Lipschitz function h is called regular (in the sense of Clarke) at x∈X if for all v∈X the onesided directional derivative h′(x;v) exists and satisfies h ^{0}(x;v)=h′(x;v) for all v∈X.
Finally we state results needed in a sequel whose proofs can be found in Kulig [18].
Lemma 1
 (i)
φ(⋅,y) is continuous for all y∈Y;
 (ii)
φ(x,⋅) is locally Lipschitz on Y for all x∈X;
 (iii)there is a constant c>0 such that for all η∈∂φ(x,y), we havewhere ∂φ denotes the generalized gradient of φ(x,⋅).$$\ \eta\_{Y^*} \le c \bigl(1 + \ x \_X + \ y\_Y \bigr) \quad \mbox{\textit{for\ all}} \ x \in X, \ y \in Y,$$
Proposition 2
Let X be a separable reflexive Banach space, 0<T<∞ and φ:(0,T)×X→ℝ be a function such that φ(⋅,x) is measurable for all x∈X and φ(t,⋅) is locally Lipschitz for all t∈(0,T). Then the multifunction (0,T)×X∋(t,x)↦∂φ(t,x)⊂X ^{∗} is measurable, where ∂φ denotes the Clarke generalized gradient of φ(t,⋅).
3 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.
3.1 Physical Setting of the Problem
The physical setting and the process are as follows. The set Ω 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 Ω is a bounded domain with Lipschitz boundary Γ which is divided into three mutually disjoint measurable parts Γ_{ D }, Γ_{ N } and Γ_{ C } with m(Γ_{ 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) \(\varepsilon_{ij} = \varepsilon_{ij}(u) = \frac{1}{2} (u_{i,j} + u_{j,i})\), where i, j=1,…,d. In cases where an index appears twice, we use the summation convention. We also put Q=Ω×(0,T).
3.2 Weak Formulation of the Problem
 \(\underline{H({\mathcal{A}})}\):

The viscosity operator \({\mathcal{A}} \colon Q \times{ \mathbb {S}^{d}} \to{ \mathbb {S}^{d}}\) is such that
 (i)
\({\mathcal{A}}(\cdot,\cdot, \varepsilon)\) is measurable on Q for all \(\varepsilon\in{ \mathbb {S}^{d}}\);
 (ii)
\({\mathcal{A}}(x, t, \cdot)\) is continuous on \({\mathbb {S}^{d}}\) for a.e. (x,t)∈Q;
 (iii)
\(\ {\mathcal{A}}(x,t,\varepsilon) \_{{\mathbb {S}^{d}}} \le {\widetilde{a_{1}}}(x,t) + {\widetilde{a_{2}}} \ \varepsilon\_{{\mathbb {S}^{d}}}\) for all \(\varepsilon\in{ \mathbb {S}^{d}}\), a.e. (x,t)∈Q with \({\widetilde{a_{1}}} \in L^{2}(Q)\), \({\widetilde {a_{1}}}\), \({\widetilde{a_{2}}} \ge0\);
 (iv)
\(( {\mathcal{A}}(x, t, \varepsilon_{1})  {\mathcal{A}} (x, t, \varepsilon_{2}) ): (\varepsilon_{1}  \varepsilon_{2}) \ge{\widetilde{a_{4}}}\ \varepsilon_{1}  \varepsilon_{2} \^{2}_{\mathbb {S}^{d}}\) for all ε _{1}, \(\varepsilon_{2} \in{ \mathbb {S}^{d}}\), a.e. (x,t)∈Q with \({\widetilde{a_{4}}} > 0\);
 (v)
\({\mathcal{A}}(x,t,\varepsilon) : \varepsilon\ge {\widetilde{a_{3}}} \ \varepsilon\^{2}_{{\mathbb {S}^{d}}}\) for all \(\varepsilon\in{ \mathbb {S}^{d}}\), a.e. (x,t)∈Q with \({\widetilde{a_{3}}} > 0\).
 (i)
Remark 3
It should be remarked that the growth condition \(H({\mathcal{A}})\)(iii) excludes terms with power greater than one, but is satisfied within linearized viscoelasticity, and is satisfied by truncated operators, cf. [11, 41].
 \(\underline{H({\mathcal{B}})}\):

The elasticity operator \({\mathcal{B}} \colon Q \times{ \mathbb {S}^{d}} \to{ \mathbb {S}^{d}}\) is such that
 (i)
\({\mathcal{B}}(\cdot,\cdot, \varepsilon)\) is measurable on Q for all ε∈S _{ d };
 (ii)
\(\ {\mathcal{B}}(x,t,\varepsilon) \_{{\mathbb {S}^{d}}} \le {\widetilde{b_{1}}}(x,t) + {\widetilde{b_{2}}} \ \varepsilon\_{{\mathbb {S}^{d}}}\) for all \(\varepsilon\in{ \mathbb {S}^{d}}\), a.e. (x,t)∈Q with \({\widetilde{b_{1}}} \in L^{2}(Q)\), \({\widetilde {b_{1}}}\), \({\widetilde{b_{2}}} \ge0\);
 (iii)
\(\ {\mathcal{B}}(x,t,\varepsilon_{1})  {\mathcal{B}} (x,t,\varepsilon_{2})\_{{\mathbb {S}^{d}}}\le L_{\mathcal{B}} \ \varepsilon_{1}  \varepsilon_{2} \_{\mathbb {S}^{d}}\) for all ε _{1}, \(\varepsilon_{2} \in{ \mathbb {S}^{d}}\), a.e. (x,t)∈Q with \(L_{\mathcal{B}} > 0\).
 (i)
Remark 4
If \({\mathcal{B}} (x, t, \cdot) \in{\mathcal{L}}(\mathbb {S}^{d}, \mathbb {S}^{d})\) for a.e. (x,t)∈Q, the conditions \(H({\mathcal{B}})\)(ii) and (iii) hold. Thus the hypothesis \(H({\mathcal{B}})\) 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).
 \(\underline{H({\mathcal{C}})}\):

The relaxation operator \({\mathcal{C}} \colon Q \times{ \mathbb {S}^{d}} \to{ \mathbb {S}^{d}}\) is of the form \({\mathcal{C}}(x,t,\varepsilon) = c(x,t) \varepsilon\) and c(x,t)={c _{ ijkl }(x,t)} with c _{ ijkl }=c _{ jikl }=c _{ lkij }∈L ^{∞}(Q).
 \(\underline{H(f)}\):

f _{0}∈L ^{2}(0,T;H), f _{1}∈L ^{2}(0,T;L ^{2}(Γ_{ N };ℝ^{ d })), u _{0}∈V, u _{1}∈H.
 \(\underline{H(j_{k})}\):

The function j _{ k }:Γ_{ C }×(0,T)×(ℝ^{ d })^{2}×ℝ→ℝ is such that
 (i)
j _{ k }(⋅,⋅,ζ,ρ,r) is measurable for all ζ, ρ∈ℝ^{ d }, r∈ℝ,
j _{ k }(⋅,⋅,v(⋅),w(⋅),0)∈L ^{1}(Γ_{ C }×(0,T)) for all v, w∈L ^{2}(Γ_{ C };ℝ^{ d });
 (ii)
j _{ k }(x,t,⋅,⋅,r) is continuous for all r∈ℝ, a.e. (x,t)∈Γ_{ C }×(0,T),
j _{ k }(x,t,ζ,ρ,⋅) is locally Lipschitz for all ζ, ρ∈ℝ^{ d }, a.e. (x,t)∈Γ_{ C }×(0,T);
 (iii)
∂j _{ k }(x,t,ζ,ρ,r)≤c _{ k0}+c _{ k1}∥ζ∥+c _{ k2}∥ρ∥+c _{ k3}r for all ζ, ρ∈ℝ^{ d }, r∈ℝ, a.e. (x,t)∈Γ_{ C }×(0,T) with c _{ kj }≥0, j=0, 1, 2, 3, where ∂j _{ k } denotes the Clarke subdifferential of j _{ k }(x,t,ζ,ρ,⋅);
 (iv)
\(j^{0}_{k} (x,t, \cdot, \cdot, \cdot; s)\) is upper semicontinuous on (ℝ^{ d })^{2}×ℝ for all s∈ℝ, a.e. (x,t)∈Γ_{ C }×(0,T), where \(j^{0}_{k}\) denotes the generalized directional derivative of Clarke of j _{ k }(x,t,ζ,ρ,⋅) in the direction s.
 (i)
The functions j _{ k } for k=3, 4 satisfy the corresponding conditions with the last variable being in ℝ^{ d }.
 \(\underline{H(j)_{reg}}\):

The functions j _{ k } for k=1,…,4 are such that for all ζ, ρ∈ℝ^{ d }, a.e. (x,t)∈Γ_{ C }×(0,T), either all j _{ k }(x,t,ζ,ρ,⋅) are regular or all −j _{ k }(x,t,ζ,ρ,⋅) are regular for k=1,…,4.
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.
3.3 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 Z=H ^{ δ }(Ω;ℝ^{ d }), δ∈(1/2,1) and γ:Z→L ^{2}(Γ_{ C };ℝ^{ d }) be the trace operator. Let γ ^{∗}:L ^{2}(Γ_{ C };ℝ^{ d })→Z ^{∗} stand for the adjoint operator to γ.
Definition 5
 \(\underline{H(A)}\):

The operator A:(0,T)×V→V ^{∗} is such that
 (i)
A(⋅,v) is measurable on (0,T) for all v∈V;
 (ii)
A(t,⋅) is strongly monotone for a.e. t∈(0,T), i.e. there exists m _{1}>0 such that 〈A(t,v)−A(t,u),v−u〉≥m _{1}∥v−u∥^{2} for all u, v∈V, a.e. t∈(0,T);
 (iii)
\(\ A(t, v) \_{V^{*}} \le a_{0}(t) + a_{1} \ v \\) for all v∈V, a.e. t∈(0,T) with a _{0}∈L ^{2}(0,T), a _{0}≥0 and a _{1}>0;
 (iv)
〈A(t,v),v〉≥α∥v∥^{2} for all v∈V, a.e. t∈(0,T) with α>0;
 (v)
A(t,⋅) is hemicontinuous for a.e. t∈(0,T).
 (i)
Remark 6
The hypothesis H(A) implies the operator A is pseudomonotone. Indeed, strong monotonicity clearly implies monotonicity which with hemicontinuity entails (cf. Proposition 27.6(a), p. 586, of Zeidler [44]) pseudomonotonicity. We also recall (cf. Remark 1.1.13 of [8]) that for monotone operators, demicontinuity and hemicontinuity are equivalent notions.
 \(\underline{H(B)}\):

The operator B:(0,T)×V→V ^{∗} is such that
 (i)
B(⋅,v) is measurable on (0,T) for all v∈V;
 (ii)
B(t,⋅) is Lipschitz continuous for a.e. t∈(0,T), i.e. \(\ B(t, u)  B(t, v) \_{V^{*}} \le L_{B} \ u  v \\) for all u, v∈V, a.e. t∈(0,T) with L _{ B }>0;
 (iii)
\(\ B(t, v) \_{V^{*}} \le b_{0}(t) + b_{1} \ v \\) for all v∈V, a.e. t∈(0,T) with b _{0}∈L ^{2}(0,T) and b _{0}, b _{1}≥0.
 (i)
 \(\underline{H(C)}\):

The operator C satisfies \(C\in L^{2}(0,T; {\mathcal{L}}(V, V^{*}))\).
 \(\underline{H(F)}\):

The multifunction \(F \colon(0,T) \times V \times V \to{\mathcal{P}}_{fc}(Z^{*})\) is such that
 (i)
F(⋅,u,v) is measurable on (0,T) for all u, v∈V;
 (ii)
F(t,⋅,⋅) is upper semicontinuous from V×V into wZ ^{∗} for a.e. t∈(0,T), where V×V is endowed with (Z×Z)topology;
 (iii)
\(\ F(t, u, v) \_{Z^{*}} \le d_{0}(t) + d_{1} \ u \ + d_{2} \ v \\) for all u, v∈V, a.e. t∈(0,T) with d _{0}∈L ^{2}(0,T) and d _{0}, d _{1}, d _{2}≥0.
 (i)
 \(\underline{H(F)_{1}}\):

The multifunction \(F \colon(0,T) \times V \times V \to{\mathcal{P}}_{fc}(Z^{*})\) satisfies H(F) and
 (iv)
\(\langle F(t, u_{1}, v_{1})  F(t, u_{2}, v_{2}), v_{1}  v_{2} \rangle_{Z^{*}\times Z} \ge m_{2} \v_{1}  v_{2}\^{2}  m_{3} \v_{1}  v_{2}\ \u_{1}u_{2}\\) for all u _{ i }, v _{ i }∈V, i=1, 2, a.e. t∈(0,T) with m _{2}, m _{3}≥0.
 (iv)
 \(\underline{(H_{0})}\):

\(f \in{\mathcal{V}}^{*}\), u _{0}∈V, u _{1}∈H.
 \(\underline{(H_{1})}\):

\(\alpha> 2 \sqrt{3} c_{e} (d_{1} T + d_{2})\), where c _{ e }>0 is the embedding constant of V into Z, i.e., ∥⋅∥_{ Z }≤c _{ e }∥⋅∥.
 \(\underline{(H_{2})}\):

\(m_{1} > m_{2} + \frac{1}{\sqrt{2}} m_{3}T\).
Theorem 7
Under the hypotheses H(A), H(B), H(C), H(F)_{1}, (H _{0}), (H _{1}) and (H _{2}), Problem \({\mathcal{P}}\) admits a unique solution.
For the proof, we refer to Theorem 8 of [19].
Lemma 8
Under the hypothesis \(H({\mathcal{A}})\), the operator A:(0,T)×V→V ^{∗} defined by (9) satisfies H(A) with \(a_{0}(t) = \sqrt{2} \ {\widetilde{a_{1}}} (t) \_{L^{2}(\Omega)}\), \(a_{1} = \sqrt{2} {\widetilde{a_{2}}}\), \(\alpha= {\widetilde{a_{3}}}\), and \(m_{1} = {\widetilde{a_{4}}}\).
Proof
Lemma 9
Under the hypothesis \(H({\mathcal{B}})\), the operator B:(0,T)×V→V ^{∗} defined by (10) satisfies H(B) with \(L_{B} = L_{\mathcal{B}}\), \(b_{0}(t) = \sqrt{2} \ {\widetilde{b}}_{1}(t) \_{L^{2}(\Omega)}\) and \(b_{1} = \sqrt{2} {\widetilde{b}}_{2}\).
Proof
Lemma 10
Under the hypothesis \(H({\mathcal{C}})\), the operator C defined by (11) satisfies H(C).
Proof
 \(\underline{H(g)}\):

The function g:Γ_{ C }×(0,T)×(ℝ^{ d })^{4}→ℝ satisfies the following
 (i)
g(⋅,⋅,ζ,ρ,ξ,η) is measurable for all ζ, ρ, ξ, η∈ℝ^{ d },
g(⋅,⋅,v(⋅),w(⋅),0,0)∈L ^{1}(Γ_{ C }×(0,T)) for all v, w∈L ^{2}(Γ_{ C };ℝ^{ d });
 (ii)
g(x,t,⋅,⋅,ξ,η) is continuous for all ξ, η∈ℝ^{ d }, a.e. (x,t)∈Γ_{ C }×(0,T),
g(x,t,ζ,ρ,⋅,⋅) is locally Lipschitz for all ζ, ρ∈ℝ^{ d }, a.e. (x,t)∈Γ_{ C }×(0,T);
 (iii)
\(\ \partial g (x, t,\zeta, \rho, \xi, \eta) \_{(\mathbb{R}^{d})^{2}} \le c_{g0} + c_{g1} (\ \zeta\ + \ \xi\) + c_{g2} (\ \rho\ + \\eta\)\) for all ζ, ρ, ξ, η∈ℝ^{ d }, a.e. (x,t)∈Γ_{ C }×(0,T) with c _{ g0}, c _{ g1}, c _{ g2}≥0, where ∂g denotes the Clarke subdifferential of g(x,t,ζ,ρ,⋅,⋅);
 (iv)
g ^{0}(x,t,⋅,⋅,⋅,⋅;χ,σ) is upper semicontinuous on (ℝ^{ d })^{4} for a.e. (x,t)∈Γ_{ C }×(0,T) and all χ, σ∈ℝ^{ d } where g ^{0} denotes the generalized directional derivative of Clarke of g(x,t,ζ,ρ,⋅,⋅) in the direction (χ,σ).
 (i)
 \(\underline{H(g)_{reg}}\):

The function g:Γ_{ C }×(0,T)×(ℝ^{ d })^{4}→ℝ is such that either g(x,t,ζ,ρ,⋅,⋅) or −g(x,t,ζ,ρ,⋅,⋅) is regular for all ζ,ρ∈ℝ^{ d }, a.e. (x,t)∈Γ_{ C }×(0,T).
Lemma 11
Proof
If for k=1,…,k the functions −j _{ k } are regular in their last variables, then we proceed in the same way as above and deduce the regularity of −g(x,t,ζ,ρ,⋅,⋅) for (x,t)∈Γ_{ C }×(0,T) and ζ, ρ∈ℝ^{ d }. We use the property (−g)^{0}(x,t,ζ,ρ,ξ,η;χ,σ)=g ^{0}(x,t,ζ,ρ,ξ,η;−χ,−σ) for all ζ, ρ, ξ, η, χ, σ∈ℝ^{ d }, a.e. (x,t)∈Γ_{ C }×(0,T) (cf. Proposition 2.1.1 of [6]), and again deduce (15).
 \(\underline{H(G)}\):

The functional G:(0,T)×L ^{2}(Γ_{ C };ℝ^{ d })^{4}→ℝ is such that
 (i)
G(⋅,w,z,u,v) is measurable for all w, z, u, v∈L ^{2}(Γ_{ C };ℝ^{ d }),
G(⋅,w,z,0,0)∈L ^{1}(0,T) for all w, z∈L ^{2}(Γ_{ C };ℝ^{ d });
 (ii)
G(t,w,z,⋅,⋅) is Lipschitz continuous on bounded subsets of L ^{2}(Γ_{ C };ℝ^{ d })^{2} for all w, z∈L ^{2}(Γ_{ C };ℝ^{ d }), a.e. t∈(0,T);
 (iii)
\(\ \partial G (t, w, z, u, v) \_{L^{2}(\Gamma_{C};\mathbb{R}^{d})^{2}} \le c_{G0} + c_{G1} (\ w \ + \ u \) + c_{G2} (\ z \ + \ v \ )\) for all w, z, u, v∈L ^{2}(Γ_{ C };ℝ^{ d }), a.e. t∈(0,T) with c _{ G0}, c _{ G1}, c _{ G2}≥0, where ∂G denotes the Clarke subdifferential of G(t,w,z,⋅,⋅);
 (iv)For all w, z, u, v, \({\overline {u}}\), \({\overline {v}} \in L^{2}(\Gamma_{C};\mathbb{R}^{d})\), a.e. t∈(0,T), we havewhere G ^{0} denotes the generalized directional derivative of G(t,w,z,⋅,⋅) at a point (u,v) in the direction \(({\overline {u}}, {\overline {v}})\);$$ G^0(t, w, z, u, v; {\overline {u}}, {\overline {v}}) = \int _{\Gamma_C} g^0 \bigl(x, t, w(x), z(x), u(x), v(x); {\overline {u}}(x), {\overline {v}}(x) \bigr) \,d\Gamma,$$(20)
 (v)
\(G^{0} (t, \cdot, \cdot, \cdot, \cdot; {\overline {u}}, {\overline {v}})\) is upper semicontinuous on L ^{2}(Γ_{ C };ℝ^{ d })^{4} for all \({\overline {u}}, {\overline {v}} \in L^{2}(\Gamma_{C};\mathbb{R}^{d})\), a.e. t∈(0,T).
 (i)
 \(\underline{H(G)_{reg}}\):

The functional G:(0,T)×L ^{2}(Γ_{ C };ℝ^{ d })^{4}→ℝ is such that either G(t,w,z,⋅,⋅) or −G(t,w,z,⋅,⋅) is regular for all w, z∈L ^{2}(Γ_{ C };ℝ^{ d }), a.e. t∈(0,T).
Lemma 12
Under the hypotheses H(g) and H(g)_{ reg } hold the functional G defined by (19) satisfies H(G) with \(c_{G0} = c_{g0} \sqrt{5 m(\Gamma_{C})}\), \(c_{G1} = c_{g1} \sqrt{5}\), \(c_{G2} = c_{g2} \sqrt{5}\) and H(G)_{ reg }.
Proof
First, from H(g)(ii) and Lemma 1, it follows that g(x,t,⋅,⋅,⋅,⋅) is continuous on (ℝ^{ d })^{4} which together with H(g)(i) implies that g is a Carathéodory function. Hence (x,t)↦g(x,t,w(x),z(x),u(x),v(x)) is measurable for all w, z, u, v∈L ^{2}(Γ_{ C };ℝ^{ d }) and subsequently the integrand of (19) is a measurable function of x.
Before we establish the properties of the multifunction F given by (22), we need the following auxiliary lemma (cf. [18]).
Lemma 13
Let (Ω,Σ) be a measurable space, Y _{1}, Y _{2} be separable Banach spaces, \(A \in{\mathcal{L}}(Y_{1}, Y_{2})\) and let \(G \colon\Omega\to{\mathcal{P}}_{wkc}(Y_{1})\) be measurable. Then the multifunction \(F \colon\Omega\to{\mathcal{P}}_{wkc}(Y_{2})\) given by F(ω)=AG(ω) for ω∈Ω is measurable.
Lemma 14
Under H(G) and H(G)_{ reg }, the multifunction \(F \colon(0,T) \times V \times V \to2^{Z^{*}}\) defined by (22) satisfies H(F) with d _{0}(t)=c _{ G0}∥γ∥, d _{1}=2c _{ e } c _{ G1}∥γ∥^{2} and d _{2}=2c _{ e } c _{ G2}∥γ∥^{2}.
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}(Γ_{ C };ℝ^{ d }), using H(G)_{1}, we can also easily check that the mapping F has closed values in Z ^{∗}.
 \(\underline{H(j_{1})_{1}}\):

j _{1}:Γ_{ C }×(0,T)×(ℝ^{ d })^{2}×ℝ→ℝ is such thatfor all ζ_{1}, ζ_{2}, ρ _{1}, ρ _{2}∈ℝ^{ d }, r _{1}, r _{2}∈ℝ, a.e. (x,t)∈Γ_{ C }×(0,T) with a constant L _{1}≥0.$$\bigl \partial j_1 (x, t,\zeta_1, \rho_1,r_1)  \partial j_1 (x, t,\zeta_2,\rho_2, r_2)\bigr \le L_1 \bigl( \ \zeta_1 \zeta_2 \ + \ \rho_1  \rho_2 \ +r_1  r_2 \bigr)$$
 \(\underline{H(j_{2})_{1}}\):
 \(\underline{H(j_{3})_{1}}\):

j _{3}:Γ_{ C }×(0,T)×(ℝ^{ d })^{3}→ℝ is such thatfor all ζ_{1}, ζ_{2}, ρ _{1}, ρ _{2}, θ _{1}, θ _{2}∈ℝ^{ d }, a.e. (x,t)∈Γ_{ C }×(0,T) with a constant L _{3}≥0.$$\bigl\ \partial j_3 (x, t,\zeta_1, \rho_1,\theta_1)  \partial j_3 (x, t,\zeta_2,\rho_2, \theta_2)\bigr\ \le L_3 \bigl( \\zeta_1  \zeta_2 \ + \ \rho_1 \rho_2 \ + \ \theta_1  \theta_2 \ \bigr)$$
Remark 15
The hypothesis H(j _{2})_{1} (and H(j _{4})_{1}) has been introduced and used earlier in [23] (under the name of relaxed monotonicity condition) in the case when j _{2} (and j _{4}) does not depend on the variables ζ and ρ.
Lemma 16
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).
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.
Lemma 17
Under hypotheses \(H({\mathcal{A}})\), \(H({\mathcal{B}})\), \(H({\mathcal{C}})\), H(f) and H(j _{ k }) for k=1,…,4, every solution of the Problem \(\mathcal{P}\) with the multivalued mapping of the form (22), with G:(0,T)×L ^{2}(Γ_{ C };ℝ^{ d })^{4}→ℝ of the form (19) and its integrand g:Γ_{ C }×(0,T)×(ℝ^{ d })^{4}→ℝ given by (14), and the operators A, B and C defined by (9), (10) and (11) respectively, is a solution to Problem (HVI).
Proof
Lemma 18
Assume the function definitions and hypotheses of Lemma 17. If either j _{1}=j _{3}=0 or j _{2}=j _{4}=0, then u is a solution to Problem (HVI) if and only if u is a solution to Problem \(\mathcal{P}\).
Proof
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.
Theorem 19
Theorem 20
Assume the hypotheses of Theorem 19. If, in addition, either j _{1}=j _{3}=0 or j _{2}=j _{4}=0, then the hemivariational inequality in Problem (HVI) admits a unique solution.
4 Applications to Viscoelastic Mechanical Problems
4.1 Prescribed Normal Stress and Nonmonotone Friction Laws
4.1.1 Nonmonotone Friction Independent of Slip and Slip Rate
4.1.2 Contact with Nonmonotone Normal Damped Response
4.2 Viscous Contact with Tresca’s Friction Law
4.3 Other Nonmonotone Friction Contact Laws
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 cementless) demonstrate that the adhesion is important at the boneimplant 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].
Plate tectonics and earthquakes predictions
Results may be applicable to models with nonmonotone strainstress laws in rock layers. Frictional contact between rocks are described by several models, cf. Dumont et al. [9], Ionescu et al. [12, 14], Ionescu and Nguyen [13], Ionescu and Paumier [15, 16] and Rabinowicz [36].
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].
Acknowledgements
Research partially supported by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme under Grant Agreement No. 295118.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.