Well-posedness of evolutionary differential variational–hemivariational inequalities and applications to frictional contact mechanics

In this paper, we study the well-posedness of a class of evolutionary variational–hemivariational inequalities coupled with a nonlinear ordinary differential equation in Banach spaces. The proof is based on an iterative approximation scheme showing that the problem has a unique mild solution. In addition, we established the continuity of the flow map with respect to the initial data. Under the general framework, we consider two new applications for modeling of frictional contact for viscoelastic materials. In the first application, we consider Coulomb’s friction with normal compliance, and in the second, normal damped response. The structure of the friction coefficient μ is new with motivation from geophysical applications in earth sciences with dependence on an external state variable α and the slip rate | u · τ | .


Introduction
This work concerns the study of an evolutionary differential variational-hemivariational inequality modelling mathematical problems from contact mechanics.These systems are relevant for many physical phenomena ranging from engineering to biology (see, e.g., [6,9,33] and the references therein).We are interested in frictional contact phenomena for viscoelastic materials with the linearized strain tensor, which have been studied intensively, see, e.g., some of the relevant books [9,23,33].
Here, A and R are nonlinear operators related to the viscoelastic constitutive laws.Further, j • is a generalized directional derivative of a functional j.The functionals φ and j are determined by contact boundary conditions.We require φ to be convex in its last argument, while j may be nonconvex with appropriate structures given later (see Section 3).The operators S φ and S j relate to the contact conditions, and G is assumed to be a nonlinear operator related to the change in the external state variable α.The data f is related to the given body forces and surface traction, and w 0 and α 0 represent the initial data.Lastly, M , N , and K are bounded linear operators related to the tangential and normal trace operators.The Cauchy problem (1.1b)-(1.1c) is called a hemivariational inequality if φ ≡ 0 and variational inequality if j • ≡ 0.Moreover, a solution to (1.1) is understood in the mild sense.
The main purpose of this paper is to extend the results from [19,29] to prove wellposedness of (1.1) with applications to rate-and-state frictional contact problems.We prove that the pair (w, α) is a solution to (1.1) in the sense of Definition 1.1 and that the flow map depends continuously on the initial data.The problem setting is motivated by [26,29,33,35], and the techniques have taken inspiration from [20].
1.1.Former well-posedness results.Special cases of (1.1) have been investigated in literature.The recent work [19] is closest to our setting.They prove well-posedness for an ordinary differential equation coupled with a variational-hemivariational inequality with applications to viscoplastic material and viscoelasticity with adhesion.In fact, if we let φ be independent of M w in its third argument and relax the more generalized structure of φ (see Remark 3.2 for more details), then (1.1) reduces to the problem studied in [19].However, keeping the dependence of M w in φ and a generalized structure of φ (see Remark 3.2) allows us to include applications with a new structure of the friction coefficient.On the other hand, neglecting α and M w in φ and α in j • , existence and uniqueness is provided in [35,Section 10.3].If we let φ ≡ 0 and j • be independent of α and S j w, existence and uniqueness was proved in [14, Section 6].
In the quasi-static case tackled in [29] with j • ≡ 0 and a simplified structure of φ (see Remark 3.2), they proved existence and uniqueness of the solution pair by an implicit method, where they rewrite (1.1a) to only depend on w.However, the setting of [29] is not applicable in our case as the inertial term restricts the space-time regularity for w.We refer to [19, p.2] for further discussion.1.2.Physical setting.A mathematical model in contact mechanics needs several relations: a constitutive law, a balance equation, boundary conditions, interface laws, and initial conditions.The constitutive laws help us describe the material's mechanical reactions (stress-strain type).In most cases, constitutive laws originate from experiments, though they are verified to satisfy certain invariance principles.We refer to [13,Chapter 6] for a general description of several diagnostic experiments which provide the needed information to construct constitutive laws for specific materials.The interface laws are prescribed on the possible contact surface.We refer to the interface laws in tangential direction as friction laws and in normal direction as contact conditions.The mathematical treatment of these problems gives rise to the variational-hemivariational inequalities of the form (1.1b)- (1.1c) where we put appropriate constraints on the operators to fit the applications of interest.
We are mainly interested in studying frictional problems with the following dependencies: One application with the dependencies seen in (1.2) is a memory-dependent friction coefficient (see, e.g., [24,Section 5.3]), which is also referred to as rate-and-state friction law.This is modelled via an ODE, where the state variable α tracks information of the contact surface using the slip rate | uτ (t)| found from solving (1.1b)-(1.1c)and then updates the friction coefficient.Under certain constraints we may consider α as the surface temperature or humidity on the contact surface.
In [29], they assume that µ(| uτ (t)|, α(t)) is bounded and Lipschitz with respect to both arguments.Additionally to the boundeness assumption on the friction coefficient, [19] considers applications in the frictionless setting and µ = µ(u ν (t)).Our framework is therefore an extension of the frameworks in [19,29], where we need to use different techniques to prove the well-posedness of (1.1) in the sense of Definition 1.1.Lastly, µ = µ(| uτ |) is covered in [23,Section 6.3 and 8.1] (see also [14, p.185-187]).A discussion on many different friction models can be found in [39].1.3.Contributions and outline.The novelties of this paper are: • Well-posedness of (1.1) in the sense of Definition 1.1, where the proof is based on an iterative decoupling approach that directly gives rise to a numerical method.• A more complicated structure of φ (see Remark 3.2) that allows for a larger set of conditions on the contact surface.• Two new applications with Coulomb's friction.One contact problem is with normal compliance, and the latter is with normal damped response.The general framework allows a new structure for the friction coefficient that can be unbounded.
The paper is organized as follows.In Section 2, we introduce the function spaces and some basics of nonsmooth analysis in order to better understand the problem setting.In Section 3, we present our problem statement and the assumptions on the data.The section ends with our main result, Theorem 3.3, that summarizes the well-posedness of (1.1) in the sense of Definition 1.1.The proof of the theorem is presented in Section 5 utilizing a preliminary result stated in Section 4. Next, two applications fitting our framework will be introduced in Section 6.1-6.2.In Section 6.3, we introduce an application motivated by earth sciences.Lastly, in Appendix A, we include remarks on the assumptions needed in the proof of Theorem 3.3, 6.3, and 6.8.Appendix B-G contains proofs of results that are similar to ones found elsewhere but needed throughout the paper.
1.4.Notation.We now present some notations that will be used in this paper.
• Let d denote the dimension.In the applications, d = 2, 3.
• A point in R d is denoted by x = (x i ), 1 ≤ i ≤ d.
• S d denotes the space of second order symmetric tensors on R d .
• Ω ⊂ R d is a bounded open connected subset with a Lipschitz boundary Γ = ∂Ω.We split Γ into three disjoint parts; Γ D , Γ N , and Γ C with meas(Γ D ) > 0, meas(Γ C ) > 0, i.e., nonzero Lebesgue measure, but Γ N is allowed to be empty.• In the applications, Ω is the reference configuration of a viscoelastic deformable body sliding on a foundation.Moreover, Γ D denotes the Dirichlet boundary, Γ N the Neumann boundary, and Γ C is the contact boundary.• ν denotes the outward normal on Γ.
• C ∞ c (Ω) denotes the space of infinitely differentiable functions with compact support.• We will denote c as a positive constant, which might change from line to line.
• Let h(t) be a function, then we denote the time derivative of h(t) by ḣ(t) and the double time derivative as ḧ(t).Assuming here that h(t) has enough regularity such that it makes sense to take the time derivative of it twice.• ( X, ∥•∥ X ) and ( Ỹ , ∥•∥ Ỹ ) denote arbitrary (separable reflexive) Banach spaces.For the convenience of the reader, this notation will be used when introducing general theory.If the theory is only needed for a specific (separable reflexive) Banach space, we write the specific space.• The dual space of X will be denoted by X * .
• 2 X * denotes the set containing all subsets of X * .
• The dual product between the spaces X and X * will be denoted by ⟨•, •⟩ X * × X .
• In the particular case, when X is an inner product space, we denote •, • X as its inner product.3.4.4]).• For simplicity of notation, the dual product between V * and V is denoted by ⟨ ) denotes the set of all bounded linear maps from X into Ỹ .
• We denote the operator norm of the operators M : V → U , N : V → X, and

Function spaces and basics of nonsmooth analysis
In this section, we present the function spaces and fundamental results.For further information, we refer to standard textbooks, e.g., [4,10,25,31].
2.1.Sobolev spaces.This section defines the solution spaces and the usual Sobolev spaces, which will become useful in Section 3 and in the applications, i.e., Section 6.We define The associated norms will be denoted ∥•∥ L p (Ω;R d ) and ∥•∥ L p (Ω;S d ) , respectively.For p = 2, (2.1)-(2.2) are Hilbert spaces with the canonical inner products ).Moreover, we define the spaces With abuse of notation, the trace of functions v ∈ H 1 (Ω; R d ) on Γ will still be denoted by v.For the displacement, we use the space As a consequence of meas(Γ D ) > 0, it follows by Korn's inequality, i.e., ∥ε(v)∥ L 2 (Ω;S d ) ≥ c∥v∥ H 1 (Ω;R d ) (see, e.g., [17,Lemma 6.2]), that V is a Hilbert space with the canonical inner product Here, ε : We denote the associated norm on V by ∥•∥ V .Moreover, if σ is a regular function, say σ ∈ C 1 ( Ω; S d ), the following Green's formula holds We also need the following trace theorem from [1, Theorem 4.12].Theorem 2.1.Let Ω ⊂ R d be bounded with Lipschitz boundary Γ and Γ C ⊂ Γ be such that meas(Γ C ) > 0. Then there exists a linear continuous operator γ : We denote X = ℓ i=1 Xi as a Cartesian product space, for some ℓ ∈ Z + , where ( Xi , ∥•∥ Xi ) are normed spaces for i = 1, . . ., ℓ.Then, X is equipped with the norm Equivalently, we may equip X with the norm Xi for all v i ∈ Xi , i = 1, . . ., ℓ.
Definition 2.2.Let X be a Banach space, and T > 0. The space L p (0, T ; X), 1 ≤ p ≤ ∞, consists of all measurable functions v : [0, T ] → X such that With the usual modifications for L ∞ (0, T ; X).For brevity, we use the standard short-hand notation We also introduce the solution space We denote the space of continuous functions defined on [0, T ] with values in X by The next proposition can be found in, e.g., [7,Proposition 3.4.14]or [31,Lemma 7.3] and will help us provide estimates.
We lastly introduce the following Bochner space needed in the applications, i.e., in Section 6: Definition 2.4.Let V be a real Banach space, then the Bochner space W 1,2 (0, T ; V ) consists of all functions u ∈ L 2 T V such that u exists in the weak sense and belongs to L 2 T V .The space W 1,2 (0, T ; V ) is equipped with the norm 2.3.Generalized gradients.Let X be a reflexive Banach space.In contact mechanics, we are often interested in contact conditions of the form ζ ν ∈ ∂h(u ν ), where ζ ν represents an interface force, u ν = u • ν the normal component of the displacement, and ∂h(u ν ) being the Clarke subdifferential of h defined as below.
Definition 2.5.Let h : X → R be a locally Lipschitz function.The generalized (Clarke) directional derivative of h at x ∈ X in the direction v ∈ X, denoted h • (x; v), is defined by Moreover, the subdifferential in the sense of Clarke of h at x, denoted ∂h(x), is a subset of X * of the form Remark 2.6.To say that a function h : X → R is locally Lipschitz on X means that h(x) is Lipschitz continuous in a neighborhood of x ∈ X.
Proposition 2.8.Let X be a Banach space and h : X → R be locally Lipschitz on X.
The proof of the above proposition is found in [5,Proposition 2.2.7].Lastly, to show that (w, α) is indeed a solution to Problem 1 (see Section 3), we require the following result found in [23,Lemma 3.43].
Lemma 2.12.Let X and Ỹ be two Banach spaces and h : X × Ỹ → R be such that (1) h(•, ỹ) is continuous on X for all ỹ ∈ Ỹ .
(3) There exists c > 0 such that for all ζ ∈ ∂h(x, ỹ) we have where ∂h denotes the generalized gradient of h(x, •).Then h is continuous on X × Ỹ .

Problem statement and main result
In this section, we first introduce the problem and then present the main result.
3.1.Problem statement.Let (V, H, V * ) be an evolution triple, and U , X, Y , Z real separable reflexive Banach spaces, with the other function spaces defined in Section 2.2.We only seek a solution of (1.1) in the sense of Definition 1.1.We are therefore interested in the following evolutionary differential variational-hemivariational inequality: T and α ∈ C([0, T ]; Y ) such that We require the following assumptions on the operators and data: V for all v i ∈ V , i = 1, 2, a.e.t ∈ (0, T ).
T X is such that (i) S φ is a history-dependent operator, i.e., We also assume the following regularity on the source term and initial data: Lastly, we require the following smallness-condition: Remark 3.1.Similar assumptions can be found in, e.g., [19,20,29,33,35].The same type of condition as H(j)(iii) is found in, e.g., [21,22].If j 0 is independent of α and S j w in (1.1b), we may relax the assumption H(j)(iii), and H(j)(ii) and Proposition 2.8 are enough.
We make a brief remark on the assumptions in Appendix A.
3.2.Main result.We will now state the main result, i.e., Theorem 3.3; the first part is an existence and uniqueness result, and the latter provides that the flow map depends continuously on the initial data.The proof of Theorem 3.3 is deferred to Section 5 after the preparation in Section 4.
T ⊂ C([0, T ]; H) and α ∈ C([0, T ]; Y ) is a unqiue solution to Problem 1.(b) Moreover, there exists a neighborhood around (w 0 , α 0 ) so that the flow map F : , we obtain global time of existence, i.e., the existence of a solution holds for any finite time T > 0.
Remark 3.4.The theorem can easily be extended to include more than three historydependent operators without needing any additional assumptions other than the once put on R, S φ , and S j , i.e., H(R), H(S φ ), and H(S j ), respectively.

3.3.
Strategy of the proof of Theorem 3.3.The proof of the theorem is divided into six steps.In the first step, we introduce an auxiliary problem to Problem 1, calling this Problem 3. Specifically, we fix five of the functions in (3.1b) and leave (3.1a) intact.We recast the auxiliary problem as a differential inclusion (introduced in Section 4) and use existing results to prove that Problem 3 has a unique solution (Step 1-4).Next, we define an iterative scheme for Problem 1 using Problem 3.This iterative scheme decouples (3.1a) and Problem 3 at each step.Then, we study the difference between two successive iterates and show that these iterates are Cauchy sequences.We then pass to the limit to show that the iterative scheme converges to Problem 1 (Step 5).Finally, we show that the flow map continuously depends on the initial data (Step 6).

Preliminary result
Before proving Theorem 3.3, we present an existence and uniqueness result for a differential inclusion problem see, e.g., [2].The forthcoming result will be used to prove existence of a solution to an auxiliary problem of (3.1b)-(3.1c) in Problem 1.To utilize this result, we need to introduce a differential inclusion which we relate to the auxiliary problem of (3.1b)- (3.1c).This will be made clear in Step 1-2 in the proof of Theorem 3.3.We begin by introducing the inclusion problem.
For clarity on how to work with the preceding problem, we include the following definition: for a.e.t ∈ (0, T ) with w(0) = w 0 .
In the preliminary existence and uniqueness result, we consider the following assumptions: We further assume that the operator A : (0, T ) × V → V * satisfies H(A), and the source term f and the initial data w 0 satisfy (3.3a).Additionally, we assume that the following smallness-condition holds: Theorem 4.2.Assume that H(A), H(ψ), (3.3a), and (4.1) hold.Then Problem 2 has a unique solution w ∈ W 1,2 T in the sense of Definition 4.1 for any T > 0. The theorem was proved in [20,Theorem 3].This result will be used in Step 2 of the proof of Theorem 3.3.

Proof of Theorem 3.3
With the preparation in Section 2-4, we proceed to the proof of Theorem 3.3.For the convenience of the reader, the proof is established in several steps, and some of the proofs have been moved to the appendix.We recall that the function spaces are defined in Section 2.2.
Step 1 (Auxiliary problem to the evolutionary hemivariational-variational inequality T X be given, then we define an auxiliary problem to (3.1b)-(3.1c) in Problem 1.
Remark 5.1.A glance at Problem 3 and (3.1b) lets us see that the auxiliary problem keeps ξ = Rw, α (still denoted by α), η = S φ w, g = w, and χ = S j w known in contrast to (3.1b).We find it worth mentioning that we use the subscripts on w to emphasize that a solution This also helps to distinguish between a solution to Problem 1 and a solution to Problem 3.
Step 2 (Existence of a solution to Problem 3).
T X be given.We wish to utilize Theorem 4.2 in order to prove that Problem 3 has a solution.We therefore define the functional for all v ∈ V , a.e.t ∈ (0, T ).Verification of the hypothesis of Theorem 4.2 follows from the same approach as the first part of the proof in [20,Theorem 5].We investigate the assumption H(ψ) and the smallness-condition (4.1), as there are some modifications in comparison to [20,Theorem 5].Keeping Proposition 2.9 in mind, we only comment on the changes and leave the reader to visit [20,Theorem 5] for a detailed verification.Using (3.2), we find that H(ψ) holds with c 0 and m ψ = m j ∥N ∥ 2 .This, together with the smallness-condition (3.4) leads to (4.1).Thus, we conclude by Theorem 4.2 that there exists a solution w αξηgχ ∈ W 1,2  T of Problem 2 with ψ αξηgχ defined in (5.1).It remains to show that the existence of a solution to Problem 2 implies the existence of a solution to Problem 3.This is a consequence of Definition 2.5 and 2.10, and basic results of the generalized gradients, see, e.g., [14, Theorem 3.7, Proposition 3.10-3.12],where they have summarized these properties, and [35,Lemma 7,p.124].A more detailed approach to this part can be found in, e.g., [14,Section 6] or [35, p.190-192].
Step 3 (Uniqueness of a solution to Problem 3).Uniqueness is immediate from the proof in [35,Theorem 98] with α j = m j ∥N ∥ 2 and the smallness-condition (3.4), i.e., m A > m j ∥N ∥ 2 .
Step 4 (Estimate on the solution to Problem 3, that is, We now find an estimate on the solution to Problem 3, which will come in handy later: Proposition 5.2.Under the assumptions of Theorem 3.3, for given T X, let w αξηgχ be a solution to Problem 3.Then, there exists a constant c > 0 independent of w αξηgχ such that and Step 5 (Scheme for the approximated solution to Problem 1).and w T be known.We construct the approximated solutions , where (w n , α n ) is a solution of the scheme: for all v ∈ V , a.e.t ∈ (0, T ), and (5.3b) w n (0) = w 0 . (5.3c) Step 5.1 (Existence and uniqueness of ( We establish existence and uniqueness by induction on n.First, applying Minkowski's inequality, Young's inequality, integrating over the time interval (0, t ′ ) ⊂ (0, T ), and lastly applying the Cauchy-Schwarz inequality to hypothesis H(R), H(S φ ), and H(S j ), respectively, yield for all t ′ ∈ [0, T ].We combine (5.4)-(5.6)with the estimates (5.2a)-(5.2b) in Proposition 5.2 for , and χ = S j w n−1 .This implies for a.e.t ∈ (0, T ).We observe that by Minkowski's inequality, H(G)(ii) and H(M N K) for a.e.s ∈ (0, t) ⊂ (0, T ).Accordingly, the Cauchy-Schwarz inequality and H(G)(iii) implies for a.e t ∈ (0, T ).By Grönwall's inequality (see, e.g., [10]), we have (5.9a) and from Young's inequality for k ≥ 1/2.We will show the uniform bound by induction on n.
For n = 1, with the initial guesses; w 0 = w 0 and α 0 = α 0 , (5.7a)-(5.7b)becomes From the smallness-assumption (3.4), it follows that We next define the complete metric space T .We verify that α 1 is indeed a solution to (5.3c) for n = 1 in the next lemma.
T be a solution of (5.3a)-(5.3b)with w 0 = w 0 ∈ V and α 0 = α 0 ∈ Y .Under the assumptions of Theorem 3.3, the operator Λ : X T (a) → X T (a), defined by (5.11), has a unique fixed-point, i.e., there exists a constant 0 ≤ L < 1 such that . The proof of Lemma 5.3 is moved to Appendix C as it follows from the standard ODE arguments combined with the estimate (5.10), and the assumptions H(G) and H(M N K).
Next, investigating n = 2.This implies T V .From the estimate (5.9a) for n = 1, we have that small enough for some k ≥ 1/2.From the smallness-condition (3.4) and the choice of T > 0, we have that √ Gathering the above and using (5.10) implies Moreover, verifying that α 2 ∈ C([0, T ]; Y ) is indeed a solution to (5.3c) for n = 2 follows by the same approach as for n = 1.The induction step follows the same procedure as for n = 2. Consequently, (w n , α n ) ∈ W 1,2  T × C([0, T ]; Y ) is the approximated solution of (5.3a)-(5.3c).Further, we obtain the following uniform bound Step 5.2 (Convergence of the approximated solution).We first show that . This is summarized in the proposition below.
In addition, {Rw n } n≥1 , {S φ w n } n≥1 , and {S j w n } n≥1 are Cauchy sequences in L 2 T V * , L 2 T X and L 2 T X, respectively.
Remark 5.5.To cover the case where we obtain global time of existence when β 1φ = β 4φ = β 5φ = β 6φ = β 7φ = 0, the proof needs to be slightly modified.This case is included in Corollary 5.6.
The proof of Corollary 5.6 can be found in Appendix D.

By
Step 5.2, we have that (w n , α n ) → (w, α) strongly in U when n → ∞.Consequently, B 1 and B 2 must at least satisfy the estimate While for B 3 , we need the continuity of the flow map with respect to (5.3a)-(5.3c).We add the two inequalities and choose v = w n k (t) for a.e.t ∈ (0, T /2) and k T /2 × C([0, T /2]; Y ) be given, and for a.e.t ∈ (0, T /2).To find the desired estimates, we need the following lemma.
The proof of Lemma 5.7 is postponed to Appendix E. From (5.12) and (5.13), the estimate (5.28) becomes In a similar manner as we obtained (5.9b), we see from (5.27c), a standard Grönwall argument, the Cauchy-Schwarz inequality, and Young's inequality that By induction of n, it follows the same procedure as in Step 5.1 that for all n ∈ Z + .Consequently, we may choose a δ > 0 in (5.25a) to obtain (5.25b).
Step 7 (Proof of Theorem 3.3).We now have all the tools to prove the main theorem.

Viscoelastic frictional contact problems
We will present two applications to frictional contact; the first considering contact with normal compliance, and the second contact with normal damped response.Moreover, in Section 6.3, we introduce a first-order approximation of a rate-and-state friction law that is covered by our framework.Let u : Ω × [0, T ] → R d denote the displacement, σ : Ω × [0, T ] → S d the stress tensor, and α : Γ C × [0, T ] → R the external state variable.In addition, f 0 denotes the body forces, f N the surface traction, and ρ the density.We let the spaces H = L 2 (Ω; R d ), Q = L 2 (Ω; S d ), V , and W 1,2 T be defined by (2.1), (2.2), (2.3), and (2.5), respectively.We refer to Section 2.1-2.2 for further definitions of the function spaces.Further, let X = L 4 (Γ C ) and U = L 4 (Γ C ; R d ).Let γ ν : V → X denote the normal trace operator, and γ τ : V → U denote the tangential trace operator.It then follows by Theorem 2.1 that γ τ and γ ν are well-defined for d = 2, 3.For all v ∈ V , we let v ν = γ ν v = v • ν denote the normal components on Γ, and v τ = γ τ v = v − v ν ν the tangential components on Γ.Similarly, let σ ν = (σν) • ν, and σ τ = σν − σ ν ν be the normal and tangential components of the tensor σ on Γ, respectively.6.1.Dynamic frictional contact problem with normal compliance.In this section, we present a system of equations describing the evolution of a viscoelastic body in frictional contact with a foundation.Viscoelastic contact problems with normal compliance and friction are discussed in, e.g., [33,Section 8.3].The normal compliance condition is used as an approximation of the Signorini non-penetration condition.More on this can be found in [13,Chapter 5], [17,Chapter 11] and [34].We wish to study the following problem: on Ω × (0, T ) (6.1a) on Ω × (0, T ) (6.1b) with the initial conditions u(0) = u 0 , u(0) = w 0 on Ω (6.1i) In the above problem, (6.1a) is a general viscoelastic constitutive law, where A is a viscosity operator, B an elasticity operator, and C is referred to as a relaxation tensor.We note that Aε( u) and Bε(u) are short-hand notation for A(x, ε( u)) and B(x, ε(u)), respectively.Moreover, (6.1b) is a momentum balance equation, (6.1c) denotes the Dirichlet boundary conditions, and (6.1d) the traction applied to the surface.The equation (6.1e) is a contact condition, where p is a prescribed function describing the penetration condition.Next, (6.1f)-(6.1g)denotes a generalized Coulomb's friction law, and (6.1h) describes the evolution of the external state variable, see Section 1.2 for a discussion on this equation.Lastly, (6.1i)-(6.1j)are initial conditions.We wish to investigate (6.1a)-(6.1j)under the following assumptions: x ∈ Ω. (iii) There exists m A > 0 such that (A(x, ε 1 ) x ∈ Γ C .(iii) There exists p * > 0 such that p(x, r) ≤ p * for all r ∈ R, a.e.x ∈ Γ C .
We refer the reader to Appendix A for a discussion on applications under these assumptions.
6.1.1.Variational formulation.We find a formal derivation of the variational formulation of Problem 4, i.e., assuming sufficiently regular functions, as we only are interested in a mild solution (see Definition 1.1).We refer to, e.g., [33,Section 5.2] for a more detailed derivation, especially how to deal with the contact conditions.Inserting (6.1b) into Green's formula (2.4) yields for all v ∈ V , a.e.t ∈ (0, T ).From the terms on Γ C , we deduce for all v ∈ V , a.e.t ∈ (0, T ).We write the above inequality slightly more compactly.We observe that the map t → Γ N f N (t) • vda + Ω f 0 (t) • vdx is linear and bounded in V .Consequently, the Riesz representation theorem implies the existence of f (t) ∈ V * such that As mentioned, we are interested in a mild solution of (6.1h) (see Definition 1.1), so we integrate (6.1h) over the time interval (0, t) and use the initial condition (6.1j) to obtain this equation on the desired form.We may now formulate a variational inequality of Problem 4.
6.1.2.Proof of Theorem 6.3.Our aim is to use Theorem 3.3 to prove Theorem 6.3.Our first task is to rewrite Problem 5 in the same form as Problem 1.Then, we will verify the hypothesis of Theorem 3.3.
Remark 6.6.Since Problem 5 is contained in Problem 6, it suffices to show well-posedness of Problem 6.
Lemma 6.7.Under the assumptions of Theorem 6.3, the hypothesis of Theorem 3.3 holds for (6.8a)-(6.8g).Here, To maintain the flow of the article, the proof of Lemma 6.7 is placed in Appendix F. We are now ready to prove Theorem 6.3.
6.2.2.Proof of Theorem 6.8.We use the same approach as in Section 6.1, meaning that we will use Theorem 3.3 to prove Theorem 6.8.But to use this theorem, we need to first rewrite Problem 8 into the same form as Problem 1. Then we will verify the hypothesis of Theorem 3.3.
and f : (0, T ) → V * as in (6.8a)-(6.8d)and (6.5), respectively.We choose M : V → U and N : V → X to be as in (6.8e) and K ≡ M .Moreover, we define the functional and the functional j : (0, T ) × X → R by j(t, w) = Γ C j ν ( w)da for w ∈ X, a.e.t ∈ (0, T ).(6.19b)The problem is then on the following form.
To see that it suffices to prove existence of a solution to Problem 9 in order for Problem 8 to have a solution, we introduce the following result, which is of a similar form as found in [35,Lemma 8,p.126] (see also [23,Theorem 3.47]).The result will also be useful to prove uniqueness.Corollary 6.11.Assume that H(j ν ) holds.Then, the functional j defined by (6.19b) has the following properties: ν ( w; v)da.Lemma 6.12.Under the assumptions of Theorem 6.8, H(φ) holds for φ defined by (6.19a) Moreover, j defined by (6.19b) satisfies H(j) The proof of Lemma 6.12 is placed in Appendix G.We may now prove Theorem 6.8.
Formal augmentations of (6.26a)-(6.26b).For simplicity in notation, we let y = α(t) and r = uτ (t).We wish to have the same order of error for the approximation of (6.23a) and (6.24a) as in (6.25).For (6.26a), we are considering ĉ = −1.We directly obtain Next, considering (6.26b), we are interested in the approximation (6.25) for ĉ = b a .We let g = 1 2v 0 e µ 0 +by a and f = 1 2v 0 e µ 0 +bα 0 a (1 + b a (y − α 0 )).Then, by the mean value theorem with z ∈ (|r|f, |r|g) where f ≤ g.So, we have We also note that , where f ∼ c if y is close to α 0 .Consequently, we have the same order of error as in (6.25) as desired.□ Remark 6.13.Above, we gave formal arguments that our model is a first-order expansion around the initial value α 0 .We will now investigate if our approximated model has the same qualitative behavior as the original model.Following [30], the key restriction on G is that when the slip rate | uτ | is constant, the equation α = G(α, | uτ |) has a stable solution that evolves monotonically towards a definite value of α, denoted α * = α * (| uτ |), at which This holds true if ∂G ∂α < 0, which is easily verified for the approximation of G given by (6.26a).
For the friction term, again following [30], we seek ∂µ ∂| uτ | > 0 and ∂µ ∂α > 0. The first condition is consistent with the experimental observations and holds if α is close to α 0 .The latter condition agrees with the established convention for the state variable; larger values mean greater strength.This is also consistent with the usual interpretation of α as a measure of contact maturity and the fact that more mature contact is stronger.Consequently, this shows that qualitatively our approximate model has the same behavior as the original model problem.One can also see in Figure 2-3 that there is a neighborhood where the first-order approximations (6.26a)-(6.26b)are close to the original equations for the values used in Table 1.
Remark 6.17.We observe from (6.28) and (6.29) that either is small enough, or we must compensate by adding more viscosity.
Remark 6.18.In [27,28], they study (6.23)-(6.24) in a time-discrete setting with S φ w ≡ constant (the normal stresses are constant -referred to as Tresca friction), φ being independent of M w in its third argument, relaxing the structure of φ (see Remark 3.2), and putting j • ≡ 0 in Problem 1.
Step i (The operator Λ is well-defined on X T (a)).Indeed, Λα 1 ∈ X T (a).We first prove that for given α 1 ∈ X T (a), then ∥Λα 1 ∥ L ∞ T Y ≤ a.We apply Minkowski's inequality to (5.11), then we utilize the estimate (5.8) with α n = α 1 and w n = w 1 together with H(M N K) and H(G)(iii).This yields for a.e.t ∈ (0, T ).From Hölder's inequality and the Cauchy-Schwarz inequality, we obtain We then see from Young's inequality and the estimate (5.10) that Choosing a such that it provides the desired upper bound concludes this part.

Figure 1 .
Figure 1.A standard illustration of a sliding block.

Problem 4 .
Find the displacement u : Ω × [0, T ] → R d and the external state variable α