Modeling and analysis of a contact problem for a viscoelastic rod

We consider a nonlinear viscoelastic rod which is in contact with a foundation along its length and is in contact with an obstacle at its end. The rod is acted up by body forces and, as a result, its mechanical state evolves. Our aim in this paper is twofold. The first one is to construct an appropriate mathematical model which describes the evolution of the rod. The second one is to prove the weak solvability of the problem. To this end, we use arguments on second-order inclusions with multivalued pseudomonotone operators. Mathematics Subject Classification. 49J40, 47J20, 47J22.


Introduction
Vibration processes and objects or media arise in engineering and everyday life. The vibration of a bridge, of a window, of a spring or an automotive platform with a low-power active suspension represents only four simple examples, among others. Vibration produces sounds and could lead to resonance phenomena which can have destructive effects on the mechanical systems. For this reason, there is a considerable interest in the modeling, analysis and numerical simulation of such processes. The literature in the field is extensive.
In the engineering literature, the vibrations are very often studied by considering mechanical systems based on finite collections of masses, springs, dampers, and rods in frictional or frictionless contact. The analysis of such systems leads often to nonlinear differential equations or inclusions of second order. The nonlinearity arises either from the nonlinearity of the constitutive laws involved into the model or from the contact conditions.
In this work, we study the evolution of a simple mechanical system, consisting of a vibrating rod in contact with a support along its length, the so-called foundation. The interaction between the rod and the foundation is modeled with specific interface conditions. The rod is clamped at one end and, in addition, is in contact with an obstacle, at the other end. The contact is modeled with a subdifferential condition involving a possible nonconvex potential. We derive a mathematical model describing the above physical setting which leads to a new and nonstandard problem, expressed in terms of a second-order differential inclusion. Solving this inclusion, which involves strongly nonlinearities, represents the main trait of novelty of this paper. In this way, we show how one can apply the rapidly developing theory of differential inclusions to describe contact processes with rods. We do it in a simple setting that avoids some complications related to higher dimensions, making the mathematical approach more transparent. Moreover, these simple settings are of importance since they allow for easier experimental measurements and identification of the system parameters. These parameter functions then may be used in more realistic applications.

Preliminaries
In this section, we briefly present the notation and some preliminary material to be used later in this paper. More details on the material presented below can be found in the books [10,11,19,20,26].
First, we precise that all linear spaces used in this paper are assumed to be real. Unless it is stated otherwise, below in this section we denote by X a normed space with the norm · X , by X * its topological dual, and ·, · X * ×X will represent the duality pairing of X and X * . The symbol 2 X * is used to represent the set of all subsets of X * . We start with definition of the generalized directional derivative and the subdifferential in sense of Clarke. Definition 2.1. Let ϕ : X → R be a locally Lipschitz function. The Clarke generalized directional derivative of ϕ at the point x ∈ X in the direction v ∈ X is defined by The Clarke subdifferential of ϕ at x is a subset of X * given by In what follows we introduce the notion of coercivity.

Definition 2.2.
Let X be a real Banach space and A : X → 2 X * be a multivalued operator. We say that A is coercive if either the domain D(A) of A is bounded or D(A) is unbounded and We now proceed with the definition of a pseudomonotone operator in both single valued and multivalued case.
We now recall two important results concerning properties of pseudomonotone operators.
Proposition 2.5. Let X be a reflexive Banach space and A 1 , A 2 : X → 2 X * be pseudomonotone operators.
Theorem 2.6. Let X be a reflexive Banach space and let A : X → 2 X * be a coercive, bounded and pseudomonotone multivalued operator. Then A is surjective, i.e., R(A) = X * .
Let X be a Banach space and T > 0. We introduce the space BV (0, T ; X) of functions of bounded total variation on [0, T ]. Let π denote any finite partition of [0, T ] by a family of disjoint subintervals Let F denote the family of all such partitions. Then for a function x : [0, T ] → X and for 1 ≤ q < ∞, we define a seminorm For 1 ≤ p ≤ ∞, 1 ≤ q < ∞ and Banach spaces X, Z such that X ⊂ Z, we introduce a vector space Then the space M p,q (0, T ; X, Z) is also a Banach space with the norm given by · L p (0,T ;X) + · BV q (0,T ;Z) . Next we recall a compactness result, which will be used in the sequel. For its proof, we refer to [15].
Proposition 2.7. Let 1 p, q < ∞. Let X 1 ⊂ X 2 ⊂ X 3 be real Banach spaces such that X 1 is reflexive, the embedding X 1 ⊂ X 2 is compact and the embedding X 2 ⊂ X 3 is continuous. Then the embedding The following version of Aubin-Celina convergence theorem (see [4]) will be used in what follows.
Proposition 2.8. Let X and Y be Banach spaces, and F : X → 2 Y be a multifunction such that (a) the values of F are nonempty closed and convex subsets of Y ; (b) F is upper semicontinuous from X into Y endowed with a weak topology. Let x n , x: (0, T ) → X, y n , y : (0, T ) → Y , n ∈ N, be measurable functions such that x n (t) → x(t) for a.e. t ∈ (0, T ) and y n → y weakly in L 1 (0, T ; Y ). If y n (t) ∈ F (x n (t)) for all n ∈ N and a.e. t ∈ (0, T ), then y(t) ∈ F (x(t)) for a.e. t ∈ (0, T ).

The model
We consider a viscoelastic rod which occupies, in its reference configuration, the interval (0, L) with L > 0. The rod is attached at its end x = 0 and is in contact with an obstacle at x = L. In addition, it is contact with a reactive foundation along its length that opposes its deformation. The rod is acted up by time-dependent body forces of density f . The physical setting is depicted in Fig. 1.
We are interested in the description of the dynamic evolution of the rod in the physical setting above and in providing the analysis of the corresponding mathematical model. To this end, we denote in what follows by x and t the spatial and the time variables, respectively. Note that x ∈ [0, L] and t ∈ [0, T ], where T represents the length of the time interval of interest. Moreover, for a function G = G(x, t), we use the subscripts x and t for the derivatives with respect to x and t, i.e., Everywhere in this paper, we denote by u = u(x, t) and σ = σ(x, t) the displacement and the stress function, respectively. We also denote by ε = ε(x, t) the deformation function defined by ε = u x .
We turn now to the construction of our mathematical model, which gathers the equation of motion, the constitutive law, the boundary conditions and the initial conditions, that we describe in what follows.
First, the equation of motion of the rod is given by Here ρ = ρ(x) represents the density of mass in the reference configuration and F = F(x, t) represents the total force acting on the rod, i.e., the sum of the applied force and the reaction of the foundation. We assume that the reaction of the foundation has an additive decomposition of the form ψ + ξ, where the functions ψ and ξ will be described below. Therefore Next we assume that where g and h are given nonlinear functions, assumed to be positive for positive argument and negative for negative argument. This restriction guarantees that the forces ψ and ξ are opposite to the velocity and the displacement fields, respectively. Note that assumption (3.3) shows that the force ψ depends only on the velocity field u t which mimics the behavior of a nonlinear viscous damper. Therefore it could be used to model the friction between the rod and the foundation. In contrast, assumption (3.4) shows that the force ξ depends only on the displacement field u, which mimics the behavior of a nonlinear elastic spring. It could be used to model the adhesion between the rod and the foundation. We now gather the Eqs. (3.1)-(3.4) and assume, for simplicity, that ρ ≡ 1. As a result, we obtain the balance equation The next step is to prescribe the constitutive law. We assume that the rod is viscoelastic and its behavior is described with the equation Here η > 0 is a viscosity coefficient, E > 0 represents the Young modulus and p ≥ 2 is a given coefficient. Note that (3.6) represents a nonlinear version of the well-known Kelvin-Voigt linear viscoelastic constitutive law σ = ηε t + Eε.
(3.7) Actually, (3.6) could be recovered from (3.7) by assuming that the viscosity coefficient η depends on the strain rate ε t , i.e., η = η(ε t ) = η|ε t | p−2 . Such kind of dependence is justified from physical point of view since it can be observed to various materials like polymers and pastes, as explained in [24], for instance. In addition, it makes the resulting boundary value problem more difficult from mathematical point of view, since it introduces a strong nonlinearity into the model.
We now replace ε = u x in (3.6) to see that then we substitute this equality in the balance Eq. (3.5) to find that We now describe the boundary conditions. First of all, since the rod is fixed in x = 0, the displacement field vanishes there, i.e., u(0, t) = 0 for all t ∈ [0, T ]. (3.10) Next we assume that the rod is in contact at x = L with an obstacle and we model the contact with a subdifferential inclusion of the form Here j is a prescribed possible nonconvex function and ∂j represents its Clarke subdifferential. Examples of contact conditions which can be cast in the form (3.11), can be found in [13,19,25], for instance. Here we restrict ourselves to recall that they include the so-called normal damped response condition and various viscous-type contact conditions. We combine now (3.8) and (3.11) to deduce that Finally, we prescribe the initial displacement and the initial velocity of the rod, i.e., where u 0 and v 0 are given functions. We are now in a position to formulate the mathematical model which describes the dynamic evolution of the rod, in the physical setting described above.
The existence of weak solution of Problem P will be provided in Sect. 4. It is based on technique used recently in [5]. Here we mention that the main difficulty in the analysis of Problem P arises in the nonlinearities of this problem, which appear both in the second-order Eq. (3.9) and in the multivalued boundary condition (3.12). We also note that, if the displacement function u represents a solution to Problem P , then the corresponding stress field can be easily computed by using the constitutive law (3.8).

Main result
In this section, we state our main result in the study of Problem P , Theorem 4.9. Here and below, we take 2 ≤ p < ∞ and 1 < q ≤ 2 satisfying 1 p + 1 q = 1. We use notation R + for a set of nonnegative real numbers. We impose the following assumptions on the functions g, h, j and f .
We denote by ·, · W * ×W and ·, · V * ×V the duality in spaces W and V , respectively. The inner product in H is denoted by (·, ·) H . Identifying H with its dual, we remark that the above spaces form the evolution ZAMP Modeling and analysis of a contact problem Page 7 of 21 127 with all embeddings dense and continuous. We also recall that the embeddings W ⊂ H and V ⊂ H are compact. It is well known that V ⊂ C(0, L) and the following inequality holds: Here We also define the functional F : We remark that the weak formulation used in Definition 4.1 can be obtained from equation in Problem P by multiplying it by a test function v ∈ W and using an integration by parts formula.
In what follows we will deal with the existence of weak solutions of Problem P . To this end, we define the multifunction M : R → 2 R by M (s) = ∂j(s) for all s ∈ R. We also define the operator γ : W → R given by γv = v(L) for all v ∈ W . We recall that W ⊂ C(0, L) and v(L) is understood as a value of a continuous representant of v ∈ W at L. Thus operator γ is well defined. We use notation γ := γ L(W,R) . Next we formulate the following auxiliary problem.
Problem P. Find u ∈ W with u t ∈ W and u tt ∈ W * such that Next we state and prove several properties of the operators A, B, C, M and γ which will be used in Sect. 5.

Lemma 4.3. If the assumptions H(g) hold, then the operator
Proof. Condition (i) follows from H(g)(iii) and (4.2). Condition (ii) follows directly from the definition of A and H(g)(ii). Finally for the proof of (iii), we refer to the proof of Proposition 27.9 in [26].
The proof of Lemma 4.4 follows directly from the definition of B.

Lemma 4.5. If the assumptions H(h) hold, then the operator
Proof. We start with the proof of (i). Using H(h)(i) and Hölder inequality, and (4.2), we calculate Thus it follows that Cv W * ≤ c h L(L 1 q + v 2 q V ) and (i) holds with β C = c h L max{1, L 1 q }. Now we prove (ii). Let u, v ∈ V and w ∈ W . Applying again Hölder inequality and (4.2), we get and, in a consequence, it follows that Since function h is nondecreasing, using (4.1), we obtain Moreover using again Hölder inequality, we obtain Thus condition (ii) holds with the function C given by C(s) = L 1 2q +1 h( √ Ls) for all s ∈ R + . It remains to show (iii). Let v n → v weakly in V . It is enough to show that Cv n → Cv in W * for a subsequence. Since the embedding V ⊂ H is compact then, for a subsequence again denoted v n , we have v n → v in H. Thus by (ii), it follows that Cv n → Cv in W * . This completes the proof. The proof of Lemma 4.6 follows directly from the properties of Clarke subdifferential (see [10]).

Lemma 4.7. The operator γ : W → R is linear and strongly continuous.
Proof. The linearity of γ is obvious. We also observe, that for all v ∈ C(0, L), we have |γv| = |v(L)| ≤ max x∈[0,L] |v(x)| = v C(0,L) , which means, that γ ∈ C(0, L) * . Let v n → v weakly in W . Since the embedding W ⊂ C(0, L) is continuous, we also have v n → v weakly in C(0, L), so, in particular γv n → γv in R, which completes the proof. The proof of Lemma 4.8 exploits Lemma 4.7 and follows the lines of the proof of Proposition 1.6 in [6] and, therefore, we omit it.
We now impose the following additional assumption on the constants of the problem.
Our existence result in the study of Problem P that we state here and prove in Sect. 5 is the following.

Theorem 4.9. Let the assumptions H(g), H(h), H(j)
, H(f ) and H 0 hold. Moreover assume that either p = 2 or H const holds. Then Problem P admits a weak solution.

The Rothe method
In this section, we study a time semidiscrete scheme corresponding to Problem P. We provide the existence result for the approximate problem, and we study the convergence of its solution to the solution of Problem P, when the discretization parameter converges to 0. In this way, we will prove Theorem 4.9.
The technique presented below is referred as the Rothe method and was already used in many references, including [5,7,8].

Discrete problem
We divide the time interval [0, T ] by means of a sequence {t k } Nn k=0 ⊂ [0, T ] defined as follows t k = kτ n , where τ n = T /N n for k = 0, . . . , N n .
In the above notation, N n denotes the number of time steps in nth division of [0, T ], so we have N n → ∞ and τ n → 0, as n → ∞. For the convenience we will omit the subscript n and write N, τ instead of N n , τ n . We approximate the initial condition u 0 and v 0 by elements of W . Namely, let {u 0 For a given τ > 0 we formulate the following Rothe problem.
In what follows we will study the existence of solution to Problem P τ . To this end, we define an auxiliary multivalued operator T : for all r ∈ R + , y, z, w ∈ W.
The significance of operator T in the study of Problem P τ is explained below.
Remark 5.1. It is easy to observe, that Problem P τ is equivalent with finding a sequence {w k and, for k = 2, . . . , N, w k τ satisfies The following lemmata provide properties of operator T . H(j) hold. Moreover assume that either p = 2 or H const holds. Then, there exists τ 0 > 0 such that for all 0 < τ < τ 0 , operator T (τ, y, z, ·): W → 2 W * is coercive for all y, z ∈ W .

Lemma 5.2. Let the assumptions H(g), H(h) and
Proof. Let y, z ∈ W be fixed. In the whole proof, we will denote by c a positive function, which may change from line to line and may have a various set of arguments. Suppose that u ∈ T (τ, y, z, w), where ZAMP Modeling and analysis of a contact problem Page 11 of 21 127 w ∈ W is given. Then we have, u = 1 τ w + Aw + τ Bw + τ C(y + τ z + τ w) + γ * ξ, where ξ ∈ M (γw). In order to show the coercivity of T , we calculate u, w W * ×W = 1 τ w 2 H + Aw, w W * ×W + τ Bw, w V * ×V + C(y + τ z + τ w), w W * ×W + ξγw.

Lemma 5.3. Let the assumptions H(g), H(h) and H(j) hold. Then operator T is bounded with respect to the last variable.
Proof. The boundedness of T follows directly from Lemma 4.3 (i), Lemmas 4.4, 4.5 (i) and 4.6 (iii).

Lemma 5.4. Let the assumptions H(g), H(h) and H(j) hold. Then operator T is pseudomonotone with respect to the last variable.
Proof. We examine the pseudomonotonicity of each components of T . The operator W w → 1 τ w ∈ W * is pseudomonotone, since it is linear and monotone. The pseudomonotonicity of A is provided by Lemma 4.3 (iii). Operator τ B is pseudomonotone, since it is linear and monotone. By Lemma 4.5 (iii), and the continuity of embedding W ⊂ V , we claim, that the operator C is strongly continuous from W to W * and, in a consequence, it is pseudomonotone. Finally the multivalued term of T is pseudomonotone due to Lemma 4.8. Thus from Proposition 2.5, it follows, that T is pseudomonotone. H(g), H(h) and H(j) hold. Moreover assume that either p = 2 or H const holds. Then there exists τ 0 > 0 such that for all 0 < τ < τ 0 the mapping T (τ, y, z, ·): W → 2 W * is surjective for all y, z ∈ W , i.e., for every f ∈ W * , there exists w ∈ W such that T (τ, y, z, w) f.

Corollary 5.5. Let the assumptions
Proof. The proof is a consequence of Lemmas 5.2-5.4 and Theorem 2.6. Now we are in a position to formulate an existence result for Problem P τ . Theorem 5.6. Let the assumptions H(g), H(h), H(j) and H 0 hold. Moreover assume that either p = 2 or H const holds. Then there exists τ 0 > 0 such that for all 0 < τ < τ 0 Problem P τ has a solution.
Proof. We have to provide the existence of a sequence {w k τ } N k=0 , that is a solution of Problem P τ . First, we define w 0 τ = v 0 τ . By Corollary 5.5, we know that for τ > 0 small enough, operator T is surjective with respect to the last variable, and, in a consequence, there exists w 1 τ that satisfies (5.4). Then we proceed by induction. Suppose that elements w j τ , j = 0, . . . , k −1 are already found for a fixed k = 2, . . . , N. Using again surjectivity of T , we deduce that there exists w k τ ∈ W that satisfies (5.5). Proceeding recursively, we provide existence of the entire sequence {w k τ } N k=0 . Applying Remark 5.1, we state that it is a solution of Problem P τ .

A-priori estimates
In this subsection, we provide a priori estimates for the solution of Problem P τ .
Let where the constant c does not depend on τ .
Proof. We take v = w k τ in (5.1) and obtain By a property of scalar product in Hilbert space, we have By the properties of operator B (see Lemma 4.4), we get By Lemma 4.5 (i), we get Moreover we claim that Finally we estimate the term ξ k τ γw k τ . To this end, we consider two cases. If p = 2, then, analogously to (5.10), we get If p > 2, we use estimate analogous to (5.8) to obtain where the constants c 1 , c 2 , c 3 , c may vary from line to line. Moreover from (5.26), it follows that F τ is bounded in W * . Thus applying (5.39)-(5.42) to (5.38), we obtain (5.36).
Next we pass to the proof of (5.37). Taking into account (5.31), it is enough to estimate the seminorm w τ BV q (0,T ;W * ) . Since the function w τ is piecewise constant, the seminorm will be measured by means of jumps between elements of sequence The proof of Lemma 5.9 follows the lines of the proof of Lemma 1 in [15] and exploits Lemma 4.3. We remark that every operator, which satisfies H(A), is said to be pseudomonotone with respect to the space M p,q (0, T ; W, W * ). Proof. Let the sequence {v n } be bounded in L ∞ (0, T ; V ) and let v n → v in L 1 (0, T ; H). It follows from Lemma 4.5 (ii) that Since the function C is nondecreasing, and {v n } is bounded in L ∞ (0, T ; V ), we conclude that for a.e. t ∈ (0, T ) Combining it with (5.45), we have Cv n − Cv q W * ≤ c v n − v L 1 (0,T ;H) . Since v n → v in L 1 (0, T ; H), we found that Cv n − Cv W * → 0, which completes the proof. Proof. Let the sequence {v n } be bounded in M p,q (0, T ; W, W * ). We recall that W ⊂ C(0, L) ⊂ W * , where the first embedding is compact and the second one is continuous. Thus applying Proposition 2.7, we claim that for a subsequence (still denoted by v n ) we have v n → v in L p (0, T ; C(0, L)).