Waves in flexural beams\with nonlinear adhesive interaction

The paper studies the initial boundary value problem related to the dynamic evolution of an elastic beam interacting with a substrate through an elastic-breakable forcing term. This discontinuous interaction is aimed to model the phenomenon of attachement-detachement of the beam occurring in adhesion phenomena. We prove existence of solutions in energy space and exhibit various counterexamples to uniqueness. Furthermore we characterize some relavant features of the solutions, ruling the main effectes of the nonlinearity due to the elasic-breakable term on the dynamical evolution, by proving the linearization property according to \cite{G96} and an asymtotic result pertaining the long time behavior.


Introduction
In a broader sense the term adhesion refers to a physical situation in which two material bodies, during their mechanical evolution, experience a contact interaction which fails in (possibly bounded) regions of space-time and restores after a while. The phenomenon strongly depends on the constitutive properties of the involved materials and a crucial problem relies in understanding the nature of the interaction. The manifestation of such such phenomenon occurs at every scale, ranging from DNA molecules to the structural engineering works. Mathematics has looked at these problems, at least in the stationary case, since the seminal paper [1] and in some recent works (see, e.g., [7,8,9,10,11]) one of the authors contributed to the study of the static problem of adhesion of elastic structures by exploiting different constitutive assumptions to the aim of characterizing, in a variational framework, the interplay of the of debonding with other constitutive properties.
However the dynamical problem is a different story since at the heart of the question stays the understanding of the attachement-detachement occurrence and how this affects the whole evolution problem. This is a subtle problem since the analytical tools at disposal, such as spatio-temporal estimates in some norms, seem too rough to catch exhaustive quantitative informations, even in short time.
In [3,4] the simplest mechanical model consisting in elastic string was considered, whereas a discontinuous forcing term was assumed to model the adhesive interaction of the string with a rigid substrate. The resulting mathematical problem is then ruled by an initial boundary value problem for a semilinear second order wave equation and the results in [3,4] show some tricky peculiarities of the problem. As it is well known in continuum mechanics, the other basic model for one dimensional elastic structures is represented by the so called Bernoulli-Navier beam governing the flexural deformations of a slender material body. In the linear elastic framework the equation expressing the balance of momentum is ruled by a fourth order spatial differential operator. Analogously to [3], we assume a discontinuous forcing term to model the adhesive interaction of the beam with the external environment. One can visualize as a physical situation an elastic beam connected to a rigid substrate through a foundation made of continuous distribution of elastic-breakable springs. We study the well posedness of the mathematical problem proving existence of global in time solutions in the natural energy space and exploit some features of the solutions to obtain information about the role played by the attachment-detachment occurrence on the dynamical evolution. Indeed, in [2] it was proved that the main effects induced by the nonlinearity at the transition from attached to detached states consist in a loss of regularity of the solution and in a migration of the total energy through the scales. Here we deepen the analysis by using the linearization condition introduced by Patrick Gérard in [5] according to which one can conclude that if the semilinear evolution problem satisfies such condition, then the nonlinear forcing term does not induce any new oscillation or energy concentrations ( [5] ). Furthermore we prove an asymptotic result for the long time behavior in the case of bounded solutions, asserting the occurrence of three mutually exclusive states: the trivial one, the totally detached state, the totally attached state. The paper is organized as follows. In Section 2 we formulate the initial boundary value problem. In Section 3 we state the main results of the paper consisting in Theorem 3.1 (Existence of solutions), Theorem 3.2 (Characerization of adhesive states), Theorem 3.3 (Long time behavior), Theorem 3.4 (Linearization property). The proofs of these theorems are given respectively in Sections 4, 5, 6, 7. In Section 8 we provide some examples showing non-uniqueness and lack ofsmoothness of the solutions.

Statement of the problem
Let us consider an elastic beam under Bernoulli-Navier constitutive assumption, occupying in the reference configuration the interval [0, L] ⊂ R, the balance of linear momentum delivers the semilinear initial boundary value problem We shall assume that As a consequence of (H.1), Φ ′ has a jump discontinuity in u = ±1 and To fix ideas, a function satisfying such assumption is In particular we have for all u = ±1 The forcing term Φ ′ (u) is thought to model the elastic breakable interaction between the beam and the external environment. It confers to the problem a localized nonlinearity which affects the evolution in a significant way. The natural energy associated to the Problem (2.1) (i.e. to any solution u to (2.1)), is given at time t by the quantity In general, the lack of Lipschitz continuity in the nonlinear term Φ ′ suggests we cannot expect the existence of conservative solutions, i.e. solutions that preserve energy. Also the physics underlying the problem, foresees a kind of dissipation when the material is detaching from the substrate and this is accompanied by hysteresis cycles (see e.g. [8,Appendix B]).

Main results
We begin by studying the well-posedness of Problem (2.1) and the regularity of its solutions (see Definition 3.1 below). We prove the existence of Lipshitz continuous dissipative solutions and give examples of distinct solutions to (2.1) which do not depend continuously on the initial conditions. Definition 3.1. We say that a function u : where h u ∈ ∂Φ ′ (u), that is the subdifferential of Φ ′ (u); (iv) u may dissipate energy, i.e. for almost every t > 0: Let us state the following theorem asserting the existence of dissipative solutions. The following result provides a sufficient condition ruling the non-detachment of dissipative solutions in dependence on the initial data.
Theorem 3.2. Let u be a dissipative solution of (2.1). If where κ is defined in (H.1), then The long time behavior is a very subtle problem for evolutionary partial differential equations, so by restricting the focus on bounded dissipative solutions, we are able to prove the following statement.
then there exist a subsequence {t n k } k∈N and two constants a, b ∈ R such that Moreover, only one of the following statements can occur The question at the basis of the last subsequent result can be formulated as follows: How the nonlinearity characterizing the forcing term Φ ′ affects the evolution problem? We retain that Gérard's linearization condition provides a precise mathematical tool to answer to the previous rather vague question. Indeed the absence of further energy concentrations or oscillations due to the nonlinearity, suspected to arise in correspondence of the attachment-detacment process, constitutes an interesting property in itself, also considering the nonuniqueness of the solutions {u n } n below.
Theorem 3.4 (Linearization Property). Let {u 0,n } n ⊂ H 2 (0, L), {u 1,n } n ⊂ L 2 (0, L), {u n } n be a sequence of dissipative solutions of (2.1) in correspondence of such initial data, and v n be the dissipative solution of the linearized problem where k is defined in (H.1), then the following linearization condition holds true

Existence of Dissipative Solutions
This section is dedicated to the proof of Theorem 3.1. Our argument is based on the approximation of the Neumann problem (2.1) with a sequence of Neumann problems (4.1) characterized by smooth source terms and smooth initial data. More precisely, , for every n ∈ N consider the approximating problems where {u 0,n } n∈N , {u 1,n } n∈N , {Φ n } n∈N are sequences of smooth approximations of u 0 , u 1 , and Φ respectively, i.e. they satisfy the following requirements where C is a positive constant which does not depend on n. For any n ∈ N, (4.1) admits a classical solution for short time thanks to the Cauchy-Kowaleskaya Theorem (see [14]). Furthermore, for such a problem, solutions are indeed global in time thanks to the following results. Let u n be the unique classical solution to (4.1). Proof. Set E n := E[u n ] the energy corresponding to u n . We have to prove that the function t → E n (t) is constant (with constant value E n (0)). Indeed, we have that As a consequence of energy conservation, since the functions Φ n are positive, we have the following boundedness result.      We have yet to verify that u is a weak solution of (2.1) i.e. Definition 3.1 -item (iii). Let ϕ ∈ C ∞ (R 2 ) be a test function with compact support, since u n is a solution to (4.1), we have that for every n ∞ 0ˆL 0 Then, by taking the limit as n → ∞, (3.1) follows by using (4.2) and (4.3). Finally, due to (4.3) and (4.2) we have ∂ t u n ⇀ ∂ t u in L p (0, T ; L 2 (0, L)) for each T ≥ 0 and 1 ≤ p < ∞, Therefore, Definition 3.1 -item (iv) follows by the lower semicontinuity of the L 2 norm with respect to the weak convergence by taking into account Lemma 4.1.

Adhesive states
This section is dedicated to the proof of Theorem 3.2.
Proof of Theorem 3.2. Since u is continuous by (3.3) there exists τ > 0 such that in the short time So we can define τ * as follows We claim that Observe that, due to (H.1), and, in particular, Since u is dissipative, using the Sobolev embedding H 1 (0, L) ⊂ L ∞ (0, L) (see [6,Theorem 8.5]) and Lemma 4.3, we have from every t ∈ [0, τ * ) (where the last inequality holds thanks to (3.3)) that proves (5.1).

Long Time Behavior
This section is dedicated to the proof of Theorem 3.3. Let u be a dissipative solution of (2.1) satisfying (3.5). STEP 1. We begin by deducing the effective asymptotic problem.
Consider the functions u τ is a dissipative solution of the initial boundary value problem in the sense of Definition 3.1, namely (iii) for every test function ϕ ∈ C ∞ (R 2 ) with compact support where h τ ∈ ∂Φ ′ (u τ ), that is the subdifferential of Φ ′ (u τ ); (iv) u τ may dissipate energy, i.e. for almost every t > 0: there exists two functions U ∈ L ∞ (0, ∞; H 2 (0, L)), H ∈ L ∞ ((0, ∞) × (0, L)) such that, passing to a subsequence, therefore sending τ → ∞ in (6.2) we get We exploit more subtle characterizations of the limit functions U and H. To this aim we fix a sequence {t n } n∈N ⊂ (0, ∞) such that t n → ∞ and study the convergence of the sequence {u(t n , ·)} n∈N .
Since we have the dissipation inequality (3.2) and the assumption (3.5), we gain Therefore there exist two functions u ∞ ∈ H 2 (0, L), h ∞ ∈ L ∞ (0, L) such that passing to a subsequence u(t n , ·) ⇀ u ∞ weakly in H 2 (0, L) as n → ∞, Due to the result in STEP 1, we know that the functions u ∞ and h ∞ must satisfy the effective problem Moreover, by (6.6) we have also that By multiplying (6.7) by u ∞ , integrating over (0, L), and recalling (6.8), we get it follows that ∂ 2 xx u ∞ ≡ 0. Therefore, we can conclude that (3.6) holds.
In addition, the function w n = u n − v n is a dissipative solution of Multiplying (7.2) by ∂ t w n we gain and applying the Gronwall Lemma Thanks to Theorem 3.1 and [13, Theorem 5], (7.1) there exists a function u satisfying items (i) and (ii) in Definition 3.1 such that, passing to a subsequence, L)) and in L 2 (0, T ; H 2 (0, L)), for each T ≥ 0, , for each T ≥ 0 and 1 ≤ p < ∞.

(7.4)
Therefore u is a distributional solution of (7.5) Since u takes values in [−1, 1] and Φ is C 2 therein we can differentiate (7.5) and get Multiplying by ∂ 2 tt u, using (H.1) and (ii) in Definition 3.1 through a regularization argument we get Thanks to the Gronwall Lemmâ As a consequence u is an energy preserving solution of (7.5) and then it must be the trivial one. Eventually, (7.3) concludes the proof.

Non-uniqueness and lack of smoothness
This section is devoted to exploit some qualitative properties of (2.1) through explicit analytical examples evidencing the lack of uniqueness and smoothness of solutions. A key mechanism ruling these phenomena relies in the transition between the two configurations induced by the discontinuity affecting the forcing term Φ ′ . In particular, the first two examples show the lack of uniqueness while the last one and the numerical experiments enlighten the occurrences of lack of smoothness.
We have As ε → 0 we have and u and v provides two different solutions of (2.1) in correspondence of the initial data u 0 (x) = 1, u 1 (x) = 0.
The energies associated to (8.1) and (8.2) are respectively.