On a Laminated Timoshenko Beam with Nonlinear Structural Damping

: In the present work, we study a one-dimensional laminated Timoshenko beam with a single nonlinear structural damping due to interfacial slip. We use the multiplier method and some properties of convex functions to establish an explicit and general decay result. Interestingly, the result is established without any additional internal or boundary damping term and without imposing any restrictive growth assumption on the nonlinear term, provided the wave speeds of the ﬁrst equations of the system are equal.


Introduction
In the late 1990s, a model for structure of two identical beams with uniform thickness was developed by Hansen and Spies [1]. This model is called a laminated Timoshenko beam and is given by where the terms w, ψ, s represent the transverse displacement, the rotation angle, and the amount of slip along the interface, respectively. The positive parameters ρ, G, I ρ , D, γ, and β are known as the density, shear stiffness, mass moment of inertia, flexural rigidity, adhesive stiffness, and adhesive (structural) damping at the interface, respectively. It is very close to the well-known classical Timoshenko system because the equations of motion modeling the system are derived under the assumption of the Timoshenko beam theory. In addition, the third equation (which contains the structural damping s t ) is coupled with the first two describing the dynamic of the interfacial slip. These structures are highly important and have gained massive popularity in engineering fields. Stabilization of the system with various internal or boundary damping mechanisms has been the subject of research over the years. Specifically, an increasing interest has been developed to determine the asymptotic behavior of the system which are paramount in classifying the empirical observations of the engineers.
Let us start with the boundary stabilization. To the best of our knowledge, the first known result in this case was established by Wang et al. [2]. They considered (1) [2] provided ρG < I ρ . Mustafa [4] also obtained a similar result under the condition of equality between velocities of wave propagation, that is We refer the reader to [5][6][7][8] and the references cited therein for some other results on boundary stabilization.
For viscoelastic damping (memory term), we mention the work of Lo and Tatar [9].
They considered system (1) with viscoelastic damping of the form t 0 g(t − r)(3s xx − ψ xx )(r)dr on the second equation and proved that the resulting system is exponentially stable provided the relaxation function g decays exponentially and (2) is satisfied, in addition to some conditions on the parameter G.
Mustafa [10] improved the result in [9] by adopting minimal and general conditions on g. Consequently, he established explicit energy decay result. Some other results can be found in [11][12][13][14][15] and the references cited therein. The thermal effect (classical or second sound) is another way of stabilizing the laminated beams. Apalara in [16] proved that the heat effect is sufficiently strong enough to stabilize laminated beams exponentially without any other damping term, provided (2) holds. A similar result was obtained by Apalara [17] for the second sound in the presence of a structural damping term.
For frictional damping on the three equations, we mention the work of Raposo [18]. Recently, Alves et al. [19] established an exponential decay for the laminated system with only structural damping, provided the wave velocities are equal. It is imperative to mention that when s ≡ 0, then the laminated beams reduced to the standard Timoshenko system, see [17,18]. We refer the reader to [19][20][21][22][23][24] for some other varios forms of damping mechanisms In this paper, we consider a laminated Timoshenko beam with only a single source of dissipation in the form of a nonlinear interfacial slip and discuss the general decay of the energy of the system under suitable assumption on the nonlinear term and coefficients of wave propagation speed. On the nonlinear term h, we assume, as in Lasiecka and Tataru [25], that it satisfies the following hypotheses: (A1) h : R → R is a non-decreasing C 0 -function such that there exist positive constants c 1 , c 2 , , and a strictly increasing function H ∈ C 1 ([0, +∞)), with H(0) = 0, and H is linear or strictly convex C 2 -function on (0, ] such that Remark 1. 1. Hypothesis (A1) implies that yh(y) > 0, for all y = 0. [25] used the monotonicity and continuity of h to establish the existence of H as defined in (A1).

Lasiecka and Tataru in
For completeness purpose, we introduce the following spaces and state, without proof, the following existence and regularity result: then the solution satisfies

Remark 2.
This result can be proved using standard arguments such as nonlinear semi-group method (see [26]) or the Faedo-Galerkin method (see [27]).
The rest of the paper is organized as follows. In Section 2, we state and prove some essential technical lemmas. We give our stability result in Sections 3. We use c throughout this paper to denote a generic positive constant, which may be different from line to line (even in the same line). Our work gives an adequate answer to the possibility of stabilizing a Timoshenko Laminated beam with single non-linear damping present only in the third equation with some appropriate conditions.

Technical Lemmas
This section is devoted to the statements and proofs of some essential technical lemmas, which are highly influential in proving our main result. Lemma 1. Let (w, ψ, s) be the solution of system (3) and assume (A1) holds. Then the energy functional defined by Proof. By multiplying (3) 1 by w t , (3) 2 by (3s t − ψ t ), and (3) 3 by s t , then integrating over (0, 1), using integration by parts and the boundary conditions, we obtain The combination of (6) to (8) bearing in mind (4) and the fact that gives (5).
Lemma 7. Let (w, ψ, s) be the solution of system (3). Then, for N, N i (i = 1 · · · 3) > 0 sufficiently large, the Lyapunov functional defined by satisfies, for some positive constant k, Proof. Direct computations using (5), (9), (13), (18), (19), (21), and let We now carefully choose the rest of the constants. First, we choose N 3 large so that Next, we pick N 2 large so that Finally, we select N 1 large so that Thus, we end up with for some α > 0. Using the energy functional defined by (4), we obtain (23), for some k > 0.

Remark 3.
Choosing N large, it can easily be shown that F ∼ E in the sense that there exist two positive constants a and b such that
Using (A1) and the continuity of h along the fact that |h(y)| > 0, for s = 0, we deduce that We now shift our attention to the last integral in (23): To estimate this integral, we consider, as in [28], the following partition: Thus, with I(t) defined by we obtain, thanks to Jensen inequality and the fact that H −1 is concave Hence, using (30) and (31), we end up with 1 0 The substitution of (32) into (23) and using (28), we have Now, for 0 < and δ 0 > 0, using (33) and the following properties of E and H: we deduce that the functional F 1 , defined by satisfies, for some c 1 , c 2 > 0, and In order to estimate T in (35), we let H * be the convex conjugate of H defined by then, using the general Young's inequality By using the energy functional (4) and (36), we end up with Combining (35) and (37), we obtain Letting 0 = k 2c E(0), δ 0 = 2c, and using the fact that E (t) ≤ 0, we get where a 1 > 0 and H 0 (t) = tH ( 0 t). So, using the fact that H is strictly convex on (0, ], we find that H 0 (t), H 0 (t) > 0 on (0, 1]. Hence, with F (t) = c 1 F 1 (t) E(0) and using (34) and (38), we have and, for some k 2 > 0, Inequality (40) implies that H 1 F (t) ≥ k 2 , where Thus, by integrating over [0, t], bearing in mind the properties of H 0 , and the fact that H 1 is strictly decreasing on (0, 1] we obtain, for some k 3 > 0, We complete the proof of Theorem 1 by using (39) and (41).

Concluding Remarks
In this work, as in [19], we show (see (29)) that the structural damping is strong enough to exponentially stabilize the laminated Timoshenko beam system provided the wave speeds of the first two equations of the system are equal. A similar result was recently obtained in [29] when frictional damping is acting on the second equation. For the nonlinear case, the result is more general. For this case (nonlinear), it is an interesting open problem to investigate the case when the wave speeds of the first two equations of the system are not equal. Funding: The first and third authors are supported by UHB under project number #G-106-2020. The second author thank FUD for continuous support.

Conflicts of Interest:
The authors declare no conflict of interest.