Uniform stability result of laminated beams with thermoelasticity of type Ⅲ

1.   Introduction

Our objective in this present study is to consider laminated Timoshenko beams with thermoelasticity of type Ⅲ and establish an exponential stability result without structural (interfacial slip) damping or any other internal or boundary dampings. The heat conduction governing thermoelasticity of type Ⅲ is popularly called the Green and Naghdi theory. In contrast with the classical Fourier's law of heat conduction, the Green and Naghdi theory propounds a finite speed of heat propagation. See ^[1,2,3,4] for detailed historical reviews of the theory.

Towards the end of the nineteenth century, Hansen and Spies ^[5] introduced a model describing the vibrations in a structure made up of two uniform layered beams stuck together by a thin adhesive layer such that interfacial slip is possible when the beams are in continuous contact. Mathematically, the model comprises of the following three differential equations

The first two equations in (1.1), derived based on the assumptions of Timoshenko beam theory, are coupled with the third equation describing the slip dynamics. The dependent variables $u, v,$ and $w$ represent the transverse displacement, the proportionality to the volume of slip along the interface, and the rotation angle. The constants $\rho, G, I_\rho, D, \gamma,$ and $\beta$ are positive parameters representing the density, the shear stiffness, the mass moment of inertia, the flexural rigidity, the adhesive stiffness, and the adhesive damping, respectively.

The first stability result concerning (1.1) was carried out by Wang et al. ^[6]. They opined that the structural damping is not strong enough to exponentially stabilize the system. As a result, they only proved an exponential stability result for (1.1) by adding additional boundary damping terms. However, the structural damping was demonstrated to stabilize the system exponentially, provided the wave velocities of the system are equal. The result was independently established by Apalara in ^[7] and Alves and Monteiro in ^[8]. When the three equations in System (1.1) are damped, the system is exponentially stable regardless of the wave velocities or any other relationship between the coefficients $\rho, G, I_\rho,$ and $D$ of the system. In this regard, we cite, among others, Raposo ^[9], Liu and Zhao ^[10], Lo and Tatar ^[11]. For other results, we invite the reader to see ^[12,13,14]. However, Mustafa ^[15,16] showed that if the damping terms are only on one or two of the equations of the system, then an exponential stability result is only possible if some relationship exists among the coefficients of the system. A similar result can be found in ^{[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]} and references therein.

In the present study, we consider the laminated beams system given by (1.1) with thermoelasticity of type Ⅲ acting on the effective rotation angle, but with negligible structural damping ($\beta = 0$). That is,

The parameter $\alpha\ne0$ denotes the thermoelastic coefficient. The dependent variable $\Theta$ represents the empirical temperature. The variable $q$ denotes the heat flux with constitutive law given by

where $\omega$ is the thermal displacement given by (see ^[32])

From (1.4), it follows that the time derivative of the thermal displacement yields the empirical temperature, that is, $\omega_t = \Theta.$ The coefficients $k > 0, \, \rho_2 > 0,$ and $\kappa > 0$ represent thermal and elastic properties. We differentiate the fourth equation in (1.2) with respect to time, perform simple computations using (1.3) while bearing in mind that $\omega_t = \Theta$ and then supplement the resulting system with suitable initial and boundary conditions. Thus, we obtain the following laminated beams system with thermoelasticity of type Ⅲ

where $x\in(0, 1)$ and $t\geq 0$. Liu et al. ^[33] considered (1.5) in the presence of structural damping (i.e., $4\beta v_t$ added to the third equation) and established exponential stability of the system provided that the wave velocities of the system are equal and assuming a positive thermoelastic coefficient $(\alpha > 0)$.

Remark 1.1. (i) In the result of Liu et al. ^[33], the structural damping $(4\beta v_t)$ freely provided the needed negative term for $v^2_t$; it was very crucial and unavoidable in the proof of their stability result.

(ii) In the coupling, they assumed that $\alpha > 0$.

It is very important to mention here that, as far as the coupling is concerned, in the current work $\alpha$ is not necessarily positive; it is only required to be different from zero.

The novelty of this paper lies in addressing the two weaknesses presented by the result of Liu et al. ^[33], as encapsulated in Remark 1.1. We consider System (1.5) without structural damping (i.e., without the term $4\beta v_t$ in the third equation) and establish an exponential stability result under the condition of equal wave velocities of the system. The work is more challenging than that of Liu et al. ^[33] due to the absence of structural damping. In other words, instead of two dampings in ^[33], we have only one dissipation source via heat conduction. Furthermore, our result covers a broader range of $\alpha$; we only assumed that $\alpha$ is different from zero instead of applying $\alpha > 0$, as in ^[33]. Consequently, we extend the result obtained by Liu et al. ^[33] and some other results in the literature.

It is paramount to discuss the dissipative nature of System (1.5). It is not obvious at the level of the energy that System (1.5) is dissipative. In fact, the energy functional $\mathcal{E}_0(t)$ of System (1.5) given as

has the derivative

which is not necessarily decreasing (because $\alpha\neq 0$ is any real number). To overcome this problem, as in ^[34], we introduce a new variable

where $\zeta(x)$ is the solution of

Here, the superscripts denote the first and second derivatives in the variable $x$, respectively. By integrating the fourth equation in (1.5) and using (1.8) and (1.9), System (1.5) becomes

for $x\in\left(0, 1\right), \ t\ge 0$. Using the perturbed energy method, we consider (1.10) and proceed to show an exponential decay result provided that

holds. Throughout this paper, $\|.\|$ and $\langle\cdot, \cdot\rangle$ represent the usual $L^2(0, 1)-$norm and inner product, respectively. The letter "$c > 0$" represents a generic constant. Concerning the well-posedness result for (1.10), we refer the reader to ^[33].

2.   Stability result

In this section, we state and prove our stability result using the multiplier method. To achieve this task, we need to introduce some auxiliary functionals together with the classical energy defined by (2.2) in order to construct a suitable Lyapunov functional (of course, equivalent to the energy). In the lemma below, we establish the dissipativity of System (1.10) via the energy functional $\mathcal{E}$.

Lemma 2.1. $\mathcal{E}(u, w, v, \theta, t)$ is the energy given by (1.6), and it satisfies

where

Proof. We multiply (1.10)$_1$ by $u_{t}$ and integrate (by parts) over $(0, 1)$; then, the use of the boundary conditions leads to

Similarly, by multiplying the second, third, and fourth equations in (1.10) by $3 v_t- w_t, v_t,$ and $\theta_t$, respectively, we get

Summing up (2.3)–(2.6), we end up with (2.1).

The next seven lemmas have been created to produce the needed auxiliary functionals and their derivatives.

Lemma 2.2. Let (1.11) hold; the functional

along the solution of (1.10), satisfies for any $\varepsilon_{1} > 0$

Proof. Clearly, taking the derivative of $\mathcal{D}_{1}$ and incorporating the boundary conditions in (1.10) yields

Using the first two equations in (1.10), the identity $- w_t = (3 v_t- w_t)-3 v_t,$ and Eq (1.11), we end up with

The last two terms on the right of Eq (2.8) is estimated using Young's inequality; thus, for any $\varepsilon_{1} > 0,$ we have

Substituting (2.9) and (2.10) into (2.8) and taking $\delta_1 = \dfrac{G^2}{2}$ yields (2.7).

Lemma 2.3. The derivative of the functional

along solutions of (1.10) and for any $\varepsilon_{2}, \varepsilon_{3} > 0,$ satisfies

Proof. Performing the same routine derivative of $\mathcal{D}_{2}$, integrating by parts, and incorporating the boundary conditions in (1.10), we get

Now, some terms on the right of Eq (2.12) will be estimated. For any $\varepsilon_{2} > 0$, we use Young's inequality and Poincaré's inequality with the constant $c_p$, to perform the estimation as follows:

To estimate the last term (on the right) of Eq (2.12), using Young's inequality for any $\varepsilon_{3} > 0$, we get

where we have applied the Cauchy-Schwarz inequality and Poincaré's inequality to have

We end up with the estimate (2.11) by merely substituting (2.13)–(2.16) into (2.12).

Remark 2.4. The next lemma caters for the missing structural (interfacial slip) damping.

Lemma 2.5. Let (1.11) hold; the functional

along the solution of (1.10) satisfies

Proof. As done in previous lemmas, by differentiating, integrating by parts and then using boundary conditions as usual we get

Using (1.11) and the identity $w_t = -(3 v_t- w_t)+3 v_t$ leads to

By Young's inequality, we have

Consequently, we arrive at (2.17).

Lemma 2.6. The derivative of the functional

along the solution of (1.10) satisfies

Proof. Applying the fourth equation in (1.10) with the boundary conditions, it follows that

Indeed, applying Young's and Poincaré's inequalities, the estimate (2.19) is reached.

Lemma 2.7. The functional

along the solution of (1.10) satisfies

Proof. After differentiating $\mathcal{D}_{5}(t)$, we integrate by parts and apply boundary conditions; then, we administer the identity $u_x = -(w- u_x)-(3v- w)+3v.$ Consequently, we deduce

The last two terms on the right of Eq (2.21) are estimated to get

courtesy of Poincaré's and Young's inequalities. As usual, we substitute (2.22) and (2.23) into (2.21) to get (2.20).

Lemma 2.8. The functional

along the solution of (1.10) satisfies

Proof. By using the second equation in (1.10), it is obvious that for all $t\ge0$,

Then, application of Young's and Poincaré's inequalities brings about the estimate (2.24).

Lemma 2.9. The functional

along solutions of (1.10) satisfies

Proof. By exploiting (1.10) together with Poincaré's and Young's inequalities, (2.25) is deduced.

Having introduced all the needed auxiliary functionals, we now define

which is a Lyapunov functional, where $\mathcal{N} > 0, \ \mathcal{N}_1 > 0$, and $\mathcal{N}_2 > 0$ are constants to be carefully chosen later. Of course, we can verify (applying Poincaré's and Young's inequalities) that $\mathcal{P}$ is equivalent to the energy functional $\mathcal{E}$ given that $\mathcal{N}$ is sufficiently large. Summarily, for a large $\mathcal{N}$,

for some $\ell_1, \ \ell_2 > 0.$

Next, our stability result is stated and proved.

Theorem 2.10. Let (1.11) hold; then, there exist $\mu > 0$ and $\nu > 0$ such that

Proof. Differentiating $\mathcal{P}$ and incorporating (2.1), (2.7), (2.11), (2.17), (2.19), (2.20), (2.24), and (2.25), we end up with

It is clear that by taking

we obtain

Taking $\mathcal{N}_1, \mathcal{N}_2,$ and $\mathcal{N}$ large enough, sequentially, we end up with

for some $\zeta > 0.$ Using (1.6), (2.27) and (2.29), for some $\nu > 0$, we get

Now, (2.30) is integrated over $(0, t)$ to get

which implies (2.28) with $\mu = \dfrac{\ell_{2}}{\ell_{1}}\mathcal{E}(0)$ due to (2.27).

3.   Conclusions and open problems

In this work, we establish that the thermal effect exhibited by heat conduction of type Ⅲ is strong enough to exponentially stabilize the laminated beams system without any additional internal or boundary damping mechanism. For example, instead of the two dampings used in ^[33], we used only one dissipation source to achieve the exponential stability result. Furthermore, our result covers a broader range of $\alpha$, assuming that it is only different from zero instead of $\alpha > 0$, as in ^[33]. As usual, we assume that the system's wave propagation velocities are equal. It is an intriguing open problem to prove that the system has no exponential stability except that the condition of equal wave velocities is imposed. In addition, the polynomial stability of the system when the wave velocities are not equal is an enthralling problem to consider.

Acknowledgments

The authors appreciate the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia, for funding this research work through institutional fund projects under the project no. IFP-A-2022-2-1-02. The second author thanks Lagos State University, Ojo, Nigeria, for continuous support.

Conflict of interest

The authors declare no potential conflicts of interests.