Stabilization of a Rao–Nakra Sandwich Beam System by Coleman–Gurtin’s Thermal Law and Nonlinear Damping of Variable-Exponent Type

Al-Gharabli, Mohammed M.; Al-Omari, Shadi; Al-Mahdi, Adel M.

doi:https://doi.org/10.1155/2024/1615178

Journal of Mathematics

On this page

Abstract Introduction Examples Data Availability Ethical Approval Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 1615178 | https://doi.org/10.1155/2024/1615178

Stabilization of a Rao–Nakra Sandwich Beam System by Coleman–Gurtin’s Thermal Law and Nonlinear Damping of Variable-Exponent Type

Mohammed M. Al-Gharabli,^1,2Shadi Al-Omari ,^1,2and Adel M. Al-Mahdi^1,2

Academic Editor: Genni Fragnelli

Received30 Nov 2023

Revised27 Dec 2023

Accepted01 Feb 2024

Published13 Feb 2024

Abstract

In this paper, we explore the asymptotic behavior of solutions in a thermoplastic Rao–Nakra (sandwich beam) beam equation featuring nonlinear damping with a variable exponent. The heat conduction in this context adheres to Coleman–Gurtin’s thermal law, encompassing linear damping, Fourier, and Gurtin–Pipkin’s laws as specific instances. By employing the multiplier approach, we establish general energy decay results, with exponential decay as a particular manifestation. These findings extend and generalize previous decay results concerning the Rao–Nakra sandwich beam equations.

1. Introduction

Partial differential equations (PDEs) featuring variable exponents have garnered considerable attention from researchers in recent times. Unlike conventional PDEs with constant exponents, equations with variable exponents entail power-law dependencies on spatial variables, with the exponents subject to variation along the spatial coordinates. This variation introduces added intricacy and complexities in both the analysis and solution of these equations.

The importance of PDEs with variable exponents is due to several reasons. They offer a more adaptable framework for modeling physical phenomena, enabling the representation of diverse behaviors in different regions of the domain by allowing spatial variation in the exponents. This type of nonlinearity within PDEs allows for a more precise representation of various physical phenomena, particularly in instances where the system’s behavior cannot be accurately depicted by linear models. The variable exponents enable a more adaptive and versatile approach to processing, contingent upon the characteristics of the image content. They come up in the research on optimal control problems, where the goal is to find strategies that make certain criteria as good as possible while considering the changes described by the PDE with variable exponents.

Moreover, heat conduction is a fundamental phenomenon with widespread applications in various physical processes, frequently described through partial differential equations (PDEs). These equations are instrumental in capturing the intricacies of heat distribution, as evidenced by their application in diverse scenarios. For instance, the temperature variation along a solid rod or bar finds representation in a 1D heat conduction equation, commonly formulated as the one-dimensional heat equation. This PDE correlates temporal changes with the distribution of temperature, offering a comprehensive understanding of the thermal dynamics. In the realm of electronics, especially for components such as computer chips and integrated circuits, a profound comprehension of heat conduction is indispensable. PDEs prove invaluable in modeling the temperature distribution within these devices, thereby contributing to the design of efficient cooling systems. Furthermore, engineering simulations leverage PDEs to analyze the thermal behavior of structures and systems, extending their application to fields such as civil engineering. The study of how buildings respond to environmental temperature changes relies on the insights gained from understanding heat conduction, showcasing the pervasive significance of PDEs in elucidating complex thermal phenomena.

For additional insights into the hierarchy of heat conduction laws, the authors refer to [1]. Motivated by the mathematical complexities presented by these equations and their efficacy in modeling intricate physical phenomena, we examine the following Rao–Nakra with a sandwich beam system:in which and denote the longitudinal displacement of the top layer and shear angle of the bottom layer, respectively, and represents the transverse displacement of the beam. The positive constants , , and are physical parameters representing, respectively, density, thickness, Young’s modulus, and the moments of inertia of the -th layer for and . Here, , , and where is the Poisson ratio. and are coupling constants. is a time-dependent coefficient and . is the temperature supposed to be known for negative times. and are the ratio between the relaxation time and the thermal conductivity. is the variable exponent function and satisfies some conditions that will be mentioned later. The functions and represent the convolution thermal kernel, nonnegative bounded convex summable function on [), belonging to a wide class of relaxation functions that satisfy the unitary total mass and additional properties specified in the paper. The system (1) consists of one Euler–Bernoulli beam equation for the transverse displacement and two wave equations for the longitudinal displacement of the top layer and the shear angle of the bottom layer. The top and bottom layers of the beam are subjected to Coleman–Gurtin’s thermal law [2], where the Fourier, Maxwell–Cattaneo’s, and Gurtin–Pipkin’s laws [3] are special cases. We subject the system (1) to the following boundary conditions:and the initial data are given by

The stabilization of Rao–Nakra beam systems has recently captured significant attention among researchers, leading to the establishment of numerous findings. The Rao–Nakra beam model involves the dynamics of two outer face plates, presumed to be relatively rigid, along with a compliant inner core layer sandwiched between them. The authors of [4–7] provide insights into Rao–Nakra, Mead–Markus, and multilayer plate or sandwich models. The fundamental equations of motion for the Rao–Nakra model are derived based on Euler–Bernoulli beam assumptions for the outer face plate layers, Timoshenko beam assumptions for the Sandwich layer, and a no-slip assumption for motion along the interface.

Let us begin by revisiting some prior works concerning multilayered sandwich beam models. Wang et al. [8] explored a sandwich beam system with boundary control using the Riesz basis approach, establishing exponential stability, exact controllability, and observability. In a different approach, Rajaram [9] utilized the multiplier approach to determine the precise controllability of a Rao–Nakra sandwich beam with boundary controls. Hansen and Imanuvilov [10, 11] investigated a multilayer plate system with locally distributed control in the boundary, employing Carleman estimations to establish precise controllability. Özer and Hansen [12, 13] achieved boundary feedback stabilization and perfect controllability for a multilayer Rao–Nakra Sandwich beam. Liu et al. [14] considered viscous damping effects on either the beam equation or one of the wave equations, establishing a polynomial decay rate using the frequency domain technique. Wang [15] analyzed a Rao–Nakra beam with boundary damping on one end, finding that the semigroup created by the system is polynomially stable of order 1/2. Mukiawa [16] recently studied system (1) with linear damping and Gurtin–Pipkin’s thermal law for heat conduction, proving the existence and establishing an exponential decay rate. Additional results on multilayer beams can be found in [17–30].

Our objective is to explore the asymptotic behavior of solutions in the context of the system (1)–(3). We aim to investigate how the thermal damping and the nonlinear damping with a variable exponent power, introduced in equation (1)₃, impact the asymptotic behavior of the energy function. Without imposing restrictions on the wave propagation speeds in the system, we employ the multiplier approach to establish both exponential and general energy decay rates for this system. These results extend and generalize previous decay findings related to the Rao–Nakra sandwich beam equation. The primary objectives are as follows:(i)Establishing an exponential decay of the system when the relaxation functions converge exponentially, and the variable exponent is set to .(ii)Establishing more generalized decay results for the system applies when the relaxation functions do not converge exponentially, and the variable exponent . In this scenario, various cases will be discussed based on the range of variable exponents and the convergence type of the relaxation functions.

To the best of our knowledge, stability results for the Rao–Nakra sandwich beam with nonlinear damping of variable exponent type have not been explored.

The rest of the paper is structured as follows: Section 2 introduces preliminary results and notations. Section 3 presents and proves technical lemmas, and Section 4 outlines and proves the stability theorem, offering a detailed proof.

2. Notations, Assumptions, and Transformations

This section is dedicated to the assumptions and specific transformations required for our problem. We make the following assumptions: . [) are nonincreasing ([)) and convex summable functions satisfying the following: Furthermore, there exists such that By setting: we obtain the following: . [) are nonincreasing ([)) and convex summable functions satisfying and there exists such that . We assume the existence of two positive constants such that where and are defined (below). We define , and , which defines a Hilbert space. . The time-dependent coefficient [) is a nonincreasing function satisfying . . The variable exponent [) is a continuous function such thatand . Moreover, the variable function satisfies the log-Hölder continuity condition; that is, for any with , there exists a constant such that

For further details on the memory kernels, refer to [31, 32].

Due to Dafermos [33], we define new functions for the relative past history of and as follows:define by

On account of the boundary conditions (2), we haveand routine calculation giveswhere and represent the history of and , respectively. Also, using integration by parts and change of variables, we have

Similarly, we get

Using (13)–(17), system (1)–(3) becomeswith the boundary conditionsand the initial datawhere and .

3. Essential Lemmas

In this section, we establish essential lemmas necessary for proving the stability of the system (18)–(20).

The energy function associated with the solution of the aforementioned system is precisely defined as follows:

Lemma 1. The energy functional (21) satisfies

Proof. Multiplying the equations (18)₁, (18)₂, (18)₃, (18)₄, and (18)₆ by , and , respectively, followed by the multiplication of (18)₅ and (18)₇ by and , respectively. Subsequently, utilizing integration by parts and incorporating the boundary conditions (19), we obtainThe summation of the aforementioned equations results inThis completes the proof.

Lemma 2 (see [34]). Assume that hold. Then, for all and , there exists a positive constant such that

Now, we present three lemmas without providing proofs; the methodology aligns with that employed in [35].

Lemma 3. Given the assumptions and , the subsequent approximations are as follows:where .

Lemma 4. Assuming that and hold, then for any , we havewhere

Lemma 5. If the assumptions – hold, then we have the following estimates:

Lemma 6. The functional defined bysatisfies the estimate

Proof. Differentiation of givesUsing equations (18)₁, (18)₂, and (18)₃, we obtainSubsequently employing integration by parts across the interval and incorporating the boundary conditions (19) results inUtilizing Young’s and Poincaré’s inequalities, we acquireThis completes the proof.

Lemma 7. The functional defined bysatisfies, for any and , the estimate

Proof. Differentiation of , using (18)₁ and (18)₄, we getNow, through the application of integration by parts and considering the boundary conditions (19), we reach the following:Employing Cauchy–Schwarz, Young’s, and Poincaré’s inequalities leads toThus, we establish (37).

Lemma 8. The functional defined bysatisfies, for any and , the estimate

Proof. Differentiation of , using (18)₂ and (18)₅, we obtainSubsequently, through integration by parts and considering the boundary conditions (19), we arrive atApplying Cauchy–Schwarz’ Young’s, and Poincaré’s inequalities, we haveHence, we get (42).

Lemma 9. Assuming that – are satisfied, the function defined bysatisfies for some of the following equivalence:and satisfies the following estimate:

Proof. Using Lemmas 6–8, and the estimate (27), we getBy choosingthen (49) takes the formNext, we determined the remaining parameters. Initially, we chose to be sufficiently large, ensuringSecond, we chose to be large enough, such thatFinally, we determined to be large enough so that (47) remains valid andThus, we obtainfor some . Recalling (21), it follows from (55) thatfor some .

4. Stability Results

Theorem 9. If the conditions hold, , and . Then, there exist constants and such that, for all and for all (], then the energy functional (21) satisfieswhere will be defined in the proof.

Proof. To prove the energy decay (57), we multiply (48) by , yieldingBy combining (58) with (21) and utilizing the estimate in (26)₁, (58) can be expressed as follows:where . Using (21), (21) and the fact that and are nonincreasing, we find thatSince , we haveMultiplying (59) by and combining it with (60), (61), and the constraint provided in hypothesis yieldswhere .
Let . Then, (62) becomesfor some . This last inequality remains true for any (]; that is,Therefore, direct integration leads toand the fact that givesWe note thatThen, integration by parts givesCombining with (66), we haveWe note thatWe have andFinally, combining (69) and (71), we obtainwhere . Thus, the proof of (57)₁ is completed, and the proof of (57)₂ and (57)₃ will be similar to taking and .

Theorem 10. If the conditions hold, , , and . Then, the energy functional (21) satisfies for a positive constant where and .

Proof. To prove (73), we first multiply (48) by , to getCombining (74), (29), and choosing small enough, we arrive atwhere . Using the estimate of in (26)₃, we have, for , the following equation:Multiplying (76) by , where , we getwhere . Choosing small enough, then we getMultiplying (78) by , using (21), and using that is nonincreasing, we getwhere and .
Use of Young’s inequality, with and , gives for some positive constant and .Multiply both sides of (109) by , , thus, we getLet which is nonincreasing, we find thatSetting and noting , one finds thatLetFrom the definition of , we havesince , then we have for all and thenThus, (85) yields, ,we can choose large enough so that , and then, we getNow, using (89) and the definition of , we get, ,Since , then there exists such that [). Hence,Thus,This givesIntegrate over , we haveTherefore, we getHence, it follows thatIf , then, using the definitions of and , we observe that, for sufficiently large ,By multiplying (97) with and recalling the definition of , we obtain the following for :Using the fact , we have two cases:
If , then and , we getIf , we haveThis establishes (73).

Theorem 11. If conditionshold,,, and. Then, the energy functional (21) satisfies for a positive constant,where .

Proof. To prove the decay (101), we first multiplying (48) by , recalling the estimate of in (26)₂, to getMultiplying (102) by where , we getwhere . Choosing sufficiently small, we getUsing (21), (21) and the fact that and are nonincreasing, we find thatSince , we haveNow, multiplying (104) by and using (105) and (106), we getwhere .
Recalling (21) and (25) and putting . Then, (107) becomesSetting which is a positive nonincreasing, then we getUsing Young’s inequality, with and , we get some positive constant and Repeating the procedures outlined in the last part of the proof of Theorem (58), substituting with , concludes the verification of the decay (101).

5. Examples and Remarks

In this section, we provide examples and remarks to illustrate and compare our stability results from Theorems 9–11 with some earlier findings in the literature.

Example 1. For, , let . Therefore, . Then, in case if , the energy decay (73) becomes for ,Thus, we achieve polynomial stability in the form ofFor and , it is evident that . In the case of , for any , we observe .
In the second scenario, when , and considering , , and , we obtain polynomial stability in the form ofNotably, for and , we have . Similarly, for and any , we find .

Remark 12. (1)If , then the decay (57) becomes(2)If and are functions of , then the decay (57) becomeswhere , , and .(3)The proofs of (114) and (115) are similar to the proof of decay in (57).

Remark 13. (1)In the case where , the energy decay estimate mirrors the decay (73), with the only difference being . The proof remains the same.(2)If and are functions of , then the energy decay will be the same as the decay (73) except where and and the proof will be the same.

Remark 14. (1)If , the energy decay aligns with the decay (101), with the only difference being . The proof remains the same.(2)If and are functions of , then the energy decay will be the same as the decay (101) except where and . The proof will be the same.

6. Use of AI Tools Declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Data Availability

No data were used to support this study.

Ethical Approval

Ethics approval was not required for this research.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors read and approved the final version of the manuscript.

Acknowledgments

The authors thank the King Fahd University of Petroleum & Minerals (KFUPM) and the Interdisciplinary Research Center in Construction and Building Materials (IRC) for their continuous support. This work was funded by King Fahd University of Petroleum and Minerals (KFUPM)-Deanship of Research Oversight and Coordination (DROC).

References

F. Dell’Oro and V. Pata, “A hierarchy of heat conduction laws,” 2022, https://arxiv.org/abs/2207.01428.
View at: Google Scholar
B. D. Coleman and M. E. Gurtin, “Equipresence and constitutive equations for rigid heat conductors,” Journal of Applied Mathematics and Physics ZAMP, vol. 18, no. 2, pp. 199–208, 1967.
View at: Publisher Site | Google Scholar
M. E. Gurtin and A. C. Pipkin, “A general theory of heat conduction with finite wave speeds,” Archive for Rational Mechanics and Analysis, vol. 31, no. 2, pp. 113–126, 1968.
View at: Publisher Site | Google Scholar
Y. Sadasiva Rao and B. Nakra, “Vibrations of unsymmetrical sandwich beams and plates with viscoelastic cores,” Journal of Sound and Vibration, vol. 34, no. 3, pp. 309–326, 1974.
View at: Publisher Site | Google Scholar
D. Mead and S. Markus, “The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions,” Journal of Sound and Vibration, vol. 10, no. 2, pp. 163–175, 1969.
View at: Publisher Site | Google Scholar
M.-J. Yan and E. Dowell, “Governing equations for vibrating constrained-layer damping sandwich plates and beams,” in Proceedings of the Winter Annual Meeting of the American Society of Mechanical Engineers, New York, NY, USA, November 1972.
View at: Google Scholar
S. W. Hansen, “Several related models for multilayer sandwich plates,” Mathematical Models and Methods in Applied Sciences, vol. 14, no. 8, pp. 1103–1132, 2004.
View at: Publisher Site | Google Scholar
J.-M. Wang, B.-Z. Guo, and B. Chentouf, “Boundary feedback stabilization of a three-layer sandwich beam: Riesz basis approach,” ESAIM: Control, Optimisation and Calculus of Variations, vol. 12, no. 1, pp. 12–34, 2006.
View at: Publisher Site | Google Scholar
R. Rajaram, “Exact boundary controllability results for a rao–nakra sandwich beam,” Systems and Control Letters, vol. 56, no. 7-8, pp. 558–567, 2007.
View at: Publisher Site | Google Scholar
S. W. Hansen and O. Imanuvilov, “Exact controllability of a multilayer rao-nakra plate with clamped boundary conditions,” ESAIM: Control, Optimisation and Calculus of Variations, vol. 17, no. 4, pp. 1101–1132, 2011.
View at: Publisher Site | Google Scholar
S. W Hansen and O. Yu Imanuvilov, “Exact controllability of a multilayer rao-nakra plate with free boundary conditions,” Mathematical Control and Related Fields, vol. 1, no. 2, pp. 189–230, 2011.
View at: Publisher Site | Google Scholar
A. O. Ozer and S. W. Hansen, “Uniform stabilization of a multilayer rao-nakra sandwich beam,” 2013, https://arxiv.org/abs/1311.2662.
View at: Google Scholar
A. O. Ozer and S. W. Hansen, “Exact boundary controllability results for a multilayer rao–nakra sandwich beam,” SIAM Journal on Control and Optimization, vol. 52, no. 2, pp. 1314–1337, 2014.
View at: Publisher Site | Google Scholar
Z. Liu, B. Rao, and Q. Zhang, “Polynomial stability of the rao-nakra beam with a single internal viscous damping,” Journal of Differential Equations, vol. 269, no. 7, pp. 6125–6162, 2020.
View at: Publisher Site | Google Scholar
Y. Wang, “Boundary feedback stabilization of a rao-nakra sandwich beam,” in Proceedings of the Journal of Physics: Conference Series, vol. 1324, Hangzhou, China, May 2019.
View at: Google Scholar
S. E. Mukiawa, “Well-posedness and stabilization of a type three layer beam system with gurtin-pipkins thermal law,” AIMS Mathematics, vol. 8, no. 12, pp. 28188–28209, 2023.
View at: Publisher Site | Google Scholar
A. A Allen and S. W Hansen, “Analyticity and optimal damping for a multilayer mead-markus sandwich beam,” Discrete and Continuous Dynamical Systems-B, vol. 14, no. 4, pp. 1279–1292, 2010.
View at: Publisher Site | Google Scholar
A. A. Allen and S. W. Hansen, “Analyticity of a multilayer mead–markus plate,” Nonlinear Analysis: Theory, Methods and Applications, vol. 71, no. 12, pp. e1835–e1842, 2009.
View at: Publisher Site | Google Scholar
R. H. Fabiano and S. W. Hansen, “Modeling and analysis of a three-layer damped sandwich beam,” Conference Publications, vol. 2001, pp. 143–155, 2013.
View at: Publisher Site | Google Scholar
S. W. Hansen and R. Rajaram, “Simultaneous boundary control of a rao-nakra sandwich beam,” in Proceedings of the 44th IEEE Conference on Decision and Control, pp. 3146–3151, IEEE, Seville, Spain, December 2005.
View at: Google Scholar
S. W. Hansen and R. Rajaram, “Riesz basis property and related results for a rao-nakra sandwich beam,” Conference Publications, vol. 2005, pp. 365–375, 2005.
View at: Publisher Site | Google Scholar
A. Ozkan Ozer and S. W. Hansen, “Exact controllability of a Rayleigh beam with a single boundary control,” Mathematics of Control, Signals, and Systems, vol. 23, no. 1-3, pp. 199–222, 2011.
View at: Publisher Site | Google Scholar
F. Dell’Oro and V. Pata, “On the stability of timoshenko systems with gurtin–pipkin thermal law,” Journal of Differential Equations, vol. 257, no. 2, pp. 523–548, 2014.
View at: Publisher Site | Google Scholar
C. Yang and J.-M. Wang, “Exponential stability of an active constrained layer beam actuated by a voltage source without magnetic effects,” Journal of Mathematical Analysis and Applications, vol. 448, no. 2, pp. 1204–1227, 2017.
View at: Publisher Site | Google Scholar
F. Dell’Oro, “On the stability of bresse and timoshenko systems with hyperbolic heat conduction,” Journal of Differential Equations, vol. 281, pp. 148–198, 2021.
View at: Publisher Site | Google Scholar
M. Akil, “Stability of piezoelectric beam with magnetic effect under (coleman or pipkin)–gurtin thermal law,” Journal of Applied Mathematics and Physics, vol. 73, no. 6, p. 236, 2022.
View at: Publisher Site | Google Scholar
M. Akil and Z. Liu, “Stabilization of the generalized rao-nakra beam by partial viscous damping,” Mathematical Methods in the Applied Sciences, vol. 46, no. 2, pp. 1479–1510, 2023.
View at: Publisher Site | Google Scholar
F. Dell’Oro, L. Paunonen, and D. Seifert, “Optimal decay for a wave-heat system with coleman–gurtin thermal law,” Journal of Mathematical Analysis and Applications, vol. 518, no. 2, Article ID 126706, 2023.
View at: Publisher Site | Google Scholar
A. Guesmia, “Well-posedness and stability results for thermoelastic bresse and timoshenko type systems with gurtin–pipkins law through the vertical displacements,” Spanish Society of Applied Mathematics Journal, pp. 1–49, 2023.
View at: Publisher Site | Google Scholar
M. Akil, H. Badawi, and A. Wehbe, “Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay,” 2020, https://arxiv.org/abs/2007.08316.
View at: Google Scholar
C. Giorgi and V. Pata, “Asymptotic behavior of a nonlinear hyperbolic heat equation with memory,” Nonlinear Differential Equations and Applications, vol. 8, no. 2, pp. 157–171, 2001.
View at: Publisher Site | Google Scholar
M. Akil, H. Badawi, S. Nicaise, and A. Wehbe, “Stability results of coupled wave models with locally memory in a past history framework via nonsmooth coefficients on the interface,” Mathematical Methods in the Applied Sciences, vol. 44, no. 8, pp. 6950–6981, 2021.
View at: Publisher Site | Google Scholar
C. M. Dafermos, “An abstract volterra equation with applications to linear viscoelasticity,” Journal of Differential Equations, vol. 7, no. 3, pp. 554–569, 1970.
View at: Publisher Site | Google Scholar
S. A. Messaoudi and M. M. Al-Gharabli, “A general stability result for a nonlinear wave equation with infinite memory,” Applied Mathematics Letters, vol. 26, no. 11, pp. 1082–1086, 2013.
View at: Publisher Site | Google Scholar
A. M. Al-Mahdi, “Long-time behavior for a nonlinear timoshenko system: thermal damping versus weak damping of variable-exponents type,” AIMS Mathematics, vol. 8, no. 12, pp. 29577–29603, 2023.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2024 Mohammed M. Al-Gharabli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

113

Downloads

133

Citations