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.
1. Introduction
In [1], Eringen developed a continuum theory for a mixture consisting of three components: an elastic solid, viscous fluid, and gas. Also, the author obtained the field equations for a heat-conducting mixture. In the theory of mixtures, the great abstraction was extended by assuming that the constituents of a mixture could be modeled as superimposed continua, for that each point in the mixture was simultaneously occupied by a material point of each constituent. A brief description concerning the details of the historical development/review related to the general theory of the mixtures is given by Bedford and Drumheller in [2].
Swelling porous media have been studied in many disparate fields including soil science, hydrology, forestry, geotechnical, chemical, and mechanical engineering, and this is due to its prevalence in nature and modern technologies. In this article, we focused on the asymptotic behavior of swelling soils that belong to the porous media theory in the case of fluid saturation. Swelling soils contain clay minerals that change volume with water content changes that result in major geological hazards and extensive damage worldwide. The swelling soils are caused by the chemical attraction of water, where water molecules are incorporated in the clay structure in between the clay plates separating and destabilizing the mineral structure. The swelling clay particles have the property of forming a unit (particle) from lattice hydrated aluminum and magnesium silicate minerals. Thus, the clay’s particle is a mixture of clay platelets and adsorbed water (vicinal water). Such a particle can be thought to define a mesoscale which is large compared to platelet, but small compared to the soil itself. A proper description of the mesoscale system behavior is critical when modeling consolidation of a swelling clay soil. As pointed out by Eringen [1], this system is the prototype for diffusion type models in swelling soils ([3–5]).
As established by Ieşan [6] and simplified by Quintanilla [7] (see also [8, 9]), the basic field equations for the linear theory of swelling porous elastic soils are mathematically given by the following equation:
Among the investigations that have been realized regarding the theory of swelling porous elastic soils, we cite the work of Quintanilla [7] when the author considered the following problem:
The author established an exponential stability result for the solution of equation (4) using the energy method in the isothermal case (
Furthermore, in the nonisothermal case and
In [10], Wang and Guo considered a problem of swelling of one-dimensional porous elastic soils given by the following equation:
In [13], the authors considered the following system:
In [8], Apalara considered a swelling porous elastic system with a viscoelastic damping given by the following equation:
Recently, in [9], the authors considered the following swelling problem in porous elastic soils with fluid saturation, viscous damping, and a time delay term.
Motivated by the above work, in this article, we considered the following problem:
The article is organized as follows: In Section 2, we gave the existence and uniqueness result of solutions of system (13) using some results from the semigroup theory. In Section 3, we use the multipliers method to prove the exponential stability result.
2. Well-Posedness
In this section, we gave the existence and uniqueness of solutions of system (13) using semigroup theory. First, we introduced the vector function
We considered the following spaces:
Then,
It is easy to see that the operator
Theorem 1.
Let
3. Exponential Decay
In this section, we stated and proved that technical lemmas are needed for the proof of our stability result.
Lemma 2.
Let
Proof.
Multiplying systems (13)1, (13)2, and (13)3 by
Using the fact that
Inserting (23) in (22), we get (20) and (21).
Lemma 3.
Let
Proof.
By differentiating
Young’s inequality leads to the following equations:
Substituting (27) and (28) in (26), we get (25).
Lemma 4.
Let
Proof.
By differentiating
Using Young’s inequality, we get the following equations:
Inserting (32)–(34) in (31), we obtain (30).
Lemma 5.
Let
Proof.
By exploiting the functional
Note that
So, equation (37) becomes as follows:
Using Young’s inequality,
Substituting (40) into (39), we get (36).
Lemma 6.
Let
Proof.
By differentiating
Using the fact that
Then, equation (43) can be rewritten as follows:
Young’s inequality leads to the following equations:
Using Young’s and Cauchy Schwarz inequalities, we find
Inserting (46)–(49) into (45), we obtain (42).
Now, we define the Lyapunov functional
Theorem 7.
Let
Proof.
From (50), we have the following equation:
By using the Young’s, Poincaré, Cauchy–Schwarz inequalities, we obtain the following equation:
Now, we select our parameters appropriately as follows:
First, we choose
Next, we select
We take
Finally, we choose
All these choices with relation (56) lead to the following equation:
On the other hand, from equation (20), we obtain equation (52).
Now, we can state and prove the following stability result.
Lemma 8.
Let
Proof.
By using estimation (52), we get the following equation:
Acknowledgments
This research has been funded by the Deputy for Research & Innovation, Ministry of Education through Initiative of Institutional Funding at University of Ha’il, Saudi Arabia, through project number IFP-22 021.
[1] A. C. Eringen, "A continuum theory of swelling porous elastic soils," International Journal of Engineering Science, vol. 32 no. 8, pp. 1337-1349, 1994.
[2] A. Bedford, D. S. Drumheller, "Theories of immiscible and structured mixtures," International Journal of Engineering Science, vol. 21 no. 8, pp. 863-960, DOI: 10.1016/0020-7225(83)90071-x, 1983.
[3] J. Parlange, "Water transport in soils," Annual Review of Fluid Mechanics, vol. 12 no. 1, pp. 77-102, DOI: 10.1146/annurev.fl.12.010180.000453, 1980.
[4] J. Philip, "Hydrostatics and hydrodynamics in swelling soils," Water Resources Research, vol. 5, pp. 1070-1077, DOI: 10.1029/wr005i005p01070, 1969.
[5] D. Smiles, M. J. Rosenthal, "The movement of water in swelling materials," Soil Research, vol. 6 no. 2, pp. 237-248, DOI: 10.1071/sr9680237, 1968.
[6] D. Ieşan, "On the theory of mixtures of thermoelastic solids," Journal of Thermal Stresses, vol. 14 no. 4, pp. 389-408, DOI: 10.1080/01495739108927075, 1991.
[7] R. Quintanilla, "Exponential stability for one-dimensional problem of swelling porous elastic soils with fluid saturation," JJournal of Computational and Applied Mathematics, vol. 145 no. 2, pp. 525-533, DOI: 10.1016/s0377-0427(02)00442-9, 2002.
[8] T. A. Apalara, "General stability result of swelling porous elastic soils with a viscoelastic damping," Zeitschrift für Angewandte Mathematik und Physik, vol. 71 no. 6,DOI: 10.1007/s00033-020-01427-0, 2020.
[9] A. J. A. Ramos, D. S. Almeida Júnior, M. M. Freitas, A. S. Noé, M. J. D. Santos, "Stabilization of swelling porous elastic soils with fluid saturation and delay time terms," Journal of Mathematical Physics, vol. 62 no. 2,DOI: 10.1063/5.0018795, 2021.
[10] J. M. Wang, B. Z. Guo, "On the stability of swelling porous elastic soils with fluid saturation by one internal damping," IMA Journal of Applied Mathematics, vol. 71 no. 4, pp. 565-582, DOI: 10.1093/imamat/hxl009, 2006.
[11] F. Bofill, R. Quintanilla, "Anti-plane shear deformations of swelling porous elastic soils," International Journal of Engineering Science, vol. 41 no. 8, pp. 801-816, DOI: 10.1016/s0020-7225(02)00281-1, 2003.
[12] M. A. Murad, J. H. Cushman, "Thermomechanical theories for swelling porous media with microstructure," International Journal of Engineering Science, vol. 38 no. 5, pp. 517-564, DOI: 10.1016/s0020-7225(99)00054-3, 2000.
[13] A. J. A. Ramos, M. M. Freitas, D. S. Almeida, A. S. Noé, M. J. D. Santos, "Stability results for elastic porous media swelling with nonlinear damping," Journal of Mathematical Physics, vol. 61 no. 10,DOI: 10.1063/5.0014121, 2020.
[14] A. M. Al-Mahdi, S. A. Messaoudi, M. M. Al-Gharabli, "A stability result for a swelling porous system with nonlinear boundary dampings," Journal of Function Spaces, vol. 2022,DOI: 10.1155/2022/8079707, 2022.
[15] A. M. Al-Mahdi, M. M. Al-Gharabli, T. A. Apalara, "On the stability result of swelling porous-elastic soils with infinite memory," Applicable Analysis, vol. 2022,DOI: 10.1080/00036811.2022.2120865, 2022.
[16] A. M. Al-Mahdi, M. M. Al-Gharabli, M. Alahyane, "Theoretical and numerical stability results for a viscoelastic swelling porous-elastic system with past history," AIMS Mathematics, vol. 6 no. 11, pp. 11921-11949, DOI: 10.3934/math.2021692, 2021.
[17] T. A. Apalara, "On the stability of porous-elastic system with microtemparatures," Journal of Thermal Stresses, vol. 42 no. 2, pp. 265-278, DOI: 10.1080/01495739.2018.1486688, 2018.
[18] H. Dridi, A. Djebabla, "On the stabilization of linear porous elastic materials by microtemperature effect and porous damping," Annali dell'Universita di Ferrara, vol. 66 no. 1, pp. 13-25, DOI: 10.1007/s11565-019-00333-2, 2020.
[19] J. R. Fernández, M. Masid, "A porous thermoelastic problem with microtemperatures," Journal of Thermal Stresses, vol. 40 no. 2, pp. 145-166, DOI: 10.1080/01495739.2016.1249038, 2016.
[20] F. Foughali, S. Zitouni, L. Bouzettouta, H. E. Khochemane, "Well-posedness and general decay for a porous-elastic system with microtemperatures effects and time-varying delay term," Zeitschrift für Angewandte Mathematik und Physik, vol. 73 no. 5,DOI: 10.1007/s00033-022-01801-0, 2022.
[21] H. E. Khochemane, "Exponential Exponential Stability for a Thermoelastic Porous System with Microtemperatures Effectstability for a thermoelastic porous system with microtemperatures effects," Acta Applicandae Mathematicae, vol. 173 no. 1,DOI: 10.1007/s10440-021-00418-1, 2021.
[22] H. E. Khochemane, "General stability result for a porous thermoelastic system with infinite history and microtemperatures effects," Mathematical Methods in the Applied Sciences, vol. 45 no. 3, pp. 1538-1557, DOI: 10.1002/mma.7872, 2022.
[23] M. Saci, H. Eddine Khochemane, A. Djebabla, "On the stability of linear porous elastic materials with microtemperatures effects and frictional dampingtability of linear porous elastic materials with microtemperatures effects and frictional damping," Applicable Analysis, vol. 101 no. 8, pp. 2922-2936, DOI: 10.1080/00036811.2020.1829602, 2020.
[24] A. Youkana, A. M. Al-Mahdi, S. A. Messaoudi, "General energy decay result for a viscoelastic swelling porous-elastic system," Zeitschrift f ür angewandte Mathematik und Physik, vol. 73 no. 3,DOI: 10.1007/s00033-022-01696-x, 2022.
[25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, 1983.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer
Copyright © 2023 Ali Rezaiguia et al. This is an open access article distributed under the Creative Commons Attribution License (the “License”), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. https://creativecommons.org/licenses/by/4.0/
Abstract
In this article, we considered the one-dimensional swelling problem in porous elastic soils with microtemperatures effects in the case of fluid saturation. First, we showed that the system is well-posed in the sens of semigroup. Then, we constructed a suitable Lyapunov functional based on the energy method and we proved that the dissipation given only by the microtemperatures is strong enough to provoke an exponential stability for the solution irrespective of the wave speeds of the system.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer
Details



1 Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il 2440, Saudi Arabia; Department of Computer Science and Mathematics, Mouhamed Cherif Messadia University, Souk Ahras, Algeria
2 Department of Computer Science and Mathematics, Mouhamed Cherif Messadia University, Souk Ahras, Algeria
3 Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il 2440, Saudi Arabia