On the Model of Isothermal Acoustics for a Two-Component Medium
DOI:
https://doi.org/10.24160/1993-6982-2017-6-146-151Keywords:
composite media, periodic structure, Lame's equations, porous elasticity, homogenization of periodic structures, two-scale convergenceAbstract
A mathematical model describing the processes of isothermal acoustics in a heterogeneous medium with two components separated by a common boundary is studied. One of the components is an elastic body, and the other one is a poroelastic medium (for example, it may be a liquid-saturated soil). The poroelastic medium is permeated with a system of pores filled with viscous weakly compressible liquid. The differential equations of the model describing the motion of an elastic body and the joint motion of a solid skeleton and liquid in the pores are based on the classical laws of continuous medium mechanics and adequately reflect the physical processes. However, these equations contain rapidly oscillating coefficients that depend on a small parameter equal to the ratio of the mean pore size to the size of the region under study. The existence of such coefficients prevents the use of the model for carrying out numerical calculations. The generalized solution of the initial boundary-value problem is given; and the theorem for existence and uniqueness of the generalized solution is presented together with its a priori estimates. For performing the homogenization procedure, the standard assumption about the periodicity of the pore space and solid skeleton is adopted. The obtained a priori estimates and the N. Nguetseng's two-scale convergence method were used as a basis for deriving the averaged equations and the initial boundary conditions (that is, the limit equations with the small parameter tending to zero). Different limiting modes depending on the continuous medium parameters are obtained. An averaged model for a special case that does not contain rapidly oscillating coefficients and can be used for numerical calculations is presented.
References
2. Жиков В.В., Козлов С.М., Олейник О.А. Усреднение дифференциальных операторов. М.: Наука, 1993.
3. Шамаев А.С., Шумилова В.В. Усреднение уравнений акустики для частично перфорированного упругого материала со слабовязкой жидкостью // Журнал Сибирского федерального ун-та. Серия «Математика и физика». 2015. Т. 8. № 3. С. 356—370.
4. Жиков В.В., Иосифьян Г.А. Введение в теорию двухмасштабной сходимости // Труды семинара им. И.Г. Петровского. 2013. Т. 29. С. 281—332.
5. Burridge R., Keller J.B. Poroelasticity Equations Derived from Micro-structure // J. Acoust. Soc. Am. 1981. V. 70. No. 4. Pp. 1140—1146.
6. Sanchez-Palencia E. Non-homogeneous Media and Vibration Theory // Lecture Notes in Physics. Berlin: Springer, 1980. V. 129.
7. Nguetseng G. A General Convergence Result for a Functional Related to the Theory of Homogenization // SIAM J. Math. Anal. 1989. V. 20. Pp. 608—623.
8. Nguetseng G. Asymptotic Analysis for a Stiff Variational Problem Arising in Mechanics // SIAM J. Math. Anal. 1990. V. 21. Pp.1394—1414.
9. Lukkassen D., Nguetseng G., Wall P. Two-scale Convergence // Int. J. Pure and Appl. Math. 2002. V. 2. No. 1. Pp. 35—86.
10. Мейрманов А.M. Уравнения акустики в упругих пористых средах // Сибирский журнал индустриальной математики. 2010. Т. XIII. № 2. C. 98—110.
11. Meirmanov A.M. Derivation of Equations of Seismic and Acoustic Wave Propagation and Equations of Filtration Via Homogenization of Periodic Structures // J. Math. Sci. 2009. V. 163. No. 2. Pp. 111—172.
12. Мейрманов А.М., Герус А.А., Гриценко С.А. Усредненные модели изотермической акустики в конфигурации упругое тело — пороупругая среда // Математическое моделирование. 2016. Т. 28. № 12. С. 3—19.
13. Conca C. On the Application of the Homogenization Theory to a Class of Problems Arising in Fluid Mechanics // Math. Pures et Appl. 1985. V. 64. Pp. 31—75.
---
Для цитирования: Гриценко С.А., Мейрманов А.М. О модели изотермической акустики для двухкомпонентной среды // Вестник МЭИ. 2017. № 6. С. 146—151. DOI: 10.24160/1993-6982-2017-6-146-151.
#
1. Bahvalov N.S., Panasenko G.P. Osrednenie Protsessov v Periodicheskih Sredah. Matematicheskie Zadachi Mekhaniki Kompozitsionnyh Materialov. M.: Nauka, 1984. (in Russian).
2. Zhikov V.V., Kozlov S.M., Oleynik O.A. Usrednenie Differentsial'nyh Operatorov. M.: Nauka, 1993. (in Russian).
3. Shamaev A.S., Shumilova V.V. Usrednenie Uravneniy Akustiki dlya Chastichno Perforirovannogo Uprugogo Materiala so Slabovyazkoy Zhidkost'yu. Zhurnal Sibirskogo Federal'nogo Un-ta. Seriya «Matematika i Fizika». 2015;8;3:356—370. (in Russian).
4. Zhikov V.V., Iosif'yan G.A. Vvedenie v Teoriyu Dvuhmasshtabnoy Skhodimosti. Trudy Seminara im. I.G. Petrovskogo. 2013;29:281—332. (in Russian).
5. Burridge, R., Keller, J.B. Poroelasticity Equations Derived from Micro-structure. J. Acoust. Soc. Am. 1981;70;4:1140—1146.
6. Sanchez-Palencia E. Non-homogeneous Media and Vibration Theory. Lecture Notes in Physics. Berlin: Springer, 1980;129.
7. Nguetseng G. A General Convergence Result for a Functional Related to the Theory of Homogenization. SIAM J. Math. Anal. 1989;20:608—623.
8. Nguetseng G. Asymptotic Analysis for a Stiff Variational Problem Arising in Mechanics. SIAM J. Math. Anal. 1990;21:1394—1414.
9. Lukkassen D., Nguetseng G., Wall P. Two-scale Convergence. Int. J. Pure and Appl. Math. 2002;2;1:35—86.
10. Meyrmanov A.M. Uravneniya Akustiki v Uprugih Poristyh Sredah. Sibirskiy Zhurnal Industrial'noy Matematiki. 2010;XIII;2:98—110. (in Russian).
11. Meirmanov A.M. Derivation of Equations of Seismic and Acoustic Wave Propagation and Equations of Filtration Via Homogenization of Periodic Structures. J. Math. Sci. 2009;163;2:111—172.
12. Meyrmanov A.M., Gerus A.A., Gritsenko S.A. Usrednennye Modeli Izotermicheskoy Akustiki v Konfiguratsii Uprugoe Telo — Porouprugaya Sreda. Matematicheskoe Modelirovanie. 2016;28;12:3—19. (in Russian).
13. Conca C. On the Application of the Homogenization Theory to a Class of Problems Arising in Fluid Mechanics. Math. Pures et Appl. 1985;64:31—75.
---
For citation: Gritsenko S.A., Meirmanov A.M. On the Model of Isothermal Acoustics for a Two-Component Medium. MPEI Vestnik. 2017; 6:146—151. (in Russian). DOI: 10.24160/1993-6982-2017-6-146-151.
