# Homogenization of Some Multiparametric Problems

• N. S. Bakhvalov
• M. E. Eglit
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 10)

## Abstract

Inhomogeneous media are considered in this paper, e.g., composites and mixtures. Let L be the length scale of the problem and d be the typical length of inhomogeneities. The ratio ε = d/L is supposed to be small. Then the averaged equations can often be obtained that describe a certain homogeneous medium and have solutions close in some sense to solutions of original equations. For periodic media an asymptotic homogenization method to obtain the averaged equations is developed making use of the presence of a small parameter ε.

In many problems there are additional small parameters γi, besides ε. For example the following parameters can be small: the ratios of different phases moduli, the ratios of coefficients determining different properties of a phase, the ratios of inhomogeneity scales in different directions. The averaged equations essentially depend on the relations between small parameters ε and γi. In some cases the homogenized equations are of another type than equations describing the process in original medium. For example, instead of differential equations we obtain integro-differential equations.

Construction of averaged equations for periodic media includes solution of the so-called cell-problems. They are boundary-value problems for partial differential equations. As a rule they can be solved only numerically. In some cases analytical approximate solutions to cell-problems and explicit formulae for effective coefficients can be obtained due to presence of additional small parameters. The explicit formulae for effective moduli are very useful, especially in optimal design of materials and constructions.

The paper is a brief review of some author’s results concerning the effect of different small parameters.

## Keywords

Homogenize Equation Periodic Medium Effective Modulus Stratify Medium Matrix Modulus
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
Bakhvalov, N.S.: Averaging of partial differential equations with the rapidly oscillating coefficients. Doklady Akademii Nauk SSSR 221, 3, (1975), 516–519.
2. [2]
Bakhvalov, N.S., Panasenko, G.P.: Homogenization. Averaging Processes in Periodic Media. Mathematical Problems in Mechanics of Composite Materials, Nauka, Moscow 1984. Kluwer Academic Publishers, Dordrecht-Boston-London 1989.Google Scholar
3. [3]
Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Methods in Periodic Structures. North Holland, Amsterdam 1978.Google Scholar
4. [4]
Sanchez-Palencia, E.: Non Homogeneous Media and Vibration Theory. Lecture Notes in Physics 127, Springer, Berlin 1980.Google Scholar
5. [5]
Bakhvalov, N.S., Eglit, M.E.: Variational properties of averaged models for periodic media. Trudy MIAN 192 (1990), 5–19.
6. [6]
Bakhvalov, N.S., Eglit, M.E.: Homogenization of dynamic problems singularly depending on small parameters. Proceedings of Second Workshop on Composite Media and Homogenization Theory ( Trieste, 1993) World Scientific, Singapore 1995, 17–35.Google Scholar
7. [7]
Bakhvalov, N.S., Eglit, M.E.: The limiting behavior of periodic media with soft media inclusions. Comp. Maths Math. Phys. 35, no 6 (1995), 719–730.
8. [8]
Bakhvalov, N.S., Eglit, M.E.: Explicit Calculation of Effective Moduli for Composites Reinforced by an Irregular System of Fibres. Doklady Mathematics 51 (1995), 46–50.
9. [9]
Bakhvalov, N.S., Saint Jean Paulin, J.: Homogenization for thermoconductivity in a porous medium with periods of different orders in the different directions. Asymptotic Analysis 13 (1996), 253–276 .Google Scholar
10. [10]
Bakhvalov, N.S., Eglit, M.E.: Effective moduli of composites reinforced by systems of plates and bars. Comp. Maths Math. Phys. 38, no 5 (1998), 783–804.
11. [11]
Bakhvalov, N.S., Eglit, M.E.: Effective equations with dispersion for waves propagation in periodic media. Doklady Math. 370 (2000), 1–4.
12. [12]
Bakhvalov, N.S., Eglit, M.E.: Long-waves asymptotics with dispersion for the waves propagation in stratified media. Part 1. Waves orthogonal to the layers. Russian J. Numer. Analys. and Math. Modelling 15 (2000), 3–14.
13. [13]
Bakhvalov, N.S., Eglit, M.E.: Long-waves asymptotics with dispersion for waves propagation in stratified media. Part 2. Waves in arbitrary direction. Russian J. Numer. Analys. And Math. Modelling 15, no 3 (2000).Google Scholar
14. [14]
Dubinskaya, V.Yu.: Asymptotic expansion for a solution of a stationary heat conduction problem in a medium with two small parameters. Doklady RAN 333 (5) (1993), 571–574.Google Scholar
15. [15]
Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. and Analysis 11 (1962), 415–448.
16. [16]
Sandrakov, G.V.: The homogenization of nonstationary problems the theory of strong nonuniform elastic media. Doklady Mathematics 355, no 5 (1997), 605608.Google Scholar
17. [17]
Yakubenko, T.A.: Averaging a periodic porous medium with periods of different orders in different directions. Russian J. Numer. Analys. and Math. Modelling 13, no 2 (1998), 149–157.
18. [18]
Yakubenko, T.A.: Averaging of periodic structures with nonsmooth data. Moss-cow State University, Mech. and Math. Dept., Preprint no 2, 1999, 30 pp.Google Scholar