Homogenization and random evolutions: applications to the mechanics of composite materials
Author:
Georges A. Bécus
Journal:
Quart. Appl. Math. 37 (1979), 209-217
MSC:
Primary 73C40
DOI:
https://doi.org/10.1090/qam/548985
MathSciNet review:
548985
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Abstract: The technique of homogenization is used to derive the effective properties of laminated composites. A new probabilistic justification for homogenization using the concept of random evolutions is provided and indicates that the effective properties of deterministic periodic composite and those of a randomly perturbed periodic composite are the same.
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I. Babuška, Homogenization and its application, mathematical and computational problems, in Numerical solution of partial differential equations—III, Synspade 1975, B. Hubbard ed., Academic Press, New York, 1976, pp. 89–116
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E. Sanchez-Palencia, Comportements local et macroscopique d’un type de milieux physiques hétérogénes, Int. J. Engng. Sci. 12, 331–351 (1974)
S. Spagnolo, Convergence in energy for elliptic operators, in Numerical solutions of partial differential equations—III, Synspade 1975, B. Hubbard ed., Academic Press, New York, 1976, pp. 469–498
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J. M. Burgers, On some problems of homogenization, Quart. Appl. Math. 35, 421–434 (1978)
Z. Hashin and B. W. Rosen, The elastic moduli of fiber-reinforced materials, J. Appl. Mech. 31, 223–232 (1964)
E. Kröner, Elastic moduli of perfectly disordered composite materials, J. Mech. Phys. Solids 15, 319–323 (1967)
M. J. Beran, Statistical continuum theories, Interscience Publ., New York, 1968, ch. 5
R. Griego and R. Hersh, Theory of random evolutions with applications to partial differential equations, Trans. Amer. Math. Soc. 156, 405–418 (1971)
J. E. White and F. A. Angona, Elastic wave velocities in laminated media, J. Acoust. Soc. Amer. 27, 310–317 (1955)
W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath, Boston, 1965
A. Bensoussan, J. L. Lions and G. Papanicolaou, Sur quelques phénomènes asymptotiques stationnaires, C. R. Acad. Sc. Paris 281 (A), 89–94 (1975)
G. A. Bécus, Wave propagation in imperfectly periodic structures: a random evolution approach, ZAMP 29, 252–261 (1978)
T. G. Kurtz, A random Trotter product formula, Proc. Amer. Math. Soc. 35, 147–154 (1972)
T. G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Func. Anal. 12, 55–67 (1973)
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Article copyright:
© Copyright 1979
American Mathematical Society
