Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



A variational approach to statistically nonhomogeneous fields

Author: A. Somoroff
Journal: Quart. Appl. Math. 28 (1970), 219-236
DOI: https://doi.org/10.1090/qam/99795
MathSciNet review: QAM99795
Full-text PDF

Abstract | References | Additional Information

Abstract: Classical variational principles of elasticity are cast in a form amenable to the treatment of certain problems concerned with statistically nonhomogeneous fields. The variational principles are utilized to obtain bounds directly upon the ensemble average values of physical quantities which in some sense characterize their problems. General formulations and applications are given for randomly heterogeneous elastic media and elastically supported beams with random axial variation of sectional rigidity and foundation stiffness. Bounds are obtained for the average displacement at the inner surface of a hollow sphere under pressure, the average elongation of a cylinder in uniform tension, and the average displacement of a beam directly under a point load. General methods are given for the selection of the deterministic admissible functions which provide the best bounds that can be obtained from classical principles for this restricted class of admissible functions. The closeness of the bounds is of the same order as the elementary bounds on effective material properties, which are of quantitative value only for small dispersion of the random coefficients. In a special statistically homogeneous case, previously obtained bounds for an effective Young's modulus are regained. Usually, however, the bounded quantities do not possess the generality of effective properties but are more closely associated with the results of particular problems. The developments of the present paper are considered an initial approach to the investigation of statistically nonhomogeneous problems.

References [Enhancements On Off] (What's this?)

  • [1] B. Paul, Prediction of elastic constants of multiphase materials, Trans. Met. Soc. AIME, 28, 36 (1960)
  • [2] W. F. Brown, Dielectric constants, permeabilities and conductivities of random media, Trans. Rheology Soc. 9, 357 (1965)
  • [3] M. Beran, Use of the variational approach to determine bounds for the effective bulk modulus in heterogeneous media, Quart. Appl. Math., 24, 107 (1966)
  • [4] Z. Hashin and S. Shtrikman, On some variational principles in anisotropic and non-homogeneous elasticity, J. Mech. Phys. Solids 10 (1962), 335–342. MR 0147031, https://doi.org/10.1016/0022-5096(62)90004-2
  • [5] I. S. Sokolnikoff, Mathematical theory of elasticity, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956. 2d ed. MR 0075755
  • [6] M. Miller, Bounds for effective dielectric constant and bulk modulus of heterogeneous materials in terms of statistical information, Ph.D. dissertation, University of Pennsylvania, Philadelphia, Pa., 1967
  • [7] A. Somoroff, An application of variational principles to statistically nonhomogeneous fields, Ph.D. dissertation, University of Pennsylvania, Philadelphia Pa., 1968

Additional Information

DOI: https://doi.org/10.1090/qam/99795
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society