Variational Estimates for the Overall Response of an Inhomogeneous Nonlinear Dielectric
Part of the
The IMA Volumes in Mathematics and its Applications
book series (IMA, volume 1)
For any problem that can be formulated as a “minimum energy” principle, a procedure is given for generating sets of upper and lower bounds for the energy. It makes use of “comparison bodies” whose energy functions may be easier to handle than those in the given problem. No structure for the energy functions is assumed in the formal development but useful results are most likely to follow when they are convex. When applied to linear field equations, the procedure yields the Hashin-Shtrikman variational principle, and so can be regarded as its generalization to nonlinear problems.
The procedure is applied explictly to a boundary value problem for an inhomogeneous, nonlinear dielectric. Then, a slight extension which describes randomly inhomogeneous media is applied, to develop bounds for the overall energy of a nonlinear composite, which reduce to the Hashin-Shtrikman bounds in the linear limit. Sample results are shown for a simple two-phase composite.
Ekeland, I. and Temam, R., “Convex Analysis and Variational Problems”. North Holland, Amsterdam (1976).MATHGoogle Scholar
Hashin, Z. and Shtrikman, S., “On some variational principles in anisotropic and nonhomogeneous elasticity”, J. Mech. Phys. Solids 10, 335–342 (1962a).MathSciNetADSCrossRefGoogle Scholar
Hashin, Z. and Shtrikman, S., “A variational approach to the theory of the elastic behaviour of polycrystals”, J. Mech. Phys. Solids 10, 343–350 (1962b).MathSciNetADSCrossRefGoogle Scholar
Hashin, Z. and Shtrikman, S., “A variational approach to the theory of the magnetic permeability of multiphase materials”, J. Appl. Phys. 33, 3125–3131 (1962c).ADSCrossRefMATHGoogle Scholar
Hashin, Z. and Shtrikman, S., “A variational approach to the theory of the elastic behaviour of multiphase materials”, J. Mech. Phys. Solids 11, 127–140 (1963).MathSciNetADSCrossRefMATHGoogle Scholar
Miksis, M.J., “Dielectric constant of a nonlinear composite material”, SIAM J. Appl. Math. 43, 1140–1155 (1983).MathSciNetCrossRefMATHGoogle Scholar
Willis, J.R., “Bounds and self-consistent estimates for the overall properties of anisotropic composites”, J. Mech. Phys. Solids 25, 185–202 (1977).ADSCrossRefMATHGoogle Scholar
Willis, J.R., “The overall elastic response of composite materials”, J. Appl. Mech. 50, 1202–1209 (1983).ADSCrossRefMATHGoogle Scholar
© Springer-Verlag New York Inc. 1986