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The energy of inclusions in linear media exact shape-independent relations

1992; Elsevier BV; Volume: 40; Issue: 5 Linguagem: Inglês

10.1016/0022-5096(92)90056-8

1873-4782

Mordehai Milgrom, S. Shtrikman,

Thermoelastic and Magnetoelastic Phenomena

We discuss the energy of an inclusion—a configuration whereby a "polarization" distribution is dictated in a bounded domain, D of a medium with linear-response properties. Such is the case when a stress-free strain is prescribed in a domain of an elastic medium, or when a magnet is introduced into a paramagnetic medium. First we treat elastic inclusions. We find that when an eigenstrain ϵ∗ij is dictated within D the energy cannot exceed W0=(12)∫Dε∗ij(r)Cijkl(r)ε∗kl(r)d3r Cijkl being the elastic tensor in the medium. The elastic energy for an inclusion with a constant eigenslrain, ϵ0ij in a homogeneous medium, is described by an energy tensor, I, such that the energy, W, is W=(12)ε0ijIijklε0kl. The components of I—which depend on the geometry (shape and orientation) of the inclusion—satisfy a linear, geometry-independent relation of the form BijklIijkl = 3 V. where B is the inverse of the elastic tensor, and V is the volume of the inclusion. For a certain class of media, which include isotropic ones, a second relation is obeyed: BikjlIijkl = V (5 + 3v2)sol[2(1-v2)] v is the Poisson ratio). As a special case. the components of the Eshelby tensor are found to obey a new linear relation of the form Sijij = 3. We also treat inclusions in a medium that responds linearly to many coupled scalar potentials, as in the magnetoelectric or thermoelectric effects. We find a bound on the energy (or entropy-generation rate when dealing with dissipative phenomena) of the form W0=(12)∫DPkλL-1kmλβPmβd3r, where L(r) is the response-matrix of the medium, and Pkxr is the αspace component of the k-type external polarization field in the inclusion. Again, when the polarization fields arc constant. W is described in terms of an energy tensor Ikmxβ. We find that its components satisfy n(n + 1)2 geometry-independent relations (n is the number of coupled fields) LkmxβImpxβ = δkpV. Analogous bounds and constraints on the energy tensor exist for inclusions in a medium that responds linearly to the most general phenomenon including coupled fields of different tcnsorial ranks, such as a piezoelectric or piezo-magneto-electric medium.