THE ENERGY OF INCLUSIONS IN LINEAR MEDIA EXACT SHAPE-INDEPENDENT RELATIONS

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 epsilon(ij)* is dictated within D the energy cannot exceed W0 = (1/2) integral-D epsilon(ij)*(r)C(ijkl)(r)epsilon(kl)*(r) d3r, C(ijkl) being the elastic tensor in the medium. The elastic energy for an inclusion with a constant eigenstrain, epsilon(ij)o, in a homogeneous medium, is described by an energy tensor, I, such that the energy, W, is W = (1/2)epsilon(ij)(o)I(ijkl)epsilon(kl)o. The components of I-which depend on the geometry (shape and orientation) of the inclusion-satisfy a linear, geometry-independent relation of the form B(ijkl)I(ijkl) = 3V, 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: B(ikjl) I(ijkl) = V(5 + 3-nu-2)/[2(1-nu-2)] (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 S(ijij) = 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 = (1/2) integral-D P(k-alpha)L(km-alpha-beta)-1P(m-beta) d3r, where L(r) is the response-matrix of the medium, and P(k-alpha)(r) is the alpha-space space component of the k-type external polarization field in the inclusion. Again, when the polarization fields are constant, W is described in terms of an energy tensor I(km-alpha-beta). We find that its components satisfy n(n + 1)/2 geometry-independent relations (n is the number of coupled fields) L(km-alpha-beta)I(mp-alpha-beta) = delta(kp)V. 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 tensorial ranks, such as a piezoelectric or piezo-magneto-electric medium.