If a neutral inclusion is inserted in a matrix containing a uniform applied electric field, it does not disturb the field outside the inclusion. The well known Hashin coated sphere is an example of a neutral coated inclusion. In this paper, we consider the problem of constructing neutral inclusions from nonlinear materials. In particular, we discuss assemblages of coated spheres and the two-dimensional analogous problem of assemblages of coated disks.
The equations of motion for an elastic laminar spatial-temporal composite are investigated. It is assumed that the composite is binary, that is, it is assembled of two original constituents capable of changing (in space-time) their material density, as well as their material stiffness. The condition of plane strain was then imposed on the composite. The paper begins by attempting to evaluate the materials' average Lagrangian (action density). In doing so, it immediately becomes apparent that expressions are needed for average momentum and stress. Both quantities are found to depend linearly on average strain and average velocity. After calculating the general Euler equations of motion, isotropy was assumed, and two additional forces (one being of a Coriolis nature) were found in the averaged equations of elastodynamics due to the presence of simultaneous change in both inertial and elastic properties of the original material constituents. The appearance of these two forces is a consequence of both dynamics and plane strain; the Coriolis type force disappears in the case of one dimensional strain that arises when longitudinal dynamic disturbances propagate along an elastic bar.
We study the propagation of dilatational and shear waves through an isotropic elastic material having the dilatation and shear moduli variable in space and time. Two isotropic materials alternate occupying rectangular cells in 1D space + time producing a double periodic checkerboard material assembly. The materials are assumed to differ in their wave velocities (dilatational and shear) and to have pairwise equal values of wave impedances for each type of wave. We show, however, that, for both types of waves traveling normally to spatial interfaces between the materials, the average velocity of propagation is the same for certain ranges of material and structural parameters. Also, the energy is accumulated in those waves, and such accumulation occurs in very narrow pulses. This is unlike the wave propagation in a uniform static material, where both types of waves propagate at different speeds. The coincidence of average speeds of propagation appears to be due to the checkerboard material geometry. It creates the "plateau effect" within the ranges of material and structural parameters mentioned above [3,4,6]. These ranges do not include the purely uniform material. In elastodynamics, the concept of dynamic materials has been previously studied in references [2,7].
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