An Introduction to the Mathematical Theory of Dynamic Materials by Konstantin A. Lurie

Author:Konstantin A. Lurie
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

4.3 A Stable G-Closure of a Set U of Two Isotropic Dielectrics with Respect to One-Dimensional Wave Propagation

The original set U will be first assumed consisting of two original materials with positive material properties ε i , μ i ,   i = 1, 2. As shown in Section 3.​7, every such material generates a set of spatio-temporal polycrystals; their effective constants fill the hyperbola for the first material, and for the second. Figure 4.1 reproduces these hyperbolas in the first quadrant of -plane, with the assumption ε 1∕μ 1 > ε 2∕μ 2. All points on each hyperbola are attainable except the point on the diagonal; the relevant polycrystals are uniformly stable, and therefore, each hyperbola represents a stable G-closure of a set of differently oriented fragments of the same original material.

Fig. 4.1A stable G-closure of a set of two isotropic dielectrics of the same sign

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