Self-Consistent Methods for Composites: Vol.2: Wave by S.K. Kanaun, V. Levin

By S.K. Kanaun, V. Levin

This designated e-book is devoted to the applying of self-consistent ways to the answer of static and dynamic difficulties of the mechanics and physics of composite fabrics. The powerful elastic, electrical, dielectric, thermo-conductive and different houses of composite fabrics bolstered via ellipsoidal, round multi-layered inclusions, skinny tough and gentle inclusions, brief fibers and unidirected multi-layered fibers are thought of. The publication comprises many concrete effects.

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86) Here the spherical Bessel functions in the vector spherical harmonics (1) (1) (3) (3) M01n , Ne1n , M01n and Ne1n have the arguments kr if 0 ≤ r ≤ r0 , the arguments k0 r if r0 < r < r1 , and k∗ r if r ≥ r1 . 86) are to be found from the boundary conditions on the two boundaries: the kernel - layer (r = r0 ) and layer - effective medium (r = r1 ). 87) where [f (r)]j = f (rj +0)−f (rj −0) is a jump of the function at the boundaries, rj ± 0 = limδ→0 (rj ± δ), δ > 0. 88) 38 3. 89) m86 = −k0 hn (k0 r1 ), = fn(4) m87 = k∗ hn (k∗ r0 ), = 0, fn(5) = jn (ke r1 ), fn(6) = k∗ Djn (k∗ r1 ), fn(7) = Djn (k∗ r1 ), fn(8) = k∗ jn (k∗ r1 ).

4), then G0 (x) − grad divG0 (x) + ω 2 ε0 · G0 (x) = −δ(x)1. 6) Here 1 is the second order unit tensor with the components δij , δ(x) is the 3D-Dirac delta-function. 7) G(x) = ω 2 G0 (x). 8) Here integration is spread over all 3D-space, E0 (x) is the incident field that would have existed in the medium without inclusions and by the given sources of the field. 9) where k0 is the wave vector for the matrix material, U0 is the polarization vector. 9) are orthogonal (E0 (x) is the transverse wave). 7) has been considered by many authors, and is in essence the equation for the field E(x) inside the inclusions (in the region V ).

135) where the function Φ(x) is equal to zero outside a certain vicinity of the origin (x = 0). 134) is a convolution equation. This fact has two consequences. First, the plane waves E(x) and E∗ (x) have the same wave vector k∗ . E(x) = Ue−ik∗ ·x , E∗ (x) = U∗ e−ik∗ ·x . 138) G(x)Φ(x)eik·x dx. 134) we obtain the following equation for the Fourier transform of the mean wave field E(k) : E(k) = E0 (k) + pG(k) · ε1 · Λ0 (k∗ ) · Π(k) · E(k) . 30): L0 (k) = ε0 k02 k 2 − k02 1 − k 2 m ⊗ m . Because L0 (k) · E0 (k) = 0, the Fourier transform of the mean electric field in the composite should satisfy the equation: [(k2 − k02 )1 − k 2 m ⊗ m − pk02 ε¯1 · Λ0 (k∗ ) · Π(k)] · E(k) = 0, 1 ε1 .

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