Asymptotic models of fields in dilute and densely packed by A. B. Movchan

By A. B. Movchan

This monograph offers a scientific research of asymptotic versions of continuum mechanics for composite buildings, that are both diluted or densely packed. textual content covers static difficulties of elasticity in dilute composites in addition to spectral difficulties.

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Let T represent Green's tensor for the system of Navier equations in two dimensions 1+ T(aj) = q *2 1+ 2 „ 2 here the constants q and x are defined by « = 5 ^ A W * =3-4'' where i> is the Poisson ratio. 86) j=i where T^\ j = 1,2, represent the columns of Green's tensor T; VJ (gj) are the first-order differential operators. Definition. 86) is said to be the dipole matrix of the void g. 2 Physical interpretation Let S denote the vector of strain defined by the formula 5 ( u ) = (en(u),e 2 2 (u), V2e 1 2 (u)) T , where the strain components tij are defined in a standard way Let u° denote the unperturbed diplacement field, before the inclusion/void is introduced in the elastic plane.

171) Discrete lattice (A) approximations _sL ^f (B) Vl v s-l us vs us+i vs+i Fig. 5 One-dimensional arrays of two layers (A) and two particles with springs (B). 1 r / 2/ 2 f i 1 ea2)p/m. 5B). Let d be a period of the array. 175) with unknown amplitudes u,v. 174) we have { (w 2 m! - 2c)u + c(l + e-ikod)v = 0, ikod )u + (w 2 m 2 - 2c)v = 0. c ( l 4. e ^ ^' ' 50 Long and close range interaction within elastic structures Fig. 6 Acoustic and optical bands for a one-dimensional array of particles connected by linear springs a = 1, m i = 1, m,2 — 2 and c = 10.

150) ~ a2(2m - 3)c_ ( 2 m _ 3 ) -aa 2 ' 2 ™" 1 ) f > _ ( 2 i _ 1 } ( 2 Z + J ™ - 3 ) S 2 m + 2 i _ 2 = 0, m > 2. 144) we obtain a finite system of linear algebraic equations. We note that the multipole coefficients decay rapidly as we increase the order of the multipole. 5 Integral approximation for the multipole Inclusions close to touching coefficients. 1 Scalar problem Here we describe the idea proposed by McPhedran, Poladian and Milton (1988) for a scalar conductivity problem in a two-dimensional domain containing circular highly conducting inclusions, which are close to touching.

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