Fundamental Solutions for Periodic Media

Abstract

Necessity for the periodic fundamental solutions arises when the periodic boundary value problems should be analyzed. The latter are naturally related to problems of finding the homogenized properties of the dispersed composites, porous media, and media with uniformly distributed microcracks or dislocations. Construction of the periodic fundamental solutions is done in terms of the convergent series in harmonic polynomials. An example of the periodic fundamental solution for the anisotropic porous medium is constructed, along with the simplified lower bound estimate.

Elastic Waves in Spatially Periodic Anisotropic Media: Diffraction by Periodic Arrays of Voids or Anisotropic Inclusions

“…The region occupied by an individual inhomogeneity in a cell Q is denoted by Ω. The two-scale asymptotic analyses being applied to such a medium produces the following expression for the corrector [12]: (11) where Y are the "fast" variables, H is the third-order tensorial field, being a solution of the following boundary value problem:…”

“…In Eqs. (11) and (12) ν Y represents a field of external unit normals to the boundary ∂Ω, and the elasticity tensor Ω is defined by:…”

“…Proof of the lemma can be found in [11,12]. Remark : Supposition that the tensor C in the left-hand side of Eq.…”

“…Periodic fundamental solutions for media with arbitrary anisotropy were developed in [11] coupled with the boundary integral equation method (BIEM); that approach was applied to solution of the cell problem for composites with anisotropic inhomogeneities and porous anisotropic media in [12,13], analysis of microstructural stresses in the matrix material was considered in [14]. Some dynamic problems were studied in [15] by the same method.…”

Polarization of the Longitudinal Pochhammer–Chree Waves

“…In [6,15] the long-wave limit c 2,lim was named as the "rod" wave speed. Dispersion curves related to higher axially symmetric modes were studied in [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. In [8] the first several roots of the dispersion equation were (numerically) obtained and it was revealed that some of the roots were complex relating to attenuating modes.…”