TAILIEUCHUNG - Waves in fluids and solids Part 7

Tham khảo tài liệu 'waves in fluids and solids part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Multiple Scattering of Elastic Waves in Granular Media Theory and Experiments 139 and K is the time-evolution operator given by the 13x13 matrix K 0 rìĩì 0 3X3 0 J Pjj 0 9 jl pT . 2U r _ pT 1 . ị c Ljk 1 v 2r r 1 20 2 ejk 0 9x9 19 1 VĨŨ where we have defined the operator p iV and introduced the third-rank tensor Ljki 1 2 pyâkl Pkõji . In Eqs. 18 and 19 in order to gain compactness in writing the vector and matrix representations the notation 0 9T is used to signify a column array consisting of nine zeros. Conversely 0 9 is employed to denote a row array of nine zeros filling the right top part of the matrix. In addition 0 3X3 and 0 9X9 indicate square arrays of 3 X 3 and 9 X 9 zeros respectively. A similar time-evolution operator to Eq. 19 was previously obtained by Trégourès van Tiggelen 2002 for elastic wave scattering and transport in heterogeneous media except for the adding term PT 2jậ 2r jk between square brackets in the middle of the right column of Eq. 19 . It arises because of the additional term that appears in the Jiang-Liu formulation of the elastic stress see Eq. 10 compared to the traditional expression given by Eq. 12 . It is this remarkable difference along with the stress-dependent moduli that allow for a theoretical description of granular features such as volume dilatancy mechanical yield and anisotropy in the stress distribution which are always absent in a pure elastic medium under deformation. 4. Multiple scattering radiative transport and diffusion approximation In the previous section we have presented the main steps to build up a theory for the propagation of elastic waves in disordered granular packings. Now we proceed to develop the rigorous basis to modeling the multiple scattering and the diffusive wave motion in granular media by employing the same mathematical framework used to describe the vibrational properties of heterogeneous materials Frisch 1968 Karal Keller 1964 Ryzhik et al. 1996 Sheng 2006 Weaver 1990 . The inclusion of .

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