Effective elastic modulus of a transverse isotropy solid with aligned inhomogeneity

Xu Song; Tang Xiao-Ming; Su Yuan-Da

doi:10.7498/aps.64.206201

Abstract

The effective modulus of transversely isotropic compound material containing aligned ellipsoidal inhomogeneity is derived using the method of sphere-equivalency of effective scattering. Based on this approach, we derive the integral solution of the Eshelby tensor for the anisotropic medium, allowing for numerically evaluating the effects of anisotropy for the solution. The numerical results show that the effective modulus of the medium decreases monotonically with increasing the concentration of the inhomogeneties. The anisotropy increases if the inhomogeneity alignment direction is perpendicular to the TI symmetry axis of the background medium. By reducing the numbers of matrix elastic modulus from 5 to 2, we calculate the slowness surfaces for the three modes of propagation in an isotropic medium containing aligned ellipsoidal inhomogeneity. The results are the same as the existing ones, which validates the exactness of our theory. The modeling results can be used to evaluate elastic property of an anisotropic medium with aligned inclusions, such as earth formation shale rocks containing aligned cracks.

Keywords:

Authors and contacts

1.
School of Geosciences, China University of Petroleum, Qingdao 266580, China
Funds: Project supported by the National Basic Research Program of China (Grant No. 2014CB239006), the National Natural Science Foundation of China (Grant Nos. 41474092, 41174088), and the Fundamental Research Funds for the Central Universities, China (Grant No. 14CX06076A).

References

[1]	Xu S, Su Y D, Chen X L, Tang X M 2014 Chin. J. Geophys. 57 1999 (in Chinese) [许松, 苏远大, 陈雪莲, 唐晓明 2014 地球物理学报 57 1999]
[2]	Hu H S 2003 Acta Phys. Sin. 52 1954 (in Chinese) [胡恒山 2003 物理学报 52 1954]
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[5]	Crampin S 1985 Geophysics 50 142
[6]	Tang X 2003 Geophysics 68 118
[7]	Sinha B K, Norris A N, Chang S K 1994 Geophysics 59 1037
[8]	He X, Hu H 2009 Geophysics 74 E149
[9]	White J E, Tongtaow C 1981 J. Acoust. Soc. Am. 70 1147
[10]	Zhang B, Dong H, Wang K 1994 J. Acoust. Soc. Am. 96 2546
[11]	Li X Q, Chen H, He X, Wang X M, Cong J S 2013 Chin. J. Geophys. 56 3212 (in Chinese) [李希强, 陈浩, 何晓, 王秀明, 丛健生 2013 地球物理学报 56 3212]
[12]	Eshelby J D 1957 Proc. R. Soc. London, Ser. A 241 376
[13]	Walsh J B 1965 J. Geophys. Res. 70 381
[14]	O'Connell R J, Budiansky B 1974 J. Geophys. Res. 79 5412
[15]	Budiansky B, O'connell R J 1976 Int. J. Solids Structures 12 81
[16]	O'Connell R J, Budiansky B 1977 J. Geophys. Res. 82 5719
[17]	Budiansky B, O'Connell R J 1980 Solid Earth Geophys. Geotech. 42 1
[18]	O'Connell R J 1984 Physics and Chemistry of Porous Media 107 166
[19]	Kuster G T, Toksöz M N 1974 Geophysics 39 587
[20]	Biot M A 1956 J. Acoust. Soc. Am. 28 168
[21]	Biot M A 1956 J. Acoust. Soc. Am. 28 179
[22]	Biot M A 1962 J. Acoust. Soc. Am. 34 1254
[23]	Tang X 2011 Sci. China: Earth Sci. 41 784 (in Chinese) [唐晓明 2011 中国科学:地球科学 41 784]
[24]	Tang X M, Chen X L, Xu X K 2012 Geophysics 77 D245
[25]	Gassmann F 1951 ber Die Elastizität Poröser Medien: Vierteljahrss-chrift Der Naturforschenden (Zurich: Gesellschaft) pp1-23
[26]	Budiansky B 1965 J. Mech. Phys. Solids 13 223
[27]	Hill R 1965 J. Mech. Phys. Solids 13 213
[28]	Cleary M P, Lee S M, Chen I W 1980 J. Engrg. Mech. Div. 106 861
[29]	Norris A N, Sheng P, Callegari A J 1985 J. Appl. Phys. 57 1990
[30]	Zimmerman R W 1991 Compressibility of Sandstones (New York: Elsevier) pp10-40
[31]	Xu S, White R E 1995 Geophys. Prospect. 43 91
[32]	Hudson J A 1980 Math. Proc. Camb. Phil. Soc. 88 371
[33]	Cheng C H 1993 J. Geophys. Res. 98 675
[34]	Schoenberg M 1980 The J. Acoust. Soc. Am. 68 1516
[35]	Schoenberg M 1983 Geophys. Prospect. 31 265
[36]	Schoenberg M, Sayers C M 1995 Geophysics 60 204
[37]	Kong L Y, Wang Y B, Yang H Z 2012 Chin. J. Geophys. 55 189 (in Chinese) [孔丽云, 王一博, 杨慧珠 2012 地球物理学报 55 189]
[38]	Kong L Y, Wang Y B, Yang H Z 2013 Acta Phys. Sin. 62 139101 (in Chinese) [孔丽云, 王一博, 杨慧珠 2013 物理学报 62 139101]
[39]	Zhang G Z, Chen H Z, Wang Q, Yin X Y 2013 Chin. J. Geophys. 56 1707 (in Chinese) [张广智, 陈怀震, 王琪, 印星耀 2013 地球物理学报 56 1707]
[40]	Chen H Z, Yin X Y, Zhang J Q, Zhang G Z 2014 Chin. J. Geophys. 57 3431 (in Chinese) [陈怀震, 印兴耀, 张金强, 张广智 2014 地球物理学报 57 3431]
[41]	Song Y J, Hu H S 2014 Acta Phys. Sin. 63 016202 (in Chinese) [宋永佳, 胡恒山 2014 物理学报 63 016202]
[42]	Hornby B E, Schwartz L M, Hudson J A 1994 Geophysics 59 1570
[43]	Brown R J S, Korringa J 1975 Geophysics 40 608
[44]	Sarout J, Guéguen Y 2008 Geophysics 73 D75
[45]	Sarout J, Guéguen Y 2008 Geophysics 73 D91
[46]	Mal A K, Knopoff L 1967 IMA J. Appl. Math. 3 376
[47]	Miles J W 1960 Geophysics 25 642
[48]	Qu J, Cherkaoui M 2006 Fundamentals of Micromechanics of Solids (New Jersey: John Wiley & Sons, Inc.) p87
[49]	Zhu Y, Liu E 2011 SEG Annual Meeting San Antonio, Texas, September 18-23, 2011 SEG-2011-2216
[50]	Kinoshita N, Mura T 1971 Phys. Stat. Sol. 5 759
[51]	Lin S C, Mura T 1973 Phys. Stat. Sol. 15 281
[52]	Walpole L J 1977 Math. Proc. Camb. Phil. Soc. 81 283
[53]	Withers P J 1989 Philos. Mag. A 59 759
[54]	Mura T 1987 Micromechanics of Defects in Solids (Springer Science & Business Media) pp129
[55]	David E C, Zimmerman R W 2011 Int. J. Solids Structures 48 680
[56]	Kachanov M L, Shafiro B, Tsukrov I 2003 Handbook of Elasticity Solutions (Berlin: Springer) pp242-243

Cited By

[1]	Xu S, Su Y D, Chen X L, Tang X M 2014 Chin. J. Geophys. 57 1999 (in Chinese) [许松, 苏远大, 陈雪莲, 唐晓明 2014 地球物理学报 57 1999]
[2]	Hu H S 2003 Acta Phys. Sin. 52 1954 (in Chinese) [胡恒山 2003 物理学报 52 1954]
[3]	Hudson J A 1981 Geophys. J. Int. 64 13
[4]	Crampin S 1984 Geophys. J. Int. 76 135
[5]	Crampin S 1985 Geophysics 50 142
[6]	Tang X 2003 Geophysics 68 118
[7]	Sinha B K, Norris A N, Chang S K 1994 Geophysics 59 1037
[8]	He X, Hu H 2009 Geophysics 74 E149
[9]	White J E, Tongtaow C 1981 J. Acoust. Soc. Am. 70 1147
[10]	Zhang B, Dong H, Wang K 1994 J. Acoust. Soc. Am. 96 2546
[11]	Li X Q, Chen H, He X, Wang X M, Cong J S 2013 Chin. J. Geophys. 56 3212 (in Chinese) [李希强, 陈浩, 何晓, 王秀明, 丛健生 2013 地球物理学报 56 3212]
[12]	Eshelby J D 1957 Proc. R. Soc. London, Ser. A 241 376
[13]	Walsh J B 1965 J. Geophys. Res. 70 381
[14]	O'Connell R J, Budiansky B 1974 J. Geophys. Res. 79 5412
[15]	Budiansky B, O'connell R J 1976 Int. J. Solids Structures 12 81
[16]	O'Connell R J, Budiansky B 1977 J. Geophys. Res. 82 5719
[17]	Budiansky B, O'Connell R J 1980 Solid Earth Geophys. Geotech. 42 1
[18]	O'Connell R J 1984 Physics and Chemistry of Porous Media 107 166
[19]	Kuster G T, Toksöz M N 1974 Geophysics 39 587
[20]	Biot M A 1956 J. Acoust. Soc. Am. 28 168
[21]	Biot M A 1956 J. Acoust. Soc. Am. 28 179
[22]	Biot M A 1962 J. Acoust. Soc. Am. 34 1254
[23]	Tang X 2011 Sci. China: Earth Sci. 41 784 (in Chinese) [唐晓明 2011 中国科学:地球科学 41 784]
[24]	Tang X M, Chen X L, Xu X K 2012 Geophysics 77 D245
[25]	Gassmann F 1951 ber Die Elastizität Poröser Medien: Vierteljahrss-chrift Der Naturforschenden (Zurich: Gesellschaft) pp1-23
[26]	Budiansky B 1965 J. Mech. Phys. Solids 13 223
[27]	Hill R 1965 J. Mech. Phys. Solids 13 213
[28]	Cleary M P, Lee S M, Chen I W 1980 J. Engrg. Mech. Div. 106 861
[29]	Norris A N, Sheng P, Callegari A J 1985 J. Appl. Phys. 57 1990
[30]	Zimmerman R W 1991 Compressibility of Sandstones (New York: Elsevier) pp10-40
[31]	Xu S, White R E 1995 Geophys. Prospect. 43 91
[32]	Hudson J A 1980 Math. Proc. Camb. Phil. Soc. 88 371
[33]	Cheng C H 1993 J. Geophys. Res. 98 675
[34]	Schoenberg M 1980 The J. Acoust. Soc. Am. 68 1516
[35]	Schoenberg M 1983 Geophys. Prospect. 31 265
[36]	Schoenberg M, Sayers C M 1995 Geophysics 60 204
[37]	Kong L Y, Wang Y B, Yang H Z 2012 Chin. J. Geophys. 55 189 (in Chinese) [孔丽云, 王一博, 杨慧珠 2012 地球物理学报 55 189]
[38]	Kong L Y, Wang Y B, Yang H Z 2013 Acta Phys. Sin. 62 139101 (in Chinese) [孔丽云, 王一博, 杨慧珠 2013 物理学报 62 139101]
[39]	Zhang G Z, Chen H Z, Wang Q, Yin X Y 2013 Chin. J. Geophys. 56 1707 (in Chinese) [张广智, 陈怀震, 王琪, 印星耀 2013 地球物理学报 56 1707]
[40]	Chen H Z, Yin X Y, Zhang J Q, Zhang G Z 2014 Chin. J. Geophys. 57 3431 (in Chinese) [陈怀震, 印兴耀, 张金强, 张广智 2014 地球物理学报 57 3431]
[41]	Song Y J, Hu H S 2014 Acta Phys. Sin. 63 016202 (in Chinese) [宋永佳, 胡恒山 2014 物理学报 63 016202]
[42]	Hornby B E, Schwartz L M, Hudson J A 1994 Geophysics 59 1570
[43]	Brown R J S, Korringa J 1975 Geophysics 40 608
[44]	Sarout J, Guéguen Y 2008 Geophysics 73 D75
[45]	Sarout J, Guéguen Y 2008 Geophysics 73 D91
[46]	Mal A K, Knopoff L 1967 IMA J. Appl. Math. 3 376
[47]	Miles J W 1960 Geophysics 25 642
[48]	Qu J, Cherkaoui M 2006 Fundamentals of Micromechanics of Solids (New Jersey: John Wiley & Sons, Inc.) p87
[49]	Zhu Y, Liu E 2011 SEG Annual Meeting San Antonio, Texas, September 18-23, 2011 SEG-2011-2216
[50]	Kinoshita N, Mura T 1971 Phys. Stat. Sol. 5 759
[51]	Lin S C, Mura T 1973 Phys. Stat. Sol. 15 281
[52]	Walpole L J 1977 Math. Proc. Camb. Phil. Soc. 81 283
[53]	Withers P J 1989 Philos. Mag. A 59 759
[54]	Mura T 1987 Micromechanics of Defects in Solids (Springer Science & Business Media) pp129
[55]	David E C, Zimmerman R W 2011 Int. J. Solids Structures 48 680
[56]	Kachanov M L, Shafiro B, Tsukrov I 2003 Handbook of Elasticity Solutions (Berlin: Springer) pp242-243

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Vol. 64, Issue 20 - 2015-10-20

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