横向各向同性固体材料中含定向非均匀体的有效弹性模量

许松; 唐晓明; 苏远大

doi:10.7498/aps.64.206201

摘要

针对含定向非均匀体的横向各向同性复合材料(即TI介质), 采用球形有效体散射等效的方法, 根据TI材料下的D, Nij表达式, 对横向各向同性条件下Eshelby 张量的积分通用表达式进行化简, 推导出了复合材料的具有横向各向同性特性的有效弹性模量的表达式, 并依此进行了数值分析. 计算结果表明: 利用本方法计算的有效模量随非均匀体含量的增大而减小; 定向排列的非均匀体影响横向各向同性介质的固有各向异性, 水平指向的非均匀体会增大材料的横向各向同性, 模拟结果对评价含非均匀体各向异性介质的特征具有指导意义.

关键词:

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:

作者及机构信息

1.
中国石油大学(华东)地球科学与技术学院, 青岛 266580

基金项目: 国家重点基础研究发展计划(批准号: 2014CB239006)、国家自然科学基金(批准号: 41474092, 41174088)和中央高校基本科研业务费(批准号: 14CX06076A)资助的课题.

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).

参考文献

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[2]	Hu H S 2003 Acta Phys. Sin. 52 1954 (in Chinese) [胡恒山 2003 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 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
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[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 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 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 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 57 3431]
[41]	Song Y J, Hu H S 2014 Acta Phys. Sin. 63 016202 (in Chinese) [宋永佳, 胡恒山 2014 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

施引文献

[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 57 1999]
[2]	Hu H S 2003 Acta Phys. Sin. 52 1954 (in Chinese) [胡恒山 2003 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 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 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 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 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 57 3431]
[41]	Song Y J, Hu H S 2014 Acta Phys. Sin. 63 016202 (in Chinese) [宋永佳, 胡恒山 2014 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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