含定向非均匀体固体材料的横观各向同性有效弹性模量

宋永佳; 胡恒山

doi:10.7498/aps.63.016202

摘要

针对非均匀体定向指向的情况，将非均匀体对弹性波的散射等效为球形有效体对弹性波的散射，推导出了呈现横观各向同性的有效弹性模量. 理论分析表明：本文得到的有效模量公式至少具有二阶精度. 数值计算表明：本文的有效模量随孔隙度增加而降低，不会出现Hudson模型中在孔隙度超过一定阈值后不减反增的问题. 另外，在非均匀体指向随机的情况，本文得到的有效模量公式归结为Kuster-Toksöz模型的公式. 对于由扁状裂隙作为非均匀体的孔隙岩石，裂隙内液体主要影响横观各向同性对称轴方向的纵波模量.

关键词:

Abstract

In this paper the effective or overall moduli of a solid containing aligned ellipsoidal inhomogeneties, are derived by making the scattered displacement field equal to that scattered by a spherical–shaped effective medium in the same matrix. It is shown that the obtained formulae of effective elastic moduli are of second-order accuracy at least. The effective moduli decrease monotonically with porosity, thus excluding the unphysical behavior in Hudson’s model in which there is an increase of moduli with porosity when the porosity goes beyond a certain threshold. By integration of inhomegeneity orientation angle, the effective moduli can be obtained for a solid with randomly orientated inhomogeneities, which are the same as those in the Kuster-Toksöz model. Numerical calculations show that a rock with fluid-saturated inhomegeneities has a higher longitudinal wave modulus in the direction of TI symmetric axis than the modulus for a rock with empty inhomegeneities.

Keywords:

作者及机构信息

1.
哈尔滨工业大学, 航天科学与力学系, 哈尔滨 150001

基金项目: 国家自然科学基金（批准号：41174110，11372091）资助的课题.

Authors and contacts

1.
Department of Astronautics and Mechanics, Harbin Institute of Technology, Harbin 150001, China

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 41174110, 11372091).

参考文献

[1]	Du S, Wang B 1998 Micromechanics of composites (Beijing: Science Press) p5–8 [杜善义, 王彪 1998 复合材料细观力学 (北京: 科学出版社) 第5–8页]
[2]	Mavko G, Mukerji T, Dvorkin J 2009 The rock physics handbook second edition (Cambridge: Cambridge University Press) pp183–205
[3]	Thomsen L 1986 Geophysics 51 1954
[4]	He X, Hu H 2009 Geophysics 74 E149
[5]	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]
[6]	Liu E, Chapman M, Zhang Z, Queen J H 2006 Wave Motion 44 44
[7]	Gurevich B, Makarynska D, Pervukhina M 2009 Geophysics 74 N25
[8]	Song Y, Hu H 2013 Chinese Journal of Theoretical and Applied Mechanics 45 395 (in Chinese) [宋永佳, 胡恒山 2013 力学学报 45 395]
[9]	Eshelby J D 1957 Proc. R. Soc. London, Ser. A 241 376
[10]	Eshelby J D 1959 Proc. R. Soc. London, Ser. A 252 561
[11]	O’Connell R J, Budiansky B 1974 J. Geophys. Res. 79 5412
[12]	Berryman J G, Berge P A 1996 Mechanics of Materials 22 149
[13]	Bruner W M 1976 J. Geophys. Res. 81 2573
[14]	Sun Z, Garboczi E J, Shah S P 2007 Cement and Concrete Composites 29 22
[15]	Kuster G T, Toksö z M N 1974 Geophysics 39 587
[16]	Cheng C H 1993 J. Geophys. Res. 98 675
[17]	Hudson J A 1994 Geophys. J. Int. 117 555
[18]	Chapman M 2003 Geophysical Prospecting 51 369
[19]	Jakobsen M, Hudson J A, Johansen T A 2003 Geophys. J. Int. 154 533
[20]	Kong L Y, Wang Y B, Yang H Z2013 Acta Phys. Sin. 62 139101 (in Chinese) [孔丽云, 王一博, 杨慧珠 2013 62 139101]
[21]	Shen Wei, Fan Qun-Bo, Wang Fu-Chi, Ma Zhuang 2013 Chin. Phys. B 22 044601
[22]	Mal A K, Knopoff L 1967 J. Inst. Math. Appl. 3 376
[23]	Miles J W 1960 Geophysics 25 642
[24]	Wu T T 1966 Int. J. Solids Structures 2 1
[25]	Qu J, Cherkaoui M 2006 Fundamentals of micromechanics of solids (New Jersey: John Wiley & Sons, Inc.) pp87
[26]	Mura T 1987 Micromechanics of defects in solids (Dordrecht: Martinus Nijhoff Publishers) pp79
[27]	David E C, Zimmerman R W 2011 Int. J. Solids Structures 48 680
[28]	Auld B A 1973 Acoustic fields and waves in solids (New York: John Wiley & Sons, Inc.) pp211–212
[29]	Hill R 1963 J. Mech. Phys. Solids 11 357
[30]	Berryman J G 1979 Appl. Phys. Lett. 35 856

施引文献

[1]	Du S, Wang B 1998 Micromechanics of composites (Beijing: Science Press) p5–8 [杜善义, 王彪 1998 复合材料细观力学 (北京: 科学出版社) 第5–8页]
[2]	Mavko G, Mukerji T, Dvorkin J 2009 The rock physics handbook second edition (Cambridge: Cambridge University Press) pp183–205
[3]	Thomsen L 1986 Geophysics 51 1954
[4]	He X, Hu H 2009 Geophysics 74 E149
[5]	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]
[6]	Liu E, Chapman M, Zhang Z, Queen J H 2006 Wave Motion 44 44
[7]	Gurevich B, Makarynska D, Pervukhina M 2009 Geophysics 74 N25
[8]	Song Y, Hu H 2013 Chinese Journal of Theoretical and Applied Mechanics 45 395 (in Chinese) [宋永佳, 胡恒山 2013 力学学报 45 395]
[9]	Eshelby J D 1957 Proc. R. Soc. London, Ser. A 241 376
[10]	Eshelby J D 1959 Proc. R. Soc. London, Ser. A 252 561
[11]	O’Connell R J, Budiansky B 1974 J. Geophys. Res. 79 5412
[12]	Berryman J G, Berge P A 1996 Mechanics of Materials 22 149
[13]	Bruner W M 1976 J. Geophys. Res. 81 2573
[14]	Sun Z, Garboczi E J, Shah S P 2007 Cement and Concrete Composites 29 22
[15]	Kuster G T, Toksö z M N 1974 Geophysics 39 587
[16]	Cheng C H 1993 J. Geophys. Res. 98 675
[17]	Hudson J A 1994 Geophys. J. Int. 117 555
[18]	Chapman M 2003 Geophysical Prospecting 51 369
[19]	Jakobsen M, Hudson J A, Johansen T A 2003 Geophys. J. Int. 154 533
[20]	Kong L Y, Wang Y B, Yang H Z2013 Acta Phys. Sin. 62 139101 (in Chinese) [孔丽云, 王一博, 杨慧珠 2013 62 139101]
[21]	Shen Wei, Fan Qun-Bo, Wang Fu-Chi, Ma Zhuang 2013 Chin. Phys. B 22 044601
[22]	Mal A K, Knopoff L 1967 J. Inst. Math. Appl. 3 376
[23]	Miles J W 1960 Geophysics 25 642
[24]	Wu T T 1966 Int. J. Solids Structures 2 1
[25]	Qu J, Cherkaoui M 2006 Fundamentals of micromechanics of solids (New Jersey: John Wiley & Sons, Inc.) pp87
[26]	Mura T 1987 Micromechanics of defects in solids (Dordrecht: Martinus Nijhoff Publishers) pp79
[27]	David E C, Zimmerman R W 2011 Int. J. Solids Structures 48 680
[28]	Auld B A 1973 Acoustic fields and waves in solids (New York: John Wiley & Sons, Inc.) pp211–212
[29]	Hill R 1963 J. Mech. Phys. Solids 11 357
[30]	Berryman J G 1979 Appl. Phys. Lett. 35 856

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