Geophys. J. Int. (2001) 145, 19–32
Seismic velocity anisotropy in mica-rich rocks: an inclusion model
O. Nishizawa1 and T. Yoshino2 1 Geological Survey of Japan, 1-1-3 Higashi, Tsukuba Ibaraki 305-8567, Japan. E-mail: [email protected] 2 Department of Civil and Environmental Engineering, Toyo University, 2100 Kujirai Kawagoe, Saitama 350-8585, Japan
Accepted 2000 September 11. Received 2000 September 7; in original form 2000 March 7
SUMMARY We calculated seismic wave velocity anisotropy caused by the preferred orientation of mica minerals by using the differential effective medium method (DEM). Spheroidal Downloaded from https://academic.oup.com/gji/article/145/1/19/620505 by guest on 30 September 2021 biotite crystals with their c-axes coinciding with the symmetry axis of the spheroid are embedded in an isotropic matrix up to a volume ratio of 30 per cent. All crystals are aligned with their c-axes parallel to the symmetry axis of the effective homogeneous medium, which shows transverse isotropy. The effect of crystal shape on anisotropy was studied by changing the aspect ratio (the ratio between the minor axis and the major axis of the spheroid) from 0.01 (flat spheroid) to 1 (sphere). The S-velocity anisotropy becomes large as the crystal shape becomes flat, whereas the P-velocity anisotropy shows only small changes with changes of the crystal shape. In particular, biotite generates large S-wave anisotropy, and the anisotropy becomes stronger as the aspect ratio of the biotite crystal becomes smaller. When the volume ratio of the mica mineral is large, the P-wave phase velocity surface shows considerable deviation from the ellipse, and the SV-wave phase velocity surface forms a large bulge and crosses the SH-wave phase velocity surface (singularity) in the plane including the symmetry axis. These results show an interesting contrast when compared with the effect of crack or pore shape on seismic velocity anisotropy: crack (or pore) shape affects the P velocity more than the S velocity. We also calculated Thomsen’s anisotropic parameters, e, c and d as functions of the crystal aspect ratio and the mica volume ratio. Key words: anisotropy, composite media, mica-rich rocks, seismic wave propagation.
& Cara 1991; Barrruol & Mainprice 1993), (2) the preferred 1 INTRODUCTION orientation of cracks and fractures (Crampin 1984), and Seismic anisotropy in the crust has been extensively studied in (3) layered structures of different elastic materials (Backus 1962; geophysics and exploration geophysics (Crampin 1987; Leary Helbig 1984). Among the three origins of crustal anisotropy, et al. 1990; Babusˇka & Cara 1991; Thomsen 1986). Seismic the effects of cracks and preferred orientations of minerals have anisotropy is important in reflection seismology because the been studied by many people because there are many variations seismic wavefront affects conventional reflection data process- of anisotropy associated with those origins. Distributions of ing such as moveout corrections (normal moveout, NMO, and crack orientation and mineral orientation control the rock dipping moveout, DMO) and amplitude variation with offset anisotropy, and the crack shape also controls the anisotropy. (AVO) (Thomsen 1988; Tsvankin 1997a). Shear wave splitting Aligned thin cracks produce strong anisotropy (Anderson et al. seen in seismograms of local earthquakes has been reported 1974; Hudson 1981; Nishizawa 1982; Douma 1988). A similar at many observation sites and used for monitoring tectonic effect may occur in rocks containing aligned thin crystals that activity and for the classification of crustal structures (Crampin have strong anisotropy. It is also interesting to compare the et al. 1984; Booth et al. 1990; Crampin et al. 1990; Kaneshima anisotropy caused by the orientation of minerals and aligned 1990). Shear wave splitting is also used for interpreting vertical cracks. seismic profiling data or for characterizing rock formations by Recently, Takanashi et al. (2000) found an interesting velocity well logging (Liu et al. 1989; Douma & Crampin 1990). anisotropy in biotite schist from Hokkaido, Japan. The aniso- A number of experimental measurements and theoretical tropy is characterized by a remarkable bulge for one of the studies have been carried out in order to understand seismic S-wave phase velocity surfaces and a non-elliptic shape of the anisotropy in rocks (Kern & Schenk 1985). The plausible P-wave phase velocity surface in the plane including the direction origins of seismic anisotropy of the crust are (1) the preferred of lineation. This may result from the preferred orientation of orientation of anisotropic rock-forming minerals (e.g. Babusˇka biotite grains because the rock shows a strong concentration
# 2001 RAS 19 20 O. Nishizawa and T. Yoshino of the biotite c-axes and a large volume ratio of biotite—over of Eshelby’s tensor following the formulations given by Mura 30 per cent. To study the origin of anisotropy in mica-rich (1968) and Mura & Mori (1976). Results were explicitly shown rocks, we modelled the rocks as composite solids containing in Lin & Mura (1973) and quoted in Nishizawa (1982), but here inclusions of mica-group minerals in an initially isotropic matrix. we describe the process for deriving the resultant formulations. The differential effective medium method (DEM) was employed to calculate elastic constants, where we assume that the composite is an effective medium with transverse isotropy due to the mica-mineral inclusions. This is a slight modification 2.2 Fictitious stress-free strain and Eshelby’s tensor of the method of Nishizawa (1982) for the case that the inclusion We assume a solid composite including spheroids of which the is an anisotropic single crystal. Mineral shapes are assumed to elastic properties are different from the surrounding solid. The be oblate spheroids and the effect of the aspect ratio (the ratio spheroid is called the ‘inclusion’ and the surrounding solid is between the minor and major axes of the spheroid) on elastic called the ‘matrix’. Our goal is to obtain elastic properties of anisotropy was investigated. The effect of oriented cracks on the composite material when we regard the composite as an anisotropy in the mica-bearing rocks was also studied. effective homogeneous medium. This assumption is valid only Here we use the notation P, SH and SV waves instead of when the medium is under a static equilibrium condition, or qP qS qS q the general expressions , 1 and 2 waves, where stands when the static approximation is applicable. The condition is Downloaded from https://academic.oup.com/gji/article/145/1/19/620505 by guest on 30 September 2021 for quasi (Winterstein 1990). Since the anisotropy considered is generally satisfied for field seismic observations since the seismic only transverse, P, SV and SH waves are clearly distinguished. wavelength is much larger than the grain sizes of rock-forming Thus, the abbreviated notation does not cause any confusion. minerals. In laboratory experiments, with the shorter wave- length, the assumption of an effective homogeneous medium may not be strictly valid, but the inclusion model is still a useful 2 COMPOSITE MEDIUM THEORY approximation except for large grain-size rocks. When the composite medium is under stress, its elastic 2.1 Eshelby’s theory of composite media energy differs from that of an inclusion-free matrix material that is under the same stress or strain condition. If we can calcu- We consider a composite material that contains inclusions late the elastic energy of the effective homogeneous medium of mineral grains inside a homogeneous material. In such a of the composite for the same strain or stress that appears in material, stress and strain are not uniform and will fluctuate the inclusion-free matrix, we can obtain the elastic constants randomly from their mean values averaged over the material. (or compliance constants) of the effective homogeneous medium The stress and strain fluctuations can be described by their by differentiating the elastic energy with respect to strain autocorrelation functions, which control the elastic properties (or stress). Therefore, calculation of the elastic energy change of a composite material (Kro¨ner 1967). However, in the present in the composite material is a key to obtaining the elastic treatment we are not concerned about autocorrelation functions properties of the effective homogeneous medium. of stress and strain inside the material, and consider a perfectly Stress and strain inside the composite material are disturbed disordered fluctuation. We regard the composite material as an by an inclusion and the disturbance causes an energy change equivalent homogeneous medium and assume homogeneous from the inclusion-free state. In the following, for simplicity, stress and strain fields in that medium. The homogeneous stress we first assume that stress and strain inside the inclusion are and strain correspond to the effective elastic constants, which A A uniform. sij and eij denote the stress and strain of the inclusion- represent the overall properties of the equivalent homogeneous C C free medium, respectively, and sij and eij , respectively, denote medium. the stress and strain disturbance of the composite material To calculate the effective elastic constants, one should first caused by the inclusion. For the inclusion-free state, we have estimate the energy change between the inclusion-free material and the composite material; the stress–strain relation can then A~ 0 A be derived by the differentiation of the energy with respect to pij cijkl ekl ,(1) homogeneous stress or strain (Eshelby 1957). Thus, we are only 0 concerned about the change in elastic energy of the composite where cijkl are the elastic constants of the inclusion-free matrix A C material and we calculate the energy as a function of the elastic material. In the inclusion, stress and strain are given by sij +sij A C constants of the matrix, the inclusion mineral and the volume and eij +eij , respectively, and ratio of the inclusion. We describe fundamental equations to calculate elastic constants, because the composite theory pAzpC~c0 eA zeC ,(2) of an anisotropic matrix is not well known by geophysicists, ij ij ijkl ð kl klÞ although basic treatments have already been given by Mura (1968) and Mura & Mori (1976). where ckijkl are the elastic constants of the inclusion. C Here we describe the following theoretical bases: (i) an To calculate eij , Eshelby (1957) introduced stress-free strain, essential part of Eshelby’s fictitious stress-free strain that is which is defined from the stress-free state of the inclusion. The used for expressing the stress–strain relationship in a com- term eigenstrain is sometimes used with the same meaning posite, (ii) the derivation of Eshelby’s tensor in general aniso- (Mura 1968; Kinoshita & Mura 1971). When the inclusion is tropic media, and (iii) the energy calculation of a composite fluid, the stress-free strain is uniform everywhere inside the medium to obtain the elastic constants of an effective homo- spheroidal inclusion. This corresponds to fluid-filled cracks or geneous medium. (i) and (iii) are described in this section but pores in rocks. When the stress-free strain of the inclusion is (ii) is described in the Appendix as it is more complex. In uniform, the elastic strain of the embedded spheroidal inclusion the Appendix we describe the basis for numerical calculation is also uniform. Eshelby demonstrated that the stress-free strain
# 2001 RAS, GJI 145, 19–32 Velocity anisotropy in mica-rich rocks 21
C 0 has a linear relationship with strain disturbance eij , inclusion, cijkl and ckijkl. We note that eq. (6) is applicable even when both the matrix and the inclusion show anisotropy. A C~ T eij Sijkl ekl ,(3) formulation for calculating the tensor Sijkl for elliptic inclusions in general elastic media is described in Appendix A2. where Sijkl is Eshelby’s tensor. Formulations to obtain Eshelby’s tensor in a general elastic medium are shown in the Appendix. The strain disturbance eC in eq. (3) includes not only elastic ij 2.3 Elastic energy and the effective homogeneous strain but other kinds of strain such as plastic strain and the medium strain caused by thermal expansion or phase transition. When we use the relationship in eq. (3), we note the uniformity of To calculate elastic constants of the composite medium, we stress-free strain, which depends on the shape and elasticity must evaluate the change of elastic energy due to inclusions. of the inclusion. If the inclusion is fluid or an isotropic solid and We regard the composite material as an effective homogeneous the shape of the inclusion is spheroidal, the stress-free strain medium of which the elastic properties are unknown. We is uniform everywhere inside the inclusion. When the inclusion is assume that the inclusion-free matrix is under uniform stress A A an anisotropic single crystal, the uniformity of stress-free strain and strain conditions of sij and eij . When spheroidal inclusions is not obvious. However, we can assume that the stress-free appear inside the matrix, the elastic field of the composite Downloaded from https://academic.oup.com/gji/article/145/1/19/620505 by guest on 30 September 2021 A A strain is close to uniform if the shape of inclusion is spheroidal material changes. The strain eij and the stress sij change C and the inclusion and matrix are transversely isotropic with accompanying the disturbances of strain and stress eij and C their symmetry axes coincident with the unique axis of the sij , respectively. The actual stress and strain distributions inside spheroid. the composite material are very complicated and difficult to C To obtain strain disturbance inside the composite, eij , determine. Instead, we consider the effective elastic constants Eshelby (1957) proposed replacing the inclusion with the matrix of the equivalent homogeneous medium. We also consider material (fictitious inclusion) and assumed a fictitious stress- that either stress or strain inside the homogeneous material T free strain eij of the fictitious inclusion. We have a linear is uniform. To obtain elastic constants, we calculate the change C T relationship between eij and eij as shown in eq. (3) for the of elastic energy in the composite material with respect to C A fictitious strain. We note that eij in eq. (2) includes only elastic the following two extreme cases: (i) uniform stress, sij , and C A strain, but eij in eq. (3) includes both elastic strain and plastic (ii) uniform strain, eij , of the effective medium. In case (i), the C strain because the origin of eij is the fictitious stress-free strain shape of the composite material changes to keep the surface of the matrix material. Since material has a discontinuity at the force constant, and the surface force gives the external work to boundary between matrix and inclusion, the fictitious stress- the material, while in case (ii), the external force should be free strain originates from the difference in elastic properties changed to keep the strain uniform. between the matrix and the inclusion. For example, if the The changes of energy in (i) and (ii) correspond to the inclusion is fluid, the fictitious stress-free strain is the plastic interaction energy between inclusions and the initial elastic field strain that corresponds to the shear strain of the fictitious of the matrix. Eshelby (1957) gave the following relations for inclusion, because there is no shear stress inside the real inclusion each change of elastic energy DE by using the interaction energy and shear stress should also be zero inside the fictitious Eint between the elastic field of the matrix and the inclusion: inclusion. Accordingly, the stress disturbance inside the real inclusion sC can be described by using the fictitious stress-free ij *E ¼ {Eint ,(8) strain. Applying Hooke’s law to the fictitious inclusion, we obtain *E ¼þEint : (9) pC~c0 ðeC {eT Þ : (4) ij ijkl kl kl Eqs (8) and (9) correspond to the conditions (i) and (ii), respectively. Eshelby (1957) and Mura (1968) found that Eqs (2), (3) and (4) are independent equations showing the the interaction energy inside the unit volume can be calculated relationships between sC, eA and eC From eqs (2), (1) and (4), ij ij ij as we have
Az C~ 0 A z C { T 1 pij pij cijkl ðekl ekl ekl Þ : (5) E ~{ pAeT ,(10) int 2 ij ij A Using eqs (3) and (5), we obtain a relationship between eij T and eij , where w is the volume ratio of the inclusion. This equation is valid only when the volume ratio w is very small and inclusions 0 T ~ A z T cijkl ekl *cijkl ekl *cijkl Sklmnemn ,(6) are uniformly distributed because we neglect the secondary disturbance of the elastic field due to other inclusions. DE of where Dcijkl is the difference in the elastic constants between the cases (i) and (ii) show the mechanical energy corresponding matrix and the inclusion, to the change of the Gibb’s and the Helmholtz free energy, respectively. ~ 0 { 0 *cijkl cijkl cijkl : (7) We denote the elastic constants of the effective medium c*ijkl. The energy of the composite medium is given by adding T Eq. (6) shows that the fictitious stress-free strain eij can be the change of the free energy to the energy of the inclusion- A obtained from the strain of the matrix material eij , Eshelby’s free state. For case (i) we have the following relationship for tensor Sijkl, and the elastic constants of the matrix and the the potential energy of the unit volume measured from the
# 2001 RAS, GJI 145, 19–32 22 O. Nishizawa and T. Yoshino stress- and strain-free states: small increase of inclusion volume,
1 {1 1 {1 1 { à A A ~ 0 A A z A T 1 ÃðnÞ 1 Aðn{1Þ Aðn{1Þ cijkl pij pkl cijkl pij pkl pij eij ,(11) c p p 2 2 2 2 ijkl ij ij x1 0 x1 where c*ijkl and c denote the inverses of the matrices ijkl 1 { {1 { { 1 { { c* and c0 , respectively. c* x1 is given by the second-order ¼ c0ðn 1Þ pAðn 1ÞpAðn 1Þz pAðn 1ÞeTðn 1Þ* ,(13) ijkl ijkl ijkl 2 ijkl ij kl 2 ij ij differentials of the potential energy with respect to stress. This is possible because we can express the second term on the right- 1 ÃðnÞ Aðn 1Þ Aðn 1Þ 0 A A cijkl en 1 ijen 1 ij hand side of eq. (11) as cijkl, ckijkl, Sijkl and eij , where eij is given 2 A 0 x1 A by eij =cijkl skl , by using the relationship given by eq. (6). 1 0 n 1 A n 1 A n 1 1 A n 1 T n 1 For case (ii), the relationship between effective elastic ¼ c ð Þe ð Þije ð Þkl p ð Þe ð Þ* ,(14) 2 ijkl n 1 n 1 2 ij ij constants and the stress is (n) (n)x1 1 1 1 where cijkl* and cijkl* are the elastic constants and their inverse à A A ~ 0 A A { A T cijkl eij ekl cijkl eij ekl pij eij : (12) (compliance) obtained at the nth step of calculation, respectively, 2 2 2 0 A A T and the superscripts (nx1) of cijkl, eij , sij and eij denote the We can also solve this equation by using the relationships T(nx1)
values of the (n 1)th step. e is calculated from Eshelby’s Downloaded from https://academic.oup.com/gji/article/145/1/19/620505 by guest on 30 September 2021 A 0 A x ij sij =cijklekl and eq. (6). tensor for the effective homogeneous medium at the (nx1)th It is important to note that the elastic constants obtained by step. DEM evaluates the stress or strain change inside the eqs (11) and (12) give two extreme values. Although the actual composite medium by regarding the composite medium as a stress and strain inside the composite material are not uniform, new homogeneous matrix material with new elastic constants. it is still worthwhile considering the two extreme cases because Since the effective homogeneous medium shows transverse iso- these two values give the lower and upper boundaries of the tropy from the second step of calculation, we need to calculate elastic constants, similar to the Reuss and Voigt boundaries of Eshelby’s tensor for the transversely isotropic medium. the elastic constants of an aggregate. We assume that the shape of mica mineral is an oblate If the difference in elastic constants between the matrix spheroid with major axis a1(=a2) and minor axis a3, and define and the inclusion is large, the difference between the upper and the aspect ratio of the spheroid a=a3/a1. Mica group minerals lower limits of the elastic constants becomes large (Hill 1965), show monoclinic symmetry. However, they are approximated and we cannot exactly determine the elastic constants of the as hexagonal when the crystallographic c-axis is selected per- composite. To solve this problem, McLaughlin (1977) and pendicular to the silicate layer of the mica mineral (Vaughan & Yamamoto et al. (1981) applied a differential method, where Guggenheim 1986). We also assume that the c-axis of the mica a small number of inclusions are embedded in the matrix, mineral corresponds to the minor axis of the oblate spheroid. which is regarded as an effective homogeneous medium deter- After mica minerals are embedded in the matrix, the composite mined by the previous step of the calculation. Starting from the material shows transverse isotropy, with its symmetry axis initially inclusion-free matrix by stepwise increments of the parallel to the c-axis of the mica minerals. inclusion volume, this method calculates the interaction energy We calculated elastic constants of the composite material by between the composite of the previous step and the newly using the following four methods for changing the volume ratio added inclusions. The interaction energy between inclusions of mica (Yamamoto et al. 1981; Nishizawa 1982): is implicitly evaluated. This is called the differential effective medium method (DEM) (e.g. Mavko et al. 1998), and its appli- (1) application of Eshelby’s method for the actual volume cability has been studied by several authors (Nishizawa 1982; ratio with a constant external force condition: inserting the Norris 1985; Hornby et al. 1994; Le Ravalec & Gueguen 1996a; inclusion into the isotropic matrix and using eq. (11) for the Le Ravalec & Gueguen 1996b; Mainprice 1997; Singh et al. mica volume ratio w; 2000). We employed DEM for numerical calculations because (2) application of Eshelby’s method for the actual volume this method can be applicable to transverse isotropy as shown ratio with a constant external displacement condition: inserting by Nishizawa (1982) and Singh et al. (2000). the inclusion into the isotropic matrix and using eq. (12) for the mica volume ratio w; (3) DEM with a constant external force condition: a step- 3 NUMERICAL CALCULATION wise calculation using eq. (13) from the initial isotropic matrix to a large volume ratio; 3.1 Differential effective medium method (DEM) for (4) DEM with a constant external displacement condition: transverse isotropy a stepwise calculation using eq. (14) from the initial isotropic matrix to a large volume ratio. Eqs (11) and (12) cannot be used for large volume ratios because those equations give only the elastic energy of the Calculation by (1) or (2) is called first-order perturbation composite in which inclusions are embedded in the stress or (Le Ravalec & Gueguen 1996a; Le Ravalec & Gueguen 1996b) strain field of the initial inclusion-free matrix, and ignore the and is easily performed because Eshelby’s tensor in an isotropic disturbance of stress or strain caused by other inclusions. The material is given by analytic functions including only the actual stress or strain field of the composite should be between Poisson’s ratio of the isotropic matrix and the aspect ratio of the two extreme cases given by eqs (11) and (12). To obtain the inclusion. Calculation by (3) or (4) needs Eshelby’s tensor elastic constants for a large volume ratio of inclusions, we first for a transversely isotropic matrix because the effective homo- apply eqs (11) and (12) for only a small volume ratio, and then geneous medium shows anisotropy depending on the shape and repeat the same method up to large volume ratios. We start elastic properties of the inclusion. When the inclusion shape is from the initial isotropic matrix and repeat calculation for a spheroidal and its elasticity is isotropic or transversely isotropic
# 2001 RAS, GJI 145, 19–32 Velocity anisotropy in mica-rich rocks 23 with a symmetry axis that coincides with the unique axis of Figs 1(a) and (b) show the change of elastic constants the spheroid, the effective homogeneous medium of the com- against the volume ratio of biotite for C*11, C*33, C*44 and posite also shows transverse isotropy. Eshelby’s tensor for a C*66 (=[C*11xC*12]/2) for the crystal aspect ratio 0.05. On transversely isotropic matrix can be obtained by numerical increasing the mica volume ratio, C*11 and C*66 increase, integration (Lin & Mura 1973; Nishizawa 1982). At each step and C*33 and C*44 decrease. The values obtained from the two of methods (3) and (4), Eshelby’s tensor for a transversely different conditions for the first-order perturbation, (1) and (2), isotropic matrix is calculated and two sets of elastic constants differ from each other, and the difference becomes larger as the are obtained corresponding to eqs (13) and (14), respectively. volume ratio of biotite increases. The first-order perturbation, We selected the volume increment of each step to be Dw=0.01. eqs (11) and (12), can be used only for small volume ratios. The The actual volume ratio of the mica mineral is given by values obtained from the methods (3) and (4) are almost equal. They both appear between the values obtained from methods ~1{ð1{* Þn ,(15) (1) and (2). The increasing elastic constants (C*11 and C*66) where n is the number of steps. We calculated elastic constants calculated by DEM are close to the lower limits obtained from up to w=0.3. method (2), whereas the decreasing elastic constants (C*33 and C*44) calculated by DEM are close to the upper limits obtained from
method (1). Downloaded from https://academic.oup.com/gji/article/145/1/19/620505 by guest on 30 September 2021 4 RESULTS
4.1 First-order perturbation and differential 4.2 Effect of aspect ratio on elastic constants calculations of the effective medium We calculated elastic constants of mica-bearing rocks by using We assume Lame’s parameter l=m=35 GPa for the initial DEM for crystal aspect ratios of 1, 0.5, 0.25, 0.1, 0.05, 0.025 isotropic matrix and use elastic constants of mica-group and 0.01 up to a volume ratio of 30 per cent. We took average minerals as shown in Table 1, which is given in Simmons & values obtained from (3) and (4) as the numerical results. Wang (1971) (original data in Aleksandrov & Ryzhova 1961). Fig. 2 shows the change of elastic constants versus the crystal In Table 1, elastic constants are shown in Voigt notation aspect ratio. C*11 and C*66 increase with decreasing crystal aspect (11, 22, 33, 23, 31, 12 of cijkl correspond to 1, 2, 3, 4, 5, 6 of Cij, ratio, indicating increasing P and SH velocities along the iso- respectively). We use the Voigt notation for representing tropic plane. On the other hand, C*33 and C*44 decrease with the observed and calculated values. The elastic constants of the decreasing aspect ratio, indicating decreases in the P- and S effective homogeneous medium are shown by C*ij velocities (both SV and SH) along the symmetry axis. In biotite, the change of C*44 with crystal aspect ratio is larger than the changes of C*11, C*33 and C*66. C*44 decreases rapidly when
Table 1. Cij (Voigt notation) of mica minerals (unit: GPa). C66= the crystal aspect ratio changes from 1 to 0.1. This suggests that (C11xC12)/2. Data are from Simmons & Wang (1971). the anisotropy of biotite-bearing rocks is characterized by a large decrease of the SV-wave velocity (the slow wave) in the Mineral C C C C C C 11 33 44 66 12 13 isotropic plane. Figs 3(a), (b) and (c) show changes of the elastic constants muscovite 178.0 54.9 12.2 67.8 42.4 14.5 biotite 186.0 54.0 5.8 76.8 32.4 11.6 with respect to the biotite volume ratio. Elastic constants are plotted for different mineral aspect ratios ranging from 0.01
Figure 1. Comparison of the elastic constants calculated by different methods. Symbols show the results obtained from the first-order perturbation of Eshelby’s effective medium model for an isotropic matrix. Open and closed symbols correspond to constant stress and strain conditions, respectively. Lines show the results obtained by DEM using Eshelby’s tensor of transverse isotropy in each calculation step. The two constraints give almost same results for DEM.
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Figure 2. Change of elastic constants as a function of crystal aspect ratio of mica minerals. (a) C*11 and C*33, (b) C*66 and C*44, (c) C*12=(C*11xC*66)/2 and C*13. Closed and open symbols indicate biotite and muscovite, respectively.
to 1. C*11, C*33 and C*66 change almost linearly with increasing phase velocity surface becomes large as the crystal aspect ratio volume ratio of biotite for all aspect ratios. However, C*44 becomes small. decreases rapidly with increasing volume ratio of biotite when Figs 4(c) and (d) show the SV-andSH-wave phase velocities in the crystal aspect ratio is small. This also suggests that the SV the same quadrant section. The SV-wave phase velocity surface velocity in the isotropic plane is strongly affected by the biotite shows a large deviation from a circle and a bulge extending volume ratio. around the direction 45u from the symmetry axis. This is a typical case of strong SV-wave anisotropy (Banik 1987). The SV-wave anisotropy is expressed by eSV, V ðn=4Þ{V ð0Þ 4.3 Phase velocity e ~ SV S ,(16) SV V ð0Þ Phase velocity in an arbitrary direction can be calculated S by solving Christoffel’s equation. We assumed a rock density where VSV(p/4) and VS(0) are the velocity of the SV wave of 2.75r103 kg mx3. Figs 4(a) and (b) show P-wave phase propagating in the direction of the polar angle p/4 (measured velocities for rocks containing 30 per cent volume ratio biotite from the z-axis) and the S velocity along the symmetry axis and muscovite with different aspect ratios. The velocity surfaces (no splitting), respectively. eSV increases with decreasing aspect are one quadrant of a planar section containing the symmetry ratio of the mica crystal, and results in remarkable S-wave axis (z-axis) and perpendicular to the isotropic plane (xy-plane). splitting around the incident angle p/4. This large bulge pro- For all aspect ratios, P-wave velocity surfaces are non-elliptical duces cusps in the group velocity surface of the SV wave (Banik for both biotite and muscovite, and the shapes are almost the 1987). A line singularity appears between the incident angle p/4 same for all aspect ratios. The non-ellipticity of the P-wave and p. The fast S wave changes its polarity at the line singularity.
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Figure 3. Change of elastic constants of biotite-bearing rock as a function of biotite volume ratio.
where the directions of the seismic wave are indicated by the 4.4 Phase velocity and Thomsen’s approximation polar angles of the wave vector, zero, p/2 or p/4. The phase We can simplify the velocity calculation in transversely isotropic velocity surfaces for P, SV and SH waves at the polar angle h media by using Thomsen’s anisotropic parameters (Thomsen are given by 1986) when anisotropy is weak. It is interesting to compare the 2 2 4 phase velocities of exact calculations and those of Thomsen’s VPðhÞ¼VPð0Þð1zd sin h cos hze sin hÞ ,(20) approximation. Thomsen’s anisotropic parameters, e, c and d 2 2 are determined by the velocities of the axial directions and the VSV ðhÞ¼VSð0Þð1 þ p sin h cos hÞ ,(21)
P velocity of the selected direction, 2 VSH ðhÞ¼VSð0Þð1zc sin hÞ ,(22) C {C V ðnÞ{V ð0Þ e ¼ 11 33 & P P ,(17) 2C33 VPð0Þ where s is given by C C V ðnÞ V ð0Þ 66 44 SH SH c ¼ & ,(18) 2 2C44 VSH ð0Þ V ~ Pð0Þ { 2 2 p ðe dÞ : (23) ðC zC Þ {ðC {C Þ VSð0Þ d ¼ 13 44 33 44 2C13ðC33{C44Þ Figs 5(a) and (b) and 6 show examples of phase velocity VPðn=4Þ VPðn=2Þ surfaces plotted on a quadrant including the symmetry axis. &4 {1 { {1 ,(19) VPð0Þ VPð0Þ Thomsen’s approximation for the P velocity is close to the exact
# 2001 RAS, GJI 145, 19–32 26 O. Nishizawa and T. Yoshino Downloaded from https://academic.oup.com/gji/article/145/1/19/620505 by guest on 30 September 2021
Figure 4. Phase velocity surface of the P, SV and SH waves for rock containing 30 per cent volume ratio of biotite and muscovite plotted on one quadrant of the planar section. calculation for all aspect ratios. On the other hand, for the Figs 7(a) and (b) show the change of Thomsen’s parameter SV wave, Thomsen’s approximation deviates from the exact as a function of volume ratio with different aspect ratios for values when the aspect ratio becomes small. biotite and muscovite, respectively. c shows a large change Although Thomsen’s anisotropic parameters can be used for both aspect ratio and volume ratio. Changes in e are not as only for weak anisotropy, it is still important to compare them large as the changes in c for both biotite- and muscovite-rich for the strong anisotropic cases because they characterize the rocks. c in biotite-rich rock is strongly affected by the crystal anisotropy. The difference between e and d shows the deviation aspect ratio and exceeds 0.5 with increasing volume ratio of the phase velocity surface from the elliptic shape, and it also and decreasing aspect ratio. For muscovite and biotite, the affects the SV-wave anisotropy, eSV. differences between e and d become large as the volume ratio of
Figure 5. Deviation from ellipticity of the P-wave phase velocity surface and an approximation by Thomsen’s anisotropic parameter. Thomsen’s anisotropic parameters are calculated by the axial P- and S-velocity data and the P velocity of the p/4 incident angle.
# 2001 RAS, GJI 145, 19–32 Velocity anisotropy in mica-rich rocks 27
Figure 6. Comparison of exact phase velocity surfaces of SV and SH waves with the phase velocity surface approximated by Thomsen’s parameters.
Thomsen’s anisotropic parameters are calculated from the velocity data. Thomsen’s approximation is applicable only for small aspect ratio biotite Downloaded from https://academic.oup.com/gji/article/145/1/19/620505 by guest on 30 September 2021 crystals.
the mica mineral increases or as the crystal aspect ratio becomes small. In the biotite-rich rocks in particular, the non- ellipticity of the P-wave phase velocity surface becomes very clear with increasing volume ratio of biotite or decreasing crystal aspect ratio.
4.5 Effect of cracks When the crack shape is spheroidal, the effect of cracks on the anisotropy of mica-bearing rock can be calculated by using the same method as described by Nishizawa (1982). In the calculation, we insert cracks into an initially transverse iso- tropic medium of the mica-rich rock, whose elastic constants have already been obtained from the previous DEM calcu- lations. Figs 8(a) and (b) show the phase velocity surfaces of the 1 per cent crack-bearing rock (the crack aspect ratio is 0.05), which contains 30 per cent volume ratio of biotite (aspect ratio 0.05). The P-velocity anisotropy increases with increasing crack porosity. The anisotropy of the SV wave becomes weak and the bulge of the SV wave becomes smaller, whereas the anisotropy of the SH wave is almost the same. Fig. 9 shows the changes of Thomsen’s anisotropic para- meter with respect to the crack porosity when the crack aspect ratio is 0.05 and the rock contains 30 per cent biotite with aspect ratio 0.05. c and d increase only a little with increasing crack porosity but e shows a greater increase with increasing crack porosity. The difference between e and d gradually increases as the crack porosity increases. Fig. 10 shows the change in Thomsen’s anisotropic para- meter with respect to the bulk modulus of the inclusion fluid. e and d increase with decreasing fluid bulk modulus, whereas c is independent of the fluid bulk modulus.
5 DISCUSSION
5.1 Differential effective medium method (DEM) When we use the first-order perturbation of Eshelby’s effective Figure 7. Change of Thomsen’s anisotropic parameter as a function medium method for calculating the elastic constants of com- of volume ratio of (a) biotite and (b) muscovite. a denotes aspect ratio. posite material, the two extreme conditions described by eqs Thomsen’s anisotropic parameters are calculated from elastic constants. (11) and (12) do not give the same value, because the actual
# 2001 RAS, GJI 145, 19–32 28 O. Nishizawa and T. Yoshino
Figure 8. Phase velocity surface of cracked rock. The rock includes 30 per cent biotite with aspect ratio 0.05, with 0.1 per cent crack porosity with an aspect ratio of 0.05. Velocity surfaces of crack-free rock are also shown. Downloaded from https://academic.oup.com/gji/article/145/1/19/620505 by guest on 30 September 2021 stress and strain fields of the composite material deviate from those of the initial inclusion-free matrix material. Eqs (11) and (12) are derived from the interaction energy between inclusions and the uniform elastic field of the initial matrix material. The first-order perturbation ignores the interaction between inclusions; that is, it ignores the interaction between inclusions and the disturbed stress or strain field. The interaction becomes larger as the volume ratio of the inclusion increases. DEM evaluates the disturbance of the elastic field inside the com- posite by considering a new matrix derived from the previous step when calculating the interaction energy. When the elastic constants of the inclusion are very different from the elastic constants of the matrix, the discrepancy between the results obtained from eqs (11) and (12) becomes large even for small volume ratios. This corresponds to open cracks (or pores) in the rock. In the present case, since the differences in the elastic constants between the matrix and the mica crystals are not large, the first-order perturbation by Eshelby’s effective
Figure 10. Change of Thomsen’s anisotropic parameter with fluid-filled cracks as a function of the fluid bulk modulus.
medium method gives only small differences in the elastic constants between the two extreme conditions, even for a large inclusion-volume ratio. Elastic constants calculated by DEM fall between the two extreme values of the first-order pertur- bation, and the values calculated from eqs (13) and (14) are close to each other. The energy calculation of DEM is divided into small steps, and in each step the interaction energy between inclusions is implicitly considered. This provides better estimates of the elastic energy compared to the first-order perturbation of Eshelby’s effective medium theory when the inclusion-volume ratio is large. Calculations of the anisotropic Eshelby tensor have been performed in order to study the elastic properties of composite media with an initially anisotropic matrix with liquid inclusions (Hornby et al. 1994; Singh et al. 2000). When both the matrix Figure 9. Change of Thomsen’s anisotropic parameter as a function and the inclusion show transverse isotropy, the effective homo- of crack volume. geneous medium also shows transverse isotropy. The concept
# 2001 RAS, GJI 145, 19–32 Velocity anisotropy in mica-rich rocks 29 of DEM is very simple and has the advantage that it can be surface. Even for thin cracks, the difference between e and d is easily applied to the case of an anisotropic matrix and an very small and does not exceed 0.1 (Douma 1988). Unlike the anisotropic inclusion. crack-induced anisotropy, the mica-induced anisotropy gives Our method is based on the energy criterion. However, there considerable differences between e and d—larger than 0.2 when is another method for calculating elastic constants of com- the volume ratio of the mica mineral is 30 per cent. This results posite media (Domany et al. 1975; Gubernatis & Krumhansl in a considerable non-elliptic anisotropy of the P-wave phase 1975). This calculates the perturbation of the displacement field velocity. The large bulge of the SV-wave phase velocity surface due to inclusions without using the energy criterion, and is is caused by two factors: the VP/VS ratio in the axial direction, called the self-consistent scheme (SC). The method is the same and the difference between e and d. For the crack-induced aniso- as that used for the scattering of elastic or electromagnetic tropy, VP decreases greatly when the crack volume increases waves. SC is also applicable to the anisotropic matrix (Hornby or the cracks become thinner, whereas VS decreases only little et al. 1994; Mainprice & Humbert 1994) and may be applicable with these changes. Therefore, the value of VP/VS does not to anisotropic inclusions. The values of DEM are close to the increase with increasing crack volume or when cracks become lower and upper boundaries for increasing and decreasing thinner. This reduces the SV-velocity anisotropy (Fig. 8b). elastic constants, respectively, and seem to underestimate the On the other hand, mica-induced anisotropy is characterized V /V change of elastic constants of mica-rich rocks (Le Ravalec & by the large P S value and the large difference between e Downloaded from https://academic.oup.com/gji/article/145/1/19/620505 by guest on 30 September 2021 Gueguen 1996b). Comparison of the results between different and d. This causes a large bulge of the SV-wave phase velocity schemes is another interesting topic that is left for future studies. surface. We thus conclude that the strong S-velocity anisotropy (large c, large bulge of the SV phase velocity surface and singularity of the S wave) and the non-ellipticity of the P-wave phase velocity surface are the most important characteristics of 5.2 Mica- and crack-induced anisotropy the mica-induced anisotropy. Here we refer to the anisotropy produced by aligned micro- cracks as crack-induced anisotropy and the anisotropy caused by aligned mica crystals as mica-induced anisotropy. It is 5.3 Velocity anisotropy in the crust interesting to note that the mica-induced anisotropy produces stronger S-velocity anisotropy than P-velocity anisotropy. The There have been many discussions about crustal seismic velocity mica-induced S-velocity anisotropy becomes large as the aspect anisotropy. The preferred orientations of olivine-, pyroxene-, ratio of the crystal decreases, especially for biotite. A line mica- and amphibole-group minerals were investigated to singularity (Crampin & Yedlin 1981) of the S wave appears for explain observed seismic velocity anisotropy in crustal rocks all crystal aspect ratios because the SV wave shows strong (Barruol & Kern 1996; Kern & Wenk 1990; Siegesmund et al. anisotropy: deviation from the circular phase velocity surface 1989; Siegesmund & Kern 1990; Siegesmund & Volbrecht 1991; is about 1 km sx1. On the other hand, the crack-induced Jones & Nur 1984; Barrruol & Mainprice 1993). However, anisotropy produces mostly P-velocity anisotropy rather than most experimental measurements are only performed in the S-velocity anisotropy, depending on the crack aspect ratio and axial directions assigned by the foliation and lineation in rocks. the bulk modulus of the fluid inclusion of the crack (Hudson Recently, Takanashi et al. (2000) measured Thomsen’s aniso- 1981; Nishizawa 1982; Douma 1988). The S-wave singularity tropic parameters for some metamorphic rocks and found a appears when the aspect ratios of the cracks are very small typical non-elliptic P-velocity anisotropy and an S-wave singu- (Douma 1988; Douma & Crampin 1990; Crampin et al. 1990). larity in biotite schist. Mica-group minerals are very common However, the bulge of the SV wave shows only a small in the shallower crust and their preferred orientation is also deviation from the circle (less than 0.25 km sx1). These different common. The volume fraction of mica-group minerals some- characteristics of the two kinds of anisotropy are important in times exceeds 30 per cent (Siegesmund & Kern 1990; Jones & interpreting observed seismic anisotropy. Nur 1984; Takanashi et al. 2000). Thus, mica-induced aniso- Thomsen’s parameter d changes from a negative value to a tropy is expected in many parts of the Earth’s crust. The non- positive value as the crack volume ratio increases. Takanashi elliptic P-wave velocity anisotropy and the S-wave splitting in et al. (2000) measured the change of d in biotite schist (Hidaka, the crust may be, in some parts, attributable to mica-induced Hokkaido, Japan, with 30 per cent volume ratio of biotite) anisotropy. under confining pressure. Since open cracks are closed with Mica-induced anisotropy is important in shear zones because increasing confining pressure, the change in the anisotropic a strong preferred orientation of mica minerals is expected parameter with increasing confining pressure corresponds to in fault rocks (Siegesmund & Kern 1990; Jones & Nur 1984). the reduction in crack volume. Their experiment showed that e S-wave splitting is often attributed to the preferred orientation decreases considerably with increasing pressure but c decreases of fractures or cracks. If the shear zone extends along the only slightly, and d changes from a positive value at zero vertical plane, large shear splitting is expected in the vertical pressure to negative values as pressure increases. In the biotite incidence of the seismic wave because the c-axes of mica minerals schist, the c-axis of biotite is mostly aligned perpendicular to are aligned horizontally and vertically to the shear plane. The the foliation plane. The experimental result can be interpreted present results also indicate that considerable shear splitting by the present model if we assume that the thin cracks are appears when the S-wave incident angle is 30–40u from the mostly aligned with their crack plane parallel to the foliation symmetry axis of transverse isotropy that is generated from plane, which is also parallel to the isotropic plane of the biotite aligned mica minerals with their c-axes parallel to the symmetry crystals. axis direction. A line singularity appears in the mica-induced The difference in Thomsen’s anisotropic parameters between anisotropy. Therefore, the origin of S-wave splitting along the e and d indicates the non-ellipticity of the P-wave phase velocity fault zone should be reconsidered, that is, whether it is generated
# 2001 RAS, GJI 145, 19–32 30 O. Nishizawa and T. Yoshino from the preferred orientation of fractures (or cracks) or the REFERENCES strong preferred orientation of mica minerals in the shear zone, or both. Alekesandrov, K.S. & Ryzhova, T.V., 1961. The elastic properties of The non-elliptic P-wave velocity anisotropy affects the inter- rock-forming minerals, II: Layered silicates, Izv. Acad. Sci. USSR, pretation of reflection seismic data as first discussed by Thomsen Geophys. Ser., 12, 186–189. (1986, 1988) and further discussed by Tsvankin (1997a,b) and Anderson, D., Minster, B. & Cole, D., 1974. The effect of oriented cracks on seismic velocities, J. geophys. Res., 79, 4011–4015. Vavrycˇuk & Psˇencˇ´ık (1998). The non-elliptic P-velocity aniso- Babusˇka, V. & Cara, M., 1991. Seismic Anisotropy in the Earth, tropy may be expected in many observations where mica- Kluwer, Dordrecht. induced anisotropy is essential. Conventional data processing Backus, G.E., 1962. Long-wave elastic anisotropy produced by in seismic exploration may need some modifications. horzontal layering, J. geophys. Res., 67, 4427–4440. The group velocity surface of the S wave has a cusp that Banik, N.C., 1987. An effective anisotropy parameter in transversely results in complicated waveforms (Banik 1987). The seismic isotropic media, Geophysics, 52, 1654–1664. wave will be strongly affected when it propagates through the Barruol, G. & Kern, H., 1996. Seismic anisotropy and shear-wave heterogeneity produced by the mica-induced anisotropy. This splitting in lower-crust and upper mantle rocks from the Ivrea will cause complexity in the seismic waveforms propagating in Zone—experimental and calculated data, Phys. Earth planet. Inter., 95, 175–194.
the crust. Downloaded from https://academic.oup.com/gji/article/145/1/19/620505 by guest on 30 September 2021 Barrruol, G. & Mainprice, D., 1993. A quantitative evaluation of the contribution of crustal rocks to the shear-wave splitting of teleseismic SKS waves, Phys. Earth planet. Inter., 78, 281–300. Booth, D., Crampin, S. & Lovell, J., 1990. Temporal changes in shear 6 CONCLUSIONS wave splitting during an earthquake swarm in Arkansas, J. geophys. Res., 95, 11 151–11 164. So far most studies on rock anisotropy have focused on the Crampin, S., 1984. Effective anisotropic elastic constants for wave anisotropy induced by cracks or the anisotropy induced by propagation through cracked solids, Geophys. J. R. astr. Soc., 76, the preferred orientation of minerals. The velocity anisotropy 135–145. caused by the mineral preferred orientation has been studied by Crampin, S., 1987. Geological and industrial implications of extensive- calculating the Voigt or Reuss averages of an aggregate based dilatancy anisotropy, Nature, 328, 491–496. Crampin, S. & Yedlin, M., 1981. Shear-wave singularities of wave on lattice preferred orientation (LPO) and single crystal elastic propagation in anisotropic media, J. Geophys., 49, 43–46. constants (Mainprice & Humbert 1994). However, the LPO- Crampin, S., Evans, R. & Atkinson, B., 1984. Earthquake prediction: a based average often ignores the effect of crystal shape. Crystal new physical basis, Geophys. J. R. astr. Soc., 76, 147–156. shape may be important for some mica-group minerals, i.e. Crampin, S., Booth, D.C., Evans, R., Peacock, S. & Fletcher, J.B., biotite and phlogopite. Mica-group minerals have strong aniso- 1990. Changes in shear wave splitting at Anza near the time of the tropy and they result in characteristic anisotropy of crustal North Palm Springs earthquake, J. geophys. Res., 95, 11 197–11 212. rocks because volume ratios of mica minerals are sometimes Domany, E., Gubernatis, J.E. & Krumhansl, J.A., 1975. The elasticity large, more than 30 per cent. In fact, natural mica-rich rocks of polycrystals and rocks, J. geophys. Res., 80, 4851–4856. show a large variation in anisotropy, and their anisotropy has Douma, J., 1988. The effect of the aspect ratio on crack-induced not been studied systematically. For further studies of aniso- anisotropy, Geophys. Prospect., 36, 614–632. Douma, J. & Crampin, S., 1990. The effect of a changing aspect ratio of tropy in mica-rich rocks, the modelling of different matrix aligned cracks on shear wave vertical seismic profiles: a theoretical materials, including initial anisotropy, will be necessary. study, J. geophys. Res., 95, 11 293–11 300. The inclusion model presented here has the advantage that it Eshelby, J., 1957. The determination of the elastic field of an can be applicable to the anisotropic inclusions and anisotropic ellipsoidal inclusion, and related problems, Proc. R. Soc. Lond., matrix. The usefulness of the anisotropic composite model A241, 376–396. has also been shown in other problems in geophysics (Singh Gubernatis, J.E.. & Krumhansl, J.A., 1975. Macroscopic engineering et al. 2000). However, DEM is merely a numerical method for properties of polycrystalline materials: elastic properties, J. appl. calculating the elastic properties of composite media. SC can be Phys., 46, 1875–1883. implemented in the same manner (Hornby et al. 1994) and may Helbig, K., 1984. Anisotropy and dispersion in periodically layered be applicable to the calculation of anisotropic inclusion and media, Geophysics, 49, 364–373. Hill, R., 1965. A self-consistent mechanics of composite materials, anisotropic matrix problems. A comparison between several J. Mech. Phys. Sol., 13, 213–222. calculation schemes is important. This is an interesting topic Hornby, B.E., Schawartz, L.M. & Hudson, J.A., 1994. Anisotropic for future studies. In conclusion, we would like to stress effective-medium modelling of the elastic properties of shales, the importance of the modelling of anisotropic inclusions and Geophysics, 59, 1570–1583. anisotropic matrices, which are not yet well understood. Hudson, J.A., 1981. Wave speeds and attenuation of elastic waves in material containing cracks, Geophys. J. R. astr. Soc., 64, 133–150. Jones, T. & Nur, A., 1984. The nature of seismic reflection from deep crustal fault zones, J. geophys. Res., 89, 3153–3171. Kaneshima, S., 1990, Origin of crustal anisotropy: shear wave splitting ACKNOWLEDGMENTS studies in Japan, J. geophys. Res., 95, 11 121–11 133. Kern, H. & Schenk, V., 1985. Elastic wave velocities in rocks from a We thank Robert Kranz, Kyuich Kanagawa and Yves Gueguen lower crustal section in southern Calabria (Italy); experiments in for their comments and discussions on the manuscript. An solid state physics relevant to lithospheric dynamics, Phys. Earth anonymous reviewer commented on important points of the planet. Inter., 40, 147–160. manuscript. This helped us to improve the manuscript. Part of Kern, H. & Wenk, H.R., 1990. Fabric-related velocity anisotropy and this work was supported by the GSJ’s research program, shear wave splitting in rocks from the Santa Rosa mylonite zone, Researches of Geophysical Exploration. California, J. geophys. Res., 95, 11 213–11 223.
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Kinoshita, N. & Mura, T., 1971. Elastic field of inclusions in Winterstein, D.F., 1990. Velocity anisotropy terminology for geo- anisotropic media, Phys. Status Solidi, Ser. A, 5, 859–768. physicists, Geophysics, 55, 1070–1088. Kro¨ner, E., 1967. Elastic moduli of perfectly disordered composite Yamamoto, K., Kosuga, M. & Hirasawa, T., 1981. A theoretical materials, J. Mech. Sol., 15, 319–329. method for determination of effective elastic constants of isotropic Leary, P.C., Crampin, S. & McEvilly, T.V., 1990. Seismic fracture composite, Sci. Rept Tohoku Univ., Ser. 5, 28, 47–67. anisotropy in the Earth’s crust: an overview, J. geophys. Res., 95, 11 105–11 114. Le Ravalec, M. & Gueguen, Y., 1996a. High and low frequency elastic moduli for a saturated porous/cracked rock (differential self consisitent and poroelastic theories), Geophysics, 61, 1080–1094. APPENDIX A: ESHELBY’S TENSOR IN Le Ravalec, M. & Gueguen, Y., 1996b. Comments on ‘The elastic ANISOTROPIC MEDIA modulus of media containing strongly interacting cracks’ by Paul M. Davis and Leon Knopoff, J. geophys. Res., 101, 25 373–25 375. A1 General formulation of the static Green’s function Lin, S. & Mura, T., 1973. Elastic fields of inclusions in anisotropic Let x be a space vector and j be a wave vector of the space. The media (II), Phys. Status Solidi, (a)15, 281–285. displacement at x is Liu, E., Crampin, S. & Booth, D., 1989. Shear-wave splitting in cross- ð hole surveys: modeling, Geophysics, 54, 57–65. ? Downloaded from https://academic.oup.com/gji/article/145/1/19/620505 by guest on 30 September 2021 ~ . Mainprice, D., 1997. Modelling the anisotropic properties of umðxÞ umðîÞ expðiî xÞdî ,(A1) {? partially molten rocks found at mid-ocean ridges, Tectonophysics, 279, 161–179. where u¯m(j) is the Fourier transform of um(x), Mainprice, D. & Humbert, M., 1994. Methods of calculating ð 3 ? petrophysical properties from lattice preferred orientation data, 1 umðîÞ~ umðxÞ expð{ix . îÞdx : (A2) Surv. Geophys., 15, 575–592. 2n {? Mavko, G., Mukerji, T. & Dvorkin, J., 1998. The rock physics handbook: tools for seismic analysis in porous media, Cambridge In what follows, the suffix appearing after a comma denotes the University Press, Cambridge. partial derivative; for example, um,n denotes a partial derivative McLaughlin, R.A., 1977. A study of the differential scheme for of the mth component of vector u with respect to the nth composite materials, Int. J. Eng. Sci., 15, 237–244. component of the coordinate. The distortion is given by ð Mura, T., 1968. The continuum theory of dislocations, in Advances in ? Material Research, Vol. 3, pp. 1–107, ed. Herman, H., Interscience. um,nðxÞ~ imnumðîÞ expðiî . xÞdî : (A3) Mura, T. & Mori, T., 1976. Micromechanics: Dislocations and Inclusions, {? Baifukan, Tokyo (in Japanese). The stress-free strain is described by the displacement uT (x), Nishizawa, O., 1982. Seismic velocity anisotropy in a medium m containing oriented cracks—transversely isotropic case, J. Phys. 1 eT ðxÞ¼ ðuT zuT Þ Earth, 30, 331–347. mn 2 m,n n,m Norris, A.N., 1985. A differential scheme for the effective moduli of ð ? composites, Mech. Mater., 4, 1–16. T . ¼ emnðîÞ expðiî xÞdî ,(A4) Siegesmund, S. & Kern, H., 1990. Velocity anisotropy and shear-wave ? splitting in rocks from the mylonite belt along the Insubric line (Ivrea T zone, Italy), Earth planet. Sci. Lett., 99, 29–47. where e¯ mn(j) is the Fourier transform of the stress-free strain T Siegesmund, S. & Volbrecht, A., 1991. Complete seismic properties emn(x). obtained from microcrack fabrics and textures in an amphibolite If stress-free strain exists in the material, the stress inside the from Ivrea zone, Western Alps, Italy, Tectonophysics, 199, 13–24. material is given by the elastic strain, which is the difference Siegesmund, T., Takeshita, T. & Kern, H., 1989. Anisotropy of Vp and between the total strain and the stress-free strain, Vs in an amphibolite of the deeper crust and its relationship to { T the mineralogical, microstructural and textural characteristics of the ppqðxÞ¼cpqmn½emnðxÞ emnðxÞ rock, Tectonophysics, 157, 25–38. ÈÉ Simmons, G. & Wang, H., 1971. Single Crystal Elastic Constants and 1 T ¼ cpqmn um;nðxÞþun;mðxÞ e ðxÞ : (A5) Calculated Aggregate Properties: A Handbook, MIT Phys. Prospect. 2 mn Singh, S.C., Taylor, M.A.J. & Montagner, J.P., 2000. On the presence of liquid in Earth’s inner core, Science, 287, 2471–2474. Since cpqmn=cpqnm, Takanashi, M., Nishizawa, O., Kanagawa, K. & Yasunaga, K., 2000. T ppqðxÞ¼cpqmn½um,nðxÞ{e ðxÞ Laboratory measurements of elastic anisotropy parameters for the mn ð exposed crustal rocks from the Hidaka Metamorphic Belt, central ? T . Hokkaido, Japan, Geophys. J. Int., 145, 33–47 (this issue). ¼ cpqmn ½imnumðîÞ emnðîÞ expðiî xÞdî : (A6) Thomsen, L., 1986. Weak elastic anisotropy, Geophysics, 51, ? 1954–1966. When the material is in elastic equilibrium, Thomsen, L., 1988. Reflection seismology over azimuthally anisotropic media, Geophysics, 53, 304–313. ppq,qðxÞ~0 , Tsvankin, I., 1997a. Reflection moveout and parameter estimation for then partial differentiation of eq. (A6) gives horizontal transverse isotropy, Geophysics, 62, 614–629. ð Tsvankin, I., 1997b. Anisotropic parameters and P-wave velocity for ? { { T . ~ orthorhombic media, Geophysics, 62, 1292–1300. cpqmn ½ mnmqumðîÞ imqemnðîÞ expðiî xÞdî 0 : (A7) { Vaughan, M.T. & Guggenheim, S., 1986. Elasticity of muscovite and its ? relationship to crystal structure, J. geophys. Res., 91, 4657–4664. This should hold in the entire space, thus Vavrycˇuk, V. & Psˇencˇ´ık, I., 1998. PP-wave reflection coefficients in z T ~ weakly anisotropic elastic media, Geophysics, 63, 2129–2141. cpqmn½mnmqumðîÞ imqemnðîÞ 0 : (A8)
# 2001 RAS, GJI 145, 19–32 32 O. Nishizawa and T. Yoshino
T We use the following notation: eq. (A16) does not affect the integration. eij can be put outside the integration. The displacement associated with the stress-free K c m m ,(A9) pm ¼ pqmn n q strain inside the elliptic domain is given by ð eT ¼ ic m eT ðîÞ : (A10) p pqmn q mn ~{ T { umðxÞ eij cklijGkm,lðx x’Þdx’ : (A18) Eq. (A7) is described as ) 9 T If we replace the elliptic inclusion with the matrix material, the K11u1 þ K12u2 þ K13u3 ¼ e > 1 => elastic constants inside the elliptic domain are the same as those T K u þ K u þ K u ¼ eT : (A11) of the matrix. We consider that stress-free strain eij is the 21 2 22 2 23 3 2 > ;> fictitious stress-free strain of the matrix material, and we obtain T K31u3 þ K32u2 þ K33u3 ¼ e3 the disturbance of strain inside the inclusion: ð By solving eq. (A11), we obtain the Fourier transform of u (x), 1 m C ~{ T 0 { z { emnðxÞ eij cklij½Gkm,lnðx x’Þ Gkn,lmðx x’Þ dx’ : ~ T {1 ) 2 umðîÞ ek NkmðîÞD ðîÞ ,(A12) x1 (A19) where D (j) is the inverse of the determinant |Kij|, Downloaded from https://academic.oup.com/gji/article/145/1/19/620505 by guest on 30 September 2021 0 {1 Note that c is used for describing the strain disturbance. D ðîÞ~1=jKijj , ijkl This gives the relationship expressed in (3). We thus obtain and Nkm(j) is the co-factor of |Kij| with respect to Kkm The Eshelby’s tensor Sijkl for general elastic media. inverse Fourier transform of eq. (A12) gives a general formula Mura & Mori (1976) and Kinoshita & Mura (1971) demon- expressing the displacement at the point x: strated that eq. (A16) can be described by a surface integral of ð ? the unit sphere, u ðxÞ¼ eTN ðîÞD{1ðîÞ expðiî . xÞdî ð m k km a a a {? ~ 1 2 3 . {3 0 T {1 ð uiðxÞ ðx îÞmlf cjlmnemnNijðîÞD ðîÞdSðîÞ , ? 4n S T 1 . ¼ i cklijmleij ðîÞNkmðîÞD ðîÞ expðiî xÞdî : (A13) ? (A20)
Here, we define Gkm(xxxk)as where j¯ is the unit vector, and S and dS are the surface and ð ? the partial surface area of the unit sphere, respectively. f is { ~ 3 {1 . { Gkmðx x’Þ ið1=2nÞ NkmðîÞD ðîÞ exp½î ðx x’Þdî : given by {? 2 2 2 2 2 2 1=2 (A14) f~ða1 m1 za2 m2 za3 m3 Þ : (A21)
Partial differentiation of Gkm(xxxk) gives Distortion is described by ð Gkm,l ðx{x’Þ a1a2a3 {3 0 T {1 ui,kðxÞ~ m m f c e NijðîÞD ðîÞdSðîÞ : (A22) ð 4n l k jlmn mn ? S ~ 1=2n 3 im N î D{1 î exp iî . x{x’ dî : (A15) ð Þ l kmð Þ ð Þ ½ ð Þ We define G¯ ijkl such that {? ð {3 {1 um(x) is given by G ~ða a a Þ m m f N ðîÞD ðîÞdSðîÞ : (A23) ð ð ijkl 1 2 3 l k ij ? ? S 3 {1 T umðxÞ¼{ið1=2nÞ cklijm NkmD ðîÞe ðx’Þ C T l ij The relation between eij and eij is given by {? {? 1 | . { C z exp½iî ðx x’Þdx’dî eik ¼ ðÞui,k uk,i ð 2 ? T 1 ¼ cklije ðx’ÞGkm;lðx x’Þdx’ : (A16) 0 0 T ij ¼ ðc Gijkl þ c Gkjil Þe ? 8n jlmn jlmn mn
This equation indicates the relationship between the displace- T ¼ Sikmnemn : (A24) ment and the stress-free strain. Gkm(xxxk) is the static Green’s function. When unit force is acting at the point xk, Gkm(xxxk) Eshelby’s tensor Sijkl is given by G¯ ijkl, which is a surface gives the kth component of the displacement at the point x integral including elastic constants of the matrix material. Eqs generating from the mth component of the unit force at xk. (A24) and (A23) are the correct expressions of (9) and (10) of Nishizawa (1982), as pointed out by Douma (1988). When the matrix is hexagonal, each component of G¯ is given by a A2 Elliptic inclusion ijkl definite integral of polynomial functions from 0 to 1 (Lin & We consider an elliptic domain V described by Mura 1973). Therefore, each component of Eshelby’s tensor S can be obtained numerically (Nishizawa 1982). When the x 2 x 2 x 2 ijkl 1 z 2 z 3 ƒ matrix material is isotropic and the inclusion is spheroidal, 2 2 2 1 (A17) a1 a2 a3 Eshelby’s tensor is given by analytic formulae. However, when in a general elastic medium. When the stress-free strain the matrix is anisotropic, Eshelby’s tensor is obtained by T is uniform everywhere inside the elliptic domain, eij (xk)in numerical integration.
# 2001 RAS, GJI 145, 19–32