Showing papers on "Transverse isotropy published in 1998"

Variation of P-wave reflectivity with offset and azimuth in anisotropic media

Abstract: P-wave amplitudes may be sensitive even to relatively weak anisotropy of rock mass. Recent results on symmetry‐plane P-wave reflection coefficients in azimuthally anisotropic media are extended to observations at arbitrary azimuth, large incidence angles, and lower symmetry systems. The approximate P-wave reflection coefficient in transversely isotropic media with a horizontal axis of symmetry (HTI) (typical for a system of parallel vertical cracks embedded in an isotropic matrix) shows that the amplitude versus offset (AVO) gradient varies as a function of the squared cosine of the azimuthal angle. This change can be inverted for the symmetry‐plane directions and a combination of the shear‐wave splitting parameter γ and the anisotropy coefficient δ(V). The reflection coefficient study is also extended to media of orthorhombic symmetry that are believed to be more realistic models of fractured reservoirs. The study shows the orthorhombic and HTI reflection coefficients are very similar and the azimuthal v...

Numerical simulation of crack growth in an isotropic solid with randomized internal cohesive bonds

Abstract: A virtual internal bond (VIB) model with randomized cohesive interactions between material particles is proposed as an integration of continuum models with cohesive surfaces and atomistic models with interatomic bonding. This approach differs from an atomistic model in that a phenomenological “cohesive force law” is assumed to act between “material particles” which are not necessarily atoms; it also differs from a cohesive surface model in that, rather than imposing a cohesive law along a prescribed set of discrete surfaces, a randomized network of cohesive bonds is statistically incorporated into the constitutive law of the material via the Cauchy-Born rule, i.e., by equating the strain energy function on the continuum level to the potential energy stored in the cohesive bonds due to an imposed deformation. This work is motivated by the notion that materials exhibit multiscale cohesive behaviors ranging from interatomic bonding to macroscopic ductile failure. It is shown that the linear elastic behavior of the VIB model is isotropic and obeys the Cauchy relation; the instantaneous elastic properties under equibiaxial stretching are transversely isotropic, with all the in-plane components of the material tangent moduli vanishing at the cohesive stress limit; the instantaneous properties under equitriaxial stretching are isotropic with a finite strain modulus. We demonstrate through two preliminary numerical examples that the VIB model can be applied in direct simulation of crack growth without a presumed fracture criterion. The prospect of this type of approach in numerical simulations of fracture seems to be highly promising.

Analytical predictions for the magnetoelectric coupling in piezomagnetic materials reinforced by piezoelectric ellipsoidal inclusions

Abstract: In this paper, a number of problems of fundamental importance in a piezoelectric-piezomagnetic composite material are solved. The problems covered range from the derivation of the analytical expressions for the magneto-electro-elastic Eshelby tensors to the analysis of the magnetoelectric coupling effect which is a new property exhibited in the piezoelectric-piezomagnetic composite. In particular, when both the matrix and the inclusions of the composite are transversely isotropic with different magneto-electro-elastic moduli, and shapes of inclusions are of elliptic cylinder, circular cylinder, disk, and ribbon, closed-form solutions for the magnetoelectric coupling coefficientS are presented compactly. The resulting solutions are a function of the shape of inclusion, phase properties, and volume fraction of the inclusions. These results could provide us with insight into how a piezoelectric-piezomagnetic composite material consisting of inclusions will perform and would be helpful in understanding the magneto-electric-elastic behavior of the composite.

A transversely isotropic biphasic model for unconfined compression of growth plate and chondroepiphysis.

Abstract: Using the biphasic theory for hydrated soft tissues (Mow et al., 1980) and a transversely isotropic elastic model for the solid matrix, an analytical solution is presented for the unconfined compression of cylindrical disks of growth plate tissues compressed between two rigid platens with a frictionless interface. The axisymmetric case where the plane of transverse isotropy is perpendicular to the cylindrical axis is studied, and the stress-relaxation response to imposed step and ramp displacements is solved. This solution is then used to analyze experimental data from unconfined compression stress-relaxation tests performed on specimens from bovine distal ulnar growth plate and chondroepiphysis to determine the biphasic material parameters. The transversely isotropic biphasic model provides an excellent agreement between theory and experimental results, better than was previously achieved with an isotropic model, and can explain the observed experimental behavior in unconfined compression of these tissues.

The measurement of stiffness anisotropy in clays with bender element tests in the triaxial apparatus

Abstract: The paper presents the results of a program of research investigating the effectiveness of bender elements when used in conjunction with the triaxial apparatus for measuring the anisotropy of small strain stiffness of fine-grained soils. Tests were carried out on both intact and reconstituted samples of London clay and on kaolin up to high stresses. The paper shows that the transverse isotropy of small-strain stiffness that commonly occurs in many soils because of a one-dimensional loading history can be fully investigated in the conventional triaxial apparatus and that London clay is an example of such a soil. The stress-induced component of anisotropy was found to be very small for axi-symmetric loading conditions common to both the appartus and the in situ state of these soils. In contrast, the inherent or structural anisotropy was much more significant and is shown to be a variable factor resulting from the plastic strain history and is not related to its natural structure. Consequently, inherent anisotropy is reversible, but the rate of change is very slow when a new regime of stresses is imposed. Inherent anisotropy of the very small strain stiffness also persists long after the plastic strains of the soil have become oriented toward the new stresses.

Magneto-electro-elastic Eshelby tensors for a piezoelectric-piezomagnetic composite reinforced by ellipsoidal inclusions

Abstract: The magneto-electro-elastic Eshelby tensors that represent the stress, electric displacement, and magnetic induction in an inclusion resulting from the constraint of the surrounding matrix of piezomagnetic-piezoelectric composites are presented. The tensors provide the basis for analysis of the magneto-electro-elastic response of the composites and have numerous applications such as to the study of defects, overall properties of multiphase composites, and fracture mechanics, etc. For a three-dimensional transversely isotropic piezoelectric/piezomagnetic composite containing spheriodal inclusions, the number of nonzero Eshelby tensors is 41 while among them only 17 are independent. These independent tensors can be divided into three categories. The first category consists of those tensors related to the elastic response, the second involves those related to the piezomagnetic response, while the third includes elastic and magnetic interactive terms. Moreover, the magneto-electro-elastic Eshelby tensors are obtained in the closed forms when both constituents of the composite are transversely isotropic and shapes of the inclusion are elliptic, rod shaped, penny shaped, and ribbon like.

PP-wave reflection coefficients in weakly anisotropic elastic media

Abstract: Approximate PP-wave reflection coefficients for weak contrast interfaces separating elastic, weakly transversely isotropic media have been derived recently by several authors. Application of these coefficients is limited because the axis of symmetry of transversely isotropic media must be either perpendicular or parallel to the reflector. In this paper, we remove this limitation by deriving a formula for the PP-wave reflection coefficient for weak contrast interfaces separating two weakly but arbitrarily anisotropic media. The formula is obtained by applying the first-order perturbation theory. The approximate coefficient consists of a sum of the PP-wave reflection coefficient for a weak contrast interface separating two background isotropic half-spaces and a perturbation attributable to the deviation of anisotropic half-spaces from their isotropic backgrounds. The coefficient depends linearly on differences of weak anisotropy parameters across the interface. This simplifies studies of sensitivity of such coefficients to the parameters of the surrounding structure, which represent a basic part of the amplitude-versus-offset (AVO) or amplitude-versus-azimuth (AVA) analysis. The reflection coefficient is reciprocal. In the same way, the formula for the PP-wave transmission coefficient can be derived. The generalization of the procedure presented for the derivation of coefficients of converted waves is also possible although slightly more complicated. Dependence of the reflection coefficient on the angle of incidence is expressed in terms of three factors, as in isotropic media. The first factor alone describes normal incidence reflection. The second yields the low-order angular variations. All three factors describe the coefficient in the whole region, in which the approximate formula is valid. In symmetry planes of weakly anisotropic media of higher symmetry, the approximate formula reduces to the formulas presented by other authors. The accuracy of the approximate formula for the PP reflection coefficient is illustrated on the model with an interface separating an isotropic half-space from a half-space filled by a transversely isotropic material with a horizontal axis of symmetry. The results show a very good fit with results of the exact formula, even in cases of strong anisotropy and strong velocity contrast.

Polarization, phase velocity, and NMO velocity of qP-waves in arbitrary weakly anisotropic media

Abstract: We present approximate formulas for the qP-wave phase velocity, polarization vector, and normal moveout velocity in an arbitrary weakly anisotropic medium obtained with first-order perturbation theory. All these quantities are expressed in terms of weak anisotropy (WA) parameters, which represent a natural generalization of parameters introduced by Thomsen. The formulas presented and the WA parameters have properties of Thomsen's formulas and parameters: (1) the approximate equations are considerably simpler than exact equations for qP waves, (2) the WA parameters are nondimensional quantities, and (3) in isotropic media, the WA parameters are zero and the corresponding equations reduce to equations for isotropic media. In contrast to Thomsen's parameters, the WA parameters are related linearly to the density normalized elastic parameters. For the transversely isotropic media with vertical axis of symmetry, the equations presented and the WA parameters reduce to the equations and linearized parameters of Thomsen. The accuracy of the formulas presented is tested on two examples of anisotropic media with relatively strong anisotropy: on a transversely isotropic medium with the horizontal axis of symmetry and on a medium with triclinic anisotropy. Although anisotropy is rather strong, the approximate formulas presented yield satisfactory results.

Elastic solutions for a transversely isotropic half-space subjected to a point load

Abstract: SUMMARY We rederive and present the complete closed-form solutions of the displacements and stresses subjected to a point load in a transversely isotropic elastic half-space. The half-space is bounded by a horizontal surface, and the plane of transverse isotropy of the medium is parallel to the horizontal surface. The solutions are obtained by superposing the solutions of two infinite spaces, one acting a point load in its interior and the other being free loading. The Fourier and Hankel transforms in a cylindrical co-ordinate system are employed for deriving the analytical solutions. These solutions are identical with the Mindlin and Boussinesq solutions if the half-space is homogeneous, linear elastic, and isotropic. Also, the Lekhnitskii solution for a transversely isotropic half-space subjected to a vertical point load on its horizontal surface is one of these solutions. Furthermore, an illustrative example is given to show the e⁄ect of degree of rock anisotropy on the vertical surface displacement and vertical stress that are induced by a single vertical concentrated force acting on the surface. The results indicate that the displacement and stress accounted for rock anisotropy are quite di⁄erent for the displacement and stress calculated from isotropic solutions. ( 1998 John Wiley & Sons, Ltd.

An approach for the stress analysis of transversely isotropic biphasic cartilage under impact load.

Abstract: Stress analysis of contact models for isotropic articular cartilage under impacting loads shows high shear stresses at the interface with the subchondral bone and normal compressive stresses near the surface of the cartilage. These stress distributions are not consistent, with lesions observed on the cartilage surface of rabbit patellae from blunt impact, for example, to the patello-femoral joint. The purpose of the present study was to analyze, using the elastic capabilities of a finite element code, the stress distribution in more morphologically realistic transversely isotropic biphasic contact models of cartilage. The elastic properties of an incompressible material, equivalent to those of the transversely isotropic biphasic material at time zero, were derived algebraically using stress-strain relations. Results of the stress analysis showed the highest shear stresses on the surface of the solid skeleton of the cartilage and tensile stresses in the zone of contact. These results can help explain the mechanisms responsible for surface injuries observed during blunt insult experiments.

An Acoustic Wave Equation For Anisotropic Media

Abstract: A wave equation, derived using the acoustic medium assumption for P-waves in transversely isotropic (TI) media with a vertical symmetry axis (VTI media), yields a good kinematic approximation to the familiar elastic wave equation for VTI media. The wavefield solutions obtained using this VTI acoustic wave equation are free of shear waves, which significantly reduces the computation time compared to the elastic wavefield solutions for exploding‐reflector type applications. From this VTI acoustic wave equation, the eikonal and transport equations that describe the ray theoretical aspects of wave propagation in a TI medium are derived. These equations, based on the acoustic assumption (shear wave velocity = 0), are much simpler than their elastic counterparts, yet they yield an accurate description of traveltimes and geometrical amplitudes. Numerical examples prove the usefulness of this acoustic equation in simulating the kinematic aspects of wave propagation in complex TI models.

Stress-Induced Seismic Anisotropy Revisited

Abstract: This summary contains formulas (***) which can not be displayed on the screenA general principle outlined by P. Curie (1894) regarding the influence of symmetry in physical phenomena states, in modern language, that the symmetry group of the causes is a sub-group of the symmetry group of the effects. For instance, regarding stress-induced seismic anisotropy, the most complex symmetry exhibited by an initially isotropic medium when tri-axially stressed is orthorhombic, or orthotropic, symmetry characterized by three symmetry planes mutually perpendicular (Nur, 1971). In other respects, Schwartz et al. (1994) demonstrated that two very different rock models, namely a cracked model and a weakly consolidated granular model, always lead to elliptical anisotropy when uniaxially stressed. The addressed questions are : Is this result true for any rock model? and more generally : Do initially isotropic rock form a well-defined sub-set of orthorhombic media when triaxially stressed?Under the hypothesis of 3rd order nonlinear isotropic hyperelasticity (i. e. , no hysteresis and existence of an elastic energy function developed to the 3rd order in the strain components) it is demonstrated that the qP-wave stress-induced anisotropy is always ellipsoidal, for any strength of anisotropy. For instance point sources generate ellipsoidal qP-wave fronts. This result is general and absolutely independent of the rock model, that is to say independent of the causes of nonlinearity, as far as the initial assumptions are verified. This constitutes the main result of this paper. Thurston (1965) pointed out that an initially isotropic elastic medium, when non-isotropically pre-stressed, is never strictly equivalent to an unstressed anisotropic crystal. For instance the components of the stressed elastic tensor lack the familiar symmetry with respect to indices permutation. This would prohibit Voigt's notation of contracted indices. However if the magnitude of the components of the stress deviator is small compared to the wave moduli, which is always verified in practical situations of seismic exploration, the perfect equivalence is re-established. Under this condition, the 9 elastic stiffnesses C'ij (in contracted notation) of an initially isotropic solid, when triaxially stressed, are always linked by 3 ellipticity conditions in the coordinate planes associated with the eigen directions of the static pre-stress, namely :(***)Thus only 6 of the 9 elastic stiffnesses of the orthorhombic stressed solid are independent (Nikitin and Chesnokov, 1981), and are simple functions of the eigen stresses, and of the 2 linear (2nd order) and the 3 nonlinear (3rd order) elastic constants of the unstressed isotropic solid. Furthermore, given the state of pre-stress, the strength of the stress-induced P- or S-wave anisotropy and S-wave birefringence (but not the magnitude of the wave moduli themselves) are determined by only 2 intrinsic parameters of the medium, one for the P-wave and one for the S-waves. Isotropic elastic media, when triaxially stressed, constitute a special sub-set of orthorhombic media, here called ellipsoidal media , verifying the above conditions. Ellipsoidal anisotropy is the natural generalization of elliptical anisotropy. Ellipsoidal anisotropy is to orthorhombic symmetry what elliptical anisotropy is to transversely isotropic (TI) symmetry. Elliptical anisotropy is a special case of ellipsoidal anisotropy restricted to TI media. In other words, ellipsoidal anisotropy degenerates in elliptical anisotropy in TI media. In ellipsoidal media the qP-wave slowness surface is always an ellipsoid. The S-wave slowness surfaces are not ellipsoidal, except in the degenerate elliptical case, and have to be considered as a single double-valued self-intersecting sheet (Helbig, 1994). The intersections of these latter surfaces with the coordinate planes are either ellipses, for the S-vave polarized out of the coordinate planes, or circles, for the qS-wave polarized in the coordinate planes. The nearly exhaustive collection of experimental data on seismic anisotropy in rocks (considered as transverse isotropic) by Thomsen (1986) show that elliptical anisotropy is more an exception than a rule. Since stress-induced anisotropy is essentially elliptical when restricted to transversely isotropic media, as a consequence this work clearly shows that stress can be practically excluded as a unique direct cause of elastic anisotropy in rocks.

Propagation of Love waves in a transversely isotropic fluid-saturated porous layered half-space

Abstract: The dispersion equation for Love wave propagation in a layer lying over a half-space is derived. Both media are assumed to be transversely isotropic fluid-saturated poroelastic solids with principal axes perpendicular to the surface. The analysis is based on the Biot’s theory. The dissipation due to fluid viscosity is considered and therefore the dispersion equation is complex and intractable analytically. An iterative procedure is developed to solve this equation. Two situations are discussed in detail: (i) an elastic layer overlying a poroelastic half-space and (ii) a poroelastic layer lying over an elastic half-space. Dispersion curves and attenuation curves of Love waves are plotted for these two cases. In addition, the upper and lower bounds of Love wave speeds are also explored.

Seismic anisotropy in the core–mantle transition zone

Abstract: Split S waves observed at Hockley, Texas from events in the Tonga–Fiji region of the southwest Pacific show predominantly vertically polarized shear-wave (SV ) energy arriving earlier than horizontally polarized (SH) energy for rays propagating horizontally through D″. After corrections are made for the effects of upper-mantle anisotropy beneath Hockley, a time lag of 1.5 to 2.0 s remains for the furthest events (93.9°–100.6° ), while the time lags of the nearer observations (90.5°–92.9° ) nearly disappear. At closer distances, the S waves from these same events do not penetrate as deeply into the lower mantle, and are not split. These observations suggest that a patch of D″ beneath the central Pacific is anisotropic, while the mantle immediately above the patch is isotropic. The thickness of the anisotropic zone appears to be of the order of 100–200 km. Observations of shear-wave splitting have previously been made for paths that traverse D″ under the Caribbean and under Alaska. SH leads SV , the reverse of the Hockley observations, but in these areas the fact that SV leads SH in the HKT data shown here suggests a different sort of anisotropy under the central Pacific from that under Alaska and the Caribbean. The case of SH travelling faster than SV is consistent with transverse isotropy with a vertical axis of symmetry (VTI) and does not require variations with azimuth. The case of SV leading SH is consistent with transverse isotropy with a horizontal axis of symmetry (HTI), an azimuthally anisotropic medium, and with a VTI medium formed by a hexagonal crystal. Given that (Mg,Fe)SiO3 perovskite appears unlikely to form anisotropic fabrics on a large scale, the presence of anisotropy may point to chemical heterogeneity in the lowermost mantle, possibly due to mantle–core interactions.

Nonhyperbolic reflection moveout for horizontal transverse isotropy

Abstract: The transversely isotropic model with a horizontal axis of symmetry (HTI) has been used extensively in studies of shear‐wave splitting to describe fractured formations with a single system of parallel vertical penny‐shaped cracks. Here, we present an analytic description of longspread reflection moveout in horizontally layered HTI media with arbitrary strength of anisotropy. The hyperbolic moveout equation parameterized by the exact normal‐moveout (NMO) velocity is sufficiently accurate for P-waves on conventional‐length spreads (close to the reflector depth), although the NMO velocity is not, in general, usable for converting time to depth. However, the influence of anisotropy leads to the deviation of the moveout curve from a hyperbola with increasing spread length, even in a single‐layer model. To account for nonhyperbolic moveout, we have derived an exact expression for the azimuthally dependent quartic term of the Taylor series traveltime expansion [t2(x2)] valid for any pure mode in an HTI layer. Th...

The Numbers of Elastic Properties and Failure Parameters for Fiber Composites

Abstract: It is shown that there is a coordination between the numbers of the elastic properties and the numbers of the failure criteria parameters for aligned fiber composites under certain realistic conditions. These conditions require a high degree of anisotropy appropriate to polymeric matrix composites such that failure decomposes into separate fiber dominated and matrix dominated modes. Failure criteria are given in both five parameter and four parameter forms. The five parameter failure form coordinates with the usual five elastic property form for transversely isotropic composites. The four parameter failure form coordinates with a reduced four elastic property form which is shown to be applicable as an approximation for the same typical fiber composites.

Complex elastic wave band structures in three-dimensional periodic elastic media

Abstract: Propagation of elastic waves can be completely inhibited, irrespective of propagation directions, if oscillation frequencies of elastic waves fall inside elastic wave band gaps—three-dimensional frequency stop bands of elastic waves generated in three-dimensional periodic elastic materials. The three-dimensional linear vector equation of motion is transformed into a matrix eigensystem and solved numerically by the plane wave expansion method. Existence of elastic wave band gaps is theoretically demonstrated for the face centered cubic lattice structure based on isotropie materials. The example structure is spherical tungsten scatterers embedded in a polyethylene matrix. Complex elastic wave band structures were also computed for tunneling evanescent waves at symmetric points at X and L points.

Misalignment of the orientation of fractures and the principal axes for P and S waves in rocks containing multiple non-orthogonal fracture sets

Abstract: SUMMARY Azimuthal anisotropy in rocks can result from the presence of one or more sets of partially aligned fractures with orientations determined by the stress history of the rock. A shear wave propagating in an azimuthally anisotropic medium splits into two components with diVerent polarizations if the source polarization is not aligned with the principal axes of the medium. In the presence of two or more non-orthogonal sets of vertical fractures, the symmetry of the medium may be approximated as monoclinic with a horizontal plane of mirror symmetry if, in the absence of fractures, the rock is transversely isotropic (TI) with the symmetry axis perpendicular to the bedding plane. For such a medium, the fast and slow polarization directions for vertically propagating shear waves are not parallel or perpendicular to any of the fracture planes but lie in directions given by the principal axes of a second-rank fracture compliance tensor. This tensor is independent of the normal compliance of the fractures. It follows that for vertical propagation of shear waves in a vertically fractured TI medium, any number of arbitrarily oriented vertical fracture sets is equivalent to two mutually perpendicular fracture sets, provided that the seismic wavelength is large compared to the size of the fractures. For oVsets typical of surface seismic acquisition, the P-wave velocity at fixed oVset varies with azimuth as cos 2(w’w 0 ), where w is the azimuth measured with respect to the fast polarization direction for a vertically polarized shear wave; w 0 depends on both the normal and the shear compliance of the fractures and may diVer from zero if the ratio of the normal to shear compliance of the fractures varies significantly between fracture sets. If this ratio is similar for all fractures, w 0 #0 and the principal axes of the variation in P-wave velocity with azimuth for fixed oVset are determined by the principal axes of the second-rank fracture compliance tensor.

Elastic rebound between an indenter and a layered specimen : part i. model

Abstract: A model approximates the compliance of elastic contact between an indenter and a layered specimen (isotropic and transversely isotropic layers) by averaging the displacements beneath a uniformly loaded area at the surface of an elastic half-space. For a uniform, isotropic material the specimen compliance is (1 - v2)/βES where S is the square root of the projected contact area, E and v are Young’s modulus and Poisson’s ratio of the specimen, and β depends on indenter shape: β = 1.044, 1.057, or 1.086 for circular, square, and triangular-shaped indenters, respectively. An expression to replace (1 - v2)/E is provided in the treatment of a transversely isotropic solid.

Kelvin Notation for Stabilizing Elastic-Constant Inversion

Abstract: Inverting a set of core-sample traveltime measurements for a complete set of 21 elastic constants is a difficult problem. If the 21 elastic constants are directly used as the inversion parameters, a few bad measurements or an unfortunate starting guess may result in the inversion converging to a physically impossible solution . Even given perfect data, multiple solutions may exist that predict the observed traveltimes equally well. We desire the inversion algorithm to converge not just to a physically possible solution, but to the best(i. e. most physically likely) solution of all those allowed. We present a new parameterization that attempts to solve these difficulties. The search space is limited to physically realizable media by making use of the Kelvin eigenstiffness-eigentensor representation of the 6 x 6 elastic stiffness matrix. Instead of 21 stiffnesses, there are 6 eigenstiffness parametersand 15 rotational parameters . The rotational parameters are defined using a Lie-algebra representation that avoids the artificial degeneracies and coordinate-system bias that can occur with standard polar representations. For any choice of these 21 real parameters, the corresponding stiffness matrix is guaranteed to be physically realizable. Furthermore, all physically realizable matrices can be represented in this way. This new parameterization still leaves considerable latitude as to which linear combinations of the Kelvin parameters to use, and how they should be ordered. We demonstrate that by careful choice and ordering of the parameters, the inversion can be relaxedfrom higher to lower symmetry simply by adding a few more parameters at a time. By starting from isotropy and relaxing to the general result in stages (isotropy, transverse isotropy, orthorhombic, general), we expect that the method should find the solution that is closest to isotropy of all those that fit the data.

A new model for the multiphase fiber–matrix composite materials

Abstract: A new model for a multiphase fiber–matrix composite material is proposed and the corresponding constitutive equations are derived. The matrix in the composite material is considered to be an isotropic continuous medium, and the reinforcing fibers are modeled by equivalent fibers using statistical averaging. The equivalent fibers are divided into those homogeneously distributed and oriented in the matrix and those having a preferred orientation. Based on the assumption of homogeneous deformation, a meso-constitutive equation for the multiphase fiber–matrix composite material is derived. The constitutive equation can be applied to the cases of isotropic, transversely isotropic or generally anisotropic properties of composite materials. As examples, several fiber–matrix composite material structures with isotropic and transversely isotropic characteristics are constructed, and the corresponding constitutive equations are obtained and analyzed.