Micromechanics of Composite Materials pp 177-220 | Cite as

# Estimates of Mechanical Properties of Composite Materials

## Abstract

Together with the methods described in the previous chapter, overall moduli and local field averages in the phases can be estimated by one of several approximate methods, which use different models of the microstructure. Among those described here are variants of the average field approximation, or AFA, which rely on strain or stress field averages in solitary ellipsoidal inhomogeneities, embedded in large volumes of different comparison media **L**_{0}. Among the most widely used procedures are the self-consistent and Mori-Tanaka methods, and the differential scheme, described in Sects. 7.1, 7.2 and 7.3. Those are followed by several double inclusion or double inhomogeneity models in Sect. 7.4, and by illustrative comparison with finite element evaluations for functionally graded materials in Sect. 7.5.

Together with the methods described in the previous chapter, overall moduli and local field averages in the phases can be estimated by one of several approximate methods, which use different models of the microstructure. Among those described here are variants of the average field approximation, or AFA, which rely on strain or stress field averages in solitary ellipsoidal inhomogeneities, embedded in large volumes of different comparison media **L**_{0}. Among the most widely used procedures are the self-consistent and Mori-Tanaka methods, and the differential scheme, described in Sects. 7.1, 7.2 and 7.3. Those are followed by several double inclusion or double inhomogeneity models in Sect. 7.4, and by illustrative comparison with finite element evaluations for functionally graded materials in Sect. 7.5.

Although the methods described here can also be applied to periodic composites, providing that their overall material symmetry is taken into account, this special and rather rare class of actual materials has been analyzed by methods not discussed herein, but revisited in Chap. 12. Extensive treatment and related references can be found in Babuska (1975), Suquet (1987), Nemat-Nasser and Hori (1999), and Walker (1993). Bensoussan et al. (1978) and Sanchez-Palencia (1980) survey basic theory of periodic homogenization problems.

All moduli estimates depend in different ways on the elastic moduli, volume fraction, shape and orientation of the phases. Spatial distribution of the phases is also reflected in certain estimates. However, absolute size of individual phases is not a factor in evaluation of overall moduli by the models. Of course, large differences in scale may cause interactions that distort the magnitudes of phase field averages in the small constituents. In such multi-scale systems, homogenization should proceed in sequence, at increasingly coarser scales, by first homogenizing the matrix and the finer scale inhomogeneities, before proceeding to the next scale. Hierarchical or multi-scale computational models had been described, for example, by Zohdi et al. (1996), Ghosh et al. (2001), Zohdi and Wriggers (2005) and Oskay and Fish (2007).

## 7.1 The Self-consistent Method (SCM)

The original idea of the method may be attributed to Einstein (1905). Evolution of the current form dates back to the work of Bruggeman (1935), who had used it to estimate dielectric, conductivity, and elastic constants of composite aggregates. That was followed by Hershey (1954), Kerner (1956), Kröner (1958), Hill (1965a) who applied the method to polycrystals, and by Budiansky (1965) and Hill (1965b, c) who applied it to composites. Laws (1973, 1974) extended the method to thermo-elastic problems, and Laws and McLaughlin (1978) used it to estimate creep compliances of linear visco-elastic solids. Other self-consistent estimates were found by Christensen and Waals (1972), Boucher (1974), Berryman (1980), and by Cleary et al. (1980), and by many other writers. Hill’s version of the method is in current use; see also the reviews by Laws (1980), Walpole (1981, 1984), Willis (1981), and Nemat-Nasser and Hori (1999) for additional results and references.

### 7.1.1 Estimates of Overall Elastic Moduli

_{r}of phase

**L**_{r}as a solitary ellipsoidal inhomogeneity in a large volume Ω of

**L**_{0}≡

*, that has the as yet unknown overall stiffness*

**L***of the aggregate. Overall uniform strain \( {{\varepsilon }^0} \) or stress \( {{\sigma }^0} \) is applied at the remote boundary \( \partial {\Omega} \), Fig. 7.1. The mechanical strain and stress concentration factors of each phase*

**L**

**L**_{r}follow from the expressions for their partial counterparts in (4.2.14) as

*is established by expanding the first term*

**L**That also confirms that the self-consistent estimate of the overall stiffness * L* follows from the general estimate (6.3.5), when one selects \( {{L}_0} = {L} \), and evaluates \( {P} = {{(}{L}^{*} + {L}{)}^{{ - 1}}} \) using the coefficients of

*. A similar proof can be derived for the overall compliance. The inclusion problem solution (4.2.14) and (6.3.5), both with identical*

**L***matrix, provide the same result.*

**P***In each application of the self-consistent method, it is necessary to select or identify the overall material symmetry* of \( {L} = {{L}^{\text{T}}} \) according to the dominant geometry of the microstructure. The as yet unknown coefficients of * L* are thereby identified and then used to construct the

*or*

**P***and*

**Q**

**A**_{r}or

**B**_{r}, for substitution into (7.1.2) or (7.1.5). The resulting system of implicit algebraic equations is solved for the magnitudes of the coefficients of

*or*

**L**

**M.**An iterative solution may start using initial values of the coefficients of **L**_{0} consistent with the bounding theorems (6.2.16), taken as one of the H-S bounds (6.3.13). The iteration then proceeds until \( {{L}_0} \to {L} \) or \( {{M}_0} \to {M} \), respectively. The resulting estimate of the overall moduli is evidently bracketed by the H-S bounds. That was demonstrated, for example, by Hill (1964) for fiber composites, and by Kröner et al. (1966) for isotropic aggregates of cubic crystals. In multiphase systems, the self-consistent estimate assigns each phase, even a matrix phase, the same shape and alignment, defined by the single * P* or

*matrix in (7.1.1).*

**Q***In a two-phase system*, \( r = 1,\,\,2 \), where \( {c6557}{{A}6557} = {I} - {c_2}{{A}_2} \), (7.1.2) is reduced to

The **L**_{1} represents the stiffness of a matrix, and **A**_{2} is the concentration factor of the inhomogeneity. In matrix-based particulate or fibrous aggregates, the overall moduli do not change if the role of the two phases is reversed, as long as their volume fractions remain unchanged, because the model treats all phases ‘on an equal footing’ (Hill 1965c).

### 7.1.2 Elastic Moduli of Two-Phase Fiber Composites

*r*= 1, 2, both transversely isotropic about and aligned in the longitudinal direction, the self-consistent estimates of overall moduli were derived by Hill (1965b) from elasticity solutions of an extended composite cylinder element with a surrounding shell made of the effective medium. The composite is transversely isotropic, with the overall stiffness matrix in (2.3.3). The plane strain bulk modulus

*k*is connected to the transverse shear modulus

*m*by

*m*is the positive root of the cubic

*p*is the positive root of the quadratic

*r*= 1 or both phases are isotropic, with moduli

*K*

_{1}and

*G*

_{1}, then according to (2.3.6), the modulus \( {k6557} = {K6557} + {G6557}/3 \), and \( {m6557} = {p6557} = {G6557} \). The overall modulus

*k*and the phase moduli are then substituted into the universal connections

*n*and

*l*. It can be verified that the above results are bracketed by the Hashin-Shtrikman bounds in Sect. 6.3.3, but not necessarily by tighter bounds.

In contrast to the Hashin-Shtrikman bounds on overall * L* or

*, derived with comparison media of constant stiffness \( {L}_0^{{{(} + {)}}}\,\,{\text{or}}\ {L}_0^{{{(} - {)}}} \), the self-consistent method relies on variable*

**M**

**L**_{0}=

*, which depends on phase volume fractions. Therefore, the self-consistent estimates of moduli are not aligned with either bound, but approach the lower or upper bound at \( {c_f} \to 0 \) or \( {c_f} \to 1 \)*

**L**_{.}

That is illustrated in Figs. 6.4 and 6.6 for a glass/epoxy fiber composite. The generalized self-consistent (GSCM) estimate of the transverse shear modulus shown in Fig. 6.6 is typically closer and aligned with the H-S lower bound, and with the Hashin and Rosen (1964) CCA upper bound \( {m^{{{(} + + {)}}}} \).

The above results may also be applied to two-phase systems reinforced with aligned discontinuous fibers that have sufficiently large length/diameter aspect ratio. For example, Laws and McLaughlin (1979) computed the overall compliances of a glass-polyester composites, where the fibers were modeled as aligned prolate spheroids. The fiber length effect receded entirely at aspect ratios exceeding 100 for any fiber concentration. Since the effect of fiber length depends also on the magnitude of phase moduli and volume fractions, their procedure has to be repeated in applications to any other system of interest. The role of phase moduli was illustrated by Russel (1973) in dilutely reinforced systems.

Self-consistent estimates of overall elastic moduli of composite materials reinforced by randomly oriented needle-like short fibers or by disk-shaped platelets were derived by Walpole (1969, eqns (60)-(61)). At a given volume fraction of reinforcement, platelets have a stronger effect in determining the overall bulk and shear moduli of the aggregate.

### 7.1.3 Elastic Moduli of Two-Phase Particulate Composites

*G*and

*K*of such composites were found by Hill (1965c) as

The latter result gives an exact value of *K* for isotropic composites of arbitrary geometry when the phases have identical shear moduli (Hill 1963a). Since *K* depends on the shear modulus *G*, given by the quartic equation (7.1.9), the self-consistent method predicts only one independent modulus of an isotropic two-phase system. As an alternative, Walpole’s (1985c) equations (6.3.26) can be iteratively solved for \( {K_0} \to K \) and \( {G_0} \to G. \)

Figure 6.4 shows an example of a self-consistent prediction \( {G_{\mathit{SCM}}} \) of the shear modulus of a S-glass/epoxy particulate composite, that is bracketed by the H-S bounds \( {G^{ + }} \) and \( {G^{ - }} \). However, it violates the tighter Hashin bound \( {G^{{ + + }}}, \) which is respected by the generalized self-consistent estimate \( {G_{\mathit{GSCM}}} \).

If the disperse phase is replaced by cavities, \( {K_2} = {G_2} = 0 \), and also when both phases are incompressible, \( {K6557},\,\,{K_2} \to \infty \), then (7.1.9) has a positive root when and only when \( {c_2} < 0.5 \), and \( G = 0 \) at \( {c_2} > 0.5 \). That prediction is contradicted, for example, by properties of closed cell foams. Budiansky and O’Connell (1976) also derived an unexpected prediction, in applications of the self-consistent method to isotropic solids containing randomly oriented circular cracks of radius *a* and density \( N/V \), the number of cracks per unit volume. Both overall moduli reach zero at \( \varepsilon = N\left\langle {{a^3}} \right\rangle /V \to 9/16, \) which might estimate a critical crack density, albeit not confirmed by later estimates by the Mori-Tanaka and double inhomogeneity models, as shown in Sects. 7.2 and 7.4 below. However, such physically improbable outcomes do not arise in fiber composites weakened by aligned slit or penny-shaped cracks, where all moduli decrease gradually to either finite or zero values with increasing crack density (Laws et al. 1983; Laws and Dvorak 1987). Similar issues arise in other applications of the method, e.g., to dielectrics (Milton 2002). Therefore, the method should not be used when the phase moduli are of different order in magnitude, or when at least one of them assumes extreme or zero magnitude. Such situations may also be encountered when the method is applied to composites with elastic-plastic, viscous, and other inelastic matrices, which may have low instantaneous tangential stiffness.

### 7.1.4 Restrictions on Constituent Shape and Alignment

Equations 7.1.3 show that the method predicts a diagonally symmetric stiffness matrix * L* =

**L**^{T}when applied to systems where each phase subvolume has the same shape and alignment described by a single

*tensor. Applications of the self-consistent method to multi-phase composites that have more than one reinforcement phase shape or alignment, yield stiffness or compliance estimates that are not diagonally symmetric (Benveniste et al. 1991b). However, numerical experiments described, in part, in (7.1.15) and (7.1.16) below, indicate that different phase shapes and alignments may be admitted for the reinforcement phase*

**P**

**L**_{2}in a matrix-based two-phase system, \( r = 1,\,\,2. \)

Each superscript \( s = 2, \ldots n, \) denotes a particular shape and/or alignment of an inhomogeneity \( {{L}_{2 }} \). The \( {{P}_s}\,{\text{or}}\,\,{{Q}_s} \) in (7.1.11) depend only on the overall * L* or

*, and not directly on*

**M**

**L**_{r}or

**M**_{r}of either phase.

_{3}Al matrix, reinforced by SiC fibers of circular crossection, and by SiC circular discs. Both the fiber axes and the normals to the disks planes are aligned with the

*x*

_{3}–axis of a Cartesian system. The phases are isotropic, and have the following elastic moduli

## 7.2 The Mori-Tanaka Method (M-T)

The presented form of this method was proposed by Benveniste (1987a), who interpreted a brief derivation by Mori and Tanaka (1973) of the average stress caused by transformed homogeneous inclusions in a large matrix volume. In the context of the procedures leading to the estimates of concentration factors in Sect. 6.3.1, their result suggests that the inclusion should embedded in a large volume of the matrix phase, and subjected to an average matrix stress. A different form of the method was suggested by Weng (1984). Numerous applications to many different problems have appeared in the literature. For example, composites with coated fibers were analyzed by Benveniste et al. (1989) and by Chen et al. (1990). Porous materials were treated by Zhao et al. (1989). Specific results for many typical composite systems were derived by Chen et al. (1992).

### 7.2.1 Elastic Moduli and Local Fields of Multiphase Composites

**L**_{r}as a solitary inhomogeneity Ω

_{r}embedded in a large volume Ω

_{1}of the matrix

**L**_{1}. The as yet unknown average strain \( {{\varepsilon }6557} \) or stress \( {{\sigma }6557} \) in the matrix phase are applied as a uniform strain or stress at a remote boundary \( \partial {{\Omega}6557}\), Fig. 7.2.

**L**_{r}

**L**^{*}tensors in (4.2.9) are evaluated in

**L**_{1}.

*uniform overall strain*\( {{\varepsilon }^0} \) is applied to the representative volume of a composite material, the matrix average strain \( {{\varepsilon }6557} \) is found by referring to (3.5.5) which indicates that \( \sum {{c_r}{{\varepsilon }_r}} = \left( {\sum {{c_r}{{T}_{r}}} } \right){{\varepsilon }6557} = {{\varepsilon }^0} \). Of course, \( {{T}6557}{{\varepsilon }6557} = {{\varepsilon }6557} \), since the partial strain concentration factor of the matrix, \( {{T}6557} = {I} \). Notice the difference from the dilute approximation result (4.4.3). Therefore, the local strain averages are

After some algebra, this form can be converted to (6.3.5)_{2}, with both **L**^{*} and * P* evaluated in the matrix,

**L**_{0}=

**L**_{1}as shown in (7.2.29).

*For a two-phase composite*\( \ r = 1,\,\,2 \), with matrix

**L**_{1}of volume fraction

*c*

_{1}, and reinforcements

**L**_{2}, the corresponding expressions are

Equation (7.2.26) below shows a more general form of the two-phase composite stiffness, which admits different shapes and alignments, or different * P* tensors for the inhomogeneities of phase

**L**_{2}.

*uniform overall stress*\(\ {{\sigma }^0} \), the above sequence is modified. It starts with the estimate of the average stress in each phase embedded in a large matrix volume loaded by a uniform stress \( {{\sigma }6557} \)

**W**_{1}=

*. Since \( \left( {\sum {{c_r}{{W}_r}} } \right){{\sigma }6557} = {{\sigma }^0} \), the local stress average are*

**I**

**M**_{1}of volume fraction

*c*

_{1}

If for all *r*, the difference \( \left( {{{L}6557} - {{L}_r}} \right) \) is positive [or negative] semi-definite, then, according to the bounding theorems (6.2.16), the method delivers the upper [or lower] Hashin-Shtrikman bound on the overall * L*. These and other connections with the bounds were examined by Norris (1989) and Weng (1990, 1992). In composites which display a large contrast between constituent moduli, the Mori-Tanaka method tends to underestimate [or overestimate], even at moderate concentrations, the actual overall elastic moduli, c. f., Fig. 6.5. However, in applications to two-phase media containing voids and/or cracks dispersed in a homogeneous matrix, the method delivers a Hashin-Shtrikman upper bound which, unlike the self-consistent estimate, approaches zero value only when so does the matrix volume fraction.

### 7.2.2 Elastic Moduli of Fibrous, Particulate and Layered Composites

*two-phase fiber composite*, with transversely isotropic fiber and matrix phases, where phase elastic moduli denoted by \( {k_f},{m_f}\,{\text{and}}\,{p_f} \) and \( {k_m},{m_m}\,{\text{and}}\,{p_m} \), the Mori-Tanaka method estimates of the corresponding overall moduli are

The above axisymmetric moduli *k*, *l*, and *n*, coincide with those of a single composite cylinder that has a fiber core and matrix shell (Hill 1964), and also with those of an assemblage of such composite cylinders shown in Fig. 6.1. It can be verified that the same results are obtained from the Walpole’s formulae (6.3.27) when the comparison medium moduli are selected there as equal to those of the matrix, i.e., \( {k_0} = {k_m},\,\,{m_0} = {m_{{m\,}}},\,\,{p_0} = {p_m}. \) Estimates of overall elastic moduli of *multi-phase aligned fiber composites* can be found in Chen et al. (1992).

Figure 6.5 shows the M-T estimate of the overall transverse shear modulus *m* and the transverse Young’s modulus *E*_{T}, for a glass fiber/epoxy composite. Since the difference between the fiber and matrix stiffness \( \left( {{{L}_f} - {{L}_m}} \right) \) is positive definite and **L**_{m} = **L**_{1} is the comparison medium in this case, the M-T estimates coincide with the H-S lower bounds \( {m^{{( - )}}} \) and \( E_T^{{{(} - {)}}} \).

*For isotropic particulate composites*with an isotropic matrix \( r = 1 \) containing isotropic spherical reinforcements \( r = 2 \), Benveniste (1987a) found the Mori-Tanaka method estimates of the bulk and shear moduli as

When \( \,{G6557} < {G_2},\,\,{K6557} < {K_2},\, \)these expressions are equivalent to the Hashin-Shtrikman lower bounds and they also follow from Walpole’s formulae (6.3.26) for \( {G_0} = {G6557},\,\,{K_0} = {K6557} \). The ratio \( G/{G6557} \) found from (7.2.15) is also equal to \( G/{G_m} \) in (6.5.13). Upper bounds are obtained by exchanging the phase subscripts in (7.2.15).

In the P100/Cu composite, Fig. 6.2, the Mori-Tanaka estimates coincide with the upper bounds \( {k^{{{(} + {)}}}} \), \( {m^{{{(} + {)}}}} \) and \( {p^{{{(} + {)}}}} \), and the lower bounds \( {n^{{{(} - {)}}}} \) and \( E_A^{{{(} - {)}}} \). In the S-glass/epoxy fiber composite considered in Fig. 6.4, the Mori Tanaka estimate of the overall shear modulus *G*_{T} coincides with the H-S lower bound \( {G^{{{(} - {)}}}} \). In both illustrations, the method delivers a fairly accurate prediction of the relevant elastic modulus.

Another measure of accuracy of the method was presented by Christensen et al. (1992). They evaluated the percent error in the shear modulus of a particulate composite, compared to the generalized self-consistent estimate of Sect. 6.5, for different combinations of phase moduli and particle volume fractions. As expected, the error was magnified by large contrast between phase moduli, but it turned out to be less than 10% for \( {c_2} \leq 0.4 \) at \( {\nu6557} = 1/2,\,\,{\nu_2} = 1/3 \), and \( 0.2 < {G_2}/{G6557} < 10. \) At higher particle concentrations, the Mori-Tanaka method tends to underestimate the effective shear modulus obtained from experiments and the generalized self-consistent method, as illustrated in Fig. 6.5.

*multi-layer composite aggregate*consisting of platelets or flat discs \( r = 2,3, \ldots, n \), all bonded to a continuous matrix \( r = 1 \), and aligned such that their normals are parallel to the

*x*

_{1}–axis. Since the in-plane size of the discs need not be limited, this composite is equivalent to flat multilayer plates or solids, without a distinct matrix, which appear in assorted technological applications, including layered coatings, soil and rock formations, and sediments. Each layer is regarded as transversely isotropic, with the

*x*

_{1}–axis of rotational symmetry normal to the layer plane. Transverse isotropy prevails on the overall scale, hence there are five independent overall elastic moduli (2.3.3), exhibiting the interrelations.

These results agree with the self-consistent estimates found by Laws (1974, eqns (42)–(46)) and also with those found by Postma (1955) for multilayered solids.

### 7.2.3 Elastic Moduli of Solids Containing Randomly Oriented Reinforcements and Cracks

*and*

**P**

**L**_{r}. Using curly brackets for the averages, the desired overall stiffness is

Analytic derivations of the scalars in the orientation average \( \left\{ {{{T}_s}} \right\} \), for different oblate and prolate spheroidal shapes, including spheres, needles, disks and penny-shaped cracks, were carried out by Kröner (1958), Wu (1966) and Berryman (1980), who also shows corrections to earlier solutions, and denotes \( a = P,\,\,b = Q \). In numerical evaluation of the orientation averages, the nonzero coefficients of the (6 × 6) matrix \( {\text{\{ }}{T_{{\alpha \beta }}}{\text{\} }} = {\{ {T_{{\alpha \beta }}}{\text{\} }}^{\text{T}}} \) follow from (2.2.30) and (2.2.33) in terms of the coefficients of \( {T_{{\alpha \beta }}} \).

**L**_{1}, reinforced by randomly oriented short fibers, distribution of which is isotropic on the macroscale. General forms of the averaged moduli are

**L**_{r}. Berryman (1980) derived expressions for their evaluation for cracks and other shapes.

*K*

_{2}and

*G*

_{2}, the effective overall bulk and shear moduli

*K*and

*G*are

For the isotropic matrix, \( {\gamma6557} = {G6557}{(3}{K6557} + {G6557}{)/(3}{K6557} + 7{G6557}{)} \).

This M-T result is identical to that predicted by the self-consistent method (Walpole 1969, eqn. (61)). At a given concentration, the platelets yield higher overall moduli than do needles or short fibers, as first observed by Wu (1966). Another analysis of overall elastic moduli of randomly reinforced composites was developed by Christensen and Waals (1972).

The above results apply only to composites reinforced by a spatially random distribution of short fibers or other reinforcements. However, preferential fiber orientations can develop, for example, in systems produced by injection of liquid polymer and short fiber mixtures into final shape moulds. Actual distributions are not easily detected, but when they are, they can be related to overall material symmetry of the composite by certain orientation distribution functions (Ferrari and Johnson 1989). Actual material symmetry can be deduced, for example, from directionally dependent velocities of ultrasonic waves (Sayers 1992; Dunn and Ledbetter 2000).

The Mori-Tanaka method also predicts overall properties of solids containing a distribution of *cavities and cracks*. As long as the cavities are approximated by spherical inhomogeneities with vanishing stiffness, the overall moduli can be obtained by letting \( {G_2} = {K_2} = 0 \) in (7.2.15).

*a*as

*N*/

*V*is the number of cracks in a unit volume

*V*. Both the ‘matrix’ and the cracked medium are isotropic, with elastic moduli \( {K6557},\,{G6557},\,{\nu6557} \) and

*K*,

*G*, respectively. In contrast to the self-consistent prediction of these moduli by Budiansky and O’Connell (1976), who found zero moduli at \( {(}\varepsilon {)} = \eta \to 9/16 \), the above expressions predict only a gradual decrease with growing \( \eta. \) At very low crack concentrations, for \( \eta \ll 1, \) there is agreement with the dilute estimate by Walsh (1965). Of course, crack configurations yielding vanishing elastic moduli can be constructed or imagined with different values of

*η*, including those obtained by the self-consistent method. However, as shown by (7.1.14), the method also predicts zero shear modulus of a matrix with 0.5 volume fraction of spherical cavities.

### 7.2.4 Restrictions on Constituent Shape and Alignment

In applications to multiphase systems, the diagonally symmetric Mori-Tanaka estimates of overall stiffness * L* =

**L**^{T}and compliance

*=*

**M**

**M**^{T}follow from (6.3.5) and (6.3.6), providing that all inhomogeneities have the same shape and alignment, described by a single

*or*

**P***tensor. Here we show that the said symmetry can also be established in applications of the method to two-phase composites, consisting of the matrix phase*

**Q**

**L**_{1}and many perfectly bonded ellipsoidal inhomogeneities \( s = 2,3, \ldots, n \) of phase

**L**_{2}, which may have different shape and/or alignment, described by

**P**_{s}or \( {L}_s^{*} \). The proof was derived by Benveniste et al. (1991b).

*r*= 1 and \( s = 2,3, \ldots, n \), while \( {{\varepsilon }6557} = {{T}6557}{{\varepsilon }6557} \) and \( {{T}6557} = {I} \). A more convenient form of the partial concentration factors

**T**_{s}is now derived, by defining a new tensor \( {P}6557^s \).

**T**_{s}yields the overall stiffness in the following diagonally symmetric form, which is then adjusted using (7.2.23), to depend only on the

**P**_{s}matrices that describe shape and alignment of different inhomogeneities

**L**_{2}.

Therefore, two-phase systems with arbitrary phase geometry are admissible in implementation of the Mori-Tanaka method, as they appear to be in the self-consistent method. In contrast, applications of either of the two procedures to multiphase systems require all inhomogeneities to have the same shape and alignment, or spatial distributions amenable to orientation averaging. As already indicated in the opening paragraph, the inhomogeneities need not be of the same size, but they should belong to the same size scale. The shape and alignment constraints are relaxed by the double-inclusion model *CB*, in Sect. 7.4.4 and Fig. 7.7.

**L**^{*}and \( {P} = \left( {{L}^{*} + {{L}6557}} \right)^{-1} \) tensors, both evaluated in the matrix

**L**_{1}. Recall from (7.2.1) that \( {{T}_{r}} = {\left[ {{I} + {P}\left( {{{L}_r} - {{L}6557}} \right)} \right]^{{ - 1}}} = {\left( {{L}^{*} + {{L}_r}} \right)^{{ - 1}}}\left( {{L}^{*} + {{L}6557}} \right) \). Substitute this into (7.2.3), written as

This confirms that the Mori-Tanaka method yields the stiffness predicted by (6.3.5), providing that both **L**^{*} and * P* are evaluated in the matrix,

**L**_{0}=

**L**_{1}. Therefore, either the self-consistent or the Mori-Tanaka predictions of overall stiffness and compliance can be generated by solving inhomogeneity problems in Figs. 7.1 or 7.2, or by using (6.3.5) and (6.3.6), with

**L**_{0}=

*or*

**L**

**L**_{0}=

**L**_{1}respectively. Both approaches require evaluation of the

*tensor, but only in the matrix phase for the M-T method. Recall that (6.3.5) also yields the H-S bounds, as described in Sect. 6.3.2.*

**P**### 7.2.5 Derivation of Effective Phase Moduli

Experimentally determined overall moduli of a two-phase composite material, and those of one phase may be used to estimate the effective moduli of the other phase. For example, mechanical testing of thin fibers and filaments is limited to simple tension and torsion, which yield the longitudinal Young’s modulus **E**_{11} and if the fiber is regarded as transversely isotropic according to (2.3.3), the longitudinal shear modulus *p.* The remaining elastic moduli of transversely isotropic fibers, such as carbon or graphite fibers, cannot be found by direct measurement. However, estimates of effective magnitudes of the unknown moduli can be derived from experimentally measured overall or macroscopic elastic moduli of two-phase aligned fiber composites reinforced by such fibers. Matrix moduli and phase volume fractions also need to be known. Composite test samples used in such tests should be fabricated in a manner that assures good fiber alignment and statistically homogeneous distribution of the fibers in the transverse plane, which may not be found in all standard fibrous plies. The same approach can yield effective moduli of reinforcements consisting of particles of irregular shape and varied composition.

Loading conditions that provide the overall moduli of fibrous systems are described in Table 2.4. The measured values are used to generate coefficients of the compliance and stiffness * M* and

*in (2.3.2) and (2.3.3). Together with the known moduli of the composite matrix and the matrix volume fraction*

**L***c*

_{1}, the overall elastic moduli

*k*,

*m*, and

*p*included in the coefficients of

*and*

**M***are substituted into expressions for self-consistent or Mori-Tanaka estimates of these moduli.*

**L**In particular, self-consistent estimates of the fiber moduli *k*_{2}, *m*_{2}, and *p*_{2} can be found by solving in sequence equations (7.1.6), (7.1.7), (7.1.8). The remaining moduli *n*_{2}, *l*_{2} then follow from the universal connections (3.9.4). Mori-Tanaka estimates of all five fiber moduli can be found by solving (7.2.11), (7.2.12), (7.2.13) for *k*_{f}, *m*_{f}, and *p*_{f}. For particulate composites, self-consistent estimates of effective reinforcement moduli, say, \( {K_2}{\text{\; and \;}}{G_2} \) follow from (7.1.9) and (7.1.10), and the Mori-Tanaka estimates from (7.2.15). The pairs of equations in (7.2.19) or (7.2.21) may also be used to estimate ‘effective’ moduli of randomly oriented fiber or platelet reinforcements.

The same approach can be used for evaluation of effective moduli of a matrix, of interest in materials with large specific surface areas, which may promote realignment of polymer chains or other interfacial reactions leading to moduli changes. Again, elastic properties of the other phase, and of the aggregate need to be known together with phase volume fractions.

## 7.3 The Differential Scheme

This averaging method employs an incremental sequence of dilute approximations discussed in Sect. 4.4, to find the stiffness or compliance matrix of a composite made of two or more phases in non-dilute concentrations. In each increment, a homogeneous matrix or a ‘backbone’ medium **L**_{0} is enriched by inserting a dilute concentration of inhomogeneities of one or more distinct phases **L**_{r}, *r* = 1, 2,…, *n*, and the mixture is homogenized. This incremental homogenization continues until it reaches final phase concentrations. The choice of **L**_{0} and of the volume fraction increments \( \Delta {c_r} \) added in each step may yield different estimates of the final stiffness. However, the sequence of dilute approximations guarantees diagonal symmetry of the predicted overall stiffness and compliance for any combination of phase properties, shapes and alignments.

First proposed by Bruggeman (1935) and Roscoe (1952), and later expanded by Boucher (1974), the method was reviewed by Cleary et al. (1980). McLaughlin (1977) had shown that stiffness estimates for two-phase dispersion of spheres and fiber reinforced materials, generated by incremental additions to a unit matrix volume, lie between the Hashin-Shtrikman bounds. Callegari, et al. (1985), Norris (1985) and Norris et al. (1985) expanded the basic theory of the method and examined how such bounds can be realized by two-phase systems. Benveniste (1987b) suggested creating a two-phase dispersion by adding composite spheres, and had recovered the Hashin and Shtrikman (1962a, b) composite sphere assemblage results in the context of heat conduction. Applications of the differential scheme to materials containing distributions of cracks were explored by Hashin (1988).

*fixed volume process*or FVP, the initial ‘backbone’ material of stiffness

**L**_{0}resides in a fixed representative volume

*V*

_{0}. Each addition of phase volumes \( \Delta {{\text{v}}6557} + { }\Delta {{\text{v}}_2} + \cdots + \Delta {{\text{v}}_n} = \Delta {\text{v}} \) is preceded by removing from

*V*

_{0}an equivalent volume \( \Delta {\text{v}} \) of the already homogenized material which always includes certain volume ratio of

**L**_{0}. After a current increment is homogenized, the phase volume fractions in the FVP procedure satisfy

This volume exchange process continues until all phase volume fractions reach their prescribed magnitudes in *V*_{0}. At the end point, some of the backbone material may be left as one of the actual phases, or it may be entirely replaced by the gradually added phases \( r = 1,2, \ldots n. \) In any event, the final stiffness prediction depends on **L**_{0}, and on the phase volume ratios added in each step.

*variable volume process*(VVP), which builds the same two or multiphase material \( r \,{=}\, 1,2, \ldots n \) by starting with volume v

_{1}of an actual phase

**L**_{1}, usually selected as the matrix material. Volume v

_{1}remains constant during homogenization. Dilute reinforcement volumes \( \Delta {{\text{v}}_r},r \,{\geqslant}\,2, \) are added in certain ratios, until all phase volumes reach their prescribed final magnitudes \( {{\text{v}}_{{1}}},{{\text{v}}_{{2}}}, \ldots {{\text{v}}_n} \). The mixture is homogenized after each such addition, hence the current total volume

*V*(

*t*) increases from

*V*(0) = v

_{1}as a function of ‘time’

*t*, until it reaches the final volume \( V = \Sigma_{{r = 1}}^n{{\text{v}}_r} \). Since all

*r*> 1 phase volumes gradually increase, their volumes and volume fractions are

**L**_{1}is now replaced by the overall

*(*

**L***t*) found in the previous step. The stiffness increment caused by addition of small reinforcement volumes in the VVP sequence is

*(*

**L***t*). In each step, the

**P**_{r}(

*t*) needs to be updated as a function of

*(*

**L***t*). This expression for \( \Delta {L}(t) \) provides a recursive formula for numerical evaluation of each next stiffness increment until the final overall stiffness

*is reached at prescribed phase concentrations.*

**L***is obtained by letting \( \Delta {{\text{v}}_r} \to 0 \) and \( V{(}t{)} \gg \Delta V \to 0 \). Then*

**L***c*follows from (7.3.2)

*(0) =*

**L**

**L**_{1}(Norris 1985). This equation also governs the removal-replacement or FVP procedure, where

*(0) =*

**L**

**L**_{0}.

**L**_{0}, typically chosen as the actual matrix of the composite system,

**L**_{0}=

**L**_{1}. Then, in a two-phase system

*r*= 1, 2, where the reinforcements have stiffness

**L**_{2}and are added by increments \( \Delta {c_2} \), there is \( {c_r} = c = {c_2} \) and (7.3.6) is reduced to

Together with \( {{T}_{2}} = {{T}_{2}}{(}{L}{)} \), this is a coupled system of ordinary differential equations which can be integrated to yield the final stiffness. McLaughlin (1977) derived (7.3.7) as his equation (4), and had shown its solutions for both an isotropic dispersion of spheres and transversely isotropic dispersion of aligned and similar spheroids. In both cases, the predicted moduli lie between the corresponding Hashin-Shtrikman bounds. However, as shown by Christensen (1990), shear moduli estimates generated by the differential scheme can be very different from those predicted by the much more rigorous generalized self-consistent method, Fig. 6.5.

The matrix **L**_{1} can function both as a ‘backbone’ and one of the incrementally added phases. Norris (1985) shows that this enables a wider selection of the path followed in adding the phase increments. For example, each phase can be added by a separate sequence while the other phases remain constant or zero, and the order of these sequences can be varied within certain restrictions. This family of differential schemes can generate many different overall stiffness estimates of uncertain value, depending on the choice of the path. Of course, it seems reasonable to add in each step all reinforcement volumes in proportion to their final densities, as they might be added in actual fabrication. This happens in the variable volume of VVP process, which gradually reduces the matrix volume fraction from unity to its final magnitude. However both FVP and VVP are governed by (7.3.6).

Since each phase *r* > 1 is recognized only while being added to the mixture, the differential scheme does not offer a direct insight into phase interactions, as reflected by the mechanical strain and stress concentration factors. Only in two-phase systems \( r = \alpha, \,\,\beta \), one can find estimates of concentration factor tensors \( {{A}_r},\,\,{{B}_r} \), in terms of the current or final overall stiffness * L* and phase stiffnesses \( {{L}_{\alpha }},\,\,{{L}_{\beta }} \), using (3.5.13). The main advantage of the differential scheme is its freedom from the restrictions on shape and alignment of the reinforcements, discussed in Sects. 7.1.4 and 7.2.4.

## 7.4 The Double Inclusion and Double Inhomogeneity Models

### 7.4.1 Field Averages in a Double Inhomogeneity

**L**_{r}resides in an ellipsoidal subvolume \( {\Omega_r} \), which is surrounded by a layer or coating of another material

**L**_{g}in a volume \( {\Omega_g} = {\Omega_2} - {\Omega_r} \). All volumes \( {\Omega_2} \) are also ellipsoids, not necessarily coaxial with \( {\Omega_r} \). Each double inhomogeneity is then embedded in a large volume \( {\Omega_0} \supset {\Omega_2} \supset {\Omega_r} \) of a comparison medium

**L**_{0}. A uniform overall strain \( {{\varepsilon }^0} \) is applied at the remote boundary \( \partial {\Omega_0} \), Fig. 7.3. Several different predictions of overall stiffness of a composite aggregate can be derived using this model, based on distinct selections of the shapes and orientations of \( {\Omega_r} \) and \( {\Omega_2} \), and of the stiffnesses

**L**_{r},

**L**_{g}and

**L**_{0}. The original form of the equivalent inclusion method was derived by Hori and Nemat-Nasser (1993), together with an extension to a configuration with multiple layers surrounding \( {\Omega_r} \), which can be useful, for example, in modeling of graded interphases.

All interfaces are assumed to be perfectly bonded, but boundary conditions at \( \partial {\Omega_2}\,\,{\text{and \;}}\partial {\Omega_r} \), and actual local fields in the phases, are not known. However, strain averages in \( {\Omega_r} \) and \( {\Omega_g} \) can be approximated by referring to the Tanaka-Mori (1972) theorem in Sect. 4.5.4, which describes those caused in the double inclusion in a homogeneous medium by uniform eigenstrains applied in \( {\Omega_r} \) and \( {\Omega_g} \). A homogeneous double inclusion is created in **L**_{0}, in parallel with the double inhomogeneity of the same geometry. The connection between the field averages in the double inclusion and inhomogeneity is established by a formal application of the equivalent inclusion method of Sect. 4.3.2, albeit to local fields that are not necessarily uniform.

**L**_{0}. In superposition with a uniform overall strain \( \varepsilon_{\textit{mn}}^0 \) applied at the remote boundary \( \partial {\Omega_0} \), the total local strain averages, denoted by top bars are

In the double inclusion model, the above eigenstrains are regarded as equivalent eigenstrains, applied in the respective volumes of the homogeneous comparison medium **L**_{0}, to generate average local fields equal to those in the corresponding double inhomogeneity. The polarization fields (6.2.1) are now generated by two distinct local eigenstrains, in an admissible **L**_{0} restricted by (6.2.16).

**L**_{0}. The local stress averages are

**L**_{r}and double inhomogeneity

**L**_{2}, while both are present in a dilute concentration in the homogeneous medium

**L**_{0}which is remotely loaded by the uniform strain \( {{\varepsilon }^0} \). The strain \( {{\bar{\mu }}_2} = {{\Phi }_2}{{\varepsilon }^0} \) is now an average equivalent eigenstrain in the total volume \( {\Omega_2} \) of the double inclusion in a homogeneous medium

**L**_{0}. Although the eigenstrains are only piecewise uniform in \( {\Omega_r} \) and \( {\Omega_g} \), the Eshelby tensor

**S**_{2}and the average equivalent eigenstrain \( {{\bar{\mu }}_2} \) provide the average strain \( {{\bar{\varepsilon }}_2} \) suggested by (4.5.47) and the Tanaka-Mori (1972) theorem.

**S**_{r}and

**L**_{r}, which may be different within each subvolume \( {\Omega_2} \). Diagonal symmetry of \( {L}_2^{{{(}r{)}}} \) requires that \( \left( {{{L}_0}{{\Phi }_2}} \right) = {\left( {{{L}_0}{{\Phi }_2}} \right)^{\text{T}}} \).

Once the effective stiffness of the double inhomogeneity is known, it can be used in modeling of composite aggregates that contain an assemblage of different inclusion pairs, each within its own outer boundary \( \partial {\Omega_2} \) defined by a single **S**_{2}, and perfectly bonded to a common comparison medium **L**_{0}. Such applications are described below. Hori and Nemat-Nasser (1993) give a proof of consistency \( {M}_2^{{{(}r{)}}} = {{(}{L}_2^{{{(}r{)}}}{)}^{{ - 1}}} \) of the (7.4.14) estimate and of many other features of the double inhomogeneity model.

### 7.4.2 Double Inhomogeneity Microstructures

The double inhomogeneity may not represent an element of the actual composite material, and the traction and displacement fields at the interface with the surrounding medium are not known. However, a double inhomogeneity with known effective stiffness \( {L}_2^{{{(}r{)}}} \) can be regarded as a single material inhomogeneity embedded in different concentrations in a large volume of a suitably selected medium **L**_{0}, in the context of one of the average field or AFA approximations of overall stiffness of an aggregate. For example, the partial strain concentration factor \( {T}_2^{{{(}r{)}}} \) derived in (7.4.12) can be used in (6.3.2) to develop a corresponding \( {A}_2^{{{(}r{)}}} \) for substitution into the overall stiffness formula (6.3.5). Since the stiffness \( {L}_2^{{{(}r{)}}} \) depends on the shape and orientation of \( {\Omega_r} \) selected for each inhomogeneity **L**_{r}, and on the stiffness **L**_{g}, a composite aggregate ‘reinforced’ by double inhomogeneities of different stiffnesses \( {L}_2^{{{(}r{)}}} \) is a multi-phase system. This implies that the enclosures \( {\Omega_2} \) need to have the same shape and alignment, described by **S**_{2}, to satisfy the restrictions outlined in Sects. 7.1.3 and 7.2.4. Interpenetration or overlap of the \( {\Omega_2} \) subvolumes are excluded. Inhomogeneities in the interior of enclosures \( {\Omega_2} \) may have different shapes, orientations and material properties, subject to the requirement that \( {{L}}_2^{{{(}r{)}}} = {{(}{{L}}_2^{{{(}r{)}}}{)}^{\text{T}}} \).

Different choices of **S**_{2} impose a spatial distribution on the subvolumes \( {\Omega_r} \) of the inhomogeneities **L**_{r} in the entire volume of a composite material. For example, in the composite sphere assemblage of Fig. 6.1, the outer surfaces of the shells surrounding the inhomogeneities impose a spherical distribution, with density \( {c_2} = {{{\Omega }}_2}/{{{\Omega }}_0} \), \( 0 < {c_2} \leq 1 \), where \( {\Omega_2} \) is the total volume of all double inhomogeneities in the representative volume \( {\Omega_0} \) of the composite. However, \( {c_2} \leq 1 \) in general, depending on the selected range of sphere diameters. If all spheres are of the same size, then their volume fraction may not exceed the upper packing limit, \( {c_2} \leq {c_{{cp}}} \) which different methods cited in Sect. 3.3.2 estimate as \( 0.6 \leq {c_{{cp}}} \leq\break 0.7405 \).

**L**_{r}have many different shapes and orientations, and are distributed with spheroidal symmetry in the representative volume. Prolate spheroids with parallel symmetry planes depict all enclosures and some inhomogeneities in this idealized image. All spheroidal enclosures have identical aspect ratios and alignment.

Each transverse plane intersects the spheroidal enclosures in an assemblage of circles, hence isotropic distribution of reinforcements is assured in this plane of the aggregate. In a similar manner, aligned ellipsoidal enclosures may impose orthotropic overall symmetry, even on aggregates reinforced by particles which have the same shape and random alignment. Overall transverse isotropy or orthotropy could also be imposed by aligned cylindrical enclosures containing ribbons of ellipsoidal crossections.

_{2}, and \( {c_2} = {{{\Omega }}_2}/{{{\Omega }}_0}\,{\leq}\,1 \) is the volume fraction occupied by the double inhomogeneities in a representative volume of the composite aggregate. While the magnitude of \( {f_r} \) may approach unity when Ω

_{2}and Ω

_{r}are of similar orientation and size, for example, in coated reinforcements, the magnitude of \( {c_r} \) is often significantly reduced, even at \( {c_2} \to 1 \), by a large difference in aspect ratios or orientations of Ω

_{2}and Ω

_{r}.

Of course, actual dimensions or volume magnitudes of either Ω_{r} or Ω_{2} can not be enforced in the double inhomogeneity model, or by other AFA models, since they all admit only volume fraction, shape and alignment information. The two volume fractions, the aspect ratios of the ellipsoids used in deriving the **P**_{2} and **P**_{r} tensors, and the orientations of these ellipsoids in the coordinates of the representative volume, are the only parameters that define the geometry of the microstructure.

_{r}and Ω

_{2}, defined by (4.6.1), Fig. 7.5. Both Ω

_{r}and Ω

_{2}share the \( {x6557} - {\text{axis}} \) of rotational symmetry. Individual aspect ratios are \( {\rho_r} = a6557^{{{(}r{)}}}/a_2^{{{(}r{)}}} \) and \( {\rho_2} = a6557^{{{(}2{)}}}/a_2^{{{(2)}}} \), where the length is \( \left| {\,{x6557}} \right| = {a6557} \) and the diameter \( \left| {\,{x_2}} \right| = {a_2} = \left| {\,{x_3}} \right| = {a_3} \). Volumes of the spheroids are \( \left| {\,{\Omega_r}} \right| = {(4}\pi {/3)}a6557^{{{(}r{)}}}{{(}a_2^{{{(}r{)}}}{)}^{{2}}} \), and \( \left| {\,{\Omega_2}} \right| = {(4}\pi {/3)}a6557^{{{(2)}}}{{(}a_2^{{{(2)}}}{)}^{{2}}} \), and their ratio is \( {f_r} = \left| {{\Omega_r}} \right|/\left| {{\Omega_0}} \right| \).

_{r}is enclosed by an ‘elongated’ Ω

_{2}. Next, let \( {\rho_2} < {\rho_r} \), so that \( a6557^{{{(}r{)}}} \leq a6557^{{{(2)}}} \) and \( a_2^{{{(}r{)}}} < a_2^{{{(2)}}} \); a ‘flatter’ Ω

_{2}surrounds and comes in contact with an ‘elongated’ Ω

_{r}when \( a6557^{{{(}r{)}}} = a6557^{{{(2)}}} \). This yields volume fraction limits

Coefficients of **P**_{2} for oblate and prolate spheroids in a transversely isotropic solid can be derived from those of the related Eshelby tensor * S*, which were determined by Withers (1989). A simpler form of

*, valid for spheroids in an isotropic solid, is given by Ponte Castaneda and Willis (1995), Sect. 4.6.5. Spherical enclosures surrounded by an isotropic matrix or an isotropic comparison medium are indicated in modeling of randomly orientated inhomogeneities or cracks. For the former, the corresponding*

**P***tensor appears in Sect. 4.6.2. Aligned penny-shaped or slit cracks can be enclosed by flat disks of ribbons, with*

**P***tensor described in Sects. 4.6.4 or 4.6.3. The effort involved in finding the required coefficients of the*

**P***tensors is reduced by selecting*

**P**

**L**_{0}=

**L**_{g}=

**L**_{1}, as suggested in (7.4.25) below. However, any differences in alignments of Ω

_{r}call for transforming their coefficients into overall coordinates attached to the representative volume, as described in Chap. 1.

### 7.4.3 Connections with the Self-consistent and Mori-Tanaka Estimates

**S**_{2}tensors, and the stiffnesses

**L**_{g}and

**L**_{0}, which complement the

**S**_{r}and

**L**_{r}characterizing the actual inhomogeneity. One such form, for a two-phase system with aligned reinforcements of the same shape, postulates that

**S**_{2}=

**S**_{r}=

*, or \( \Delta {S} = \mathbf{0} \), and it satisfies the diagonal symmetry requirement \( {{L}}_2^{{{(}r{)}}} = {{(}{{L}}_2^{{{(}r{)}}}{)}^{\text{T}}} \) by selecting the external comparison medium \( {{L}_0} = {L}_0^{\text{T}} \) to have the same stiffness \( {{L}_0} = {L}_2^{{{(}r{)}}} \). This implies that \( {T}_2^{{{(}r{)}}} = {I} \), \( {{\boldsymbol{\Phi }}_2} = {\mathbf{0}} \), and that \( {{\Phi }_g} + \gamma {{\Phi }_r} = \mathbf{0} \) in (7.4.13). The double inhomogeneity now behaves as a neutral inhomogeneity, however, the average strain in the inhomogeneity*

**S**

**L**_{r}in Ω

_{r}follows from (7.4.10) as

Although the \( \Delta {{S}} = \mathbf{0} \) also holds for the double inhomogeneity used in the derivation of the generalized self-consistent method of Sect. 6.5, the present model predicts only the self-consistent result. This is a reminder of the approximation induced by application of the uniform equivalent eigenstrains (7.4.6) in the double inclusion, which do not reproduce the nonuniform local fields, derived for spherical and cylindrical double inhomogeneities by Christensen and Lo (1979). Moreover, since neither the tractions nor the displacements create homogeneous boundary conditions on \( \partial {\Omega_2} \), the Hill lemma (3.8.19) does not apply, and the energy of the double inclusion cannot be exactly evaluated using the phase field averages, to confirm (7.4.14).

**L**_{r}. To examine the symmetry condition \( {{L}_0}{\Phi }_2 = {\Phi }_2^{\text{T}}{{L}_0} \) in (7.4.14), we denote

*CA*) of this model stipulates that the ellipsoids \( {\Omega_r} \) and \( {\Omega_2} \) are similar and have identical Eshelby tensors.

*CA*represents a double inhomogeneity consisting of a core

**L**_{r}surrounded by a layer of matrix

**L**_{1}, such that both the core and outer ellipsoidal surfaces have the same aspect ratio and alignment. The comparison medium can have a different stiffness

**L**_{0}, selected in agreement with (6.2.16) or (6.2.24). Each inhomogeneity may have a certain stiffness

**L**_{r}and volume fraction

*f*

_{r}, both different inside each double inhomogeneity, yielding different stiffnesses \( {{(}{L}_2^{{{(}r{)}}}{)}^{{CA}}} \). However, the shape and alignment of all Ω

_{2}outer envelopes is the same, hence overall stiffness of the aggregate can be derived from a standard AFA procedure. If one identifies the comparison medium with the matrix

**L**_{0}=

**L**_{1}, then \( {{L}_0}{{\Phi }_g} \to \mathbf{0} \) and

**L**_{r}, and of the same shape Ω

_{r}. Therefore, if the overall stiffness is selected as\( \ {L} = {( {{L}_2^{{{(}r{)}}}} )^{{CA}}} \), then the double inclusion model provides the Mori-Tanaka estimate of

*. This implies that an entire representative volume would be filled with double inhomogeneities of the same ellipsoidal shape and different size, arranged in the spirit of the CSA model, Fig. 7.6. Of course, the double inhomogeneities may be assigned only certain volume fractions \( c_2^{{{(}r{)}}} < 1 \) in the surrounding comparison medium or matrix*

**L**

**L**_{0}=

**L**_{1}. Since the shape of Ω

_{2}is specified by

**P**_{2}=

**P**_{r}, the restrictions noted in Sect. 7.2.4 still apply in multi-phase systems. Aggregates reinforced by randomly orientated inhomogeneities can again be analyzed using orientation averaging in Sect. 2.2.10.

### 7.4.4 Multiphase Composites with Different Constituent Shapes and Alignments

*CB*, also suggested Hu and Weng (2000)

Each inhomogeneity **L**_{r} is now embedded in a common matrix **L**_{1}, and it may have its own shape and alignment described by a different **P**_{r}. Since these three tensors follow from the known properties of a given matrix and reinforcements, one only needs to select a single **P**_{2} for all enclosures Ω_{2}, which have the same shape and alignment. This provides relief from making a ‘suitable’ choices of \( \,{{\boldsymbol{L}}_0}{\text{\; and \;}}{{\boldsymbol{L}}_g} \), required by the general double inhomogeneity model in Sect. 7.4.1. However, all * P* and

*tensors that apply to inhomogeneities of different alignment need to be transformed into a single coordinate system that is used in the representative volume.*

**L**

**L**_{r}in the matrix material

**L**_{1}. Here, the partial strain concentration factor \( {{T}_{r}} = {\left[ {{I} + {{P}_r}\left( {{{L}_r} - {{L}6557}} \right)} \right]^{{ - 1}}} \) is that of a single-material inhomogeneity

**L**_{r}in the matrix material

**L**_{1}, derived in (4.2.14). Average strains in Ω

_{r}and Ω

_{2}follow from (7.4.2). The average strain in the double inhomogeneity is

**P**_{r}and

**P**_{2}that define shapes of the ellipsoids Ω

_{r}and Ω

_{2}.

If all inhomogeneities Ω_{r} have the same shape and alignment, then all \( {( {{\boldsymbol{L}}_2^{{{(}r{)}}}} )^{{CB}}} \) are identical, and the overall stiffness of the composite aggregate can be selected as \( {L} = {(}{L}_2^{{{(}r{)}}}{)}^{{CB}} \), possibly with \( {f_r} \to {c_r} = {(1} - {c6557}{)} \). However, many distinct \( {(}{L}_2^{{{(}r{)}}}{)}^{{CB}} \) may be admitted in modeling of matrix-based composites with misaligned reinforcements, as long as the shape and alignment of all enclosures \( \Omega_2^{{{(}r{)}}} \) are described by a single matrix **P**_{2}. Short or long fibers combined with particles in a common matrix are among the systems that can be modeled in this manner. Overall stiffness or compliance can then be evaluated using an AFA procedure, based on (6.3.3), (6.3.4) and (6.3.5).

**L**_{r}, the overall stiffness

*of the composite aggregate is obtained in (7.4.29), as an orientation average of the terms associated with the inhomogeneities \( r = 2,3, \ldots n \), indicated by the \( \left\{ {} \right\} \) brackets; Sect. 2.2.10. The shape tensor*

**L**

**P**_{2}and matrix stiffness

**L**_{1}remain unchanged.

This stiffness formula was first found by a different procedure, as an estimate of Hashin-Shtrikman type, by Ponte Castaneda and Willis (1995), with variants valid for selected two-phase microstructures.

### 7.4.5 Composites Containing Distributed Voids or Cracks

When some or all reinforcements undergo complete decohesion from the matrix, they are replaced by cavities Ω_{c} that have the original inhomogeneity shape defined by \( {{P}_r} \equiv {{P}_c} \), while \( {{L}_r} = {{L}_c} \to 0 \). In some cases the overall applied strain or stress may be associated with preferential decohesion of certain orientations and volume fractions of the originally bonded reinforcements. Then, the composite system can be modeled as a mixture of double inhomogeneities with solid and vacuous cores, each with prescribed orientation and volume fraction, possibly embedded in a common matrix.

**L**_{1}is the comparison medium \( {L}_0^{{{(} + {)}}} = {{L}6557} \), and \( {{L}_r} \to 0 \), which leads to the following substitutions

*CA*in (7.4.24), with \( {f_c} = {\Omega_c}/{\Omega_2} \to {c_c} = {(1} - {c6557}{)} \). That provides an upper Hashin-Shtrikman bound on overall stiffness, identical with that in (6.3.21)

*CB*variant yields stiffness of the double inhomogeneity with a vacuous core as

**P**_{c}, and spatial distribution described by

**P**_{2}. For example, the geometry of Fig. 7.7 can be used, with some or all inhomogeneities replaced by cavities.

For both cavities and cracks, the **P**_{c} tensors are described in Sect. 4.6, together with the **P**_{2} tensors for spheres and with references to related publications. Since the \( {P} = {\left( {{L}^{*} + {{L}_0}} \right)^{{ - 1}}} \) tensors are positive definite, a comparison of (7.4.31) with (7.4.32) and (7.4.33) indicates that the latter upper bounds are tighter than the former, which agrees with the Mori-Tanaka estimate by Benveniste (1987a).

*K*,

*G*. Matrix moduli are \( 3{K6557} = 2{G6557}{(1} + {\nu6557}{)/(1} - 2{\nu6557}{)} \), and the volume fraction of the enclosures Ω

_{2}in a representative volume Ω

_{0}is denoted by \( {c_2} = {\Omega_2}/{\Omega_0} \leq 1. \) The upper bound on the two overall moduli is

*E*

_{11}and \( p = {G_{{12}}} = {G_{{13}}} \). The remaining three moduli \( {E_{{22}}} = {E_{{33}}},\,\,{G_{{23}}} = {G_{{32}}},\,\,{\nu_{{23}}} = {\nu_{{32}}} \) are not changed by introduction of cracks aligned on parallel \( {x_2}{x_3} - {\text{planes}}{.} \) For a spherical distribution

**P**_{2}is that for flat disks aligned with the cracks, can be found in Willis (1980).

### 7.4.6 Predictive Reliability of Micromechanical Methods

As one would expect in linear elasticity, methods described in Chap. 6, and especially the rigorous evaluations in Sects. 6.4 and 6.5, should deliver reliable magnitudes of overall properties. The approximate methods in Chap. 7 are useful in certain applications, as shown in Sect. 7.5. Among those, the Mori-Tanaka estimate appears to be most reliable when it coincides with the H-S lower bound. The self-consistent method requires care when there is large contrast between phase properties. The double inclusion model exhibits greater flexibility of property choices, and the *CB* form in Sect. 7.4.4, while allowing for different phase properties, shapes and alignments, also enjoys theoretical support as a H-S type estimate.

- (i)
There is a nearly perfect bond everywhere between matrix and reinforcement or polycrystal grains, and the representative sample is free of voids or cracks that may degrade overall stiffness.

- (ii)
Elastic moduli measured on a large volume specimen of the matrix material actually prevail

*in situ*, where matrix interlayers between fibers may be just few microns thick, and may locally disappear when interrupted by fiber contact, as in the “string of pearls” formations often observed on micrographs.

The first condition (*i*) should be satisfied in well-made materials, but it is useful to recall from Sect. 3.2.3 and (3.2.12) that the interface area is rather large in materials reinforced by small diameter fibers of particles. For example, \( {s}\left({{{v}_f}} \right) = {c_f} \times 0.4\,{{\text{m}}^{{2}}}/1\,{\text{c}}{{\text{m}}^{{3}}} \) in commonly used fiber composites reinforced by 10 μ diameter fibers. At the usual \( {c_f} \doteq 0.6 \) there is about 0.24 m^{2}/1 cm^{3} of interface area, which may accommodate localized interface debonds, impurities and other possible interruptions of perfect bond. The effect may not be significant under sustained loads, but it may be magnified under cyclic loading.

The second condition (*ii*) should again be satisfied in a large volume of well-made composite material, with a high volume fraction \( {c_m} = 1 - {c_f} \) of matrix. However, the matrix polymer chains may often align around and along the fiber interface, yielding somewhat different matrix moduli *in situ*. Interface reactions producing a thin but distinct coating-like layer of a different material may also be observed, both in polymer and metal matrix systems. This may have an effect on actual overall properties, which can be accounted for by measuring effective moduli of the matrix in a material with well established fiber moduli.

Therefore, it is not unusual to find that measured moduli of composite specimens or plies are somewhat different from the predicted values, or that they may change from one batch of material to another. Measured strength magnitudes, both in tension and compression, provided by different sources, are often much larger than those found in the elastic moduli. Recourse to experimentally determined magnitudes for plies, laminate and particulate mixtures, such as can be found tabulated in Herakovich (1998) or Daniel and Ishai (2006) may provide useful guidance. However, designed properties of each material or part should be verified in each application.

In analyzing an available experimental result, on a material having a certain range of volume fractions, one may predict overall moduli as function of average volume fraction fluctuation by referring to Walpole’s equations (6.3.25), (6.3.26) and (6.3.27). Those permit computation of the moduli of the underlying comparison medium from known overall and given phase properties. The expectation is that the same comparison medium properties prevail at all volume fractions in the selected range.

## 7.5 Applications of SCM and M-T to Functionally Graded Materials

Functionally graded materials (FGM) are particulate composites, with spatially variable phase volume fractions that gradually change in at least one material direction. In a typical single gradient two-phase system, *r* = *α*, *β*, particles of phase *β* are added in a selected direction and in increasing concentrations to a continuous matrix of phase *α*, until the material is divided by a percolation threshold or by a transition zone. Beyond that zone, the matrix is a continuous phase *β* that contains dispersed α–phase particles in diminishing numbers. Such materials may remain dimensionally stable under the influence of stress or thermal gradients, for example in thermal barrier coatings. They may also facilitate joining of metal/ceramic interfaces.

Early developments in modeling of functionally graded materials relied on the elementary ‘rule of mixtures’ approximation, for example, by Fukui et al. (1994), Markworth and Saunders (1995), and in stress intensity factor evaluations in graded materials by Lee and Erdogan (1994, 1995). Giannakopoulos et al. (1995) and Finot and Suresh (1996) used this approach in elastic-plastic systems. Hirano et al. (1990) introduced a fuzzy-set estimate based on the Mori-Tanaka method, with an assumed transition function to account for the effect of changing volume fractions. The method was also used in modeling of thermoelastic behavior of FGM microstructures (Tanaka et al. 1993a, b). Reviews with additional references were written by Markworth et al. (1995) and Williamson et al. (1993).

A detailed description of the actual geometry of graded microstructures is usually not available, except perhaps for information on direction and magnitude of volume fraction distribution and approximate shape of the dispersed phase or phases. Therefore, evaluation of overall response and local stresses and strains in graded materials must rely on idealized models. Those may be based on finite element analysis of selected discrete microstructures, or on estimates of locally homogenized properties of such microstructures obtained by the self-consistent or Mori-Tanaka schemes.

The major difference between application of these schemes to statistically homogeneous or graded materials is in selection of a representative volume, which has been identified in Sect. 3.3 as sufficiently large to have the properties of any larger volume. As suggested by Drugan and Willis (1996) and corroborated by numerical simulations by Gusev (1997), the diameter of the RVE should be equal to at least twice as large as that of spherical grains reinforcing isotropic matrix-based mixtures. Such representative volumes are not easily identified in systems with variable phase volume fractions, which may also be subjected to loading by nonuniform overall fields. However, application of AFA type methods appears to be justified by the slow density changes, and by the relatively small ratios of field gradients to field averages found in most graded systems. This is confirmed by the good agreement between results obtained using the two material models described below.

The present exposition is based on the work of Reiter et al. (1997) and Reiter and Dvorak (1998), designed to determine if the available analytical models can be applied with reasonable degree of confidence to prediction of homogenized properties of graded microstructures subjected to mechanical and thermal loads. To this end, selected two-phase microstructures with single composition gradients were modeled both by distributions of discrete phase subvolumes, and by a sequence of parallel homogenized layers with effective properties estimated by either the self-consistent or Mori-Tanaka methods. Overall response and phase field averages predicted by these discrete and layered models were compared under both mechanical loading, thermal changes and steady-state heat conduction. To make good agreement more difficult to achieve, a C/SiC composite system with large differences in phase properties and steep composition gradients was used in the comparisons.

### 7.5.1 Discrete and Layered Models of Graded Microstructures

Both the discrete and layered graded material models used in the comparative studies are based on planar arrays of hexagonal inhomogeneities in continuous matrices, which are more easily implemented in a discrete model.

*x*

_{2}–direction by one half width of one hexagonal cell. A series of computer generated random distributions of the hexagons in the double array indicates percolation thresholds at \( 0.6 \leq {c_r} \leq 0.73 \), much higher than the 0.5 threshold for the random distribution of uncorrelated hexagons in the adjacent single honeycomb array.

The composition gradients observed in actual microstructures are usually much smaller, equivalent to about 0.005–0.0025/row.

The double array is subdivided into thin material layers parallel to the *x*_{2}–axis. The thickness of three such layers is equal to that of one row of the hexagons. Phase volume fractions \( {c_r}{(}{x_3}{)} \) indicate the number of phase parts in each layer. The finite element Model 3 further subdivides each layer into 320 triangular elements. Convergence with respect to coarseness of the mesh was established by comparisons of overall stiffness and field averages with those found using more refined meshes.

The layered model consists of 150 thin material layers, with effective layer properties evaluated by one of the averaging methods. The fine subdivision of the mesh and the small thickness of the homogenized layers relative to particle size cause oscillations in layer volume fractions and in estimated effective properties; these were reduced by superimposing a three-layer moving average on the computed results.

*E*,

*ν*are elastic moduli,

*α*is the linear coefficient of thermal expansion and

*κ*denotes heat conductivity. Several combinations of the SCM and M-T schemes were employed in finding effective property estimates of individual layers in the layered models. At both matrix-rich regions at the upper and lower ends on the graded microstructures in Fig. 7.9, the Mori-Tanaka method was used in two versions, once with the matrix properties equal to those of phase 1, and once with those of phase 2; these estimates are labeled as MTM1 and MTM2 in the figures below. SCS denotes homogenization by the self-consistent method, which yields gradual property changes with the SiC volume fraction

*c*

_{2}.

Phase properties of the carbon/silicon carbide system

| \( \nu \) | \( \alpha \) (10 | \( \kappa \) (Wm | |
---|---|---|---|---|

C ( | 28 | 0.3 | 9.3 | 9.5 |

SiC ( | 320 | 0.3 | 4.2 | 135 |

Models COMB3.1 and COMB3.2 employed different transition functions between the three methods, to describe effective property changes with changing *c*_{2}; these can be found in Reiter and Dvorak (1998). The COMB3.1 model corresponds to the domain subdivision indicated in the right image of Fig. 7.9, where the self-consistent estimate is employed in the in the layers that have skeletal microstructures lacking a distinct matrix. Model COMB3.2 is suitable for materials with a narrow transition zone and distinct percolation threshold.

*E*

_{33}as a function of

*c*

_{2}, which is the volume fraction of SiC. Similar predictions for the coefficient of thermal expansion

*α*

_{eff}are shown in Figs. 7.11. In all three figures, the transitions are centered at \( {c_2} = 0.5 \) and \( {c_2} = 0.65 \) for model COMB3.1, and at \( {c_2} = 0.66 \) for COMB3.2. Width of the transitions is equal to 0.05 on the

*c*

_{2}scale. The modulus \( {E_{{33}}} = {E_{{22}}} \) described in (2.3.5) was estimated using the Hill’s moduli from Sect. 7.1.3 for the self-consistent method, and from Sect. 7.2.2 for the Mori-Tanaka method. The CTE is the transverse component of the eigenstrain in (3.6.18), where either the SCM or M-T estimates of the overall moduli are used to find

*The phase eigenstrains are found as \( {{\mu }_r} = {{m}_r}\Delta \theta \), where for the isotropic case, \( {{m}_r} = {[{\alpha_r},\,\,{\alpha_r},\,\,{\alpha_r},\,\,0,\,\,0,\,\,0]^{\text{T}}} \), and \( {\alpha_r} \) is the linear coefficient of thermal expansion of phase*

**M.***r.*Heat conduction in the transverse direction of a fiber composite is governed by the same equations as the longitudinal shearing deformation. This axial shearing-transverse conduction analogy (Hashin 1968, 1972) allows writing down an expression for \( {\kappa_T} \) by exchanging \( {\kappa_r} \) for \( {p_r} \)\( {(}r = 1,\,\,2) \) in the self-consistent form of \( p \) in (7.1.8), and in the Mori-Tanaka form (7.2.13), where \( {(}r = m,\,\,f). \) Results for particulate composites were derived by Hatta and Taya (1986), and for coated orthotropic fibers by Benveniste et al. (1990, 1991a). Models with homogenized layers were also analyzed by Ozisik (1968). Figure 7.12 illustrates the types of boundary conditions applied to both discrete and layered models to simulate heat conduction in the

*x*

_{3}–direction, or a constant temperature change. The solution domain was bounded in the thickness direction by two parallel planes that allowed a uniform normal strain in the thickness direction; resultants of the external forces and moments on the bounding planes were equal to zero. In the finite element Model 3, the thermal and mechanical fields were obtained from a two-dimensional solution, using ABAQUS generalized plane strain elements.

### 7.5.2 Selected Comparisons of Discrete and Homogenized Models

These comparisons indicate the complex nature of strain and stress distributions in individual phases of the functionally graded materials. In single-gradient systems considered herein, the combined self-consistent and Mori-Tanaka estimates applied to layered models of the graded materials, provide fairly accurate predictions of both overall and phase strain and stress averages in systems subjected to mechanical and/or thermal changes and gradients.

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