Estimates of Mechanical Properties of Composite Materials

  • George J. Dvorak
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 186)


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 L0. 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 L0. 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

In the original version of this method, interaction between individual subvolumes of the phases is approximated by embedding each subvolume Ωr of phase Lr as a solitary ellipsoidal inhomogeneity in a large volume Ω of L0L, 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 Lr follow from the expressions for their partial counterparts in (4.2.14) as
$$ \left. \begin{array}{llll} & {{A}_r} = {[{I} + {P}{(}{{L}_r} - {L}{)]}^{{ - 1}}} = {{(}{L}^{*} + {{L}_r}{)}^{{ - 1}}}{(}{L}^{*} + {L}{)} \\ & {{B}_r} = {[{I} + {Q}{(}{{M}_r} - {M}{)]}^{{ - 1}}} = {{(}{M}^{*} + {{M}_r}{)}^{{ - 1}}}{(}{M}^{*} + {M}{)} \end{array} \right\} $$
Fig. 7.1

The self-consistent model (SCM). Stiffness of the inhomogeneity and of the comparison medium are denoted by \( {{L}_r} \) and by \( {{L}_0} = {L} \)

Substitution of the concentration factors (7.1.1) into the first right-hand terms of (6.3.5) yields the overall stiffness
$$ {L} = \sum\limits_{{r = 1}}^n {{c_r}{{L}_r}{{A}_r}} = \left[ {\sum\limits_{{r = 1}}^n {{c_r}{{L}_r}{{\left( {{L}^{*} + {{L}_r}} \right)}^{{ - 1}}}} } \right]\left( {{L}^{*} + {L}} \right) $$
Diagonal symmetry of L is established by expanding the first term
$$ \left. \begin{array}{llll} & \sum\limits_{{r = 1}}^n {{c_r}{{L}_r}} {{(}{L}^{*} + {{L}_r}{)}^{{ - 1}}} + \sum\limits_{{r = 1}}^n {{c_r}{L}^{*}} {{(}{L}^{*} + {{L}_r}{)}^{{ - 1}}} - \sum\limits_{{r = 1}}^n {{c_r}{L}^{*}} {{(}{L}^{*} + {{L}_r}{)}^{{ - 1}}} \\ & \quad= \sum\limits_{{r = 1}}^n {{c_r}} {(}{L}^{*} + {{L}_r}{)(}{L}^{*} + {{L}_r}{{)}^{{ - 1}}} - \sum\limits_{{r = 1}}^n {{c_r}{L}^{*}} {{(}{L}^{*} + {{L}_r}{)}^{{ - 1}}} \\ &\quad= {\mathbf{I}} - {L}^{*}\sum\limits_{{r = 1}}^n {{c_r}} {{(}{L}^{*} + {{L}_r}{)}^{{ - 1}}} \end{array} \right\} $$
and by
$$ \left. \begin{array}{llll} {L} & = \sum\limits_{{r = 1}}^n {{c_r}{{L}_r}{{A}_r}} = \left[ {{I} - {L}^{*}\sum\limits_{{r = 1}}^n {{c_r}} ({L}^{*} + {{L}_r}{{)}^{{ - 1}}}} \right]\left( {{L}^{*} + {L}} \right) \\ {L} & = {L}^{*} + {L} - {L}^{*}\sum\limits_{{r = 1}}^n {{c_r}{{\left( {{L}^{*} + {{L}_r}} \right)}^{{ - 1}}}\left( {{L}^{*} + {L}} \right)} \\ \boldsymbol{0} & = {L}^{*}\left[ {{I} - \left( {\sum\limits_{{r = 1}}^n {{c_r}} {{{(}{L}^{*} + {{L}_r}{)}}^{{ - 1}}}{(}{L}^{*} + {L}{)}} \right)} \right] \\ {L} & = {\left[ {\sum\limits_{{r = 1}}^n {{c_r}} {{{(}{L}^{*} + {{L}_r}{)}}^{{ - 1}}}} \right]^{{ - 1}}} - {L}^{*} = {{L}^{\text{T}}}\quad \quad {L}^{*} \ne {\boldsymbol{0}} \end{array} \right\} $$

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 L. A similar proof can be derived for the overall compliance. The inclusion problem solution (4.2.14) and (6.3.5), both with identical P matrix, provide the same result.

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 P or Q and Ar or Br, 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 L or M.

An iterative solution may start using initial values of the coefficients of L0 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 Q matrix in (7.1.1).

In a two-phase system, \( r = 1,\,\,2 \), where \( {c6557}{{A}6557} = {I} - {c_2}{{A}_2} \), (7.1.2) is reduced to
$$ {L} = {{L}6557} + {c_2}\left( {{{L}_2} - {{L}6557}} \right){{A}_2}\quad \quad {M} = {{M}6557} + {c_2}\left( {{{M}_2} - {{M}6557}} \right){{B}_2} $$

The L1 represents the stiffness of a matrix, and A2 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

For a fiber composite consisting of two phases 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
$$ \frac{1}{{k + m}} = \frac{{{c6557}}}{{{k6557} + m}} + \frac{{{c_2}}}{{{k_2} + m}} $$
where m is the positive root of the cubic
$$ \frac{{{c6557}{k6557}}}{{{k6557} + m}} + \frac{{{c_2}{k_2}}}{{{k_2} + m}} = 2\left( {\frac{{{c6557}{m_2}}}{{{m_2} - m}} + \frac{{{c_2}{m6557}}}{{{m6557} - m}}} \right) $$
The longitudinal shear modulus p is the positive root of the quadratic
$$ \frac{{{c6557}p}}{{p - {p_2}}} + \frac{{{c_2}p}}{{p - {p6557}}} = \frac{1}{2}$$
If the matrix r = 1 or both phases are isotropic, with moduli K1 and G1, 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
$$ \frac{{k - {k6557}}}{{l - {l6557}}} = \frac{{k - {k_2}}}{{l - {l_2}}} = \frac{{l - {c6557}{l6557} - {c_2}{l_2}}}{{n - {c6557}{n6557} - {c_2}{n_2}}} = \frac{{{k6557} - {k_2}}}{{{l6557} - {l_2}}} $$
which yield the two remaining overall moduli 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.
The overall moduli can be also evaluated by solving the Walpole’s equations (6.3.27) for \( {k_0} \to k,{m_0} \to m\ {\text{ and }}\ {p_0} \to p \).
$$ \left. \begin{array}{llll} k & = {c6557}{k6557} + {c_2}{k_2} - {c6557}{c_2}{{(}{k6557} - {k_2}{)}^{{2}}}{{(}{c6557}{k_2} + {c_2}{k6557} + {m_0}{)}^{{ - 1}}} \\ m & = {c6557}{m6557} + {c_2}{m_2} - {c6557}{c_2}{{(}{m6557} - {m_2}{)}^{{2}}}{{[}{c6557}{m_2} + {c_2}{m6557} + {m_0}{k_{{0}}}{/(}{k_0}{ + 2}{m_0}{)]}^{{ - 1}}} \\ p & = {c6557}{p6557} + {c_2}{p_2} - {c6557}{c_2}{{(}{p6557} - {p_2}{)}^{{2}}}{{(}{c6557}{p_2} + {c_2}{p6557} + {p_0}{)}^{{ - 1}}} \end{array} \right\}\\ $$

In contrast to the Hashin-Shtrikman bounds on overall L or M, derived with comparison media of constant stiffness \( {L}_0^{{{(} + {)}}}\,\,{\text{or}}\ {L}_0^{{{(} - {)}}} \), the self-consistent method relies on variable L0 = L, 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 \( {c_f} \to 1 \).

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

Polycrystals, or matrix-based composites reinforced by a random or any other dispersion of spheres that provides for overall statistical isotropy are considered. The effective, or overall bulk and shear moduli G and K of such composites were found by Hill (1965c) as
$$ \frac{{{c6557}{K6557}}}{{{{{{K6557} + 4G}} \left/ {3} \right.}}} + \frac{{{c_2}{K_2}}}{{{{{{K_2} + 4G}} \left/ {3} \right.}}} + 5\left( {\frac{{{c6557}{G_2}}}{{G - {G_2}}} + \frac{{{c_2}{G6557}}}{{G - {G6557}}}} \right) + 2 = 0 $$
$$ \frac{1}{{K + 4G/3}} = \frac{{{c6557}}}{{{K6557} + 4G/3}} + \frac{{{c_2}}}{{{K_2} + 4G/3}} $$

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 = LT when applied to systems where each phase subvolume has the same shape and alignment described by a single P 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 L2 in a matrix-based two-phase system, \( r = 1,\,\,2. \)

Concentration factors (7.1.1) for a two-phase system are
$$ {A}_2^s = {\left[ {{I} + {{P}_s}{(}{{L}_2} - {L}{)}} \right]^{{ - 1}}}\quad {B}_2^s = {\left[ {{I} + {{Q}_s}{(}{{M}_2} - {M}{)}} \right]^{{ - 1}}} $$
$$ {{{P}}_s} = {\left( {{{L}}_s^{*} + {{L}}} \right)^{{ - 1}}}\quad {{{Q}}_s} = {{L}}\left( {{{I}} - {{{P}}_s}{{L}}} \right) = {\left( {{{M}}_s^{*} + {{M}}} \right)^{{ - 1}}} $$

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 M, and not directly on Lr or Mr of either phase.

Overall stiffness of such two-phase systems follows from (7.1.2) as
$$ {L} = {{{L}}6557} + \sum\limits_{{s = 2}}^n {c6557^s\left( {{{L}_2} - {{L}6557}} \right){A}_2^s} = {{L}6557} + \left( {{{L}_2} - {{L}6557}} \right)\sum\limits_{{s = 2}}^n {c_2^s{A}_2^s} $$
where the phase volume fractions \( {c6557} + \Sigma c_2^s = 1 \) and
$$ {A}_2^s = {\left[ {{I} + {{P}_s}\left( {{{L}_2} - {L}} \right)} \right]^{{ - 1}}} = {\left( {{L}_s^{*} + {{L}_2}} \right)^{{ - 1}}}\left( {{L}_s^{*} + {L}} \right) $$
As an example of the self-consistent moduli prediction for a two-phase composite with different reinforcement shapes, Benveniste et al. (1991b), selected a two-phase composite with a Ti3Al 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 x3–axis of a Cartesian system. The phases are isotropic, and have the following elastic moduli
$$ \left. \begin{array}{llll} {\text{T}}{{\text{i}}_{{3}}}{\text{Al:}}\quad {{E}_{1}} = 96.5\;{\text{GPa}},\quad {{G}6557} &= 37.1\;{\text{GPa}} \\ {\text{SiC:}}\quad {{E}_2} = {431}{\text{.0 GPa,}}\quad {{G}_2} &= 172.0\;{\text{GPa}} \end{array} \right\} $$
The self-consistent estimate (7.1.16) of the overall stiffness of this systems, was obtained as the ninth iteration of the solution of (7.1.13) and (7.1.14). It is expected that diagonal symmetry of the overall stiffness may also be found in other reinforcement shape combinations in two-phase systems. However replacement of the above elastic moduli of the circular discs by a third set of different moduli renders a stiffness estimate which is not diagonally symmetric. That can be verified as an exercise.
$$ {{(}{L}{)}_{{9}}} = \left[ {\begin{array}{*{20}{c}} {269.95} & {95.45} & {82.80} & 0 & 0 & 0 \\ {} & {269.95} & {82.80} & 0 & 0 & 0 \\ {} & {} & {249.24} & 0 & 0 & 0 \\ {} & {} & {} & {64.44} & 0 & 0 \\ {} & {\mathit{sym.}} & {} & {} & {64.44} & {} \\ {} & {} & {} & {} & {} & {87.25} \\ \end{array} } \right]{\text{GPa}} $$

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

This AFA method approximates interaction between phases in a matrix-based system by regarding each reinforcement Lr as a solitary inhomogeneity Ωr embedded in a large volume Ω1 of the matrix L1. 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.
Fig. 7.2

The Mori-Tanaka (M-T) model. Stiffnesses of the inhomogeneity and matrix are denoted by \( {{L}_r} \) and by \( {{L}6557} = {{L}_0} \)

The inclusion-based form of this method utilizes (4.2.14), for the average strain \( {{\varepsilon }_r} \) in a single inhomogeneity Lr
$$ {{\varepsilon }_r} = {{T}_{r}}{{\varepsilon }6557}\quad {{T}_{r}} = {\left[ {{I} + {P}\left( {{{L}_r} - {{L}6557}} \right)} \right]^{{ - 1}}} $$
where both the Eshelby tensor \( {S} = {P}{{L}6557} \) and the \( {P} = {\left( {{L}^{*} + {{L}6557}} \right)^{{ - 1}}} \) and L* tensors in (4.2.9) are evaluated in L1.
When a 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
$$ {{\varepsilon }6557} = {{A}6557}{{\varepsilon}^0} = {\left[ {{c6557}{I} + \sum\limits_{{s = 2}}^n {{c_s}{{T}_s}} } \right]^{{ - 1}}}{{\varepsilon }^0}\quad {{\varepsilon }_r} = {{A}_r}{{\varepsilon }^0} = {{T}_{r}}{\left[ {{c6557}{I} + \sum\limits_{{s = 2}}^n {{c_s}{{T}_s}} } \right]^{{ - 1}}}{{\varepsilon }^0} $$
Overall stiffness of a multiphase system, \( r = 1,2, \ldots, n \), is then found as
$$ \left. \begin{array}{llll}{L} & = \sum\limits_{{r = 1}}^n {{c_r}{{L}_r}{{A}_r}} = \left[ {\sum\limits_{{r = 1}}^n {{c_r}{{L}_r}{{T}_{r}}} } \right]{\left[ {\sum\limits_{{s = 1}}^n {{c_s}{{T}_s}} } \right]^{{ - 1}}} \\& = {{L}6557} + \left[ {\sum\limits_{{r = 2}}^n {{c_r}\left( {{{L}_r} - {{L}6557}} \right){{T}_{r}}} } \right]{\left[ {\sum\limits_{{s = 1}}^n {{c_s}{{{T}}_s}} } \right]^{{ - 1}}}\end{array} \right\} $$

After some algebra, this form can be converted to (6.3.5)2, with both L* and P evaluated in the matrix, L0 = L1 as shown in (7.2.29).

For a two-phase composite\( \ r = 1,\,\,2 \), with matrix L1 of volume fraction c1, and reinforcements L2, the corresponding expressions are
$$ \left. \begin{array}{llll} & {{A}6557} = {\left[ {{c6557}{I} + {(1} - {c6557}{)}{{\left[ {{I} + {P}{(}{{L}_2} - {{L}6557}{)}} \right]}^{{ - 1}}}} \right]^{{ - 1}}}\\ & {{A}_2} = {\left[ {\left( {{1} - {c6557}} \right){I} + {c6557}{P}\left( {{{L}_2} - {{L}6557}} \right)} \right]^{{ - 1}}} \end{array} \right\} $$
$$ {L} = {{L}6557} + \left( {{1} - {c6557}} \right)\left( {{{L}_2} - {{L}6557}} \right){\left[ {{I} + {c6557}{P}\left( {{{L}_2} - {{L}6557}} \right)} \right]^{{ - 1}}} $$

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 L2.

When the composite is subjected to a 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 \( {{\sigma }6557} \)
$$ {{\sigma }_r} = {{W}_r}{{\sigma }6557}\quad {{W}_r} = {\left[ {{I} + {Q}\left( {{{M}_r} - {{M}6557}} \right)} \right]^{{ - 1}}} $$
where \( {Q} = {{L}6557}\left( {{I} - {P}{{L}6557}} \right) = {\left( {{M}^{*} + {{M}6557}} \right)^{{ - 1}}} \) and again, W1 = I. Since \( \left( {\sum {{c_r}{{W}_r}} } \right){{\sigma }6557} = {{\sigma }^0} \), the local stress average are
$$ {{\sigma }_r} = {{B}_r}{{\sigma }^0} = {{W}_r}{\left[ {{c6557}{I} + \sum\limits_{{s = 2}}^n {{c_s}{{W}_s}} } \right]^{{ - 1}}}{{\sigma }^0}\quad \sum\limits_{{s = 1}}^n {{c_s}} {{B}_s} = {I} $$
The overall compliance of the composite aggregate follows from (6.3.6) as
$$ \left. \begin{array}{llll} {M} & = \sum\limits_{{r = 1}}^n {{c_r}{{M}_r}{{B}_r} = } \left[ {\sum\limits_{{r = 1}}^n {{c_r}{{M}_r}{{W}_r}} } \right]{\left[ {\sum\limits_{{s = 1}}^n {{c_s}{{W}_s}} } \right]^{{ - 1}}} \\ & = {{M}6557} + \left[ {\sum\limits_{{r = 2}}^n {{c_r}\left( {{{M}_r} - {{M}6557}} \right){{W}_r}} } \right]{\left[ {\sum\limits_{{s = 1}}^n {{c_s}{{W}_s}} } \right]^{{ - 1}}} \end{array} \right\}$$
For two-phase systems with matrix M1 of volume fraction c1
$$ {{B}6557} {=} {\left[ {{c6557}{I} + \left( {{1} - {c6557}} \right){{\left[ {{I} + {Q}\left( {{{M}_2} - {{M}6557}} \right)} \right]}^{{ - 1}}}} \right]^{{ - 1}}}\quad {{B}_2} {=\,} {\left[ {{I} + {c6557}{Q}\left( {{{M}_2} - {{M}6557}} \right)} \right]^{{ - 1}}} $$
$$ {M} = {{M}6557} + \left( {{1} - {c6557}} \right)\left( {{{M}_2} - {{M}6557}} \right){\left[ {{I} + {c6557}{Q}\left( {{{M}_2} - {{M}6557}} \right)} \right]^{{ - 1}}} $$

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

For a 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
$$ k = \frac{{{c_f}{k_f}{(}{k_m} + {m_m}{)} + {c_m}{k_m}{(}{k_f} + {m_m}{)}}}{{{c_f}{(}{k_m} + {m_m}{)} + {c_m}{(}{k_f} + {m_m}{)}}} $$
$$ m = \frac{{{m_f}{m_m}{(}{k_m} + 2{m_m}{)} + {k_m}{m_m}{(}{c_f}{m_f} + {c_m}{m_m}{)}}}{{{k_m}{m_m} + {(}{k_m} + 2{m_m}{)(}{c_f}{m_m} + {c_m}{m_f}{)}}} $$
$$ p = \frac{{2{c_f}{p_f}{p_m} + {c_m}{(}{p_f}{p_m} + p_m^2{)}}}{{2{c_f}{p_m} + {c_m}{(}{p_f} + p_m{)}}} $$
The remaining two moduli follow from the universal connections (3.9.4) as
$$ \left. \begin{array}{llll}{} & \qquad l = \frac{{{c_f}{l_f}{(}{k_m} + {m_m}{)} + {c_m}{l_m}{(}{k_f} + {m_m}{)}}}{{{c_f}{(}{k_m} + {m_m}{)} + {c_m}{(}{k_f} + {m_m}{)}}} \\ & n - {c_f}{n_f} - {c_m}{n_m} = {(}l - {c_f}{l_f} - {c_m}{l_m}{)}\frac{{{(}{l_f} - {l_m}{)}}}{{{(}{k_f} - {k_m}{)}}} \end{array} \right\} $$

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 ET, 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 Lm = L1 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
$$ \left. \begin{array}{llll} &\qquad \frac{{K - {K6557}}}{{{K_2} - {K6557}}} = \frac{{{c_2}}}{{1 + {c6557}({K_2} - {K6557}){{{(}{K6557} + 4{G6557}/3{)}}^{{ - 1}}}}} \\ & \frac{{G - {G6557}}}{{{G_2} - {G6557}}} = \frac{{{c_2}}}{{1 + {c6557}({G_2} - {G6557}){{\left( {{G6557} + \dfrac{{{G6557}{(9}{K6557} + 8{G6557}{)}}}{{6({K6557} + 2{G6557})}}} \right)}^{{ - 1}}}}} \end{array} \right\} $$

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 GT 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.

Next, we consider a 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 x1–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 x1–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.
$$ \left. \begin{array}{llll} m &= \sum\limits_{{r = 1}}^n {{c_r}{m_r}} \quad {n^{{ - 1}}} = \sum\limits_{{r = 1}}^n {{c_r}{{{(}{n_r}{)}}^{{ - 1}}}} \quad {p^{{ - 1}}} = \sum\limits_{{r = 1}}^n {{c_r}{{{(}{p_r}{)}}^{{ - 1}}}} \\ k &= {l^2}/n + \sum\limits_{{r = 1}}^n {{c_r}{(}{k_r}} - l_r^2/{n_r}{)}\quad l/n = \sum\limits_{{r = 1}}^n {{c_r}{l_r}/{n_r}} \end{array} \right\} $$

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

Here we find an estimate of overall stiffness of a multi-phase and matrix-based composite, reinforced by randomly oriented but otherwise identical ellipsoidal inhomogeneities, such that the composite is isotropic on the macroscale. As already mentioned, this result can be found in terms of averages over all orientations in Sect. 2.2.10, of the orientation dependent terms in the overall stiffness (7.2.3). Those are, of course, the strain concentration factors (7.2.1) that depend on both P and Lr. Using curly brackets for the averages, the desired overall stiffness is
$$ {L} = {{L}6557} + \left[ {\sum\limits_{{s = 2}}^n {{c_s}\left\{ {\left( {{{L}_s} - {{L}6557}} \right){{T}_s}} \right\}} } \right]{\left[ {\sum\limits_{{s = 1}}^n {{c_s}\left\{ {{{T}_s}} \right\}} } \right]^{{ - 1}}} $$

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 \( {T_{{\alpha \beta }}} \).

Chen et al. (1992) found the orientation averages of the moduli (7.2.17), for matrix based composites consisting of an isotropic matrix L1, reinforced by randomly oriented short fibers, distribution of which is isotropic on the macroscale. General forms of the averaged moduli are
$$ \left. \begin{array}{llll} K = {K6557} + \frac{1}{3}\sum\limits_{{r = 2}}^n {{c_r}\left( {{\delta_r} - 3{K6557}{\alpha_r}} \right)} {\left[ {{c6557} + \sum\limits_{{r = 2}}^n {{c_r}{\alpha_r}} } \right]^{{ - 1}}} \\ G = {G6557} + \frac{1}{2}\sum\limits_{{r = 2}}^n {{c_r}\left( {{\eta_r} - 2{G6557}{\beta_r}} \right){{\left[ {{c6557} + \sum\limits_{{r = 2}}^n {{c_r}{\beta_r}} } \right]}^{{ - 1}}}} \end{array} \right\} $$
where the parameters \( {\alpha_r},{\beta_r},{\delta_r},{\eta_r} \) depend on the moduli and geometry of the phases. \( {K6557},{G6557} \) are the bulk and shear moduli of the matrix, and each fiber orientation has a different stiffness Lr. Berryman (1980) derived expressions for their evaluation for cracks and other shapes.
When the fibers are modeled by randomly distributed very long prolate spheroids, made of an isotropic material moduli with K2 and G2, the effective overall bulk and shear moduli K and G are
$$ \left. \begin{array}{llll}{} &\qquad\qquad\quad K = {K_2} - {c6557}{(}{K_2} - {K6557}{)}{\left[ {{1} - {c_2}\frac{{{3(}{K_2} - {K6557}{)}}}{{{3(}{G6557} + {K_2}{)} + {G_2}}}} \right]^{{ - 1}}} \\ G & = {G_2} - {c6557}{(}{G_2} - {G6557}{)} \\ &\quad\times {\left[ {{1} - \frac{{{(}{G_2} - {G6557}{)}}}{{{5[3(}{G6557} + {K_2}{)} + {G_2}]}} - \frac{2}{5}{c_2}{(}{G_2} - {G6557}{)}\left( {\frac{{1}}{{{(}{G_2} + {\gamma6557}{)}}} + \frac{1}{{{(}{G_2} + {G6557}{)}}}} \right)} \right]^{{ - 1}}} \end{array} \right\}$$

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

Of interest in many applications are composites reinforced by relatively short, randomly distributed fibers which are transversely isotropic, such as carbon, with moduli \( {k_r},\,\,{l_r},\,\,{m_r},\,\,{n_r},\,\,{\text{and}}\,\,{p_r} \) in the local coordinates aligned with fiber axis. The fiber diameter is usually <20 μm, hence the fiber aspect ratio is very high. For such short fiber composite, the coefficients in (7.2.18) are
$$ \left. \begin{array}{llll} {\alpha_r} & = \frac{{3{(}{K6557} + {G6557}{)} + {k_r} - {l_r}}}{{3{(}{G6557} + {k_r}{)}}}\quad {\beta_r} = \frac{1}{5}{\left[ {\frac{{4{G6557} + 2{k_r} + {l_r}}}{{3{(}{G6557} + {k_r}{)}}} + \frac{{4{G6557}}}{{{G6557} + {p_r}}} + \frac{{2{(}{G6557} + {\gamma6557}{)}}}{{{m_r} + {\gamma6557}}}} \right]_{{}}} \\ {\delta_r} & = \frac{1}{3}{\left[ {2{l_r} + {n_r} + \frac{{{(2}{k_r} + {l_r}{)(3}{K6557} + 2{G6557} - {l_r}{)}}}{{{G6557} + {k_r}}}} \right]_{{_{{_{{}}}}}}}^{{}} \\ {\eta_r} & = \frac{1}{5}\hfill \\ & \times\left[ {\frac{2}{3}{(}{n_r} - {l_r}{)} {+} \frac{{8{G6557}{m_r}{(3}{K6557} + 4{G6557}{)}}}{{{(3}{K6557} + 7{G6557}{)}{m_r} + {(3}{K6557} + {G6557}{)}{G6557}}} + \frac{{8{G6557}{p_r}}}{{{G6557} + {p_r}}} + \frac{{{(}4{G6557}{ + 2}{l_r}{)(}{k_r} - {l_r}{)}}}{{3({G6557} + {k_r}{)}}}} \right] \end{array} \right\} $$
In addition to the above results, Chen et al. (1992) provide similar parameters for composites reinforced by randomly oriented, transversely isotropic platelets or disks of vanishing thickness. When the disks are isotropic, the overall bulk and shear moduli are
$$ \left. \begin{array}{llll}{} &\qquad\quad K = {K_2} - {c6557}{(}{K_2} - {K6557}{)}{\left[ {{1} - {c_2}\frac{{{3(}{K_2} - {K6557}{)}}}{{{3}{K_2} + 4{G_2}}}} \right]^{{ - 1}}} \\ G & = {G_2} - {c6557}{(}{G_2} - {G6557}{)}{\left[ {{1} - \frac{2}{5}{c_2}\left( {\frac{{2({G_2} - {G6557})}}{{{3}{K_2} + 4{G_2}}} + \frac{{{G_2} - {G6557}}}{{{G_2}}}} \right)} \right]^{{ - 1}}} \end{array} \right\} $$

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).

Using Berryman’s (1980) results, Benveniste (1987a) found the moduli for a solid with randomly oriented penny-shaped cracks of radius a as
$$ \frac{K}{{{K6557}}} = {\left[ {1 + \frac{{16\eta {(1} - \nu6557^2{)}}}{{{9(1} - 2\nu6557{)}}}} \right]^{{ - 1}}}\quad \frac{G}{{{G6557}}} = {\left[ {1 + \frac{{32\eta {(1} - {\nu6557}{)(5} - {\nu6557}{)}}}{{{45(2} - {\nu6557}{)}}}} \right]^{{ - 1}}} $$
where \( \eta = N{a^3}/V \), and 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 = LT and compliance M = MT follow from (6.3.5) and (6.3.6), providing that all inhomogeneities have the same shape and alignment, described by a single P or Q 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 L1 and many perfectly bonded ellipsoidal inhomogeneities \( s = 2,3, \ldots, n \) of phase L2, which may have different shape and/or alignment, described by Ps or \( {L}_s^{*} \). The proof was derived by Benveniste et al. (1991b).

According to (7.2.1), the local strain averages are \( {{\varepsilon }_s} = {{T}_s}{{\varepsilon}6557} = [{{I} + {{P}_s}\left( {{{L}_2} - {{L}6557}} \right)}]^{{ - 1}}{{\varepsilon }6557} \), where \( {{\varepsilon }6557} \) is the average strain in the matrix 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 Ts is now derived, by defining a new tensor \( {P}6557^s \).
$$ \left. \begin{array}{llll} & {{P}_s}\left( {{{L}_2} - {{L}6557}} \right){P}6557^s = {{P}_s} - {P}6557^s \to \left( {{{L}_2} - {{L}6557}} \right) = {\left( {{P}6557^s} \right)^{{ - 1}}} - {\left( {{{P}_s}} \right)^{{ - 1}}} \\ & {{P}_s}\left( {{{L}_2} - {{L}6557}} \right) = {{P}_s}{\left( {{P}6557^s} \right)^{{ - 1}}} - {I}\quad {P}6557^s\left( {{{L}_2} - {{L}6557}} \right) = {I} - {P}6557^s{\left( {{{P}_s}} \right)^{{ - 1}}} \end{array} \right\} $$
Since \( {{P}_s} {=} {\left( {{L}_s^{*} + {{L}6557}} \right)^{{ - 1}}} {=} {{P}_s}^{\text{T}} \), this shows that \( {P}6557^s {=} {[ {( {{{L}_2} - {{L}6557}} ) + {{( {{{P}_s}} )}^{{ - 1}}}} ]^{{ - 1}}}\break = {( {{P}6557^s} )^{\text{T}}} \)and also that \( {P}6557^s = {\left( {{L}_s^{*} + {{L}_2}} \right)^{{ - 1}}} \). The partial strain concentration factors can be written in a modified form
$$ {\left( {{{T}_s}} \right)^{{ - 1}}} {=} {I} + {{P}_s}\left( {{{L}_2} - {{L}6557}} \right) {=} {{P}_s}{\left( {{P}6557^s} \right)^{{ - 1}}} \to {{T}_s} {=} {P}6557^s{\left( {{{P}_s}} \right)^{{ - 1}}} {=} {I} - {P}6557^s\left( {{{L}_2} - {{L}6557}} \right) $$
The overall stiffness of a matrix-based two-phase system \( {{L}6557},\,\,{{L}_2} \), with subvolumes \( s = 1{,}2{,}3{,} \ldots {,}n \) that have different shapes and alignment for \( s \geqslant 2 \), follows from (7.2.3) as
$$ \left. \begin{array}{llll} {L} & = {{L}6557} + \sum\limits_{{s = 2}}^n {{c_s}\left( {{{L}_2} - {{L}6557}} \right){{A}_s}} \\ & = {{L}6557} - \left( {{{L}_2} - {{L}6557}} \right)\left[ {{c6557}{I} - \sum\limits_{{s = 1}}^n {{c_s}{{T}_s}} } \right]{\left[ {\sum\limits_{{s = 1}}^n {{c_s}{{T}_s}} } \right]^{{ - 1}}} \\ \end{array} \right\} $$
Substitution of the modified Ts yields the overall stiffness in the following diagonally symmetric form, which is then adjusted using (7.2.23), to depend only on the Ps matrices that describe shape and alignment of different inhomogeneities L2.
$$ \left. \begin{array}{llll} {L} & = {{L}_2} - \left( {{{L}_2} - {{L}6557}} \right){c6557}{\left[ {\sum\limits_{{s = 1}}^n {{c_s}} \left[ {{I} - {P}6557^s\left( {{{L}_2} - {{L}6557}} \right)} \right]} \right]^{{ - 1}}} \\ &= {{L}_2} - {c6557}{\left[ {\sum\limits_{{s = 1}}^n {{c_s}} \left[ {{{\left( {{{L}_2} - {{L}6557}} \right)}^{{ - 1}}} - {P}6557^s} \right]} \right]^{{ - 1}}} = {{L}^{\text{T}}} \\ & = {{L}_2} - {c6557}{\left[ {\sum\limits_{{s = 1}}^n {{c_s}\left\{ {{{\left( {{{L}_2} - {{L}6557}} \right)}^{{ - 1}}} - {{\left[ {\left( {{{L}_2} - {{L}6557}} \right) + {{\left( {{{P}_s}} \right)}^{{ - 1}}}} \right]}^{{ - 1}}}} \right\}} } \right]^{{ - 1}}} \\ \end{array}\quad \right\} $$

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.

In closing this section, we convert (7.2.3) to a diagonally symmetric form, valid for multiphase composite systems with the same shape and alignment of the reinforcement, given by one pair of L* and \( {P} = \left( {{L}^{*} + {{L}6557}} \right)^{-1} \) tensors, both evaluated in the matrix L1. 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
$$ \left. \begin{array}{llll} {L} & = \sum\limits_{{r = 1}}^n {{c_r}{{L}_r}{{A}_r}} = \left[ {\sum\limits_{{r = 1}}^n {{c_r}{{L}_r}{{T}_{r}}} } \right]{\left[ {\sum\limits_{{r = 1}}^n {{c_r}{{T}_{r}}} } \right]^{{ - 1}}} \\ & = \left[ {\sum\limits_{{r = 1}}^n {{c_r}{{L}_r}} {{\left( {{L}^{*} + {{L}_r}} \right)}^{{ - 1}}}} \right]{\left[ {\sum\limits_{{r = 1}}^n {{c_r}} {{\left( {{L}^{*} + {{L}_r}} \right)}^{{ - 1}}}} \right]^{{ - 1}}}\end{array} \right\} $$
and use (7.1.3) to find
$$ \sum\limits_{{r = 1}}^n {{c_r}{{L}_r}} {\left( {{L}^{*} + {{L}_r}} \right)^{{ - 1}}} = {I} - {L}^{*}\sum\limits_{{r = 1}}^n {{c_r}} {\left( {{L}^{*} + {{L}_r}} \right)^{{ - 1}}} $$
Finally, take this into (7.2.27) to get the last equality in (6.3.5)
$$ {L} = {\left[ {\sum\limits_{{r = 1}}^n {{c_r}} {{\left( {{L}^{*} + {{L}_r}} \right)}^{{ - 1}}}} \right]^{{ - 1}}} - {L}^{*} = {{L}^{\text{T}}} $$

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, L0 = L1. 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 L0 = L or L0 = L1 respectively. Both approaches require evaluation of the P 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.

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 E11 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 L in (2.3.2) and (2.3.3). Together with the known moduli of the composite matrix and the matrix volume fraction c1, the overall elastic moduli k, m, and p included in the coefficients of M and L are substituted into expressions for self-consistent or Mori-Tanaka estimates of these moduli.

In particular, self-consistent estimates of the fiber moduli k2, m2, and p2 can be found by solving in sequence equations (7.1.6), (7.1.7), (7.1.8). The remaining moduli n2, l2 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 kf, mf, and pf. 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 L0 is enriched by inserting a dilute concentration of inhomogeneities of one or more distinct phases Lr, r = 1, 2,…, n, and the mixture is homogenized. This incremental homogenization continues until it reaches final phase concentrations. The choice of L0 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).

Two procedures can be employed to reach an estimate of overall stiffness (Norris et al. 1985). In the fixed volume process or FVP, the initial ‘backbone’ material of stiffness L0 resides in a fixed representative volume V0. 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 V0 an equivalent volume \( \Delta {\text{v}} \) of the already homogenized material which always includes certain volume ratio of L0. After a current increment is homogenized, the phase volume fractions in the FVP procedure satisfy
$$ \sum\limits_{{r = 0}}^n {{{\text{v}}_r}} {(}t{)} = {V_0}\quad \sum\limits_{{r = 0}}^n {\Delta {{\text{v}}_r}} {(}t{)} = 0\quad {c_r}{(}t{)} = {{\text{v}}_r}{(}t{)}/{V_0}\quad \sum\limits_{{r = 0}}^n {{c_r}{(}t{)} = 1} $$

This volume exchange process continues until all phase volume fractions reach their prescribed magnitudes in V0. 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 L0, and on the phase volume ratios added in each step.

Physically more plausible is the alternative variable volume process (VVP), which builds the same two or multiphase material \( r \,{=}\, 1,2, \ldots n \) by starting with volume v1 of an actual phase L1, usually selected as the matrix material. Volume v1 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) = v1 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
$$ {{\text{v}}_{{1}}} + \sum\limits_{{r = 2}}^n {{{\text{v}}_r}} {(}t{)} = V(t)\quad {c_r}{(}t{)} = {{\text{v}}_r}{(}t{)}/V(t)\quad \sum\limits_{{r = 2}}^n {{c_r}{(}t{)} = 1} - {c6557}{(}t{)} $$
where \( {c6557}{(}t{)} \) decreases from 1 to its final magnitude.
Corresponding changes in the overall stiffness can be derived with reference to (4.4.6), where L1 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
$$ \Delta {L}(t) = {L}\left( {t + \Delta t} \right) - {L}(t) = \sum\limits_{{r = 1}}^n {\frac{{\Delta {{\text{v}}_r}}}{{V(t)}}} \left[ {{{L}_r} - {L}{(}t{)}} \right]{{T}_{r}}{(}t{)} $$
where \( {{T}_{r}}{(}t{)} = {\left\{ {{\mathbf{I}} + {{{P}}_r}{(}t{)}\left[ {{{L}_r} - {L}{(}t{)}} \right]} \right\}^{{ - 1}}} \) denote the partial strain concentration factors for each inhomogeneity added to the currently homogenized medium L(t). In each step, the Pr(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 L is reached at prescribed phase concentrations.
A differential equation for evaluation of L is obtained by letting \( \Delta {{\text{v}}_r} \to 0 \) and \( V{(}t{)} \gg \Delta V \to 0 \). Then
$$ \frac{\partial }{{\partial t}}\frac{{{{\text{v}}_r}}}{V} = {{{\dot{\text v}}}_r}/V - {{\text{v}}_r}\dot{V}/{V^2} \doteq {{{\dot{\text v}}}_r}/V $$
The \( {{{\dot{\text{v}}}}_r} \) can be replaced by \( {\dot{c}_r} \), with the substitution \( {{\text{v}}_{{1}}}/V = {c6557} = 1 - c \), where c follows from (7.3.2)
$$ \frac{{{{\text{v}}_r}}}{{{{\text{v}}6557}}} = \frac{{{c_r}}}{{{c6557}}} \to \frac{{{{{{\dot{\text v}}}}_r}}}{{{{\text{v}}_{{1}}}}} = \frac{{{{\dot{c}}_r}}}{{{(1} - c{)}}} + {c_r}\frac{{\dot{c}}}{{{{{(1} - c{)}}^{{2}}}}} =\frac{\dot{\rm v}_r}{V(1-c)} $$
Substitution into (7.3.3) yields the following system of coupled ordinary differential equations for evaluation of the overall stiffness
$$ {\dot{L}} = \sum\limits_{{r = 1}}^n {\left( {{{L}_r} - {L}} \right){{T}_{r}}\left( {{{\dot{c}}_r} + {c_r}\frac{{\dot{c}}}{{(1 - c)}}} \right)} $$
starting at \( t = 0 \) with the initial conditions L(0) = L1 (Norris 1985). This equation also governs the removal-replacement or FVP procedure, where L(0) = L0.
Specific applications require a selection of the initial ‘backbone’ medium L0, typically chosen as the actual matrix of the composite system, L0 = L1. Then, in a two-phase system r = 1, 2, where the reinforcements have stiffness L2 and are added by increments \( \Delta {c_2} \), there is \( {c_r} = c = {c_2} \) and (7.3.6) is reduced to
$$ \frac{{{\text{d}}{L}}}{{{\text{d}}{c_2}}} = \frac{1}{{\left( {1 - {c_2}} \right)}}\left( {{{L}_2} - {L}} \right){{T}_{2}} $$

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 L1 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

In this class of models of possibly multiphase composite materials, each inhomogeneity Lr resides in an ellipsoidal subvolume \( {\Omega_r} \), which is surrounded by a layer or coating of another material Lg 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 L0. 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 Lr, Lg and L0. 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.
Fig. 7.3

Geometry of the double inclusion and the double inhomogeneity models

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 L0, 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.

To evaluate effective stiffness of a double inhomogeneity, recall from Sect. 4.5.4 the derivation of average strains caused in a homogeneous double inclusion \( {\Omega_0} \supset {\Omega_2} \supset {\Omega_r} \) by eigenstrains \( \mu_{\textit{mn}}^g\,\,{\text{and}}\,\,\mu_{\textit{mn}}^r, \) applied in the subvolumes \( {\Omega_g} = {\Omega_2} - {\Omega_r}\,\,{\text{and}}\,\,{\Omega_r}, \) respectively, all in L0. 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
$$ \bar{\varepsilon }_{\textit{mn}}^g = \varepsilon_{\textit{mn}}^0 + {S_{\textit{mnkl}}}{(}{\Omega_2}{)}\mu_{\textit{kl}}^g + \frac{{{\Omega_r}}}{{{\Omega_2} - {\Omega_r}}}\left[ {{S_{\textit{mnkl}}}{(}{\Omega_2}{)} - {S_{\textit{mnkl}}}{(}{\Omega_r}{)}} \right]{(}\mu_{\textit{kl}}^r - \mu_{\textit{kl}}^g{)} $$
$$ \bar{\varepsilon }_{\textit{mn}}^r = \varepsilon_{\textit{mn}}^0 + {S_{\textit{mnkl}}}{(}{\Omega_2}{)}\mu_{\textit{kl}}^g + {S_{\textit{mnkl}}}{(}{\Omega_r}{)(}\mu_{\textit{kl}}^r - \mu_{\textit{kl}}^g{)} $$

In the double inclusion model, the above eigenstrains are regarded as equivalent eigenstrains, applied in the respective volumes of the homogeneous comparison medium L0, 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 L0 restricted by (6.2.16).

Converting the second and fourth order tensors to contracted tensorial or engineering matrix notation, and writing them as (6 × 1) and (6 × 6) matrices, renders the above equations in the form
$$ {\bar{\varepsilon }}_g^{{{(}a{)}}} = {{\varepsilon }^0} + {{S}_2}{\mu }_g^{\textit{eq}} + \gamma \,\Delta {S}\left( {{\mu }_r^{\textit{eq}} - {\mu }_g^{\textit{eq}}} \right) $$
$$ {\bar{\varepsilon }}_r^{{{(}a{)}}} = {{\varepsilon }^0} + {{S}_2}{\mu }_g^{\textit{eq}} + {{S}_{r}}\left( {{\mu }_r^{\textit{eq}} - {\mu }_g^{\textit{eq}}} \right) $$
where \( \gamma = {\Omega_r}/\left( {{\Omega_2} - {\Omega_r}} \right) \), and \( \Delta {S} = {{S}_2} - {{S}_{r}} \) is the difference between the Eshelby tensors of the ellipsoids \( {\Omega_2} \) and \( {\Omega_r} \) in L0. The local stress averages are
$$ {\bar{\sigma }}_g^{{{(}a{)}}} = {{L}_0}\left( {{\bar{\varepsilon }}_g^{{{(}a{)}}} - {\mu }_g^{\textit{eq}}} \right)\quad \quad {\bar{\sigma }}_r^{{{(}a{)}}} = {{L}_0}\left( {{\bar{\varepsilon }}_r^{{{(}a{)}}} - {\mu }_r^{\textit{eq}}} \right) $$
Average stresses caused inside the double inhomogeneity by the overall applied strain \( {{\varepsilon }^0} \) have the form
$$ {\bar{\sigma }}_g^{{{(}b{)}}} = {{L}_g}{\bar{\varepsilon }}_g^{{{(}b{)}}}\quad {\bar{\sigma }}_r^{{{(}b{)}}} = {{L}_r}{\bar{\varepsilon }}_r^{{{(}b{)}}} $$
The equivalent eigenstrains, and the average strains and stresses in the double inhomogeneity follow from the equalities implied by the equivalent inclusion method, expressed in terms of the respective subvolume averages of local fields
$$ {\bar{\varepsilon }}_g^{{{(}a{)}}} = {\bar{\varepsilon }}_g^{{{(}b{)}}}\quad {\bar{\varepsilon }}_r^{{{(}a{)}}} = {\bar{\varepsilon }}_r^{{{(}b{)}}}\quad {\bar{\sigma }}_g^{{{(}a{)}}} = {\bar{\sigma }}_g^{{{(}b{)}}}\quad {\bar{\sigma }}_r^{{{(}a{)}}} = {\bar{\sigma }}_r^{{{(}b{)}}} $$
Notice that in contrast to the fields employed in the original application of the method in Sect. 4.3.2, the current local fields may not be uniform.Consequences appear in (7.4.17). Substitution into (7.4.3) for the strains \( {\bar{\varepsilon }}_g^{{{(}a{)}}}\,\,{\text{and}}\,\,{\bar{\varepsilon }}_r^{{{(}a{)}}} \) from (7.4.1) and (7.4.2) and implementation of (7.4.5) yields the equivalent eigenstrains
$$ {\mu }_g^{\textit{eq}} = {{\Phi }_g}{{\varepsilon }^0}\quad {\mu }_r^{\textit{eq}} = {{\Phi }_r}{{\varepsilon }^0} $$
$$ {{\Phi }_g} = - {\left[ {\Delta {S} + \left( {{{S}_{r}} + {{E}_{r}}} \right){{\left( {{{S}_{r}} - \gamma \,\Delta {S} + {{E}_{r}}} \right)}^{{ - 1}}}\left( {{{S}_{r}} - \gamma \,\Delta {S} + {{E}_g}} \right)} \right]^{{ - 1}}} $$
$$ {{\Phi }_r} = - {\left[ {\left( {{{S}_{r}} + {{E}_{r}}} \right) + \Delta {S}{{\left( {{{S}_{r}} - \gamma \,\Delta {S} + {{E}_g}} \right)}^{{ - 1}}}\left( {{{S}_{r}} - \gamma \,\Delta {S} + {{E}_{r}}} \right)} \right]^{{ - 1}}} $$
and \( {{E}_g} = {\left( {{{L}_g} - {{L}_0}} \right)^{{ - 1}}}{{L}_0},\;{{E}_{r}} = {\left( {{{L}_r} - {{L}_0}} \right)^{{ - 1}}}{{L}_0} \).
The equivalent eigenstrains (7.4.6) are now used in (7.4.1), (7.4.2), (7.4.3) to find the local strain and stress averages in the subvolumes \( {\Omega_r} \) and \( {\Omega_g} \) of the double inhomogeneity.
$$ \left. \begin{array}{llll} &{\bar{\varepsilon }}_g = \left[ {{I} + {{S}_2}{\Phi }_g + \gamma \Delta {S}\left( {{{\Phi }_r} - {{\Phi }_g}} \right)} \right]{{\varepsilon }^0} \\& {{\bar{\sigma }}_g} = {{L}_0}\left[ {{I} - \left( {{I} - {{S}_2}} \right){{\Phi }_g} + \gamma \Delta {S}\left( {{{\Phi }_r} - {{\Phi }_g}} \right)} \right]{{\boldsymbol{\varepsilon }}^0} \end{array} \right\} $$
$$ \left. \begin{array}{llll} & {{\bar{\varepsilon }}_r} = {{T}^{{{(}r{)}}}}{{\varepsilon }^0} = \left( {{I} + \Delta {S}{{\Phi }_g} + {{S}_{r}}{{\Phi }_r}} \right){{\varepsilon }^0} \\ & {{\bar{\sigma }}_r} = {{L}_0}\left[ {{I} + \Delta {S}{{\Phi }_g} - \left( {{I} - {{S}_{r}}} \right){{\Phi }_r}} \right]{{\varepsilon }^0} \end{array} \right\}$$
Corresponding field averages over the entire volume \( {\Omega_2} = {\Omega_r} + {\Omega_g} \) can be found in analogy to (3.5.5) as
$$ {{\bar{\varepsilon }}_2} = {f_r}{ }{{\bar{\varepsilon }}_r} + {\left( {{1} - f_r} \right)}{{\bar{\varepsilon }}_g}\quad \quad {{\bar{\sigma }}_2} = {f_r}\,{{\bar{\sigma }}_r} + \left( {{1} - {f_r}} \right){{\bar{\sigma }}_g} $$
$$ {{\bar{\varepsilon }}_2} = \left( {{I} + {{S}_2}\,{{\Phi }_2}} \right){{\varepsilon }^0} = {T}{_2^{{(r)}}}{{\varepsilon }^0}\quad \quad {{\bar{\sigma }}_2} = {{L}_0}\left[ {{I} - \left( {{I} - {{S}_2}} \right){{\Phi }_2}} \right]{{\varepsilon }^0} $$
$$ {{\Phi }_2} = {f_r}\,{{\Phi }_r} + \left( {1 - {f_r}} \right){{\boldsymbol{\Phi }}_g} $$
where \( {f_r} \,{=}\, {\Omega_r}/{\Omega_2} \,{=}\, \gamma (1 - {f_r}{),} \)\( \gamma \,{=}\, {\Omega_r}/\left( {{\Omega_2} - {\Omega_r}} \right) \). The partial strain concentration factors \( {{T}^{{{(}r{)}}}} \) and \( {T}{_2^{{(r)}}} = \left( {{I} + {{S}_2}{{\Phi }_2}} \right) \) in (7.4.10) and (7.4.12) describe average strains in the single inhomogeneity Lr and double inhomogeneity L2, while both are present in a dilute concentration in the homogeneous medium L0 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 L0. Although the eigenstrains are only piecewise uniform in \( {\Omega_r} \) and \( {\Omega_g} \), the Eshelby tensor S2 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.
Derivation of the overall stiffness of the double inhomogeneity is completed by relating the averages (7.4.12) by \( {{\bar{\sigma }}_2} = {L}_2^{{{(}r{)}}}{{\bar{\varepsilon }}_2} \), where
$$ {L}_2^{{{(}r{)}}} = {{L}_0}\left[ {{I} - {{\Phi }_2}{{\left( {{I} + {{S}_2}{{\Phi }_2}} \right)}^{{ - 1}}}} \right] = {{L}_0} - {\left[ {{{\left( {{{L}_0}{{\Phi }_2}} \right)}^{{ - 1}}} + {{P}_2}} \right]^{{ - 1}}} $$
and \( {{P}_2} = {{S}_2}{L}_0^{{ - 1}} = {P}_2^{\text{T}} \). The superscript \( {{(}^{{{(}r{)}}}}{)} \) reminds that this tensor depends on both Sr and Lr, 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 S2, and perfectly bonded to a common comparison medium L0. 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 L0, 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 Lr, and on the stiffness Lg, 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 S2, 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 S2 impose a spatial distribution on the subvolumes \( {\Omega_r} \) of the inhomogeneities Lr 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 \( 0.6 \leq {c_{{cp}}} \leq\break 0.7405 \).

Spherical distributions with either variable or constant enclosure diameters are useful in modeling of two-phase or multiphase statistically isotropic aggregates, but they are not well suited for materials with statistically anisotropic distributions of reinforcements. Overall transverse isotropy, due to particle alignment or distribution in a layered texture, can be reproduced by prolate or oblate spheroidal enclosures, which promote such textures with either enhanced or reduced layer spacing, respectively. Figure 7.4 shows such arrangement, where possibly anisotropic spheroidal inhomogeneities Lr have many different shapes and orientations, and are distributed with spheroidal symmetry in the representative volume.
Fig. 7.4

Spheroidal inhomogeneities distributed with spheroidal spatial symmetry that promotes layered texture and isotropy in the transverse plane

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.

While the above mentioned choices of enclosure shapes allow modeling of different statistically isotropic or anisotropic aggregates, they also impose restrictions on the total volume fraction \( {c_r} = {{{\Omega }}_r}/{{{\Omega }}_0} \) of inhomogeneities that can be accommodated by the double inclusion model. In particular,
$$ {c_r} = {{{\Omega }}_r}/{{{\Omega }}_0} = {f_r}{c_2} = {(}{{{\Omega }}_r}/{{{\Omega }}_2}){(}{{{\Omega }}_2}/{{{\Omega }}_0}{)} $$
where \( {f_r} = {{{\Omega }}_r}/{{{\Omega }}_2} < 1 \) is the volume fraction of each inhomogeneity inside Ω2, 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 Ω2 and Ωr 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 Ω2 and Ωr.

Of course, actual dimensions or volume magnitudes of either Ωr or Ω2 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 P2 and Pr 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.

The effect of aspect ratio differences on the respective volume fractions was illustrated by Ponte Castaneda and Willis (1995), who considered two double inhomogeneities, each formed by a pair of coaxial spheroids Ωr and Ω2, defined by (4.6.1), Fig. 7.5. Both Ωr and Ω2 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 \( {f_r} = \left| {{\Omega_r}} \right|/\left| {{\Omega_0}} \right| \).
Fig. 7.5

Pairs of coaxial oblate and prolate spheroids

First, let \( {\rho_2} > {\rho_r} \), and \( a_2^{{{(}r{)}}} = a_2^{{{(2)}}} \), \( a6557^{{{(}r{)}}} < a6557^{{{(2)}}} \), so that a ‘flatter’ Ωr is enclosed by an ‘elongated’ Ω2. Next, let \( {\rho_2} < {\rho_r} \), so that \( a6557^{{{(}r{)}}} \leq a6557^{{{(2)}}} \) and \( a_2^{{{(}r{)}}} < a_2^{{{(2)}}} \); a ‘flatter’ Ω2 surrounds and comes in contact with an ‘elongated’ Ωr when \( a6557^{{{(}r{)}}} = a6557^{{{(2)}}} \). This yields volume fraction limits
$$ \left. \begin{array}{llll} {f_r} =& \left( {\frac{{{\rho_r}}}{{{\rho_2}}}} \right)\quad {\left. {{\rho_r}} \right|_{{\min }}} = {\rho_2}{f_r}\quad {\left. {{\rho_2}} \right|_{{\max }}} = {\rho_r}/{f_r}\quad {\text{for \;}}{\rho_2} > {\rho_r} \\ {f_r} =& {\left( {\frac{{{\rho_2}}}{{{\rho_r}}}} \right)^2}\quad {\left. {{\rho_r}} \right|_{{\max }}} = {\rho_2}/\sqrt {{{f_r}}} \quad {\left. {{\rho_2}} \right|_{{\min }}} = {\rho_r}\sqrt {{{f_r}}} \quad {\text{for \;}}{\rho_2} < {\rho_r} \end{array} \right\} $$

Coefficients of P2 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 P, 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 L0 = Lg = L1, as suggested in (7.4.25) below. However, any differences in alignments of Ωr 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

The double inhomogeneity model may assume several different forms, each determined by a particular choice of the S2 tensors, and the stiffnesses Lg and L0, which complement the Sr and Lr characterizing the actual inhomogeneity. One such form, for a two-phase system with aligned reinforcements of the same shape, postulates that S2 = Sr = S, 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 Lr in Ωr follows from (7.4.10) as
$$ {{\bar{\varepsilon }}_r} = \left[ {{I} + \frac{1}{{1 - {f_r}}}\left( {{{S}_{r}} - {f_r}{{S}_2}} \right){{\Phi }_r}} \right]{{\varepsilon }^0} = {{T}^{{{(}r{)}}}}{{\varepsilon }^0}\quad {\text{for}}\,\,{{L}_0} = {{L}_2} $$
where \( {{\Phi }_r} = - {\left[ {{{S}_{r}} + {{{(}{{L}_r} - {{L}_0}{)}}^{{ - 1}}}{{L}_0}} \right]^{{ - 1}}} \). The strain concentration factor can be reduced to the form
$$ {{T}^{{{(}r{)}}}} = {\left[ {{I} + {P}\left( {{{L}_r} - {L}} \right)} \right]^{{ - 1}}}\,\,\,{\text{for \;}}{{L}_0} = {{L}_2} = {L},\,\,\,\Delta {S} = {\mathbf{0}} $$
where \( {P} = {S}{{L}^{{ - 1}}} \). The same result is provided by the self-consistent method in (7.1.1).

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).

Alternate forms of the double inhomogeneity model, suggested by Hu and Weng (2000), are based on selections of comparison medium stiffness as \( {{L}_0} \ne {L}_2^{{{(}r{)}}} \) and \( {{L}_g} = {{L}6557} \), the stiffness of the actual matrix surrounding the inhomogeneities Lr. To examine the symmetry condition \( {{L}_0}{\Phi }_2 = {\Phi }_2^{\text{T}}{{L}_0} \) in (7.4.14), we denote
$$ \left. \begin{array}{llll}{}& {{F}6557} = {\left( {{{L}6557} - {{L}_0}} \right)^{{ - 1}}} = {{E}_{1}}{L}_0^{{ - 1}} = {F}6557^{\text{T}}\quad {{F}_{r}} = {\left( {{{L}_r} - {{L}_0}} \right)^{{ - 1}}} = {{E}_{r}}{L}_0^{{ - 1}} = {F}_r^{\text{T}} \\ &\qquad\qquad\qquad\qquad\quad{{S}_i} = {{P}_{i}}{{L}_0}\quad {\text{for}}\;i = r,1,2 \end{array} \right\}\\ $$
Then, (7.4.7), (7.4.8) change to
$$ {{L}_0}{{\Phi }_g} = - {\left[ {\Delta {P} + \left( {{{P}_r} + {{F}_{r}}} \right){{\left( {{{P}_r} - \gamma \,\Delta {P} + {{F}_{r}}} \right)}^{{ - 1}}}\left( {{{P}_r} - \gamma \,\Delta {P} + {{F}6557}} \right)} \right]^{{ - 1}}}\\ $$
$$ {{L}_0}{{\Phi }_r} = - {\left[ {\left( {{{P}_r} + {{F}_{r}}} \right) + \Delta {P}{{\left( {{{P}_r} - \gamma \,\Delta {P} + {{F}6557}} \right)}^{{ - 1}}}\left( {{{P}_r} - \gamma \,\Delta {P} + {{F}_{r}}} \right)} \right]^{{ - 1}}}\\ $$
where again \( \gamma = {f_r}/(1 - {f_r}{)} \), \( {f_r} = {{{\Omega }}_r}/{{{\Omega }}_0} \). Both above terms need to be diagonally symmetric to assure that \( {{L}_0}{\Phi }_2 = {\Phi }_2^{\text{T}}{{L}_0} \) in (7.4.14), where
$$ {{\Phi }_2} = {f_r}\,{{\Phi }_r} + \left( {1 - {f_r}} \right){{\Phi }_g} $$
One configuration (CA) of this model stipulates that the ellipsoids \( {\Omega_r} \) and \( {\Omega_2} \) are similar and have identical Eshelby tensors.
$$ \left. \begin{array}{llll}{} & \textit{CA} \equiv \left( {\Delta {P} = \mathbf{0}} \right) \to {{P}_2} = {{P}_r},\;\;\;\quad {{L}_0}{{\Phi }_g} = - {\left[ {{{P}_2} + \left( {{{L}6557} - {{L}_0}} \right)} \right]^{{ - 1}}}, \\& {{L}_0}{{\Phi }_r} = - {\left[ {{{P}_2} + \left( {{{L}_r} - {{L}_0}} \right)} \right]^{{ - 1}}} \end{array} \right\}$$
Average strains in \( {\Omega_r} \), \( {\Omega_g} \) and \( {\Omega_2} \) follow from (7.4.11), (7.4.12), (7.4.13). According to (7.4.14), effective stiffness of the double inhomogeneity is
$$ \left. \begin{array}{llll} \left( {{L}_2^{(r)}} \right)^{CA} & = {{L}_0} + \left({\left[ {f_r}{\left[{{P}_2} + {\left( {{L}_r} - {{L}_0} \right)}^{- 1} \right]}^{ - 1} - {(1 - {f_r})} \right.}\right. \\& \quad\times\left.\left. {\left[{{P}_2} + {\left( {{L}6557} - {{L}_0} \right)}^{- 1} \right]}^{ - 1} \right]^{- 1} - {{P}_2} \right)^{ - 1} = {\left( {\left( {{L}_2^{(r)}} \right)}^{\textit{CA}} \right)^{\text{T}}} \end{array} \right\} $$
Configuration CA represents a double inhomogeneity consisting of a core Lr surrounded by a layer of matrix L1, such that both the core and outer ellipsoidal surfaces have the same aspect ratio and alignment. The comparison medium can have a different stiffness L0, selected in agreement with (6.2.16) or (6.2.24). Each inhomogeneity may have a certain stiffness Lr and volume fractionfr, both different inside each double inhomogeneity, yielding different stiffnesses \( {{(}{L}_2^{{{(}r{)}}}{)}^{{CA}}} \). However, the shape and alignment of all Ω2 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 L0=L1, then \( {{L}_0}{{\Phi }_g} \to \mathbf{0} \) and
$$ {\left( {{L}_2^{{{(}r{)}}}} \right)^{{CA}}} = {{L}_{{1`}}} + {f_r}{\left[ {{{\left( {{{L}_r} - {{L}6557}} \right)}^{{ - 1}}} + \left( {{1} - {f_r}} \right){{P}_r}} \right]^{{ - 1}}} $$
which for \( {f_r} \to {c_r} = {(1} - {c6557}{)} \) is the Mori-Tanaka estimate (7.2.5) of the stiffness of a two-phase composite reinforced by aligned inhomogeneities of single stiffness Lr, and of the same shape Ωr. Therefore, if the overall stiffness is selected as\( \ {L} = {( {{L}_2^{{{(}r{)}}}} )^{{CA}}} \), then the double inclusion model provides the Mori-Tanaka estimate of L. 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 L0 = L1. Since the shape of Ω2 is specified by P2 = Pr, 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.
Fig. 7.6

The CA configuration of the double inhomogeneity model

7.4.4 Multiphase Composites with Different Constituent Shapes and Alignments

Another form of the double inhomogeneity model, that extends the capability of standard AFA models, is offered by the configuration CB, also suggested Hu and Weng (2000)
$$ \left. \begin{array}{llll}{} & CB \equiv \left( {\Delta {P} \ne {\mathbf{0}},\,\,{{L}_0} = {{L}_g} = {{L}6557}} \right) \to {{L}_0}{{\Phi }_g} \to {\mathbf{0}}, \\ & {{L}_0}{{\Phi }_r} = - {\left[ {{{P}_r} + \left( {{{L}_r} - {{L}_0}} \right)} \right]^{{ - 1}}} \end{array} \right\}$$

Each inhomogeneity Lr is now embedded in a common matrix L1, and it may have its own shape and alignment described by a different Pr. Since these three tensors follow from the known properties of a given matrix and reinforcements, one only needs to select a single P2 for all enclosures Ω2, 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 L tensors that apply to inhomogeneities of different alignment need to be transformed into a single coordinate system that is used in the representative volume.

From (7.4.13) and (7.4.21)
$$ {{L}_0}{{\Phi }_2} = {c_r}\,{{L}_0}{{\Phi }_r} = - {c_r}{\left[ {{{P}_r} + {{\left( {{{L}_r} - {{L}6557}} \right)}^{{ - 1}}}} \right]^{{ - 1}}} = - {c_r}\left( {{{L}_r} - {{L}6557}} \right){{T}_{r}} $$
where \( {c_r} \,{=}\, {(1} - {c6557}{)} \) is the actual volume fraction of inhomogeneities Lr in the matrix material L1. 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 Lr in the matrix material L1, derived in (4.2.14). Average strains in Ωr and Ω2 follow from (7.4.2). The average strain in the double inhomogeneity is
$$ {\bar{\varepsilon }}{_2^{{(r)}}} = \left( {{I} - {{P}_2}{{\left[ {{{P}_r} + {{\left( {{{L}_r} - {{L}6557}} \right)}^{{ - 1}}}} \right]}^{{ - 1}}}} \right){{\varepsilon }^0} = {T}{_2^{{(r)}}}{{\varepsilon }^0} $$
Equation (7.4.14) yields the stiffness of the double inhomogeneity \( \Omega_2^{{{(}r{)}}} \)
$$ \left.\begin{array}{llll} {\left( {{L}_2^{{{(}r{)}}}} \right)^{{CB}}} & = {{L}6557} + {f_r} {\left[ {{{\left( {{{L}_r} - {{\mathbf{L}}6557}} \right)}^{{ - 1}}} + {{P}_r} - {f_r}{{P}_2}} \right]^{{ - 1}}} \\ & = {{L}6557} + {f_r}{\Big[ {{I} - {f_r}\left( {{{L}_r} - {{L}6557}} \right){{T}_{r}}{{P}_2}} \Big]^{{ - 1}}}\left( {{{L}_r} - {{L}6557}} \right){{T}_{r}}\end{array}\right\} $$
where \( {f_r} = {\Omega_r}/\Omega_2^{{{(}r{)}}} \). Also, \( {( {{L}_2^{{{(}r{)}}}} )^{{CB}}} = \left( ( {{L}}_2^{{{(}r{)}}})^{CB} \right)^{\text{T}} \) for any Pr and P2 that define shapes of the ellipsoids Ωr and Ω2.

If all inhomogeneities Ωr 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 P2. 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).

For a random distribution of orientations of inhomogeneities Lr, the overall stiffness L 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 P2 and matrix stiffness L1 remain unchanged.
$$ {L} = \left\{ {{(}{L}_2^{{{(}r{)}}}{)}^{{CB}}} \right\} = {{L}6557} + (1 - {c6557}){\left[ {{I} - (1 - {c6557})\left\{ {{(}{{L}_r} - {{L}6557}{)}{{T}_{r}}} \right\}{{P}_2}} \right]^{{ - 1}}}{\left\{ {{(}{{L}_r} - {{L}6557}{)}{{T}_{r}}} \right\}} $$

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 Ωc 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.

In a completely debonded or porous aggregate, the matrix L1 is the comparison medium \( {L}_0^{{{(} + {)}}} = {{L}6557} \), and \( {{L}_r} \to 0 \), which leads to the following substitutions
$$ \left. \begin{array}{llll}{} & \left[ {{{P}_r} + {{\left( {{{L}_r} - {{L}6557}} \right)}^{{ - 1}}}} \right] \to {{P}_r} - {{M}6557}\quad {{T}_{r}} \to {\left( {{I} - {{P}_r}{{L}6557}} \right)^{{ - 1}}} \\&\quad \left( {{{L}_r} - {{L}6557}} \right){{T}_{r}} \to {\left( {{{P}_r} - {{M}6557}} \right)^{{ - 1}}} \end{array} \right\}$$
Two geometries of porous media are of particular interest. When all cavities or cracks have the same shape and alignment, with \( {{P}_r} \equiv {{P}_2} \), and the entire representative volume is filled with the double inhomogeneities, overall stiffness is predicted by the double inhomogeneity model 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)
$$ {{L}^{{{(} + {)}}}} = {{L}6557} + \left( {{1} - {c6557}} \right){\left( {{c6557}{P} - {{M}6557}} \right)^{{ - 1}}} $$
The CB variant yields stiffness of the double inhomogeneity with a vacuous core as
$$ {{L}^{{{(} + {)}}}} = {\left( {{L}_2^{{{(}c{)}}}} \right)^{{CB}}} = {{L}6557} + {(1} - {c6557}{)}\left[ {{{P}_c} - {(1} - {c6557}{)}{{P}_2} - {{M}6557}} \right]^{{ - 1}} $$
This represents another bound on the overall stiffness of a porous medium with different cavity shapes and orientations described by Pc, and spatial distribution described by P2. For example, the geometry of Fig. 7.7 can be used, with some or all inhomogeneities replaced by cavities.
Fig. 7.7

The CB configuration of the double inhomogeneity model

For randomly distributed cavities or cracks, orientation averaging in (7.4.29) provides the upper bound on overall stiffness
$$ {{L}^{{( + )}}} = {{L}6557} + \left( {1 - {c6557}} \right){\left[ {{I} - \left( {1 - {c6557}} \right)\left\{ {{{\left( {{{P}_c} - {{M}6557}} \right)}^{{ - 1}}}} \right\}{{P}_2}} \right]^{{ - 1}}}\left\{ {{{{(}{{P}_c} - {{M}6557}{)}}^{{ - 1}}}} \right\} $$

For both cavities and cracks, the Pc tensors are described in Sect. 4.6, together with the P2 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).

Simple forms of the bounds (7.4.33) for cracked solids have been derived by Ponte Castaneda and Willis (1995). A medium with a spherical distribution of randomly oriented cracks generates an isotropic solid with overall bulk and shear moduli K, G. Matrix moduli are \( 3{K6557} = 2{G6557}{(1} + {\nu6557}{)/(1} - 2{\nu6557}{)} \), and the volume fraction of the enclosures Ω2 in a representative volume Ω0 is denoted by \( {c_2} = {\Omega_2}/{\Omega_0} \leq 1. \) The upper bound on the two overall moduli is
$$ \left. \begin{array}{llll} & \frac{{{K^{{{(} + {)}}}}}}{{{K6557}}} = 1 - \frac{{12{c_2}{(1} - \nu6557^2{)}}}{{{9}\pi {(1} - 2\nu6557{)} + 4{c_2}{{{(1} + \nu6557{)}}^{{2}}}}} \\ & \frac{{{G^{{{(} + {)}}}}}}{{{G6557}}} = 1 - \frac{{120{c_2}{(1} - {\nu6557}{)(5} - \nu6557{)}}}{{{225}\pi {(2} - {\nu6557}{)} + {16}{c_2}{(4} - 5\nu6557{)(5} - \nu6557{)}}} \end{array} \right\} $$
Distributions of aligned circular cracks on planes perpendicular to the \( {x6557} - {\text{axis,}} \) in an isotropic matrix with moduli \( E^{{{(1)}}} = 2{(1} + {\nu^{{{(1)}}}}{)}{G^{{{(1)}}}} \), create a transversely isotropic material, where the cracks change only the longitudinal Young’s and shear moduli E11 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
$$ \left. \begin{array}{llll} & \frac{{E_{{11}}^{{{(} + {)}}}}}{{E_{{11}}^{{{(1)}}}}} = 1 - \frac{{60{c_2}{(1} - \nu6557^2{)}}}{{{15}\pi + 4{c_2}{{{(7} - 15\nu6557^2{)}}_{{}}}}} \\ & \frac{{{p^{{{(} + {)}}}}}}{{{p^{{{(1)}}}}}} = 1 - {\frac{{60{c_2}{(1} - {\nu6557}{)}}}{{15\pi {(2} - {\nu6557}{)} + 8{c_2}{(4} - 5{\nu6557}{)}}}} \end{array} \right\} $$
where elastic moduli of the isotropic matrix are \( E_{{11}}^{{{(1)}}} = 2{p^{{{(1)}}}}{(1} - {\nu6557}{)} \). Results for a flat distribution of cracks, where P2 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.

Of course, experimental verification of the different predictions is the ultimate test of their utility. That can be expected only when the following two conditions are satisfied:
  1. (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.

  2. (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 m2/1 cm3 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.

Figure 7.8 shows the double array used in generating both graded material models. It is created by two overlapping honeycomb arrays, which have been separated in the x2–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.
Fig. 7.8

Double and single honeycomb arrays. The double array is subdivided by 24 triangular elements per hexagonal cell, or by parallel layers that have effective properties estimated by the M-T or SCM procedures

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 x2–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.

Figure 7.9 provides examples of three discrete microstructures, designated as Model 1_2 with a distinct percolation threshold, Model 2 with a wide skeletal transition zone, and Model 3 which has both a wide transition region and a threshold. Each micro-structure has 50 rows of hexagonal cells, with 40 cells per row. Five rows at both ends are filled with either homogeneous carbon or silicon carbide, then one hexagonal cell of the other phase is added in each next row, hence each new cell was added to a sufficiently large volume of constant composition. The resulting gradient is uniform, equal to 0.025/row, and of the same magnitude in all three models shown.
Fig. 7.9

Models of graded materials with the same linear composition gradient in the \( {x_3} - \)direction. (a) Model 1-2. (b) One of computer-generated random micro-structures of Model 2. (c) Microstructure used in the finite element analysis as Model 3 and as a layered model COMB3.1

The graded microstructures were realized by the C/SiC system (Sasaki and Hirai 1991). Both phases were regarded as isotropic with the thermo-mechanical properties in Table 7.1, where 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 c2.
Table 7.1

Phase properties of the carbon/silicon carbide system


E (GPa)

\( \nu \)

\( \alpha \) (10−4/°C)

\( \kappa \) (Wm−1/°C)

C (r = 1)





SiC (r = 2)





Models COMB3.1 and COMB3.2 employed different transition functions between the three methods, to describe effective property changes with changing c2; 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.

Figure 7.10 shows predictions by these models of the transverse Young’s modulus E33 as a function of c2, which is the volume fraction of SiC. Similar predictions for the coefficient of thermal expansion αeff 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 c2 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 M. 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 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 x3–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.
Fig. 7.10

Estimates of the overall transverse Young’s modulus \( {E_{{22}}} = {E_{{33}}} \) as functions of the SiC volume fraction \( {c_2} \)

Fig. 7.11

Estimates of the overall coefficient of thermal expansion as functions of the SiC volume fraction \( {c_2} \)

Fig. 7.12

Boundary conditions applied to the graded material subjected to a temperature gradient. Similar boundary conditions, with T = 100°C, were applied to impose the uniform change in temperature

Fig. 7.13

Comparisons of average transverse strains \( {\varepsilon_{{22}}}{(}{x_3}{)} \) caused by a uniform change in temperature. Phase 0 denotes the effective medium, Phase 1 is carbon, and Phase 2 is silicone carbide

7.5.2 Selected Comparisons of Discrete and Homogenized Models

Graded materials are often used in thermal barrier coatings that are subjected to both uniform changes in temperature and to thermal gradients. Of interest in such applications are temperature distributions in individual layers, as well as the overall and phase field averages of strain and stress fields, which may be useful in estimates of dimensional changes and life expectancy. Here we compare predictions obtained under the said conditions between the discrete finite element model Model 3 shown in Fig. 7.9c, and the layered model COMB3.1, which has layer overall properties of the kind shown in Figs. 7.10 and 7.11. It turns out that properties of layers with a distinct matrix phase are closely approximated by one of the two Mori-Tanaka estimates. The skeletal zone separating the distinct matrix layers is well represented by the self-consistent estimate and by the transition functions. Narrower transition zones in model COMB3.2 are indicated for systems with sharper boundaries, shown for example in Fig. 7.9a. Figure 7.13, shows close agreement between all phase strain averages computed with the discrete finite element Model 3 and the homogenized layer model COMB3.1.
Fig. 7.14

Temperature distributions in the graded layer subjected to different surface temperatures, evaluated using discrete and homogenized layer models

Next, the graded layer was subjected to a thermal gradient. Assuming that the thermal and mechanical responses are not coupled, the steady state temperature distribution caused by the prescribed heat flow is evaluated first, and then applied together with the mechanical constraints to the solution domain. Figure 7.14 compares predictions of temperature distribution through the thickness of the graded layer under an applied temperature gradient. In this case, a close agreement is found between the finite element Model 3 and COMB3.1, while other layered models show small deviations.
Fig. 7.15

Comparisons of average transverse strains \( {\varepsilon_{{22}}}{(}{x_3}{)} \) caused by a temperature gradient of 100°C. Phase 0 denotes the effective medium, Phase 1 is carbon, and Phase 2 is silicone carbide

Figure 7.15 shows the overall and phase transverse strain \( {\varepsilon_{{22}}}{(}{x_3}{)} \) averages predicted by the two models. Notice that the overall transverse strain is near zero through the thickness of the layer, a desirable if fortuitous outcome, albeit under a small temperature difference. Steeper thermal gradients would generate overall transverse deformation. Finally, Fig. 7.16 shows the overall and phase transverse stress \( {\sigma_{{22}}}{(}{x_3}{)} \) averages caused by the thermal gradient. Changes in material properties through the thickness cause very dissimilar trends in phase averages of transverse strain and stress, not observed in statistically homogeneous system.
Fig. 7.16

Comparisons of average transverse stresses \( {\sigma_{{22}}}{(}{x_3}{)} \) caused by a temperature gradient of 100°C. Phase 0 denotes the effective medium, Phase 1 is carbon, and Phase 2 is silicone carbide

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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Copyright information

© Springer Science+Business Media B.V. 2013

Authors and Affiliations

  • George J. Dvorak
    • 1
  1. 1.Mechanical, Aeronautical and Nuclear EngineeringRensselaer Polytechnic InstituteTroyUSA

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