Effective elastic properties of a composite containing multiple types of anisotropic ellipsoidal inclusions, with the application to the attachment of tendon to bone

Abstract

Estimates of the effective stiffness of a composite containing multiple types of inclusions are needed for the design and study of functionally graded systems in engineering and physiology. While excellent estimates and tight bounds exist for composite systems containing specific classes and distributions of identical inclusions, these are not easily generalized to complex systems with multiple types of inclusions. For example, three-point parameters are known for only a few inclusion shapes and orientations. The best estimate available for a composite containing multiple classes of inclusions arises from the Kanaun-Jeulin approach. However, this method is analogous to a generalized Benveniste approach, and therefore suffers from the same limitations: while excellent for low volume fractions of inclusions, the Kanaun-Jeullin and Benveniste estimates lie outside of three-point bounds at higher volume fractions. Here, we present an estimate for composites containing multiple classes of aligned ellipsoidal inclusions that lies within known three-point bounds at relatively higher volume fractions of inclusions and that is applicable to many engineering and biological composites.

1. Introduction

Composite structural materials offer the potential for efficient solutions to engineering problems and are central to many mechanical functions in physiology. Extensive effort has been devoted to predicting the overall mechanical responses of composites based on knowledge of the mechanical properties, geometries, and distributions of their constituent phases (e.g.,(Torquato, 2001; Milton, 2002)). However, many challenges remain, especially for composites having inclusions of varying shapes, sizes, and mechanical properties. An additional challenge is that most homogenization techniques provide the best estimates for relatively low volume fractions of inclusions. Here, we present a homogenization scheme that appears to provide reasonable predictions under these conditions.

To highlight the need for this method and the ways that it differs from existing methods, we briefly review literature on both homogenization of multiphase composites as well as the accuracy of micromechanical methods at high volume fraction of inclusions. Predicting effective moduli of composites containing relatively high volume fractions of inclusions is challenging even for materials containing only a single class of inclusions Ferrari (1991); Roumi and Shodja (2007); Ju and Yanase (2010, 2011); Kwon and Dharan (1995). The best of these render good correlation with experimental data for inclusion volume fractions below about 60% or for special distributions of the inclusions. Numerous homogenization techniques use the effective medium method, which builds from the work of Eshelby (1957) by considering an equivalent effective medium embedded with non-interacting inhomogeneities. These include the method of cells Aboudi (1989), the Mori and Tanaka (1973) approach ,the generalized self-consistent method Budiansky (1965); Kerner (1956); Hill (1965), the models of Castañeda and Willis (1995), and the Kuster and Toksöoz (1974) model. These approximations depend only upon the volume fractions and geometries of inclusions and are independent of the spatial or probabilistic distributions of inclusions. Accordingly, they are pertinent to heterogeneous solids containing a low concentration of identical inclusions and cannot be easily generalized to composites with multiple classes of inclusions. Higher-order bounds such as those of Weng (1992); Silnutzer (1972); Torquato and Lado (1986) and Sen et al. (1987) and the three-point bounding technique (e.g., Milton and Phan-Thien (1982)) are far tighter than classic Voigt and Reuss (Hill, 1952) or Hashin and Shtrikman (1963) bounds. However, the parameters needed for three-point bounds and estimates are known only for a few classes of inclusion shapes and orientations (Torquato, 2001). While these can be combined to estimate the mechanics of composites containing multiple class of inclusions (Genin and Birman, 2009), they are limited to cases for which the three-point parameters are known. More recent homogenization procedures (e.g., Tucker III and Liang (1999); Torquato (2001); Kakavas and Kontoni (2006); Sevostianov and Giraud (2013)) are also limited to low inclusion volume fraction.

Although a wealth of literature exists on the estimation of effective elastic properties of composites containing a single class of inclusions, few methods are available for analysis of composites with multiple types of inclusions (Torquato, 2002; Hashin, 1983; Tucker III and Liang, 1999; Hu and Weng, 2000; Christensen, 2012). For example, Lim (2002) presented a generalized mechanics of materials approach to evaluating the moduli of three-phase composites consisting of two types of reinforcement (fibers and/or particles) embedded within the matrix. Kanaun and Jeulin (2001) developed a model allowing for the special statistical distribution of the centers of inclusions of various orientations and shapes. The Maxwell scheme was modified and extended to the case of anisotropic multiphase composites by Sevostianov (2014).

Challenges involved in the homogenization of multiphase composite materials were outlined in a recent paper by Sevostianov and Kachanov (2014). In particular, they pointed that the Mori-Tanaka approach applied to multiphase composites may violate the Hashin-Shtrikman bounds as well as compromise the symmetry of the stiffness tensor. The latter problem was resolved by imposing a symmetrization of the stiffness tensor as demonstrated in their paper. Although the Kanaun and Jeulin (2001) effective field estimate applied to multiphase composites avoids the violation of the symmetry of the stiffness tensor by assuming that different effective fields affect different composite phases, this results in the violation of the principle of superposition. Without the adaptation of different effective fields, the Kanaun-Jeulin estimate is analogous to a generalized formula of Benveniste (1987) that represents a modified Mori-Tanaka approach, and therefore suffers from the same limitations (Genin and Birman, 2009). While excellent for low volume fractions, both the Kanaun-Jeulin and Benveniste estimates lie outside three-point bounds at higher volume fractions. and therefore sufferings from the same limitations (Genin and Birman, 2009). While excellent for low volume fractions, the Kanaun-Jeulin and Benveniste estimates lie outside of three-point bounds at higher volume fractions in cases for which one would desire them to lie within the three-point bounds.

As follows from this discussion, the estimation of effective properties of composites containing high volume fractions of inclusions continues to present a challenge, especially in cases of multiple classes of inclusions. In engineering structures, these issues arise in the study of functionally graded materials (Birman et al., 2013; Suresh et al., 1998; Byrd and Birman, 2007), especially for aerospace applications, that employ high-performance polymers with high volume fractions of inclusions. Hence, it is necessary to predict the effective moduli of composites featuring multiple classes of inclusions over the full range of volume fractions and geometries.

Besides homogenization methods applicable to a broad range of engineering structural materials, our specific interest is the modeling of partially mineralized tissue. Bone, a highly mineralized tissue, is a composite material made up primarily of flexible collagen fibers and a high (50%) volume fraction of stiff mineral inclusions (Alexander et al., 2012). The shapes of these inclusions have been variably described as plate-like (Eppell et al., 2001; Kim et al., 1995), needle-like (Traub et al., 1992), and amorphous (Mahamid et al., 2008; Boonrungsiman et al., 2012). At the attachment of tendon to bone, the volume fraction of the mineral inclusions increases from 0% in tendon to 50% in bone over a distance on the order of tens of micrometers Genin et al. (2009). One important mechanism of toughening in biologic systems such as the tendon-to-bone attachment is stress redistribution associated with the progressive stiffening of collagen fibers by mineral (Liu et al., 2012, 2014; Thomopoulos et al., 2006). Predicting this is particularly important for defining design criteria for tissue engineered constructs needed for improving surgical outcomes (Thomopoulos et al., 2010, 2011). In spite of the progress reflected in the above-mentioned publications, modeling the mineralized collagen tissues remains challenging. Physiologically, debate continues about where mineral lies within collagen, how it accumulates, and the shape it has (Schwartz et al., 2012; Thurner et al., 2007; Pasteris et al., 2008).

The present paper presents an approximation that is applicable to composites with multiple classes of inclusions and that retains high accuracy at higher volume fractions. Expressions for effective moduli were derived under two conditions: (i) we required the total strain energy stored in a homogenized material to equal that in the actual material, with the average inclusion and the average matrix strains evaluated based on Eshelbys approach; and (ii) we modeled the effective matrix and inclusions as acting in parallel. Contrary to the Mori-Tanaka technique, we replaced the evaluation of the strain in the representative volume element using the rule of mixtures with a strain energy equivalence requirement; this leads to a formulation that therefore differs from the Kanaun-Jeulin estimate. Our results suggest that the new formulation is useful at higher volume fractions, although it is less useful at lower volume fractions.

Subsequently, the developed theory was specified for the case of a three-phase composite. For verification of the proposed theory, we compared predictions to experimental data and to other estimates. Our method provided good accuracy for a simultaneously fiber- and particulate-reinforced composite, and lay within three-point bounds at relatively high volume fractions for all cases studied. The method was also applied to study stiffening of tissue at the tendon-to-bone attachment.

2. Estimate of overall elastic properties of a composite

Our model emanates from the Kanaun-Jeulin methodology (Kanaun and Jeulin, 2001) for estimating the stiffness tensor of a three-phase composite. This was shown by Genin and Birman (2009) to arise as an extrapolation of the method of Benveniste (1987) for multiple types of inclusions. These methods under-predict the stiffness of a great number of composites of interest. In this section, we review the Kanaun-Jeulin methodology, then present our adaptation of it.

Like the Kanaun-Jeulin estimate, the adaptation was conducted within the framework of linear elasticity assuming that the perturbed strain in the matrix, due to the presence of inclusions, was not affected by interactions between different types of inclusions. In other words, each type of inclusion affects the strains in the matrix, but the perturbed matrix strain, due to the interaction between these inclusions, was assumed to be of second order. We considered a certain representative volume V₀ of an infinite three-dimensional linear elastic medium embedded with N types of linear elastic inclusions that are aligned but randomly distributed. The properties of the matrix and of the different types of inclusions will be identified by the subscripts j = 1 and j > 1, respectively. We assumed perfect adhesion between the matrix and each inclusion. Each type of inclusion had uniform elastic properties and possessed a known geometry.

2.1. Kanaun-Jeulin methodology

The Kanaun-Jeulin methodology expanded upon the Mori-Tanaka approach, in which the tensors of average strain in each type of inclusion can be written in terms of the tensor of the average strain in the matrix by:

$${\overline{\mathit{∊}}}_j={\mathit{T}}_j{\overline{\mathit{∊}}}_1,$$ (1)

where

$${\mathit{T}}_j={[\mathit{I}+{\mathit{S}}_j{{\mathit{L}}_1}^{-1}({\mathit{L}}_j-{\mathit{L}}_1)]}^{-1}$$ (2)

and ${\overline{\mathit{∊}}}_j=\frac{1}{V_j}\underset{V_j}{\mathit{\int }}\mathit{∊}d{V}_j$. Here, S_j are fourth-order Eshelby tensors corresponding to the j^th type of inclusions, and I is a fourth-order identity tensor. Explicit examples of the Eshelby tensor are presented in the appendix for the cases of spheroidal inclusions, penny-shaped inclusions, and long cylindrical inclusions (fibers). Equation (1) can be used jointly with the condition that the stresses on the homogenized and actual representative volume are equal, with matrix and inclusions resisting deformation in parallel:

$${\overline{\mathit{\sigma }}}_0=f_1{\overline{\mathit{\sigma }}}_1+\sum _{j=2}^N{f}_j{\overline{\mathit{\sigma }}}_j.$$ (3)

Noting that σ̄_j = L_j∊̄_j and combining Eq.(3) and Eq.(1), the resulting system of (j+1) linear algebraic equations can be solved to express the tensor of strains in the matrix and inclusions in terms of the tensor of strains in the homogeneous material:

$$\begin{array}{l}{\overline{\mathit{∊}}}_j={\mathit{T}}_j{\overline{\mathit{∊}}}_1={\mathit{T}}_j{(f_1{\mathit{L}}_1{\mathit{T}}_1+\sum _{i=2}^N{f}_i{\mathit{L}}_i{\mathit{T}}_i)}^{-1}{\mathit{L}}_e{\overline{\mathit{∊}}}_0\\ \equiv {{\mathit{N}}_j}^s{\mathit{L}}_e{\overline{\mathit{∊}}}_0\equiv {\mathit{Q}}_j{\overline{\mathit{∊}}}_0,\end{array}$$ (4)

where j = 1, 2, …N and T₁ = I.

This estimate has been shown to provide an underestimate of the strain energy stored by a composite material under a given loading; that is, the estimate is too compliant (Genin and Birman, 2009). This is expected because the approach uses both averaged stresses and averaged strains, which are bound to combine to yield a strain energy lower than that stored by the composite except in special cases.

2.2. Estimate based upon energy-based effective strains

To improve upon this approximation, we replaced the average strains ∊̂_i in the phases with effective strains ∊̄_j that better represented the energy stored in the composite. That is, we identified effective strains so that the strain energy in the homogenized material equaled that in an actual material, under the above assumptions. We partitioned the energy amongst the phases:

$$\underset{V_0}{\mathit{\int }}{\mathit{\sigma }}_0^T:{\mathit{∊}}_{0}d{V}_0=\underset{V_1}{\mathit{\int }}{\mathit{\sigma }}_1^T:{\mathit{∊}}_{1}d{V}_1+\sum _{j=2}^N\underset{V_j}{\mathit{\int }}{\mathit{\sigma }}_j^T:{\mathit{∊}}_{j}d{V}_j,$$ (5)

where the sum of the volumes of the matrix and of all phases equals that of the representative volume ( $V_0=V_1+{\sum }_{J=2}^N{V}_j$, and σ and ∊ are the tensors of engineering stress and linearized strain, respectively. We defined the effective strain ∊̂_j for each phase j within a representative volume so that each term in Eq.(5) can be written as:

$$\begin{array}{l}\underset{V_j}{\mathit{\int }}{\mathit{\sigma }}_j^T:{\mathit{∊}}_{j}d{V}_j=\underset{V_j}{\mathit{\int }}{({\mathit{L}}_j{\mathit{∊}}_j)}^T:{\mathit{∊}}_{j}d{V}_j\\ =\underset{V_j}{\mathit{\int }}({\mathit{L}}_j{\mathit{∊}}_j):{\mathit{∊}}_j^{T}d{V}_j\equiv ({\mathit{L}}_j{\widehat{\mathit{∊}}}_j):{\widehat{\mathit{∊}}}_j^T{V}_j.\end{array}$$ (6)

Note that ∥ ∊̂_j ∥ ≤ ∥ ∊̄_j ∥. Then Eq.(5) can be rewritten:

$$({\mathit{L}}_e{\widehat{\mathit{∊}}}_0):{\widehat{\mathit{∊}}}_0^T{V}_0=({\mathit{L}}_1{\widehat{\mathit{∊}}}_1):{\widehat{\mathit{∊}}}_1^T{V}_1+\sum _{i=2}^N({\mathit{L}}_i{\widehat{\mathit{∊}}}_i):{\widehat{\mathit{∊}}}_i^T{V}_i.$$ (7)

Substituting Eq.(4) into Eq.(7) yields the following explicit expression for the effective stiffness tensor:

$${\mathit{L}}_e=\sum _{j=1}^N{f}_j{\mathit{Q}}_j^T{\mathit{L}}_j{\mathit{Q}}_j.$$ (8)

Substituting for Q and solving for L_e,

$${\mathit{L}}_e={\left(\sum _{j=1}^{\mathit{N}}{f}_j{\mathit{N}}_j^s{\mathit{L}}_j{\mathit{N}}_j^s\right)}^{-1},$$ (9)

where in the last step we employed the fact that L and N^s are symmetric.

2.3. Specific result for effective elastic stiffness of three-phase composites

For the simple case of two types of inclusions within a matrix,

$$\begin{array}{lll}{\mathit{N}}_1^s& =& {\mathit{T}}_1{(f_1{\mathit{L}}_1{\mathit{T}}_1+f_2{\mathit{L}}_2{\mathit{T}}_2+f_3{\mathit{L}}_3{\mathit{T}}_3)}^{-1}\\ {\mathit{N}}_2^s& =& {\mathit{T}}_2{\mathit{N}}_1^s\\ {\mathit{N}}_3^s& =& {\mathit{T}}_3{\mathit{N}}_1^s\end{array}$$

Hence, the effective stiffness tensor is:

$${\mathit{L}}_e={(f_1{\mathit{N}}_1^s{\mathit{L}}_1{\mathit{N}}_1^s+f_2{\mathit{N}}_2^s{\mathit{L}}_2{\mathit{N}}_2^s+f_3{\mathit{N}}_3^s{\mathit{L}}_3{\mathit{N}}_3^s)}^{-1}$$ (10)

3. Results and discussion

We studied Eqs. 9 and 10 in two ways. First, predictions were compared to available experimental data and to several estimates and bounding techniques for the case of glass inclusions in an epoxy matrix, including inclusion volume fractions beyond the range over which the Mori-Tanaka and Benveniste estimates are deemed reliable. Second, we checked predictions against existing models and predictions for three-phase composites. Thereafter, we simulated the spatial variation of the mechanics of partially mineralized collagen fibers.

3.1. Epoxy with glass inclusions

Analytical predictions compared well to two well known experimental datasets for two-phase composites containing randomly distributed elastic spheres: the data of Smith (1976) (E₁ = 3:01 GPa, ν₁ = 0:394, E₂ = 76:0 GPa, ν₂ = 0:23), and the data of Richard (1975) (E₁ = 1:69 GPa, ν₁ = 0:444, E₂ = 70:3 GPa, ν₂ = 0:21). Predictions were close for effective shear and Young’s moduli for particle volume fractions ranging from 0 to ~ 0:6 (dense) (Figs. 1 and 2). Minor deviations of predictions for effective bulk modulus K and Poisson’s ratio ν were expected due to both datasets relying on uniaxial tests for ν. Predictions lay within three-point bounds at high volume fractions (Figs. 1 and 2). Significantly, this remained the case for the effective Poisson’s ratio and the effective bulk modulus. Like the Benveniste (and hence Kanaun-Jeulin) estimate, the new method lay slightly outside of the three-point bounds at very low volume fractions. The Benveniste estimate follows the Hashin-Shtrikman lower bound and drops increasingly beneath the three-point bounds at larger particle volume fractions (the inverse of this trend occurs for the effective Poisson’s ratio) (cf. (Genin and Birman, 2009)). Like the Mori-Tanaka theory, the Benveniste approach is accurate only for materials containing dilute or low volume fraction distributions of inhomogeneities.

Figure 1.

Estimates and bounds for the effective elastic properties of an isotropic epoxy matrix (E₁ = 3:01 GPa, ν₁ = 0:394) embedded with spherical glass inclusions (E₂ = 76:0 GPa , ν₂ = 0:23), compared with the experimental data of. All moduli are normalized by the corresponding moduli of the matrix.

Figure 2.

Estimates and bounds for the effective elastic properties of an isotropic epoxy matrix (E₁ = 1:69 GPa, ν₁ = 0:444) embedded with spherical glass inclusions (E₂ = 70:3 GPa, ν₂ = 0:21), compared to the experimental data of Richard, 1975. All moduli are normalized by the corresponding moduli of the matrix

The effect of the aspect ratio of inclusions on effective elastic properties was then evaluated for two-phase composites consisting of an epoxy matrix (E₁ = 3:12 GPa, ν₁ = 0:38) embedded with 30% aligned glass (E₂ = 76 GPa, ν₂ = 0:25) ellipsoidal particles of a variable aspect ratio (Fig. 3). As expected, estimates lay within the Hashin-Shtrikman bounds for nearly spherical particles, and within the Voigt-Reuss bounds for both large aspect ratio (continuous fiber-like) and small aspect ratio (disc-like inclusions. Expected trends (e.g., Tucker III and Liang (1999)) were reflected in Fig. 3, including an increase of longitudinal elastic modulus (E_L) and a reduction of transverse modulus (E_T) for increasing aspect ratios of inclusions that were greater than 1. Asymptotes in all moduli were reached at very small inclusion aspect ratios, when the composite behaves as a matrix embedded with discs oriented perpendicular to the longitudinal axis, and at very high inclusion aspect ratios, when the composite behaves as a matrix embedded with aligned fibers. The Kanaun-Jeulin estimate predicts a more compliant material for all inclusion aspect ratios. The current method appears fully adequate for two-phase composites.

Figure 3.

The effect of the aspect ratio of ellipsoidal inclusions (volume fraction f₂ = 0:3) on the effective elastic properties of two-phase composites. Note that the volume fraction of the inclusions is considered to be 30%. E₁ = 3:12 GPa and ν₁ = 0:38 for the epoxy matrix, and E₂ = 76 GPa and ν₂ = 0:25 for the glass particles. All moduli are normalized by the corresponding moduli of the matrix.

Although much discussion of them can be found, experimental characterizations of three-phase composites are not available in the literature. To assess the validity of the proposed method in predicting effective elastic properties of three-phase composites, we instead compared the current approach to bounds and estimates for a model three-phase system that we studied previously (Genin and Birman, 2009). The model system studied had a matrix of epoxy (E₁ = 3:12 GPa, ν₁ = 0:38) that was reinforced by both spherical particles (phase 2) and long fibers (phase 3) made of glass (E₂ = E₃ = 76 GPa, ν₂ = ν₃ = 0:25). Three types of analyses were compared to the current approach, as presented by Genin and Birman (2009) (Fig. 4). The first were “bounds” that serve not as true bounds but rather as guidelines for appropriate trends. These were obtained by using the three-point bounds (cf. Torquato (2001)) for a matrix that contained a volume fraction f₃ of glass fibers; the matrix had properties obtained from the three-point bounds for epoxy reinforced with a volume fraction f₂/(f₁+f₂) of randomly distributed glass spheres. The second were “two-step” estimates obtained using three-point estimates (cf. Torquato (2001)) in the same way (Genin and Birman, 2009). The third was the Kanaun-Jeulin estimate.

Figure 4.

Theoretical predictions and bounds for the effective elastic properties of an epoxy matrix (E₁ = 3:12 GPa, ν₁ = 0:38) containing inclusions of glass (E₂ = 76 GPa, ν₂ = 0:25). Two types of inclusions are considered: infinitely long fibers (volume fraction f₃) and glass spheres (volume fraction f₂, ranging from 0 to the dense packing of 0.63). The upper and lower bounds are calculated using the three-point bounds discussed by Genin and Birman, 2009. Moduli are normalized by those of epoxy.

For the range of particle and fiber volume fractions examined, the Kanaun-Jeulin estimate lay just outside of the “bounds,” with smaller deviations at lower volume fractions of inclusions (Fig. 4). This is expected because the Kanaun-Jeulin approach is acceptable only for relatively low particle volume fractions. In all cases, the properties evaluated by the proposed approach were within the bounds, except for the longitudinal elastic modulus (E^L), which was slightly outside the bounds for larger particle volume fractions. Note that, except for estimates obtained for Poisson’s ratio, the “two-step” estimate and the current method were close.

3.2. Partially mineralized collagen fibers

The current approach was then used to estimate the stiffening of collagen fibers by mineral. The accommodation of bioapatite mineral within fibers and fibrils follows a sequence that is debated. A sequence supported by our own work involves mineralization inside of collagen fibrils (“intrafibrillar”) preceding mineralization outside of collagen fibrils (“extrafibrillar”) (Alexander et al., 2012; Schwartz et al., 2012). In this interpretation, bioapatite mineral accumulates inside collagen fibrils at tissue-level volume fractions 0% ≤ ϕ_m ≤ 21%, displacing water, and then accumulates outside of fibrils, displacing extra-fibrillar matrix proteins (“EFM”). The stochastic finite element estimates of Liu et al. (2014) for the stiffening of fibers associated with this mineralization sequence (symbols in Fig. 5) showed relatively minor stiffening for intrafibrillar mineralization (ϕ_m ≤ 21%) followed by much higher stiffening beyond a percolation threshold for extrafibrillar mineralization. Due to the long, slender shapes of collagen fibrils, the percolation threshold is higher for the longitudinal modulus of a fiber (ϕ_m ≈ 30%) than for its transverse modulus (ϕ_m ≈ 21%). Note that in Fig. 5, the bioapatite volume fraction was plotted for consistency with Liu et al. (2014) as a tissue level volume fraction, ϕ_m; the fiber-level volume fraction is 25% higher than ϕ_m. The volume fraction therefore ranged from 0 vol% (typical for tendon) to 41 vol% (typical for bone) (Hellmich et al., 2008; Hellmich and Ulm, 2002; Hamed et al., 2010).

Figure 5.

Linear estimation of longitudinal (a) and transverse (b) elastic moduli of partially mineralized collagen fibers (solid lines), plotted against stochastic finite element results of Liu et al. (2014) (gray circles). The darker shaded regions (orange in the online version) correspond to Hashin-Shtrikman type bounds that account for the continuous, fibrous nature of collagen fibrils, and the lighter shaded regions (yellow in the online version) correspond to Hashin-Shtrikman bounds that do not account for this (cf. Liu et al. (2014)).

The modeling of this stiffening was achieved in two stages. In the first stage, mineral accumulated on the inside of collagen fibrils, replacing water. The composite considered in this stage was a three-phase composite consisting of cross-linked collagen fibrils (treated as a transversely isotropic solid), bioapatite mineral, and water. Random spheres of bioapatite mineral replaced random spheres of water within the fibrils in this stage. Each phase was assigned moduli consistent with those used by Liu et al. (2014). The cross-linked collagen fibrils, transversely isotropic and rotationally symmetric about the x₃-axis, were assigned Young’s moduli E₃ = 814 MPa and E₁ = E₂ = 8.27 MPa, Poisson’s ratios ν₁₃ = 0.1 and ν₁₂ = 0:3, and shear modulus G₁₃ = 6.53 MPa. Bioapatite mineral was treated as isotropic and much stiffer than the cross-linked collagen fibrils (E=110 GPa, ν=0.27), and water was treated as isotropic, nearly incompressible (ν=0.4999), and of negligible shear resistance. The predictions of the current approach followed the stochastic finite element predictions of Liu et al. (2014) closely in this regime, and was near a Hashin-Shtrikman lower bound accounting for the fibrillar nature of collagen fibrils, as described by Liu et al. (2014) (Fig. 5).

In the second phase, bioapatite mineral accrued on the outside of fibrils, and the nature of the relatively compliant, anisotropic EFM had to be accounted for explicitly. For extrafibrillar mineralization, we note debate in the literature about the shape of mineral inclusions within bone and partially mineralized tissues, and emphasize that uncertainty exists with both the current predictions and with the finite element predictions to which they were compared. We focussed on the following single example as a means of testing the range of validity of the current approach for multi-phase composites. The three-phase composite considered consisted of mineralized fibrils (infinitely long, with moduli calculated as above for ϕ_m = 0:21), bioapatite mineral (moduli as above), and EFM (isotropic with E = 280 kPa and ν = 0:3). The example chosen was one in which mineral accumulates on the outside of fibrils (within and upon fibers) through plate-like segments that align transverse to the collagen fiber axes (Alexander et al., 2012). Plate-like segments were modeled as randomly distributed mineral platelets oriented perpendicular to the collagen fibril axes. Results were consistent with the finite element simulations of Liu et al. (2014)) at all volume fractions for the transverse moduli, and up to ϕ_m ≈ 27% for the longitudinal modulus. However, the longitudinal modulus predictions highlight a limitation of the current approach, namely that it does not account for the spatial disposition of the different phases and can miss a percolation threshold. Nevertheless, the current approach was effective at sufficiently high volume fraction to represent the full range of mineralization: higher degrees of mineralization could be modeled by beginning with an extrafibrillar space that was fully dense with bioapatite mineral, then replacing mineral with EFM. This trend, illustrated in the region from 27 vol% to 41 vol% in Fig. 5, matched finite element data very well, and intersected the previous predictions at 27 vol% to provide estimates of mechanical behavior over the full range of mineralization. This is visible as a change in slope in the plots for both longitudinal and transverse moduli.

4. Conclusions

A method was proposed to predict the effective elastic moduli of composites containing multiple types of anisotropic ellipsoidal inclusions. The method employed effective rather than average strains within phases to ensure that the energy stored within phases summed to that stored in the composite. The special cases of two- and three-phase composites were predicted. The method reproduced available experimental data with good accuracy, especially at the higher volume fractions at which the Kanaun-Juelin estimate is expected to perform poorly. The proposed method produced physically consistent results and showed good predictive accuracy. Accordingly, the method is a candidate for the characterization of composites with multiple types of anisotropic inclusions, even if these inclusions have moderate volume fractions and a variety of aspect ratios.

Appendix A. Eshelby tensors for various ellipsoidal shapes

For a general ellipsoid, components S_ijkl are expressed in terms of (incomplete) elliptic integrals (Eshelby, 1957). For a spheroid, S_ijkl are elementary functions of the spheroid’s aspect ratio.

Appendix A.1. General ellipsoid

Components S_ijkl are as follows:

$$S_{1111}=\frac{3}{8\pi (1-\nu )}{a_1}^2{I}_{11}+\frac{1-2\nu }{8\pi (1-\nu )}{I}_1$$ (A.1)

$$S_{1122}=\frac{1}{8\pi (1-\nu )}{a_2}^2{I}_{12}-\frac{1-2\nu }{8\pi (1-\nu )}{I}_1$$ (A.2)

$$S_{1133}=\frac{1}{8\pi (1-\nu )}{a_3}^2{I}_{13}-\frac{1-2\nu }{8\pi (1-\nu )}{I}_1$$ (A.3)

$$S_{1212}=\frac{{a_1}^2+{a_2}^2}{16\pi (1-\nu )}{I}_{12}+\frac{1-2\nu }{16\pi (1-\nu )}(I_1+I_2)$$ (A.4)

where a₁, a₂ and a₃ are general ellipsoid semiaxes. Other S_ijkl are found from symmetry relations S_ijkl = S_jikl = S_ijlk and by the cyclic permutation of indices (1,2,3) in quantities S_ijkl, a_i, I_i and I_ij. Those components that cannot be obtained by the cyclic permutation of the above are zeros. Assuming that a₁ ≥ a₂ ≥ a₃, we have:

$$\begin{array}{lll}{I}_1& =& \frac{4\pi a_1{a}_2{a}_3}{({a_1}^2-{a_2}^2)\sqrt{{a_1}^2-{a_3}^2}}[F(\theta ,k)-E(\theta ,k)]\\ I_3& =& \frac{4\pi a_1{a}_2{a}_3}{({a_1}^2-{a_2}^2)\sqrt{{a_1}^2-{a_3}^2}}[\frac{a_2\sqrt{{a_1}^2-{a_3}^2}}{a_1{a}_3}-E(\theta ,k)]\\ I_2& =& 4\pi -I_1-I_3\end{array}$$ (A.5)

Functions F(θ, k) and E(θ, k) are the incomplete elliptical integrals:

$$\begin{array}{lll}F(\theta ,k)& =& \underset{0}{\overset{\theta }{\mathit{\int }}}\frac{dw}{\sqrt{1-k^2{\sin }^{2}w}},\\ E(\theta ,k)& =& \underset{0}{\overset{\theta }{\mathit{\int }}}\sqrt{1-k^2{\sin }^{2}wdw}\end{array}$$ (A.6)

and where it is denoted:

$$\begin{array}{lll}\theta & =& \arcsin \sqrt{1-{(\frac{a_3}{a_1})}^2},\\ k& =& \sqrt{\frac{({a_1}^2-{a_2}^2)}{({a_1}^2-{a_3}^2)}}\end{array}$$ (A.7)

The following equations (and the equations obtained by their cyclic permutations) express all I_ij in terms of I₁, I₂, I₃:

$$3I_{11}+I_{12}+I_{13}=\frac{4\pi }{{a_1}^2}$$ (A.8)

$$3{a_1}^2{I}_{11}+{a_2}^2{I}_{12}+{a_3}^2{I}_{13}=3I_1$$ (A.9)

$$I_{12}=\frac{I_2-I_1}{{a_1}^2-{a_2}^2}$$ (A.10)

Appendix A.2. Sphere

In this case (a₁ = a₂ = a₃ ≡ a) we have

$$\begin{array}{lll}{I}_1& =& I_2=I_3=\frac{4\varphi }{3}\\ I_{11}& =& I_{22}=I_{33}=I_{12}=I_{23}=I_{31}=\frac{4\pi }{5({a_1}^2)}\end{array}$$ (A.11)

$$S_{1111}=S_{2222}=S_{3333}=\frac{7-5\nu }{15(1-\nu )}$$ (A.12)

$$S_{1212}=S_{2323}=S_{3131}=\frac{4-5\nu }{15(1-\nu )}$$ (A.13)

$$S_{1122}=S_{2233}=S_{3311}=S_{1133}=S_{2211}=S_{3322}=\frac{5\nu -1}{15(1-\nu )}$$ (A.14)

Appendix A.3. Needle

In this case (a₁ = a₂ = a ≪ a₃), the terms a/a₃ are negligible, so that, to within higher order terms, S_ijkl do not depend on ratio a/a₃ and coincide with their limiting values at a/a₃→0:

$$S_{1111}=S_{2222}=\frac{5-4\nu }{8(1-\nu )}$$ (A.15)

$$S_{1122}=S_{2211}=\frac{4\nu -1}{8(1-\nu )}$$ (A.16)

$$S_{1133}=S_{2233}=\frac{\nu }{2(1-\nu )}$$ (A.17)

$$S_{1313}=S_{2323}=\frac{1}{4}$$ (A.18)

$$S_{1212}=\frac{3-4\nu }{8(1-\nu )}$$ (A.19)

$$S_{3333}=S_{3311}=S_{3322}=0$$ (A.20)