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Spectral Analysis of Inhomogeneities Shows that the Elastic Stiffness of Random Composites Decreases with Increasing Heterogeneity

Investigation of inhomogeneities has wide applications in different areas of mechanics including the study of composite materials. Here, we analytically study an arbitrarily-shaped isotropic inhomogeneity embedded in a finite-sized heterogeneous medium. By modal decomposition of the influence of the inhomogeneity on the deformation of the composite, a relation is presented that determines the variation of effective elastic stiffness caused by the presence of the inhomogeneity. This relation indicates that the effective elastic stiffness of a composite is always a concave function of the properties of the inhomogeneity, embedded inside the composite. Therefore, as the heterogeneity of elastic random composites increases, the rate of increase in effective stiffness caused by the stiffer constituents is smaller than the rate of its decrease due to the softer constitutions. So, weakly heterogeneous random composites become softer and less conductive with increasing heterogeneity at the same mean of constituent properties. We numerically evaluated the effective properties of about ten thousand composites to empirically support these results and extend them to conductive materials. This article presents a generalization of our recent theoretical study on the influence of the stiffness of a single fiber on the elastic stiffness of a network of fibers to arbitrarily-shaped inhomogeneities and different transport phenomena.


Introduction
The study of inhomogeneities has wide applications in mechanics. For instance, in determination of the effects of healthy and cancerous cells on the deformation of soft tissues (Schwarz and Safran, 2013;Mills et al., 2014;Shin et al., 2016) as well as predicting the properties of rubber in the presence of filler particles (Smallwood, 1944;Lopez-Pamies et al., 2013). Such composite materials are ubiquitously found in nature, whereas they can also be engineered to produce materials with desired properties. Stiff, lightweight composites are used as structural materials in cars and airplanes (Matthews and Rawlings, 1999); as for composites with specific constituent geometries and composites with specific distributions of constituents (Milton, 2002;Torquato, 1991). The estimates for random composites can be improved by accounting for the statistical moments of the spatial distribution of constituents, represented by multipoint correlation functions (Torquato, 2005(Torquato, , 1991. Similar approaches have been taken in studies of weakly heterogeneous composites where the differences between the constituent properties are small. Such methods have been utilized to study the bulk modulus of weakly heterogeneous random composites (Molyneux, J and Beran, M, 1965), evaluate the effective thermal conductivity of polycrystals using weak contrast expansions (Avellaneda and Bruno, 1990), and provide solutions for the effective stiffness and toughness of biomimetic random composites (Dimas et al., 2015a(Dimas et al., , 2015b. Similar approaches have been proposed to evaluate the effective nonlinear properties of composites (Castañeda, 1991).
In this paper, we first present the analytical spectral decomposition of the influence of an arbitrarily-shaped inhomogeneity on the effective elastic stiffness of a finite-sized elastic medium. The modal relation is illustrated and validated using the example of a spherical inhomogeneity embedded in a volumetrically expanding composite. Next, we used the analytical inhomogeneity results together with a weak contrast series expansion to study the effective properties of random elastic and conductive composites. Finally, numerical calculations are presented that empirically support the theoretical results, characterize the displacement of the random composites and its variability, and extend the presented results to the study of the thermal conductivity of composites.

Theory
This section describes an exact relation for the variation of the effective elastic stiffness, eff , of a finite-sized heterogeneous material caused by the addition of an arbitrarily-shaped isotropic inhomogeneity using the modal decomposition of the influence of the inhomogeneity. The inhomogeneity relation is then validated and illustrated using the example case of a spherical inhomogeneity embedded in a volumetrically expanding composite. The modal relation is used together with a weak contrast expansion to study the change in effective properties of random composites at various degrees of the microstructural heterogeneity of the composite.

Uniaxial stretch tests of a heterogeneous sample including an isotropic inhomogeneity
Uniaxial tests are considered to quantify the elastic stiffness of heterogeneous samples. A cubic heterogeneous elastic sample of dimensions 0 and volume = 0 3 is considered. The effective elastic stiffness is measured by applying a small constant displacement 0 in the direction 1 to one of the cube face, R , while holding the opposite face fixed in 1 . Few other degrees of freedom are constrained to eliminate rigid body motions. The application of the boundary conditions produces a distribution of boundary traction 3 = { 1 , 2 , 3 }. The 1 component of the traction . 1 = 1 can be integrated over R to evaluate the total boundary reaction 4 . Here, 1 = {1,0,0} denotes the unit vector in the direction 1 . The effective small-strain elastic stiffness resisting uniaxial deformation in the 1 direction, can be evaluated as eff = ∫ . 1 R /( 0 0 ) = ∫ 0 . 1 R /( 0 2 ), where 0 = 0 / 0 is the applied normal strain in the 1 direction, and 0 is the area of R at rest. We denote this state as the reference state, state 0, with the boundary traction (0) and effective elastic stiffness eff (0) .
In this state, the deformation of the sample is described by the displacement field, (0) (Fig. 1a). Throughout the text, parenthesized superscripts refer to the state at which the deformation 0 is applied. State 0 ( Fig. 1a)

Spectral decomposition of the influence of an isotropic inhomogeneity on effective elastic stiffness
In this section, the influence of an inhomogeneity on effective elastic stiffness is evaluated using a modal decomposition and superposition operations. We begin by considering the case where an inhomogeneity with the elastic modulus inh (1) is added to the heterogeneous cubic sample (Fig. 1b). If the boundary conditions described in state 0 ( Fig. 1a) are applied to this sample, the boundary traction changes to (1) . The effective elastic stiffness and the displacement field in this case are denoted by eff (1) and (1) , respectively. We refer to this state of the system as state 1. If we consider the free body diagram of the inhomogeneity in state 1, its boundary is deformed by the displacement (1) , ̅ 2 (1) , ̅ 3 (1) } and the traction force (1) = { 1 (1) , 2 (1) , 3 (1) }. As commonly practiced in small-strain elasticity, we replace the inhomogeneity by the forces that it applies to the surrounding medium, − (1) . Using the reciprocal theorem (Love, 2011) between the states 0 and 1 yields (1) Therefore, the change in the effective elastic stiffness can be expressed as (2) In the next section, we use the modal decomposition of the force and displacement in the inhomogeneity to rewrite Eq. (2) in terms of inh .

Spectral decomposition of the traction and displacement at the boundary of the inhomogeneity
Spectral decompositions have been thoroughly studied in problems of elasticity (Ciarlet, 2013;Lebedev and Vorovich, 2003) and are commonly used in studying the normal modes of elastic vibrations (Thomson, 1996). While, eigenvectors form an orthonormal basis in the spectral form of stiffness matrices for frames and discrete structures (Bathe, 1982), eigenfunctions are present in the spectral analysis of continua. In this section, we use eigenfunctions to analyze the spectral decomposition of the displacement and traction at the boundary of an isotropic inhomogeneity in the absence of body forces (Fig. 1c).
We begin by considering the tractions and boundary displacement of an isotropic inhomogeneity with an elastic modulus inh = 1 and Poisson's ratio .
We denote the traction of the inhomogeneity with unit elastic modulus by ′.
Following the work of Eshelby (Eshelby, 1959), the Green's function can be used to find displacement at the boundary of the inhomogeneity due to the boundary traction as ̅ = ∫ ′ Γ Γ . Here, the functions ′ and ̅ are elements of a Hilbert space. The integral equation relating ′ and ̅ can be represented by ̅ = ′ where is a linear operator associated with the inner product ( ′ (1) , ′ (2) ) = ∫ ′ (1) . ′ (2) Γ Γ (Reed and Simon, 1981). is positive because is twice the external work expended to deform the inhomogeneity by ̅, which is stored as elastic strain energy and is non-negative. The reciprocal theorem of elasticity shows that ( ′ (1) , ′ (2) ) = ( ′ (1) , ′ (2) ), and therefore is also self-adjoint. Furthermore, over the compact domain Ω the integral representation of indicates that is compact (Reed and Simon, 1981). Therefore, the spectral theorem of compact self-adjoint operators can be used to decompose the boundary traction and displacements as ′ = ∑ =1 and ̅ = ∑ =1 (Reed and Simon, 1981). Here, and are eigenvalues and eigenfunctions, = ( ′, ), and is the number of eigenfunctions, which is not necessarily finite in this case. The basis set, functions, can be chosen such that the eigenfunctions are orthonormal, i.e.
Next, we consider the change in the orthonormal decomposition when the inhomogeneity's elastic modulus is changed to inh at the same Poisson's ratio.
We recall that boundary traction is related to stress as = , where is the stress tensor, and denotes the unit outward normal at the boundary. For an isotropic material, Hook's law can be used to express stress as This relation also shows that These equations indicate that any traction and boundary displacement ̅ for an inhomogeneity with modulus inh can be expressed as = inh ′ and ̅, where ′ and ̅ correspond to an inhomogeneity with inh = 1 and the same . Therefore, using the spectral decomposition of ′ and ̅, for an inhomogeneity with modulus inh , and ̅ can be expressed as = inh ′ = inh ∑ =1 and ̅ = ∑ =1 .

Influence of a single mode on effective properties
In this Section, we use Eq.
(2) to find eff (1) at state 1. If we consider that only the reaction to the traction related to the 'th eigenfunction, , is present in the stretch tests, then we can rewrite Eq. (1) as where (1) is the traction at the boundary R due to the displacement 0 and traction − (1) . The orthonormal property of the eigenmodes, , yields that Now, we consider another state where the inhomogeneity has a modulus inh (2) at the same Poisson's ratio, and the reaction to the traction from the 'th mode is . We denote this state as state 2. If the reciprocal theorem is applied between states 1 and 2 when we apply the traction from the 'th mode, , we obtain By using the orthogonality of the modes, we can rewrite Eq. (5) as If the inhomogeneity is not deformed in the 'th mode in state 0, increasing its stiffness does not alter the effective properties. However, if the element deforms, combining Eqs. (4 and 6) and the counterpart of Eq. (4) corresponding to states 2 and 0, gives Equation (7) indicates that the quantity We denote this invariant by . We note that = 0 when the deformation of the inhomogeneity is prescribed by the boundary conditions and will not change by the application of (1) . It can be noted that is invariant of inh ( ) , whereas it depends on the prescribed boundary conditions. For a nonzero , by replacing (1) in Eq. (4) and using the definition of from Eq. (7) we obtain Equations (3 and 8) indicate that ∫ (1) . ̅ (0) Γ Γ = ( (0) ) 2 /( + 1/ inh (1) ).

A modal relation for effective elastic stiffness
The integral ∫ (1) .
(2) can be rewritten as where eff (0) and eff (1) are the effective stiffness in states 0 and 1. Equation (9) provides a relation between the change of the effective elastic stiffness of the composite and the variation of elastic modulus of an inhomogeneity. We note that is necessarily non-negative since otherwise, it is possible to find a variation of the modulus inh (1) = −1/ , that leads to the divergence of eff (1) , which is physically impossible.
Equation (9) indicates that effective elastic modulus is, in general, a concave function of the properties of the inhomogeneity: The exceptional case where equality holds is described in the Discussion Section.
The inhomogeneity relation of Eq. (9) is different from most previous results by the fact that we did not assume a specific shape for the inhomogeneity. Many previous exact theories hold for inhomogeneities with specific geometric shapes, for example, spheres, ellipsoids, or polygons.
Furthermore, we did not assume a homogeneous medium for the sample containing the inhomogeneity. Our theoretical derivations hold even if the inhomogeneity is part of a finite-sized composite consisting of phases with highly different properties. This assumption distinguishes our derivations from the results obtained using the T-matrix approaches (Gubernatis and Krumhansl, 1975) which are based on the assumption of small fluctuations of the strain field.
Additionally, we assumed a finite-sized domain for the composite. Finally, the condition of the continuity of the medium is not required for our analysis. The reciprocal theorem for frame structures (Maxwell, 1864) can be used to derive Eq. (9) in the case of discrete structures such as cellular solids, networks of fibers (Ban et al., 2016), and metamaterials. The spectral theorems linear algebra and the eigendecomposition of compliance matrices can be used in that case. Special cases of Eq. (9) for inhomogeneities with one and two modes of deformation such as elastic rods and Euler-Bernoulli beams were explored in networks of beams (Ban et al., 2016).

The illustrative example of a spherical inhomogeneity in a volumetrically expanding composite
To validate the modal approach and apply it to a specific case, we evaluated the change in the effective bulk modulus of a spherical composite that includes a spherical inhomogeneity. The inhomogeneity and the surrounding matrix have elastic moduli inh and m and Poisson's ratio . The two spheres have radii i and o and are centered at the same point (Fig. 2).  A planar representation of the traction force is shown for simplicity.
Similar to the case of the arbitrarily-shaped inhomogeneity, we consider the two states 0 and 1 where the inhomogeneity is absent (Fig 2a) and where an inhomogeneity of elastic modulus inh (1) is present, respectively (Fig. 2b). The freebody diagram of the inhomogeneity shows that at state 1 it will have uniform surface traction and displacement of (1) and ̅ (1) , respectively (Fig. 2c). Without a loss of generality, we assume that the composite is fixed at the point . We and ̅ (1) = ̅ 1 (1) . In this case, the modal decomposition, presented in Section 2.2.1, can be employed to express (1) in terms of an eigenfunction, (1) = inh (1) 1 (1) 1 , where 1 is the only eigenfunction. The normal property of the modes requires that the inner product ( 1 , 1 ) = 1. Therefore, 1 = (1/2 i √ ) . This eigenfunction represents the volumetric expansion of the inhomogeneity, and the factor (1/2 i √ ) ensures that the inner product equals 1. The eigendecomposition also indicates that ̅ (1) = 1 1 (1) 1 , where 1 is the only present eigenvalue.
Equation (8) can be in this case rewritten as where 1 (0) corresponds to the deformation of the inhomogeneity in state 0.
1 (0) and the invariant 1 can be evaluated for this problem using the solution for a pressurized hollow sphere (Bower, 2009). In general, the solution for the displacement in the radial direction takes the form ( ) = + / 2 where and depend on the geometry, material properties, and boundary conditions. We begin by 1 (0) in the absence of internal pressure, or traction in the inner surface of the hollow sphere. By prescribing the displacement 0 at the outer boundary of the composite and considering the constitutive equation, we evaluate the constants and . We then find the displacement at the inner surface of the sphere ̅ 1 (0) , and evaluate 1 (0) = 2 ̅ 1 (0) √ /(1 − 2 ) as Similarly, we prescribed a uniform radial traction force of magnitude 1 (1) at the interior surface of the hollow sphere and evaluated the displacement. We then used the expression 1 = ( 1 (1) − 1 (0) )/( 1 1 (1) ) and evaluated the invariant 1 as Equations (11-13) and the relation between the boundary tractions and eff can be used to evaluate the change in eff by the addition of the inhomogeneity as eff (1) = eff (0) + 1 0 2 Equation (14) demonstrates that eff is a concave function of inh (1) . Equation (14) was evaluated for inhomogeneities of different stiffnesses and radii (Fig. 3). The numerical values used to plot Eq. (14) are described in Section 3.2. dilute approximation, using Eshelby's approach (Christensen, 2005), and the presented spectral method (Eq. 14) are plotted for inhomogeneity over composite radii of (a) 0.2, (b) 0.5, and (c) 0.9.
The values of the effective bulk moduli are normalized by the bulk modulus of the corresponding homogeneous solid, where inh = m .

Application to composites
We used Eq. (10) along with a series expansion to show that random composites become softer and less thermally conductive with increasing heterogeneity. In our analysis composites were made of a large number of inhomogeneities.

The weak contrast series expansion
We studied composites made by the coalescence of S homogenous constituents. We denote the properties (elastic modulus or thermal conductivity) of the 'th constituent by . The values can be considered as components of a vector, , which describes the microstructural properties of the composite. We take the case of a homogenous material where all 's are the same, as a reference. We refer to the effective properties and the properties of the constituents of the homogeneous material by eff (0) and 0 , respectively. In a generic composite, we express the effective properties and microstructural properties by eff and 1 , respectively. We can express the effective properties of a composite with respect to the homogeneous sample using a series expansion where 1 = 1 − 0 . Similar weak contrast series expansions have been used in studying composite systems (Milton, 2002) and random networks of resistors (Luck, 1991). properties over an ensemble of composites with the same geometry, but with different realizations of microstructural properties, 1 . Equation (10)(10) indicates that ∇ 2 eff is negative in elastic composites. Therefore, 〈 eff 〉 < eff (0) , and the effective elastic stiffness of the composites decrease by increasing heterogeneity. Intuitively, the concave characteristic of the effective stiffness indicates that if the composite is comprised of two types of materials, with increasing the variance of the microstructural properties, the rate of gain in effective elastic stiffness from the stiffer constituents is smaller than the rate of its loss from the softer constituents.

Comparison with numerical results
This section presents a comparison of the theoretical results with numerical calculations. First, the numerical methods are presented. Next, the illustrative example of the effective bulk modulus of a composite containing an individual spherical inhomogeneity is described. Finally, two random composites are investigated: composites made by the coalescence of cubic blocks, and composites consisting of spherical inclusions in a homogenous matrix.

Numerical calculation method
The composite problems can be expressed as partial differential equations with spatially varying constants. The partial differential equations of elasticity and Fourier heat conduction were solved using the Abaqus (Hibbsett et al., 1998) and COMSOL Multiphysics software packages (COMSOL, 2014). Geometric models and unstructured meshes were generated using Abaqus and COMSOL. The problem domains were discretized into tetrahedral and hexahedral elements.
Small enough elements were used to eliminate the mesh sensitivity of the reported results. Small-strain elasticity and steady state Fourier heat conduction were modeled by prescribing small displacement and temperature differences at the boundaries of the samples. The resulting forces and heat fluxes were integrated at the boundaries of the composites and were used for the evaluation of the effective properties. Implicit solution methods were used, and geometric nonlinearities were not considered.

The illustrative example of a spherical inhomogeneity in a volumetrically expanding composite
Numerical calculations were performed to verify the modal relation for the bulk modulus of a composite containing a spherical inhomogeneity. The effective bulk modulus increased with increasing the stiffness of the inhomogeneity and exhibited a concave curve (Fig. 3). The same trend was observed when testing inhomogeneities of various radii. As expected, the change in effective bulk modulus was larger in composites with larger inhomogeneities.
These results are compared with the theoretical results in the Discussion Section.

Random composites made by the coalescence of cubic blocks
To support the theoretical results on elastic random composites (Section 2.4) and  4a). In addition, the variance of effective properties increases linearly with the variance of microstructural properties: eff 2~2 (Fig. 4b).  We also tested the bulk modulus of the random composites made by the coalescence of cubic blocks (Fig. 5). The mean bulk modulus of the composites decreased with increasing microstructural heterogeneity, and the amount of decrease was proportional to the variance of the microstructural properties (Fig. 5). The resulting mean effective bulk moduli resided between the upper and lower Voigt-Reuss bounds (Hill, 1963;Milton, 2002) and also the upper and lower Hashin-Shtrikman bounds (Hashin and Shtrikman, 1963) (Fig. 5). Moreover, we tested random composites made from different numbers of constituents. The numerical results indicated that increasing the number constituents in the composite, leaves the decrease of effective properties unchanged (Fig. 4a). However, the variance of the effective properties is inversely related to the number of microstructural constituents, S (Fig. 6a). This result is in agreement with the previous observations in elastic composites (Jeulin et al., 2004) and is a consequence of the central limit theorem (Dekking et al., 2007). Since the variance of effective properties becomes smaller as composites become larger, we conclude that in composites made of a large number of constituents, the mean of the effective properties is representative of the effective properties of all of the random composites in the ensemble.
In the absence of heterogeneity, strain is spatially uniform over the composite. In contrast, in random composites, the distribution of strain is nonuniform, and the variance of the strain field grows with increasing the heterogeneity of the composites (Fig. 6b). We calculated the variance of the strain field using the metric = 〈 x ′2 〉 V / 0 2 where x ′ = − uniform is the nonuniform component of displacement in the direction of the uniaxially applied macroscopic strain. and uniform are the total and uniform components of the displacement. Here, 〈 x ′2 〉 V denotes the mean of x ′2 over the composite's volume.
In conclusion, this result indicates that the growth of heterogeneity in the displacement field with microstructural heterogeneity agrees with the reported softening effect marking the departure of effective properties from the Voigt bound as heterogeneity increases.

Random composites with spherical inclusions of random stiffnesses and thermal conductivities
We also tested the effective elastic stiffness and thermal conductivity of random  (Fig. 7).
Similarly, we tested elastic composites with inclusions that in total occupied 1% and 10% of the sample volume. In the case with 10% volume fraction, / , we tested composites with different numbers and sizes of individual inclusions. The first case had a smaller number of large inclusions, and the second case had more inclusions of a smaller size. In all cases, effective stiffness decreased with increasing the variance of the elastic moduli of the inclusions. Furthermore, at the small and moderate variances of the elastic modulus of the inclusions, the decrease in effective stiffness was proportional to the variance of the moduli of the inclusions (Fig. 7). The effective stiffness of the composites with a higher volume fraction of inclusions decreased more than the composites with a smaller volume fraction of inclusions. At the same total volume fraction, however, the relative decrease of effective stiffness was independent of the number and size of the individual inclusions. Taken together these tests demonstrate that the decrease in effective properties is also present in random composites with spherical inclusions of random properties.

Discussion
To illustrate the modal approach and validate it, we evaluated the change in the effective bulk modulus of a composite containing an individual spherical inhomogeneity. The results showed a close agreement between the numerical calculations and the modal approach (Fig. 3). We also compared the result with a dilute approximation of effective bulk modulus based on the solution for spherical inhomogeneities (Fig. 3, red dashed curves), derived using Eshelby's approach (Christensen, 2005). As expected the dilute approximation agrees with the effective bulk moduli of the composites with small volume fractions of inhomogeneities. In the systems where the inhomogeneity radius is 90% of the composite radius, however, the dilute approximation predicts effective bulk moduli that are smaller than the bulk moduli obtained using numerical calculations and the modal relation (Fig. 3). The difference between the results increases with increasing the difference between the properties of the inhomogeneity and the surrounding matrix. In contrast, the numerical calculations and the modal relation closely agree even in the tests performed at a large volume fraction of the inhomogeneity (Fig. 3). This example demonstrates the validity of the modal derivation in finite-sized composites.
The studied composites have constituents whose properties are independently sampled from identical statistical distributions. Our results are, therefore, expected to apply to composites without long-range correlations of microstructural properties. It is expected that if long-range correlations are introduced to bias the sampling of the microstructural properties, the effective elastic stiffness still decreases with increasing heterogeneity. The averaging calculations, however, will no longer be valid, and we expect that the amount of decrease in stiffness will no longer be proportional to the variance of the microstructural properties. Moreover, in cases where a gradient of material properties is present, we expect a decrease of effective stiffness with increasing heterogeneity, at the same mean of properties, whereas we do not expect the amount of decrease to be proportional to the variance of the microstructural properties. Future works may extend the present results to heterogeneous media with spatially correlated microstructural properties.
The modal relation shows that the effective stiffness is a concave function of the elastic modulus of an inhomogeneity inside a composite (Eq. (10)). The derivation also points to an exceptional case where the second derivative of effective stiffness with respect to the elastic modulus of the inhomogeneity vanishes. A system of elastic rods (or Hookean springs) connected in parallel, is an example of this exceptional case, where the deformation of the inhomogeneity is prescribed by the boundary conditions, and all 's vanish. Therefore, 2 eff / inh 2 = 0, and the effective stiffness of the system does not change with increasing the variance of the elastic modulus of the rods, at the same mean of constituent properties. In this exceptional system, after external displacement is prescribed to evaluate the effective properties, the individual constituents cannot be further deformed by the application of external forces that correspond to the modes that comprise the deformation of the constituent.
The developed modal approach presents insights into the mechanics of inhomogeneities and random composites. Its extension to different classes of composites, however, requires the determination of the invariants for specific composite geometries. The parameters are invariant of the inhomogeneity's modulus, whereas they depend on the shape of the inhomogeneity, the response function of the surrounding medium, and the prescribed boundary conditions.
Further developments of the modal approach may lead to novel solutions for inhomogeneity problems and composites.

Conclusions
This paper presented a modal analysis of the influence of an inhomogeneity on the effective elastic stiffness of a composite. The analysis showed that effective stiffness is always a concave function of the elastic modulus of an inhomogeneity inside the composite. The concave property indicates that weakly heterogenous random composites become softer with increasing microstructural heterogeneity; the loss of the stiffness by the softer constituents is larger than the gain from the stiffer constituents. The modal relation was illustrated and validated in the example case of a spherical inhomogeneity in a volumetrically expanding composite. The results on random composites were validated using numerical calculations and were compared with the Voigt-Reuss and Hashin-Shtrikman bounds. Moreover, the numerical results showed that random composites including spherical inclusions of random properties become softer with increasing the variance of the elastic moduli of the inhomogeneities. Finally, the results were numerically extended to thermally conductive composites, indicating a decrease in effective thermal conductivity with increasing microstructural heterogeneity.