## Abstract

The far-field methodology developed by Maxwell, when estimating the effective electrical conductivity of isotropic particulate composites, is used to estimate effective thermoelastic properties of multi-phase isotropic composites. In particular, Maxwell's methodology applied to the analogous thermal conduction problem is described, extending the approach to multi-phase spherical particles having different sizes and properties. The methodology is also used to estimate the effective bulk and shear moduli, and the thermal expansion coefficients, of multi-phase isotropic particulate composites. Results correspond with expressions derived in the literature, and coincide with, or lie between, variational bounds for all volume fractions. These characteristics, relating to isotropic effective properties, indicate that results obtained using the methodology, are not necessarily restricted to low volume fractions, as originally suggested by Maxwell. It is concluded that Maxwell's methodology is a unifying optimum technique to estimate the properties of multi-phase isotropic particulate composites, because it provides closed form estimates that are fully consistent with other methods, without imposing restrictions, except that the particles must be spherical (but can have a range of size and properties) and the resultant effective properties must be isotropic.

## 1. Introduction

Estimating the effective thermoelastic and conduction properties of particulate composites is a very important engineering requirement that has been studied for many years. Maxwell (1873) provided an ingenious method of estimating the effective electrical conductivity of a particle cluster embedded in an infinite medium by considering the effect of the cluster on the far-field when the system is subject to a uniform electrical field. Maxwell asserted that the sizes and distribution of the particles must be such that particle interaction effects may be neglected, and he infers that his result will be valid only for small volume fractions of reinforcing particles (see also Maxwell (1873) for an analysis of the conductivity of a special type of laminate). Landauer (1978) has given an excellent account of the historical context of Maxwell's approach, which is relevant also to thermal conductivity, permittivity and magnetic permeability, (Hashin & Shtrikman 1962; Hashin 1983; Milton 2002). In the dielectrics field (Landauer 1978; Taya 2005), the result for effective permittivity that is analogous to Maxwell's result for electrical conductivity is known as the ‘Maxwell–Garnett mixing formula’, and it has a microscopic analogue that is known as the ‘Clausius–Mossotti’ (or ‘Lorentz–Lorenz’) formula, which has been related to effective elastic property estimation by Felderhof & Iske (1992) and Cohen & Bergman (2003*a*,*b*).

Hashin (1983) has given a very detailed review of many estimation methods. The review includes a discussion of the use of the well-known composite spheres assemblage method, and methods based on variational techniques that lead to upper and lower bound estimates of properties. Variational methods of interest here, are those that estimate bounds of effective properties for multi-phase isotropic particulate composites, where extreme values depend on the geometry of the reinforcing phase through the volume fraction only. The geometry of the reinforcement can be arbitrary (including various shapes and sizes) provided that it leads to statistically isotropic effective properties. Bounds corresponding to cases where the particulate materials are all reinforcing phases will be referred to as realistic bounds as they relate to situations having most practical importance. More recent developments in the field are described by Milton (2002) and Torquato (2002).

A detailed study of the method used by Maxwell has revealed that his methodology, focusing only on the far-field, can also be applied to the estimation of other properties of composite materials. The principal objective is to show how Maxwell's methodology can estimate the effective bulk modulus, shear modulus and thermal expansion coefficient of multi-phase isotropic composites reinforced with homogeneous spherical particles. The methodology of Maxwell is naturally extended so that assemblies of multi-phase spherical particles having a range of radii and/or properties may be considered. A second objective is to show that Maxwell's methodology is one reliable technique that provides closed form estimates of effective properties and is not necessarily restricted to low volume fractions of particulate reinforcement.

Section 2 will provide a detailed description of Maxwell's methodology applied to the thermal conduction problem, extending the approach to deal with multi-phase spherical particles having different sizes. Section 3 will use a spherically symmetric formulation in conjunction with Maxwell's methodology to estimate values for the effective bulk modulus and thermal expansion coefficient of a multi-phase isotropic particulate composite. Section 4 will consider the method of applying Maxwell's methodology to the estimation of the effective shear modulus of a multi-phase isotropic particulate composite. Section 5 will discuss the results based on Maxwell's methodology, including their validity for a range of volume fractions, and conclusions will be given in §6. Results given in this paper for the effective properties of isotropic composites are compared with earlier work, where a literature study has revealed that Maxwell's pioneering methodology has largely been ignored in the context of thermoelastic properties that have been treated using more complex ‘self-consistent’, ‘mean-field’, ‘effective medium’ and ‘polarization’ concepts, often in conjunction with bounding methods.

## 2. General description of Maxwell's methodology applied to thermal conductivity

The following description will modify Maxwell's (1873) approach, when estimating the electrical conductivity of a cluster of particles embedded in an infinite matrix so that it applies to the analogous problem concerning thermal conductivity.

### (a) Description of geometry

In a well-mixed cluster of *N* types of particulate reinforcement embedded in and perfectly bonded to an infinite matrix (figure 1*a*), there are *n*_{i} spherical particles of radius *a*_{i}, *i*=1, …, *N*. Particle properties of type *i*, which may differ from those of other types, are denoted by a superscript *i*. The cluster of all particle types may be just enclosed by a sphere of radius *b* and the particle distribution is homogeneous leading to isotropic effective properties of the composite formed by the cluster and matrix lying within this sphere. The particles must not be so densely packed that effective properties are anisotropic. The volume fractions of particles of type *i* within the enclosing sphere of radius *b* are given by(2.1)where *V*_{m} is the volume fraction of matrix. For just one type of particle, with *n* particles of radius *a* within the enclosing sphere of radius *b*, the particulate volume fraction *V*_{p} is such that(2.2)

Whatever the nature and arrangement of spherical particles in the cluster, Maxwell's methodology considers the far-field when replacing the isotropic discrete particulate composite shown in figure 1*a*, that can be enclosed by a sphere of radius *b*, by a homogeneous effective composite sphere of the same radius *b* embedded in the matrix as shown in figure 1*b*. There is no restriction on sizes, properties and locations of particles provided that the equivalent effective medium is homogeneous and isotropic. Composites having statistical distributions of both size and properties can clearly be analysed.

### (b) Temperature distribution for an isolated sphere embedded in an infinite matrix

A set of spherical polar coordinates (*r*, *θ*, *ϕ*) is introduced having origin at the centre of the sphere having radius *a*. For steady-state conditions, the temperature distribution *T*(*r*, *θ*, *ϕ*) in the particle and surrounding matrix must satisfy Laplace's equation, which is expressed in terms of spherical polar coordinates for the case where the temperature is independent of *ϕ*, namely(2.3)On the external boundary *r*→∞ a temperature distribution is imposed that would lead, in a homogeneous matrix material under steady-state conditions, to the following temperature distribution having a uniform gradient *α*:(2.4)At the particle/matrix interface *r*=*a*, the following conditions are imposed:(2.5)where both the temperature and the normal heat flux are continuous across the interface, and where thermal conductivities of particles and matrix are denoted by *κ*_{p} and *κ*_{m}, respectively. The temperature distributions in the particle and matrix, satisfying (2.3), the condition at infinity and the interface conditions (2.5) are given, respectively, by(2.6)As *z*=*r* cos *θ*, the temperature gradient is uniform within the particle. Any temperature *T*_{0} can be added to (2.6) without affecting the satisfaction of the interface conditions (2.5). Because an infinite medium having a uniform temperature gradient is considered, it is inevitable that temperatures lower than absolute zero will be encountered. This is not a matter for concern as the use of an infinite medium is simply a mathematical construct, designed to enable a specific method of estimating the effective properties of the composite.

### (c) Maxwell's methodology for estimating conductivity

The first step considers the effect of embedding in the infinite matrix, an isolated cluster of spherical particles of different types that can be just contained within a sphere of radius *b*, as illustrated in figure 1*a*. The thermal conductivity of particles of type *i* is denoted by . The cluster is assumed homogeneous regarding the distribution of particles, and leads to an isotropic effective thermal conductivity *κ*_{eff} for the composite lying within the sphere of radius *b*. For a single particle, the matrix temperature distribution is perturbed from the distribution (2.4) to (2.6) that depends on particle geometry and properties. The perturbing effect in the matrix at large distances from the cluster of particles is estimated by superimposing the perturbations caused by each particle, regarded as being isolated. The second step recognizes that, at very large distances from the cluster, all the particles can be considered to be located at the origin that is chosen to be situated at the centre of one of the particles in the cluster. Thus, for the case of multiple phases, the approximate temperature distribution in the matrix at large distances from the cluster is the following generalization of the second of the relations (2.6):(2.7)On using (2.1), this relation may be expressed in terms of the volume fractions so that(2.8)

The third step replaces the composite having discrete particles lying within the sphere of radius *b* by a homogeneous spherical isotropic effective medium (figure 1*b*) having radius *b* and having the effective thermal conductivity *κ*_{eff} of the composite. On using (2.6), the temperature distribution in the matrix outside the sphere of effective medium having radius *b* is then given exactly, for a given value of *κ*_{eff}, by(2.9)If the cluster in figure 1*a* is represented accurately by the effective medium shown in figure 1*b* then, at large distances from the cluster, the temperature distributions (2.8) and (2.9) should be identical, leading to the fourth step, equating the perturbation terms in (2.8) and (2.9), so that(2.10)which is a ‘mixtures’ relation for the quantity 1/(*κ*+2*κ*_{m}). On using (2.1), the effective thermal conductivity may be estimated using(2.11)which is identical in form to the result for electrical conductivity given by Kerner (1956*a*, eqn (7)). A result related to Maxwell's methodology through the use of polarization concepts has been derived by Torquato (2002, eqn (18.8)) and it can be expressed in the simpler form (2.10) on using (2.1). For multi-phase composites, Hashin & Shtrikman (1962; equations (3.21)–(3.23)) derived bounds for magnetic permeability, pointing out that they are analogous to bounds for effective thermal conductivity. Their conductivity bounds may be expressed in the following simpler form, having the same structure as the result (2.10) derived using Maxwell's methodology:(2.12)where *κ*_{min} is the least value of conductivities for all phases, while *κ*_{max} is the greatest value.

When *N*=1, it follows from (2.2) and (2.11) that the result first derived by Maxwell (1873) for the analogous case of electrical conductivity is obtained, which may be expressed in the form of a mixtures estimate plus a correction term so that(2.13)It follows from Hashin & Shtrikman (1962), and the review by Hashin (1983), that bounds for the effective thermal conductivity of a two-phase composite, valid for arbitrary reinforcement geometries leading to statistically isotropic effective properties, may be expressed in the form(2.14)where now(2.15)The structure of (2.14) is identical to that given by Torquato (2002, eqns (21.20)–(21.22)).

## 3. Bulk modulus and thermal expansion

### (a) Spherical particle embedded in infinite matrix subject to pressure and thermal loading

Consider an isolated particle of radius *a* perfectly bonded to an infinite surrounding matrix, subject to a pressure applied at infinity and a uniform temperature change Δ*T* from the stress-free temperature at which the stresses and strains in particle and matrix are zero. The displacement field in the particle and surrounding matrix is purely radial so that displacement components, referred to an origin of spherical polar coordinates (*r*, *θ*, *ϕ*) at the particle centre, are of the form(3.1)The corresponding strain field is then given by(3.2)The stress field follows from stress–strain relations expressed in the form:(3.3)where *λ* and *μ* are Lamé's constants and *α* is now the coefficient of thermal expansion.

On using the equilibrium equations, it can be shown that, within the spherical particle of radius *a*, the resulting bounded displacement and stress fields are given by(3.4)(3.5)where *k*_{p}=*λ*_{p}+(2/3)*μ*_{p} and *μ*_{p} are the bulk and shear moduli, respectively, for the particulate reinforcement and *α*_{p} is the corresponding thermal expansion coefficient. Clearly, the strain and stress distributions within the particle are both uniform. For the matrix region, it can be shown that(3.6)(3.7)where *k*_{m}=*λ*_{m}+(2/3)*μ*_{m} and *μ*_{m} are the bulk and shear moduli, respectively, for the matrix and *α*_{m} is the corresponding thermal expansion coefficient. The stress component *σ*_{rr} is automatically continuous across *r*=*a* having the value −*p*_{0}. As the displacement component *u*_{r} must also be continuous across this interface, the value of *p*_{0} must satisfy the relation(3.8)

### (b) Applying Maxwell's methodology to isotropic multi-phase particulate composites

Owing to the use of the far-field in Maxwell's methodology, it is again possible to consider multiple types of spherical reinforcement. The perturbing effect in the matrix at large distances from the cluster of particles is estimated by superimposing the perturbations caused by each particle, regarded as being isolated, and regarding all particles to be located at the origin. The relations (3.7) for non-zero stresses in the matrix are generalized to(3.9)where is the pressure at the particle/matrix interface when an isolated particle of species *i* is placed in infinite matrix material. From (3.8), the following value of is obtained:(3.10)It then follows that the stress distribution in the matrix at large distances from the discrete cluster of particles shown in figure 1*a* is approximately given by:(3.11)where the volume fractions of particles of type *i* defined by (2.1) have been introduced.

When (3.11) is applied to a single sphere of radius *b*, having the effective properties of a composite representing the multi-phase cluster of particles embedded in matrix material (figure 1*b*), the exact matrix stress distribution, for given values of *k*_{eff} and *α*_{eff}, is(3.12)Maxwell's methodology asserts that, at large distances from the cluster, the stress distributions (3.11) and (3.12), and hence the coefficients of *p* and Δ*T*, are identical leading to the following mixtures rules for the functions 1/[1/*k*+3/(4*μ*_{m})] and *α*/[1/*k*+3/(4*μ*_{m})], respectively:(3.13)(3.14)On using (2.1), the result (3.13) may be written(3.15)so that the effective bulk modulus of the multi-phase particulate composite may instead be obtained from a mixtures relation for the quantity 1/(*k*+(4/3)*κ*_{m}). On using (2.1) and (3.15), the effective bulk modulus may be estimated using(3.16)which is identical to the result given by Kerner (1956*b*, eqn (5)) and equivalent to those given by Weng (1984, eqn (4.21) and 1990, eqn (6.2)). It follows from (3.14) and (3.15) that the corresponding relation for effective thermal expansion is(3.17)which is equivalent to the corresponding result obtained by Kerner (1956*b*). A result for bulk modulus based on polarization concepts has been derived by Torquato (2002, eqn (18.44)) and it can be expressed in the simpler form (3.15) on using (2.1). In contrast to Torquato's approach, the above analysis considers only the far-field and includes thermal expansion effects leading to the result (3.17).

The bounds for effective bulk modulus of multi-phase isotropic composites derived by Hashin & Shtrikman (1963, eqns (3.37)–(3.43)) and the bounds derived by Walpole (1966, eqn (26)) are identical and may be expressed in the following simpler form having the same structure as the result (3.15) derived using Maxwell's methodology:(3.18)where the parameters *k*_{min} and *μ*_{min} are the least values of bulk and shear moduli, respectively, of all phases in the composite, while *k*_{max} and *μ*_{max} are the greatest values.

Shapery (1968, eqns (32), (33)) derived bounds for the thermal expansion of isotropic multi-phase composites that are equivalent to those of Rosen & Hashin (1970, eqns (4.27) and (4.28)). The bounds for effective thermal expansion involve the effective bulk modulus, and the specification of bounds is complex and beyond the scope of this paper. An analysis has been undertaken showing numerically, for a very wide range of parameter values, that the effective thermal expansion obtained using Maxwell's methodology lies between the absolute bounds for all volume fractions that are consistent with isotropic properties.

When *N*=1, it follows from (3.16), derived using Maxwell's methodology, that the effective bulk modulus may be expressed as a mixtures estimate plus a correction term so that(3.19)

Walpole (1966, eqn (26)) has derived rigorous bounds for the effective bulk modulus, which can for a two-phase composite be expressed in the following two equivalent forms:(3.20)(3.21)where(3.22)The structure of (3.20) is identical to that given by Torquato (2002, eqns (21.70)–(21.72)).

When *N*=1, the effective thermal expansion resulting from (3.17), derived using Maxwell's methodology, may be written as a mixtures estimates plus a correction term so that(3.23)On using the bounds (3.20) for the bulk modulus, it can be shown that the bounds for the effective thermal expansion are such that(3.24)

(3.25)

## 4. Shear modulus

### (a) Spherical particle embedded in infinite matrix material subject to pure shear loading

For a state of pure shear, and in the absence of thermal effects, the displacement field of a homogeneous sample of material referred to a set of Cartesian coordinates (*x*, *y*, *z*) has the form(4.1)and the corresponding strain and stress components are given by(4.2)(4.3)The parameters *γ* and *τ* are, respectively, the shear strain (half the engineering shear strain) and shear stress such that , where *μ* is the shear modulus of an isotropic material. The principal values of the stress field are along (tension) and perpendicular to (compression) the line *y*=*x*.

A single spherical particle of radius *a* is now placed in, and perfectly bonded to, an infinite matrix, where the origin of spherical polar coordinates (*r*, *θ*, *ϕ*) is taken at the centre of the particle. The system is then subject only to a shear stress applied at infinity. At the particle/matrix interface, the following perfect bonding boundary conditions must be satisfied:(4.4)A displacement field equivalent to that used by Hashin (1962), based on the analysis of Love (1944, eqns (5)–(7)) that leads to a stress field satisfying the equilibrium equations and the stress–strain relations (3.3) with Δ*T*=0, can be used to solve the embedded isolated sphere problem (see electronic supplementary material). The displacement and stress fields in the particle are bounded at *r*=0 so that(4.5)(4.6)In the matrix, the displacement field and stress field (stresses bounded as *r*→∞) have the form(4.7)(4.8)The representation is identical in form to that used by Christensen & Lo (1979), although they use a definition of *ϕ* that differs from that used here by an angle of *π*/4. This difference has no effect on the approach to be followed. It follows from (4.5)–(4.8) that the continuity conditions (4.4) are satisfied if the following four independent relations are satisfied:(4.9)and it can then be shown that(4.10)As *C*_{p}=0, it follows from (4.5) and (4.6) that both the strain and stress distributions in the particle are uniform.

### (b) Application of Maxwell's methodology

To apply Maxwell's methodology to a cluster of *N* particles embedded in an infinite matrix, the stress distribution in the matrix at large distances from the cluster is considered. The perturbing effect in the matrix at large distances from the cluster of particles is estimated by superimposing the perturbations caused by each particle, regarded as being isolated, and regarding all particles to be located at the origin. The stress distribution at very large distances from the cluster is then given by the following generalization of the relations (4.8):(4.11)where from (4.10), for *i*=1, …, *N*,(4.12)For the isolated sphere of radius *b*, having the effective properties of the particulate composite cluster as illustrated in figure 1*b*, it follows that the stress field in the matrix at large distances is described exactly by relations of the type (4.8) where the coefficient *D*_{m} is replaced by having the value determined by the relation(4.13)where *μ*_{eff} is the effective shear modulus of the isotropic particulate composite. It then follows from (4.8) and (4.10) that the exact matrix stress distribution, for a given value of *μ*_{eff}, is:(4.14)

As the stress distribution given by (4.11) must be identical at large distances from the cluster with that specified by (4.14) it follows, from a consideration of terms proportional to *r*^{−3}, that:(4.15)where use has been made of (2.1). On substituting (4.12) and (4.13) in (4.15), it can be shown using (2.1) that the following mixtures result is obtained for the function :(4.16)On using (2.1), the effective shear modulus may be estimated using the following relation:(4.17)The result (4.17) is equivalent to the results given by Kerner (1956*b*; eqn (8)) and Weng (1984, eqn (4.21) and 1990, eqn (6.2)). A result, based on polarization concepts, has been derived by Torquato (2002, eqn (18.45)) and it can be expressed in the simpler form (4.16) of this paper on using (2.1). Torquato (2002) used a different representation for the stress and displacement field, where it was assumed at the outset that the strain field within the particle is uniform. By considering the far-field approach of Maxwell, this assumption is unnecessary.

It can be shown that the bounds for the effective shear modulus derived by Hashin & Shtrikman (1963, eqns (3.44)–(3.50)) and the bounds derived by Walpole (1966, eqn (26)) are identical and may be expressed in the following form that has the same structure as the result (4.16) derived using Maxwell's methodology:(4.18)where(4.19)The parameters *k*_{min} and *μ*_{min} are, respectively, the least values of the bulk and shear moduli of all phases in the composite, while *k*_{max} and *μ*_{max} are the greatest values. On writing(4.20)it follows that for all values of the bulk and shear moduli, indicating that the ‘max’ and ‘min’ subscripts are used in an appropriate sense. It should be noted that *k*_{min} and *μ*_{min} may be associated with different phases, and similarly for *k*_{max} and *μ*_{max}.

When *N*=1, the result (4.17), derived using Maxwell's methodology, may be expressed as a mixtures estimate plus a correction term so that(4.21)Walpole (1966, eqn (26)) has derived rigorous bounds for the effective shear modulus, which can, for an isotropic two-phase composite, be expressed in the following form having the same structure as the result (4.21):(4.22)where and are defined by (4.19). The structure of (4.22) is identical to that given by Torquato (2002, eqns (21.73)–(21.75)).

## 5. Discussion

### (a) Multi-phase composites

Key results, (2.10) for thermal conductivity, (3.15) for bulk modulus and (4.16) for shear modulus, derived using Maxwell's methodology, have the following simple common structure, involving mixtures formulae for the effective isotropic properties *ϕ* of the composite:(5.1)where(5.2)

The inequalities (2.12), (3.18) and (4.18), valid for all volume fractions, lead to rigorous bounds valid for any phase geometries that are statistically isotropic. They have the following common structure that is strongly related to the structure defined by (5.1) and (5.2) for effective properties determined using Maxwell's methodology:(5.3)where(5.4)

By comparing the bounds (5.3) with (5.1) when *J*=1, the result for thermal conductivity obtained using Maxwell's methodology is exactly the lower bound for *κ*_{eff} when *κ*_{min}=*κ*_{m}, and the upper bound when *κ*_{max}=*κ*_{m}. When *κ*_{min}<*κ*_{m}<*κ*_{max}, the result for effective thermal conductivity obtained using Maxwell's methodology lies between the bounds for all volume fractions. A comparison of (5.3) with (5.1) when *J*=2 shows that the result obtained for bulk modulus using Maxwell's methodology leads exactly to the lower bound for *k*_{eff} when *μ*_{min}=*μ*_{m}, and to the upper bound when *μ*_{max}=*μ*_{m}. When *μ*_{min}<*μ*_{m}<*μ*_{max}, the result for effective bulk modulus obtained using Maxwell's methodology lies between the bounds for all volume fractions. A comparison of (5.3) with (5.1) when *J*=3 shows that the result (5.1) obtained for shear modulus using Maxwell's methodology leads exactly to the lower bound for *μ*_{eff} when , and to the upper bound when . When , the result for effective shear modulus obtained using Maxwell's methodology lies between the bounds for all volume fractions. For the case of thermal expansion, it has been shown numerically, using a wide range of parameter values, that the result (3.17) derived using Maxwell's methodology always lies between the Shapery (1968) and Rosen & Hashin (1970) bounds, and is never equal to the either of the bounds.

Thus, it has been shown that effective thermoelastic properties, obtained using Maxwell's methodology, do not lie beyond rigorous bounds for properties for all volume fractions consistent with isotropic effective properties. This characteristic of Maxwell's methodology provides significant evidence that its validity is not confined to small volume fractions.

The principal results of this paper for the effective properties of multi-phase composites have been shown to be equivalent to those given by Kerner (1956*a*,*b*) and Weng (1984, 1990). While the methodology given by Kerner is not clearly expressed and difficult to appraise, his results are clearly entirely consistent with Maxwell's methodology and the mean-field approach of Weng. Kerner's method involves averaging processes over sufficiently large regions (along lines, or over areas and volumes), contrasting sharply with Maxwell's methodology where only perturbations of the far-field caused by small clusters of reinforcement are considered.

### (b) Two-phase composites

When considering the effective bulk modulus, thermal expansion coefficient and thermal conductivity for two-phase composites having spherical particles of the same size, the results obtained using Maxwell's methodology are identical to the realistic bounds. They are also identical to estimates for effective properties obtained by applying the composite spheres assemblage model (see review by Hashin 1983) for a particulate composite to a representative volume element comprising just one particle and a matrix region that is consistent with the volume fractions of the composite. In the case of shear modulus, the result (4.21) obtained using Maxwell's methodology, corresponds exactly to one of the variational bounds whenever (*k*_{p}−*k*_{m})(*μ*_{p}−*μ*_{m})≥0. For the case (*k*_{p}−*k*_{m})×(*μ*_{p}−*μ*_{m})≤0, it can be shown that and a comparison of (4.21) and (4.22) then indicates that the result (4.21) for the effective shear modulus derived using Maxwell's methodology must lie between the bounds (4.22). Also, the results (2.13), (3.19), (3.23) and (4.21), for two-phase composites arising from the use of Maxwell's methodology, are such that *κ*_{eff}→*κ*_{p}, *k*_{eff}→*k*_{p}, *μ*_{eff}→*μ*_{p} and *α*_{eff}→*α*_{p}, respectively, when *V*_{p}→1, limits requiring *V*_{p} values attained only for a range of particle sizes, as for the composite spheres assembly model.

Effective properties of two-phase composites, derived using Maxwell's methodology, may be expressed as a mixtures estimate plus a correction term, as seen from (2.13), (3.19), (3.23) and (4.21). The correction is always proportional to the product *V*_{p}*V*_{m}, and it involves the square of property differences for the case of conductivity, bulk and shear moduli, and the product of differences of the bulk compressibility and expansion coefficient for the case of thermal expansion. These results are the preferred common form for effective properties, having the advantage that conditions governing whether an extreme value is an upper or lower bound are then easily determined. In addition, such conditions determine when both upper and lower bounds coincide with each other, and with predictions based on Maxwell's methodology, leading to exact non-trivial predictions for all volume fractions. For example, when *μ*_{p}=*μ*_{m} the bounds for bulk modulus given by (3.20) are equal to the exact solution for any values of *k*_{p}, *k*_{m} and the volume fractions, and they are equal to the result (3.19) indicating that Maxwell's methodology leads, in this special non-trivial case, to an exact result for all volume fractions for which the composite is isotropic. For the case of thermal expansion, it follows from (3.24) and (3.25) that exact results are also obtained for any values of *k*_{p}, *k*_{m}, *α*_{p}, *α*_{m} and *V*_{p}, and they are equal to (3.23) indicating that Maxwell's methodology again leads, in a special non-trivial case, to an exact result for all volume fractions.

Results for effective properties of two-phase composites, are such that Maxwell's methodology, the composite spheres assemblage model when it can generate exact results, and the realistic variational bound, all lead to the same result. This suggests very strongly that the realistic bound is a much better estimate of effective properties for spherical particles than the other bound, which can be obtained simply by interchanging particle and matrix properties and the volume fractions. Further evidence of this phenomenon is provided in figures 2–4 where use has been made of conductivity results of Sangani & Acrivos (1983), based on the use of spherical harmonic expansions, and results of Arridge (1992) who used harmonics up to 11th order to estimate accurate values of bulk modulus and thermal expansion for body-centred cubic (b.c.c.) and face-centred cubic (f.c.c.) arrays of spherical particles having the same size. While Sangani & Acrivos considered simple cubic, b.c.c. and f.c.c. arrays of spheres, only the f.c.c. results are shown in figure 2, for the values of the phase contrast, as a larger range of volume fractions can be considered. It is seen that there is excellent agreement between predictions based on Maxwell's result (2.13) and results of Sangani & Acrivos for a wide range of particulate volume fractions. Results (not shown) indicate that the agreement is less good at large volume fractions of particulate, when comparing Maxwell's result with the simple cubic and b.c.c. results of Sangani & Acrivos.

It is worth noting from Bonnecaze & Brady (1990, tables 2–4), who use a multi-pole method of estimating the conductivity of cubic arrays of spherical particles, that their results for the case that retains only dipole–dipole interactions correspond almost exactly (to three significant figures) with results shown in figure 2 obtained using Maxwell's result (2.13). They did not make this comparison considering only the results of Sangani & Acrivos (1983). This result suggests that Maxwell's result, which was derived assuming particles do not interact, is in fact valid also for the case when particle interactions are represented by dipole–dipole interactions, and this might explain why Maxwell's result is found to be a good approximation for a very wide range of volume fractions. Further discussion of this issue is beyond the scope of this paper. We note that for composites used in practice, the difference in the values of the thermomechanical properties (e.g. bulk modulus, thermal expansion coefficient) of the reinforcement and matrix, seldom lead to values of phase contrast that are greater than 10 or so. The phase contrast of the transport properties (such as electrical or thermal conductivity) can be very much greater. It follows that in practical situations, greater confidence may be placed in the Maxwell formulation being accurate at relatively large volume fractions for the thermomechanical properties when compared with the case of transport properties.

The results of Arridge (1992) are based on the following properties for silicon carbide spheres in an aluminium matrixThe corresponding values of bulk and shear moduli are *k*_{p}=259.68 GPa, *k*_{m}=80.56 GPa, *μ*_{p}=202.94 GPa, *μ*_{m}=26.85 GPa, and are such that *k*_{p}>*k*_{m} and *μ*_{p}>*μ*_{m}. It should be noted that an array of spheres in a b.c.c. or in an f.c.c. arrangement possesses cubic symmetry. The space group of such an arrangement is cubic and the point group *m*3*m* represents the symmetry of physical properties (Kelly & Groves 1970). The thermal expansion coefficient of such an array is, therefore, isotropic. Arridge's results are given as mean values implying that the expansion coefficients differ slightly in various directions, a situation that could arise because an insufficient number of harmonics has been included in the representation.

Additional evidence concerning the accuracy of realistic bounds is given by Torquato (1990) who considered the effect on bounds of geometrical factors relating to the reinforcement, and developed three-point bounds that are more restrictive than the conventional two-point bounds (equivalent to the Hashin & Shtrikman (1963) bounds) for the case of bulk modulus and thermal expansion of suspensions of spheres. The definition (Torquato (1990), eqn (25)) has in fact been replaced by . Torquato's (1990) three-point bounds are compared in figures 3 and 4 with the almost exact results of Arridge (1992), and results obtained from Maxwell's methodology and the two-point variational bounds. From (3.19) and (3.20), bulk modulus results using the Hashin & Shtrikman (1963) lower bound and Maxwell's methodology are identical as *μ*_{p}>*μ*_{m} for Arridge's properties. These results are very close to those obtained using the Arridge model for both f.c.c. and b.c.c. particle arrangements, and to the three-point lower bound estimate of Torquato (1990). The results of Arridge are shown for all volume fractions up to the closest packing value for f.c.c. and b.c.c. configurations of spherical particles. The f.c.c. and b.c.c. packing configurations lead to bulk moduli that are very close together for particulate volume fractions in the range 0<*V*_{p}<0.6. Furthermore, the results obtained using Maxwell's methodology lie between the f.c.c. and b.c.c. estimates for volume fractions in the range 0<*V*_{p}<0.4. For a significant range of volume fractions, the Hashin–Shtrikman upper bound is seen in figure 3 to be significantly different to the corresponding lower bound, and to the three-point upper bound of Torquato.

For the case of thermal expansion, the Hashin & Shtrikman (1963) upper bound and Maxwell's methodology result are identical as seen from (3.23) and (3.25), since for Arridge's properties (*k*_{p}−*k*_{m})(*μ*_{p}−*μ*_{m})(*α*_{p}−*α*_{m})≤0. These results are seen in figure 4 to be very close to those obtained using the Arridge model for both f.c.c. and b.c.c. particle arrangements, and to the three-point upper bound estimate of Torquato. The results of Arridge are again shown for all volume fractions up to the closest packing value for f.c.c. and b.c.c. configurations of spherical particles. The f.c.c. and b.c.c. packing configurations lead to expansion coefficients that are very close together, and very close to results obtained using Maxwell's methodology, for particulate volume fractions in the range 0<*V*_{p}<0.5. For a significant range of volume fractions, the Hashin–Shtrikman lower bound is seen in figure 4 to be significantly different to the corresponding upper bound, and to the three-point lower bound of Torquato. In view of the almost exact results of Arridge, and the observation that the three-point bounds for bulk modulus and thermal expansion derived by Torquato are reasonably close, it is deduced that Maxwell's methodology provides accurate estimates of bulk modulus and thermal expansion coefficient for a wide range of volume fractions.

For the case of a simple cubic array of spherical particles with volume fractions in the range 0<*V*_{p}<0.4, Cohen & Bergman (2003*a*; figure 4) have shown that bounds for shear modulus, obtained using a Fourier representation of an integro-differential equation for the displacement field, are very close to the Hashin–Shtrikman lower bound when using properties for a glass–epoxy composite. The results of this paper indicate that their bounds will also be very close to the result obtained using Maxwell's methodology, showing again that its validity is not restricted to low volume fractions, as might be expected from the approximations made.

For the cases of bulk modulus and thermal expansion, Maxwell's methodology is based on a stress distribution (3.12) in the matrix outside the sphere having radius *b* of effective medium, which is exact everywhere in the matrix (i.e. *b*<*r*<∞) and involves an *r* dependence only through terms proportional to *r*^{−3}. It follows from (3.11) that, for the discrete particle model (figure 1*a*), the asymptotic form for the stress field in the matrix as *r*→∞ has the same form as the exact solution for the equivalent effective medium model (figure 1*b*). The matching of the discrete and effective medium models at large distances, leading to an exact solution in the matrix (*b*<*r*<∞) of the effective medium model, is thought to be one reason why estimates for bulk modulus and thermal expansion coefficient of two-phase composites are accurate for a wide range of volume fractions. When estimating thermal conductivity using Maxwell's methodology, the relations (2.8) and (2.9) show that a similar situation arises. The *r* dependence of the temperature gradient in the *r*-direction is through a term again proportional to *r*^{−3} and, as discussed above, estimates of thermal conductivity based on Maxwell's methodology are again accurate for a wide range of volume fractions. For the case of shear modulus, the exact solution for the stress field (4.14) in the matrix lying outside the sphere having radius *b* of effective medium (figure 1*b*) involves terms proportional to *r*^{−3} and *r*^{−5}, but only terms involving *r*^{−3} are used when applying Maxwell's methodology, as seen from (4.11). This means that, in contrast to the cases for the effective bulk modulus, thermal expansion coefficient and thermal conductivity, the resulting estimate for the effective shear modulus does not lead to an exact matrix stress distribution in the region *b*<*r*<∞ outside the sphere of effective medium, and consequently estimates for effective shear modulus are likely to be less accurate than those for other effective properties.

The results, discussed above for various effective properties, are remarkable as one might expect Maxwell's methodology to be accurate only for sufficiently low volume fractions of reinforcement. The reason is that the methodology involves the examination of the stress, displacement or temperature fields in the matrix at large distances from the cluster of particles, and assumes that the perturbing effect of each particle can be approximated by locating them at the same point. The nature of this approximation is such that interactions between particles are negligible, and it would be expected that resulting effective properties will be accurate only for low volume fractions, as originally suggested by Maxwell (1873). In view of compelling evidence presented in this paper, based on a wide variety of considerations, a major conclusion is that results for two-phase composites derived using Maxwell's methodology are not limited to small particulate volume fractions.

## 6. Conclusions

Maxwell's far-field methodology for estimating the effective electrical conductivity of isotropic particulate composites may also be used to estimate the effective bulk and shear moduli and the thermal expansion coefficient. The approach enables the consideration of multi-phase composites having distributions of both particle sizes and properties, and has been shown to be valid for a wide range of volume fractions.

Results for effective properties of multi-phase composites, derived using Maxwell's methodology, do not lie beyond the bounds, for arbitrary statistically isotropic phase geometry, obtained using variational methods. If matrix properties are the least values of all the phase properties, then the effective thermal conduction, bulk modulus and shear modulus derived using Maxwell's methodology correspond exactly, for all volume fractions, to the lower bound, and if matrix properties are the greatest values of all the phase properties, then the effective properties correspond to the upper bound. For these special cases, corresponding results for the effective thermal expansion of a multi-phase isotropic composite lie between the variational bounds.

Results derived by Kerner (1956

*a*,*b*) for the effective properties of a multi-phase composite are identical to the results obtained using Maxwell's methodology, which thus provides a more acceptable justification for Kerner's results.Expressions for the effective properties of a two-phase isotropic particulate composite may be written in the form of a mixtures estimate plus a correction term that has a particular structure. The resulting form enables the derivation of useful conditions that determine whether an extreme value obtained using variational methods is an upper or a lower bound.

Maxwell's methodology, based on a consideration of the far-field, is a very powerful and optimum technique for estimating, for a wide range of volume fractions, many effective properties of multi-phase isotropic particulate composites, because it provides just one estimate of properties, rather than bounds, which is consistent with other estimates for that property given in the literature, and it does not impose restrictions, except that the particles must be spherical (but can have a range of sizes and properties), and the resultant effective properties must be isotropic.

## Acknowledgments

The authors acknowledge Prof. J. R. Willis FRS, Cambridge University, for introducing us to the paper of Walpole, leading to a more thorough consideration of bounds for effective elastic properties, and also Dr L. Weber, EPFL, Switzerland, for referring us to the book by S. Torquato. We are also grateful to a referee for referring us to several very useful references. A.K. is grateful to the Leverhulme Foundation for the award of a grant during the tenure of which this work was undertaken.

© Crown Copyright 2007. Reproduced by permission of the Controller of HMSO and the Queen's printer for Scotland.

## Footnotes

Our work is based on a legacy of James Clerk Maxwell (1831–1879) to the composites community, and we would like to dedicate this paper to his memory.

Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2007.0071 or via http://journals.royalsociety.org.

- Received June 11, 2007.
- Accepted November 5, 2007.

- © 2007 The Royal Society