Critical Assessment 18: elastic and thermal properties of porous materials – rigorous bounds and cross-property relations

Abstract

A critical assessment of model relations describing the porosity dependence of elastic properties (Young's modulus) and thermal properties (thermal conductivity) is given. It is shown that there are essentially five types of admissible predictive model relations for the relative Young's modulus and thermal conductivity of isotropic porous materials. The cross-property relations resulting from the complete analogy between the model relations for the elastic moduli and thermal conductivity of isotropic porous materials are reviewed and compared. Finally, it is shown that the fact that relative Young's moduli are not equal to relative thermal conductivities except for materials with translational symmetry, i.e. the mere existence and necessity of non-trivial cross-property relations, proves so-called minimum solid area models to be wrong.

Keywords:

• Porosity
• Elasticity (bulk, shear, tensile modulus)
• Thermal conductivity
• Exponential relation
• Cross-property relations
• Minimum solid/contact area models
• Review

Introduction

Many materials of engineering interest are porous, and in many cases porosity is intentionally introduced in order to meet certain structural or functional requirements, e.g. light weight, low heat capacity, increased flexibility, good insulating performance (thermal, electrical or acoustic), high fluid permeability (in filters), high surface area (in catalysts and catalyst supports), etc.^1–4 Porous materials are heterogeneous materials for which the salient feature of the microstructure is the existence of a pore space. Thus, porous materials may be considered as a special case of heterogeneous materials that may be modelled as two-phase systems, consisting of a solid ‘phase’ (which may, of course, be itself a heterogeneous material, e.g. a phase mixture or composite consisting of several solid phases or a single-phase polycrystalline material) and a void phase (either vacuum voids or voids filled with fluids, usually a gas). Properties sensu stricto are coefficients in linear constitutive equations, e.g. Hooke's law or Fourier's law. Among these are the elastic constants (tensile, shear and bulk modulus as well as Poisson ratio), thermal conductivity, specific heat and thermal expansion coefficients.⁵ The properties of heterogeneous materials, including porous materials, are usually called ‘effective properties’.⁶ The theory to calculate effective properties from the constituent phase properties and the microstructure is often called ‘micromechanics’⁷ or ‘theory of composites’,⁸ although ‘theory of heterogeneous materials’, including ‘theory of porous media’ as a special field, is probably the most appropriate term.

Since the aforementioned elastic and thermal properties of a porous material of a given composition are primarily determined by the porosity (i.e. the volume fraction of pores), and only to a minor degree by ot1her microstructural features (e.g. pore size, shape and connectivity), and since porosity is usually easy to measure, it is reasonable to consider property–porosity dependences in the first place and then possibly take into account other microstructural details if desired, according to their relative importance. Now it can be shown that some of these effective properties of porous materials are indeed independent of the porosity, e.g. the specific heat per unit mass and the thermal expansion coefficients.⁹ Among the more challenging effective properties are the elastic constants and thermal conductivity, which are strongly dependent on the porosity and other microstructural features of the porous materials. Therefore it is not surprising that the literature is full of traditional model relations that have been proposed to describe the porosity dependence of these properties and are summarised both in monographs and journal papers.^10–15 However, many of these relations are redundant (since they have been ‘rediscovered’ several times and denoted by different names), and not all of these relations are admissible from the physical point of view.

In particular, in the field of thermal conductivity, it seems to be still widely believed that the parallel and series models, the Maxwell-type models and the self-consistent model (sometimes called Bruggeman model or, misleadingly, ‘effective medium theory’ model/EMT model) are the only parameter-free predictive model relations for conductivity prediction,¹⁴ although it is well known that Coble–Kingery-type relations, power–law relations and exponential relations have a quite similar theoretical status,¹⁶ the parallel, series and Maxwell-type models just correspond to the micromechanical bounds and in the case of vacuum pores the self-consistent model degenerates to the linear approximation and predicts a spurious percolation threshold,⁶ i.e. describes non-linear behaviour only when the thermal conductivity of the pore phase is sufficiently large, but not as an intrinsic consequence of the microstructure (as it should be).

On the other hand, the field of elastic moduli description is dominated by a completely unacceptable relation, the simple exponential, apart from a large amount of other traditional relations that are cited under different names. This has led to the unpleasant situation that traditional modulus–porosity relations are cited over and over again, even in quite recent papers,¹⁵ although it is well known for quite some time that many of these relations are inadmissible for predictive purposes, and some of them are misleading or inconvenient even when used for fitting. A look into the current literature shows that this practice, in which trivial mixture rules are listed side by side with modified relations and special results that hold only for very special geometries, and rigorous results are listed side by side with empirical fit relations, all this using inconsequent terminology, is bound to confuse not only the beginner. Moreover, it seems that the complete formal analogy of relations for describing elastic moduli and thermal conductivity, described in one of our earlier papers,¹⁶ is not yet generally known outside the ceramics community.

The present critical assessment is an attempt to clarify this situation by recalling the rational criteria based on which the admissibility of any relation can be tested and by presenting the most important non-empirical model relations that are at least admissible for predictive purposes and have turned out to be successful for porous real-world materials. Among other things, it is emphasised that specific differences between the relations usable for predicting the porosity dependence of thermal conductivity and Young's modulus imply the existence of non-trivial cross-property relations. Since all of these non-trivial cross-property relations, which are reviewed in this paper for the first time, directly contradict so-called minimum-solid area models,^{10,11,17–25} it is necessary to abandon the latter.

Rigorous bounds for elastic moduli and thermal conductivity

The elastic moduli and thermal conductivity of multiphase materials are strongly dependent on the microstructure, in particular on the volume fractions of the constituent phases. These properties are generally restricted from above and below by upper and lower bounds. The simplest of these bounds are the so-called one-point bounds, corresponding to the (volume-weighted) arithmetic and harmonic means, respectively, called Wiener bounds²⁶ in conductivity context (and several other context such as electrical conductivity, dielectric permittivity and magnetic permeability) and Voigt–Reuss bounds^27,28 or Paul bounds* ²⁹ in elasticity context. In the sequel these bounds are denoted as Wiener–Paul bounds (WP bounds).

In the special case of isotropic microstructures narrower bounds, so-called two-point bounds, are available, or more precisely, reduce to bounds requiring only volume fraction (one-point) information.⁶ These are called Hashin–Shtrikman bounds (HS bounds),^6–8 both for the thermal conductivity (and several other second-order tensor properties such as electrical conductivity, dielectric permittivity and magnetic permeability³⁰) and the elastic moduli^31,32 (properties based on fourth-order tensors defined for the special case of isotropic materials). When the phase contrast between the properties of the constituent phases is not too high, these bounds are fully sufficient for estimating the effective properties (e.g. elastic moduli or thermal conductivity) of multiphase materials.^{† 6–9,32}

When the phase contrast is higher, however, the bounds become wider and thus the estimates of the effective properties become very imprecise. In particular, it can easily be shown^32,34 that when one of the phases has a zero (or negligible) property value, as in the case of elastic moduli and thermal conductivity of many common porous materials, the lower bounds degenerate to zero (or negligibly small values) in the whole range of pore volume fractions (porosities) from zero to one (and thus become useless), while the upper one-point bounds (WP bounds) reduce to¹⁶ (1)

where denotes the porosity (volume fraction of pores) and we have defined the relative property as the (dimensionless) ratio of the effective property of the porous material as a whole and the property of the dense, pore-free, solid (which can be a single solid phase, usually polycrystalline, or a solid phase mixture). Note that the solid portion of the porous material can form a matrix (in the case of a matrix-inclusion or closed-cell microstructure containing closed isolated pores) or a skeleton (in the case of a bicontinuous or open-cell microstructure). Of course, the solid property itself may have to be calculated by an appropriate averaging of tensor components of the individual crystallites^6–9 and (in the case of a solid phase mixture) by applying the aforementioned rigorous bounds. Note that here and in the sequel denotes any of the elastic moduli (tensile or Young's modulus , shear modulus and bulk modulus ), or thermal conductivity .

While the upper one-point bound (upper WP bound), equation (1), holds generally for arbitrary microstructures, including anisotropic ones, in the special case of isotropic porous materials the upper two-point bounds (upper HS bounds) are (2) for the tensile modulus (Young's modulus), at least to an excellent approximation,³² and (3) for the thermal conductivity (exactly).^9,16,34

It has to be emphasised that, when the phase contrast attains several orders of magnitude or even approaches infinity (as in the case of porous materials), the simple arithmetic mean of the upper and lower bounds (either WP or HS bounds) cannot be expected to provide a realistic estimate of the effective property anymore, because it leads to a spurious value of 0.5 for the relative property in the limit . Although weighted means can be defined that avoid this difficulty,^35–37 it is clear that these empirical means do not contribute anything essential to a physical understanding of property–porosity dependences. Therefore, for porous materials (in contrast to composites with a sufficiently low phase property contrast) model-based predictive relations are needed in addition to the aforementioned rigorous bounds (which are principally not model-based, i.e. have not been derived for specific microstructures, but have been derived on the basis of very universal variational principles^30,31).

Model relations for elastic moduli and thermal conductivity

Once the solid property is reliably known (e.g. for a single polycrystalline solid phase from orientational averaging of the monocrystal tensor components,^{6–9,38–41} for multiphase materials by calculating arithmetic or other means of upper and lower one- or two-point bounds^32,34,36,37), several model relations are available that can be invoked for predicting the porosity dependence of the effective property of the corresponding porous material. In fact, the parallel model, (4)

which corresponds formally to the upper one-point bound (upper WP bound), equation (1), and which exactly predicts the axial property component of porous materials with translational symmetry (translational invariance), is such a model. The remarkable feature of this relation is the fact that it has the same form for all elastic moduli and thermal conductivity and is independent of the cross-section of the pore channels. However, for the same reasons it is clear that equation (4) can never be a good prediction for other directions than the axial one or for materials without translational symmetry, e.g. isotropic porous materials.

All model relations for isotropic materials must include a specific coefficient that depends on the property in question and takes into account pore shape and, in the case of elastic moduli, the influence of the Poisson ratio (at least that of the solid phase, in the sequel called ‘solid Poisson ratio’ ). In contrast to widespread belief, however, the value of this coefficient is not freely assignable, but must be coupled to the exact solution of the corresponding single-inclusion problem.^{6–8,42–47} The ensuing linear relation (5)

with the coefficient dependent on the property, pore shape and solid Poisson ratio (in the case of elastic moduli) may be considered as the exact solution of the single-inclusion problem, i.e. a single pore in an infinitely extended solid matrix. With respect to the properties of real-world materials this exact solution is called the ‘non-interaction approximation’,⁷ since the pore is assumed to be isolated and infinitely distant from other pores. Although this linear model relation can be expected to hold only for materials with very low porosity values, and predicts a spurious percolation threshold for higher porosity,⁶ it serves as a benchmark relation to which all other (non-linear) admissible relations should reduce in the case of low porosity. We note in passing that so-called self-consistent approximations,^6–8 in conductivity context attributed to Bruggeman,⁴⁸ in elasticity context due to Hill⁴⁹ and Budiansky,⁵⁰ reduce to the linear relation, equation (5), as soon as the pore phase has a zero (or negligibly small) property value, the case that is the main focus of this paper. That means the non-linearity of self-consistent models is a consequence of the pore phase property and not, as should be, an intrinsic feature of the porous microstructure itself. In particular, according to the self-consistent approximation, materials with vacuum pores would exhibit a linear dependence of the elastic moduli and thermal conductivity. This is in contrast to what is observed in reality. It is thus clear that, when the pore phase has a zero or negligibly small property value, self-consistent models, including the popular Bruggeman model for thermal conductivity (sometimes, misleadingly, called ‘EMT’ model), are not only unrealistic but simply redundant, because they are in this case identical to the linear relation (non-interaction approximation), equation (5). Therefore they may be ignored in this context completely.

Non-linear model relations that obey the rigorous bound, equations (1)–(3), and at the same time reduce to the linear relation, equation (4), in the low-porosity case are the Maxwell-type model, (6) in conductivity context often called ‘Maxwell–Eucken model’¹⁴ which is formally identical to the upper two-point bound (upper HS bound) and the Hasselman relation for elastic moduli,⁵¹ the Coble–Kingery relation,^52–54 (7) which is usually incorrectly cited in textbooks, the power–law relation,⁵⁵ (8) which corresponds to the differential scheme approximation^56–59 and includes the Gibson–Ashby model for open-cell foams⁶⁰ as a special case) and our exponential relation,^16,32,61,62 (9)

In all these relations the value of depends on the property in question and the pore shape and may therefore be called a ‘property-pore-shape coefficient’. Moreover, in the case of elastic moduli, this coefficient depends also on the solid Poisson ratio, but for Young's modulus this dependence is completely negligible (at least for solid Poisson ratios in the range 0.1–0.4).³² For spherical pores (and approximately for other isometric convex pore shapes with aspect ratios around unity) the value of this coefficient is 2 for Young's modulus and 3/2 for thermal conductivity.¹⁶ For strongly anisometric pores these values are different (tending to infinity for infinitely thin, oblate pores, i.e. microcracks, and to 2.34 and 1.67 for prolate pores in the case of and , respectively^46,47). It has to be emphasised that the values can never be identical for and (at least as long as the solid phase is non-auxetic³²). The set of non-linear model relations given above for isotropic porous materials (equations (6)–(9)) is exhaustive in the sense that these relations are the only empirical-parameter-free model relations that reduce to the exact solution, equation (5), for vanishingly small porosity and at the same time do not violate the upper bounds (i.e. inter alia, they ensure that when , when ). All other relations occurring in the literature do not fulfil a least one of these criteria and are therefore ignored here. In particular, there are many more or less useful fit relations, both for the elastic moduli and for thermal conductivity. Among the most popular of these are two-parameter fit relations (generalised power-law relations), which are attributed to Phani and Niyogi⁶³ in elasticity context and to McLachlan⁶⁴ in conductivity context. Less well known, but equally useful, are our one-parameter fit relations, which are based on the Coble–Kingery-type relations, equation (7), but include a parameter that can be interpreted as a percolation threshold.^65,66 On the other hand, simple (Spriggs) exponential relations,⁶⁷ which are preferentially used in the context of so-called minimum solid area models,^{10,11,17–25} necessarily violate the upper bounds for sufficiently high porosity and should therefore be avoided altogether, both for predictive and fitting purposes.

Porosity dependence of the Poisson ratio

Due to the tensorial character of elastic properties (note that the proportionality coefficient in Hooke's law is a fourth-order tensor, so that even after considering all property symmetries the stiffness matrix has 21 elastic constants in the case of triclinic monocrystals, which are, of course, further reduced for materials of higher symmetry³³) two elastic constants are needed for a complete description of the elastic behaviour even in the simplest case of isotropic materials. Therefore it would be extremely convenient to dispose of a simple estimate for the Poisson ratio of porous materials. Unfortunately, however, the porosity dependence of the Poisson ratio is an unsolved problem so far. Nevertheless, when auxetic behaviour is ignored, most authors agree on the fact that with increasing porosity the effective Poisson ratio approaches an asymptotic value. This value has been differently identified to be 0.333, 0.25 or 0.2.^{1, 7, 68} Although experimental evidence is not unambiguous,⁶⁹ there are strong arguments in favour of the latter value.⁷⁰ In particular, both the differential scheme approximation⁵⁹ and the self-consistent approximation⁵⁰ predict the Poisson ratio of porous materials to approach 0.2 with increasing porosity.⁷⁰ Moreover, according to the Mori–Tanaka approach,⁷¹ also materials with microstructures realising the upper HS bounds tend to this value.⁷⁰ In particular, all these models and also the model relations mentioned above (linear, power-law and exponential relation) lead to the conclusion that the effective Poisson ratio of porous materials with solid Poisson ratio does not dependent on porosity at all.⁷⁰

Cross-property relations

Cross-property relations connect the effective or relative values of one property to those of another, different one. They can take the form of equations or inequalities and may be universally valid or not. The most well-known cross-property relation is probably the Levin relation⁷² for the calculation of the effective thermal expansion coefficient of two-phase materials (composites).^6–8 For porous materials, this relation leads to the conclusion that the effective thermal expansion coefficient is independent of porosity.⁹

Another relatively well-known cross-property relation (in the form of an inequality) is the Milton–Torquato cross-property bound (MT bound), a so-called ‘elementary’ bound between the relative bulk modulus and the relative thermal conductivity.^6,8 For isotropic porous materials with non-negative solid Poisson ratio (, i.e. when the solid phase or phase mixture itself is assumed to be non-auxetic) the MT bound can be written as⁷³ (10)

In this relation is the relative bulk modulus ( being the effective bulk modulus of the porous material and the bulk modulus of the dense, pore-free solid) and is the relative thermal conductivity. Invoking the elasticity standard relations³³ (11) and(12) and their counterparts for the dense solid (completely analogous, just replacing , , and by , , and , respectively) the corresponding inequality (here called MT bound) for the shear modulus can be calculated as(13) and for the tensile modulus (Young's modulus) as(14)

Note that, for these cross-property bounds to be valid, non-auxeticity of the porous materials as a whole is not required (only of the solid, see above). However, simpler, but also weaker (i.e. less restrictive, less useful), bounds are available when the porous material is non-auxetic.⁷³

Less well known, but more restrictive (and thus more useful) are the so-called ‘translation’ bounds, which have been derived by Berryman and Milton⁷⁴ for porous materials and by Gibiansky and Torquato⁷⁵ for general isotropic two-phase composites (here called BMGT bounds). For isotropic porous materials with a non-auxetic solid phase (or phase mixture), i.e. , the BMGT bound for the bulk modulus can be written as⁷³ (15)

The corresponding results for the shear modulus and the tensile modulus are obtained in an obvious way by applying the elasticity standard relations, equations (11) and (12), i.e.(16) (17)

In the extreme case the BMGT bound for the bulk modulus, equation (15), reduces to the MT bound for the bulk modulus, equation (10), and the bounds for and are simplified accordingly, whereas for (i.e. incompressible materials) the right hand sides of the bounds for and approach infinity (∞) and thus become useless in practice. Note that for incompressible materials the bulk modulus is infinity per definitionem and not bounded by the thermal conductivity at all.

It should be emphasised that the cross-property bounds for the bulk modulus depend only on the solid Poisson ratio , i.e. the Poisson ratio of the solid phase (or phase mixture), whereas the cross-property bounds for the shear and tensile moduli ( and ) depend also on the effective Poisson ratio of the porous material as a whole. However, assuming that the effective Poisson ratio usually approaches the value of 0.2 (see the discussion above), it is clear that the effective Poisson ratio cannot decrease to values below 0.2 as long as the solid Poisson ratio is higher than 0.2 (and, vice versa, cannot increase to values above 0.2, as long as is lower than 0.2).

Both the (very wide) MT cross-property bounds and the (more restrictive) BMGT cross-property bounds allow the calculation of upper bounds on the elastic moduli of isotropic porous materials as soon as the thermal conductivity of these materials is known (e.g. from measurements or finite element calculations). Cross-property bounds of this type are generally better (i.e. more precise, more restrictive or, in other words, lower) than the upper HS bounds, i.e. the best bounds that can be calculated based on the knowledge of porosity alone (for isotropic porous materials). The reason is that prior knowledge of one (effective or relative) property, here the thermal conductivity, which is a result not only of the porosity, but implicitly reflects all microstructural details, allows one to elegantly circumvent the explicit quantification of these microstructural details for determining another property (here elastic moduli). At the same time both MT and BMGT bounds may serve as a cross-check for the admissibility of the following cross-property relations.

In contrast to the cross-property bounds above, which are inequalities, cross-property relations in the form of equations (equalities) principally do not claim to be of universal validity, but their advantage is that they are extremely useful in practice, because they may lead to very concrete predictions of effective property values, as soon as some very rudimentary qualitative information on the type of microstructure beyond volume fractions and isotropy is available.

The most trivial cross-property relation of this type is valid for anisotropic porous materials with translational symmetry: in the direction of translation the relative axial Young's modulus is indeed equal to the relative axial thermal conductivity, i.e.(18) (cross-property relation based on the parallel model, equation (4), which is formally identical to the upper one-point bound = upper WP bound, see equation (1)). Of course, as discussed above, for isotropic porous materials this relation cannot be valid. Probably the earliest attempt to set up a cross-property relation of this type for isotropic porous materials has been undertaken by Sevostianov et al.^{76, 77} using the Mori–Tanaka approach.⁷¹ Their cross-property relation is(19)

(Sevostianov, Kovácˇik and Simancˇík/SKS cross-property relation). More recently, we have shown that an excellent approximation to this SKS cross-property relation can be derived in a simple way from the Maxwell-type model, equation (6) (which is formally identical to the upper two-point bound = upper HS bound, equations (2) and (3)).⁷³ Thus, for practical purposes, the SKS cross-property relation can be replaced by the much simpler cross-property relation,(20)

It can easily be shown that this cross-property relation deviates from the exact SKS cross-property relation by less than 4.3% for solid Poisson ratios in the range (difference in relative properties < 0.0011) and by less than 1.4% for (difference in relative properties <0.0003).⁷³

Nevertheless, although the upper HS bounds correspond theoretically to realisable microstructures,^6–8 such microstructures are at best extremely rare among real-world materials. In fact, in the field of inorganic materials we are not aware of a single example reported in the literature for which the property–porosity dependence is well described by the upper HS bound (Maxwell model). Therefore, for the vast majority of real-world material microstructures the SKS cross-property relation and its excellent approximation, equation (20), cannot be expected to provide realistic property predictions. Similar conclusions hold for the cross-property relations (21) and(22) which can be derived in an elementary way from the linear relations and the Coble–Kingery relations, equations (5) and (7), for Young's modulus and thermal conductivity, respectively.

On the other hand, in a previous paper⁷⁸ we have shown that the cross-property relation for isotropic porous materials obeying either the power–law relation, equation (8), or our exponential relation, equation (9), is the same, viz. (23)

This extremely simple cross-property relation is valid for microstructures exhibiting either a power-law or an exponential dependence of the properties on porosity and can therefore be expected to be realistic at least for all microstructures in between these two limits (actually, recent research results indicate its validity far beyond these limits). This fact makes this cross-property relation more general than any of the other cross-property relations, although the detailed determination of its limits of ‘validity beyond’ is a desideratum of future research.

A detailed comparison of these cross-property relations with experimental values is beyond the scope of this paper, but it should have become clear to the reader that all cross-property relations for isotropic porous materials, irrespective of one or the other is more realistic for the material in question, lead to the conclusion that the porosity dependence of the (relative) thermal conductivity is never identical with that of the (relative) Young's modulus. The same is true for other elastic moduli. In other words, all cross-property relations lead to the conclusion that for isotropic porous materials (and most anisotropic materials as well, except for those with translational symmetry in the axial direction) we have . Of course, this conclusion is in direct contradiction with minimum solid area models which claim that .^10,11,17,18 This claim is wrong. Therefore it is clear that minimum solid area models (also called minimum contact area models)^16–25 represent a blind alley of engineering science that should be abandoned.

Summary, conclusion and outlook

Based on a rational micromechanical framework and using the rigorous one-point (Voigt-Wiener) and two-point (HS) bounds, a critical assessment of model relations describing the porosity dependence of elastic properties (in particular Young's modulus) and thermal properties (in particular thermal conductivity) has been given. It has been shown that – in contrast to common belief and the impression evoked by many publications – the set of admissible and useful model relations for calculating the Young's modulus and thermal conductivity of isotropic porous materials is not very large. More specifically, when empirical fit relations with freely assignable fit parameters are ignored, the set of admissible predictive model relations reduces essentially to five types: linear relations (for small porosities), Maxwell-type relations, Coble–Kingery-type relations, power–law relations (corresponding to the well-known Gibson–Ashby relations for open-cell foams) and our (Pabst–Gregorová-type) exponential relations. Note that the popular simple (Spriggs-type) exponential relations have to be a priori discarded from this set of potentially predictive relations, because they necessarily violate the upper bounds for sufficiently high porosity. Finally, it is shown that – due to the same underlying geometrical models – there is a complete formal analogy between the model relations for the elastic moduli and thermal conductivity of porous materials and that, as a consequence, non-trivial cross-property relations exist between the relative Young's modulus and the relative thermal conductivity of isotropic porous materials. These cross-property relations arise due to rigorous mathematics and are not to be considered as empirical findings. Irrespective of the question as to which of the cross-property relations is optimal for a certain microstructure or type of porous material, all cross-property relations for isotropic porous materials lead to the conclusion that the relative Young's modulus (or any other elastic modulus) is not equal to the relative thermal conductivity. Thus, the mere existence of non-trivial cross-property relations of this type provides a refutation of the simplistic assumption of so-called minimum solid area models (also called minimum contact area models) that the relative Young's modulus equals the relative thermal conductivity. As a consequence, minimum solid or contact area models represent a blind alley of engineering science that should be abandoned.

Acknowledgement

This work is part of the project ‘Preparation and characterisation of oxide and silicate ceramics with controlled microstructure and modelling of microstructure–property relations’ (GA15-18513S), supported by the Czech Science Foundation (GACˇR).

Notes

*Paul²⁹ was apparently the first to recognise the significance of Voigt- and Reuss-type relations, i.e. arithmetic and harmonic volume-weighted means, respectively, as upper and lower bounds for multiphase materials.

^†Of course, these ‘microstructure-insensitive’ bounds and all model relations mentioned below remain valid only as long as elastic moduli and thermal conductivity are well defined via their corresponding linear constitutive equations, i.e. as long as the materials behave in a linear elastic manner according to Hooke's law,³³ i.e. do not exhibit non-linear elastic or elastoplastic behaviour, and heat transfer obeys Fourier's law,⁵ i.e. does not involve significant radiative or convective mechanisms.³⁴