Shear and bulk moduli of isotropic porous and cellular alumina ceramics predicted from thermal conductivity via cross-property relations

Ceramics International 44 (2018) 8100–8108

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Shear and bulk moduli of isotropic porous and cellular alumina ceramics predicted from thermal conductivity via cross-property relations

T

⁎

Willi Pabst , Tereza Uhlířová, Eva Gregorová Department of Glass and Ceramics, University of Chemistry and Technology, Prague (UCT Prague), Technická 5, 166 28 Prague 6, Czech Republic

A R T I C L E I N F O

A B S T R A C T

Keywords: Microstructures (A) Voids (A) Elastic material (B) Porous material (B) Shear and bulk modulus

Based on alumina as a paradigmatic example, shear and bulk moduli have been numerically calculated on computer-generated digital random microstructures representing isotropic porous materials with convex or concave pores and for cellular materials with closed or open cells. On the other hand, the relative elastic moduli cross-property relations (CPRs) have been predicted from the numerically calculated relative thermal conductivities. It has been shown that the Pabst-Gregorová cross-property relation (PG-CPR) with constant CPR exponent 4/3 or its generalized version with the correct Poisson-ratio-dependent CPR exponents (for alumina 1.316 and 1.426, respectively, for the shear and bulk modulus) provides the best prediction currently available, although its accuracy is significantly worse for the bulk modulus (maximum differences between predicted and calculated values ranging from – 0.05 to + 0.09 relative property units/RPU) than for the shear modulus, where the accuracy is excellent for all microstructures (maximum difference smaller than ± 0.02 RPU).

1. Introduction The prediction of the effective elastic properties of porous materials is a long-standing problem in ceramic materials science. It is known that purely model-based approaches do not provide reliable predictions, because the effective elastic properties are not determined by porosity, i.e. the volume fraction of the pore space, alone. Although the correlation function approach, masterly exposed in the authoritative treatments by Milton [1] and Torquato [2], may be an alternative for multiphase materials (composites) with small to moderate phase contrast, in the case of porous materials, where the phase contrast is extremely large, it has but limited success. This is due to the fact that from the well-known rigorous micromechanical bounds only upper bounds are available, because the lower bounds degenerate to zero when the pore phase properties are negligible compared to the solid phase properties. Therefore any attempt to develop alternative models based on porosity alone must necessarily result (at best) in nothing more than another relation for tentative prediction. On the other hand, despite the availability of systematic approaches to determine complete sets of microstructural descriptors of porous ceramics, e.g. in the form of Minkowski-functional-based parameters, [3,4] there is currently no theory available that would allow an implementation of these parameters into microstructure-property relations. Therefore the application of cross-property relations is currently the only way to go beyond volume fraction information and at the same time to circumvent the

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aforementioned problems. Cross-property relations (CPRs) are relations that connect one relative property, e.g. the thermal or electrical conductivity, with another relative property, e.g. elastic moduli. Rigorous CPRs between conductivity and elastic moduli are usually available only in the form of inequalities (so-called cross-property bounds), e.g. the “elementary“ cross-property bounds described by Milton [1] and Torquato [2] and the (more restrictive) so-called “translational“ cross-property bounds derived by Berryman and Milton [5] for isotropic porous materials and generalized by Gibiansky and Torquato [6] to isotropic two-phase composites (MT and BMGT cross-property bounds, respectively [7]). These cross-property bounds are usually more restrictive than the Hashin-Shtrikman bounds, [8,9] but in the case of porous materials, where lower Hashin-Shtrikman bounds are not available, also these crossproperty provide nothing more than useful tools for checking the general admissibility of calculated property values. It is exactly the non-availability of lower Hashin-Shtrikman bounds that makes modeling indispensable in the case of porous materials (in contrast to two-phase composites with lower phase contrast, where the Hashin-Shtrikman variational principle alone is often able to yield sufficiently accurate predictions of effective properties). Of the model relations available, power-law and our (Pabst-Gregorová-type) exponential relations [7] are the most successful ones in practice, in the sense that their effective property predictions are often close to experimental findings obtained for real-world materials. While the

Corresponding author. E-mail address: [email protected] (W. Pabst).

https://doi.org/10.1016/j.ceramint.2018.01.254 Received 17 December 2017; Received in revised form 29 January 2018; Accepted 29 January 2018 Available online 01 February 2018 0272-8842/ © 2018 Elsevier Ltd and Techna Group S.r.l. All rights reserved.

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former, also known under the names “differential (scheme) approximation” [10] or “Gibson-Ashby relation for open-cell foams”, [11] has been extensively used in the literature of cellular materials (ceramic foams), the latter has shown superior performance for the prediction of effective properties of porous materials with convex pores. [12,13] Compared to the power-law relation and our exponential relation, all other model-based relations and effective medium approximations are irrelevant, because they are either inadmissible from a general physical point of view (e.g. the simple Spriggs-type exponential, [14] which violates the upper Hashin-Shtrikman bound), contained as special cases in the relations above (e.g. the Maxwell-type models [2] for spherical pores are formally identical to the upper Hashin-Shtrikman bounds) or redundant and leading in the case of porous materials to spurious predictions of percolation thresholds [2] (e.g. self-consistent models, [15,16] which are for porous materials identical to the linear “noninteraction” approximations [7]). Now, based both on power-law relations and our exponential relations, it is possible to derive CPRs between elastic moduli and conductivity in the form of equations [17,18] that can be used for predicting the relative elastic modulus when the relative conductivity is known and vice versa (relative properties being defined as nondimensional ratios of the effective properties of porous materials and the properties of the dense, i.e. pore-free solid [7]). A CPR of similar type has been derived by Sevostianov et al. [19,20] However, the derivation of the latter is implicitly based on the assumption that the porosity dependence of the elastic modulus and conductvity follows exactly the upper Hashin-Shtrikman bound (or, equivalently, the Maxwell-type model relation for spherical pores [2]). However, materials with a microstructure realizing the upper Hashin-Shtrikman bound, albeit thinkable in theory, [1,2,10] are very rare in practice (to say the very least; as far as we know, not a single case of that type of porosity dependence has been reported in the ceramic literature so far). On the other hand, materials with a porosity dependence close to the powerlaw or our exponential relations (or somewhere in the range between these two) are very frequent, and for all these materials our CPR should be the CPR of choice. So far, a systematic investigation of our CPRs, in particular a comparison between our CPR predictions and numerical data for the shear and bulk modulus has not yet been performed. Therefore it is the primary aim of this paper to fill this gap and to present such a comparison on the basis of numerically calculated effective property data for computer-generated digital microstructures. In contrast to real-world materials, digital microstructures have the advantage that they can be “prepared without defects” and therefore represent “ideal” microstructures. Based on five paradigmatic types of microstructures (porous materials with isolated convex pores, overlapping convex pores or concave pores between partially sintered particles, as well as closedcell/wall-based and open-cell/strut-based foams) and taking the property values of dense polycrystalline alumina as input parameters, the numerically calculated effective shear and bulk moduli are confronted with rigorous bounds, model predictions and CPR predictions (based on independently calculated effective thermal conductivities). Although obtained for alumina-like materials, in a qualitative sense, the conclusions of this paper are valid for all materials with a non-auxetic solid phase.

conductivity, shear modulus and bulk modulus of the porous material as a whole and k 0 , G0 and K 0 the thermal conductivity, shear modulus and bulk modulus of the dense, pore-free solid; for example, for polycrystalline alumina ceramics at room temperature the latter are 33 W/ mK, 163 GPa and 247 GPa, respectively [21,22]). These bounds are called upper Wiener bounds [23] in the case of thermal conductivity and upper Paul bounds [24] or Voigt bounds [10] in the case of elastic moduli. Isotropic porous materials cannot realize these upper bounds, i.e. no isotropic microstructures are thinkable for which the aforementioned inequalities, Eq. (1a)–(1c), become equalities. However, for translationally invariant microstrutures (i.e. materials with pore channels of arbitrary cross section), these inequalities would indeed degenerate to equalities and thus for these specific (anisotropic) microstructures the relative property values in the direction of the translational axis would indeed be identical, i.e.

(in the sequel called Wiener-Paul cross-property relation and abbreviated WP-CPR). For isotropic porous materials more restrictive upper bounds are available for the relative properties, namely the upper Hashin-Shtrikman bounds, [8,9] which can be written as kr ≤

1−ϕ , 1 + ([k ] − 1) ϕ

Kr ≤

1−ϕ , 1 + ([K ] − 1)⋅ϕ

(3a,b,c)

15 (1 − ν0) , 7 − 5ν0

(4)

[K ] =

3 (1 − ν0) . 2 (1 − 2ν0)

(5)

For example, in the case of alumina, where the solid Poisson ratio ν0 is 0.23, [21] these coefficients are 1.974 and 2.139 for the shear and bulk modulus, respectively. Although the Hashin-Shtrikman bounds are universally valid laws of nature, derived on the basis of a variational approach [8,9] and thus principally model-independent, they are formally identical with the so-called Maxwell-type effective medium approximations. [2] Therefore the coefficients [k ], [G] and [K ] occurring in Eq. (3a)–(3c) correspond to the coefficients occurring in Maxwell-type effective medium approximations for the conductivity of materials with spherical pores [25] and the corresponding exact single-inclusion solutions for the elastic moduli that have been derived for spherical pores by Dewey [26] and Mackenzie. [27] Generalizations of these solutions are available for ellipsoidal shapes, [28] but not for other pore shapes. [29] For the implementation of the exact single-inclusion solutions for spheroidal pores into the effective medium approximations, the reader may refer to our previous papers. [30,31] When the solid Poisson ratio ν0 is 0.2 (sometimes called “magic” Poisson ratio [10]), it follows from Eqs. (4) and (5) that the coefficients [G] and [K ] for the elastic constants are both exactly 2, and the HashinShtrikman bounds for isotropic porous materials can be written as

1−ϕ . 1+ϕ

(6)

(identical for the shear and bulk moduli). For isotropic porous materials with microstructures realizing the upper Hashin-Shtrikman bound, corresponding e.g. to coated-spheres assemblages or hierarchical laminates, [1,2,10] the relative elastic moduli are related to the relative thermal conductivity via the relation [7]

The relative thermal conductivity, as well as the relative shear and bulk moduli, of porous materials with negligible pore phase property values are all bounded from above by the upper bounds

Kr ≤ 1 − ϕ ,

1−ϕ , 1 + ([G] − 1)⋅ϕ

[G] =

2. Theory – analytical relations

Gr ≤ 1 − ϕ,

Gr ≤

when the pore phase property values are zero or negligibly small compared to the solid property values. In these inequalities [k ] is 3/2, while [G] and [K ] are related to the solid Poisson ratio ν0 via the relations

Gr = Kr ≤

kr ≤ 1 − ϕ,

(2)

Gr = Kr = kr ,

(1a,b,c)

Gr = Kr =

where ϕ is the porosity (volume fraction of pores) and kr = k / k 0 , Gr = G / G0 and Kr = K / K 0 are the relative thermal conductivity and elastic moduli (with k , G and K being the effective thermal

3kr , 4 − kr

(7)

(in the sequel called Hashin-Shtrikman cross-property relation and 8101

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the digital model microstructures, and the isotropy of the microstructure has been checked throughout. The creation of random foams (closed-cell/wall-based or open-cell/strut-based) via the FoamGeo module of GeoDict® was based on random packings of 30 µm spheres and used a thinning factor of 0.5, i.e. the thickness and diameter, respectively, of walls and struts decreased to 0.5 of the original value in the middle between two edges or nodes. Solid walls and solid struts with circular cross section have been used in this work. All microstructures thus created have periodic boundaries on opposite faces of the representative volume elements, in order to allow the application of periodic boundary conditions for the numerical calculation of the effective properties.

abbreviated HS-CPR), obtained by combining Eq. (7) with Eq. (7a). Generalized versions of Eq. (7) for other solid Poisson ratios are obtained in an obvious way by combining Eq. (3b) or (3c) with Eq. (3a). The results are then identical to those obtained by Sevostianov et al. [19,20] Since the lower Wiener-Paul and Hashin-Shtrikman bounds degenerate to zero and thus become useless for porous materials, more concrete predictions can only be made via model relations. The most successful of these are the power-law relations

kr = (1 − ϕ)[k],

Gr = (1 − ϕ)[G],

Kr = (1 − ϕ)[K ],

(8a,b,c)

and our exponential relations −[k ] ϕ ⎞ kr = exp ⎜⎛ ⎟, ⎝1 − ϕ⎠

−[G] ϕ ⎞ ⎟, Gr = exp ⎜⎛ ⎝1−ϕ⎠

−[K ] ϕ ⎞ ⎟, Kr = exp ⎜⎛ ⎝1−ϕ⎠

3.2. Numerical calculation of effective properties (9a,b,c) Also the numerical calculations of effective properties have been performed using the commerical software package GeoDict® (Math2Market, Germany). [32] In particular, thermal conductivity tensor calculations have been performed via the ConductoDict module of GeoDict® (with a solver based on the so-called explicit jump immersed interface method [33,34]), whereas elastic tensor (stiffness matrix) calculations have been performed using the ElastoDict module of GeoDict® (based on an iterative solution, assisted by fast Fourier transform, of the Lippmann-Schwinger equation [35,36]). For both types of calculations periodic boundary conditions were adopted. The final effective property values have been obtained by orientational averaging of the corresponding tensors (thermal conductivity tensor and stiffness matrix). The input parameters used correspond to dense polycrystalline alumina with a random orientation of crystallites and air-filled pores at room temperature and atmospheric pressure (i.e. assuming for the solid phase Young's modulus 400 GPa, Poisson ratio 0.23, thermal conductivity 33 W/mK [21,22] and for the pore phase a thermal conductivity of 0.026 W/mK [37] and zero elastic moduli).

where for materials with spherical pores the value of [k ] is 3/2 and the values of [G] and [K ] are given by Eqs. (4) and (5) (e.g. for alumina with ν0 = 0.23 [G] = 1.974 and [K ] = 2.139). Remarkably from the physical point of view, for both types of dependences (power-law and exponential) one obtains the same type of cross-property relation between the elastic moduli and the thermal conductivity (in the sequel called PG-CPR), viz.

Gr = kr2[G]/3,

Kr = kr2[K ]/3,

(10a,b)

and for materials with solid Poisson ratio 0.2 this PG-CPR becomes [17,18]

Gr = Kr = kr4/3.

(11)

It should be emphasized that in the case of the shear modulus Eq. (11) is a good approximation to Eq. (10a) for all materials with solid Poisson ratios in the range from 0 to close to 0.5, whereas in the case of the bulk modulus Eq. (11) is a reasonable approximation to Eq. (10b) only as long as the solid Poisson ratio is sufficiently close to 0.2. Actually, when the solid Poisson ratio is zero (ν0 = 0, corresponding to an ideally compressible solid), the PG-CPR for the bulk modulus, Eq. (10b), degenerates to the WP-CPR, Eq. (2), whereas for essentially incompressible solids (i.e. with ν0 approaching 0.5) the PG-CPR (similar to the HS-CPR) tends to predict zero bulk moduli irrespective of the thermal conductivity. For alumina (with solid Poisson ratio ν0 = 0.23) the PG-CPR for the shear modulus is practically indistinguishable from Eq. (11), whereas for the bulk modulus there is a small, but non-negligible, difference between the two. The latter will be taken into account below. The CPR exponents for alumina that follow from inserting [G] = 1.974 and [K ] = 2.139 into Eq. (10a) and (10b) are 1.316 and 1.426, respectively.

4. Results and discussion Fig. 1 shows representative examples of the digital model microstructures generated in this work. From this figure it is evident that the present work covers a wide range of different microstructures, ranging from matrix-inclusion to bicontinuous microstructures and from convex pores to concave pores and saddle surfaces. Of course, the porosity range covered depends on the microstructure. While for foams and overlapping convex (spherical) pores the attainable porosity range is essentially from 0 to 1 (i.e. until the body disintegrates into fragments), that of the microstructures with concave pores is limited by the space complementary to the packing fraction of monosized spherical particles in point contact (i.e. 1–0.524 = 0.476, 1–0.680 = 0.320 and 1–0.741 = 0.259, for periodic SC, BCC and FCC packing, respectively, and approximately 1–0.64 = 0.36 for random packing of spherical particles in point contact) and that of isolated spherical pores is limited here by the algorithm available, which does not achieve 50% porosity without pore contact or overlap (therefore, and due to the 10% porosity steps chosen in this case, the porosities considered here for isolated spherical pores do not exceed 40%, although slightly higher values are thinkable). Figs. 2–4 show numerically calculated data for the effective thermal conductivity, shear modulus and bulk modulus, respectively, of isotropic porous alumina ceramics with a random arrangement of convex (spherical) pores (isolated or overlapping, see Fig. 1a and b, respectively) in comparison to the upper bounds (upper Wiener or Paul bounds, and upper Hashin-Shtrikman bounds) and the model predictions (power-law and exponential). With respect to the fact that the Poisson ratio of alumina is 0.23, i.e. rather close to the “magic“ value of 0.2, the model-based predictions for the shear and bulk moduli are similar. Actually, in terms of relative elastic moduli the differences between the predictions for the shear and bulk moduli are almost indistinguishable. This is due to the fact that the coefficients [G] and [K ], see

3. Modeling approach 3.1. Generation of digital model microstructures Digital model microstructures of different types have been generated using the commerical software package GeoDict® (Math2Market, Germany) [32] in the form of representative volume elements with 200 × 200 × 200 voxels (voxel edge length 1 µm). Microstructures with convex pores (spherical, isolated or overlapping) have been prepared using the GrainGeo module of GeoDict®, with pore diameters chosen to be 15 µm. Microstructures with randomly arranged concave pores (i.e. intergranular voids between convex isometric grains) have been obtained using the GrainGeo module of GeoDict® by creating a random close packing (with packing density 64%) of spherical particles with diameter 30 µm and letting the particle diameters grow uniformly, thus transforming point contacts into contact areas and reducing the porosity down to values approaching zero. The spatial distribution of pores or particles has been chosen to be random but uniform in all cases, periodic boundary conditions have been used throughout for creating 8102

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Fig. 1. Computer-generated digital model microstructures; (a) top left: spherical isolated pores (porosity 30%), (b) bottom left: spherical overlapping pores (porosity 30%), (c) top middle: closedcell (wall-based) foam (porosity 68%), (d) bottom middle: open-cell (strut-based) foam (porosity 85%), (e) top right: concave pores (intergranular voids, porosity 28%), (f) bottom right: concave pores (intergranular voids, porosity 10%).

35

180 Paul +

160

30 HS +

Shear modulus [GPa]

Thermal conductivity [W/mK]

Wiener +

25 power

20

expon

15 10

Convex random isolated

5

140

HS +

120 power

100 80

expon

60 Convex random isolated

40 20

Convex random overlap

0

Convex random overlap

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

Porosity [1]

0.4

0.5

0.6

0.7

0.8

0.9

1

Porosity [1]

Fig. 2. Porosity dependence of the thermal conductivity of isotropic porous alumina ceramics with a random arrangement of convex (spherical) pores (isolated/full triangles or overlapping/empty triangles); data points numerically calculated on digital model microstructures, compared to the upper Wiener bound (Wiener +), upper HashinShtrikman bound (HS +), power-law prediction (power) and exponential prediction (expon).

Fig. 3. Porosity dependence of the shear modulus of isotropic porous alumina ceramics with a random arrangement of convex (spherical) pores (isolated/full circles or overlapping/empty circles); data points numerically calculated on digital model microstructures, compared to the upper Paul bound (Paul +), upper Hashin-Shtrikman bound (HS +), power-law prediction (power) and exponential prediction (expon).

250 Paul +

Eqs. (5) and (6), are very similar and rather close to 2.0. It is evident that the effective thermal conductivity and elastic moduli of microstructures with isolated pores are rather close to the power-law predictions, whereas for overlapping pores the effective properties are slightly lower (albeit still higher than the exponential prediction). This finding is interesting, because it contradicts the common belief, based on cellular materials research, that the powerlaw prediction (which corresponds to the Gibson-Ashby prediction for open-cell foams) should generally be more useful for open-cell materials than for closed-cell materials. Actually, the present results clearly show, that in the case of convex (spherical) pores the power-law relation provides a better prediction for materials with closed (isolated) pores than for materials with open (overlapping) pores. Nevertheless, it has to be recalled that both the power-law and the exponential prediction (or any other effective medium approximation) provide only tentative predictions. That means, the relative success of any of these (model-based) predictions can be assessed only after the “true“ values are known (either from measurements on real-world materials or, as in the present case, from numerical calculations on digital microstructures).

Bulk modulus [GPa]

200

HS +

150

power

expon

100

Convex random isolated

50

Convex random overlap

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Porosity [1] Fig. 4. Porosity dependence of the bulk modulus of isotropic porous alumina ceramics with a random arrangement of convex (spherical) pores (isolated/full squares or overlapping/empty squares); data points numerically calculated on digital model microstructures, compared to the upper Paul bound (Paul +), upper Hashin-Shtrikman bound (HS +), power-law prediction (power) and exponential prediction (expon).

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Table 1 Maximum deviations of different cross-property relation (CPR) predictions for the shear modulus of digital microstructures with randomly arranged convex pores (isolated or overlapping), cellular materials (closed-cell and open-cell random foams/RF) and random microstructures with concave pores from the numerically calculated values; all values in relative property units (RPU).

Convex isolated Convex overlap RF closed-cell RF open-cell Concave

HS-CPR

PG-CPR (4/3)

PG-CPR (1.316)

+ + + + +

− + − − +

− + − + +

0.02 0.05 0.03 0.04 0.05

0.01 0.01 0.02 0.01 0.02

0.01 0.01 0.02 0.01 0.02

Table 2 Maximum deviations of different cross-property relation (CPR) predictions for the bulk modulus of digital microstructures with randomly arranged convex pores (isolated or overlapping), cellular materials (closed-cell and open-cell random foams/RF) and random microstructures with concave pores from the numerically calculated values; all values in relative property units (RPU).

Fig. 5. Porosity dependence of the shear modulus of isotropic porous alumina ceramics with a random arrangement of convex (spherical) pores (isolated/full circles or overlapping/empty circles); data points numerically calculated on digital model microstructures, compared to the Pabst-Gregorová cross-property (PG-CPR) predictions with CPR exponent 4/3.

Convex isolated Convex overlap RF closed-cell RF open-cell Concave

Predictions via cross-property relations (CPRs) are of a completely different character, since they use available information on one (relative) property to predict the value of another (relative) property. In particular, when the thermal conductivity is known (e.g. from previous numerical calculations), the elastic moduli can be predicted. Figs. 5 and 6 show the Pabst-Gregorová cross-property relation (PG-CPR) predictions of shear and bulk moduli for random microstructures with convex (spherical) pores (isolated or overlapping), calculated via Eqs. (10) and (11), in comparison to the directly calculated values of the elastic moduli. From these figures it is evident that the PG-CPRs provide very accurate predictions for the shear and bulk moduli of microstructures with convex (spherical) pores, irrespective of whether they are isolated (closed) or overlapping (open) and irrespective of whether the CPR exponent is assumed to be 4/3 (amounting to assuming the solid Poisson ratio to be approximately 0.2) or specified according to the true Poisson ratio of the solid in question. In the case of the shear modulus the PG-CPR prediction with CPR exponent 4/3 is almost identical to that with CPR exponent 1.316 (for materials with solid Poissson ratio 0.23, e.g. alumina) and therefore not shown here, while for the bulk modulus the former is slightly higher than the PG-CPR prediction with CPR exponent 1.426 (for alumina); therefore both predictions are shown for the bulk modulus in Fig. 6. It is evident that the difference between the PG-CPR predictions and the true values (i.e. the

HS-CPR

PG-CPR (4/3)

PG-CPR (1.426)

+ + + + +

− + − + +

− − − − +

0.03 0.06 0.03 0.05 0.12

0.01 0.01 0.03 0.03 0.09

0.03 0.01 0.05 0.02 0.07

numerically calculated data) is very small. In terms of relative property units (RPU), taking the property value of the fully dense solid (alumina) as unity, the maximum difference for these microstructures is ± 0.01 (i.e. 1%) for the shear modulus and ranges from – 0.03 to + 0.01 RPU for the bulk modulus, see Tables 1 and 2. It may be noted that in the present case (alumina) the predictions are not critically dependent on the exact value of the CPR coefficient, because the solid Poisson ratio of alumina (0.23) is close to the “magic“ Poisson ratio of 0.2. Obviously, for solids with a Poisson ratio far away from 0.2, the prediction via the “ideal“ CPR exponent 4/3 (correct for solids with Poisson ratio 0.2) becomes less reliable, especially for the bulk modulus, and should in any case be replaced by a prediction using the correct CPR coefficient for the Poisson ratio of the solid in question. Naturally this statement applies equally to all other results below. Figs. 7–9 show the porosity dependences of the effective thermal conductivity, shear modulus and bulk modulus, respectively, of isotropic cellular alumina ceramics with random microstructure (i.e. closed-cell/wall-based or open-cell/strut-based random foams, see Fig. 1c and d), again in comparison to the upper bounds (upper Wiener or Paul bounds, and upper Hashin-Shtrikman bounds) and the model predictions (power-law and exponential). It is evident that in this case the range of values significantly exceeds the power-law prediction for closed-cell foams with high porosities and is slightly lower than the exponential prediction for open-cell foams with low porosities. Nevertheless, Figs. 10 and 11 show that also in this case the PG-CPRs provide very accurate predictions for the shear modulus (maximum difference between predictions and true values ranging from – 0.02 to + 0.01 RPU, see Tables 1 and 2) and slightly worse, but still satisfactory, predictions for the bulk modulus of random closed-cell or open-cell foams (maximum difference between predictions and true values ranging from – 0.05 to + 0.03 RPU). Finally, Figs. 12–14 show the porosity dependences of the effective thermal conductivity, shear modulus and bulk modulus, respectively, of isotropic porous alumina ceramics with a random arrangement of concave pores, i.e. intergranular voids between spherical particles (convex grains, see Fig. 1e and f), also here in comparison to the upper bounds (upper Wiener or Paul bounds and upper Hashin-Shtrikman bounds) and the model predictions (power-law and exponential). In both cases (shear and bulk moduli) the effective property values are

Fig. 6. Porosity dependence of the bulk modulus of isotropic porous alumina ceramics with a random arrangement of convex (spherical) pores (isolated/full or overlapping/ empty squares); data points numerically calculated on digital model microstructures, compared to the Pabst-Gregorová cross-property (PG-CPR) predictions with CPR exponents 4/3 and 1.426 (bold and slender curves, respectively).

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Shear modulus [GPa]

160

RF closed-cell

140 120

RF open-cell

100 80 RF closed-cell PG-CPR

60 40

RF open-cell PG-CPR

20 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Porosity [1] Fig. 10. Porosity dependence of the shear modulus of isotropic cellular alumina ceramics (closed-cell, i.e. wall-based, and open-cell, i.e. strut-based random foams/RF, denoted by full and empty circles, respectively); data points numerically calculated on digital model microstructures, compared to the Pabst-Gregorová cross-property (PG-CPR) predictions with CPR exponent 4/3.

Fig. 7. Porosity dependence of the thermal conductvity of isotropic cellular alumina ceramics (closed-cell, i.e. wall-based, and open-cell, i.e. strut-based random foams/RF, denoted by full and empty triangles, respectively); data points numerically calculated on digital model microstructures, compared to the upper Wiener bound (Wiener +), upper Hashin-Shtrikman bound (HS +), power-law prediction (power) and exponential prediction (expon).

250 RF closed-cell

180 Paul +

200

140

Bulk modulus [GPa]

Shear modulus [GPa]

160

HS +

120 power

100 80

expon

60

20

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

100

RF closed-cell PG-CPR (1.426)

RF open-cell PG-CPR (4/3)

0

0 0.1

RF closed-cell PG-CPR (4/3)

RF open-cell PG-CPR (1.426)

RF open-cell

0

150

50

RF closed-cell

40

RF open-cell

0

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Porosity [1]

Porosity [1]

Fig. 11. Porosity dependence of the shear modulus of isotropic cellular alumina ceramics (closed-cell, i.e. wall-based, and open-cell, i.e. strut-based random foams/RF, denoted by full and empty squares, respectively); data points numerically calculated on digital model microstructures, compared to the Pabst-Gregorová cross-property (PG-CPR) predictions with CPR exponents 4/3 and 1.426 (bold and slender curves, respectively).

Fig. 8. Porosity dependence of the shear modulus of isotropic cellular alumina ceramics (closed-cell, i.e. wall-based, and open-cell, i.e. strut-based random foams/RF, denoted by full and empty circles, respectively); data points numerically calculated on digital model microstructures, compared to the upper Paul bound (Paul +), upper Hashin-Shtrikman bound (HS +), power-law prediction (power) and exponential prediction (expon). 250 Paul +

Bulk modulus [GPa]

200 HS +

150

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expon

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50 RF open-cell

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0.1

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Porosity [1]

Fig. 12. Porosity dependence of the thermal conductivity of isotropic porous alumina ceramics with a random arrangement of concave pores, i.e. intergranular voids between spherical particles (convex grains); data points numerically calculated on digital model microstructures, compared to the upper Wiener bound (Wiener +), upper HashinShtrikman bound (HS +), power-law prediction (power) and exponential prediction (expon).

Fig. 9. Porosity dependence of the bulk modulus of isotropic cellular alumina ceramics (closed-cell, i.e. wall-based, and open-cell, i.e. strut-based random foams/RF, denoted by full and empty squares, respectively); data points numerically calculated on digital model microstructures, compared to the upper Paul bound (Paul +), upper Hashin-Shtrikman bound (HS +), power-law prediction (power) and exponential prediction (expon).

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Porosity [1] Fig. 13. Porosity dependence of the shear modulus of isotropic porous alumina ceramics with a random arrangement of concave pores, i.e. intergranular voids between spherical particles (convex grains); data points numerically calculated on digital model microstructures, compared to the upper Paul bound (Paul +), upper Hashin-Shtrikman bound (HS +), power-law prediction (power) and exponential prediction (expon).

Fig. 16. Porosity dependence of the bulk modulus of isotropic porous alumina ceramics with a random arrangement of concave pores, i.e. intergranular voids between spherical particles (convex grains); data points numerically calculated on digital model microstructures, compared to the Pabst-Gregorová cross-property (PG-CPR) predictions with CPR exponents 4/3 and 1.426 (bold and slender curve, respectively).

Fig. 14. Porosity dependence of the bulk modulus of isotropic porous alumina ceramics with a random arrangement of concave pores, i.e. intergranular voids between spherical particles (convex grains); data points numerically calculated on digital model microstructures, compared to the upper Paul bound (Paul +), upper Hashin-Shtrikman bound (HS +), power-law prediction (power) and exponential prediction (expon).

significantly below the exponential predictions and thus even more below the power-law predictions (and of course the upper bounds). In fact, since minimum solid area models (minimum contact area models) have turned out to be wrong and useless, [38] there is actually no analytical model available that would allow any – albeit tentative – prediction based on the knowledge of the porosity (pore volume fraction) alone. It is therefore quite remarkable that even in this unfavorable situation the PG-CPR provides an excellent prediction for the shear modulus (maximum difference between predictions and true values + 0.02 RPU), see Fig. 15. For the bulk modulus the PG-CPR prediction is admittedly less accurate (maximum difference between predictions and true values up to + 0.09 RPU), see Fig. 16, but still better than any other prediction currently available, see also Tables 1 and 2. This is also nicely seen on Figs. 17 and 18, which show the correlation between the relative elastic moduli (shear and bulk modulus, respectively) and the relative thermal conductivity, as calculated numerically, in comparison to the different CPRs (WP-CPR, HS-CPR and PG-CPR). In particular, Fig. 17 shows that for all microstructures treated in this paper the shear modulus is most accurately predicted by the PG-CPR (with CPR exponent 4/3, the difference to the correct value 1 WP-CPR

Relative shear modulus [1]

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Relative thermal conductivity [1] Fig. 15. Porosity dependence of the shear modulus of isotropic porous alumina ceramics with a random arrangement of concave pores, i.e. intergranular voids between spherical particles (convex grains); data points numerically calculated on digital model microstructures, compared to the Pabst-Gregorová cross-property (PG-CPR) prediction with CPR exponent 4/3.

0.9

1 Concave random

Fig. 17. Correlation between the relative shear modulus and the relative thermal conductivity for isotropic porous and cellular alumina ceramics with isometric pores or cells (convex isolated/full circles, convex overlapping/empty circles, open-cell/full squares, closed-cell/empty squares, concave/full triangles); data points numerically calculated on digital model microstructures, compared to the three different cross-property relations mentioned in the text (WP-CPR, HS-CPR and PG-CPR with CPR exponent 4/3, the PG-CPR with CPR exponent 1.316 being indistinguishable from the latter).

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can also be used to predict the effective Poisson ratio of isotropic porous materials. From the results of this work it is clear, however, that the possibilities to predict the bulk modulus, even via PG-CPRs, are much more limited than for the shear modulus. Since it is known from previous work that the PG-CPR prediction for the tensile modulus (Young's modulus) is even more accurate that for the shear modulus, estimates of the effective Poisson ratio of isotropic porous materials should always be calculated from the effective tensile and shear moduli. Finally let us mention that using our recently proposed generalized version of the PGCPR for spheroidal pores, [18] it is possible to apply the present approach to isotropic porous materials with microstructures containing randomly oriented anisometric pores of spheroidal (prolate or oblate) shape.

WP-CPR

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5. Summary and conclusions

Concave random

Based on alumina as a paradigmatic example, effective thermal conductivities, shear moduli and bulk moduli have been numerically calculated on computer-generated digital random microstructures representing isotropic porous materials with convex pores (isolated or overlapping) or concave pores and for cellular materials (random foams) with closed cells (i.e. wall-based cellular materials) or open cells (i.e. strut-based cellular materials). The numerically calculated elastic moduli have been compared to the rigorous micromechanical bounds (upper Wiener or Paul bounds and upper Hashin-Shtrikman bounds), the two most important model predictions (power-law and exponential) and, in particular, the cross-property relations (CPRs) that use the knowledge of the (previously calculated) relative thermal conductivity for predicting the relative elastic moduli. The results show that all numerically calculated effective properties are below the upper Hashin-Shtrikman bounds (and thus much below the upper Wiener and Paul bounds), as required for isotropic porous materials. The effective elastic moduli of microstructures with overlapping convex pores or open cells are generally lower than those for isolated pores or closed cells, although this difference tends to be overestimated in the literature. While for microstructures with convex pores (isolated or overlapping) and for random foams (closed-cell or open-cell) the effective elastic moduli are relatively close to the powerlaw or exponential predictions, they are significantly below the latter for microstructures with concave pores. From the CPRs currently available for the prediction of relative elastic moduli from the relative thermal conductivity, the PabstGregorová cross-property relation (PG-CPR) with constant CPR exponent 4/3 or its generalized version with the correct Poisson-ratio-dependent CPR exponent (1.316 and 1.426, respectively, for the shear and bulk modulus of materials with solid Poisson ratio 0.23), which has been applied here for the first time to shear and bulk moduli, provides the most realistic prediction of relative elastic moduli when the relative thermal conductivity is known. This prediction is highly accurate for all microstructures in the case of the shear modulus (maximum difference between predicted and calculated values smaller than ± 0.02 relative property units/RPU). In the case of the bulk modulus the prediction is of a similar quality for microstructures with convex pores and for random foams with open cells (maximum difference smaller than ± 0.03 RPU), but significantly worse for random foams with closed cells (underestimating the true values by up to – 0.05 RPU) and for microstructures with concave pores (overestimating the true values by up to + 0.09 RPU). Although in the latter case the PG-CPR clearly overestimates the true values, it is still the best prediction currently available.

Fig. 18. Correlation between the relative bulk modulus and the relative thermal conductivity for isotropic porous and cellular alumina ceramics with isometric pores or cells (convex isolated/full circles, convex overlapping/empty circles, open-cell/full squares, closed-cell/empty squares, concave/full triangles); data points numerically calculated on digital model microstructures, compared to the three different cross-property relations mentioned in the text (WP-CPR, HS-CPR and PG-CPR with CPR exponents 4/3 and 1.426).

of 1.316 for alumina being completely negligible). Fig. 18, on the other hand, shows that for the bulk modulus there can be major differences between predicted values and true values, especially in the case of closed-cell random foams (where the PG-CPR underestimates the true values) and in the case of concave pores (where all CPRs, including the PG-CPR, grossly overestimate the true values). Nevertheless, in the absence of more specific microstructural information the PG-CPR, preferably with the CPR exponent corresponding to the solid Poisson ratio in question (i.e. 1.416 for alumina with solid Poisson ratio 0.23), is still the safest choice for estimating the effective bulk moduli of porous materials. It has to be emphasized that the qualitative conclusions of this paper are independent of the fact that alumina (with solid Poisson ratio 0.23) has been chosen as a model material. In particular, for the shear modulus both the analytical and the numerical results are only very weakly dependent on the solid Poisson ratio (as long as it is non-negative), see Eq. (4). In the case of the bulk modulus the PG-CPR prediction is significantly less accurate for some microstructures, especially for closed-cell foams (for which the HS-CPR can be better, especially when the porosity is high, because the relative bulk moduli of this specific microstructure are indeed relatively close to the upper HashinShtrikman bound, see Fig. 8) and microstructures with concave pores (for which the PG-CPR is still better than any other prediction). The reason of these problems for the bulk modulus is obviously related to the fact that the coefficients for the bulk modulus are not only strongly dependent on the solid Poisson ratio, see Eq. (5), but also the pore shape dependence is much stronger for the bulk modulus than for the shear (and tensile) modulus, as shown in our recent paper. [18] Therefore, for the bulk modulus any deviation from spherical pore shape results in a more significant deviation from our CPR predictions (which have been derived from model relations that are related to the exact single-inclusion solution for spherical pores) than for the shear (and tensile) modulus. We would like to remind the reader in this context that the effective Poisson ratio of isotropic porous materials is not bounded by any rigorous micromechanical bounds and currently no reliable theory exists for its prediction by analytical relations. The only bound on the effective Poisson ratio is the thermodynamic bound – 1 < ν < 0.5. Nevertheless, it is clear that when two independent elastic moduli are known (or reliable estimates for them available, e.g. using the PG-CPR), any other elastic constant of isotropic materials, including the Poisson ratio, can be calculated. That means, in an indirect way, the PG-CPRs

Acknowledgement This work is part of the project “Preparation and characterization of oxide and silicate ceramics with controlled microstructure and modeling of microstructure-property relations” (GA15-18513S), supported by the Czech Science Foundation (GAČR). 8107

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References

[18] W. Pabst, E. Gregorová, A generalized cross-property relation between the elastic moduli and conductivity of isotropic porous materials with spheroidal pores, Ceram. Silik. 61 (1) (2017) 74–80. [19] I. Sevostianov, J. Kováčik, F. Simančík, Correlation between elastic and electric properties for metal foams: theory and experiment, Int. J. Fract. 114 (2002) L23–L28. [20] I. Sevostianov, J. Kováčik, F. Simančík, Elastic and electric properties of closed-cell aluminum foams cross-property connection, Mater. Sci. Eng. A 420 (2006) 87–99. [21] W. Pabst, G. Tichá, E. Gregorová, Effective elastic properties of alumina-zirconia composite ceramics − Part III: calculation of elastic moduli for polycrystalline alumina and zirconia from monocrystal data, Ceram. Silik. 48 (2) (2004) 41–48. [22] W. Pabst, E. Gregorová, Effective thermal and thermoelastic moduli of alumina, zirconia and alumina-zirconia composite ceramics, in: B.M. Caruta (Ed.), New Developments in Materials Science Research. Nova Science Publishers, New York, 2007, pp. 77–137. [23] O. Wiener, Die Theorie des Mischkörpers für das Feld der stationären Strömung, Abh. Math. Phys. Kl. Königl. Sächs. Ges. Wissen. 32 (1912) 509–604. [24] B. Paul, Prediction of elastic constants of multiphase materials, Trans. Metall. Soc. ASME 218 (1960) 36–41. [25] J.C. Maxwell, A Treatise on Electricity and Magnetism 1 Clarendon Press, Oxford, 1873, pp. 435–449 (reprint Dover, New York 1954). [26] J.M. Dewey, The elastic constants of materials loaded with non-rigid fillers, J. Appl. Phys. 18 (1947) 578–581. [27] J.K. Mackenzie, Elastic constants of a solid containing spherical holes, Proc. Phys. Soc. (Lond.) B 63 (1950) 2–11. [28] J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. Roy. Soc. Lond. A 241 (1957) 376–396. [29] V.A. Lumbrarda, X. Markenscoff, On the absence of Eshelby's property for nonellipsoidal inclusions, Int. J. Solids Struct. 35 (1998) 3405–3411. [30] W. Pabst, E. Gregorová, Conductivity of porous materials with spheroidal pores, J. Eur. Ceram. Soc. 34 (11) (2014) 2757–2766. [31] W. Pabst, E. Gregorová, Young's modulus of isotropic porous materials with spheroidal pores, J. Eur. Ceram. Soc. 34 (13) (2014) 3195–3207. [32] Math2Market: GeoDict® software, version, 2015 and 2017. [33] A. Wiegmann, K.P. Bube, The explicit-jump immersed interface method: finite difference methods for PDEs with piecewise smooth solutions, SIAM J. Numer. Anal. 37 (3) (2000) 827–862. [34] A. Wiegmann, A. Zemitis, A fast explicit jump harmonic averaging solver for the effective heat conductivity of composite materials. Technical Report No. 94. Fraunhofer-Institut für Techno- und Wirtschaftsmathematik (Fraunhofer ITWM), Kaiserslautern, 2006. [35] H. Andrä, N. Combaret, J. Dvorkin, E. Glatt, J. Han, M. Kabel, Y. Keehm, F. Krzikalla, M. Lee, C. Madonna, M. Marsh, T. Mukerji, E.H. Saenger, R. Sain, N. Saxena, S. Ricker, A. Wiegmann, X. Zhan, Digital rock physics benchmarks – Part II: computing effective properties, Comput. Geosci. 50 (2013) 33–43. [36] M. Schneider, F. Ospald, M. Kabel, Computational homogenization of elasticity on a staggered grid, Int. J. Numer. Methods Eng. 105 (2016) 693–720. [37] W. Pabst, J. Hostaša, Thermal conductivity of ceramics – from monolithic to multiphase, from dense to porous, from micro to nano, in: M.C. Wythers (Ed.), Advances in Materials Science Research, 7 Nova Science Publishers, New York, 2011, pp. 1–112. [38] W. Pabst, E. Gregorová, Minimum solid area models for the effective properties of porous materials – a refutation, Ceram. Silik. 59 (3) (2015) 244–249.

[1] G.W. Milton, The Theory of Composites, Cambridge University Press, Cambridge, 2002, pp. 75–498. [2] S. Torquato, Random Heterogeneous Materials – Microstructure and Macroscopic Properties (Ch. 16–23), Springer, New York, 2002, pp. 403–646. [3] J. Ohser, F. Mücklich, Statistical Analysis of Microstructures in Materials Science, John Wiley & Sons, Chichester, 2000, pp. 57–106. [4] W. Pabst, T. Uhlířová, E. Gregorová, Microstructure characterization of porous ceramics via Minkowski functionals, in: D. Singh, M. Fukushima, Y.-W. Kim, K. Shimamura, N. Imanaka, T. Ohji, J. Amoroso, M. Lanagan (Eds.), Proceedings of the 12th Pacific Rim Conference on Ceramic and Glass Technology (Ceramic Transactions, Volume 264 – A Collection of Papers Presented at the 12th Pacific Rim Conference on Ceramic and Glass Technology, Waikoloa/Hawaii, USA, 21-26 May 2017), The American Ceramic Society/John Wiley & Sons, Inc., Hoboken / NJ, USA, 2018, pp. 53–64. [5] J.G. Berryman, G.W. Milton, Microgeometry of random composites and porous media, J. Phys. D: Appl. Phys. 21 (1988) 87–94. [6] L.V. Gibiansky, S. Torquato, Connection between the conductivity and bulk modulus of isotropic composite materials, Proc. R. Soc. Lond. A 452 (1996) 253–283. [7] W. Pabst, E. Gregorová, Critical assessment 18: elastic and thermal properties of porous materials – rigorous bounds and cross-property relations, Mater. Sci. Technol. 31 (15) (2015) 1801–1808. [8] Z. Hashin, S. Shtrikman, A variational approach to the theory of the effective magnetic permeability of multiphase materials, J. Appl. Phys. 33 (1962) 3125–3131. [9] Z. Hashin, S. Shtrikman, A variational approach to the theory of the elastic behavior of multiphase materials, J. Mech. Phys. Solids 11 (1963) 127–140. [10] K. Markov, Micromechanics of heterogeneous media, in: K. Markov, L. Preziosi (Eds.), Heterogeneous Media – Micromechanics Modeling and Simulations. Boston, Birkhäuser, Boston, 2000, pp. 1–162. [11] L.J. Gibson, M.F. Ashby, (and 283–308), Cellular Solids – Structure and Properties, second edition, Cambridge University Press, Cambridge, 1997, pp. 175–234. [12] W. Pabst, E. Gregorová, Exponential porosity dependence of thermal conductivity, in: J.R. Koenig, H. Ban (Eds.), Thermal Conductivity 29/Thermal Expansion 17, DEStech Publications Inc., Lancaster, 2008, pp. 487–498. [13] W. Pabst, E. Gregorová, M. Černý, Isothermal and adiabatic Young's moduli of alumina and zirconia ceramics at elevated temperatures, J. Eur. Ceram. Soc. 33 (2013) 3085–3093. [14] W. Pabst, T. Uhlířová, E. Gregorová, Microstructure characterization of porous ceramics via Minkowski functionals, in: D. Singh, M. Fukushima, Y.-W. Kim, K. Shimamura, N. Imanaka, T. Ohji, J. Amoroso, M. Lanagan (Eds.), Proceedings of the 12th Pacific Rim Conference on Ceramic and Glass Technology (Ceramic Transactions, Volume 264 – A Collection of Papers Presented at the 12th Pacific Rim Conference on Ceramic and Glass Technology, Waikoloa/Hawaii, USA, 21–26 May 2017), 44 The American Ceramic Society / John Wiley & Sons, Inc., Hoboken / NJ, USA, 2018, pp. 53–64. [15] D.A.G. Bruggeman, Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielectrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen, Ann. Phys. Leipz. 24 (1935) 636–679. [16] R. Landauer, The electrical resistance of binary metallic mixtures, J. Appl. Phys. 23 (1952) 779–784. [17] W. Pabst, E. Gregorová, A cross-property relation between the tensile modulus and the thermal conductivity of porous materials, Ceram. Int. 33 (2007) 9–12.

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