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Relative Young’s modulus and thermal conductivity of isotropic porous ceramics with randomly oriented spheroidal pores – Model-based relations, cross-property predictions and numerical calculations
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Original Article
Relative Young’s modulus and thermal conductivity of isotropic porous ceramics with randomly oriented spheroidal pores – Model-based relations, cross-property predictions and numerical calculations ⁎
Willi Pabsta, , Tereza Uhlířováa, Eva Gregorováa, Andreas Wiegmannb a b
Department of Glass and Ceramics, University of Chemistry and Technology, Prague (UCT Prague), Technická 5, 166 28 Prague 6, Czech Republic Math2Market GmbH, Stiftsplatz 5, 67655 Kaiserlautern, Germany
A R T I C LE I N FO
A B S T R A C T
Keywords: Elastic modulus (Young’s modulus) Porosity Thermal conductivity Anisometric pores Spheroidal pore shape (prolate, oblate)
Based on computer-generated digital microstructures with spherical or spheroidal pores (isolated or overlapping) of aspect ratio 1 (spherical), 10 (prolate) and 0.1 (oblate), the relative thermal conductivity and Young’s modulus is numerically calculated and compared to predictions based on generalized effective medium approximations (EMAs) and a recently proposed generalized cross-property relation (CPR). It is shown that, when the aspect ratio is known, generalized power-law and exponential relations provide satisfactory predictions of these two properties without the use of empirical fit parameters. The maximum deviation of these two types of EMA predictions from the true values ranges from −0.04 to +0.06 relative property units (RPU) for the powerlaw relation and from −0.02 to −0.10 RPU for the exponential relation. However, the generalized CPR results in an even higher accuracy of the predictions, with maximum deviations smaller than 0.01 RPU for the Young’s modulus when the thermal conductivity is known.
1. Introduction Elastic moduli and thermal conductivity are basic properties of ceramics that play a role in almost all practical applications of these materials. For a ceramic material of given composition these properties can be efficiently controlled via the microstructure, in particular the porosity, i.e. the volume fraction of the pore space. For this reason the investigation of the porosity dependence of effective properties has been a mainstay of ceramic research for decades, and considerable efforts have been devoted to the description, restriction, estimation and prediction of this porosity dependence. However, the number of predictive tools is relatively limited, more limited than commonly assumed. Indeed the effective properties of porous materials usually depend on porosity, but often they cannot be reliably predicted based on porosity information alone. In principle, the majority of effective properties depend on all details of the microstructure, the systematic characterization of which requires great efforts and does not always lead to satisfactory results. For example, the correlation function approach (using higher-order correlation functions up to fourth order) [1,2], albeit very elegant and perfectly general, does in the case of porous materials not provide more than slight improvements of the upper bounds (up to fourth order) of the (thermal or electrical) conductivity
⁎
and elastic moduli. Similarly, although a systematic characterization of the global microstructural features of porous materials can be done using Minkowski-functional-based descriptors [3,4], the implementation of the resulting parameters into microstructure-property relations is still a desideratum of future research. Therefore it is useful to investigate alternative ways that can be used to improve the reliability of estimates and predictions concerning the effective properties of porous materials. There are principally at least two ways to do this: either the tentative application of effective medium approximations (EMAs) that have turned out to be successful in practice, e.g. power-law and exponential relations [5–7], or the application of cross-property relations (CPRs) [8–10]. In the case of spherical or isometric pores the former approach has the advantage that a priori estimates or predictions are available “right from the desk”, without any previous property measurements and without more than basic qualitative microstructural information, while the latter has the advantage that no microstructural information is required at all, but measurements of one property are required to estimate another property. In the case of anisometric pores the situation becomes more complicated. Even in the simplest case of randomly oriented anisometric pores, where the overall material symmetry remains isotropic, both the effective medium approximations and the cross-property relations
Corresponding author. E-mail address: [email protected] (W. Pabst).
https://doi.org/10.1016/j.jeurceramsoc.2018.04.051 Received 14 January 2018; Received in revised form 19 April 2018; Accepted 21 April 2018 0955-2219/ © 2018 Elsevier Ltd. All rights reserved.
Please cite this article as: Pabst, W., Journal of the European Ceramic Society (2018), https://doi.org/10.1016/j.jeurceramsoc.2018.04.051
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spheroidal pores, for which a single aspect ratio is sufficient to uniquely define the pore shape. In two recent papers [15,16] Pabst and Gregorová have shown how these single-inclusion solutions have to be implemented into the EMAs for thermal conductivity and elastic constants, including Young’s modulus. For materials with monosized spheroidal pores the procedure can be summarized as follows: insert the Maxwell coefficient [k ] and the Eshelby-Wu coefficient [E ], respectively, as defined in the two aforementioned papers [15,16] for the aspect ratio in question (larger than unity for prolate and smaller than unity for oblate pore shape), into the generalized Maxwell relations,
depend on the pore shape. Fortunately, exact analytical solutions are available for spheroidal pores in the low-porosity limit [11–14], and these solutions can be implemented into the nonlinear effective medium approximations [15,16] and cross-property relations [10]. The present paper makes use of these solutions and compares, for the first time, the resulting aspect-ratio-based predictions for the effective thermal conductivity and Young’s modulus with the results of numerical calculations for computer-generated digital microstructures with isolated or overlapping monosized pores (prolate or oblate). 2. Theoretical
kr = It is well known that effective and relative properties such as conductivities and elastic moduli of porous materials with negligible pore phase property values are bounded from above by rigorous bounds, viz. the upper Wiener and Paul bounds for all microstructures (including anisotropic ones) [17,18] and the upper Hashin-Shtrikman bounds [19,20] for isotropic microstructures. However, all lower bounds of this contrast type [1] degenerate to zero when the pore phase property approaches zero. Therefore model-based predictions are indispensable tools for the a priori estimation of the effective or relative properties of porous materials. There are essentially four types of effective medium approximations (EMAs) that can be used for this purpose, viz. Maxwell-type relations [1,21], self-consistent relations [1,22,23], power-law (or differential scheme) approximations [24–26] and exponential relations of the Pabst-Gregorová type [6,7,9]. All these relations are well known and their spherical pore variants have been more or less (some more, some less) successful in predicting the effective properties of real isotropic materials with isometric pores. However, some isotropic materials have (randomly oriented) anisometric pores, either prolate (elongated) or oblate (flattened). Extreme cases of these anisometric pores would be elongated pore channels and microcracks, respectively. For these cases, as well as their more moderate counterparts, it can be useful or even necessary to include pore shape information into the relations for property prediction. In practice, for real materials, such shape information is usually extracted by microscopic image analysis. However, modern computer technology enables one to create digital microstructures with isolated or overlapping pores of specified shape, so that virtual microstructure control and material design is possible. Moreover, the numerical algorithms of commercial software packages available today for the numerical calculation of effective properties are mature to a degree that allows one to take the calculated numerical values as excellent approximations to the “true “values (much more accurate and precise than any experimentally accessible data), with the advantage that virtual materials with welldefined and defect-free digital microstructures can be used for these calculations. From the viewpoint of theory, any model-based prediction of porous materials properties must be based on the exact solutions of the corresponding single-inclusion problems. Analytical solutions of this type, in the case of elastic properties going back to Eshelby’s classical paper [12], are available for ellipsoidal pores, but not for other pore shapes [27]. (In fact, only numerical solutions exist for non-ellipsoidal pore shapes.) Relatively simple are the single-inclusion solutions for
1−ϕ , 1 + ([k ] − 1) ϕ
Er =
1−ϕ , 1 + ([E ] − 1) ϕ
(1a,b)
the generalized self-consistent relations,
kr = 1 − [k ] ϕ, Er = 1 − [E ] ϕ,
(2a,b)
the generalized power-law relations,
kr = (1 − ϕ)[k] , Er = (1 − ϕ)[E ] ,
(3a,b)
or the generalized exponential relations,
kr = exp
( ), −[k ] ϕ 1−ϕ
Er = exp
(
−[E ] ϕ 1−ϕ
),
(4a,b)
where ϕ is the porosity (=volume fraction of the pore space), and kr = k k 0 and Er = E E0 are the relative thermal conductivity and Young’s modulus, i.e. the non-dimensional ratios of the effective conductivity and modulus of the porous materials (k and E ) and the corresponding quantities of the fully dense solid (k 0 and E0 ). Note that in the case of porous materials with negligible pore property values the self-consistent relation automatically degenerates to the linear approximation (also called dilute or non-interaction approximation [1,2,24]), i.e. the single-inclusion solution itself. It can therefore not be expected to provide realistic predictions of porous materials properties for any other than very low porosities. Based on the latter two relations, which are the most common and successful predictive relations occurring in the literature, a generalized cross-property relation (CPR) between the relative thermal conductivity and the relative Young’s modulus can be derived [10]:
Er = krΩ,
(5)
where Ω = [E ] [k ] may be called the CPR exponent. Using this CPR it is possible to predict the relative Young’s modulus when the relative thermal conductivity is known and vice versa. For the sake of completeness it has to be noted that, apart from the aspect ratio of the pores, the Eshelby-Wu coefficient of Young’s modulus is (very slightly) dependent on the Poisson ratio of the solid phase. In contrast to other elastic moduli, however, it is an advantage of Young’s modulus that its Eshelby-Wu coefficient is rather insensitive vis-a-vis any changes in the solid Poisson ratio, as long as the solid phase does not become auxetic [16]. Table 1 lists Maxwell coefficients and Eshelby-Wu coefficients (the latter for different solid Poisson ratios from 0.1 to 0.4). It is evident that the Eshelby-Wu coefficients for Young’s modulus are rather insensitive to significant changes of the Poisson ratio. In fact the predictions are practically indistinguishable for porous materials based on a different solid phase, as long as the solid phase is non-auxetic.
Table 1 Maxwell and Eshelby-Wu coefficients (the latter for different values of solid Poisson ratios). Maxwell coefficient
Spherical (1) Prolate (10) Oblate (0.1)
1.500 1.647 3.111
Eshelby-Wu coefficient for solid Poisson ratio … 0.1
0.2
0.23
0.33
0.4
1.973 2.304 5.251
2.000 2.304 5.250
2.004 2.299 5.233
2.000 2.270 5.111
1.980 2.234 4.964
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materials to 264–306 isolated spheroidal pores in the 20% porosity materials and up to 529–917 overlapping spheroidal pores in the 40–60% porosity materials, respectively. For spherical pores the numbers per volume are higher, because of the smaller size, resulting in 45 spherical isolated pores in the 1% porosity materials up to 4077 spherical overlapping pores in the 90% porosity materials.
Therefore the results obtained for a special material (here alumina with a solid Poisson ratio of 0.23 [28]) are essentially of general validity, as can be readily verified. Although not explicitly shown in this paper, the same is true for the results of numerical calculations. The Maxwell coefficients for the thermal conductivity are 1.500, 1.647 and 3.111 for spherical, prolate and oblate pores with aspect ratios 1, 10 and 0.1, respectively [15], while the Eshelby-Wu coefficients of Young’s modulus are (for a material with solid Poisson ratio 0.23) 2.004, 2.299 and 5.233, respectively [16]. Based on these values the corresponding CPR exponents are 1.336, 1.396 and 1.682, respectively [10].
3.2. Numerical calculation of effective properties Thermal conductivity tensor calculations have been performed via the GeoDict® ConductoDict module (with a solver based on the so-called explicit jump immersed interface method [30,31]), whereas elastic tensor (stiffness matrix) calculations have been performed using the Geo-Dict® ElastoDict module (based on an iterative solution, assisted by fast Fourier transform, of the Lippmann-Schwinger equation [32,33]). For both types of calculations periodic boundary conditions were assumed. For the thermal conductivity tensor calculations a temperature gradient of 2 K and for the stiffness matrix calculations a strain of 0.005 has been imposed across the cubic box in either direction (x, y, z), and calculations have been performed separately in all three directions. To give an order-of-magnitude estimate, calculation times range from a few seconds to a few minutes for thermal conductivity and from a few minutes to approximately 2–3 h for the elastic properties. For the random microstructures the thermal conductivity values in the three principal directions have just been averaged to obtain the final value, while orientationally averaged Young’s modulus values have been obtained in the isotropic approximation by orientational averaging of the stiffness matrix. The solid phase of the model material is assumed to be polycrystalline alumina with randomly oriented crystallites, i.e. the solid phase is assumed to be statistically isotropic with room temperature values of thermal conductivity, Young’s modulus and Poisson ratio of 33 W/mK, 400 GPa and 0.23, respectively, for the homogenized dense solid phase [28,34]. The pore space is assumed to be filled with air at atmospheric pressure, i.e. the elastic moduli of the pore phase are zero and the thermal conductivity is 0.026 W/mK at room temperature [34]. Since the thermal conductivity of the pore phase is smaller than the solid conductivity by more than three orders of magnitude, the former is negligible and the results of this paper are essentially the same as for vacuum void space. Therefore the results of this paper are
3. Modeling procedure 3.1. Creation of digital microstructures The digital microstructures in this work were created using the GeoDict® software package [29] in cubic boxes with 200 × 200 × 200 cubic voxels (voxel edge length 1 μm). Representative volume elements of microstructures with randomly arranged spherical pores and randomly arranged and randomly oriented spheroidal pores (prolate and oblate) have been generated using the GeoDict® GrainGeo module. In all cases the pores were chosen to be monosized and with fixed aspect ratios of 1, 10 and 0.1 for spherical, prolate and oblate pores, respectively (dimensions 15 μm, 100 × 10 × 10 μm and 50 × 50 × 5 μm, respectively). Microstructures with porosities of 1%, 2%, 5%, 10% and subsequent steps of 5 or 10% were created up to values approaching the highest porosity attainable, which was restricted to around 40% porosity in the case of isolated spherical pores and around 20% in the case of isolated prolate and oblate pores (for overlapping pores the attainable porosities are much higher, viz. up to 90%, 60% and 40%). The spatial distribution of the pores has been chosen to be uniform in all cases, and periodic boundaries have been created in such a way that opposite faces of the cubic boxes are compatible, in order to enable effective property calculations with periodic boundary conditions. The orientation of the pores has been chosen to be random, i.e. without any degree of preferential orientation, so that the resulting microstructures are as isotropic as possible. The number of pores in the cubic boxes ranges from 13 to 15 oblate or prolate isolated pores in the 1% porosity
Fig. 1. Computer-generated digital microstructures of isotropic porous materials with randomly oriented spheroidal pores (prolate and oblate with aspect ratios 10 and 0.1, respectively) with different porosity ϕ : prolate isolated with ϕ = 10% (A) and ϕ = 20% (B), prolate overlapping with ϕ = 20% (C) and ϕ = 40% (D), oblate isolated with ϕ = 10% (E) and ϕ = 20% (F), oblate overlapping with ϕ = 20% (G) and ϕ = 40% (H). 3
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theoretical maximum porosity for random packing is 64% [35], while the maximum porosity that can be achieved for digital microstructures without loosing integrity is much higher (e.g. higher than 70% when using the GrainGeo module of GeoDict®). In the case of microstructures with anisometric pores the situation is qualitatively similar, but the maximum porosities are lower. For isolated prolate pores with aspect ratios 0.1 and 0.05 the maximum porosities achieved using the GrainGeo module of GeoDict® are lower than 30 and 20%, respectively, while for overlapping prolate pores porosities as high as 60 and 40%, respectively, can be readily achieved. Similarly, for isolated oblate pores with aspect ratios 10 and 20 the maximum porosities achieved using the GrainGeo module of GeoDict® are lower than 30 and 5%, respectively, while for overlapping prolate pores porosities as high as 40 and 20%, respectively, can be readily achieved. It is evident that oblate pores lead to disintegration of the material at lower porosities than prolate pores. This is of course expected, since
essentially independent of the choice of alumina as a model material. More precisely, they are valid for any porous material with negligible pore phase property values and solid Poisson ratio 0.23 (and, more than that, they are excellent approximations for any porous material with non-negative solid Poisson ratio, as indicated above). 4. Results and discussion Fig. 1 shows examples of the computer-generated digital microstructures of isotropic porous materials with randomly oriented anisometric pores (spheroidal prolate and oblate with aspect ratios 0.1 and 10, respectively) with different porosity ϕ (10 and 20% for isolated pores, 20 and 40% for overlapping pores). It is clear that the maximum porosity that can be achieved is higher for overlapping pores than for isolated pores. This is similar in the case of isometric pores. For example, in the case of identical (i.e. monosized) spherical pores the
Fig. 2. Porosity dependence of the relative thermal conductivity (top) and Young’s modulus (bottom) of porous materials with spherical (aspect ratio 1) pores, isolated or overlapping; numerically calculated relative property data compared to the Maxwell-type, self-consistent (=linear), power-law and exponential effective medium approximations (EMAs). 4
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lower than the Maxwell-type EMAs. In other words, the Maxwell-type EMAs seem to provide practical upper bounds for these data. For spherical pores this is not surprising, since the Maxwell-type EMAs for spherical pores are formally identical to the upper Hashin-Shtrikman bounds for all isotropic porous materials. Therefore it is clear that in this case all effective conductivity and elastic modulus values must always be below (or at best equal to) this bound. However, a corresponding rigorous argument does not exist for spheroidal pores. Nevertheless, since the Maxwell and Eshelby-Wu coefficients exhibit minimum values for spherical pores [15,16], it is clear that the relative conductivity and elastic modulus values of materials with spheroidal pores must always be lower than the corresponding values for materials with spherical pores. It is therefore plausible (although a rigorous proof does not exist), that the Maxwell-type EMA for materials with
oblate pores with high aspect ratio can be viewed as model shapes for cracks or microcracks, which can lead to the cleavage or segmentation of a specimen into separate parts (fragments) even at volume fractions approaching zero. Figs. 2–4 show the porosity dependences of the relative thermal conductivity (Figs. 2a, Figure 3a and Figure 4a) and Young’s modulus (Figs. 2b, Figure 3b and Figure 4b) of porous materials with spherical (aspect ratio 1), spheroidal prolate (aspect ratio 10) and spheroidal oblate (aspect ratio 0.1) pores, either isolated or overlapping. In these graphs numerically calculated relative property data are compared to the Maxwell-type, self-consistent (=linear), power-law and exponential effective medium approximations (EMAs), using the Maxwell and Eshelby-Wu coefficients listed in Table 1. It is evident that in all cases the numerically calculated values are
Fig. 3. Porosity dependence of the relative thermal conductivity (top) and Young’s modulus (bottom) of porous materials with spheroidal prolate (aspect ratio 10) pores, isolated or overlapping; numerically calculated relative property data compared to the Maxwell-type, self-consistent (=linear), power-law and exponential effective medium approximations (EMAs).
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Fig. 4. Porosity dependence of the relative thermal conductivity (top) and Young’s modulus (bottom) of porous materials with spheroidal oblate (aspect ratio 0.1) pores, isolated or overlapping; numerically calculated relative property data compared to the Maxwell-type, self-consistent (=linear), power-law and exponential effective medium approximations (EMAs).
The closest agreement between predictions and numerically calculated relative property values is found for the power-law (also called differential scheme) EMAs and our exponential EMAs. More precisely, the power-law EMA predictions are usually closest to the numerically calculated values in the case of isolated pores (overestimating these values by up to 0.02 RPU for spherical and prolate pores and underestimating them by up to −0.04 RPU for oblate pores, see Table 2), while the exponential EMA predictions seems to provide a practical lower bound in all cases and closely approximate the relative property values in the case of overlapping oblate pores (underestimating the values by up to –0.10, –0.08 and –0.04 RPU for spherical, prolate and oblate pores, respectively, see Table 2). It has to be admitted that the agreement between any of these two
spheroidal pores provides some kind of upper bound at least for the case of monosized spheroidal pores. Notwithstanding these details, however, it should be emphasized that in neither case (neither spherical nor spheroidal pores) does the Maxwell-type EMA provide a satisfactory prediction of the relative property values. As mentioned in Section 2 above, the self-consistent EMAs degenerate to linear predictions when the pores are empty voids (vacuum pore space). Therefore they principally fail to provide even a qualitatively correct prediction of the intrinsically nonlinear curve shape. What is worse, they predict very concrete but spurious percolation thresholds, as has been pointed out by Torquato [1] and is nicely seen in Fig. 2 through Fig. 4. For this reason it is fully justified to ignore them in the sequel.
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Table 2 Maximum deviation (in relative property units / RPU) of the power-law predictions, exponential predictions and CPR predictions from the numerically calculated data for porous materials with spherical (aspect ratio 1), prolate (aspect ratio 10) or oblate (aspect ratio 0.1) pores; in parentheses the approximate porosities at which this maximum deviation occurs (for other porosities the deviation is smaller). Pore shape and connectivity
Deviation of the power-law prediction for kr
Deviation of the power-law prediction for Er
Deviation of the exponential prediction for kr
Deviation of the exponential prediction for Er
Deviation of the CPR prediction for Er
Spherical isolated Spherical overlapping Prolate isolated Prolate overlapping Oblate isolated Oblate overlapping
+0.022 (40 %) +0.050 (55 %) +0.015 (20 %) + 0.054 (50 %) –0.033 (20 %) +0.055 (35 %)
+0.011 (15 %) +0.063 (75 %) +0.008 (10 %) + 0.051 (50 %) –0.043 (20 %) +0.032 (35 %)
–0.075 (40 %) –0.098 (60 %) –0.015 (20 %) – 0.084 (60 %) –0.073 (20 %) –0.036 (40 %)
–0.086 (40 %) –0.065 (50 %) –0.030 (20 %) – 0.052 (50 %) –0.084 (20 %) –0.021 (25 %)
–0.013 (35 %) +0.010 (70 %) –0.012 (20 %) + 0.007 (60 %) –0.007 (20 %) –0.012 (15 %)
Fig. 5. Porosity dependence of the relative Young’s modulus of porous materials with spherical pores of aspect ratio 1; comparison of numerically calculated data and CPR predictions (please note that the curves are predictions, without any fitting involved).
independent method, e.g. by microscopic image analysis. From this point of view the situation of the generalized CPR is not different from that of generalized EMAs (where also independent aspect ratio information is required, when these EMAs are to be used as parameterfree predictions). However, the CPR has the advantage of a much higher accuracy of the resulting predictions. We would like to emphasize that, in contrast to other elastic moduli, the results for the Young’s modulus are practically independent of the Poisson ratio, as can easily be shown by numerical calculations with other Poisson ratios and is evident also from the fact that the EshelbyWu coefficient for the Young’s modulus are practically independent of the Poisson ratio [16]. Therefore the results of this paper are practically independent of the specific material parameters that have been used here (taking alumina with air-filled pores as a paradigmatic example). In other words, the results obtained in this paper are valid for any porous ceramic material with a non-auxetic solid phase. More than that, they are valid also for electrical (instead of thermal) conductivity (as long as Ohm’s law is valid), i.e. they can directly be used e.g. for predicting the relative Young’s modulus of porous metals from their relative electrical conductivity.
types of predictions (power-law and exponential EMAs) and the numerically calculated values is not too bad, given the fact that the comparison of these model predictions and the “true” values (which are for random microstructures accessible only via numerical calculations) does not involve any kind of fitting. (Especially in the practically most important case of oblate pores the agreement between predictions and true values is surprisingly good.) That means, in contrast to many works by other authors, the present procedure does not rely upon the use of an empirical fit parameter for predicting the porosity dependence. In this sense all our predictions are “parameter-free” and can be used as a priori estimates in cases where an independent aspect ratio information is available, e.g. from microscopic image analysis. Nevertheless, we will see now that the predictions can be considerably improved by the use of our generalized cross-property relation (CPR) [10]. Figs. 5–7 show the porosity dependences of the relative Young’s modulus of porous materials with spherical (aspect ratio 1), spheroidal prolate (aspect ratio 10) and spheroidal oblate (aspect ratio 0.1) pores, respectively, either isolated or overlapping, in compared with the corresponding CPR prediction. It is immediately evident that in all cases the agreement between the predictions and the true values is much better than for any EMA prediction. Indeed using our generalized CPR it is possible to predict the relative Young’s modulus with accuracy better than ± 0.01 RPU, when the relative thermal conductivity is known or vice versa. Of course, in order to use this generalized CPR [10] in the context of real materials, the aspect ratio must be determined by an
5. Summary and conclusion Based on computer-generated digital microstructures with isolated or overlapping spherical or spheroidal pores of aspect ratio
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Fig. 6. Porosity dependence of the relative Young’s modulus of porous materials with prolate spheroidal pores of aspect ratio 10; comparison of numerically calculated data and CPR predictions (please note that the curves are predictions, without any fitting involved).
Fig. 7. Porosity dependence of the relative Young’s modulus of porous materials with oblate spheroidal pores of aspect ratio 0.1; comparison of numerically calculated data and CPR predictions (please note that the curves are predictions, without any fitting involved).
Acknowledgement
1 (spherical), 10 (prolate) and 0.1 (oblate), the relative thermal conductivity and Young’s modulus have been numerically calculated for materials with different porosity and compared to predictions based on generalized effective medium approximations (EMAs) and a recently proposed generalized cross-property relation (CPR). It has been shown that, when the aspect ratio is known, the generalized power-law relation and the generalized exponential relation provide relatively satisfactory predictions of these two properties without the use of empirical fit parameters. In terms of relative property units (RPU), the maximum deviation of these two types of EMA predictions from the “true “values (which can be calculated only numerically) ranges from −0.04 to +0.06 RPU for the power- law relation and from −0.02 to −0.10 RPU for the exponential relation. However, using the generalized CPR has been shown to result in an even higher accuracy of the predictions, with maximum deviations smaller than 0.01 RPU for the Young’s modulus when the thermal conductivity is known.
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heterogenen Substanzen. I. Dielectrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen, Ann. Phys. Leipzig 24 (1935) 636–679. R. Landauer, The electrical resistance of binary metallic mixtures, J. Appl. Phys. 23 (1952) 779–784. K. Markov, Micromechanics of heterogeneous media, in: K. Markov, L. Preziosi (Eds.), Heterogeneous Media – Micromechanics Modeling and Simulations, Birkhäuser, Boston, 2000, pp. 1–162. J.G. Berryman, Long-wavelength propagation in composite elastic media II. Ellipsoidal inclusions, J. Acoust. Soc. Am. 68 (1980) 1820–1831. N. Phan-Thien, D.C. Pham, Differential multiphase models for polydispersed spheroidal inclusions: thermal conductivity and effective viscosity, Int. J. Eng. Sci. 38 (2000) 73–88. V.A. Lubarda, X. Markenscoff, On the absence of Eshelby’s property for non-ellipsoidal inclusions, Int. J. Solids Struct. 35 (1998) 3405–3411. 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. GeoDict® - The Digital Material Laboratory (software package, versions 2015 and 2017). Math2Market, Kaiserslautern 2015 and 2017 (www.geodict.com, info@ math2market.de). 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. 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. 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. M. Schneider, F. Ospald, M. Kabel, Computational homogenization of elasticity on a staggered grid, Int. J. Numer. Methods Eng. 105 (2016) 693–720. 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, vol. 7, Nova Science Publishers, New York, 2011, pp. 1–112. S. Torquato, T.M. Truskett, P.G. Debenedetti, Is random close packing of spheres well defined? Phys. Rev. Lett. 84 (2000) 2064–2067.
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Journal of the European Ceramic Society journal homepage: www.elsevier.com/locate/jeurceramsoc
Original Article
Relative Young’s modulus and thermal conductivity of isotropic porous ceramics with randomly oriented spheroidal pores – Model-based relations, cross-property predictions and numerical calculations ⁎
Willi Pabsta, , Tereza Uhlířováa, Eva Gregorováa, Andreas Wiegmannb a b
Department of Glass and Ceramics, University of Chemistry and Technology, Prague (UCT Prague), Technická 5, 166 28 Prague 6, Czech Republic Math2Market GmbH, Stiftsplatz 5, 67655 Kaiserlautern, Germany
A R T I C LE I N FO
A B S T R A C T
Keywords: Elastic modulus (Young’s modulus) Porosity Thermal conductivity Anisometric pores Spheroidal pore shape (prolate, oblate)
Based on computer-generated digital microstructures with spherical or spheroidal pores (isolated or overlapping) of aspect ratio 1 (spherical), 10 (prolate) and 0.1 (oblate), the relative thermal conductivity and Young’s modulus is numerically calculated and compared to predictions based on generalized effective medium approximations (EMAs) and a recently proposed generalized cross-property relation (CPR). It is shown that, when the aspect ratio is known, generalized power-law and exponential relations provide satisfactory predictions of these two properties without the use of empirical fit parameters. The maximum deviation of these two types of EMA predictions from the true values ranges from −0.04 to +0.06 relative property units (RPU) for the powerlaw relation and from −0.02 to −0.10 RPU for the exponential relation. However, the generalized CPR results in an even higher accuracy of the predictions, with maximum deviations smaller than 0.01 RPU for the Young’s modulus when the thermal conductivity is known.
1. Introduction Elastic moduli and thermal conductivity are basic properties of ceramics that play a role in almost all practical applications of these materials. For a ceramic material of given composition these properties can be efficiently controlled via the microstructure, in particular the porosity, i.e. the volume fraction of the pore space. For this reason the investigation of the porosity dependence of effective properties has been a mainstay of ceramic research for decades, and considerable efforts have been devoted to the description, restriction, estimation and prediction of this porosity dependence. However, the number of predictive tools is relatively limited, more limited than commonly assumed. Indeed the effective properties of porous materials usually depend on porosity, but often they cannot be reliably predicted based on porosity information alone. In principle, the majority of effective properties depend on all details of the microstructure, the systematic characterization of which requires great efforts and does not always lead to satisfactory results. For example, the correlation function approach (using higher-order correlation functions up to fourth order) [1,2], albeit very elegant and perfectly general, does in the case of porous materials not provide more than slight improvements of the upper bounds (up to fourth order) of the (thermal or electrical) conductivity
⁎
and elastic moduli. Similarly, although a systematic characterization of the global microstructural features of porous materials can be done using Minkowski-functional-based descriptors [3,4], the implementation of the resulting parameters into microstructure-property relations is still a desideratum of future research. Therefore it is useful to investigate alternative ways that can be used to improve the reliability of estimates and predictions concerning the effective properties of porous materials. There are principally at least two ways to do this: either the tentative application of effective medium approximations (EMAs) that have turned out to be successful in practice, e.g. power-law and exponential relations [5–7], or the application of cross-property relations (CPRs) [8–10]. In the case of spherical or isometric pores the former approach has the advantage that a priori estimates or predictions are available “right from the desk”, without any previous property measurements and without more than basic qualitative microstructural information, while the latter has the advantage that no microstructural information is required at all, but measurements of one property are required to estimate another property. In the case of anisometric pores the situation becomes more complicated. Even in the simplest case of randomly oriented anisometric pores, where the overall material symmetry remains isotropic, both the effective medium approximations and the cross-property relations
Corresponding author. E-mail address: [email protected] (W. Pabst).
https://doi.org/10.1016/j.jeurceramsoc.2018.04.051 Received 14 January 2018; Received in revised form 19 April 2018; Accepted 21 April 2018 0955-2219/ © 2018 Elsevier Ltd. All rights reserved.
Please cite this article as: Pabst, W., Journal of the European Ceramic Society (2018), https://doi.org/10.1016/j.jeurceramsoc.2018.04.051
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spheroidal pores, for which a single aspect ratio is sufficient to uniquely define the pore shape. In two recent papers [15,16] Pabst and Gregorová have shown how these single-inclusion solutions have to be implemented into the EMAs for thermal conductivity and elastic constants, including Young’s modulus. For materials with monosized spheroidal pores the procedure can be summarized as follows: insert the Maxwell coefficient [k ] and the Eshelby-Wu coefficient [E ], respectively, as defined in the two aforementioned papers [15,16] for the aspect ratio in question (larger than unity for prolate and smaller than unity for oblate pore shape), into the generalized Maxwell relations,
depend on the pore shape. Fortunately, exact analytical solutions are available for spheroidal pores in the low-porosity limit [11–14], and these solutions can be implemented into the nonlinear effective medium approximations [15,16] and cross-property relations [10]. The present paper makes use of these solutions and compares, for the first time, the resulting aspect-ratio-based predictions for the effective thermal conductivity and Young’s modulus with the results of numerical calculations for computer-generated digital microstructures with isolated or overlapping monosized pores (prolate or oblate). 2. Theoretical
kr = It is well known that effective and relative properties such as conductivities and elastic moduli of porous materials with negligible pore phase property values are bounded from above by rigorous bounds, viz. the upper Wiener and Paul bounds for all microstructures (including anisotropic ones) [17,18] and the upper Hashin-Shtrikman bounds [19,20] for isotropic microstructures. However, all lower bounds of this contrast type [1] degenerate to zero when the pore phase property approaches zero. Therefore model-based predictions are indispensable tools for the a priori estimation of the effective or relative properties of porous materials. There are essentially four types of effective medium approximations (EMAs) that can be used for this purpose, viz. Maxwell-type relations [1,21], self-consistent relations [1,22,23], power-law (or differential scheme) approximations [24–26] and exponential relations of the Pabst-Gregorová type [6,7,9]. All these relations are well known and their spherical pore variants have been more or less (some more, some less) successful in predicting the effective properties of real isotropic materials with isometric pores. However, some isotropic materials have (randomly oriented) anisometric pores, either prolate (elongated) or oblate (flattened). Extreme cases of these anisometric pores would be elongated pore channels and microcracks, respectively. For these cases, as well as their more moderate counterparts, it can be useful or even necessary to include pore shape information into the relations for property prediction. In practice, for real materials, such shape information is usually extracted by microscopic image analysis. However, modern computer technology enables one to create digital microstructures with isolated or overlapping pores of specified shape, so that virtual microstructure control and material design is possible. Moreover, the numerical algorithms of commercial software packages available today for the numerical calculation of effective properties are mature to a degree that allows one to take the calculated numerical values as excellent approximations to the “true “values (much more accurate and precise than any experimentally accessible data), with the advantage that virtual materials with welldefined and defect-free digital microstructures can be used for these calculations. From the viewpoint of theory, any model-based prediction of porous materials properties must be based on the exact solutions of the corresponding single-inclusion problems. Analytical solutions of this type, in the case of elastic properties going back to Eshelby’s classical paper [12], are available for ellipsoidal pores, but not for other pore shapes [27]. (In fact, only numerical solutions exist for non-ellipsoidal pore shapes.) Relatively simple are the single-inclusion solutions for
1−ϕ , 1 + ([k ] − 1) ϕ
Er =
1−ϕ , 1 + ([E ] − 1) ϕ
(1a,b)
the generalized self-consistent relations,
kr = 1 − [k ] ϕ, Er = 1 − [E ] ϕ,
(2a,b)
the generalized power-law relations,
kr = (1 − ϕ)[k] , Er = (1 − ϕ)[E ] ,
(3a,b)
or the generalized exponential relations,
kr = exp
( ), −[k ] ϕ 1−ϕ
Er = exp
(
−[E ] ϕ 1−ϕ
),
(4a,b)
where ϕ is the porosity (=volume fraction of the pore space), and kr = k k 0 and Er = E E0 are the relative thermal conductivity and Young’s modulus, i.e. the non-dimensional ratios of the effective conductivity and modulus of the porous materials (k and E ) and the corresponding quantities of the fully dense solid (k 0 and E0 ). Note that in the case of porous materials with negligible pore property values the self-consistent relation automatically degenerates to the linear approximation (also called dilute or non-interaction approximation [1,2,24]), i.e. the single-inclusion solution itself. It can therefore not be expected to provide realistic predictions of porous materials properties for any other than very low porosities. Based on the latter two relations, which are the most common and successful predictive relations occurring in the literature, a generalized cross-property relation (CPR) between the relative thermal conductivity and the relative Young’s modulus can be derived [10]:
Er = krΩ,
(5)
where Ω = [E ] [k ] may be called the CPR exponent. Using this CPR it is possible to predict the relative Young’s modulus when the relative thermal conductivity is known and vice versa. For the sake of completeness it has to be noted that, apart from the aspect ratio of the pores, the Eshelby-Wu coefficient of Young’s modulus is (very slightly) dependent on the Poisson ratio of the solid phase. In contrast to other elastic moduli, however, it is an advantage of Young’s modulus that its Eshelby-Wu coefficient is rather insensitive vis-a-vis any changes in the solid Poisson ratio, as long as the solid phase does not become auxetic [16]. Table 1 lists Maxwell coefficients and Eshelby-Wu coefficients (the latter for different solid Poisson ratios from 0.1 to 0.4). It is evident that the Eshelby-Wu coefficients for Young’s modulus are rather insensitive to significant changes of the Poisson ratio. In fact the predictions are practically indistinguishable for porous materials based on a different solid phase, as long as the solid phase is non-auxetic.
Table 1 Maxwell and Eshelby-Wu coefficients (the latter for different values of solid Poisson ratios). Maxwell coefficient
Spherical (1) Prolate (10) Oblate (0.1)
1.500 1.647 3.111
Eshelby-Wu coefficient for solid Poisson ratio … 0.1
0.2
0.23
0.33
0.4
1.973 2.304 5.251
2.000 2.304 5.250
2.004 2.299 5.233
2.000 2.270 5.111
1.980 2.234 4.964
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materials to 264–306 isolated spheroidal pores in the 20% porosity materials and up to 529–917 overlapping spheroidal pores in the 40–60% porosity materials, respectively. For spherical pores the numbers per volume are higher, because of the smaller size, resulting in 45 spherical isolated pores in the 1% porosity materials up to 4077 spherical overlapping pores in the 90% porosity materials.
Therefore the results obtained for a special material (here alumina with a solid Poisson ratio of 0.23 [28]) are essentially of general validity, as can be readily verified. Although not explicitly shown in this paper, the same is true for the results of numerical calculations. The Maxwell coefficients for the thermal conductivity are 1.500, 1.647 and 3.111 for spherical, prolate and oblate pores with aspect ratios 1, 10 and 0.1, respectively [15], while the Eshelby-Wu coefficients of Young’s modulus are (for a material with solid Poisson ratio 0.23) 2.004, 2.299 and 5.233, respectively [16]. Based on these values the corresponding CPR exponents are 1.336, 1.396 and 1.682, respectively [10].
3.2. Numerical calculation of effective properties Thermal conductivity tensor calculations have been performed via the GeoDict® ConductoDict module (with a solver based on the so-called explicit jump immersed interface method [30,31]), whereas elastic tensor (stiffness matrix) calculations have been performed using the Geo-Dict® ElastoDict module (based on an iterative solution, assisted by fast Fourier transform, of the Lippmann-Schwinger equation [32,33]). For both types of calculations periodic boundary conditions were assumed. For the thermal conductivity tensor calculations a temperature gradient of 2 K and for the stiffness matrix calculations a strain of 0.005 has been imposed across the cubic box in either direction (x, y, z), and calculations have been performed separately in all three directions. To give an order-of-magnitude estimate, calculation times range from a few seconds to a few minutes for thermal conductivity and from a few minutes to approximately 2–3 h for the elastic properties. For the random microstructures the thermal conductivity values in the three principal directions have just been averaged to obtain the final value, while orientationally averaged Young’s modulus values have been obtained in the isotropic approximation by orientational averaging of the stiffness matrix. The solid phase of the model material is assumed to be polycrystalline alumina with randomly oriented crystallites, i.e. the solid phase is assumed to be statistically isotropic with room temperature values of thermal conductivity, Young’s modulus and Poisson ratio of 33 W/mK, 400 GPa and 0.23, respectively, for the homogenized dense solid phase [28,34]. The pore space is assumed to be filled with air at atmospheric pressure, i.e. the elastic moduli of the pore phase are zero and the thermal conductivity is 0.026 W/mK at room temperature [34]. Since the thermal conductivity of the pore phase is smaller than the solid conductivity by more than three orders of magnitude, the former is negligible and the results of this paper are essentially the same as for vacuum void space. Therefore the results of this paper are
3. Modeling procedure 3.1. Creation of digital microstructures The digital microstructures in this work were created using the GeoDict® software package [29] in cubic boxes with 200 × 200 × 200 cubic voxels (voxel edge length 1 μm). Representative volume elements of microstructures with randomly arranged spherical pores and randomly arranged and randomly oriented spheroidal pores (prolate and oblate) have been generated using the GeoDict® GrainGeo module. In all cases the pores were chosen to be monosized and with fixed aspect ratios of 1, 10 and 0.1 for spherical, prolate and oblate pores, respectively (dimensions 15 μm, 100 × 10 × 10 μm and 50 × 50 × 5 μm, respectively). Microstructures with porosities of 1%, 2%, 5%, 10% and subsequent steps of 5 or 10% were created up to values approaching the highest porosity attainable, which was restricted to around 40% porosity in the case of isolated spherical pores and around 20% in the case of isolated prolate and oblate pores (for overlapping pores the attainable porosities are much higher, viz. up to 90%, 60% and 40%). The spatial distribution of the pores has been chosen to be uniform in all cases, and periodic boundaries have been created in such a way that opposite faces of the cubic boxes are compatible, in order to enable effective property calculations with periodic boundary conditions. The orientation of the pores has been chosen to be random, i.e. without any degree of preferential orientation, so that the resulting microstructures are as isotropic as possible. The number of pores in the cubic boxes ranges from 13 to 15 oblate or prolate isolated pores in the 1% porosity
Fig. 1. Computer-generated digital microstructures of isotropic porous materials with randomly oriented spheroidal pores (prolate and oblate with aspect ratios 10 and 0.1, respectively) with different porosity ϕ : prolate isolated with ϕ = 10% (A) and ϕ = 20% (B), prolate overlapping with ϕ = 20% (C) and ϕ = 40% (D), oblate isolated with ϕ = 10% (E) and ϕ = 20% (F), oblate overlapping with ϕ = 20% (G) and ϕ = 40% (H). 3
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theoretical maximum porosity for random packing is 64% [35], while the maximum porosity that can be achieved for digital microstructures without loosing integrity is much higher (e.g. higher than 70% when using the GrainGeo module of GeoDict®). In the case of microstructures with anisometric pores the situation is qualitatively similar, but the maximum porosities are lower. For isolated prolate pores with aspect ratios 0.1 and 0.05 the maximum porosities achieved using the GrainGeo module of GeoDict® are lower than 30 and 20%, respectively, while for overlapping prolate pores porosities as high as 60 and 40%, respectively, can be readily achieved. Similarly, for isolated oblate pores with aspect ratios 10 and 20 the maximum porosities achieved using the GrainGeo module of GeoDict® are lower than 30 and 5%, respectively, while for overlapping prolate pores porosities as high as 40 and 20%, respectively, can be readily achieved. It is evident that oblate pores lead to disintegration of the material at lower porosities than prolate pores. This is of course expected, since
essentially independent of the choice of alumina as a model material. More precisely, they are valid for any porous material with negligible pore phase property values and solid Poisson ratio 0.23 (and, more than that, they are excellent approximations for any porous material with non-negative solid Poisson ratio, as indicated above). 4. Results and discussion Fig. 1 shows examples of the computer-generated digital microstructures of isotropic porous materials with randomly oriented anisometric pores (spheroidal prolate and oblate with aspect ratios 0.1 and 10, respectively) with different porosity ϕ (10 and 20% for isolated pores, 20 and 40% for overlapping pores). It is clear that the maximum porosity that can be achieved is higher for overlapping pores than for isolated pores. This is similar in the case of isometric pores. For example, in the case of identical (i.e. monosized) spherical pores the
Fig. 2. Porosity dependence of the relative thermal conductivity (top) and Young’s modulus (bottom) of porous materials with spherical (aspect ratio 1) pores, isolated or overlapping; numerically calculated relative property data compared to the Maxwell-type, self-consistent (=linear), power-law and exponential effective medium approximations (EMAs). 4
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lower than the Maxwell-type EMAs. In other words, the Maxwell-type EMAs seem to provide practical upper bounds for these data. For spherical pores this is not surprising, since the Maxwell-type EMAs for spherical pores are formally identical to the upper Hashin-Shtrikman bounds for all isotropic porous materials. Therefore it is clear that in this case all effective conductivity and elastic modulus values must always be below (or at best equal to) this bound. However, a corresponding rigorous argument does not exist for spheroidal pores. Nevertheless, since the Maxwell and Eshelby-Wu coefficients exhibit minimum values for spherical pores [15,16], it is clear that the relative conductivity and elastic modulus values of materials with spheroidal pores must always be lower than the corresponding values for materials with spherical pores. It is therefore plausible (although a rigorous proof does not exist), that the Maxwell-type EMA for materials with
oblate pores with high aspect ratio can be viewed as model shapes for cracks or microcracks, which can lead to the cleavage or segmentation of a specimen into separate parts (fragments) even at volume fractions approaching zero. Figs. 2–4 show the porosity dependences of the relative thermal conductivity (Figs. 2a, Figure 3a and Figure 4a) and Young’s modulus (Figs. 2b, Figure 3b and Figure 4b) of porous materials with spherical (aspect ratio 1), spheroidal prolate (aspect ratio 10) and spheroidal oblate (aspect ratio 0.1) pores, either isolated or overlapping. In these graphs numerically calculated relative property data are compared to the Maxwell-type, self-consistent (=linear), power-law and exponential effective medium approximations (EMAs), using the Maxwell and Eshelby-Wu coefficients listed in Table 1. It is evident that in all cases the numerically calculated values are
Fig. 3. Porosity dependence of the relative thermal conductivity (top) and Young’s modulus (bottom) of porous materials with spheroidal prolate (aspect ratio 10) pores, isolated or overlapping; numerically calculated relative property data compared to the Maxwell-type, self-consistent (=linear), power-law and exponential effective medium approximations (EMAs).
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Fig. 4. Porosity dependence of the relative thermal conductivity (top) and Young’s modulus (bottom) of porous materials with spheroidal oblate (aspect ratio 0.1) pores, isolated or overlapping; numerically calculated relative property data compared to the Maxwell-type, self-consistent (=linear), power-law and exponential effective medium approximations (EMAs).
The closest agreement between predictions and numerically calculated relative property values is found for the power-law (also called differential scheme) EMAs and our exponential EMAs. More precisely, the power-law EMA predictions are usually closest to the numerically calculated values in the case of isolated pores (overestimating these values by up to 0.02 RPU for spherical and prolate pores and underestimating them by up to −0.04 RPU for oblate pores, see Table 2), while the exponential EMA predictions seems to provide a practical lower bound in all cases and closely approximate the relative property values in the case of overlapping oblate pores (underestimating the values by up to –0.10, –0.08 and –0.04 RPU for spherical, prolate and oblate pores, respectively, see Table 2). It has to be admitted that the agreement between any of these two
spheroidal pores provides some kind of upper bound at least for the case of monosized spheroidal pores. Notwithstanding these details, however, it should be emphasized that in neither case (neither spherical nor spheroidal pores) does the Maxwell-type EMA provide a satisfactory prediction of the relative property values. As mentioned in Section 2 above, the self-consistent EMAs degenerate to linear predictions when the pores are empty voids (vacuum pore space). Therefore they principally fail to provide even a qualitatively correct prediction of the intrinsically nonlinear curve shape. What is worse, they predict very concrete but spurious percolation thresholds, as has been pointed out by Torquato [1] and is nicely seen in Fig. 2 through Fig. 4. For this reason it is fully justified to ignore them in the sequel.
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Table 2 Maximum deviation (in relative property units / RPU) of the power-law predictions, exponential predictions and CPR predictions from the numerically calculated data for porous materials with spherical (aspect ratio 1), prolate (aspect ratio 10) or oblate (aspect ratio 0.1) pores; in parentheses the approximate porosities at which this maximum deviation occurs (for other porosities the deviation is smaller). Pore shape and connectivity
Deviation of the power-law prediction for kr
Deviation of the power-law prediction for Er
Deviation of the exponential prediction for kr
Deviation of the exponential prediction for Er
Deviation of the CPR prediction for Er
Spherical isolated Spherical overlapping Prolate isolated Prolate overlapping Oblate isolated Oblate overlapping
+0.022 (40 %) +0.050 (55 %) +0.015 (20 %) + 0.054 (50 %) –0.033 (20 %) +0.055 (35 %)
+0.011 (15 %) +0.063 (75 %) +0.008 (10 %) + 0.051 (50 %) –0.043 (20 %) +0.032 (35 %)
–0.075 (40 %) –0.098 (60 %) –0.015 (20 %) – 0.084 (60 %) –0.073 (20 %) –0.036 (40 %)
–0.086 (40 %) –0.065 (50 %) –0.030 (20 %) – 0.052 (50 %) –0.084 (20 %) –0.021 (25 %)
–0.013 (35 %) +0.010 (70 %) –0.012 (20 %) + 0.007 (60 %) –0.007 (20 %) –0.012 (15 %)
Fig. 5. Porosity dependence of the relative Young’s modulus of porous materials with spherical pores of aspect ratio 1; comparison of numerically calculated data and CPR predictions (please note that the curves are predictions, without any fitting involved).
independent method, e.g. by microscopic image analysis. From this point of view the situation of the generalized CPR is not different from that of generalized EMAs (where also independent aspect ratio information is required, when these EMAs are to be used as parameterfree predictions). However, the CPR has the advantage of a much higher accuracy of the resulting predictions. We would like to emphasize that, in contrast to other elastic moduli, the results for the Young’s modulus are practically independent of the Poisson ratio, as can easily be shown by numerical calculations with other Poisson ratios and is evident also from the fact that the EshelbyWu coefficient for the Young’s modulus are practically independent of the Poisson ratio [16]. Therefore the results of this paper are practically independent of the specific material parameters that have been used here (taking alumina with air-filled pores as a paradigmatic example). In other words, the results obtained in this paper are valid for any porous ceramic material with a non-auxetic solid phase. More than that, they are valid also for electrical (instead of thermal) conductivity (as long as Ohm’s law is valid), i.e. they can directly be used e.g. for predicting the relative Young’s modulus of porous metals from their relative electrical conductivity.
types of predictions (power-law and exponential EMAs) and the numerically calculated values is not too bad, given the fact that the comparison of these model predictions and the “true” values (which are for random microstructures accessible only via numerical calculations) does not involve any kind of fitting. (Especially in the practically most important case of oblate pores the agreement between predictions and true values is surprisingly good.) That means, in contrast to many works by other authors, the present procedure does not rely upon the use of an empirical fit parameter for predicting the porosity dependence. In this sense all our predictions are “parameter-free” and can be used as a priori estimates in cases where an independent aspect ratio information is available, e.g. from microscopic image analysis. Nevertheless, we will see now that the predictions can be considerably improved by the use of our generalized cross-property relation (CPR) [10]. Figs. 5–7 show the porosity dependences of the relative Young’s modulus of porous materials with spherical (aspect ratio 1), spheroidal prolate (aspect ratio 10) and spheroidal oblate (aspect ratio 0.1) pores, respectively, either isolated or overlapping, in compared with the corresponding CPR prediction. It is immediately evident that in all cases the agreement between the predictions and the true values is much better than for any EMA prediction. Indeed using our generalized CPR it is possible to predict the relative Young’s modulus with accuracy better than ± 0.01 RPU, when the relative thermal conductivity is known or vice versa. Of course, in order to use this generalized CPR [10] in the context of real materials, the aspect ratio must be determined by an
5. Summary and conclusion Based on computer-generated digital microstructures with isolated or overlapping spherical or spheroidal pores of aspect ratio
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Fig. 6. Porosity dependence of the relative Young’s modulus of porous materials with prolate spheroidal pores of aspect ratio 10; comparison of numerically calculated data and CPR predictions (please note that the curves are predictions, without any fitting involved).
Fig. 7. Porosity dependence of the relative Young’s modulus of porous materials with oblate spheroidal pores of aspect ratio 0.1; comparison of numerically calculated data and CPR predictions (please note that the curves are predictions, without any fitting involved).
Acknowledgement
1 (spherical), 10 (prolate) and 0.1 (oblate), the relative thermal conductivity and Young’s modulus have been numerically calculated for materials with different porosity and compared to predictions based on generalized effective medium approximations (EMAs) and a recently proposed generalized cross-property relation (CPR). It has been shown that, when the aspect ratio is known, the generalized power-law relation and the generalized exponential relation provide relatively satisfactory predictions of these two properties without the use of empirical fit parameters. In terms of relative property units (RPU), the maximum deviation of these two types of EMA predictions from the “true “values (which can be calculated only numerically) ranges from −0.04 to +0.06 RPU for the power- law relation and from −0.02 to −0.10 RPU for the exponential relation. However, using the generalized CPR has been shown to result in an even higher accuracy of the predictions, with maximum deviations smaller than 0.01 RPU for the Young’s modulus when the thermal conductivity is known.
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). References [1] S. Torquato, Random Heterogeneous Materials – Microstructure and Macroscopic Properties, Springer, New York, 2002, pp. 403–646 Ch. 16–23. [2] G.W. Milton, The Theory of Composites, Cambridge University Press, Cambridge, 2002, pp. 113–568 Ch. 7–26. [3] J. Ohser, F. Mücklich, Statistical Analysis of Microstructures in Materials Science, John Wiley and Sons, Chichester, 2000, pp. 57–106 Ch. 3. [4] W. Pabst, E. Gregorová, T. Uhlířová, Microstructure characterization via stereological relations – a shortcut for beginners, Mater. Charact. 105 (1) (2015) 1–12. [5] L.J. Gibson, M.F. Ashby, Cellular Solids – Structure and Properties, second edition,
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