Porous Talcum-Based Steatite Ceramics Fabricated by the Admixture of Organic Particles: Experimental Characterization and Effective Medium/Field Modeling of Thermo-Mechanical Properties

Abstract

In this paper, an experimental campaign, as regards the thermo-mechanical properties (heat capacity, thermal conductivity, Young’s modulus, and tensile (bending) strength) of talcum-based steatite ceramics with artificially introduced porosity, is presented. The latter has been created by adding various amounts of an organic pore-forming agent, almond shell granulate, prior to compaction and sintering of the green bodies. The so-obtained porosity-dependent material parameters have been represented by homogenization schemes from effective medium/effective field theory. As regards the latter, thermal conductivity and elastic properties are well described by the self-consistent estimate, with effective material properties scaling in a linear manner with porosity, with the latter in the range of 1.5 vol-%, representing the intrinsic porosity of the ceramic material, to 30 vol-% in this study. On the other hand, strength properties are, due to the localization of the failure mechanism in the quasi-brittle material, characterized by a higher-order power-law dependency on porosity.

1. Introduction

Steatite is a constituent present in a variety of modern ceramic materials (see, e.g., [1,2]) and is frequently used as filler in composite materials (see, e.g., [3,4,5]). Steatite is characterized by low dielectric loss properties and high thermal and mechanical resistance. Combined with good workability and low shrinkage of the green bodies, steatite ceramics are predominantly used in electrical applications and as thermal insulation material. Recent, further applications were suggested in nuclear power applications [6] and medical engineering [7,8]. Recently, we described the synthesizing of a low-to-medium porosity talcum-based steatite (TBS) ceramic including a detailed pore space characterization and analysis of solid phase microstructure and composition [9]. Ceramic materials of this type are frequently denoted as steatite or steatite ceramic in the open literature. This may be ambiguous, as both the sintered product and the main raw material (talcum, soapstone), respectively, are denoted as “steatite”. Hence, the denomination “talcum-based steatite (TBS) ceramic” is chosen in this paper. [9] also contains a detailed description of the pore-forming agent—almond shell granulate—that has been intermixed within the green bodies prior to compaction and sintering. The so-obtained artificially introduced porosity may foster novel applications of the well-established ceramic material, as thermo-mechanical properties may be tailored to customer’s specifications.

The current paper focuses on the experimental characterization and discusses microstructure-based modeling of thermo-mechanical properties: heat capacity, thermal conductivity, Young’s modulus, and tensile (bending) strength of prism-shaped and plate-shaped specimens (obtained by cutting off from these prisms), respectively, of the porous TBS ceramic samples investigated in [9]. Thereby, Youngs’s modulus and bending (tensile) strength were determined from three-point bending experiments; in addition, elastic behavior was also assessed by ultrasonic compressive wave speed measurements in the longitudinal direction of the prisms. For the determination of thermal properties in the scopes of the laser flash method, the plate-shaped specimens were employed.

The material dealt with in this paper is characterized by a maximum porosity of about 30%. Hence, as regards modeling, we restrict the discussion to analytical modeling schemes pertinent for the low and medium porosity range. As regards modeling of thermo-mechanical properties of highly-porous materials, see, e.g., [10,11,12,13].

2. Material and Sample Preparation

For the manufacture of the dense ceramic samples, a commercially available granulated barium/talcum Mg${}_3$(Si${}_4$O${}_{10}$)(OH)${}_2$ mass (Quarzsandwerke Weißenbrunn, Bauer & Co., Weißenbrunn, Germany, labeled “C 221” and in accordance with IEC 672), composed by the raw materials, talcum, clay, and barium carbonate (with the latter serving as fluxing agent), was compacted (in dry state) with a force exceeding 50 kN (see

Table 1. Sample notation and preparation.

Sample Notation	Organic Material Added Prior to Compaction and Sintering	Organic Content [m-%]	Sample Compaction Force [kN]
SC-0	none, dense sample	0	50
	with intrinsic
	porosity
SC-350-02	Rehofix MS 0–350 $\mathsf{\mu }$m	2	65
SC-350-04	Rehofix MS 0–350 $\mathsf{\mu }$m	4	69
SC-350-06	Rehofix MS 0–350 $\mathsf{\mu }$m	6	71
SC-350-08	Rehofix MS 0–350 $\mathsf{\mu }$m	8	88
SC-350-10	Rehofix MS 0–350 $\mathsf{\mu }$m	10	95
SC-200-02	Rehofix MS 0–200 $\mathsf{\mu }$m	2	73
SC-200-04	Rehofix MS 0–200 $\mathsf{\mu }$m	4	82
SC-200-06	Rehofix MS 0–200 $\mathsf{\mu }$m	6	88
SC-200-08	Rehofix MS 0–200 $\mathsf{\mu }$m	8	95
SC-200-10	Rehofix MS 0–200 $\mathsf{\mu }$m	10	106

) into green bodies (prisms with 10 mm × 10 mm × 120 mm) and sintered at temperatures between 1300 and 1350 °C for 2 h and subsequently cooled to 100 °C with a cooling rate of 100 K/h. The resulting TBS ceramic is composed of proto- and clinoenstatite crystals (Mg${}_2$Si${}_2$O${}_6$), i.e., pyroxenes, and a small amount of cristobalite, all embedded in a barium containing glassy phase. As described in detail in [9], XRD gave access to the crystalline mineral phases of the material (see

Table 2. Ceramic composition (m%) as determined by XRD (crystalline phases only).

protoenstatite	75.38
clinoenstatite	22.85
cristobalite	1.77
	100.00

, listing mass fractions related to the sum of crystalline phases only), with enstatite as the main phase. Additionally, a glassy phase, serving as a matrix in the material system, is present with a mass fraction of approximately 30 m% (related to total mass).

High-resolution imaging combined with backscattered electron microscopy was employed to visualize the prevalent material phases, pore structure, and morphology. Chemical analyses of the TBS ceramic samples were obtained by energy dispersive X-ray (EDX) spectroscopy. Figure 1 shows the backscattered electron microscopy image of a polished sample together with associated EDX spectra. As indicated by the various grey scale values, TBS ceramic is composed by different material phases, mainly enstatite crystals (Mg${}_2$Si${}_2$O${}_6$, medium grey), embedded in a barium containing glassy phase (light grey); porosity is also visible (dark grey). A detailed chemical analysis of the glassy phase, containing mainly oxygen, silicon, barium, aluminum, and magnesium, can be found in [9].

In addition to the dense samples (with their intrinsic porosity), same-size prismatic specimens with artificially introduced porosity were manufactured by intermixing organic particles in the green bodies prior to compaction and sintering (see Table 1). Almond shell granulate was employed as the pore-forming agent (Rehofix MS, diameter 0–200 $\mathsf{\mu }$m and 0–350 $\mathsf{\mu }$m, respectively, J. Rettenmaier and Söhne GmbH, Rosenberg, Germany) in various contents: 2, 4, 6, 8, and 10 m-%. SEM imaging (Figure 2) reveals the morphology of the pore-forming agent, and, hence, the morphology of the artificially introduced porosity in the sintered material. The particle shape ranges from few-micrometer-sized chips to hundred-micrometer, roughly equiaxed blocks. Looking forward to modeling, the granules may be approximated as oblate to prolate ellipsoids, respectively, with aspect ratios ranging from 1/4 to 2. Prior to thermo-mechanical characterization, the mass and geometric dimensions (length, width, and height) of the dried, prismatic ceramic specimens were determined with an analytical balance (Mettler Toledo ME204T) with an accuracy of 0.0001 g and a digital caliper (Preisser) with a resolution of ≥0.01 mm, respectively, giving access to the sample density ${\rho }_{\mathrm{eff}}$ (

Table 3. Mean values of length, width, height, mass, and sample density of investigated ceramic samples.

Sample	Length	Width	Height	Mass	${\mathit{\rho }}_{\mathbf{eff}}$
	(mm)	(mm)	(mm)	(mm)	(kg/m${}^3$)
SC-0	119.55	9.78	9.99	32.53	2790
SC-350-02	119.54	10.00	10.04	30.86	2570
SC-350-04	119.70	10.02	10.14	29.88	2460
SC-350-06	119.60	10.01	10.13	27.85	2300
SC-350-08	120.26	10.07	9.99	26.05	2150
SC-350-10	120.43	10.09	10.03	25.65	2110
SC-200-02	120.04	10.03	9.82	30.94	2620
SC-200-04	120.17	10.04	9.93	29.51	2460
SC-200-06	120.17	10.05	9.97	27.79	2310
SC-200-08	120.62	10.08	10.12	26.48	2150
SC-200-10	120.91	10.10	10.23	25.18	2020

). The dense samples with intrinsic porosity (SC-0) exhibit an average density and mass of 2790 kg/m${}^3$ and 32.53 g, respectively. With an increase in artificially introduced porosity, the density and mass of the porous TBS ceramic prisms continuously decrease, with the porous samples SC-350-10 and SC-200-10 characterized by the lowest densities of 2110 and 2020 kg/m${}^3$, respectively.

In this paper, porosity of the (dried) sample $f_{\mathrm{p}}$ [–], related to the volume fraction of the solid material matrix $f_{\mathrm{m}}$ as $f_{\mathrm{p}}=1-f_{\mathrm{m}}$, is associated with a (medium) density of the solid material phases ${\rho }_{\mathrm{m}}$ [kg/m${}^3$] (mainly protoenstatite, clinoenstatite, and the glassy phase, see Table 2), as $f_{\mathrm{m}}=(1-f_{\mathrm{p}})={\rho }_{eff}/{\rho }_{\mathrm{m}}$, with ${\rho }_{eff}$ [kg/m${}^3$] denoting measured sample density. With an intrinsic porosity of the dense samples determined as $f_{\mathrm{p}}=0.015$ in [9], ${\rho }_{\mathrm{m}}$ = 2790/(1 − 0.015) = 2830 kg/m${}^3$. The latter value has been used in the figures throughout this paper to determine $f_{\mathrm{m}}$ from measured sample densities.

3. Experimental Characterization

3.1. Thermal Properties

Laser flash analyses (LFA) gave access to the (effective) thermal diffusivity of the samples D [m${}^2$/s], with the latter reflecting both steady state heat flow, i.e., thermal conductivity k [W/(m K)], as well as thermal retention, i.e., (volume) heat capacity $\left(\rho c\right)$ [J/(m${}^3$K)], of a solid/porous material as $D=k/\left(\rho c\right)$. For an LFA measurement, one side of a disc or plate-shaped specimen with thickness d [m] is exposed to an energy pulse from a light source (laser or xenon flash lamp); on the opposite side of the specimen, the temperature history is measured [14,15,16]. The analytical solution of the related one-dimensional, adiabatic heat-transfer problem gives access to thermal diffusivity as $D=\delta d^2/{\mathrm{t}}_{1/2}$ with $\delta =0.139$ and ${\mathrm{t}}_{1/2}$ denoting the time lag for the temperature to reach half of the final temperature rise. The higher the thermal diffusivity, the faster energy is dispersed within the sample.

A Netzsch LFA 447 NanoFlash system equipped with a Xenon flash lamp ($\lambda$ = 150–2000 nm) and an infrared detector (InSb) with integrated dewar enabling measurements from room temperature up to 300 °C with a repeatability and accuracy of ±2–3% was employed in the present study. Plates with a thickness of 2 mm (surface area of 10 × 10 mm${}^2$) were cut from the ceramic prisms with a diamond saw. Thereafter, both surfaces of the cleaned and dried plates were coated with graphite spray twice (Graphit 33, Kontakt Chemie, CRC Industries Europe BV, Zele, Begium). For each graphite-coated sample, three laser flash shots were conducted at 25, 50, and 75 °C (see Figure 3a,b). At the lowest temperature, the dense TBS ceramic sample (SC-0 with ${\rho }_{\mathrm{eff}}$ = 2790 kg/m${}^3$) exhibits the highest thermal diffusivity with D = 1.48 mm${}^2$/s, and the sample with the highest porosity (SC-200-10, ${\rho }_{\mathrm{eff}}$ = 2020 kg/m${}^3$), with the lowest as 1.12 mm${}^2$/s. The specific heat capacity $c_{\mathrm{p}}$ of the specimens was determined using a Pyroceram 9606 reference sample.

3.2. Mechanical Properties

Tensile bending strength was determined from the failure load in three-point experiments on the 120 mm long samples (span of 100 mm in bending experiments) with a cross-section of 10 × 10 mm${}^2$. Experiments were conducted on a Shimadzu AG-X 10 kN table-top model. This universal testing machine enables loading up to 10 kN. Data collection and assignment of the experimental conditions were accomplished by the TrapeziumX software (version 1.4.0). The specimens were loaded with a rate of 35 N/s until failure. These experiments also gave access to Young’s modulus as determined from the initial, linear part of the force-deflection graph.

Furthermore, elastic behavior was also assessed by ultrasonic (US) compressive wave speed measurements. Specimens with the same dimensions as in the bending experiments were employed, with US waves conveyed in the longitudinal direction of the specimens. For the propagation of elastic “bulk” compressive (or longitudinal) waves in an isotropic material (i.e., in the bulk continuum), the component $C_{\mathrm{iiii}}$ of the fourth order elasticity tensor is related to wave speed v [m/s] and density $\rho$ [kg/m${}^3$] as [17]:

$${\displaystyle C_{\mathrm{iiii}}=\frac{(1-\nu )}{(1+\nu )(1-2\nu )}E=\rho v^2}$$

(1)

with E [Pa] and $\nu$ [–] denoting Young’s modulus and Poisson’s ratio, respectively. Thereby, the displacement of material points (particle polarization) and wave propagation direction is equi-directional.

This theory of bulk wave propagation still holds for pulse propagation through finite bars (in the longitudinal direction) in case the wavelength $\lambda$ is much smaller than the cross-sectional dimension of the bar [18]. For the present investigation, the measured velocities ranged from 5050 to 6900 m/s and the frequency of the employed transducers is given as f = 220 kHz. Hence, the minimum wavelength is given as $\lambda$ = 5050/220,000 = 30 × 10${}^{-3}$ m, i.e., the condition is not met with a cross-sectional dimension of 10 × 10${}^{-3}$ m. On the other hand, when $\lambda$≫ than the cross sectional dimension, “extensional” or “bar” wave propagation with a (as opposed to bulk wave propagation) one-dimensional state of normal stress is predominant with [19]:

$${\displaystyle v=\sqrt{{\displaystyle \frac{E}{\rho }}}}$$

(2)

In the present case, $\lambda$ = 30 mm is, to an extent, larger than the cross-sectional dimension of 10 mm, and Equation (2) was employed for the backcalculation of Young’s modulus from ultrasonic compressive wave speed measurements.

4. Discussion: Effective Medium/Field Modeling of Structure/Property-Relations

4.1. Thermal Conductivity

Well established schemes from effective media and effective field theory [20] may be employed to relate the thermal conductivity of the porous material $k_{\mathrm{eff}}$ [W/(m K)] to the conductivity and volume fraction of the solid material matrix, $k_{\mathrm{m}}$ [W/(m K)] and $f_{\mathrm{m}}$ [–], respectively (for details, see Appendix A). Pore space morphology may be taken into account, with extensions of classical schemes accessible for spheroidal pore shapes (for details, see Appendix B). Hence, $k_{\mathrm{eff}}=k_{\mathrm{eff}}(k_{\mathrm{m}},f_{\mathrm{m}}=1-f_{\mathrm{p}},\mathrm{morphology})$. When accounting for (i) matrix/spherical pore morphology and (ii) zero conductivity in the pore space, where the latter is a reasonable simplification in case the contrast between conductivity of matrix material and pore space, $k_{\mathrm{m}}$ and $k_{\mathrm{p}}$, respectively, is large (that is the case for most ceramic materials), engineering approximations are available, as listed below (see also, Figure 4). As regards porous ceramics, the assumption of $k_{\mathrm{p}}\approx 0$ is usually acceptable, as the contrast between conductivity of matrix material and pore space, $k_{\mathrm{m}}$ and $k_{\mathrm{p}}$, respectively, is large, i.e., $k_{\mathrm{p}}/k_{\mathrm{m}}<1/100$. For porous TBS ceramics, with $k_{\mathrm{p}}$ = 0.018 W/(m K) as the respective value for air, the contrast is approximately 0.018/3 = 1/170.

• Maxwell–Eucken expression [21], Kanaun–Levin method [22,23,24,25], or Hashin–Shtrikman upper bound [26] (in continuum micromechanics, the related homogenization method is referred to as the Mori–Tanaka scheme [27,28], see Appendix A):

  $${\displaystyle \frac{k_{\mathrm{eff}}^{\mathrm{ME}}}{k_{\mathrm{m}}}=\frac{k_{\mathrm{eff}}^{\mathrm{KL}}}{k_{\mathrm{m}}}=\frac{2f_{\mathrm{m}}}{3-f_{\mathrm{m}}},}$$

  (3)

  derived for a spatial configuration where the local temperature distortion in the vicinity of a pore does not affect the temperature field in the vicinity of neighboring pores [29,30]. The expression may be employed for moderate porosity values, e.g., for spherical pores for $f_{\mathrm{p}}<\sim$0.2.
• The differential scheme (DS) [31,32,33] departs from this restriction, i.e., is suitable for materials with porosity in the medium range. This scheme follows from a recurring application of the dilute distribution estimation [34,35], the latter representing a spatial configuration where pores are diluted in the matrix material and, hence, their interaction can be neglected; starting with the homogeneous matrix material, pore volume is embedded in the matrix material in infinitesimal steps and the dilute distribution estimation is employed for the determination of effective behavior in each step, etc., finally giving:

  $${\displaystyle \frac{k_{\mathrm{eff}}^{\mathrm{DS}}}{k_{\mathrm{m}}}=f_{\mathrm{m}}^{3/2}.}$$

  (4)
• One may consider spheroidal pore shape for the Maxwell–Eucken expression, the Kanaun–Levin method, and the differential scheme, with the influence of oblate and prolate geometry on effective behavior derived in [20,36]. For an isotropic distribution of spheroidal pores, the differential scheme reads:

  $${\displaystyle \frac{k_{\mathrm{eff}}^{\mathrm{DS}}}{k_{\mathrm{m}}}=f_{\mathrm{m}}^{\eta },}$$

  (5)
  (see Figure 5b) with shape factor $\eta \ge 3/2$ a function of the aspect ratio $\gamma$ of the spheroidal pores and $\gamma <1$ corresponding to oblate, $\gamma >1$ to prolate pore shape, respectively (see Appendix B). Figure 5a depicts the shape factor $\eta$ as a function of aspect ratio $\gamma$. Note that whereas a prolate shape only minorly influences the shape factor, oblate shape with $\gamma <0.2$ significantly alters the effective thermal behavior (as compared to a composite with spherical pores), as $\eta$ gets excessively big. Even a small porosity may lead to a significant reduction of thermal conductivity.
  Further note that for non-spherical pore shape, the Kanaun–Levin method and the Maxwell–Euken scheme do not coincide (for spherical pores, they do), with:

  $${\displaystyle \frac{k_{\mathrm{eff}}^{\mathrm{KL}}}{k_{\mathrm{m}}}=\frac{1-2/3\eta (1-f_{\mathrm{m}})}{1+1/3\eta (1-f_{\mathrm{m}})}}$$

  (6)

  and

  $${\displaystyle \frac{k_{\mathrm{eff}}^{\mathrm{ME}}}{k_{\mathrm{m}}}=\frac{f_{\mathrm{m}}}{f_{\mathrm{m}}(1-\eta )+\eta },}$$

  (7)

  respectively (see Figure 5c,d).
• The self-consistent scheme (SCS) [37,38] has originally been derived for so-called polycrytalline microstructures, where none of the material phases play a distinguished role. If one material phase is represented by porosity with $k_{\mathrm{p}}=0$, the SCS reads:

  $${\displaystyle \frac{k_{\mathrm{eff}}^{\mathrm{SCS}}}{k_{\mathrm{m}}}=\frac{3f_{\mathrm{m}}-1}{2}.}$$

  (8)

Based on the measured diffusivity and heat capacity (see Section 3.1), the thermal conductivity is obtained as $k=D/\left(\rho c\right)$ and plotted as a function of solid matrix volume fraction (see Figure 6). When assessing the representation of data for the three major homogenization schemes described previously (ME, DS, SCS; spherical pore shape), one may select the SCS as the most appropriate scheme. However, the scatter of data, with the variation obtained in this paper in the typical range for ceramic materials, makes that choice rather sketchy. Also note that for a given porosity, i.e., $f_{\mathrm{m}}=$ const., thermal conductivity decreases with temperature (in the investigated temperature range from 25 to 75 °C). The (theoretical) value for the solid phase conductivity may be approximated to decrease from 3.3 to 3.1 W/(m K) in this temperature range (see Figure 6). Data quality also prevents one from backcalculating an (average) pore shape factor based on Equations (5)–(7). The average shape of the artificial pores seems to be too compact, or equiaxed, to have a significant influence on the shape of the material function $k_{\mathrm{eff}}/k_{\mathrm{m}}=k_{\mathrm{eff}}/k_{\mathrm{m}}\left(f_{\mathrm{m}}\right)$.

4.2. Mechanical Properties

Figure 7 depicts a comparison of experimental data for Young’s modulus (see Section 3.2) with classical schemes from effective medium theory (see Appendix A), with the self-consistent scheme giving a fairly reasonable representation of data.

As regards the modeling of input parameters, spherical pore shape is underlain, as well as the density and elastic properties of the solid material phase, summarized in Figure 7.

In order to employ the self-consistent scheme for upscaling of strength, further assumptions are necessary. (i) For incompressible matrix behavior, i.e., ${\mu }_{\mathrm{m}}=E_{\mathrm{m}}/3$, with ${\mu }_{\mathrm{m}}$ denoting shear modulus of the matrix material, the self-consistent estimate can be written as:

$$\begin{array}{ccccc}\hfill {\mu }_{\mathrm{eff}}& =& \hfill \mathcal{M}\left(f_{\mathrm{m}}\right){\mu }_{\mathrm{m}}\hfill & =& \hfill {\displaystyle \frac{3(2f_{\mathrm{m}}-1)}{2+f_{\mathrm{m}}}{\mu }_{\mathrm{m}}\mathrm{and}}\hfill \\ \hfill K_{\mathrm{eff}}& =& \hfill \mathcal{K}\left(f_{\mathrm{m}}\right){\mu }_{\mathrm{m}}\hfill & =& \hfill {\displaystyle \frac{4(2f_{\mathrm{m}}-1)f_{\mathrm{m}}}{(1-f_{\mathrm{m}})(2+f_{\mathrm{m}})}{\mu }_{\mathrm{m}}.}\hfill \end{array}$$

(9)

(ii) For ductile, von Mises-type behavior of the matrix material, with $c_{\mathrm{m}}$ as the cohesion of the matrix material, the domain of admissible macroscopic stress states is given as (see Equation (7.70) in [39]):

$${\displaystyle f_{\mathrm{m}}=\frac{1}{\mathcal{K}\left(f_{\mathrm{m}}\right)}{\left({\displaystyle \frac{{\mathsf{\Sigma }}_{\mathrm{vol}}/3}{c_{\mathrm{m}}}}\right)}^2+\frac{1}{\mathcal{M}\left(f_{\mathrm{m}}\right)}{\left({\displaystyle \frac{{\mathsf{\Sigma }}_{\mathrm{dev}}}{c_{\mathrm{m}}}}\right)}^2}$$

(10)

where ${\mathsf{\Sigma }}_{\mathrm{vol}}$ and ${\mathsf{\Sigma }}_{\mathrm{dev}}$ denote the volumetric and deviatoric macroscopic stress, respectively. Setting ${\mathsf{\Sigma }}_{\mathrm{vol}}$=0, the effective cohesive strength $c_{\mathrm{eff}}={\mathsf{\Sigma }}_{\mathrm{dev}}({\mathsf{\Sigma }}_{\mathrm{vol}}=0)$ is obtained as:

$${\displaystyle \frac{c_{\mathrm{eff}}}{c_{\mathrm{m}}}=\sqrt{{\displaystyle \frac{3f_{\mathrm{m}}(2f_{\mathrm{m}}-1)}{2+f_{\mathrm{m}}}}}}$$

(11)

Assuming that the (tensile) bending strength ${\sigma }_{\mathrm{t}}$ scales in the same way as cohesion, Figure 8 shows the self-consistent estimate by the scaling matrix strength of 125 MPa with the r.h.s of Equation (11). The model crudely overestimates data (see Figure 8). Employing the reasoning from Equations (9)–(11) for the Mori–Tanaka schemes leads with $\mathcal{M}\left(f_{\mathrm{m}}\right)={\mu }_{\mathrm{eff}}/{\mu }_{\mathrm{m}}=f_{\mathrm{m}}/(1+2/3(1-f_{\mathrm{m}}))$ to ${\displaystyle c_{\mathrm{eff}}/c_{\mathrm{m}}=f_{\mathrm{m}}/\sqrt{1+2/3(1-f_{\mathrm{m}})}}$, see Figure 8, see also [40]. For the differential scheme, $\mathcal{M}$ has to be determined numerically and $c_{\mathrm{eff}}/c_{\mathrm{m}}$ is determined as ${\displaystyle \sqrt{f_{\mathrm{m}}\mathcal{M}}}$. For the green graph in Figure 8, $\mathcal{M}\left(f_{\mathrm{m}}\right)$ has been determined for a Poisson’s ratio of $\nu$ = 0.4999.

This discrepancy may be explained by the localized brittle failure behavior of the material. When plotting data in a double-logarithmic diagram (Figure 9a), a power law approximation with bending strength proportional to the fourth power, the solid volume fraction of the solid material matrix, ${\sigma }_{\mathrm{t},\mathrm{eff}}\propto f_{\mathrm{m}}^4$, reasonably represents data. Thereby, the strength of the solid phase $(f_{\mathrm{m}}=1)$ is given as approximately ${\sigma }_{\mathrm{t},\mathrm{m}}$ = 125 MPa. A power-law dependency with an exponent of approximately 4 has also been reported previously for data on the fracture toughness of porous alumina ceramics [41], see Figure 9b.

The influence of spheroidal pores on fracture behavior of brittle materials may be characterized by the stress concentration factor p relating a far field uniaxial (tensile) stress to the stress in circumferential direction (so-called hoop stress) in the vicinity of a spherical (or ellipsoidal) pore. For a spherical pore, the maximum tensile stress is approximately twice the far field stress, i.e., $p\approx 2$ [44], see Appendix C. The stress concentration factor has been identified as the major parameter influencing fracture strength of brittle materials with a power-law scaling function [45]:

$${\displaystyle \frac{{\sigma }_{\mathrm{t},\mathrm{eff}}}{{\sigma }_{\mathrm{t},\mathrm{m}}}={(1-f_{\mathrm{p}})}^p={\left(f_{\mathrm{m}}\right)}^p.}$$

(12)

As regards ellipsoidal pores, ref. [45] investigated oblate and prolate shapes. Extreme stresses are associated with the maximum mean curvature of the pore surface which are located at the poles (prolate) or the equator (prolate) of the ellipsoidal pores. The stress concentration factor p may be, also depending on the inclination of the axes of the ellipsoid with regards to the direction of the far field stress, larger (and also smaller) than two. For example, for an oblate pore with an aspect ratio of 0.1 with poles aligned in the direction of the far field stress, p is approximately 13.5. Stress concentration factors have also been reported for the case that the rotational axis of the spheroidal pore (axis running through poles) is inclined with an angle $0\le \alpha \le \pi$ with regard to the direction of the far field stress [45]. In case of an assembly of ellipsoidal pores with uniform geometry randomly positioned in the solid material (with regard to inclination of ellipsoid axes), pores, however, separated sufficiently for limited mutual interaction (as regards the locally disturbed stress/strain fields); an effective angle of $\alpha$ = 54$°$ represents the assembly [45,46]. For oblate pore shape and the mentioned effective angle, a stress concentration factor p of approximately four, which is what has been observed in our experimental program (see Figure 9a), is obtained for an aspect ratio between 0.1 and 0.2 (see Figure 1a in [45]).

5. Concluding Remarks

In this paper, the thermo-mechanical properties of talcum-based steatite (TBS) ceramics with artificially introduced porosity have been (i) characterized experimentally and (ii) represented by homogenization schemes from effective medium/effective field theory. Porosity ranged from 1.5 vol-%, the intrinsic porosity of TBS ceramics, to 30 vol-% in this study. Thermal conductivity and elastic properties are well described by the self-consistent estimate, with effective material properties scaling in a linear manner with porosity as $\propto (1-f_{\mathrm{p}})$. By contrast, the strength properties are, due to the localization of the failure mechanism in the quasi-brittle material, characterized by a higher-order power-law dependency with strength scaling with a proportionality of $\propto {(1-f_{\mathrm{p}})}^4$. The latter dependency has been reasoned by the stress concentration factor for spheroidal pore shapes.

Future work will be devoted to the microstructural characterization of the solid material phases (protoenstatite, clinoenstatite, cristobalite, and the glassy phase) as regards their morphology and micro-mechanical behavior, e.g., via nanoindentation. This may allow the application of more advanced multiscale models for the prediction of thermo-mechanical properties of the sintered material [47,48].