Mechanical Properties of Porous Ceramics

Abstract

It is widely known that increasing interest in porous ceramics is due to their special properties, which comprise high volumetric porosity (up to 90%) with open or closed pores, and a broad range of pore sizes (micropores: d < 2 nm; mesopores: 50 nm > d > 2 nm and macropores: d > 50 nm). These properties have many uses comprehending macroscaled devices, mesoscaled materials and microscaled pieces. During their usage, these materials are usually submitted to thermal and/or mechanical loading stresses. Therefore, it is a premise to understand how these porous structures behave under thermomechanical stresses to design materials that show adequate properties for the required application. In this context, the aim of this chapter is to review the mechanical properties of macroporous ceramics.

1. Introduction

It is widely known that increasing interest in porous ceramics is due to their special properties, which comprise high volumetric porosity (up to 90%) with open and interconnected or closed and isolated pores, and a broad range of pore sizes (micropores: d < 2 nm; mesopores: 50 nm > d > 2 nm and macropores: d > 50 nm). These properties have many uses comprehending macroscaled devices (filters for liquid metals [1, 2], thermal insulating refractories [3, 4], bioceramics for bone regeneration [5, 6, 7], filters for water treatment [8], acoustic insulating tiles [9]), mesoscaled materials (membranes for catalysis [10], drug release substrates [11, 12]) and microscaled pieces (e.g., multifunctional materials where gravimetric properties are critical as batteries [13] and electronic sensors [14]).

During their usage, these materials are usually submitted to thermal and/or mechanical loading stresses. Therefore, it is a premise to understand how these porous structures behave under thermomechanical stresses to design materials that show adequate properties for the required application.

Despite the importance of porosity for application of these materials, there is not a general consensus about the dependence of mechanical properties on porosity parameters. In other words, the real data of the mechanical properties of these materials indicate that their mechanical behavior depends on more than just porosity of the materials.

Since its introduction to the ceramic community in the 1970s, the area of fracture mechanics has made significant contributions to improving ceramics. As an example, the combination of fracture toughness, fracture statistics and fractography has made it possible to identify critical flaws in material and, consequently develop better and reliable advanced ceramics. In addition, the contribution of fracture mechanics was fundamental in understanding the fracture process in brittle materials.

Recognizing that the area behind a crack was responsible for the increase in the R-curve in ceramics was particularly relevant. One issue concerning the uses of brittle ceramics is associated with the statistical and size-dependence of their fracture properties, which can make reliable prediction, a difficult task. Two other problems are the absence of design methodology for brittle ceramics and the high costs of producing the ceramic components [15].

Nowadays, the ceramic community is witnessing a “boom” of nature inspired materials using hierarchical structures that should have the same behavior or qualities as the natural ones. Various papers [16, 17, 18, 19, 20] in the literature show beautiful structures of natural materials and their mimicked copies by researchers. The capacity of a human being’s observation, also a characteristic controlled by nature, has been the driving force to imitate natural hierarchical structures and their qualities.

In this context, the aim of this chapter is to review the mechanical properties of macroporous ceramics. The following issues are of particular interest to this chapter:

1. Which microstructural parameters affect the mechanical strength of the porous ceramic material besides its porosity?

2. To what extent do the pores affect the fracture toughness of the porous ceramic material? Does it make sense to measure the fracture toughness of porous material knowing that the stress intensity factor at the notch tip is decreased by the presence of surrounding pores? Or, instead of this, would the total fracture energy be a more realistic measure?

3. What is the elastic modulus behavior of porous ceramics as a function of temperature?

All these questions need to be considered in order to advance not only the processing of porous ceramic materials but also to design their structures for specific applications.

2. Influence of microstructural parameters on the mechanical strength of porous ceramic materials

The objective here is not to carry out an extensive revision of the fracture of brittle porous materials, but to present results which serve as a basis to the authors´ proposal in this chapter.

First of all, fracture of porous ceramics is better described by the quasi-brittle behavior as their ultimate fracture is triggered by many local events (different from essentially brittle behavior of glass ceramics), yet they are not preceded by highly dissipative processes associated with plastic deformation and strain hardening (as observed in ductile metals) [21]. Quasi-brittle fracture behavior is also observed in rocks, bones and ceramic composites.

As mentioned earlier, the real data of the mechanical properties of porous materials indicate that their mechanical behavior depends on more than just porosity of the materials.

Questions have been raised about the models proposed by Gibson and Ashby (GA) [22, 23], which indicate that the relative strength of a porous material is a function of its relative density as follows:

$$\frac{\mathrm{\sigma }}{{\mathrm{\sigma }}_{\mathrm{s}}}=\mathrm{C}{\left(\frac{\mathrm{\rho }}{{\mathrm{\rho }}_{\mathrm{s}}}\right)}^{\mathrm{m}}$$ E1

where σ and ρ are, respectively, the fracture strength and the density of porous material; σ_S and ρ_S are the fracture strength and the density of solid material, respectively; C is a dimensionless constant and the exponent m depends on the pore morphology (m = 3/2 for open pores or m = 2 for closed ones). The Gibson and Ashby (GA) models are based on the bending or buckling of cell edges.

Figure 1 shows the relative strength predicted by the Gibson and Ashby models plotted together with experimental data of porous ceramics from different researchers. It can be seen a disagreement between the theoretical curves of GA models and the experimental results.

Figure 1.

Plot of relative strength as a function of relative porosity for Gibson and Ashby (GA) models [22, 23] and for experimental data obtained by Salvini et al. [27], Colombo et al. [24] and Seeber et al. [25].

Colombo et al. [24] attributed microstructural factors for the lack of fitting data to the Gibson and Ashby models, as shown in Figure 1, as they do not consider the distribution of pore sizes, neither have mixed pores (open and closed) nor flaws in the pore wall (struts).

Seeber et al. [25] also noted that the drop in mechanical properties of foamed ceramics was disproportionately greater than what was to be expected solely from increasing the porosity. These authors suggested this behavior must be an influence of the pore size or the strut thickness, as reported by Brezny and Green [26] in a previous paper.

Nevertheless, Salvini et al. [27] suggested that a parameter which expresses the processing method to produce the porous structure should be considered by the mechanical models. For instance, porous ceramics with similar porosity and density ranges can be produced using different ceramic methods such as sacrificial fugitives, replica templates and directing foaming. However, each method provides a different number of struts (ligaments) of distinct solid particle packing, which influences the final mechanical behavior of the material.

In this context, Lichtner et al. [28] produced porous ceramics of same porosity but different pore arrangement using the freeze casting and the slip casting processing methods. They have noted a strong influence of processing on the mechanical properties, and attributed to the differences of orientation and connectivity of macropores.

Brezny et al. [29] also reported that strength of the struts is an important parameter controlling the properties of porous ceramics. According to them, an increase in the strut strength would be expected as a result of the reduced probability of finding a critical flaw in a smaller volume of material as predicted by Weibull, the weakest link hypothesis for strength variability.

Additionally, Genet et al. [21] pointed out that the Gibson and Ashby’s approach cannot directly deal with the statistical and size-dependent aspects of fracture.

In 1996, Rice [30] had already drawn attention to this debate considering some problems with micromechanics-based models. The first concern mentioned by Rice was the assumption that porous bodies are represented by packing of hollow spherical particles of an infinite range of sizes. Then, it is assumed the application of a hydrostatic pressure is uniformly distributed in all particles so the resulting strain response can be calculated. Moreover, a common approach to improving the agreement between these models is to let some parameter, for example, the Poisson ratio, depends on the porosity, that is, using it as an adjustable parameter. Another concern is that these models assume that porosity will remain fixed during applications.

Then, Rice has proposed that the mechanical strength of porous ceramics should depend not only on the relative density but also on the minimum solid area fraction, as depicted in Figure 2. That is because the solid area is required for transmission of mechanical stresses and thermal and electrical fluxes. This concept is schematically presented in Figure 3.

Figure 2.

Models from the literature showing the effects of the minimum solid area and porosity on the strength for three basic sphere stackings (cubic, rhombohedral and a mixture of them) and for cylindrical or spherical pores, as well as their mixture, in cubic stacking. The nearly linear portion of curves is represented by the semilogarithmic expression $\sigma ={\sigma }_S{e}^{-\mathit{bP}}$ , with b values ranging from 1 to 3 for stacked pores and from 3 to 9 for stacked particles [30].

Figure 3.

Diagram of the minimum solid area concept. (a) Cross section of a dense material showing the uniform transmission among layers normal to a uniform mechanical stress or conductive flux. (b) Cross section of material showing some layers were removed, leaving only small continuous areas (MSA) for transmission of stress or flux. (c) Cross section of stacked particles and (d) cross section of stacked pores where again the minimum areas of solid will control the transmission of stress or flux normal to the plane of slab [30].

As can be seen in Figure 2, each specific model has the following three characteristics: (1) a nearly linear slope of the first half to three-quarters of the porosity range, (2) the approximate porosity value where properties start to be damaged more significantly than the linear slope and (3) the critical porosity P_C where properties go to zero.

However, as pointed out by Rice, these characteristics are useful in distinguishing the basic porosity character of each stacking model, but their utility varies. For instance, P_C values can be accurately defined theoretically, but obtaining reliable experimental data can be a difficult task.

Moreover, the approximated linear slopes are unique for the basic stacking models, and they have been the most widely available factor for polycrystalline materials. However, as clearly shown in Figure 2, they can be applied for the restricted porosity range only.

More recently, Bruno et al. [31] reviewed micromechanics aiming the development of microstructure-property relations for porous microcracked ceramics. They focused on specific issues for porous ceramics as the nonlinear stress–strain behavior and the thermal-induced microcracking.

Zheng et al. [32] considered that the fracture strength of brittle porous material is nonlinear and, therefore, there is a percolation failure phenomenon at the fracture of these materials. Therefore, their model considers the porosity (P) and the elastic percolation (ϕ), which depends on the Poisson’s ratio of material, as shown:

$$\frac{\mathrm{\sigma }}{{\mathrm{\sigma }}_{\mathrm{S}}}={\left[{\left(\frac{\mathrm{\varphi }-\mathrm{P}}{\mathrm{\varphi }}\right)}^{1+\mathrm{\nu }}.\left(1-{\mathrm{\varphi }}^{2/3}\right)\right]}^{1/2}$$ E2

$$\mathrm{\varphi }=1-\left[\frac{1+\mathrm{\upsilon }}{3\left(1-\mathrm{\upsilon }\right)}\right]=\frac{2\left(1-2\mathrm{\upsilon }\right)}{3\left(1-\mathrm{\upsilon }\right)}$$ E3

where ν is the scaling exponent for tridimensional solids and υ is the Poisson ratio of materials. In general, for ceramics υ = 0.2 and ϕ = 0.5, for metals υ = 0.3 and ϕ = 0.38, and for polymers υ = 0.33 and ϕ = 0.338. Zheng et al. [32] validated the proposed model for polymers using different porosities and experimentally measured them using the three-point-bending strength test.

In order to check the validity of Zheng’s model, Salvini et al. [27] considered the fracture flexure strength data of foamed Al₂O₃ with porosity of 76%. Nevertheless, the results must be interpreted with caution, as Zheng’s model overestimated the flexure strength indicating values around 25 MPa, while the average experimental value obtained by Salvini et al. [27] was 10 MPa.

Regarding the failure patterns, Genet et al. [21] investigated the fracture mechanism across scales of porous ceramic scaffold applying a computational method. Figure 4 shows the scanning electron microscopy (SEM) image of the porous scaffold made by freeze casting and the respective idealized geometry.

Figure 4.

SEM image of a porous ceramic scaffold made by the freeze casting method and its associated idealized geometry, which consists of walls connected by bridges positioned in staggered rows. The geometrical parameters are as follows: The distance between the walls, d_w (d_w = 25 μm), the distance between the bridges, d_b (d_b = 75 μm), and the thickness of the walls and bridges, e (e = 5 μm). A microcell is defined by r × r RVEs (representative volume elements) [21].

Genet et al. [21] found that for very small-sized samples (r = 1 × 1 RVEs), the fracture is brittle and is triggered by the first strut to break. For intermediate-sized samples (r = 5 × 5 RVEs), however, the fracture is controlled by the percolation of several strut breaks, and is mainly governed by the stress redistribution after each break. For large-sized samples (r = 256 × 256 RVEs), they found that the failure process appears to be different, and it is divided into two stages.

The initial stage consists of a widespread development of damage due to the failure of the weakest local defects. But, the stress redistribution caused by these failures is not high enough to make the neighboring cells break or initiate a macrocrack. Instead, a critical defect is activated, rapidly leading to the development of a macrocrack, which leads to the final fracture of the material. Then, the “fatal” macrocrack in large samples does not result from the percolation of previously damaged cells/pores.

Nevertheless, there are some limitations in this study. Firstly, the computational analyses were based on the assumption that porous scaffold material presents isotropic Young’s modulus and Poisson’s ratio. Another limitation of this study is that the fracture behavior was evaluated under one loading direction only (pure traction).

Although using high resolution tomography to evaluate the failure behavior of porous ceramics, Berek et al. [33] and Petit et al. [34] independently identified the same fracture pattern as proposed by Genet et al. [21]. Recently, Cui et al. [35] also found the nonlinear mechanical behavior due to the accumulation of local damage in porous ceramics.

In different works, Brezny et al. [26] and Morgan et al. [36] studied the effect of the cell size of glassy porous ceramics on their mechanical properties. Both authors reported the mechanical strength of cellular ceramics increased with decreasing cell sizes. They attributed this behavior to a reduction in the critical flaw size as well as the increasing strut strength in smaller cell sizes.

Meanwhile, Deng et al. [37] investigated the reinforcement mechanisms of fine- and coarse-grain porous SiC and found that the crack-tip blunting mechanism in porous material, as shown in Figure 5, increases the fracture toughness of the material. They also noted that the larger the pore size in front of the crack, the more the fracture toughness of the porous ceramic is relative to its fracture strength. However, a detailed discussion about the fracture toughness of porous ceramics is provided in the next section.

Figure 5.

Representation of a crack propagated in (a) dense ceramic with a sharp crack-tip and in (b) porous ceramic where the crack-tip becomes blunt [37].

Based on the Stress Concentration Design Factors, Peterson [38], Deng et al. [37] and Rice [39] attributed this behavior to the interactions between pores. Rice [39] combined Peterson’s data in Figure 6 to show that the stress concentration diminishes significantly as the pores become closer and the pore interactions begin to no longer be negligible when the center-to-center distance between them is around two times their diameter. Specifically, for the case where pores touch each other, in porosity of 78%, Figure 6 shows that the stress concentrations become very low.

Figure 6.

Stress concentration factors for a chain of holes in either uniaxial or in biaxial tension [38].

In reviewing the literature about mechanical properties of porous ceramics, it can be summarized which parameters affect, and to what extent, the strength of this class of ceramic materials.

On the one hand, there are researches considering the pores as the stress concentrations for the material fracture. Therefore, these researches consider the mechanical properties of porous ceramics depend only on the relative porosity.

On the other hand, other researches have considered the interactions among pores as low as 10% of porosity, which reduce the stress concentration factor for fracture. Then, they take into account the mechanical properties of porous ceramics depend on pores-stress interactions.

Regarding pore-stress interactions, there is strong evidence in the literature [40, 41, 42] concerning the mechanical behavior of human bones (also a quasi-brittle material) to support the hypothesis that microstructural changes in material may be essential in controlling its strength. The considered microstructural parameters are the porosity, size of pores, number and thickness of struts connecting the pores.

In human bodies, the mechanical properties of natural bone change with their biological location because the crystallinity, porosity and composition of bone adjust to the biological and biomechanical environment. For these materials, the bone volume fraction (bone volume BV/total volume TV) is given as a function of the thickness-to-length ratio, that is, $t/\ell$ , as shown in the following expression:

$$\frac{\mathrm{BV}}{\mathrm{TV}}=\frac{33\mathrm{\pi }}{80\sqrt{2}}{\left(\frac{\mathrm{t}}{\ell }\right)}^2$$ E4

Changes in the microstructure of the vertebral trabecular bone with aging have been quantified by histomorphometric analysis and also simulated by a computer using finite element software in two-dimensional (2D) [41] and three-dimensional (3D) [42] microstructural models. The microstructural changes of the bone include reductions in the trabecular thickness ( $t)$ and number (N), as shown in Figure 7. Both changes are strongly correlated with reductions in bone volume fraction and represent the two fundamental changes in microstructure associated with reduced bone volume [40, 41, 42].

Figure 7.

Simulated images of the trabecular bone showing two types of bone loss. Top right: thinning of trabeculae (ligaments) and bottom right: loss of trabeculae [42].

Silva et al. [41] showed, using a 2D model, that once significant numbers of trabeculae (ligaments) are lost, it is impossible to recover the original mechanical properties of bone merely by increasing the trabeculae thickness, indicating the importance of the trabeculae number (N) and the integrity of its microstructure.

Considering the different types of trabeculae microstructures in 3D (rod-like and plate-like as shown in Figure 8), Guo et al. [42] quantified the changes in the Young’s modulus and mechanical strength due to the trabecular bone loss. They considered the arrangement of tetrakaidecahedral cells as a 3D model for trabecular bones as shown in Figure 8. The cells filled in the 3D space were connected by either all beams (rod-like model) or all plates (plate-like model).

Figure 8.

The 3D microstructural model of the trabecular bone. Each single cell has an edge of equal length, ℓ. θ = 45° for an isotropic structure. The entire model consists of arrays of 5 × 5 × 5 cells [42].

For each case of trabeculae loss simulation, the apparent Young’s modulus and mechanical strength were normalized by the values of corresponding initial intact models (E₀ = 15 GPa, σ₀ = 100 MPa, Poisson’s ratio of 0.3 at $t/\ell =0.1$ ).

Quantitative relationships between mechanical strength and bone volume fraction (BV/TV) for the two types of bone loss in rod-like and plate-like models are presented in Table 1 and Figure 9.

Model type	Bone loss type	Mechanical strength
Rod-like	Trabeculae thinning	$\frac{{\sigma }^{\ast }}{{\sigma }_0}=0.592{\left(\frac{\mathit{BV}}{\mathit{TV}}\right)}^{1.60}$ , r^2 = 0.99
Plate-like	Trabeculae thinning	$\frac{{\sigma }^{\ast }}{{\sigma }_0}=0.378{\left(\frac{\mathit{BV}}{\mathit{TV}}\right)}^{1.10}$ , r^2 = 0.99
Rod-like	Horizontal trabeculae loss	$\frac{{\sigma }^{\ast }}{{\sigma }_0}=1.65x{10}^3{\left(\frac{\mathit{BV}}{\mathit{TV}}\right)}^{3.27}$ , r^2 = 0.85
	Oblique trabeculae loss	$\frac{{\sigma }^{\ast }}{{\sigma }_0}=4.69x{10}^{10}{\left(\frac{\mathit{BV}}{\mathit{TV}}\right)}^{6.94}$ , r^2 = 0.99
Plate-like	Trabeculae loss	$\frac{{\sigma }^{\ast }}{{\sigma }_0}=1.36x{10}^2{\left(\frac{\mathit{BV}}{\mathit{TV}}\right)}^{3.36}$ , r^2 = 0.99

Table 1.

Relationships between mechanical strength and bone volume (BV/TV) for two types of bone loss in rod-like and plate-like models [42].

Figure 9.

Reduction in the strength (a) of the rod-like model and (b) plate-like model due to bone loss (trabeculae thinning and trabeculae loss) represented by solid lines. Strength recovery by subsequent treatment with thickening trabeculae is represented by the dotted lines [42].

In the case of the rod-like model (Figure 9a), the loss of oblique trabeculae showed that the reductions in mechanical strength were more severe when the trabeculae thickness was reduced. In addition, after 13% loss of bone volume fraction (BV/TV), there was a dramatic reduction in strength due to the loss of the horizontal trabeculae.

For the plate-like model (Figure 9b), the reductions in mechanical strength due to trabeculae loss were also much more significant than those due to uniform trabeculae thinning. The quantitative relationships between the mechanical strength and bone volume fraction (BV/TV) due to trabeculae loss were dramatically different from those for trabeculae thinning (see Table 1).

These results suggest the importance of microstructural integrity such as the connectivity of the trabeculae bone architecture to maintain the mechanical integrity of bones. Besides, the extent of reduction in the mechanical properties due to trabeculae loss depends on the types of microstructures in the bone.

Thus, for the natural bones, the number of ligaments (trabeculae) between pores appears to be much more effective to increase the strength of the material in comparison to the ligament thickness.

Lichtner et al. [28] also found the mechanical properties of freeze casting porous ceramics that are controlled by the connectivity of pore walls.

Similar mechanical behavior was reported by Salvini [43] for SiC ceramic filters of the same porosity (85%) but with different pores per inch (ppi), number of struts and average pore sizes (Figure 10). It can be seen in Figure 10 that there is a rise in the number of connecting struts when the pore per inch (ppi) increases.

Figure 10.

Physical characteristics of SiC ceramic filters of the same porosity (85%) but with different pores per inch (ppi) [43].

Moreover, differences were found between the strut number and pore size tendencies as a function of pores per inch (ppi). Interestingly, the number of connecting struts is sensitive to the variation of the number of pores (ppi).

The high number of connecting struts in the SiC filters of high pores per inch (ppi) is more probably contributed to their increasing mechanical strength, as depicted in Figure 11.

Figure 11.

Compressive mechanical strength of SiC ceramic filters of the same porosity level (85%), but with different average pore sizes and number of connecting struts [43].

A possible explanation for this mechanical behavior in SiC ceramic filters, as well as in trabeculae bones, might be the diminution of the stress concentration factor due to the stress-pore interactions, which is very low for porosity higher than 78% (see Figure 6).

One of the more significant findings to emerge from this section is that, in addition to the porosity of porous ceramics, the number of connecting struts between the cells/pores in the microstructure plays a fundamental role in their mechanical strength behavior.

3. Fracture toughness of porous ceramics

A requirement for almost structural materials is that they are both strong and tough yet invariably, in most materials, the properties of strength and toughness are mutually exclusive [44]. Whereas strength of material is a stress indicating its resistance to nonrecovery deformation, toughness is the resistance of material to the propagation of a crack and, then is measured as the energy required to cause fracture.

The ability of material to experience limited plastic deformation is a critical aspect to toughness, as this characteristic enables the local dissipation of high stress that would otherwise cause the fracture of material. That is the reason why the design methodology based on yield strength of materials is a common practice in the engineering [45].

Concerning fracture toughness (K_IC) methods for brittle ceramics, there are several tests which have been used such as the precrack by indentation, the single-edge notch beam (SENB) and Chevron notch (CV) methods, which are the most common tests. All methods involve application of force to a beam test specimen in three or four point flexures.

Each method has limitations and the major problem is that significant effects due to the microstructure, for example, grain size and porosity, are not addressed in the continuum mechanics basis of these tests.

As proposed by Rice [46], the microstructural dependence of fracture energy ( $\gamma$ ) comprehends the grain size (G), the composition, the porosity (P) and the combined effects of them.

Concerning the grain size effect, the crystalline structure of material is important when evaluating behavior between G and $\gamma$ . Overall, the cubic materials present less variation of $\gamma$ with G.

On the other hand, as shown in Figure 12 for some noncubic ceramics, they present rising values of fracture energy ( $\gamma$ ) until the maximum and, then, there is a decrease as the grain size (G) increases. This effect is attributed to the thermal expansion anisotropy, which occurs only in noncubic materials, but the intensity of it varies among materials [47].

Figure 12.

Fracture energy ( $\gamma$ ) as a function of grain size (G) for noncubic oxide ceramics at 22°C [47].

Changes in the ceramic composition can also lead to a similar effect related to the grain size. These changes can be due to chemical composition or by introducing a more compliant grain boundary phase; both changes can modify the thermal expansion anisotropy affecting the fracture energy ( $\gamma$ ) behavior of the material. There are several examples in the literature showing this behavior for Al₂O₃-ZrO₂ system, and SiC and Si₃N₄ with additions of oxides.

Although there are more limited data about the relation between fracture energy and porosity ( $\gamma$ × P) in comparison to fracture energy and grain size $(\gamma$ × G), some tendencies can be mentioned.

At first, in single-phase ceramics with fine-to-medium-grain size, the fracture energy ( $\gamma )$ decreases as porosity (P) increases, following a similar tendency to the one found for mechanical strength (see Figure 2). Overall, this tendency has been shown in many advanced ceramics with porosity up to 50% [47].

Nevertheless, there are some works indicating the rate of fracture energy decrease can be reduced or even reversed [37, 47]. For instance, porous-fused SiO₂ (P~13–20%) provided fracture energy ( $\gamma )$ equal to or greater than the obtained value for dense SiO₂ glass.

Additional examples comprehend porous composite SiC-Al₂O₃-C (P~30–40%) and reaction sintered Si₃N₄ (P~45%). However, in all cases, the fracture energy ( $\gamma )$ decreased again at high porosity level (P) because the $\gamma$ value goes to zero as P achieves 100%.

Considering there is not a consensus concerning the relation between fracture energy and porosity, we would like to propose a discussion about the influence of porosity on the fracture energy measurements of porous ceramics and, consequently, on their fracture toughness values.

Recently, Salvini et al.⁴³ have shown that fracture energy values for macroporous foamed Al₂O₃ can vary significantly depending on the test conditions. According to them, this is because of the following two factors: (1) the stress intensity factor at the notch tip may be decreased by the presence of surrounding pores (crack-tip blunting mechanism) as shown in Figure 13 and (2) there is a strong interaction of cracks with the pores in the microstructure.

Figure 13.

Macroporous foamed Al₂O₃ sample notched with a 300-μm thick diamond blade where the crack-tip blunting mechanism is shown [58].

Although many researchers use a single test sample, Salvini et al. [48] carried out tests on separated notched macroporous samples for the two energy measurements ${\mathrm{\gamma }}_{\mathrm{eff}}$ and ${\mathrm{\gamma }}_{\mathrm{WOF}}$ . The ${\mathrm{\gamma }}_{\mathrm{eff}}$ represents the fracture surface energy to initiate crack propagation, as proposed by Evans [49], whereas the total work-of-fracture ${\mathrm{\gamma }}_{\mathrm{WOF}}$ expresses the energy to propagate a crack through the specimen thickness, as suggested by Nakayma [50]. A similar approach has been applied in evaluating fracture energy in materials with coarse microstructures, for example, in refractory ceramics [51, 52].

The fracture toughness (K_IC) was determined according to the ASTM E-399 at room temperature in a three-point bend test applying the equation:

$${\mathrm{K}}_{\mathrm{IC}}=\frac{\mathrm{F}\times \mathrm{S}}{\mathrm{B}\times {\mathrm{W}}^{3/2}}\times \mathrm{f}\left(\frac{\mathrm{a}}{\mathrm{W}}\right)$$ E5

where F is the fracture load (N), S is the span (m), B is the specimen width, W is the specimen thickness, a corresponds to the crack size, and f (a/W) is obtained by the following expression:

$$\mathrm{f}\left(\frac{\mathrm{a}}{\mathrm{W}}\right)=3\sqrt{\frac{\mathrm{a}}{\mathrm{W}}}\times \frac{1.99-\left(\frac{\mathrm{a}}{\mathrm{W}}\right)\times \left(1-\frac{\mathrm{a}}{\mathrm{W}}\right)\left[2.15-3.93\frac{\mathrm{a}}{\mathrm{W}}+2.7{\left(\frac{\mathrm{a}}{\mathrm{W}}\right)}^2\right]}{2\left(1+2\frac{\mathrm{a}}{\mathrm{W}}\right){\left(1-\frac{\mathrm{a}}{\mathrm{W}}\right)}^{3/2}}$$ E6

The bar samples were center-notched to one-half of their thickness $\left(\mathrm{a}/\mathrm{W}\approx 0.5\right)$ with a 300-μm thick diamond blade for the K_IC measurements. All fracture energy tests were performed on a MTS 180 machine in three-point bending over a span of 125 mm. For K_IC, fracture surface energy ( ${\mathrm{\gamma }}_{\mathrm{eff}}$ ) measurements were carried out at a stress loading rate of 1.5 kN/s.

This measured value of K_IC was then used to calculate the energy for crack initiation ( ${\mathrm{\gamma }}_{\mathrm{eff}})$ by the expression:

$${\mathrm{K}}_{\mathrm{IC}}={\left(2{\mathrm{\gamma }}_{\mathrm{eff}}\mathrm{E}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$ E7

where E is Young’s elastic modulus measured by a sonic technique in foamed Al₂O₃ bar samples (E = 18 GPa). Further details about this technique can be found in Ref. [53].

For the total work-of-fracture ( ${\mathrm{\gamma }}_{\mathrm{WOF}})$ measurements, the foamed Al₂O₃ bar samples were also center-notched with a 300-μm thick diamond blade so that one-half of their thickness cross section remained. The samples were loaded in three-point bending at a crosshead speed of 0.001 mm/min to ensure stable crack growth (ASTM C1368–10). The total work-of-fracture ( ${\mathrm{\gamma }}_{\mathrm{WOF}})$ values were then calculated by:

$${\mathrm{\gamma }}_{\mathrm{WOF}}=\frac{\int \mathrm{F}\mathrm{dx}}{2\mathrm{A}}$$ E8

where $\int \mathrm{F}\mathrm{dx}$ represents the required work for new surfaces’ generation and A is the projected area of the new fracture surfaces, as determined directly from the individual specimen notched areas.

Table 2 shows the obtained experimental results of the fracture energies ( ${\mathrm{\gamma }}_{\mathrm{eff}}$ and ${\mathrm{\gamma }}_{\mathrm{WOF}}$ ) and the fracture toughness (K_IC) for the foamed Al₂O₃.

Density, $\mathit{\rho }$ (g/cm^3)	${\mathbf{\gamma }}_{\mathbf{eff}}$ (J/m^2)	${\mathbf{\gamma }}_{\mathbf{WOF}}$ (J/m^2)	KIC (MPa.m^1/2)
0.86 ± 0.02	4.68 ± 0.5	15.81 ± 0.8	0.42 ± 0.03
0.87 ± 0.01	3.62 ± 0.3	14.46 ± 0.95	0.35 ± 0.05
0.90 ± 0.02	3.72 ± 0.6	15.39 ± 1.0	0.37 ± 0.02
0.92 ± 0.02	4.15 ± 0.8	12.27 ± 1.1	0.36 ± 0.02
0.94 ± 0.03	3.55 ± 0.35	13.32 ± 0.9	0.34 ± 0.09

Table 2.

Results of fracture energies ( ${\mathrm{\gamma }}_{\mathrm{eff}}$ and ${\mathrm{\gamma }}_{\mathrm{WOF}}$ ) and the fracture toughness (K_IC) for macroporous foamed Al₂O₃ [58].

The K_IC values in Table 2 were the expected ones for highly porous ceramics and they were compatible with data from the literature [24, 54, 55]. However, these results must be interpreted carefully, because of the crack-tip blunting mechanism due to the presence of surrounding pores (see Figure 13). This mechanism may decrease the stress intensity factor at the notch tip. Consequently, the crack propagation should not follow the required linear-elastic conditions for the fracture toughness measurement.

Based on that, we would like to raise the following issues: Does it make sense to measure the fracture toughness (K_IC) of porous ceramics knowing that the stress intensity factor at the notch tip is decreased by the crack-tip blunting? Or, instead of this, would the total work-of-fracture energy be a more realistic measure for these materials?

The results of total work-of-fracture energy ( ${\mathrm{\gamma }}_{\mathrm{WOF}}$ ) presented in Table 2 agree with the values obtained for Al₂O₃ produced using intermediate grain sizes (10–50 μm) [56].

Nevertheless, there is few data in the literature concerning fracture energies of porous ceramics. Kanhed et al. [57] found fracture toughness values in the range of 0.55–0.86 $\mathit{MPa}{m}^{1/2}$ for porous hydroxyapatite.

Considering the fracture surface microstructure of foamed Al₂O₃ presented in Figure 14, it seems that the fracture energy increase induced by porosity and the fracture energy reduction caused by decreased solid concentration are combined to describe the total work-of-fracture energy of material. However, more research on this topic is therefore recommended.

Figure 14.

Fracture surface of macroporous foamed Al₂O₃ showing the characteristics of the fracture and notched surfaces [58].

4. Elastic modulus behavior of porous ceramics

As well known, accurate elastic moduli are measured dynamically by measuring the frequency of natural vibrations of a beam, or by measuring the velocity of sound waves in the material. Both depend on $\sqrt{\mathrm{E}/\mathrm{\rho }}$ , so if density (ρ) is known, E can be determined. These properties (ρ, E) reflect the mass of atoms, the way they are packed in material and the stiffness of the bonds that hold them together. For instance, Table 3 presents the literature data of elastic modulus for several synthetic and natural porous brittle materials.

Material	Porosity (%)	Elastic modulus (GPa)
Porous hydroxyapatite (HA) [5]	82–86	0.002–0.83
Cortical bone [5, 59, 60]	5–15	7–18
Cancellous bone [5, 59, 60]	~90	0.1–5
Cordierite diesel particulate filter (DPF) [54]	~50	12–13
Porous clay ceramics [55]	35–50	1–3
Silicon oxycarbide ceramic foams [24]	70–85	1–7
Porous SiC preforms [61]	30–65	30–120
Porous Si3N4 [62, 63]	35–55	45–105
Gelcasting Al2O3 foams [63]	60–85	10–65
Foamed Al2O3 [48]	76–80	15–18
SiC filters for metals [64]	85–92	2–3

Table 3.

Elastic modulus values for porous and brittle materials.

Considering that microscaled damage in ceramics can be caused in processing as well as during their application, a technique that detects the in situ microcracks evolution is important to estimate the life operation of them [65].

The in situ elastic modulus measurement as a function of temperature may identify the causes of microcracks in material, which is very helpful to adjust ceramic processing and design the material microstructure for an extended period of use.

In this context, this section is divided into two parts. In the first part, a review about theoretical models to describe the effect of porosity on the elastic modulus of porous ceramics is presented. Then, the second part presents and discusses the elastic modulus behavior as a function of temperature for the foamed Al₂O₃ ceramics.

Much research over the last decades was dedicated to understanding the influence of the porosity on the elastic modulus of ceramics. Table 4 shows the most common theoretical models concerning the elastic modulus (E) and the porosity (P) correlations.

Model	Characteristics
$\mathit{E}={\mathit{E}}_{\mathrm{0}}{\mathit{e}}^{-\mathit{bP}}$; Refs. [28, 66]	Dependent on P and on extent of contact between solids	Valid for $P<0.5$ and isotropic material
$\mathit{E}={\mathit{E}}_{\mathrm{0}}\left(\mathrm{1}-\mathrm{1.9}\mathit{P}+\mathrm{0.9}{\mathit{P}}^{\mathrm{2}}\right)\mathit{,}\mathit{P}\le \mathrm{0.5}$; Refs. [67, 68]	Dependent on the closed pores’ concentration	Valid for $P<0.5$ and isotropic material
$\mathit{E}=\mathit{C}{\mathit{E}}_{\mathrm{0}}{\left(\frac{\mathit{\rho }}{{\mathit{\rho }}_{\mathit{s}}}\right)}^{\mathit{n}}\mathit{,}\mathit{C}\approx \mathrm{1}\mathit{\text{and}}\mathrm{1}<\mathit{n}<\mathrm{2}$; Refs. [22, 23]	Dependent on relative density and pore type	Valid for open porous and isotropic material
$\mathit{E}={\mathit{E}}_{\mathrm{0}}{\left(\mathrm{1}-{\mathit{P}}^{\mathrm{2}/\mathrm{3}}\right)}^{\mathit{s}}$; $\mathit{s}=\mathrm{1.21}{\left(\frac{\mathit{z}}{\mathit{x}}\right)}^{\mathrm{1}/\mathrm{3}}{\left\{\mathrm{1}+\left[{\left(\frac{\mathit{z}}{\mathit{x}}\right)}^{-\mathrm{2}}-\mathrm{1}\right]\right\}}^{\mathrm{1}/\mathrm{2}}{\mathit{cos}}^{\mathrm{2}}\mathit{\theta }$; Ref. [67]	Dependent on P and on porosity geometry of closed pores	Valid for $P<0.4$ and anisotropic material
$\mathit{E}={\mathit{E}}_{\mathrm{0}}\frac{{\left(\mathrm{1}-\mathit{P}\right)}^{\mathrm{2}}\mathit{R}}{\mathit{P}+\left(\mathrm{1}-\mathit{P}\right)\mathit{R}}$; $\mathit{R}=\frac{{\mathit{d}}_{\mathit{\beta }}}{{\mathit{d}}_{\mathit{\alpha }}}$ , size ratio; Ref. [69]	Dependent on P, on geometry and on topology of porosity	Valid for $R<1$ and anisotropic material

Table 4.

Summary of the elastic modulus (E) and porosity (P) models, where E₀ refers to the elastic modulus of solid material.

The Knudsen [66] and Rice [28] models fit well with real data of materials with porosity lower than 50%, as reported by Boccaccini et al. [67] and Ohji et al. [62, 63].

The model proposed by MacKenzie [68] and Kingery [69] is also defined for a lower level of closed pores.

Gibson and Ashby (GA) models [22, 23] indicated that the elastic modulus of a porous material depends only on its relative density and pore morphology.

Boccaccini model [67] introduced the s parameter to indicate the porosity geometry effect on the elastic modulus, that is, the pore shape and its orientation. However, it is valid only for low porosity level (P < 0.4) of closed pores.

In another work Boccaccini et al. [70] also included the topological parameters of highly porous microstructure to the model, besides the geometrical ones. Topological characterization comprehends the separation, the separated volume and the degree of contact and separation in two-phase microstructures. However, an issue that was addressed by the authors in this model is the necessity of a well-characterized porous structure of material, besides its porosity, to achieve rigorous verification of the model.

Overall, little research in the literature considers porous ceramics as anisotropic material in the evaluation of their elastic modulus behavior.

For instance, Rodrigues et al. [71] compared experimental data of the elastic moduli of Al₂O₃ foams (P: 60–90%) to several models proposed in the literature, as depicted in Figure 15. These authors considered the Al₂O₃ foams as isotropic material and, therefore, the experimental data fitted well with the MacKenzie [68] and Gibson and Ashby (GA) [22, 23] models.

Figure 15.

Theoretical models predicting the effect of porosity on the relative Young’s modulus (continuous lines) and experimental data of gelcasting Al₂O₃ foams [71].

This analysis, however, needs to be considered carefully because the foamed ceramics of high porosity (P > 60%) are usually anisotropic material and the mentioned models are valid only for homogeneous and isotropic ones.

Roy et al. [61], Wu et al. [59] and Lichtner [28] clearly showed that the extent of anisotropy, and its effect on the elastic modulus measurements, is strongly dependent on the porosity level, mainly for porosities higher than 40%, which is the case of the most foamed ceramics and natural porous materials.

Moving to the elastic modulus behavior at heating, Salvini et al. [48] evaluated the in situ elastic modulus behavior of foamed Al₂O₃ to identify the changes at the curing, drying and sintering stages of material.

The tests of elastic modulus were carried out in the range of temperature from 50 to 1400°C in air with heating and cooling rates of 2°C/min and a holding time of 4 h at 1400°C. After that, additional thermal cycles of elastic modulus measurements were carried out up to 1400°C.

Green bar samples (25 mm × 25 mm × 150 mm) of foamed Al₂O₃ containing 5 wt% of high alumina cement were considered for the in situ elastic modulus evaluation. The measurements were carried out according to ASTM C1198–91 using the resonance bar technique (Scanelastic equipment, ATCP, Brazil).

Figure 16 depicts the in situ elastic modulus evolution (first and second cycles) up to 1400°C, in addition to results of the as-sintered foamed Al₂O₃ at 1500°C/4 h. Table 5 presents the crystalline phase changes obtained by X-ray diffraction as a function of temperature for ceramic composition.

Figure 16.

In situ elastic modulus of macroporous Al₂O₃: blue curve corresponds to the first cycle up to 1400°C of the green sample; the red curve is the measurement after the first cycle, and the green curve corresponds to the measurement after sintering up to 1500°C/4 h. The arrows indicate the discontinuities in the curve caused by decomposition or formation reactions of specific ceramic phases [48].

Crystalline phases	110°C	1000°C	1200°C	1400°C	1500°C
α-Al2O3	******	******	******	******	******
Al2O3.3H2O, AH3	**
CaO. Al2O3.6H2O, C3AH6	**
CaO.Al2O3, CA		***	*
CaO.2Al2O3, CA2		*	***	****	*
CaO.6Al2O3, CA6					***

Table 5.

Phase changes obtained by X-ray diffraction in alumina composition containing 5 wt% of high alumina cement. The concentration of phases is qualitatively defined by the number of asterisks (*) displayed [48].

The foaming and casting processes of Al₂O₃ suspension were carried out at room temperature (~25°C). As reported by the literature [72], the main cement hydrate phase formed at room temperature is CAH₁₀ (CaO.Al₂O₃.10H₂O). When the temperature increases, this phase partially dehydrates ~110°C into a mixture of gibbsite, AH₃ (Al₂O₃.3H₂O) and tricalcium aluminate hydrate, C₃AH₆ (CaO.Al₂O₃.6H₂O). This suggests that the drop of modulus curve observed in the initial stage of heating of the first cycle was caused by the conversion of CAH_10.

At higher temperatures, the discontinuities in the temperature range of 200–400°C were probably due to dehydration of the phases AH₃ (Al₂O₃.3H₂O) and C₃AH₆ (CaO.Al₂O₃.6H₂O), besides the decomposition of the organic additives (surfactants).

After dehydration, the elastic modulus remains stable at a low value until sintering starts at ~900°C. After 1100°C, the modulus increases rapidly due to sintering involving phase changes and formation of strong atomic bonds, which are characteristics of ceramic compositions.

At ~900°C, the CA (CaO.Al₂O₃) is the first crystalline phase formed, then it reacts with Al₂O₃ giving CA₂ (CaO.2Al₂O₃) at around 1100°C. The formation of CA₂ from CA and Al₂O₃ is expansive as a result of anisotropic growth of crystals [72].

At higher temperatures, the following two competitive phenomena occur: (1) expansion due to the formation of CA₂ phase and (2) shrinkage due to sintering of Al₂O₃ particles.

In the second cycle of measurement, no significant difference in the modulus curve was found in comparison with the first one.

In contrast, in case of sintering at 1500°C/4 h, the calcium hexa-aluminate CA₆ (CaO.6Al₂O₃) phase is additionally formed due to a reaction of CA₂ with Al₂O₃, which starts at ~1450°C. The formation of CA₆ is also expansive, leading to a superior elastic modulus of material.

Finally, when cooling the modulus curves remained stable without discontinuities, indicating an absence of microcracking of material.

These findings enhance the understanding not only about the role of specific additives (surfactants and inorganic binders) but also about the crystalline phase transformations and corresponding dimensional changes at sintering of macroporous ceramics. These results have important implications for developing porous ceramics with superior mechanical properties.

5. Summary

This chapter reviews the mechanical properties of porous ceramics with special interest on the mechanical strength, fracture toughness and elastic modulus of these materials.

One of the more significant findings to emerge from analysis of mechanical strength section is that, in addition to the porosity of porous ceramics, the number of connecting struts between the cells/pores in the microstructure plays a fundamental role in their mechanical strength behavior. Data from the literature support that once the connecting struts are lost in the porous structure, it is impossible to recover the original mechanical strength of it by merely increasing the struts thickness.

Regarding to fracture toughness of porous ceramics, two factors appear to control this property: the presence of surrounding pores at the crack front and the interaction of cracks with the pores in the microstructure.

Finally, the in situ hot elastic modulus analysis appears as an important method to better understand the processing steps as well as for predicting the life operation of this class of ceramic materials.