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

mesoscaled materials (membranes for catalysis [10], drug release substrates [11, 12]) and microscaled pieces (e.g., multifunctional materials where gravimetric properties are critical as

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

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 mechan-

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

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

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 [16–20] 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 struc-

In this context, the aim of this chapter is to review the mechanical properties of macroporous

1. Which microstructural parameters affect the mechanical strength of the porous ceramic

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?

porous ceramic materials but also to design their structures for specific applications.

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

for brittle ceramics and the high costs of producing the ceramic components [15].

ceramics. The following issues are of particular interest to this chapter:

ical behavior depends on more than just porosity of the materials.

batteries [13] and electronic sensors [14]).

172 Recent Advances in Porous Ceramics

application.

process in brittle materials.

tures and their qualities.

material besides its porosity?

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:

$$\frac{\sigma}{\sigma\_s} = \mathbf{C} \left( \frac{\rho}{\rho\_s} \right)^{\mathbf{m}} \tag{1}$$

where σ and r are, respectively, the fracture strength and the density of porous material; σ<sup>S</sup> and r<sup>S</sup> 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.

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.

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].

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.

porosity value where properties start to be damaged more significantly than the linear slope

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

<sup>σ</sup> <sup>¼</sup> <sup>σ</sup><sup>S</sup> <sup>e</sup>�bP, with b values ranging from 1 to 3 for stacked pores and from 3 to 9 for stacked particles [30].

0.001

0.01

**(-)**

0.1

**Relative Crushing Strength,** 

1- Solid spheres in rhombohedral stacking

Mechanical Properties of Porous Ceramics http://dx.doi.org/10.5772/intechopen.71612 175

2- Solid spheres in cubic stacking

3- Spherical pores in cubic stacking

4- Cilindral pores in cubic stacking

σ**/**σ**s**

1

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, PC 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

More recently, Bruno et al. [31] reviewed micromechanics aiming the development of microstructureproperty relations for porous microcracked ceramics. They focused on specific issues for porous

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

: <sup>1</sup> � <sup>ϕ</sup><sup>2</sup>=<sup>3</sup> � � " #<sup>1</sup>=<sup>2</sup>

<sup>¼</sup> 2 1ð Þ � <sup>2</sup><sup>υ</sup>

3 1ð Þ � <sup>υ</sup> (3)

(2)

ceramics as the nonlinear stress–strain behavior and the thermal-induced microcracking.

shown in Figure 2, they can be applied for the restricted porosity range only.

<sup>¼</sup> <sup>ϕ</sup> � <sup>P</sup> ϕ � �<sup>1</sup>þ<sup>ν</sup>

<sup>ϕ</sup> <sup>¼</sup> <sup>1</sup> � <sup>1</sup> <sup>þ</sup> <sup>υ</sup>

3 1ð Þ � υ � �

and (3) the critical porosity PC where properties go to zero.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

b=5 b=3

b=1.4

b=9

**Volumetric Porosity, P (-)**

0.001

0.01

0.1

**(-)**

**Minimum Solid Area Fraction** 

1

depends on the Poisson's ratio of material, as shown:

σ σS

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.

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

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 <sup>σ</sup> <sup>¼</sup> <sup>σ</sup><sup>S</sup> <sup>e</sup>�bP, with b values ranging from 1 to 3 for stacked pores and from 3 to 9 for stacked particles [30].

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

Figure 1. Plot of relative strength as a function of relative porosity for Gibson and Ashby (GA) models [22, 23] and for

(σ/σs) ~(ρ/ρs)3/2 open-cell

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

flexure compression Colombo et al. - flexure Colombo et al. - compression Seeber et al. - flexure Seeber et al. - compression

(σ/σs) ~(ρ/ρs)2 closed-cell

> foamed Al 2O3

**Relative density,** ρ/ρ**<sup>s</sup> (-)**

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

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

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

thermal and electrical fluxes. This concept is schematically presented in Figure 3.

differences of orientation and connectivity of macropores.

0.0010

0.0100

**Relative strength,** 

σ**/**σ**s (-)**

174 Recent Advances in Porous Ceramics

0.1000

directly deal with the statistical and size-dependent aspects of fracture.

experimental data obtained by Salvini et al. [27], Colombo et al. [24] and Seeber et al. [25].

porosity value where properties start to be damaged more significantly than the linear slope and (3) the critical porosity PC 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, PC 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 microstructureproperty 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:

$$\frac{\sigma}{\sigma\_{\rm S}} = \left[ \left( \frac{\phi - \mathbf{P}}{\phi} \right)^{1+\mathbf{v}} . \left( 1 - \phi^{2/3} \right) \right]^{1/2} \tag{2}$$

$$\phi = 1 - \left[\frac{1+\upsilon}{\Im(1-\upsilon)}\right] = \frac{2(1-2\upsilon)}{\Im(1-\upsilon)}\tag{3}$$

scanning electron microscopy (SEM) image of the porous scaffold made by freeze casting and

and bridges, e (e = 5 μm). A microcell is defined by r r RVEs (representative volume elements) [21].

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, dw (dw = 25 μm), the distance between the bridges, db (db = 75 μm), and the thickness of the walls

Mechanical Properties of Porous Ceramics http://dx.doi.org/10.5772/intechopen.71612 177

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

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 perco-

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

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

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

mechanical behavior due to the accumulation of local damage in porous ceramics.

the respective idealized geometry.

lation of previously damaged cells/pores.

under one loading direction only (pure traction).

stages.

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].

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

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, dw (dw = 25 μm), the distance between the bridges, db (db = 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].

scanning electron microscopy (SEM) image of the porous scaffold made by freeze casting and the respective idealized geometry.

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.

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

control the transmission of stress or flux normal to the plane of slab [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

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]

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

strength test.

176 Recent Advances in Porous Ceramics

was 10 MPa.

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 coarsegrain 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.

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.

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.

Regarding pore-stress interactions, there is strong evidence in the literature [40–42] 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

1 2 3 4 5 6 7 8 9 10 11

K=

σ

b b σ a σ

σ

σmax σ

Mechanical Properties of Porous Ceramics http://dx.doi.org/10.5772/intechopen.71612 179

**b/a**

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=ℓ, as

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 (tÞ 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

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)

t ℓ � �<sup>2</sup>

(4)

BV TV <sup>¼</sup> <sup>33</sup> <sup>π</sup> 80 ffiffiffi 2 p

Figure 6. Stress concentration factors for a chain of holes in either uniaxial or in biaxial tension [38].

struts connecting the pores.

0.5

1.0

1.5

2.0

**Stress concentration factor (K)**

2.5

3.0

3.5

b b a

σmax σ σ

shown in the following expression:

associated with reduced bone volume [40–42].

and the integrity of its microstructure.

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.

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].

Figure 6. Stress concentration factors for a chain of holes in either uniaxial or in biaxial tension [38].

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 coarsegrain 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

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

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

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.

Figure 5. Representation of a crack propagated in (a) dense ceramic with a sharp crack-tip and in (b) porous ceramic

of porous ceramics is provided in the next section.

ceramics depend only on the relative porosity.

where the crack-tip becomes blunt [37].

become very low.

178 Recent Advances in Porous Ceramics

Regarding pore-stress interactions, there is strong evidence in the literature [40–42] 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=ℓ, as shown in the following expression:

$$\frac{\text{BV}}{\text{TV}} = \frac{\text{33 } \pi}{80 \sqrt{2}} \left( \frac{\text{t}}{\ell} \right)^2 \tag{4}$$

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 (tÞ 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 [40–42].

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.

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].

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).

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, <sup>σ</sup><sup>0</sup> = 100 MPa, Poisson's ratio of 0.3 at <sup>t</sup>=<sup>ℓ</sup> <sup>¼</sup> <sup>0</sup>: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.

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

Table 1 Relationships between mechanical strength and bone volume (BV/TV) for two types of bone loss in rod-like and

Figure 8. The 3D microstructural model of the trabecular bone. Each single cell has an edge of equal length, ℓ. θ = 45� for

<sup>σ</sup><sup>0</sup> <sup>¼</sup> <sup>0</sup>:<sup>592</sup> BV

<sup>σ</sup><sup>0</sup> <sup>¼</sup> <sup>0</sup>:<sup>378</sup> BV

<sup>σ</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup>:65x103 BV

<sup>σ</sup><sup>0</sup> <sup>¼</sup> <sup>4</sup>:69x1010 BV

<sup>σ</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup>:36x102 BV

TV <sup>1</sup>:60, r2 = 0.99

TV <sup>1</sup>:10, r2 = 0.99

TV <sup>3</sup>:27, r2 = 0.85

Mechanical Properties of Porous Ceramics http://dx.doi.org/10.5772/intechopen.71612 181

TV <sup>6</sup>:94, r2 = 0.99

TV <sup>3</sup>:36, r2 = 0.99

an isotropic structure. The entire model consists of arrays of 5 � 5 � 5 cells [42].

Rod-like Trabeculae thinning <sup>σ</sup><sup>∗</sup>

Plate-like Trabeculae thinning <sup>σ</sup><sup>∗</sup>

Rod-like Horizontal trabeculae loss <sup>σ</sup><sup>∗</sup>

Plate-like Trabeculae loss <sup>σ</sup><sup>∗</sup>

Model type Bone loss type Mechanical strength

Oblique trabeculae loss <sup>σ</sup><sup>∗</sup>

thickness.

plate-like models [42].

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].

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 7. Simulated images of the trabecular bone showing two types of bone loss. Top right: thinning of trabeculae

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,

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. 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

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 trabec-

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

ulae loss were dramatically different from those for trabeculae thinning (see Table 1).

<sup>σ</sup><sup>0</sup> = 100 MPa, Poisson's ratio of 0.3 at <sup>t</sup>=<sup>ℓ</sup> <sup>¼</sup> <sup>0</sup>:1).

(ligaments) and bottom right: loss of trabeculae [42].

180 Recent Advances in Porous Ceramics

microstructures in the bone.

reduction in strength due to the loss of the horizontal trabeculae.


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].

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.

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].

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.

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

Figure 10. Physical characteristics of SiC ceramic filters of the same porosity (85%) but with different pores per inch (ppi)

0 5 10 15 20 25

**Pores per inch (ppi)**

8ppi 10ppi

15ppi 20ppi

Number of struts

Pore size

0.0

0.5

1.0

1.5

**Average pore size (mm)**

2.0

2.5

3.0

Mechanical Properties of Porous Ceramics http://dx.doi.org/10.5772/intechopen.71612 183

the variation of the number of pores (ppi).

**Average number of struts (1/cm**

[43].

 **2)**

Figure 10. Physical characteristics of SiC ceramic filters of the same porosity (85%) but with different pores per inch (ppi) [43].

Lichtner et al. [28] also found the mechanical properties of freeze casting porous ceramics that

0 10 20 30 40 50

**BV/TV Reduction (%)**

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

0 5 10 15 20 25 30

Trabeculae Thinning Horizontal Trabeculae Loss Oblique Trabeculae Loss

> Trabeculae Thinning Trabeculae Loss

**BV/TV Reduction (%)**

(a)

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

are controlled by the connectivity of pore walls.

when the pore per inch (ppi) increases.

trabeculae is represented by the dotted lines [42].

0.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

σ**\*/**

σ**0**

0.2

0.4

0.6

σ**\*/**

σ**0**

0.8

1.0

1.2

182 Recent Advances in Porous Ceramics

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).

Chevron notch (CV) methods, which are the most common tests. All methods involve appli-

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

As proposed by Rice [46], the microstructural dependence of fracture energy (γ) comprehends

Concerning the grain size effect, the crystalline structure of material is important when evaluating behavior between G and γ. Overall, the cubic materials present less variation of γ with G. On the other hand, as shown in Figure 12 for some noncubic ceramics, they present rising values of fracture energy (γ) 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

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 (γ) 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 (γ � P) in comparison to fracture energy and grain sizeð � γ G), some tendencies can be

1 10 100 1000

Al2O3 MgTi2O5 BeO TiO2 Fe2TiO5 Nb2O5 Al2TiO5

Mechanical Properties of Porous Ceramics http://dx.doi.org/10.5772/intechopen.71612 185

Al 2O3 MgTi 2O5 BeO TiO <sup>2</sup> Fe2TiO <sup>5</sup> Nb2O5 Al 2TiO <sup>5</sup>

**Grain size (** μ**m)**

Figure 12. Fracture energy (γ) as a function of grain size (G) for noncubic oxide ceramics at 22�C [47].

the grain size (G), the composition, the porosity (P) and the combined effects of them.

cation of force to a beam test specimen in three or four point flexures.

in noncubic materials, but the intensity of it varies among materials [47].

mechanics basis of these tests.

mentioned.

0

10

20

30

**Fracture energy (J/m**

 **2)**

40

50

60

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].

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.

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 stresspore 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 (KIC) 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 (γ) 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 γ. Overall, the cubic materials present less variation of γ with G.

On the other hand, as shown in Figure 12 for some noncubic ceramics, they present rising values of fracture energy (γ) 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].

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 (γ) 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.

The high number of connecting struts in the SiC filters of high pores per inch (ppi) is more

Figure 11. Compressive mechanical strength of SiC ceramic filters of the same porosity level (85%), but with different

0 5 10 15 20 25 30 35 40 45

15ppi 20ppi

**Average number of struts (1/cm 2)**

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-

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

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

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

Concerning fracture toughness (KIC) 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

probably contributed to their increasing mechanical strength, as depicted in Figure 11.

pore interactions, which is very low for porosity higher than 78% (see Figure 6).

8ppi 10ppi

microstructure plays a fundamental role in their mechanical strength behavior.

3. Fracture toughness of porous ceramics

strength of materials is a common practice in the engineering [45].

as the energy required to cause fracture.

0.0

average pore sizes and number of connecting struts [43].

0.2

0.4

0.6

**Compressive mechanical strength** 

184 Recent Advances in Porous Ceramics

**(MPa)**

0.8

1.0

1.2

1.4

Although there are more limited data about the relation between fracture energy and porosity (γ � P) in comparison to fracture energy and grain sizeð � γ G), some tendencies can be mentioned.

Figure 12. Fracture energy (γ) as a function of grain size (G) for noncubic oxide ceramics at 22�C [47].

At first, in single-phase ceramics with fine-to-medium-grain size, the fracture energy (γÞ 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].

γeff represents the fracture surface energy to initiate crack propagation, as proposed by Evans [49], whereas the total work-of-fracture γWOF 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

The fracture toughness (KIC) was determined according to the ASTM E-399 at room tempera-

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:

The bar samples were center-notched to one-half of their thickness að Þ =W≈ 0:5 with a 300-μm thick diamond blade for the KIC measurements. All fracture energy tests were performed on a MTS 180 machine in three-point bending over a span of 125 mm. For KIC, fracture surface

This measured value of KIC was then used to calculate the energy for crack initiation (γeffÞ by

where E is Young's elastic modulus measured by a sonic technique in foamed Al2O3 bar

For the total work-of-fracture (γWOFÞ 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

> Ð F dx

area of the new fracture surfaces, as determined directly from the individual specimen notched

Table 2 shows the obtained experimental results of the fracture energies (γeff and γWOF) and

F dx represents the required work for new surfaces' generation and A is the projected

samples (E = 18 GPa). Further details about this technique can be found in Ref. [53].

γWOF ¼

<sup>B</sup> � W3=<sup>2</sup> � <sup>f</sup> <sup>a</sup>

W

2 1 <sup>þ</sup> <sup>2</sup> <sup>a</sup> W � � <sup>1</sup> � <sup>a</sup>

W

� � <sup>2</sup>:<sup>15</sup> � <sup>3</sup>:<sup>93</sup> <sup>a</sup>

� � (5)

Mechanical Properties of Porous Ceramics http://dx.doi.org/10.5772/intechopen.71612 187

<sup>W</sup> <sup>þ</sup> <sup>2</sup>:<sup>7</sup> <sup>a</sup> W

� �<sup>3</sup>=<sup>2</sup> (6)

� �<sup>2</sup> h i

KIC <sup>¼</sup> <sup>2</sup>γeff<sup>E</sup> � �1=<sup>2</sup> (7)

2 A (8)

W

KIC <sup>¼</sup> <sup>F</sup> � <sup>S</sup>

refractory ceramics [51, 52].

f a W � � <sup>¼</sup> <sup>3</sup>

(γWOFÞ values were then calculated by:

the fracture toughness (KIC) for the foamed Al2O3.

the expression:

where Ð

areas.

ture in a three-point bend test applying the equation:

ffiffiffiffiffi a W r

�

<sup>1</sup>:<sup>99</sup> � <sup>a</sup> W � � � <sup>1</sup> � <sup>a</sup>

energy (γeff) measurements were carried out at a stress loading rate of 1.5 kN/s.

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 (γÞ 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 (γÞ decreased again at high porosity level (P) because the γ 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.<sup>43</sup> 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.

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 γeff and γWOF. The

Figure 13. Macroporous foamed Al2O3 sample notched with a 300-μm thick diamond blade where the crack-tip blunting mechanism is shown [58].

γeff represents the fracture surface energy to initiate crack propagation, as proposed by Evans [49], whereas the total work-of-fracture γWOF 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].

At first, in single-phase ceramics with fine-to-medium-grain size, the fracture energy (γÞ 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

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

Additional examples comprehend porous composite SiC-Al2O3-C (P~30–40%) and reaction sintered Si3N4 (P~45%). However, in all cases, the fracture energy (γÞ decreased again at high

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

Recently, Salvini et al.<sup>43</sup> 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.

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 γeff and γWOF. The

Figure 13. Macroporous foamed Al2O3 sample notched with a 300-μm thick diamond blade where the crack-tip blunting

fracture energy (γÞ equal to or greater than the obtained value for dense SiO2 glass.

porosity level (P) because the γ value goes to zero as P achieves 100%.

with porosity up to 50% [47].

186 Recent Advances in Porous Ceramics

values.

mechanism is shown [58].

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

$$\mathbf{K}\_{\rm IC} = \frac{\mathbf{F} \times \mathbf{S}}{\mathbf{B} \times \mathbf{W}^{3/2}} \times \mathbf{f}\left(\frac{\mathbf{a}}{\mathbf{W}}\right) \tag{5}$$

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:

$$\text{rf}\left(\frac{\text{a}}{\text{W}}\right) = 3\sqrt{\frac{\text{a}}{\text{W}}} \times \frac{1.99 - \left(\frac{\text{a}}{\text{W}}\right) \times \left(1 - \frac{\text{a}}{\text{W}}\right) \left[2.15 - 3.93\frac{\text{a}}{\text{W}} + 2.7\left(\frac{\text{a}}{\text{W}}\right)^2\right]}{2\left(1 + 2\frac{\text{a}}{\text{W}}\right)\left(1 - \frac{\text{a}}{\text{W}}\right)^{3/2}}\tag{6}$$

The bar samples were center-notched to one-half of their thickness að Þ =W≈ 0:5 with a 300-μm thick diamond blade for the KIC measurements. All fracture energy tests were performed on a MTS 180 machine in three-point bending over a span of 125 mm. For KIC, fracture surface energy (γeff) measurements were carried out at a stress loading rate of 1.5 kN/s.

This measured value of KIC was then used to calculate the energy for crack initiation (γeffÞ by the expression:

$$\mathbf{K}\_{\rm IC} = \left(\mathfrak{D}\gamma\_{\rm eff}\mathbf{E}\right)^{\circ} \tag{7}$$

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 (γWOFÞ 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 (γWOFÞ values were then calculated by:

$$
\gamma\_{\rm WOF} = \frac{\int \mathbf{F} \, \mathbf{dx}}{\mathbf{2} \, \mathbf{A}} \tag{8}
$$

where Ð F 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 (γeff and γWOF) and the fracture toughness (KIC) for the foamed Al2O3.


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 MPa m<sup>1</sup>=<sup>2</sup> for porous

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

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

properties (r, 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

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

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 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

E=r p , so if density (r) is known, E can be determined. These

Mechanical Properties of Porous Ceramics http://dx.doi.org/10.5772/intechopen.71612 189

energy of material. However, more research on this topic is therefore recommended.

4. Elastic modulus behavior of porous ceramics

modulus for several synthetic and natural porous brittle materials.

material. Both depend on ffiffiffiffiffiffiffiffi

to estimate the life operation of them [65].

material microstructure for an extended period of use.

Table 3. Elastic modulus values for porous and brittle materials.

hydroxyapatite.

Table 2 Results of fracture energies (γeff and γWOF) and the fracture toughness (KIC) for macroporous foamed Al2O3 [58].

The KIC 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 (KIC) 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-offracture energy be a more realistic measure for these materials?

The results of total work-of-fracture energy (γWOF) presented in Table 2 agree with the values obtained for Al2O3 produced using intermediate grain sizes (10–50 μm) [56].

Figure 14. Fracture surface of macroporous foamed Al2O3 showing the characteristics of the fracture and notched surfaces [58].

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 MPa m<sup>1</sup>=<sup>2</sup> 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.
