**4.1.1 Anode microstructure**

The von Mises stress contour plots for the anode at ΔT = 100⁰C, 500⁰C, and 800⁰C are shown considering elastic-plastic behavior of nickel in Figure 8. The stress values are in units of N/m2 (i.e., Pa).

Continuum Mechanics of Solid Oxide Fuel Cells

a tensile stress *σ* is given by (Laurencin et al., 2008):

Material Weibull modulus,

*m*

the action of each individual principal stress (Laurencin et al., 2008):

Table 4. Weibull parameters of ceramic materials (room temperature values)

*s j*

*P V*

, exp

**4.2 Probability of failure analysis** 

strength *σ0* of the material.

in this study.

Also,

Using Three-Dimensional Reconstructed Microstructures 81

Ceramic materials exhibit brittle behavior under tensile stress. Also, unlike metals, they show wide variability in tensile strength values and follow a statistical strength distribution. Thus, the Weibull method of analysis (Weibull, 1951; Laurencin et al., 2008) was used to calculate the probability of failure of each SOFC component (anode/cathode). According to the Weibull method, the survival probability of a particular component *j* under the action of

0 0

*dV*

(1)

Reference volume, *V0* (mm3)

*V*

Characteristic strength, *σ0* (MPa)

*m*

*j*

*V*

*j j*

where *j* = anode or cathode, *Vj* is the volume of component *j*, *V0* is a characteristic specimen volume (reference volume) for the material of component *j*, *σ0* is the characteristic strength of the material of component *j*, and *m* is the Weibull modulus of the material. The characteristic strength *σ0* is also the scale parameter for the distribution, while the Weibull modulus *m* is the shape parameter. The reference volume *V0* is related to the characteristic

In our case, however, the Weibull method was slightly modified to account for the fact that the anode and cathode materials are composites made up of two different components (Ni-YSZ for the anode and LSM-YSZ for the cathode). The method employed is described next. The Weibull parameters used for the ceramic materials (LSM and YSZ) are shown in Table 4 (Laurencin et al., 2008). Only room temperature values of the Weibull parameters were used

LSM 7.0 52.0 1.21 YSZ 7.0 446.0 0.35

The results of each stress analysis case were post-processed by writing programs to extract the three principal stress values from each element in the anode and cathode FE models. These principal stresses were then used to perform a Weibull analysis to determine the probability of failure of the anode and cathode at each ∆T value. Since the SOFC component materials are subjected to a multi-axial state of stress, the total survival probability of each ceramic phase of the anode/cathode under the action of the three principal stresses (*σ*1, *σ*2, and *σ*3) was calculated. The principal stresses were assumed to act independently, and the total survival probability was calculated as the product of the survival probabilities under

> <sup>3</sup> 1 , , *j j s s <sup>j</sup> <sup>i</sup> <sup>j</sup> <sup>i</sup> P P*

(2)

*V V*

Fig. 8. Von Mises stress contour plots for anode considering elastic-plastic behavior of nickel: (left to right) ΔT = 100⁰C, ΔT = 500⁰C, ΔT = 800⁰C

Figure 8 shows that as ΔT increases from 100⁰C to 800⁰C, the stresses in the anode also increase. This happens because thermal stress is proportional to the CTE, and the CTEs of both nickel and YSZ increase with temperature, as seen from Figure 4. Also, the stress plots show that the stresses are greater near the regions of pores due to stress concentration, as expected. Similar results are obtained for the cases with temperature-independent material properties and temperature-dependent CTEs. The effect of the elastic-plastic behavior of nickel on the principal tensile stress values (as compared with the linear elastic behavior assumed in the cases with temperature-independent material properties and temperature-dependent CTEs) is discussed in section 4.2.1, which deals with failure probability calculations for the anode.

#### **4.1.2 Cathode microstructure**

The von Mises stress contour plots for the cathode at ∆T = 100⁰C, 500⁰C, and 800⁰C are shown in Figure 9 considering temperature-dependent material properties.

Fig. 9. Von Mises stress contour plots for cathode considering temperature-dependent material properties: (left to right) ∆T = 100⁰C, ∆T = 500⁰C, ∆T = 800⁰C

Figure 9 shows that as ΔT increases from 100⁰C to 800⁰C, the stresses in the cathode also increase. This result can be explained, just as in the case of the anode, by the fact that thermal stress is proportional to the CTE, and the CTE of YSZ increases with temperature while the CTE of LSM is assumed constant over the temperature range considered. Also, the plots show that the stresses are greater near the regions of pores due to stress concentration. Similar results were obtained for the case with temperature-independent material properties. The effect of temperature-independent versus temperature-dependent material properties on the principal tensile stresses induced in the cathode is discussed in section 4.2.2, which deals with failure probability calculations for the cathode.

### **4.2 Probability of failure analysis**

80 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

Figure 8 shows that as ΔT increases from 100⁰C to 800⁰C, the stresses in the anode also increase. This happens because thermal stress is proportional to the CTE, and the CTEs of both nickel and YSZ increase with temperature, as seen from Figure 4. Also, the stress plots show that the stresses are greater near the regions of pores due to stress concentration, as expected. Similar results are obtained for the cases with temperature-independent material properties and temperature-dependent CTEs. The effect of the elastic-plastic behavior of nickel on the principal tensile stress values (as compared with the linear elastic behavior assumed in the cases with temperature-independent material properties and temperature-dependent CTEs) is discussed in section 4.2.1, which deals with failure probability calculations for the anode.

The von Mises stress contour plots for the cathode at ∆T = 100⁰C, 500⁰C, and 800⁰C are

Figure 9 shows that as ΔT increases from 100⁰C to 800⁰C, the stresses in the cathode also increase. This result can be explained, just as in the case of the anode, by the fact that thermal stress is proportional to the CTE, and the CTE of YSZ increases with temperature while the CTE of LSM is assumed constant over the temperature range considered. Also, the plots show that the stresses are greater near the regions of pores due to stress concentration. Similar results were obtained for the case with temperature-independent material properties. The effect of temperature-independent versus temperature-dependent material properties on the principal tensile stresses induced in the cathode is discussed in section

Fig. 9. Von Mises stress contour plots for cathode considering temperature-dependent

material properties: (left to right) ∆T = 100⁰C, ∆T = 500⁰C, ∆T = 800⁰C

4.2.2, which deals with failure probability calculations for the cathode.

shown in Figure 9 considering temperature-dependent material properties.

Fig. 8. Von Mises stress contour plots for anode considering elastic-plastic behavior of

nickel: (left to right) ΔT = 100⁰C, ΔT = 500⁰C, ΔT = 800⁰C

**4.1.2 Cathode microstructure** 

Ceramic materials exhibit brittle behavior under tensile stress. Also, unlike metals, they show wide variability in tensile strength values and follow a statistical strength distribution. Thus, the Weibull method of analysis (Weibull, 1951; Laurencin et al., 2008) was used to calculate the probability of failure of each SOFC component (anode/cathode). According to the Weibull method, the survival probability of a particular component *j* under the action of a tensile stress *σ* is given by (Laurencin et al., 2008):

$$P\_s^j(\sigma\_\prime \mathcal{V}\_j) = \exp\left(-\int\_{V\_j} \left(\frac{\sigma}{\sigma\_0}\right)^m \frac{dV\_j}{V\_0}\right) \tag{1}$$

where *j* = anode or cathode, *Vj* is the volume of component *j*, *V0* is a characteristic specimen volume (reference volume) for the material of component *j*, *σ0* is the characteristic strength of the material of component *j*, and *m* is the Weibull modulus of the material. The characteristic strength *σ0* is also the scale parameter for the distribution, while the Weibull modulus *m* is the shape parameter. The reference volume *V0* is related to the characteristic strength *σ0* of the material.

In our case, however, the Weibull method was slightly modified to account for the fact that the anode and cathode materials are composites made up of two different components (Ni-YSZ for the anode and LSM-YSZ for the cathode). The method employed is described next. The Weibull parameters used for the ceramic materials (LSM and YSZ) are shown in Table 4 (Laurencin et al., 2008). Only room temperature values of the Weibull parameters were used in this study.


Table 4. Weibull parameters of ceramic materials (room temperature values)

The results of each stress analysis case were post-processed by writing programs to extract the three principal stress values from each element in the anode and cathode FE models. These principal stresses were then used to perform a Weibull analysis to determine the probability of failure of the anode and cathode at each ∆T value. Since the SOFC component materials are subjected to a multi-axial state of stress, the total survival probability of each ceramic phase of the anode/cathode under the action of the three principal stresses (*σ*1, *σ*2, and *σ*3) was calculated. The principal stresses were assumed to act independently, and the total survival probability was calculated as the product of the survival probabilities under the action of each individual principal stress (Laurencin et al., 2008):

$$P\_s^j(\overline{\sigma\_\prime}, V\_j) = \prod\_{i=1}^3 P\_s^j(\sigma\_{i\prime}, V\_j) \tag{2}$$

Also,

$$P\_s^j(\sigma\_{i'} V\_j) = \exp\left(-\int\_{V\_j} \left(\frac{\sigma\_i}{\sigma\_0}\right)^m \frac{dV\_j}{V\_0}\right) \tag{3}$$

Continuum Mechanics of Solid Oxide Fuel Cells

Fig. 10. Probability of failure values for anode

Fig. 11. Maximum principal tensile stress in the YSZ phase of the anode

values of the YSZ phase), as compared with the model that considers nonlinear elasticplastic behavior of nickel, especially at high temperatures. This may be explained by referring to Figure 11, which shows that the maximum principal tensile stress (MPTS) in the YSZ phase of the anode increases with increasing ∆T values for all three cases, as expected. Figure 11 also shows that when the elastic-plastic behavior of Ni is taken into account, the

Using Three-Dimensional Reconstructed Microstructures 83

where, *j* = YSZ for the anode, *j* = LSM or YSZ for the cathode, and *i* = 1, 2, and 3. Only tensile values of the three principal stresses were used in the Weibull analysis. The probability of failure of each phase was then calculated as follows (Anandakumar et al., 2010):

$$\mathbf{p}\_f = \mathbf{1}.0 - \mathbf{p}\_s^j(\overline{\sigma\_\mathbf{v}} \mathbf{v}\_f) \tag{4}$$

The probability of failure of the anode was calculated as the failure probability of the YSZ phase, keeping in mind that the anode material is a cermet composite (Ni-YSZ), and that the Weibull distribution is more appropriate for calculating the failure probability of ceramics (such as YSZ) (Laurencin et al., 2008). The strength distribution for metals such as nickel is closer to a normal distribution (Meyers & Chawla, 1999). Since the cathode is a composite of two different ceramic materials (LSM-YSZ), the probability of failure of the cathode was calculated by extracting positive (tensile) values of the three principal stresses from each element in the LSM and YSZ element sets of the cathode FE model, and subjecting these to the Weibull analysis. This resulted in two different failure probability values for the LSM and YSZ phases of the cathode, which were combined into a single probability of failure value for the cathode by assuming that the cathode fails when either phase fails (or when both phases fail simultaneously). The probability that both phases fail simultaneously was calculated by assuming that the failures of the two phases are independent events, and hence the probability of simultaneous failure of the two phases is just the product of the probabilities of failure of LSM and YSZ:

$$P\_f^{\text{cathode}} = P\_f\{\text{LSM} \cup \text{YSZ}\}$$

$$\implies P\_f^{\text{cathode}} = P\_f\{\text{LSM}\} + P\_f\{\text{YSZ}\} - P\_f\{\text{LSM} \cap \text{YSZ}\}$$

$$\implies P\_f^{\text{cathode}} = P\_f\{\text{LSM}\} + P\_f\{\text{YSZ}\} - P\_f\{\text{LSM}\}P\_f\{\text{YSZ}\}$$

#### **4.2.1 Anode**

The probability of failure (*Pf*) value for the YSZ phase of the anode was calculated at each ΔT value (100⁰C, 200⁰C, ..., 800⁰C) for each case described in Table 1. These values are plotted in Figure 10. Since these *Pf* values are calculated on the basis of the tensile principal stresses in the YSZ phase, the variation of the maximum principal tensile stress (MPTS) in the YSZ phase of the anode with temperature in all three cases (temperature-independent material properties, temperature-dependent CTEs, and elastic-plastic behavior of Ni) is shown in Figure 11.

The *Pf* plot for the anode (Figure 10) shows that the probability of failure increases with increasing ∆T values (and hence increasing stresses), for each of the three cases considered (temperature-independent material properties, temperature-dependent CTEs, and elasticplastic behavior of nickel). Also, the plot shows that the linear elastic material behavior models significantly underestimate the probability of failure of the anode (defined by the *Pf*

*<sup>j</sup> <sup>j</sup> <sup>i</sup> <sup>s</sup> i j V*

where, *j* = YSZ for the anode, *j* = LSM or YSZ for the cathode, and *i* = 1, 2, and 3. Only tensile values of the three principal stresses were used in the Weibull analysis. The probability of

> 1.0 , *<sup>j</sup> P P f s*

The probability of failure of the anode was calculated as the failure probability of the YSZ phase, keeping in mind that the anode material is a cermet composite (Ni-YSZ), and that the Weibull distribution is more appropriate for calculating the failure probability of ceramics (such as YSZ) (Laurencin et al., 2008). The strength distribution for metals such as nickel is closer to a normal distribution (Meyers & Chawla, 1999). Since the cathode is a composite of two different ceramic materials (LSM-YSZ), the probability of failure of the cathode was calculated by extracting positive (tensile) values of the three principal stresses from each element in the LSM and YSZ element sets of the cathode FE model, and subjecting these to the Weibull analysis. This resulted in two different failure probability values for the LSM and YSZ phases of the cathode, which were combined into a single probability of failure value for the cathode by assuming that the cathode fails when either phase fails (or when both phases fail simultaneously). The probability that both phases fail simultaneously was calculated by assuming that the failures of the two phases are independent events, and hence the probability of simultaneous failure of the two phases is just the product of the

*cathode P P <sup>f</sup> <sup>f</sup> LSM YSZ*

*cathode PP P P <sup>f</sup> f ff LSM YSZ LSM YSZ*

*cathode PP P P P <sup>f</sup> f f ff LSM YSZ LSM YSZ*

The probability of failure (*Pf*) value for the YSZ phase of the anode was calculated at each ΔT value (100⁰C, 200⁰C, ..., 800⁰C) for each case described in Table 1. These values are plotted in Figure 10. Since these *Pf* values are calculated on the basis of the tensile principal stresses in the YSZ phase, the variation of the maximum principal tensile stress (MPTS) in the YSZ phase of the anode with temperature in all three cases (temperature-independent material properties, temperature-dependent CTEs, and elastic-plastic behavior of Ni) is shown in

The *Pf* plot for the anode (Figure 10) shows that the probability of failure increases with increasing ∆T values (and hence increasing stresses), for each of the three cases considered (temperature-independent material properties, temperature-dependent CTEs, and elasticplastic behavior of nickel). Also, the plot shows that the linear elastic material behavior models significantly underestimate the probability of failure of the anode (defined by the *Pf*

*j*

 

0 0

*dV*

(3)

*V <sup>j</sup>* (4)

*V*

*m*

*P V*

probabilities of failure of LSM and YSZ:

**4.2.1 Anode** 

Figure 11.

, exp

failure of each phase was then calculated as follows (Anandakumar et al., 2010):

Fig. 10. Probability of failure values for anode

Fig. 11. Maximum principal tensile stress in the YSZ phase of the anode

values of the YSZ phase), as compared with the model that considers nonlinear elasticplastic behavior of nickel, especially at high temperatures. This may be explained by referring to Figure 11, which shows that the maximum principal tensile stress (MPTS) in the YSZ phase of the anode increases with increasing ∆T values for all three cases, as expected. Figure 11 also shows that when the elastic-plastic behavior of Ni is taken into account, the

Continuum Mechanics of Solid Oxide Fuel Cells

material properties.

Using Three-Dimensional Reconstructed Microstructures 85

The *Pf* plot for the cathode shows that the probability of failure of the cathode increases with increasing ∆T values (and hence increasing stresses), for both temperature-independent and temperature-dependent material properties, as expected. Higher *Pf* values are obtained when temperature-independent material properties are considered. A physical explanation for this observation is suggested by the temperature variation of the Young's modulus of YSZ. For YSZ, E decreases from a value of 205 GPa at T = 20⁰C to a value of 147.5 GPa at T = 800⁰C, as shown in Figure 5. On the other hand, when temperature-independent material properties are considered, the Young's modulus of YSZ has a constant value of 205 GPa. Thus, because of the large decrease in the Young's modulus of YSZ with increasing temperature, lower stresses are induced in the cathode in the case with temperaturedependent material properties than in the case with temperature-independent material properties. This in turn leads to lower *Pf* values in the case with temperature-dependent material properties as compared with the case that considers temperature-independent material properties. This is confirmed by the MPTS plot for the cathode shown below (Figure 13), which compares the maximum principal tensile stress induced in the YSZ and LSM phases of the cathode for temperature-independent and temperature-dependent

Fig. 13. Maximum principal tensile stress in LSM and YSZ phases of cathode

The plot above shows that the MPTS induced in the LSM phase for temperature-dependent material properties is lower than the MPTS in the LSM phase for temperature-independent material properties over the entire temperature range. Similarly, the MPTS induced in the YSZ phase for temperature-dependent material properties is lower than the MPTS induced in the YSZ phase for temperature-independent material properties over the entire temperature range. This implies that the cathode *Pf* values, which are calculated on the basis of the positive (tensile) principal stresses in the LSM and YSZ phases, will be higher for the

MPTS in the YSZ phase attains higher values than when linear elastic behavior is assumed, especially at high temperatures. This can be explained as follows: when the Ni phase enters the nonlinear (plastic) part of its stress-strain curve at higher temperatures (and hence higher strains), lower stresses are induced in the Ni phase than if its stress-strain curve had been purely linear elastic with the same value of Young's modulus. Thus, when the Ni phase starts showing nonlinear behavior, a higher proportion of the temperature-induced stresses are redistributed into the YSZ phase, resulting in higher MPTS values in the YSZ phase (and hence higher *Pf* values for the anode).

Figure 10 also shows that the case with temperature-dependent CTEs shows higher *Pf* values than the case with temperature-independent material properties at intermediate and high temperatures. Again, Figure 11 shows that with temperature-dependent CTE values, higher tensile stresses are induced in the YSZ phase of the anode than with temperatureindependent material properties, especially at intermediate and high temperatures. This can be explained by referring to Figure 4, which shows that the CTEs of both Ni and YSZ increase with temperature. Since thermal stresses are proportional to CTE values, it can be expected that the case with temperature-dependent CTEs will show higher MPTS values (and hence higher *Pf* values) than the case with temperature-independent material properties, which uses constant (room-temperature) values of the CTEs.

#### **4.2.2 Cathode**

The probability of failure (*Pf*) values for the LSM and YSZ phases of the cathode were calculated and combined, as described above, at each ΔT value (100⁰C, 200⁰C, ..., 800⁰C) for both the cases described in Table 2. These values are plotted in Figure 12.

Fig. 12. Failure probability values for cathode

MPTS in the YSZ phase attains higher values than when linear elastic behavior is assumed, especially at high temperatures. This can be explained as follows: when the Ni phase enters the nonlinear (plastic) part of its stress-strain curve at higher temperatures (and hence higher strains), lower stresses are induced in the Ni phase than if its stress-strain curve had been purely linear elastic with the same value of Young's modulus. Thus, when the Ni phase starts showing nonlinear behavior, a higher proportion of the temperature-induced stresses are redistributed into the YSZ phase, resulting in higher MPTS values in the YSZ

Figure 10 also shows that the case with temperature-dependent CTEs shows higher *Pf* values than the case with temperature-independent material properties at intermediate and high temperatures. Again, Figure 11 shows that with temperature-dependent CTE values, higher tensile stresses are induced in the YSZ phase of the anode than with temperatureindependent material properties, especially at intermediate and high temperatures. This can be explained by referring to Figure 4, which shows that the CTEs of both Ni and YSZ increase with temperature. Since thermal stresses are proportional to CTE values, it can be expected that the case with temperature-dependent CTEs will show higher MPTS values (and hence higher *Pf* values) than the case with temperature-independent material

The probability of failure (*Pf*) values for the LSM and YSZ phases of the cathode were calculated and combined, as described above, at each ΔT value (100⁰C, 200⁰C, ..., 800⁰C) for

properties, which uses constant (room-temperature) values of the CTEs.

both the cases described in Table 2. These values are plotted in Figure 12.

phase (and hence higher *Pf* values for the anode).

Fig. 12. Failure probability values for cathode

**4.2.2 Cathode** 

The *Pf* plot for the cathode shows that the probability of failure of the cathode increases with increasing ∆T values (and hence increasing stresses), for both temperature-independent and temperature-dependent material properties, as expected. Higher *Pf* values are obtained when temperature-independent material properties are considered. A physical explanation for this observation is suggested by the temperature variation of the Young's modulus of YSZ. For YSZ, E decreases from a value of 205 GPa at T = 20⁰C to a value of 147.5 GPa at T = 800⁰C, as shown in Figure 5. On the other hand, when temperature-independent material properties are considered, the Young's modulus of YSZ has a constant value of 205 GPa. Thus, because of the large decrease in the Young's modulus of YSZ with increasing temperature, lower stresses are induced in the cathode in the case with temperaturedependent material properties than in the case with temperature-independent material properties. This in turn leads to lower *Pf* values in the case with temperature-dependent material properties as compared with the case that considers temperature-independent material properties. This is confirmed by the MPTS plot for the cathode shown below (Figure 13), which compares the maximum principal tensile stress induced in the YSZ and LSM phases of the cathode for temperature-independent and temperature-dependent material properties.

Fig. 13. Maximum principal tensile stress in LSM and YSZ phases of cathode

The plot above shows that the MPTS induced in the LSM phase for temperature-dependent material properties is lower than the MPTS in the LSM phase for temperature-independent material properties over the entire temperature range. Similarly, the MPTS induced in the YSZ phase for temperature-dependent material properties is lower than the MPTS induced in the YSZ phase for temperature-independent material properties over the entire temperature range. This implies that the cathode *Pf* values, which are calculated on the basis of the positive (tensile) principal stresses in the LSM and YSZ phases, will be higher for the

Continuum Mechanics of Solid Oxide Fuel Cells

pp. (71-76).

pp. (287-294).

1532).

2035).

Using Three-Dimensional Reconstructed Microstructures 87

Kramer, J., Mastronarde, D., & McIntosh, J. (1996). Computer visualization of three-

Laurencin, J., Delette, G., Lefebvre-Joud, F., & Dupeux, M. (2008). A numerical tool to

Nakajo, A., Stiller, C., Harkegard, G., & Bolland, O. (2006). Modeling of thermal stresses and

Pihlatie, M., Kaiser, A., & Mogensen, M. (2009). Mechanical properties of NiO/Ni-YSZ

Pitakthapanaphong, S., & Busso, E. (2005). Finite element analysis of the fracture behaviour

*Simulation in Materials Science and Engineering*, Vol. 13, (2005), pp. (531-540). Selcuk, A., & Atkinson, A. (1997). Elastic properties of ceramic oxides used in solid oxide

Selcuk, A., & Atkinson, A. (2000). Strength and toughness of tape-cast yttria-stabilized

Singhal, S., & Kendall, K. (Eds.). (2003). *High Temperature Solid Oxide Fuel Cells: Fundamentals,* 

Toftegaard, H., Sorensen, B., Linderoth, S., Lundberg, M., & Feih, S. (2009). Effects of heat-

Weibull, W. (1951). A statistical distribution function of wide applicability. *ASME Journal of* 

Wilson, J., Kobsiriphat, W., Mendoza, R., Chen, H.-Y., Hiller, J., Miller, D., Thornton, K.,

Wilson, J., Duong, A., Gameiro, M., Chen, H.-Y., Thornton, K., Mumm, D., & Barnett, S.

cathode. *Electrochemistry Communications,* Vol. 11, (2009), pp. (1052-1056).

solid-oxide fuel-cell anode. *Nature Materials*, Vol. 5, (2006), pp. (541-544). Wilson, J., & Barnett, S. (2008). Solid oxide fuel cell Ni-YSZ anodes: effect of composition on

*Journal of the European Ceramic Society*, Vol. 28, (2008), pp. (1857-1869). Meyers, M., & Chawla, K. (1999). *Mechanical Behavior of Materials*, Prentice Hall, ISBN

0132628171, Upper Saddle River, New Jersey.

*European Ceramic Society*, Vol. 29, (2009), pp. (1657-1664).

*Design and Applications*, Elsevier, ISBN 1856173879, Oxford.

Vol. 92, No. 11, (2009), pp. (2704-2712).

*Applied Mechanics,* (1951), pp. (293-297).

10, (2008), pp. (B181-B185).

dimensional image data using IMOD. *Journal of Structural Biology*, Vol. 116, (1996),

estimate SOFC mechanical degradation: case of the planar cell configuration.

probability of survival of tubular SOFC. *Journal of Power Sources*, Vol. 158, (2006),

composites depending on temperature, porosity and redox cycling. *Journal of the* 

of multi-layered systems used in solid oxide fuel cell applications. *Modelling and* 

fuel cells (SOFC). *Journal of the European Ceramic Society*, Vol. 17, (1997), pp. (1523-

zirconia. *Journal of the American Ceramic Society*, Vol. 83, No. 8, (2000), pp. (2029-

treatments on the mechanical strength of coated YSZ: an experimental assessment. *Journal of the American Ceramic Society*, Vol. 92, No. 11, (2009), pp. (2704-2712). Toftegaard, H., & Sorensen, B. (2009). Effects of heat-treatments on the mechanical strength

of coated YSZ: an experimental assessment. *Journal of the American Ceramic Society*,

Voorhees, P., Adler, S., & Barnett, S. (2006). Three-dimensional reconstruction of a

microstructure and performance. *Electrochemical and Solid-State Letters,* Vol. 11, No.

(2009). Quantitative three-dimensional microstructure of a solid oxide fuel cell

temperature-independent material properties case than for the temperature-dependent material properties case, as is indeed observed.
