**1. Introduction**

72 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

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A solid oxide fuel cell (SOFC) is a device that converts the chemical energy of fuels into electrical energy (Singhal & Kendall, 2003). SOFCs have received much attention from researchers due to their promise of delivering relatively clean energy at high efficiencies (Singhal & Kendall, 2003). An SOFC consists of a few basic parts: an anode, a cathode, an electrolyte, and interconnect wires (Singhal & Kendall, 2003). The electrolyte in an SOFC is a solid oxide such as Yttria-Stabilized Zirconia (YSZ). The porous anode is usually a ceramicmetal composite (so called cermet) such as the nickel-zirconia cermet (Ni-YSZ). The porous cathode is usually a composite of strontium-doped lanthanum manganite (LSM) and Yttria-Stabilized Zirconia (LSM-YSZ) (Singhal & Kendall, 2003) or a composite such as gadolinium-doped ceria-lanthanum strontium cobalt ferrite (GDC-LSCF) (Anandakumar et al., 2010). Oxygen atoms undergo reduction on the porous cathode surface, and the resulting oxide ions are transported through the electrolyte to the porous anode. Here, the oxide ions react with the fuel (such as hydrogen). Hydrogen is oxidized, and the electrons of the oxide ions are liberated. The free electrons give rise to electric current (Singhal & Kendall, 2003).

Research on SOFCs has concentrated on many different aspects, including anode, cathode, and electrolyte materials; investigating the behavior of different SOFC configurations; modeling and simulating electrochemical, thermal, and flow phenomena; and performing thermal stress and probability of failure analyses. Researchers have employed experimental, analytical, and computational approaches in their investigations. For example, Selcuk and Atkinson (1997, 2000) conducted a number of experimental studies to estimate various mechanical properties of SOFC ceramic materials such as YSZ and NiO-YSZ. They determined the biaxial flexural strength and fracture toughness of YSZ both at room temperature and an operating temperature of 900⁰C (Selcuk & Atkinson, 2000). They also experimentally studied the dependence of the Young's modulus, shear modulus, and Poisson's ratio of YSZ and NiO-YSZ (amongst other ceramic materials) on porosity (Selcuk & Atkinson, 1997). The results of these studies were summarized by the authors (Atkinson & Selcuk, 2000) where they also suggested techniques for improving the mechanical

<sup>\*</sup> Corresponding Author

Continuum Mechanics of Solid Oxide Fuel Cells

Using Three-Dimensional Reconstructed Microstructures 75

lower than those induced in conventional layered SOFCs. They also found that functionally

In this work, three-dimensional micromechanical finite element (FE) models for real solid oxide fuel cell (SOFC) anode and cathode microstructures are generated from a stack of twodimensional image-based FE models of anode and cathode microstructures. Finite element analysis (FEA) of the models is carried out to determine their mechanical response to a steady-state temperature change from room temperature up to an operating temperature. The resulting stress distribution is determined in each case, and the stresses are analyzed using the Weibull method to calculate the probability of failure. The anode material is Ni-YSZ, while the cathode material is LSM-YSZ. Both linear elastic and elastic-plastic (nonlinear) behaviors are considered for nickel in the analysis of the anode. It is observed that the linear elastic models underestimate the probability of failure of the anode. The effect of temperature-dependent material properties on the probability of failure of the anode and cathode is also investigated. *The novelties of this work include micromechanical finite element analysis of the mechanical response of anode and cathode microstructural models considering temperature-dependent material properties and nonlinear elastic-plastic behavior of the nickel phase.* 

The first step in this work was to digitally reconstruct a three-dimensional (3-D) anode microstructure from two-dimensional (2-D) images of anode cross-sections obtained using focused ion beam-scanning electron microscopy (FIB-SEM). The 2-D images of the anode and cathode microstructures were obtained from Dr. Scott Barnett's research group at Northwestern University (Wilson et al., 2006, 2009). The initial 3-D reconstruction was achieved using IMOD (Kramer et al., 1996), a free collection of image processing programs developed by scientists at the Boulder Laboratory for 3-D Electron Microscopy of Cells. IMOD is capable of creating a stack of 2-D images, interpolating the gaps between consecutive images, and creating and displaying the 3-D model. A few representative 2-D images and the 3-D reconstruction of the anode microstructure are shown in Figure 1. The in-plane dimensions of the reconstructed anode are 5 μm x 6 μm, while the thickness is 3.54

Fig. 1. Two-dimensional SEM images of anode (left) and cathode (center) cross-sections

In the SEM images of the anode, white (pixel value 255) corresponds to nickel, gray (pixel value 127) to YSZ, and black (pixel value 0) to the pores. In the images of the cathode, white (pixel value 255) represents LSM, gray (pixel value 127) represents YSZ, and black (pixel value 0) represents the pores. The reconstructed 3-D model was used as a check on the

(Wilson et al., 2006, 2009) and a reconstructed three-dimensional anode (right)

graded SOFCs show a lower probability of failure than other types of SOFCs.

**2. Image-based microstructural finite element models** 

geometry of the 3-D FE model.

μm.

behavior of SOFC ceramic materials under certain operating conditions. Toftegaard et al. (2009) conducted uniaxial tensile tests on pure YSZ specimens and YSZ specimens coated with porous NiO-YSZ layers. They heat-treated the coated YSZ specimens at various temperatures to study the effect of heat treatment at different temperatures on the strength. Pihlatie et al. (2009) experimentally determined the Young's modulus (amongst other mechanical properties) of Ni-YSZ and NiO-YSZ composites as a function of porosity using the Impulse Excitation Technique (IET). They also used IET to study the dependency of the Young's modulus of these materials on temperature. Giraud and Canel (2008) also conducted experimental studies using IET to determine the variation of the Young's modulus of YSZ, LSM, and Ni-YSZ with temperature. Wilson and Barnett (2008) conducted experimental studies on Ni-YSZ/YSZ/LSM-YSZ SOFCs with Ni-YSZ anodes of different compositions to investigate the effect of varying composition of the anodes on their performance and microstructure. Their studies involved serial-sectioning using a focused ion beam scanning electron microscope (FIB-SEM) to obtain images of the microstructures of the different samples. They conducted stereological analyses on these images to calculate volume fractions and triple-phase boundary (TPB) densities for their samples. Zhang et al. (2008) proposed an analytical model for calculating residual stresses in a single SOFC with NiO-YSZ/YSZ/LSM composition. They used their model to estimate the residual stresses in an SOFC at room temperature and to study the variation of the stresses in the different components with changes in component thicknesses. They also carried out a Weibull analysis to calculate the probability of failure of the anode, and they studied the variation of the failure probability of the anode with changes in component thicknesses.

Laurencin et al. (2008) have proposed a numerical (finite element analysis-based) tool for studying the degradation of anode-supported and electrolyte-supported planar SOFCs under several types of mechanical loads, including residual stresses. They have also calculated the failure probabilities of the SOFCs using Weibull analysis. Pitakthapanaphong and Busso (2005) carried out finite element analyses to investigate the fracture of multilayered systems used in SOFCs, such as LSM films on a YSZ substrate. They have pointed out that fracture is caused by large residual stresses induced during the SOFC manufacturing process due to thermal expansion coefficient (TEC) mismatch between different layers. They observed different cracking patterns (surface cracks, channeling cracks, and interfacial cracks) in physical samples of multi-layered systems. Their study involved FE simulations to determine the crack driving force (energy release rate) for the three observed cracking patterns. Johnson and Qu (2008) used a three-dimensional stochastic reconstruction method to create multiple realizations of the microstructure of porous Ni-YSZ cermet used as SOFC anode material. They analyzed these microstructure realizations using finite element software to determine the effective elastic modulus and effective coefficient of thermal expansion (CTE) of Ni-YSZ as a function of temperature. Anandakumar et al. (2010) carried out FE analyses to estimate thermal stresses and probability of failure in functionally graded SOFCs. They employed a continuum mechanics approach and used graded finite elements to discretize effective media consisting of NiO-YSZ/YSZ/LSM as well as NiO-YSZ/YSZ/GDC-LSCF. They used the Weibull method to determine the failure probability of the individual components of the SOFC, as well as the failure probability of the whole SOFC. They found that the thermal stresses developed in functionally graded SOFCs under spatially uniform and non-uniform temperature loads are

behavior of SOFC ceramic materials under certain operating conditions. Toftegaard et al. (2009) conducted uniaxial tensile tests on pure YSZ specimens and YSZ specimens coated with porous NiO-YSZ layers. They heat-treated the coated YSZ specimens at various temperatures to study the effect of heat treatment at different temperatures on the strength. Pihlatie et al. (2009) experimentally determined the Young's modulus (amongst other mechanical properties) of Ni-YSZ and NiO-YSZ composites as a function of porosity using the Impulse Excitation Technique (IET). They also used IET to study the dependency of the Young's modulus of these materials on temperature. Giraud and Canel (2008) also conducted experimental studies using IET to determine the variation of the Young's modulus of YSZ, LSM, and Ni-YSZ with temperature. Wilson and Barnett (2008) conducted experimental studies on Ni-YSZ/YSZ/LSM-YSZ SOFCs with Ni-YSZ anodes of different compositions to investigate the effect of varying composition of the anodes on their performance and microstructure. Their studies involved serial-sectioning using a focused ion beam scanning electron microscope (FIB-SEM) to obtain images of the microstructures of the different samples. They conducted stereological analyses on these images to calculate volume fractions and triple-phase boundary (TPB) densities for their samples. Zhang et al. (2008) proposed an analytical model for calculating residual stresses in a single SOFC with NiO-YSZ/YSZ/LSM composition. They used their model to estimate the residual stresses in an SOFC at room temperature and to study the variation of the stresses in the different components with changes in component thicknesses. They also carried out a Weibull analysis to calculate the probability of failure of the anode, and they studied the variation of

the failure probability of the anode with changes in component thicknesses.

Laurencin et al. (2008) have proposed a numerical (finite element analysis-based) tool for studying the degradation of anode-supported and electrolyte-supported planar SOFCs under several types of mechanical loads, including residual stresses. They have also calculated the failure probabilities of the SOFCs using Weibull analysis. Pitakthapanaphong and Busso (2005) carried out finite element analyses to investigate the fracture of multilayered systems used in SOFCs, such as LSM films on a YSZ substrate. They have pointed out that fracture is caused by large residual stresses induced during the SOFC manufacturing process due to thermal expansion coefficient (TEC) mismatch between different layers. They observed different cracking patterns (surface cracks, channeling cracks, and interfacial cracks) in physical samples of multi-layered systems. Their study involved FE simulations to determine the crack driving force (energy release rate) for the three observed cracking patterns. Johnson and Qu (2008) used a three-dimensional stochastic reconstruction method to create multiple realizations of the microstructure of porous Ni-YSZ cermet used as SOFC anode material. They analyzed these microstructure realizations using finite element software to determine the effective elastic modulus and effective coefficient of thermal expansion (CTE) of Ni-YSZ as a function of temperature. Anandakumar et al. (2010) carried out FE analyses to estimate thermal stresses and probability of failure in functionally graded SOFCs. They employed a continuum mechanics approach and used graded finite elements to discretize effective media consisting of NiO-YSZ/YSZ/LSM as well as NiO-YSZ/YSZ/GDC-LSCF. They used the Weibull method to determine the failure probability of the individual components of the SOFC, as well as the failure probability of the whole SOFC. They found that the thermal stresses developed in functionally graded SOFCs under spatially uniform and non-uniform temperature loads are lower than those induced in conventional layered SOFCs. They also found that functionally graded SOFCs show a lower probability of failure than other types of SOFCs.

In this work, three-dimensional micromechanical finite element (FE) models for real solid oxide fuel cell (SOFC) anode and cathode microstructures are generated from a stack of twodimensional image-based FE models of anode and cathode microstructures. Finite element analysis (FEA) of the models is carried out to determine their mechanical response to a steady-state temperature change from room temperature up to an operating temperature. The resulting stress distribution is determined in each case, and the stresses are analyzed using the Weibull method to calculate the probability of failure. The anode material is Ni-YSZ, while the cathode material is LSM-YSZ. Both linear elastic and elastic-plastic (nonlinear) behaviors are considered for nickel in the analysis of the anode. It is observed that the linear elastic models underestimate the probability of failure of the anode. The effect of temperature-dependent material properties on the probability of failure of the anode and cathode is also investigated. *The novelties of this work include micromechanical finite element analysis of the mechanical response of anode and cathode microstructural models considering temperature-dependent material properties and nonlinear elastic-plastic behavior of the nickel phase.* 
