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material, anode, that, lower, Ni-YSZ, strain, level, stress–strain, SOFC, fuel, oxide, temperature., result, higher, high, this, same, brittle, e-mail:, Aerospace, Engineering, model, Mechanical, elastic, damage, developed, provide, mechanical, strength, positive–electrolyte–negative

3 e-mail: giqbal@mix.wvu.edu

4 Bruce Kang

5 e-mail: Bruce.Kang@mail.wvu.edu

7 Mechanical and Aerospace Engineering

8 Department West Virginia University

9 Mechanical and Aerospace Engineering Building

10 G-70 Morgantown, WV 26506-6106, USA

11 Elastic Brittle Damage Model

12 of Ni-YSZ and Predicted Stress–

13 Strain Relations as a Function

14 of Temperature and Porosity156

18 Nickel-yttria stabilized zirconia (Ni-YSZ) is the most widely used material for solid oxide

fuel cell (SOFC) anodes. Anode-supported SOFCs rely on the anode to provide mechanical strength to the positive–electrolyte–negative (PEN) structure. The stresses generated

in the anode can result in the formation of microcracks that degrade its structural properties and electrochemical performance. In this paper, a brittle elastic damage model is

developed for Ni-YSZ and implemented in finite element analysis with the help of a userdefined subroutine. The model is exploited to predict Ni-YSZ stress–strain relations at

temperatures and porosities that are difficult to generate experimentally. It is observed

that the anode material degradation depends on the level of strain regardless of the temperature at the same porosity: at higher temperature, lower load is required to produce a

specified level of strain than at lower temperature. Conversely, the anode material

degrades and fails at a lower level of strain at higher porosity at the same temperature.

The information obtained from this research will be useful to establish material parameters to achieve optimal robustness of SOFC stacks. [DOI: 10.1115/1.4003751]

19 Keywords: SOFC, anode, brittle elastic damage model, stress–strain relation

20 1 Introduction

21 Solid oxide fuel cells (SOFCs) are promising candidates for

22 future clean power generation due to their high efficiency and

23 ability to use a variety of fuels including hydrogen, carbon mon24 oxide, natural gas, biofuels, and coal-derived syngas [1,2]. Fuel

25 flexibility is primarily a result of SOFC high operating tempera26 tures (<600 C) which imposes stringent requirements for SOFC

27 material selection in order to have long-term acceptable electro28 chemical performance and component structural integrity [3–5].

29 Positive–electrolyte–negative (PEN) is the electrochemically

30 active composite structure of an SOFC [6]. It consists of three ce31 ramic components: (1) a fully dense electrolyte (YSZ) layer which

32 conducts oxide ions (O2) between the electrodes and hinders the

33 fuel and oxidant from mixing directly, (2) a porous cathode (e.g.

34 LSM) layer which allows the oxidant to reach the reaction sites,

35 and (3) a porous anode (e.g. Ni-YSZ) layer which enables fuel to

36 diffuse to the triple-phase boundary (TPB) and provides mechanical

37 support to the PEN structure in the case of an anode-supported cell.

38 Many types of materials have been studied in the literature for

39 SOFC anodes, such as LaxSr1xVO3d (LSV) [2], CoS1.035, WS2,

40 and Li2S=CoS1.035 [7], Ni=scandia-stabilized zirconia (Ni–ScSZ)

41 [8], and Cu–CeO2–YSZ [9]. Nickel-yttria stabilized zirconia (Ni42 YSZ), however, is the current best choice for SOFC anode mate43 rial [4,5,10]. Ni-YSZ is preferred because of its chemical and

44 structure stability, good thermal and electrical conductivity, cata45 lytic performance, cost, and compatibility with the other materials

46 of SOFCs [10]. However, the service life of a Ni-based anode is

47 deteriorated by a variety of microstructural changes which mainly

48 result from material transport, deactivation, and thermomechani49 cal processes [11,12]. The mechanical behavior of the Ni-YSZ

50 depends on its constituents, porosity, and operating temperature

51[4,13,14]. Radovic and Curzio [14] studied the effects of the frac52tion of reduction of NiO on the porosity and elastic modulus of

53Ni-YSZ. They found that the elastic modulus of NiO-YSZ

54decreased significantly with an increased fraction of reduced NiO

55due to the corresponding increase in sample porosity. Nguyen

56et al. [3] conducted failure analyses for ceramic materials and also

57developed a continuum damage mechanics (CDM) model for G5818, a glass ceramic material for SOFC applications. Liu et al. [15]

59studied the temperature-dependent Young’s modulus of G-18 and

60adopted a CDM model to capture the effects of cooling-induced

61microvoids and reheating-induced self-healing. However, there is

62no model available in the literature that can predict Ni-YSZ

63stress–strain relations under SOFC operating conditions.

64From room temperature to the operating temperature, Ni-YSZ

65behaves as a brittle elastic material [16]. Likewise, Mora et al.

66[17] found that the deformation behavior of an anode-supported

67bilayer consisting of a Ni-YSZ substrate and a YSZ layer is con68trolled by the deformation of the nickel phase in the temperature

69range of 1000 C–1200 C; however, it behaves as a brittle mate70rial below 1100 C. Since Ni-YSZ is a heterogeneous porous ma71terial that consists of nickel (Ni) and yttria-stabilized zirconia

72(YSZ), its mechanical properties are widely scattered and hard to

73define. However, for the convenience of analysis and design, it is

74often considered a homogenous material at the macroscopic level

75[4,18]. In this paper, a brittle elastic damage model is developed

76for the SOFC anode material and implemented in finite element

77analysis (FEA) software ABAQUSTM (ABAQUS Inc., Providence,

78RI) through a user-defined subroutine [19]. The proposed model is

79also used to predict Ni-YSZ stress–strain relations under compres80sion as a function of temperature and porosity.

812 Model Development

82When continuum mechanics formulation is adopted, elastic

83damage models are generally the standard approaches to describe

84the behavior of brittle material degradation [20,21]. In this

85approach, the effects of progressive microcracks and strain

1Corresponding author.

Contributed by the Advanced Energy Systems Division of ASME for publication

in the JOURNAL OF FUEL CELL SCIENCE AND TECHNOLOGY. Manuscript received December 22, 2009; Final manuscript received February 9, 2011; Published online XX XX

XXXX. Assoc. Editor Ken Reifsnider.

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Copyright VC 2011 by ASME

86 softening=hardening on the mechanical properties of a material

87 are represented by a set of internal state variables [23]. These state

88 variables govern the elastic behavior of the material, for example

89 stiffness reduction, at the macroscopic level. Material degradation

90 can be expressed in a phenomenological way through a damage

91 variable (D) that can be related to the reduction of the elastic mod92 ulus [3,23–26] at the macroscopic level

Ei

93 where Ei is the initial elastic modulus and E is the degraded elastic

94 modulus. The current state of energy in a material is described by

95 the thermodynamics potential (e.g., Helmholtz free energy) which

96 is function of the observable state variables and the internal state

97 variables under consideration [21,23]. The proposed definition of

98 Helmholtz free energy (w) allows the derivation of the constitu99 tive equations and the internal dissipation

w ¼ weðeij epij;DÞ þ wpðaÞ þ wdðjÞ (2)

100 where we, wp, and wd are free energies due to elastic deformation,

101 plastic hardening, and damage hardening=softening respectively,

102 eij is the total strain tensor, and D is the damage variable. From

103 room temperature to the operating temperature, the SOFC anode

104 material behaves as a brittle material [16,17]. Brittle elastic mate105 rials are those materials that degrade by the formation of micro106 cracks and their final fracture takes place by the coalescence of

107 microdefects without significant inelastic deformation [21].

108 Therefore, the free energy associated with the plastic hardening

109 wp(a) is negligible for brittle materials. For this research, we pro110 pose the Helmholtz free energy in the form of Eq. (3). A detailed

111 description of the formulation of Helmholtz free energy is given

112 elsewhere [21,26].

qw ¼ 1

ð1 DÞ Cijkleeijeekl þ

h

j2 (3)

113 where Cijkl is the initial stiffness matrix, q is the density, h is a ma114 terial-dependent constant related to the damage hardening=soften115 ing, and j is an internal variable that characterizes the damage

116 hardening=softening. Within the elastic limit, j is zero and it

117 evolves with the material damage. The second principle of ther118 modynamics gives the elasticity law coupled with damage

119 [20,21,23], as obtained in Eq. (4a)

rij ¼ q

@w @eeij

¼ ð1 DÞ Cijkleekl (4a)

K ¼ q @w

@j ¼ hj (4b)

Y ¼ q @w

Cijkle

e ije

e kl (4c)

120 where rij is the stress tensor, K is the damage hardening=softening

121 variable associated with the thermodynamics conjugate force, and

122 Y is the thermodynamics conjugate force associated with the dam123 age. In order to work with a positive quantity for the convenience

124 of further formulization, we define the following substitution

Cijkle

e ije

e kl (4d)

125 Eq. (4d) allows Y to be defined as the strain energy release rate.

126 By the definition of an associated variable in thermodynamics,

127 it is the principal variable governing the phenomenon of degra128 dation. This is the energy release by the loss of stiffness of the

129 representative element in which the degradation occurred. It is

130 equivalent to the energy release rate in fracture mechanics. The

131evolution of the thermodynamic conjugate forces can be obtained

132through the evolution of the internal state variables, which are

133found by assuming the physical existence of a dissipation poten134tial Fd. The energy dissipation due to degradation is found by sub135stituting the thermodynamics state laws into the Clausius–Duhem

136inequality and is, thus, given as the product of the thermodynamic

137conjugate forces with the respective flux variables [20,21,23], as

138shown in Eq. (5a)

_D ¼ _kd

@Fd

(5a)

_j ¼ _kd

@Fd

(5b)

139where kd is the Lagrange multiplier. It is determined using the

140consistency condition ð _Fd ¼ 0Þ and is assumed to obey the fol141lowing Kuhn–Tucker conditions [21,23,25].

_kd 0; Fd ¼ 0; _kdFd ¼ 0 (6)

142Similar to Murakami et al. [21] and Chow et al. [25], we assume a

143homogenous function for the degradation surface

Fd ¼ Ye ðKo þ K Þ (7)

144where Ye is the equivalent damage strain energy release rate AQ1

145(Y.Y=2)1=2 and Ko is a material constant which represents the

146threshold of degradation similar to the initial yield stress in plas147ticity theory. For isotropic degradation, the strain energy release

148rate is a scalar parameter to be used for isotropic degradation for149mulation. From Eqs. (2)–(7), the material tangential stiffness Kijkl

150(Jacobian) can be derived in the form of Eq. (8)

Kijkl ¼ ð1 DÞ Cijkl þ

h @rij

@Fd

@eekl

@Fd

151where Cijkl is the material initial stiffness matrix and h is the material

152hardening=softening variable. The proposed constitutive model is

153implemented in finite element analysis software ABAQUSTM (Abaqus

154Inc., Providence, RI) through a user-defined subroutine UMAT [19].

155User defined mechanical material behavior in ABAQUSTM is provided

156by means of an interface whereby a constitutive model can be added

157to the library. It requires that a constitutive model is programmed in

158the user-defined subroutine UMAT (ABAQUS=Standard). While

TM performs the standard finite element analysis, the subrou160tine governs the material behavior during different stages of loading.

1613 Results and Discussion

1623.1 Model Validation. The constitutive relations of the pro163posed model are validated by comparing the model results with

164the experimental [22] and an independent model [21] results for

165brittle elastic materials. The model formulation is validated with

166another brittle elastic material as Ni-YSZ behaves similarly under

167the specified conditions [16,17].The material parameters used for

168the model validation are given in Table 1.

169In order to validate the brittle elastic model, a column specimen

170is analyzed under a uniaxial state of stress. Fig. 1 to Fig. 3 show

171comparisons of the present model results with those of experimen172tal [20] and another model [21] results for a brittle elastic

173material.

Table 1 Material parameters for a brittle elastic material [21]

E (GPa) Poisson Ratio (t) Ko (MPa) h

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174 As can be seen from Fig. 1, the constitutive and evolution for175 mulation express the model results with sufficient accuracy for a

176 brittle elastic material. Material behavior is purely elastic within

177 the damage threshold as can be seen from Eqs. (5b)–(7). Therefore,

178 the stress–strain curves obtained experimentally and from the two

179 models are represented by straight lines and match with great accu180 racy within this region. As the material is loaded beyond the dam181 age threshold, it starts degrading as a result of the formation of

182 microcracks, and the stress–strain curves become nonlinear. For

183 small deformations beyond the damage threshold the stress–strain

184 curves are in good agreement, but at higher deformations they start

185 deviating. This could be a result of the assumption of the small de186 formation theory in deriving the constituting relations. Another

187 reason for the deviation is anisotropy that develops in the material

188 because of degradation, known as damage-induced anisotropy

189 [23,25]. As shown in Fig. 1, the present model and the experimen190 tal results seem to be converging again in the strain softening

191 stage. The stress–strain relations are within 4% of error between

192 the present model and the experimental results.

193 Figures 2 and 3 illustrate the evolution of material degradation

194 versus strain and the change in the material stiffness versus strain,

195 respectively. The change in Young’s modulus and the damage

196 variable with strain is due to the formation of microcracks, as also

197 shown in Eq. (4a). Similar to Fig. 1, these relations can be consid198 ered in two different regions. For a small deformation, the mate199 rial deforms purely elastically without material damage, as con200 strained by Eq. (7). Again, good agreement is observed between

201 the model results and the experiment results at the lower strain

202 value as a result of the small deformation theory adopted in deriv203 ing the constitutive relations. The deviation at higher strain could

204be due to the anisotropy developed in the material because of the

205material degradation at large deformations known as damage206induced anisotropy, as explained before. For brittle materials, the

207plastic deformation is assumed to be negligible as compared to the

208elastic deformation; therefore the material softening is mainly

209attributed to the microcracking [21,23].

2103.2 Predicted Stress–Strain Relation of Ni-YSZ. The an211ode material behaves as a brittle elastic material from room

212temperature to the operating temperature [16,17]. After the model

213validation, the model formulation is used to predict the stress–

214strain relations of Ni-YSZ as a function of temperature and poros215ity. Additionally some of the material parameters are obtained

216with the help of a user-defined subroutine implemented in FEA.

217At ambient temperature and 22% porosity, the average elastic

218modulus and biaxial strength of Ni-YSZ are inferred from Ref. [4]

219to be 105.83 GPa and 153.5 MPa, respectively. The predicted

220stress–strain relation obtained under these conditions is shown in

221Fig. 4. The damage threshold and degradation hardening=soften222ing variables are determined with the help of a user subroutine for

223this relation to be 0.0046 MPa and 0.059, respectively.

2243.3 Parametric Study. Numerous expressions have been

225proposed in the literature for the relation between the Young’s

226modulus and porosity of a material [13]. However, it has long

Fig. 1 Stress–strain relation of a brittle elastic material under

uniaxial compression Fig. 3 Predicted degradation of Young’s modulus under uniaxial compression

Fig. 2 Strain–damage relation of high strength concrete under

uniaxial stress

Fig. 4 Predicted stress–strain curve of Ni-YSZ at room

temperature

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227 been accepted that the Young’s modulus can be represented by

228 the following empirical relation, in terms of the material porosity

E ¼ Eo expðapÞ (9)

230 where p is the material porosity, Eo is the Young’s modulus at

231 zero porosity, and a is a material constant. Moreover, Rice [28,29]

232 showed that Eq. (9) can also be applied to describe the strength–

233 porosity relation for a wide range of materials for which the

234 strength and elastic modulus decrease proportionally with porosity

235 according to the minimum solid area model. Hence the strength–

236 porosity relation can be expressed as follows

S ¼ So expðbpÞ (10)

237 where p is the material porosity, So(MPa) is the material strength

238 at zero porosity, and b is a material constant. Eq. (9) and Eq. (10)

239 are known as the Ryshkewitch–Duckworth equation and Spriggs’s

240 equation [27], respectively, and generate straight lines when plot241 ted on a semi logarithmic paper. Radovic and Lara-Curzio [4]

242 determined the elastic modulus and biaxial strength of Ni-YSZ at

243 zero porosity and the material constants a¼ 3.166 0.03 and

244 b¼ 5.126 0.67. The temperature-dependent elastic modulus of

245 the anode material (Ni-YSZ) is given by Weil et al. [30], as shown

246 in Eq. (11).

247 where E(T) is the anode stiffness in GPa, and T is the absolute

248 temperature. Eqs. (9)–(11) can be combined in the normalized

249 form to obtain the stiffness and strength relations in terms of

250 porosity and temperature, in the form of Eq. (12) and Eq. (13),

251 respectively.

EðT; pÞ ¼ Eoð1:1562 5:159 104TÞ expð3:16 pÞ (12)

SðT; pÞ ¼ Soð1:1562 5:159 104TÞ expð5:12 pÞ (13)

252Equation (12) and Eq. (13) are incorporated in the model with the

253help of a user subroutine to predict the stress–strain relations of

254Ni-YSZ as a function of temperature and porosity. Figure 5 pre255dicts the stress–strain relations of Ni-YSZ at room temperature

256and at 1073 K as a function of material porosity.

257It can be seen that the material degrades and fails at a lower

258level of strain at higher porosity if the temperature is kept con259stant. This phenomenon can be explained with the help of the

260strain energy release rate, i.e., Eq. (4d). As the material porosity

261increases, it requires a lower level of strain to initiate and coales262cence microcracks in the material. Thus, a lower load causes

263higher material degradation at higher material porosity. The mate264rial parameters obtained from the subroutine for these relations

265are shown in Table 2.

266Similarly, Fig. 6 depicts the stress–strain relations of Ni-YSZ at

267various temperatures at p¼ 30%. It is found that at the same po268rosity, the material degradation depends on the level of strain,

269regardless of the temperature This paradoxical phenomenon also

270can be explained with the help of the strain energy release rate,

271i.e., Eq. (4d): at a higher temperature, a lower load is required to

272produce a specified level of strain than at a lower temperature.

Fig. 5 Predicted stress–strain curves of Ni-YSZ at (a) room temperature and (b) 1073 K, as a

function of porosity

Table 2 Predicted material parameters for Ni-YSZ at different

temperatures and porosities

Temperature p (%) E (GPa) S (MPa) Ko (MPa) h

Ambient Temperature 20 112.90 170.27 0.0050 0.068

40 36.11 36.80 0.0008 0.010 Fig. 6 Predicted stress–strain curves of Ni-YSZ at p5 30% as

a function of temperature

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273 4 Summary

274 A brittle elastic damage model is developed for anode materials

275 and implemented in finite element analysis with the help of a user276 defined subroutine. The model is exploited to predict the stress–

277 strain relations of nickel-yttria stabilized zirconia at temperatures

278 and porosities which are difficult to generate experimentally. The

279 model is also used to obtain the damage threshold variable and

280 damage hardening=softening variable at the specified temperatures

281 and porosities. At the same porosity, the material degradation

282 depends on the level of strain, regardless of the temperature: at a

283 higher temperature, a lower load is required to produce a specified

284 level of strain than at a lower temperature. On the other hand, the

285 anode material degrades and fails at a lower level of strain at higher

286 porosity at the same temperature. The knowledge obtained from

287 this research will be useful to establish material parameters to

288 achieve optimal robustness of solid oxide fuel cell stacks.

289 Acknowledgment

290 This work was supported by the Basic Energy Sciences Division,

291 U.S. Department of Energy (DOE), National Energy Technology

292 Laboratory (NETL), West Virginia State EPSCoR Office, and the

293 West Virginia University under Grant No. DE-FG02-06ER46299

295 Nomenclature

296 Cijkl ¼7 initial stiffness matrix

298 D ¼9 damage variable

300 Ei ¼1 reference Young’s modulus

302 Eo ¼3 Young’s modulus at zero porosity

304 Fd ¼5 damage function

306 h ¼7 damage hardening=softening parameter

308 K ¼9 damage hardening variable associated with thermodynam310 ics conjugate force

311 Kijkl ¼2 degraded stiffness matrix

313 Ko ¼4 material constant that represents the damage threshold

315 p ¼6 material porosity

317 So ¼8 material strength at zero porosity

319 T ¼20 absolute temperature

321 Y ¼2 strain energy release rate

323 Ye ¼4 equivalent strain energy rate

325 a ¼6 internal variable that characterizes isotropic hardening

327 ij ¼8 total strain tensor

329 eij ¼30 elastic strain tensor

331 pij ¼2 plastic strain tensor

333 j ¼4 internal variable that characterizes the degradation

335 hardening=softening

336 kd ¼7 Lagrange multiplier

338 m ¼9 Poisson ratio

340 q ¼1 material density

342 w ¼3 Helmholtz free energy

344 we ¼5 Helmholtz free energy associated with the elastic

346 deformation

347 wd ¼8 Helmholtz free energy associated with the damage

349 hardening=softening

350 wp ¼1 Helmholtz free energy associated with the plastic

352 hardening

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431[29] Rice, R. W., 1996, “Comparison of Physical Property-Porosity Behaviour with

432Minimum Solid Area Models,” J. Mater. Sci., 31(6), pp. 1509–1528.

433[30] Weil, K. S., Deibler, J. E., Hardy, J. S., Chick, L. A., Coyle, C. A., Kim, D. S.,

434and Xia, G., 2004, “Rupture Testing as a Tool for Developing Planar Solid

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J_ID: FCT DOI: 10.1115/1.4003751 Date: 18-April-11 Stage: Page: 5 Total Pages: 6

ID: ghomathys Time: 17:19 I Path: //xinchnasjn/AIP/3b2/FCT#/Vol00000/110002/APPFile/AI-FCT#110002

Journal of Fuel Cell Science and Technology APRIL 2011, Vol. 00 / 000000-5

AQ1: Please ensure that the term Y.Y=2 is correct here: “equivalent damage strain energy release rate (Y.Y=2)1=2” or

should this be YY=2, or something else?

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