Elastic Brittle Damage Model Of Ni-YSZ And ...

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Keywords

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

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2 Gulfam Iqbal1
3 e-mail: giqbal@mix.wvu.edu
4 Bruce Kang
5 e-mail: Bruce.Kang@mail.wvu.edu
6
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
17
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
D ¼ 1  E
Ei ; 0  D  1 (1)
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
2
ð1  DÞ Cijkleeijeekl þ
h 2
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
@D
¼  1
2
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
Y ¼ Y ¼ 1
2
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
@Y
(5a)
_j ¼  _kd
@Fd
@K
(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
1
h @rij
@D
@Fd
@Y
@Y
@eekl
@Fd
@Y
(8)
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
159ABAQUS
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
21.4 0.2 2.6103 0.04
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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
229 [4,13,27]
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).
EðTÞ1:213  102  5:460  102T (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
25 96.40 131.81 0.0045 0.047
30 82.31 102.04 0.0036 0.033
35 70.28 79.00 0.0027 0.024
40 60.00 61.15 0.0018 0.017
1073 K 20 67.94 102.46 0.0034 0.041
25 58.01 79.32 0.0027 0.029
30 49.53 61.40 0.0018 0.021
35 42.30 47.54 0.0013 0.014
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
353
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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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