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Al2O3 composites: a preliminary study
Solid State Ionics 148 (2002) 527 – 531 www.elsevier.com/locate/ssi
Modelling and simulation of the mechanical properties of YSZ/Al2O3 composites: a preliminary study G. Dotelli a,*, C.M. Mari b a
Dipartimento di Chimica, Materiali e Ingegneria Chimica ‘‘Giulio Natta’’, Politecnico di Milano, P.za L. da Vinci 32, 20133 Milan, Italy b Dipartimento di Scienza dei Materiali, Universita` degli Studi di Milano-Bicocca, Via Cozzi 53, 20125 Milan, Italy
Abstract In recent years, a model based on the Voronoi tessellation and the random electrical networks (EN) has been developed to simulate the influence of the microstructure on the electrical properties of polycrystalline ionic conductor composites and encouraging results have been obtained. By the same approach, the elastic modulus of yttria-stabilized cubic zirconia (YSZ)/ Al2O3 has been tentatively simulated exploiting the analogy between the electrical and mechanical dynamic responses. A random network constituted of discrete elastic elements (springs) was used to sketch the mechanical behaviour of these polycrystalline composites. Preliminary results and remarks are presented and discussed. D 2002 Elsevier Science B.V. All rights reserved. Keywords: SOFC; Composite materials; Mechanical properties; Simulation; Rheological models
1. Introduction Research activities on the design and fabrication of solid oxide fuel cells (SOFCs) have been generally devoted to optimising electrochemical, thermal and microstructural properties of different materials, while little research has been carried out on the mechanical properties of the components comparatively. On the other hand, SOFC’s reliability not only depends on the chemical and electrochemical stability of its components but also on the capability of the whole device to withstand mechanical stresses. At present, yttria-stabilized cubic zirconia (YSZ) is the most widely used solid electrolyte in high temperature SOFCs [1]. Unfortunately, it suffers from rela*
Corresponding author. Tel.: +39-2-2399-3232; fax: +39-27063-8173. E-mail address: [email protected] (G. Dotelli).
tively poor bending strength and its use in planar cells is still uncertain. Indeed, the stacking of the electrolyte sheets and interconnector plates induces high mechanical stresses, which sometimes leads to fracture, compromising the cell yields. Design requirements for fracture strength are assumed to be at least 300 MPa at room temperature, even if a safer target might be 500 MPa [2]. Bending strength measurements on YSZ at room temperature quote a widespread range of mean strength values [2 – 6], most of which around 300 MPa. Moreover, at higher temperatures, the values have a dramatic drop, with an average decrease of about 100 MPa [3,5]. Nevertheless, some authors observed an anomalous behaviour [4], the temperature dependence of bending strength shows a minimum of 500 jC to reach, again, at 1000 jC, the room temperature strength value. Recently, Bhargava and Patterson [7] and Selcuk and Atkinson [8,9] reported on the
0167-2738/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 3 8 ( 0 2 ) 0 0 11 3 - 3
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G. Dotelli, C.M. Mari / Solid State Ionics 148 (2002) 527–531
mechanical properties of the individual materials constituting the multiple electrode assembly in planar self-supporting electrolyte SOFCs. In particular, they observed that the porosity and impurity content strongly affect the Young’s modulus of the electrolyte. In order to increase the mechanical strength of cubic 8 mol% Y2O3 – ZrO2 electrolyte, it was suggested to add a-Al2O3 [3– 6,10– 13] as a secondary phase because of its high elastic modulus, little partial solubility in ZrO2 (< 1%) and high temperature phase stability. Indeed, the addition of alumina hinders the grain growth process and allows the formation of finegrained ceramic, which might enhance the mechanical properties. However, the amount of Al2O3 in the composite has to be in the proper quantity to improve the mechanical resistance without significantly reducing the electrical conductivity of the material. The mechanical failure of planar SOFCs is not a minor concern in designing these devices and a renewed interest in the study of toughening mechanism is rising. Kishimoto et al. [14,15] showed a nonlinear toughening effect by adding (up to 30 mol%) fine alumina (0.38 Am in diameter) to YSZ;
the mechanical strength of samples sintered with a rapid heating rate was larger than expected on the load sharing effect only. Bhargava and Patterson [16] studied the interactions between crack paths and microstructural features (i.e. grain boundaries, particles) in YSZ/10 vol.% alumina composites. They observed that the thermal expansion and elastic modulus mismatches have a strong influence on crack propagation. A paper on alumina –whisker zirconia composites [6] pointed out the presence of a larger amount of porosity than in particle – alumina composites with the same composition. Indeed, by increasing the alumina whisker content, the strength decreases from 400 MPa (at 5 vol.%) to 240 MPa (at 20 vol.%). It is evident that a theoretical approach to forecast the mechanical properties of composites as a function of composition may certainly be a powerful tool for the material design; a preliminary approach is described here. A recent model able to simulate the electrical behaviour of ionic conductor composites [17 – 20] was the starting point. In such a simulation, the material is
Fig. 1. Schematic representation of a 2D Voronoi tessellation. Pixels labelled with even and odd integers stand for phase A and phase B, respectively; the dots are used to evidence the grain boundaries.
G. Dotelli, C.M. Mari / Solid State Ionics 148 (2002) 527–531
529
converted into a 2D or 3D network of discrete electrical elements, and then, using a transfer-matrix algorithm, the impedance of the sample was calculated. In the present case, the discrete electrical elements are substituted with mechanical ones (e.g. springs and dashpots) and the elastic modulus calculated.
2. Modelling and simulation In the electrical model [17 – 20], the computer simulation starts by reproducing the microstructure of the composite using the Voronoi tessellation technique [21]. The polycrystalline material is represented by a two-phase polyhedral microstructure, which generates a tri-dimensional digital image (having onepixel thickness). In Fig. 1, a digital representation of the output generated by the simulation relative to a planar section of a two-phase polycrystalline composite is reported. The microstructure has been generated through an original simulator based on genetic algorithms [22]. Adopting the pixel as a reference unit of volume, the digital image can be easily converted into a 2D array of figures (all pixels of each grain are Fig. 3. Schematic view of the mechanical network obtained by conversion of the digital representation of the material, as obtained by the Voronoi tessellation. The elastic behaviour of the two phases is represented by bold and light springs, respectively; no viscous effects are considered.
Fig. 2. Graphical representation of the conversion of a pixel into an electrical circuit by using only resistors for simplicity.
labelled with the same integer, see Fig. 1). A welldefined topological characteristic (i.e. grain boundary, bulk, etc.) is assigned to each pixel (representing a cubic portion of the simulated microstructure) and then converted into a discrete electrical circuit. Thus, the digital image representation becomes (see Fig. 2) a three-dimensional electrical network (EN), with a node at the centre of each pixel and six others at the centre of each face of the pixel. The EN is successively solved by the transfer-matrix method, using nonlinear matrix recursion relations [23]. Simulations of the electrical properties obtained with this approach are in good agreement with experimental results [17 – 20]. Following the same procedure, the digital image can also be used to generate a mechanical network if the pixels are converted into discrete mechanical elements, which are commonly applied to describe the
530
G. Dotelli, C.M. Mari / Solid State Ionics 148 (2002) 527–531
viscoelastic behaviour of the materials [24]. In such a way, the composite becomes a mechanical network (MN) of elements which are represented by springs (energy stored potentially) and dashpots (energy dissipated as heat). The usual assumptions to represent the linear viscoelastic behaviour are also assumed:
the spring is linear (i.e. Hookiean) and purely elastic; the dashpot is linear (i.e. Newtonian) and purely viscous; the mass of both elements is negligible. Following the same procedure adopted in the electrical simulations, the mechanical network was solved by node analysis. The stimulus, the response and the transfer function are the force, the displacement and the elastic (or Young’s) modulus (E), respectively.
3. Results and discussion The modelling procedure described above was used to simulate the mechanical properties of YSZ/ Al2O3 composites, calculating its elastic modulus. As a first approximation, assuming a constant tensile
stress and a purely elastic behaviour of the material, springs were the only constituent of the mechanical network, as schematically shown in Fig. 3. Using the experimental values of YSZ (220 GPa) [8] and alumina (440 GPa) [25], single crystal elastic modulus and assuming zirconia and alumina average grain size equal to 5 and 2.5 Am, respectively, the Young’s modulus of YSZ/Al2O3 composites with different amounts of alumina (from 15 to 50 vol.%) was calculated and the results are plotted in Fig. 4. In the simulations, the grain size of zirconia varies from 5.8 to 4.6 Am as the alumina content increases, according to the fact that the second phase hinders the crystal growth during sintering; simulated alumina grain particle size is in the order of 2.5 Am. The average grain sizes are necessary to calculate the values of the elastic constants of the mechanical elements representing the zirconia and alumina phases, respectively. Young’s modulus of pure materials is also calculated and the perfect agreement with the experimental input data testifies the internal consistency of the modelling procedure, as clearly shown in Fig. 4. Moreover, the dependence of the elastic modulus of YSZ/Al2O3 composites from the composition appears to be nonlinear, it seems to vary according to a polynomial law of second order (r2 = 0.998).
Fig. 4. Elastic modulus (E) of YSZ/Al2O3 composites vs. alumina amount.
G. Dotelli, C.M. Mari / Solid State Ionics 148 (2002) 527–531
A comparison with the experimental results is not possible due to the lack of available literature data. In fact, only the bending strength [3 –6,14,15] and the fracture toughness (KIC) [14,15] are reported, and unfortunately, they cannot be evaluated by the present model, and moreover, the comparison of Young’s modulus with the abovementioned properties is not trivial [14]. On the other hand, this paper simply meant to give a preliminary approach to evaluate if a discrete medium model may be used as a reliable tool to forecast some mechanical properties; certainly, further developments and refinements will be necessary to improve its present capability. However, at the present status, the model offers the possibility to take the influence of grain size on the mechanical behaviour explicitly into account, as already shown for the electrical properties [20]. In principle, porosity can be accounted for in the same manner by considering it a second phase, as it is usually done in literature [26]. 4. Conclusions A preliminary approach to evaluate if a discrete medium model may be used as a reliable tool to forecast some mechanical properties of composite materials has been presented. The model has been developed in analogy to a successful approach developed to study the electrical properties of the same materials. Even if at the present state of the art, the model is still in its early stages, the very preliminary results appear to be encouraging; certainly, further developments and refinements will be necessary to improve its capability. Acknowledgements The research has been partially supported by the Ministero dell’Universita` e della Ricerca Scientifica through the project COFIN 2000– 2001. We also thank Andrea Limido and Roberto Zanchettin for their help in the numerical experiments. References [1] B.C.H. Steele, J. Mater. Sci. 36 (2001) 1053. [2] B.C.H. Steele, Science and Technology of Zirconia V, in: S.P.S. Badwal, M.J. Bannister, R.H.J. Hannink (Eds.), Technomic, Lancaster, 1993, p. 713.
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[3] M. Fukuya, K. Hirota, O. Yamaguchi, H. Kume, S. Inamura, H. Miyamoto, N. Shiokawa, R. Shikata, Mater. Res. Bull. 29 (1994) 619. [4] M. Mori, T. Abe, H. Itoh, O. Yamamoto, Y. Takeda, T. Kawahara, Solid State Ionics 74 (1994) 157. [5] L.M. Navarro, P. Recio, J.R. Jurado, P. Duran, J. Mater. Sci. 30 (1995) 1949. [6] C.S. Montross, B.A. Van Hassel, T. Kawada, H. Yokokawa, J. Mater. Sci. 30 (1995) 3285. [7] P. Bhargava, B.R. Patterson, J. Am. Ceram. Soc. 80 (1997) 1863. [8] A. Selcuk, A. Atkinson, J. Eur. Ceram. Soc. 17 (1997) 1523. [9] A. Selcuk, A. Atkinson, J. Am. Ceram. Soc. 83 (2000) 2029. [10] I. Natali Sora, C. Schmid, C.M. Mari, Proceedings of the 17th Risø International Symposium on Materials Science ‘‘High Temperature Electrochemistry: Ceramics and Metals’’, in: F.W. Poulsen, N. Bonanos, S. Linderoth, M. Mogensen, B. Zachau-Chriastiansen (Eds.), Risø National Laboratory, Roskilde, 1996, p. 369. [11] A.J. Feighery, J.T.S. Irvine, Solid State Ionics 121 (1999) 209. [12] A. Yuzaki, A. Kishimoto, Solid State Ionics 116 (1999) 47. [13] Y. Ji, J. Liu, Z. Lu¨, X. Zhao, T. He, W. Su, Solid State Ionics 126 (1999) 277. [14] K. Oe, K. Kikkawa, A. Kishimoto, Y. Nakamura, H. Yanagida, Solid State Ionics 91 (1996) 131. [15] A. Yuzaki, A. Kishimoto, Y. Nakamura, Solid State Ionics 109 (1998) 273. [16] P. Bhargava, B.R. Patterson, J. Am. Ceram. Soc. 80 (1997) 1863. [17] G. Dotelli, R. Volpe, I. Natali Sora, C.M. Mari, Solid State Ionics 113 – 115 (1998) 325. [18] G. Dotelli, F. Casartelli, I. Natali Sora, C.M. Mari, 9th Cimtec — World Forum on New Materials — Symposium I — Computational Modelling and Simulation of Materials, in: P. Vincenzini, A. Degli Esposti (Eds.), Techna, Faenza, 1999, p. 417. [19] G. Dotelli, F. Casartelli, I. Natali Sora, C. Schmid, C.M. Mari, New Materials for Batteries and Fuel Cells, in: MRS Symposium Proceedings, vol. 575, MRS, Warrendale, 2000, p. 331. [20] C.M. Mari, G. Dotelli, Solid State Ionics 136 – 137 (2000) 1315. [21] M. de Berg, M. van Kreveld, M. Overmars, O. Schwarrzkopf, Computational Geometry, Springer-Verlag, Berlin, 1997, p. 145. [22] P. Tzionas, A. Thainalakis, Ph. Tsalides, Image Vis. Comput. 15 (1997) 35. [23] B. Derrida, J.G. Zabolitzky, J. Vannimenus, D. Stauffer, J. Stat. Phys. 36 (1984) 31. [24] N.W. Tschoegl, The Phenomenological Theory of Viscoelastic Behavior. An Introduction, Springer-Verlag, Berlin, 1989, pp. 69 – 156. [25] P. Bhargava, B.R. Patterson, J. Am. Ceram. Soc. 80 (1997) 1863. [26] N. Katsube, Int. J. Solids Struct. 32 (1995) 79.
Modelling and simulation of the mechanical properties of YSZ/Al2O3 composites: a preliminary study G. Dotelli a,*, C.M. Mari b a
Dipartimento di Chimica, Materiali e Ingegneria Chimica ‘‘Giulio Natta’’, Politecnico di Milano, P.za L. da Vinci 32, 20133 Milan, Italy b Dipartimento di Scienza dei Materiali, Universita` degli Studi di Milano-Bicocca, Via Cozzi 53, 20125 Milan, Italy
Abstract In recent years, a model based on the Voronoi tessellation and the random electrical networks (EN) has been developed to simulate the influence of the microstructure on the electrical properties of polycrystalline ionic conductor composites and encouraging results have been obtained. By the same approach, the elastic modulus of yttria-stabilized cubic zirconia (YSZ)/ Al2O3 has been tentatively simulated exploiting the analogy between the electrical and mechanical dynamic responses. A random network constituted of discrete elastic elements (springs) was used to sketch the mechanical behaviour of these polycrystalline composites. Preliminary results and remarks are presented and discussed. D 2002 Elsevier Science B.V. All rights reserved. Keywords: SOFC; Composite materials; Mechanical properties; Simulation; Rheological models
1. Introduction Research activities on the design and fabrication of solid oxide fuel cells (SOFCs) have been generally devoted to optimising electrochemical, thermal and microstructural properties of different materials, while little research has been carried out on the mechanical properties of the components comparatively. On the other hand, SOFC’s reliability not only depends on the chemical and electrochemical stability of its components but also on the capability of the whole device to withstand mechanical stresses. At present, yttria-stabilized cubic zirconia (YSZ) is the most widely used solid electrolyte in high temperature SOFCs [1]. Unfortunately, it suffers from rela*
Corresponding author. Tel.: +39-2-2399-3232; fax: +39-27063-8173. E-mail address: [email protected] (G. Dotelli).
tively poor bending strength and its use in planar cells is still uncertain. Indeed, the stacking of the electrolyte sheets and interconnector plates induces high mechanical stresses, which sometimes leads to fracture, compromising the cell yields. Design requirements for fracture strength are assumed to be at least 300 MPa at room temperature, even if a safer target might be 500 MPa [2]. Bending strength measurements on YSZ at room temperature quote a widespread range of mean strength values [2 – 6], most of which around 300 MPa. Moreover, at higher temperatures, the values have a dramatic drop, with an average decrease of about 100 MPa [3,5]. Nevertheless, some authors observed an anomalous behaviour [4], the temperature dependence of bending strength shows a minimum of 500 jC to reach, again, at 1000 jC, the room temperature strength value. Recently, Bhargava and Patterson [7] and Selcuk and Atkinson [8,9] reported on the
0167-2738/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 3 8 ( 0 2 ) 0 0 11 3 - 3
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G. Dotelli, C.M. Mari / Solid State Ionics 148 (2002) 527–531
mechanical properties of the individual materials constituting the multiple electrode assembly in planar self-supporting electrolyte SOFCs. In particular, they observed that the porosity and impurity content strongly affect the Young’s modulus of the electrolyte. In order to increase the mechanical strength of cubic 8 mol% Y2O3 – ZrO2 electrolyte, it was suggested to add a-Al2O3 [3– 6,10– 13] as a secondary phase because of its high elastic modulus, little partial solubility in ZrO2 (< 1%) and high temperature phase stability. Indeed, the addition of alumina hinders the grain growth process and allows the formation of finegrained ceramic, which might enhance the mechanical properties. However, the amount of Al2O3 in the composite has to be in the proper quantity to improve the mechanical resistance without significantly reducing the electrical conductivity of the material. The mechanical failure of planar SOFCs is not a minor concern in designing these devices and a renewed interest in the study of toughening mechanism is rising. Kishimoto et al. [14,15] showed a nonlinear toughening effect by adding (up to 30 mol%) fine alumina (0.38 Am in diameter) to YSZ;
the mechanical strength of samples sintered with a rapid heating rate was larger than expected on the load sharing effect only. Bhargava and Patterson [16] studied the interactions between crack paths and microstructural features (i.e. grain boundaries, particles) in YSZ/10 vol.% alumina composites. They observed that the thermal expansion and elastic modulus mismatches have a strong influence on crack propagation. A paper on alumina –whisker zirconia composites [6] pointed out the presence of a larger amount of porosity than in particle – alumina composites with the same composition. Indeed, by increasing the alumina whisker content, the strength decreases from 400 MPa (at 5 vol.%) to 240 MPa (at 20 vol.%). It is evident that a theoretical approach to forecast the mechanical properties of composites as a function of composition may certainly be a powerful tool for the material design; a preliminary approach is described here. A recent model able to simulate the electrical behaviour of ionic conductor composites [17 – 20] was the starting point. In such a simulation, the material is
Fig. 1. Schematic representation of a 2D Voronoi tessellation. Pixels labelled with even and odd integers stand for phase A and phase B, respectively; the dots are used to evidence the grain boundaries.
G. Dotelli, C.M. Mari / Solid State Ionics 148 (2002) 527–531
529
converted into a 2D or 3D network of discrete electrical elements, and then, using a transfer-matrix algorithm, the impedance of the sample was calculated. In the present case, the discrete electrical elements are substituted with mechanical ones (e.g. springs and dashpots) and the elastic modulus calculated.
2. Modelling and simulation In the electrical model [17 – 20], the computer simulation starts by reproducing the microstructure of the composite using the Voronoi tessellation technique [21]. The polycrystalline material is represented by a two-phase polyhedral microstructure, which generates a tri-dimensional digital image (having onepixel thickness). In Fig. 1, a digital representation of the output generated by the simulation relative to a planar section of a two-phase polycrystalline composite is reported. The microstructure has been generated through an original simulator based on genetic algorithms [22]. Adopting the pixel as a reference unit of volume, the digital image can be easily converted into a 2D array of figures (all pixels of each grain are Fig. 3. Schematic view of the mechanical network obtained by conversion of the digital representation of the material, as obtained by the Voronoi tessellation. The elastic behaviour of the two phases is represented by bold and light springs, respectively; no viscous effects are considered.
Fig. 2. Graphical representation of the conversion of a pixel into an electrical circuit by using only resistors for simplicity.
labelled with the same integer, see Fig. 1). A welldefined topological characteristic (i.e. grain boundary, bulk, etc.) is assigned to each pixel (representing a cubic portion of the simulated microstructure) and then converted into a discrete electrical circuit. Thus, the digital image representation becomes (see Fig. 2) a three-dimensional electrical network (EN), with a node at the centre of each pixel and six others at the centre of each face of the pixel. The EN is successively solved by the transfer-matrix method, using nonlinear matrix recursion relations [23]. Simulations of the electrical properties obtained with this approach are in good agreement with experimental results [17 – 20]. Following the same procedure, the digital image can also be used to generate a mechanical network if the pixels are converted into discrete mechanical elements, which are commonly applied to describe the
530
G. Dotelli, C.M. Mari / Solid State Ionics 148 (2002) 527–531
viscoelastic behaviour of the materials [24]. In such a way, the composite becomes a mechanical network (MN) of elements which are represented by springs (energy stored potentially) and dashpots (energy dissipated as heat). The usual assumptions to represent the linear viscoelastic behaviour are also assumed:
the spring is linear (i.e. Hookiean) and purely elastic; the dashpot is linear (i.e. Newtonian) and purely viscous; the mass of both elements is negligible. Following the same procedure adopted in the electrical simulations, the mechanical network was solved by node analysis. The stimulus, the response and the transfer function are the force, the displacement and the elastic (or Young’s) modulus (E), respectively.
3. Results and discussion The modelling procedure described above was used to simulate the mechanical properties of YSZ/ Al2O3 composites, calculating its elastic modulus. As a first approximation, assuming a constant tensile
stress and a purely elastic behaviour of the material, springs were the only constituent of the mechanical network, as schematically shown in Fig. 3. Using the experimental values of YSZ (220 GPa) [8] and alumina (440 GPa) [25], single crystal elastic modulus and assuming zirconia and alumina average grain size equal to 5 and 2.5 Am, respectively, the Young’s modulus of YSZ/Al2O3 composites with different amounts of alumina (from 15 to 50 vol.%) was calculated and the results are plotted in Fig. 4. In the simulations, the grain size of zirconia varies from 5.8 to 4.6 Am as the alumina content increases, according to the fact that the second phase hinders the crystal growth during sintering; simulated alumina grain particle size is in the order of 2.5 Am. The average grain sizes are necessary to calculate the values of the elastic constants of the mechanical elements representing the zirconia and alumina phases, respectively. Young’s modulus of pure materials is also calculated and the perfect agreement with the experimental input data testifies the internal consistency of the modelling procedure, as clearly shown in Fig. 4. Moreover, the dependence of the elastic modulus of YSZ/Al2O3 composites from the composition appears to be nonlinear, it seems to vary according to a polynomial law of second order (r2 = 0.998).
Fig. 4. Elastic modulus (E) of YSZ/Al2O3 composites vs. alumina amount.
G. Dotelli, C.M. Mari / Solid State Ionics 148 (2002) 527–531
A comparison with the experimental results is not possible due to the lack of available literature data. In fact, only the bending strength [3 –6,14,15] and the fracture toughness (KIC) [14,15] are reported, and unfortunately, they cannot be evaluated by the present model, and moreover, the comparison of Young’s modulus with the abovementioned properties is not trivial [14]. On the other hand, this paper simply meant to give a preliminary approach to evaluate if a discrete medium model may be used as a reliable tool to forecast some mechanical properties; certainly, further developments and refinements will be necessary to improve its present capability. However, at the present status, the model offers the possibility to take the influence of grain size on the mechanical behaviour explicitly into account, as already shown for the electrical properties [20]. In principle, porosity can be accounted for in the same manner by considering it a second phase, as it is usually done in literature [26]. 4. Conclusions A preliminary approach to evaluate if a discrete medium model may be used as a reliable tool to forecast some mechanical properties of composite materials has been presented. The model has been developed in analogy to a successful approach developed to study the electrical properties of the same materials. Even if at the present state of the art, the model is still in its early stages, the very preliminary results appear to be encouraging; certainly, further developments and refinements will be necessary to improve its capability. Acknowledgements The research has been partially supported by the Ministero dell’Universita` e della Ricerca Scientifica through the project COFIN 2000– 2001. We also thank Andrea Limido and Roberto Zanchettin for their help in the numerical experiments. References [1] B.C.H. Steele, J. Mater. Sci. 36 (2001) 1053. [2] B.C.H. Steele, Science and Technology of Zirconia V, in: S.P.S. Badwal, M.J. Bannister, R.H.J. Hannink (Eds.), Technomic, Lancaster, 1993, p. 713.
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[3] M. Fukuya, K. Hirota, O. Yamaguchi, H. Kume, S. Inamura, H. Miyamoto, N. Shiokawa, R. Shikata, Mater. Res. Bull. 29 (1994) 619. [4] M. Mori, T. Abe, H. Itoh, O. Yamamoto, Y. Takeda, T. Kawahara, Solid State Ionics 74 (1994) 157. [5] L.M. Navarro, P. Recio, J.R. Jurado, P. Duran, J. Mater. Sci. 30 (1995) 1949. [6] C.S. Montross, B.A. Van Hassel, T. Kawada, H. Yokokawa, J. Mater. Sci. 30 (1995) 3285. [7] P. Bhargava, B.R. Patterson, J. Am. Ceram. Soc. 80 (1997) 1863. [8] A. Selcuk, A. Atkinson, J. Eur. Ceram. Soc. 17 (1997) 1523. [9] A. Selcuk, A. Atkinson, J. Am. Ceram. Soc. 83 (2000) 2029. [10] I. Natali Sora, C. Schmid, C.M. Mari, Proceedings of the 17th Risø International Symposium on Materials Science ‘‘High Temperature Electrochemistry: Ceramics and Metals’’, in: F.W. Poulsen, N. Bonanos, S. Linderoth, M. Mogensen, B. Zachau-Chriastiansen (Eds.), Risø National Laboratory, Roskilde, 1996, p. 369. [11] A.J. Feighery, J.T.S. Irvine, Solid State Ionics 121 (1999) 209. [12] A. Yuzaki, A. Kishimoto, Solid State Ionics 116 (1999) 47. [13] Y. Ji, J. Liu, Z. Lu¨, X. Zhao, T. He, W. Su, Solid State Ionics 126 (1999) 277. [14] K. Oe, K. Kikkawa, A. Kishimoto, Y. Nakamura, H. Yanagida, Solid State Ionics 91 (1996) 131. [15] A. Yuzaki, A. Kishimoto, Y. Nakamura, Solid State Ionics 109 (1998) 273. [16] P. Bhargava, B.R. Patterson, J. Am. Ceram. Soc. 80 (1997) 1863. [17] G. Dotelli, R. Volpe, I. Natali Sora, C.M. Mari, Solid State Ionics 113 – 115 (1998) 325. [18] G. Dotelli, F. Casartelli, I. Natali Sora, C.M. Mari, 9th Cimtec — World Forum on New Materials — Symposium I — Computational Modelling and Simulation of Materials, in: P. Vincenzini, A. Degli Esposti (Eds.), Techna, Faenza, 1999, p. 417. [19] G. Dotelli, F. Casartelli, I. Natali Sora, C. Schmid, C.M. Mari, New Materials for Batteries and Fuel Cells, in: MRS Symposium Proceedings, vol. 575, MRS, Warrendale, 2000, p. 331. [20] C.M. Mari, G. Dotelli, Solid State Ionics 136 – 137 (2000) 1315. [21] M. de Berg, M. van Kreveld, M. Overmars, O. Schwarrzkopf, Computational Geometry, Springer-Verlag, Berlin, 1997, p. 145. [22] P. Tzionas, A. Thainalakis, Ph. Tsalides, Image Vis. Comput. 15 (1997) 35. [23] B. Derrida, J.G. Zabolitzky, J. Vannimenus, D. Stauffer, J. Stat. Phys. 36 (1984) 31. [24] N.W. Tschoegl, The Phenomenological Theory of Viscoelastic Behavior. An Introduction, Springer-Verlag, Berlin, 1989, pp. 69 – 156. [25] P. Bhargava, B.R. Patterson, J. Am. Ceram. Soc. 80 (1997) 1863. [26] N. Katsube, Int. J. Solids Struct. 32 (1995) 79.