P W. Dondl, J. Zimmer/J. Mech. Phys. Solids 52(2004)2057-2077 Tetragonal Monoclinic Orthorhombic Fig. 1. Schematic phase diagram(see Fadda et al, 2002: Ondik and McMurdie, 1998). The triple point is near 1.8 GPa and 840K We remark that these ideas not only apply to multiphase crystals, but also to much more complicated situations, for example, energetic landscapes arising in molecular dynamics. Applications of our presented ideas in that context will be an area of future research. Zirconia is chosen as a prototype of a material with a triple point, due to its relevance for applications. Extraordinary mechanical properties like high corrosion resistance and a melting point at high temperature make Zirconia a potentially attractive material in engineering ceramics. Zirconia exhibits several solid-solid phase transitions that are responsible for the internal formation of microstructures. The phase changes are also the source of transformation toughening. That is considered a milestone in oughening agent for ceramics. Yet the high pressure and temperature at the triple point (see Fig. 1) render experimental investigations of the phase transformations difficult Theoretical modeling and numerical simulations, as presented here, can provide valuable Insights Zirconia also proves particularly challenging for the orbit space methods described in Section 3. The orthorhombic phases are much closer to the tetragonal phase than monoclinic ones. This scaling has to be resolved correctly The numerical simulations explore the pattern formation and nucleation in Zirconia We study a dynamic theory of phase transformations in a two-dimensional elastic lid, where the phenomenological energy for Zirconia, as developed before, is used The main purpose of the simulations is to show that, given the piecewise energy defined in Section 3, the three phases of Zirconia can be recovered correctly in a numerical setup. At the same time, a lowest-order polynomial energy fails to exhibit clearly distinguishable phases. That is due to the different heights of the energy barriers obtainable with this approach( Fig. 4 and the simulations in Section 4.4) A secondary theme of the simulation is to demonstrate the flexibility of a three-phase material, as opposed to a two-phase material. It is shown that the size of the boundary layer with high potential energy is significantly smaller for a three-phase material This indicates higher flexibility of such a material (regarding the accommodation of boundary conditions). Moreover, it suggests that the set of recoverable strains might be larger for a three-phase material than for a comparable two-phase material2060 P.W. Dondl, J. Zimmer / J. Mech. Phys. Solids 52 (2004) 2057 – 2077 Fig. 1. Schematic phase diagram (see Fadda et al., 2002; Ondik and McMurdie, 1998). The triple point is near 1:8 GPa and 840K. We remark that these ideas not only apply to multiphase crystals, but also to much more complicated situations, for example, energetic landscapes arising in molecular dynamics. Applications of our presented ideas in that context will be an area of future research. Zirconia is chosen as a prototype of a material with a triple point, due to its relevance for applications. Extraordinary mechanical properties like high corrosion resistance and a melting point at high temperature make Zirconia a potentially attractive material in engineering ceramics. Zirconia exhibits several solid–solid phase transitions that are responsible for the internal formation of microstructures. The phase changes are also the source of transformation toughening. That is considered a milestone in achieving high strength ceramics of high toughness. Zirconia is the most important toughening agent for ceramics. Yet the high pressure and temperature at the triple point (see Fig. 1) render experimental investigations of the phase transformations diFcult. Theoretical modeling and numerical simulations, as presented here, can provide valuable insights. Zirconia also proves particularly challenging for the orbit space methods described in Section 3. The orthorhombic phases are much closer to the tetragonal phase than monoclinic ones. This scaling has to be resolved correctly. The numerical simulations explore the pattern formation and nucleation in Zirconia. We study a dynamic theory of phase transformations in a two-dimensional elastic solid, where the phenomenological energy for Zirconia, as developed before, is used. The main purpose of the simulations is to show that, given the piecewise energy de3ned in Section 3, the three phases of Zirconia can be recovered correctly in a numerical setup. At the same time, a lowest-order polynomial energy fails to exhibit clearly distinguishable phases. That is due to the diIerent heights of the energy barriers obtainable with this approach (Fig. 4 and the simulations in Section 4.4). A secondary theme of the simulation is to demonstrate the Hexibility of a three-phase material, as opposed to a two-phase material. It is shown that the size of the boundary layer with high potential energy is signi3cantly smaller for a three-phase material. This indicates higher Hexibility of such a material (regarding the accommodation of boundary conditions). Moreover, it suggests that the set of recoverable strains might be larger for a three-phase material than for a comparable two-phase material