P W. Dondl, J. Zimmer/J. Mech. Phys. Solids 52(2004)2057-2077 0.14 0.15 002001 Fig. 5. The energy a, plotted as a function of e1-e2 and e6. The cut through the strain space has been hosen such that all three minima are visible. The well marked with a 't'is tetragonal, the one marked with n 'o' is orthorhombic. while,stands for the monoclinic well Fig. 5 shows the resulting function on different planes in the strain space. We note that the energy adequately captures the phenomenological structure of a multiphase with 4. simulations The simulations in this section will use the phenomenological energy defined in Section 3 and methods described in that section. The pattern formation in this two- dimensional model of Zirconia will be investigated and contrasted with that of a two-phase material. We first demonstrate in 4.1 that it is theoretically possible for any o of the three share a phase boundary. The simulations in 4.3 will indeed exhibit all three n a clearly distinguishable way, as well as all combinations of neighboring phases, even with the interface penalization introduced in 4.2 1. Compatibility of We first show that the three phases are mutually compatible. Two phases are com- patible if there is a continuous deformation exhibiting their gradients Fl and F2, say, in adjacent domains. It can be shown that this is equivalent to the requirement thatP.W. Dondl, J. Zimmer / J. Mech. Phys. Solids 52 (2004) 2057 – 2077 2069 0.02 0.01 0 0.01 0.02 0.03 0.04 0.05 0 0.05 0.1 0.15 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 e6 e1−e2 Φ t o m Fig. 5. The energy , plotted as a function of e1 − e2 and e6. The cut through the strain space has been chosen such that all three minima are visible. The well marked with a ‘t’ is tetragonal, the one marked with an ‘o’ is orthorhombic, while ‘m’ stands for the monoclinic well. Fig. 5 shows the resulting function on diIerent planes in the strain space. We note that the energy adequately captures the phenomenological structure of a multiphase energy with minima and energy barriers. 4. Simulations The simulations in this section will use the phenomenological energy de3ned in Section 3 and methods described in that section. The pattern formation in this two￾dimensional model of Zirconia will be investigated and contrasted with that of a two-phase material. We 3rst demonstrate in 4.1 that it is theoretically possible for any two of the three phases to share a phase boundary. The simulations in 4.3 will indeed exhibit all three phases in a clearly distinguishable way, as well as all combinations of neighboring phases, even with the interface penalization introduced in 4.2. 4.1. Compatibility of phases We 3rst show that the three phases are mutually compatible. Two phases are com￾patible if there is a continuous deformation exhibiting their gradients F1 and F2, say, in adjacent domains. It can be shown that this is equivalent to the requirement that