P w. Dond, J. Zimmer/J. Mech Phys. Solids 52(2004)2057-2077 For the readers convenience, the deformation gradients for the different variants ar listed below; see Truskinovsky and Zanzotto(2002). In particular, it can be seen that symmetry breaking takes place in the CIC2-plane shown in Fig. 2, to which we there fore devote our attention. Consequently, the third row and column of the deformation gradients are always given by(0, 0, 1 +u33)and will be suppressed from notation. For )123, there are two variants 1+l2 and F 1+ Similarly, for M3, there are four variants. It is easy to see that the corresponding deformation gradients F are given by the four matrices 1+uy ±12 1+u22±l 1+l2 Finally, deformation gradients preserving the tetragonal symmetry are of the form In the cIC2-plane depicted in Fig. 2, the tetragonal phase T3 is characterized by a C4 symmetry(the symmetry of a square ). This group is generated by an anti-clockwise rotation by 90, which will be denoted by o. The two orthorhombic phases have a planar C2 symmetry, since their restriction to he CIC2-plane is a rectangle. Finally, monoclinic variants reduce in the CIC2-plane to parallelograms, which also have C2 as the (orientation-preserving) planar point group. But for monoclinic phases, three-dimensional rotations by 180 along any axis in the cIC2-plane are no longer a self-mapping. Restricted to the cIC2-plane, this means that for monoclinic phases, reflections are no longer self-mappings. In sum- mary, our definition of the three phases (tetragonal, orthorhombic, monoclinic) is the standard one in a three-dimensional framework. There, phases can be defined by their orientation-preserving symmetry group. In equivalent terms, in a purely two-dimensional setting, we can define the phases by their symmetry subgroup in O(2), 1.e orientation preserving and orientation-reversing self-mappings. We think of the two-dimensiona framework studied here as a model reduction of three-dimensional phase transitions in Zirconia. Consequently, the groups operating on the phases will be the restrictions of he three-dimensional symmetry groups. Therefore, they are orientation-preserving 3. Derivation of a phenomenological free energy density The main input to the finite-el simulation will be a phenomenological en- ergy function modeling the phase ions in a two-dimensional setting. Fadda et al. (2002 ) Truskinovsky and Zanzotto(2002)have shown that, for the traditional approach based on invariant polynomials of lowest order, it is not possible to fit all availableP.W. Dondl, J. Zimmer / J. Mech. Phys. Solids 52 (2004) 2057 – 2077 2063 For the reader’s convenience, the deformation gradients for the diIerent variants are listed below; see Truskinovsky and Zanzotto (2002). In particular, it can be seen that symmetry breaking takes place in the c1c2-plane shown in Fig. 2, to which we there￾fore devote our attention. Consequently, the third row and column of the deformation gradients are always given by (0; 0; 1 + u33) and will be suppressed from notation. For O1;2;3, there are two variants, F =
1 + u11 1 + u22 and F =
1 + u22 1 + u11 : Similarly, for M3, there are four variants. It is easy to see that the corresponding deformation gradients F are given by the four matrices
1 + u11 ±u12 ±u12 1 + u22 and
1 + u22 ±u12 ±u12 1 + u11 : Finally, deformation gradients preserving the tetragonal symmetry are of the form F =
1 + u11 1 + u11 : In the c1c2-plane depicted in Fig. 2, the tetragonal phase T3 is characterized by a C4 symmetry (the symmetry of a square). This group is generated by an anti-clockwise rotation by 90◦, which will be denoted by . The two orthorhombic phases have a planar C2 symmetry, since their restriction to the c1c2-plane is a rectangle. Finally, monoclinic variants reduce in the c1c2-plane to parallelograms, which also have C2 as the (orientation-preserving) planar point group. But for monoclinic phases, three-dimensional rotations by 180◦ along any axis in the c1c2-plane are no longer a self-mapping. Restricted to the c1c2-plane, this means that for monoclinic phases, reHections are no longer self-mappings. In sum￾mary, our de3nition of the three phases (tetragonal, orthorhombic, monoclinic) is the standard one in a three-dimensional framework. There, phases can be de3ned by their orientation-preserving symmetry group. In equivalent terms, in a purely two-dimensional setting, we can de3ne the phases by their symmetry subgroup in O(2), i.e., orientation￾preserving and orientation-reversing self-mappings. We think of the two-dimensional framework studied here as a model reduction of three-dimensional phase transitions in Zirconia. Consequently, the groups operating on the phases will be the restrictions of the three-dimensional symmetry groups. Therefore, they are orientation-preserving. 3. Derivation of a phenomenological free energy density The main input to the 3nite-element simulation will be a phenomenological en￾ergy function modeling the phase transitions in a two-dimensional setting. Fadda et al. (2002); Truskinovsky and Zanzotto (2002) have shown that, for the traditional approach based on invariant polynomials of lowest order, it is not possible to 3t all available