P W. Dondl, J. Zimmer/J. Mech. Phys. Solids 52(2004)2057-2077 Fig. 2. Tetragonal reference configuration. The axes c1, c2 and c3 of rotations in the tetragonal point group Fig. 3. Schematic of the three-dimensional P3=0 is shown. The bo tetragonal minimum is marked with a t. the orthorhombic is marked with an‘o’.Th ce in scaling of Pi and p2 in the real esh is too large to be accurately displayed here. The mor is not in the plan We study this phase transformation using a continuum theory by invoking the Cauchy-Born rule(Ericksen in Gurtin, 1984). Let 22 CR3 reference configura tion. The deformation of the crystal is given by y(r). The displacement is defined as u(x): =y(x)-x. The deformation gradient is F According to the Cauchy-Born rule, this deformation gradient serves as a measure of the deformation of the lattice It is well known that there are several variants of the low-symmetry phases, where the number of variants is given by the quotient of the order of the high symmet group and the low symmetry group(see, e.g., Bhattacharya, 2003, Section 4.3)
2062 P.W. Dondl, J. Zimmer / J. Mech. Phys. Solids 52 (2004) 2057 – 2077 Fig. 2. Tetragonal reference con3guration. The axes c1, c2 and c3 of rotations in the tetragonal point group are shown. Fig. 3. Schematic representation of the three-dimensional mesh used for the interpolation of the energy. A two-dimensional cut in the plane 3 = 0 is shown. The box with the tetragonal minimum is marked with a ‘t’, the orthorhombic minimum is marked with an ‘o’. The diIerence in scaling of 1 and 2 in the real mesh is too large to be accurately displayed here. The monoclinic minimum is not in the plane 3 = 0. We study this phase transformation using a continuum theory by invoking the Cauchy–Born rule (Ericksen in Gurtin, 1984). Let ⊂ R3 be the reference con3guration. The deformation of the crystal is given by y(x). The displacement is de3ned as u(x) := y(x) − x. The deformation gradient is Fij := @yj @xi : According to the Cauchy–Born rule, this deformation gradient serves as a measure of the deformation of the lattice. It is well known that there are several variants of the low-symmetry phases, where the number of variants is given by the quotient of the order of the high symmetry group and the low symmetry group (see, e.g., Bhattacharya, 2003, Section 4.3)
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 available
P.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 therefore 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 summary, 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., orientationpreserving 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 energy 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