P W. Dondl, J. Zimmer/J. Mech. Phys. Solids 52(2004)2057-2077 the matrix D F2TFTF1F= has three eigenvalues H1 21,#2=1and Bhattacharya, 2003, Chapter 5. 4). Since we work in the two-dimensional framework presented in Section 2, we define compatibility as the reduction of full compatibility That is, two phases are compatible if 1≥1,p≤1,or, equivalently,det(D-d)≤0. To show that two phases are compatible, it suffices to demonstrate that two ar- bitrary deformation gradients reproducing the Green-St. Venant strain tensors fulfill condition (2). The values D-o, D,-m and Do-m, for the tetragonal-orthorhombic tetragonal-monoclinic and orthorhombic-monoclinic phases are easily computed from the data for E=,(F F-ld)in Table 1. It is a straightforward calculation to ver depends in a sensitive way on the signs involved. E.g., for a positive value of e of the orthorhombic phase, the orthorhombic and the tetragonal phase are no longer 4.2. Equations of motion and numerical setup We investigate the dynamic behavior of martensitic phase transitions in Zirconia. The two-dimensional theory of phase transformations in Zirconia presented in the previous sections will be applied here. As mentioned in Section 2, the martensitic transformation under consideration can be modeled within the framework of continuum mechanics The quations of motion are, as usual, given by the inertial Hamiltonian dynamics of the elastic deformation field u: 52-R2, where 12 C R2 is the reference configuration In the following, x will denote the Lagrangian coordinate of a material point. The deformation gradient with respect to the material coordinate x is given by F(x, 6): Vu(x, t). As mentioned before, we will not consider thermal effects. The non-viscous part of the Piola-Kirchhoff stress tensor is defined as 0中p(F) (to avoid confusion with the rotation o defined in Section 2, we always write the argument of the stress tensor). Here, will be the phenomenological energy of Section 3. The differentiation in Eq (3)is carried out in MATLAB. In particular, the energy p defined in Section 3 is accessible for analytic manipulatio To resolve the non-uniqueness stemming from the nonconvexity of the energy we in- troduce a strain-gradient term( compare Reid and Gooding(1997); Kloucek and Luskin (1994)). This term serves as a penalization of the formation of interfaces; it is often coined capillarity. In a variational setting, this corresponds to an augmentation of the free energy density defined in Section 3 by a nonlocal Ginzburg term 2 Au(x, t)l,with 7>0. This term prevents the formation of infinitely fine microstructure by introducing a length scale (Kohn and Muller, 1993). We denote the reciprocal mass density by x>0. The energy minimization problem reads ovu(x,t)+|△(x,)2dx.2070 P.W. Dondl, J. Zimmer / J. Mech. Phys. Solids 52 (2004) 2057 – 2077 the matrix D := F−T 2 FT 1 F1F−1 2 has three eigenvalues 1 ≥ 1, 2 = 1 and 3 6 1 Bhattacharya, 2003, Chapter 5.4). Since we work in the two-dimensional framework presented in Section 2, we de3ne compatibility as the reduction of full compatibility. That is, two phases are compatible if 1 ≥ 1; 2 6 1; or; equivalently; det(D − Id) 6 0: (2) To show that two phases are compatible, it suFces to demonstrate that two ar￾bitrary deformation gradients reproducing the Green-St. Venant strain tensors ful3ll condition (2). The values Dt−o, Dt−m and Do−m, for the tetragonal-orthorhombic, tetragonal-monoclinic and orthorhombic-monoclinic phases are easily computed from the data for E = 1 2 (FTF − Id) in Table 1. It is a straightforward calculation to ver￾ify that the three phases are mutually compatible. We remark that the compatibility depends in a sensitive way on the signs involved. E.g., for a positive value of e2 of the orthorhombic phase, the orthorhombic and the tetragonal phase are no longer compatible. 4.2. Equations of motion and numerical setup We investigate the dynamic behavior of martensitic phase transitions in Zirconia. The two-dimensional theory of phase transformations in Zirconia presented in the previous sections will be applied here. As mentioned in Section 2, the martensitic transformation under consideration can be modeled within the framework of continuum mechanics. The equations of motion are, as usual, given by the inertial Hamiltonian dynamics of the elastic deformation 3eld u :  → R2, where  ⊂ R2 is the reference con3guration. In the following, x will denote the Lagrangian coordinate of a material point. The deformation gradient with respect to the material coordinate x is given by F(x; t) := ∇u(x; t). As mentioned before, we will not consider thermal eIects. The non-viscous part of the Piola–KirchhoI stress tensor is de3ned as (F) := @(F) @F (3) (to avoid confusion with the rotation  de3ned in Section 2, we always write the argument of the stress tensor). Here,  will be the phenomenological energy of Section 3. The diIerentiation in Eq. (3) is carried out in MATLAB. In particular, the energy  de3ned in Section 3 is accessible for analytic manipulations. To resolve the non-uniqueness stemming from the nonconvexity of the energy we in￾troduce a strain-gradient term (compare Reid and Gooding (1997); KlouScek and Luskin (1994)). This term serves as a penalization of the formation of interfaces; it is often coined capillarity. In a variational setting, this corresponds to an augmentation of the free energy density de3ned in Section 3 by a nonlocal Ginzburg term ! 2 |Tu(x; t)| 2, with ! ¿0. This term prevents the formation of in3nitely 3ne microstructure by introducing a length scale (Kohn and Muller, 1993 L ). We denote the reciprocal mass density by ¿0. The energy minimization problem reads inf   (∇u(x; t)) + ! 2 |Tu(x; t)| 2  dx: (4)