P W. Dondl, J. Zimmer/J. Mech. Phys. Solids 52(2004)2057-2077 8 ape. The path from the tetragonal phase (left comer)to an orthorhombic minimum is shown. The scaling on the y-axis is both times 10. Left: the piecewise energy defined in Section 3, with an energy barrier modeled by the biharmonic equation. Right: a lowest order polynomial ansatz determines the energy barriers via analyticity and results in a much shallower energy barrier. Note hat the minimum of the energy on the right is negative. scaled according to the size of the elements on the strain space rather than the orbit space. This means that in direction of p2 and P3, where the Hilbert map is quadratic, the difference of the square roots of the coordinates is used to scale the element matrix (see Table 3). This results in an essentially equidistant scaling on the strain space. It is practical to define on a neighborhood of the orbit space (i.e, for certain negative values of p2 and P3 as well), rather than the orbit space itself. In this way, fitting parameters at the energy wells takes place in an open domain. However, this is a merely convenient method and exploits the fact that the orbit space automatically cuts out the relevant domain. Specifically, the boundaries of the domain of definition of p are given by p2=0 and p3=0. It can be shown that the definition of outside the orbit space is immaterial. Indeed, all symmetry-related variants are mapped to one point in the orbit space. Moreover, the corresponding parts of the boundary of a fundamental domain are identified; and no further identifications on the orbit space are necessary to obtain a smooth function. This greatly facilitates the construction of an energy function. The definition of large( the values for large strains) is arbitrary, since the physically correct growth rate is unknown. For the simulations, it is important to have a moderate growth for large strains to prevent numerical instabilities(this is one of the reasons why we have chosen not to use a polynomial energy function ). The simulations will show that strains in these regions of instability disappear after a suficiently large relaxation time We are mainly interested in the behavior of Zirconia at the triple point. Conseque we will not consider thermal efTects. Should this be desirable, however, a temperature dependence could be added in the same way as for the lowest order polynomial method Fadda et al. 2002). Nonetheless, it seems advisable to pursue the piecewise approach re advocate for the dependence on temperature as well. This might prevent the cre ation of additional stable phases reported by Fadda et al. (2002) for the polynomial2068 P.W. Dondl, J. Zimmer / J. Mech. Phys. Solids 52 (2004) 2057 – 2077 0 1 2 3 4 5 6 7 x 10−3 Φ Φ −10 −9.5 −9 −8.5 −8 −7.5 x 10−3 Fig. 4. Section of the energy landscape. The path from the tetragonal phase (left corner) to an orthorhombic minimum is shown. The scaling on the y-axis is both times 10−3. Left: the piecewise energy de3ned in Section 3, with an energy barrier modeled by the biharmonic equation. Right: a lowest order polynomial ansatz determines the energy barriers via analyticity and results in a much shallower energy barrier. Note that the minimum of the energy on the right is negative. scaled according to the size of the elements on the strain space rather than the orbit space. This means that in direction of 2 and 3, where the Hilbert map is quadratic, the diIerence of the square roots of the coordinates is used to scale the element matrix (see Table 3). This results in an essentially equidistant scaling on the strain space. It is practical to de3ne  on a neighborhood of the orbit space (i.e., for certain negative values of 2 and 3 as well), rather than the orbit space itself. In this way, 3tting parameters at the energy wells takes place in an open domain. However, this is a merely convenient method and exploits the fact that the orbit space automatically cuts out the relevant domain. Speci3cally, the boundaries of the domain of de3nition of  are given by 2=0 and 3=0. It can be shown that the de3nition of  outside the orbit space is immaterial. Indeed, all symmetry-related variants are mapped to one point in the orbit space. Moreover, the corresponding parts of the boundary of a fundamental domain are identi3ed; and no further identi3cations on the orbit space are necessary to obtain a smooth function. This greatly facilitates the construction of an energy function. The de3nition of large (the values for large strains) is arbitrary, since the physically correct growth rate is unknown. For the simulations, it is important to have a moderate growth for large strains to prevent numerical instabilities (this is one of the reasons why we have chosen not to use a polynomial energy function). The simulations will show that strains in these regions of instability disappear after a suFciently large relaxation time. We are mainly interested in the behavior of Zirconia at the triple point. Consequently, we will not consider thermal eIects. Should this be desirable, however, a temperature dependence could be added in the same way as for the lowest order polynomial method (Fadda et al. 2002). Nonetheless, it seems advisable to pursue the piecewise approach we advocate for the dependence on temperature as well. This might prevent the cre￾ation of additional stable phases reported by Fadda et al. (2002) for the polynomial approach