P W. Dondl, J. Zimmer/J. Mech. Phys. Solids 52(2004)2057-2077 We point out, however, that the methods presented in Section 3 are also in the three-dimensional context. Also, the framework presented here is locality, general enough to accommodate data obtained from ab initio calculations, for example energy barriers. In this case, the representation of the energy will merely be substantially longer. We choose Zirconia as a suitable material for explaining the ideas, although admittedly the data from ab initio calculations for Zirconia are not available The method of deriving energy functions described here is not only advantageous if physical data need to be fitted, but it may also be of interest from a theoretical point of view. The Landau-Ericksen theory(Landau, 1967; Ericksen, 1980)commonly used was originally designed for a local analysis. There, the aim is to catch the structure of the energy in the vicinity of bifurcation points only. Polynomials have proven to be an appropriate choice. However, he global energy pic In this case, there is no justification to rely on polynomials alone. Rather, the ideas we present appear as a natural extension of the original ideas put forward by Landau Gluing together piecewise polynomials, as they appropriately describe the local picture of the energy landscape, leads to a global picture. From that point, the idea of defining the energy as a piecewise function seems to be quite natural We observe that a purely polynomial approach may result in further stable phases, as reported by Fadda et al. (2002) for an additional orthorhombic phase for Zirconia in a certain temperature regime. It is entirely plausible that an additional phase is ust an artifact stemming from the rigidity of polynomials. It may well disappear in the piecewise framework described here. Since we focus on the isothermal situation around the triple point, we will not pursue this question further. The methods presented in Section 3 will facilitate such an investigation. It is true that any flexibility gained by adopting this piecewise approach comes at a rice. First of all, there is a drop in smoothness from polynomials to the energy function derived here, which will only be C. In principle, one could use Hermite splines of an arbitrary order to obtain an arbitrarily smooth energy. For the simulations, a continuously differentiable function will suffice(see for example Balk et al. (2001); Huo and Muller(2003)for engineering and physics literature with piecewise defined Cor C energy densities ). In our numerical study of boundary value problems, no spurious effect stemming from the discontinuity in the elastic moduli were ever observed Secondly, the energy is not represented so compactly as a polynomial one. The class of functions considered here comprises the polynomials. One could express this in a fairly by using base functions (e.g, splines).We such expressions here, since we focus on the energy just as an input of the finite element simulation. The fact that the expression of the energy is lengthy is more or less irrelevant for such simulations. In Section 4. the simulations will show that the energy derived here is very well suited to scientific computations. Thirdly, the method implies a significant number of parameters. We minimize the arbitrary nature of choosing parameters by fitting the elastic moduli of the different variants and by interpolating through solving the biharmonic equation. In that way, only the parameters in the biharmonic equation determine the interpolation(e. g, energy barriers). Suitable variations of other parameters, such as the domain of the interpolation, do not change he qualitative behavior of the energy landscapeP.W. Dondl, J. Zimmer / J. Mech. Phys. Solids 52 (2004) 2057 – 2077 2059 We point out, however, that the methods presented in Section 3 are also applicable in the three-dimensional context. Also, the framework presented here is, due to its locality, general enough to accommodate data obtained from ab initio calculations, for example energy barriers. In this case, the representation of the energy will merely be substantially longer. We choose Zirconia as a suitable material for explaining the ideas, although admittedly the data from ab initio calculations for Zirconia are not available. The method of deriving energy functions described here is not only advantageous if physical data need to be 3tted, but it may also be of interest from a theoretical point of view. The Landau–Ericksen theory (Landau, 1967; Ericksen, 1980) commonly used was originally designed for a local analysis. There, the aim is to catch the structure of the energy in the vicinity of bifurcation points only. Polynomials have proven to be an appropriate choice. However, we aim to reconstruct the global energy picture. In this case, there is no justi3cation to rely on polynomials alone. Rather, the ideas we present appear as a natural extension of the original ideas put forward by Landau. Gluing together piecewise polynomials, as they appropriately describe the local picture of the energy landscape, leads to a global picture. From that point, the idea of de3ning the energy as a piecewise function seems to be quite natural. We observe that a purely polynomial approach may result in further stable phases, as reported by Fadda et al. (2002) for an additional orthorhombic phase for Zirconia in a certain temperature regime. It is entirely plausible that an additional phase is just an artifact stemming from the rigidity of polynomials. It may well disappear in the piecewise framework described here. Since we focus on the isothermal situation around the triple point, we will not pursue this question further. The methods presented in Section 3 will facilitate such an investigation. It is true that any Hexibility gained by adopting this piecewise approach comes at a price. First of all, there is a drop in smoothness from polynomials to the energy function derived here, which will only be C1. In principle, one could use Hermite splines of an arbitrary order to obtain an arbitrarily smooth energy. For the simulations, a continuously diIerentiable function will suFce (see for example Balk et al. (2001); Huo and Muller (2003) L for engineering and physics literature with piecewise de3ned C0 or C1 energy densities). In our numerical study of boundary value problems, no spurious eIect stemming from the discontinuity in the elastic moduli were ever observed. Secondly, the energy is not represented so compactly as a polynomial one. The class of functions considered here comprises the polynomials. One could express this in a fairly neat sense by using base functions (e.g., splines). We have no use for such expressions here, since we focus on the energy just as an input of the 3nite￾element simulation. The fact that the expression of the energy is lengthy is more or less irrelevant for such simulations. In Section 4, the simulations will show that the energy derived here is very well suited to scienti3c computations. Thirdly, the method implies a signi3cant number of parameters. We minimize the arbitrary nature of choosing parameters by 3tting the elastic moduli of the diIerent variants and by interpolating through solving the biharmonic equation. In that way, only the parameters in the biharmonic equation determine the interpolation (e.g., energy barriers). Suitable variations of other parameters, such as the domain of the interpolation, do not change the qualitative behavior of the energy landscape