P W. Dondl, J. Zimmer/J. Mech. Phys. Solids 52(2004)2057-2077 2065 ons of the minima tetragonal Orthorhombic Monoclinic 0.0479 p1(e1,e2,e6) p2(e1,e2,e6) 0.0001 The minima in the eleze6-space are calculated from the data given by Fadda et al. (2002, Appendix). We sed p=1.8155 GPa and T=838.9K. The values in the orbit space follow by evaluating the Hilbert map easily be computed automatically, for example with Singular( Greuel et al., 2001) Here it is even possible to guess a basis PI(e1, e2, e6):=e1 +e2 (the trace of E), P2(e1, e2, e6):=e+e?(the radius squared ) p3(e1e2,e6):=e (1) It is easy to see that none of these invariants can be expressed as a combination of the two remaining invariants. Therefore, they are independent. We need to show that they form a basis. According to Chevalley (1955, Theorem(A)), there is a basis of 3 invariants. Since the polynomials listed above are of the lowest possible degree, they The fact that these three polynomials form a basis of the algebra of polynomials invariant under C2 means that every such polynomial p=p(e1, e2, e])can be written as p=P(P1, P2, P3), where P is a polynomial. Such polynomial bases have been given by Smith and Rivlin(1958)for the different crystal classes, where polynomial energy functions were considered. We proceed by demonstrating how to use these bases to define more general multiphase energy functions that model given mechanical properties (such as location of minimizers and elastic moduli). To do so, we introduce the hilbert map p, which is defined (e,e,6)→(p1(e,e2,6),p2(e1,e2,6),p3(e1,e2,e6) The image of R3 under the Hilbert map is the orbit space. See Zimmer(2004a)for an explanation and more background Next, we locate the position of the difierent phases of Zirconia in the orbit space P(Sym(2, R)). Consider, for example, the orthorhombic phase. In Table 1, the data of one orthorhombic variant are given as e1=0.01, e2=0, e6=0. By applying the tetragona generator o to this element, we find the other variant as e1=0, e2=0.01, e6=0 Both variants are mapped to the same point in the orbit space, namely (0.01, 0.0001, 0)P.W. Dondl, J. Zimmer / J. Mech. Phys. Solids 52 (2004) 2057 – 2077 2065 Table 1 Locations of the minima Tetragonal Orthorhombic Monoclinic e1 0 0.01 0.0479 e2 0 0 0.0055 e6 0 0 0.1600 1(e1; e2; e6) 0 0.01 0.0534 2(e1; e2; e6) 0 0.0001 0.00232 3(e1; e2; e6) 0 0 0.0256 The minima in the e1e2e6-space are calculated from the data given by Fadda et al. (2002, Appendix). We used p = 1:8155 GPa and T = 838:9K. The values in the orbit space follow by evaluating the Hilbert map  = (1; 2; 3) at these points. easily be computed automatically, for example with Singular (Greuel et al., 2001). Here it is even possible to guess a basis 1(e1; e2; e6) := e1 + e2 (the trace of E); 2(e1; e2; e6) := e2 1 + e2 2 (the radius squared); 3(e1; e2; e6) := e2 6: (1) It is easy to see that none of these invariants can be expressed as a combination of the two remaining invariants. Therefore, they are independent. We need to show that they form a basis. According to Chevalley (1955, Theorem (A)), there is a basis of 3 invariants. Since the polynomials listed above are of the lowest possible degree, they form such a basis. The fact that these three polynomials form a basis of the algebra of polynomials invariant under C2 means that every such polynomial ˜ = ˜(e1; e2; e6) can be written as ˜ = P(1; 2; 3), where P is a polynomial. Such polynomial bases have been given by Smith and Rivlin (1958) for the diIerent crystal classes, where polynomial energy functions were considered. We proceed by demonstrating how to use these bases to de3ne more general multiphase energy functions that model given mechanical properties (such as location of minimizers and elastic moduli). To do so, we introduce the Hilbert map , which is de3ned as : R3 → R3 (e1; e2; e6) → (1(e1; e2; e6); 2(e1; e2; e6); 3(e1; e2; e6)): The image of R3 under the Hilbert map is the orbit space. See Zimmer (2004a) for an explanation and more background. Next, we locate the position of the diIerent phases of Zirconia in the orbit space (Sym(2; R)). Consider, for example, the orthorhombic phase. In Table 1, the data of one orthorhombic variant are given as e1=0:01; e2=0; e6=0. By applying the tetragonal generator  to this element, we 3nd the other variant as e1 = 0; e2 = 0:01; e6 = 0. Both variants are mapped to the same point in the orbit space, namely (0:01; 0:0001; 0)