2064 P W. Dondl, J. Zimmer/J. Mech. Phys. Solids 52(2004)2057-2077 elastic moduli of Zirconia exactly. We will use the orbit space approach, where local geometrical considerations allow for a comparatively simple construction. By giving up the restriction to polynomials, flexibility is gained. Therefore, the method proposed here has the potential to fit elastic moduli which cannot be fitted with a lowest-order polynomial. We refer the reader to Zimmer(2004a) for a detailed presentation of the orbit space method. It suffices to note that the orbit space' is a quotient that may intuitively be seen as a map to identify all variants of the same phase, whilst separating unrelated variants. We introduce two new ideas to fit elastic moduli and control the growth of energy at the energy barriers and infinity. The first idea is to define the energy as a piecewise function. This turns the problem to fit elastic moduli and other data into finding the solutions of several local problems which need to be interpolated appropriately. The second idea is to interpolate between the minima by solving the biharmonic equation with a finite-element code. Again, locality makes it easy to adjust the energy barriers between the minima to a desired height. The biharmonic equation has been chosen for its resemblance to the variational principle of minimal curvature. In this way, only few parameters need to be controlled. Also, the finite-element simulation of the biharmonic equation automatically returns he of frame indifference and the polar decomposition imply that the energy function can be written as a function of E: =5(F F-ld)E Sym(2, R). Here, E is the Green-St. Venant strain tensor, and Sym(2, R) is the space of symmetric real matrices They are henceforth identified with R3. Point groups act on this set by conjugation, P×Sym(2,R)→Sym(2,) (P, E)- The Green-St. Venant tensor E will be written in the Voigt notation, i.e. with e;ER. A short calculation shows that the representation of o on R=(el, e2, e6) is given by Since a2=ld, it is immediate that the action of the point group on E is isomorphic to C2. The orthorhombic and monoclinic subgroups coincide on this space and both act The next step is to find invariant polynomials in e1, e2 and e under the action of the high symmetry point group. It is a classic theorem of Hilbert that for compact Lie groups, the algebra of invariant polynomials( that is, multiplication of invariant poly nomials is defined)is finitely generated. See, for example, Theorem 2.1.3 in Sturmfels (1993). Alternatively, Weyl (1997) is the classic reference. An invariant basis can2064 P.W. Dondl, J. Zimmer / J. Mech. Phys. Solids 52 (2004) 2057 – 2077 elastic moduli of Zirconia exactly. We will use the orbit space approach, where local geometrical considerations allow for a comparatively simple construction. By giving up the restriction to polynomials, Hexibility is gained. Therefore, the method proposed here has the potential to 3t elastic moduli which cannot be 3tted with a lowest-order polynomial. We refer the reader to Zimmer (2004a) for a detailed presentation of the orbit space method. It suFces to note that the ‘orbit space’ is a quotient that may intuitively be seen as a map to identify all variants of the same phase, whilst separating unrelated variants. We introduce two new ideas to 3t elastic moduli and control the growth of energy at the energy barriers and in3nity. The 3rst idea is to de3ne the energy as a piecewise function. This turns the problem to 3t elastic moduli and other data into 3nding the solutions of several local problems which need to be interpolated appropriately. The second idea is to interpolate between the minima by solving the biharmonic equation with a 3nite-element code. Again, locality makes it easy to adjust the energy barriers between the minima to a desired height. The biharmonic equation has been chosen for its resemblance to the variational principle of minimal curvature. In this way, only few parameters need to be controlled. Also, the 3nite-element simulation of the biharmonic equation automatically returns splines. The axiom of frame indiIerence and the polar decomposition imply that the energy function can be written as a function of E := 1 2 (FTF −Id)∈Sym(2; R). Here, E is the Green-St. Venant strain tensor, and Sym(2; R) is the space of symmetric real matrices. They are henceforth identi3ed with R3. Point groups act on this set by conjugation, P × Sym(2; R) → Sym(2; R) (P; E) → PEP−1 : The Green-St. Venant tensor E will be written in the Voigt notation, i.e., E =
e1 1 2 e6 1 2 e6 e2 with ei ∈ R. A short calculation shows that the representation of  on R3 = (e1, e2, e6) is given by ˜ =   01 0 10 0 0 0 −1   : Since ˜2 = Id, it is immediate that the action of the point group on E is isomorphic to C2. The orthorhombic and monoclinic subgroups coincide on this space and both act as identity. The next step is to 3nd invariant polynomials in e1, e2 and e6 under the action of the high symmetry point group. It is a classic theorem of Hilbert that for compact Lie groups, the algebra of invariant polynomials (that is, multiplication of invariant poly￾nomials is de3ned) is 3nitely generated. See, for example, Theorem 2.1.3 in Sturmfels (1993). Alternatively, Weyl (1997) is the classic reference. An invariant basis can