Stored Energy Functions for Phase Transitions in Crystals

Abstract

A method is presented to construct nonconvex free energies that are invariant under a symmetry group. Algebraic and geometric methods are used to determine invariant functions with the right location of minimizers. The methods are illustrated for symmetry-breaking martensitic phase transformations. Computer algebra is used to compute a basis of the corresponding class of invariant functions. Several phase transitions, such as cubic-to-orthorhombic, are discussed. An explicit example of an energy for the cubic-to-tetragonal phase transition is given.

Global existence for a nonlinear system in thermoviscoelasticity with nonconvex energy

“…Using the ideas of [11,27,28], it is clear to see that every function in C with tetragonal symmetry can be written as…”

“…However, a general method to derive energy functions meeting all physical requirements (and arbitrary growth conditions) is presented in [27,28]. For illustrational purposes, we give an example of the tetragonal-orthorhombic (orthoI) symmetry breaking, as it occurs in zirconia (ZrO 2 ).…”

On Landau theory and symmetric energy landscapes for phase transitions

“…Note that here the labelling is different. To transfer to the notation we use here, it is necessary to apply the permutation ðe 4 ; e 6 ; e 5 Þ to the invariants given by Zimmer (2004).…”

“…Even within the realm of polynomials, there are bases with a different number of elements. The Hilbert basis chosen here has the advantage that a geometric interpretation of the location of the phases can be given (Zimmer, 2004). Zimmer (2004) computed the following basis for the cubic symmetry group.…”

“…See Section 4 for the interpolated data, where an energy is derived that accurately interpolates all elastic moduli. Zimmer (2004) describes an alternative approach to derive energy functions with a given symmetry. The method is briefly sketched in Section 2.…”