P W. Dondl, J. Zimmer/J. Mech. Phys. Solids 52(2004)2057-2077 Until recently, only a few physical parameters(such as elastic moduli) were known in these cases and a simple polynomial interpolation scheme was found sufficient to fit this data. However, ab initio calculations and improved experimental techniques give a considerable wealth of data that cannot be matched easily by a polynomial approach. We present a simple framework to model these energetic landscapes, accounting for symmetry constraints and fitting a greater potential number of parameters. In particular, our framework has scope to fit important physical quantities like energy barriers, which may be difficult to resolve correctly by a polynomial approach. We choose to demonstrate the modeling framework with Zirconia(ZrO2)as a non trivial example. From a modeling viewpoint, one of the specific difficulties of Zirconia (as well as any other material undergoing phase transformations) is its complex energy ndscape: it is invariant under the high symmetry point group in the space of symmet ric strains. We propose a theoretical framework as an(isothermal) phenomenological energy density for the tetragonal-orthorhombic (ortho)-monoclinic(t-o-m) triple point of Zirconia using piecewise functions. Numerical simulations demonstrate the feasi- bility of this approach. The flexibility of such a triple point material is compared to that of a two phase solid. Modeling and simulation of martensitic transformations, i.e diffusion-less first-order solid-solid transformations, is known to be demanding even for two-phase materials(Luskin, 1996; Swart and Holmes, 1992; Reid and Gooding, 1997). We are not aware of comparable simulations of a triple point material Fadda et al.(2002)use the ansatz of lowest order invariant polynomials to obtain an energy function and fit most of the elastic moduli. They show that it is impossible to fit all elastic moduli of the tetragonal and the monoclinic phase accurately within this framework. Two elastic moduli of the monoclinic phase, C25 and C35, are too high by an order of magnitude and by about 150%, respectively. For this comparison, we have chosen the closest available experimental and theoretical data ( Elastic moduli for the orthorhombic phase were not considered; and no experimental data seem to be available here. This aspect again reflects the fact that lowest-order polynomials are often unsuitable as correct descriptions of the energetic landscape. Gooding et al. (1991) pointed out that the minimal set of order parameters may lead to unrealistically high estimates for the thermal activation energy. To determine the energy barrier correctly, they use non-symmetry-breaking order parameters or, more specifically, invariant polynomials of higher order. This approach is often difficult and results in steeply growing energy functions. We introduce a related, yet novel approach to define elastic energies in terms of piecewise functions. Within the framework of piecewise defined functions, the task of fitting elastic moduli and other parameters is essentially that of solving local problems and appropriate interpolation. As demonstrated below for Zirconia, an accurate fitting of the energy to given values for the elastic moduli of the different phases(tetragonal, monoclinic and orthorhom- bic)becomes a relatively simple task. Since the derived phenomenological energy will serve as input of the two-dimensional simulations in Section 4, we limit ourselves to a suitable plane describing the tetragonal-orthorhombic-monoclinic phase transition. Therefore, we can only fit the moduli visible in this plane. In particular, the moduli that cannot be fitted accurately with the lowest order polynomial ansatz are invisible2058 P.W. Dondl, J. Zimmer / J. Mech. Phys. Solids 52 (2004) 2057 – 2077 Until recently, only a few physical parameters (such as elastic moduli) were known in these cases and a simple polynomial interpolation scheme was found suFcient to 3t this data. However, ab initio calculations and improved experimental techniques give a considerable wealth of data that cannot be matched easily by a polynomial approach. We present a simple framework to model these energetic landscapes, accounting for symmetry constraints and 3tting a greater potential number of parameters. In particular, our framework has scope to 3t important physical quantities like energy barriers, which may be diFcult to resolve correctly by a polynomial approach. We choose to demonstrate the modeling framework with Zirconia (ZrO2) as a non￾trivial example. From a modeling viewpoint, one of the speci3c diFculties of Zirconia (as well as any other material undergoing phase transformations) is its complex energy landscape: it is invariant under the high symmetry point group in the space of symmet￾ric strains. We propose a theoretical framework as an (isothermal) phenomenological energy density for the tetragonal-orthorhombic (orthoI)-monoclinic (t-o-m) triple point of Zirconia using piecewise functions. Numerical simulations demonstrate the feasi￾bility of this approach. The Hexibility of such a triple point material is compared to that of a two phase solid. Modeling and simulation of martensitic transformations, i.e., diIusion-less 3rst-order solid-solid transformations, is known to be demanding even for two-phase materials (Luskin, 1996; Swart and Holmes, 1992; Reid and Gooding, 1997). We are not aware of comparable simulations of a triple point material. Fadda et al. (2002) use the ansatz of lowest order invariant polynomials to obtain an energy function and 3t most of the elastic moduli. They show that it is impossible to 3t all elastic moduli of the tetragonal and the monoclinic phase accurately within this framework. Two elastic moduli of the monoclinic phase, Cm 25 and Cm 35, are too high by an order of magnitude and by about 150%, respectively. For this comparison, we have chosen the closest available experimental and theoretical data. (Elastic moduli for the orthorhombic phase were not considered; and no experimental data seem to be available here.) This aspect again reHects the fact that lowest-order polynomials are often unsuitable as correct descriptions of the energetic landscape. Gooding et al. (1991) pointed out that the minimal set of order parameters may lead to unrealistically high estimates for the thermal activation energy. To determine the energy barrier correctly, they use non-symmetry-breaking order parameters or, more speci3cally, invariant polynomials of higher order. This approach is often diFcult and results in steeply growing energy functions. We introduce a related, yet novel approach to de3ne elastic energies in terms of piecewise functions. Within the framework of piecewise de3ned functions, the task of 3tting elastic moduli and other parameters is essentially that of solving local problems and appropriate interpolation. As demonstrated below for Zirconia, an accurate 3tting of the energy to given values for the elastic moduli of the diIerent phases (tetragonal, monoclinic and orthorhom￾bic) becomes a relatively simple task. Since the derived phenomenological energy will serve as input of the two-dimensional simulations in Section 4, we limit ourselves to a suitable plane describing the tetragonal-orthorhombic-monoclinic phase transition. Therefore, we can only 3t the moduli visible in this plane. In particular, the moduli that cannot be 3tted accurately with the lowest order polynomial ansatz are invisible