JOURNAL OF THE Journal of the Mechanics and Physics of Solids PHYSICS OF SOLID ELSEVIER 52(2004)2057-2077 Modeling and simulation of martensitic phase transitions with a triple point Patrick W. donda. Johannes zimmer a Dinision of Engineering and Applied Science, California Institute of Technology, Mail Stop 104-4 Pasadena ca 9/125. USa b Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig. Germany Received 24 November 2003: received in revised form 17 February 2004; accepted 2 March 2004 A framework for modeling complex global energy landscapes in a piecewise manner is pre- sented. Specifically, a class of strain-dependent energy functions is derived for the triple point of Zirconia(ZrO2), where tetragonal, orthorhombic(ortho) and monoclinic phases are stable. A simple two-dimensional framework is presented to deal with this symmetry breaking. An explicit energy is then fitted to the available elastic moduli of Zirconia in this two-dimensional setting. First, we use the orbit space method to deal with symmetry constraints in an easy way. Sec- ond, we introduce a modular(piecewise) approach to reproduce or model elastic moduli, energy barriers and other characteristics independently of each other in a sequence of local steps. This allows for more general results than the classical Landau theory (understood in the sense that the energy is a polynomial of invariant polynomials ) The class of functions considered here is strictly larger. Finite-element simulations for the energy constructed here demonstrate the pattern formation in Zirconia at the triple point. C 2004 Elsevier Ltd. All rights reserved PACS61.50Ks;,62.20.-x;81.30.Kf Keywords: A. Microstructures; A. Phase transformations; B. Elastic material 1. Introduction This paper provides a framework for modeling complex energetic landscapes, such as atomistic potentials or energies describing materials that undergo phase transitions. Corresponding author.Tel:+49-341-99-59-545;fax:+49-341-99-59-633 E-mail addresses: pwd(@caltech.edu(P w. DondI), zimmer(@mis. mpg. de (J. Zimmer ) URL //www.mis.mpg.de/zimme 0022-5096/S-see front matter e 2004 Elsevier Ltd. All rights reserved doi:10.1016/jmps200403.001
Journal of the Mechanics and Physics of Solids 52 (2004) 2057 – 2077 www.elsevier.com/locate/jmps Modeling and simulation of martensitic phase transitions with a triple point Patrick W. Dondla, Johannes Zimmerb;∗ aDivision of Engineering and Applied Science, California Institute of Technology, Mail Stop 104-44, Pasadena, CA 91125, USA bMax-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany Received 24 November 2003; received in revised form 17 February 2004; accepted 2 March 2004 Abstract A framework for modeling complex global energy landscapes in a piecewise manner is presented. Speci3cally, a class of strain-dependent energy functions is derived for the triple point of Zirconia (ZrO2), where tetragonal, orthorhombic (orthoI) and monoclinic phases are stable. A simple two-dimensional framework is presented to deal with this symmetry breaking. An explicit energy is then 3tted to the available elastic moduli of Zirconia in this two-dimensional setting. First, we use the orbit space method to deal with symmetry constraints in an easy way. Second, we introduce a modular (piecewise) approach to reproduce or model elastic moduli, energy barriers and other characteristics independently of each other in a sequence of local steps. This allows for more general results than the classical Landau theory (understood in the sense that the energy is a polynomial of invariant polynomials). The class of functions considered here is strictly larger. Finite-element simulations for the energy constructed here demonstrate the pattern formation in Zirconia at the triple point. ? 2004 Elsevier Ltd. All rights reserved. PACS: 61.50.Ks; 62.20.−x; 81.30.Kf Keywords: A. Microstructures; A. Phase transformations; B. Elastic material 1. Introduction This paper provides a framework for modeling complex energetic landscapes, such as atomistic potentials or energies describing materials that undergo phase transitions. ∗ Corresponding author. Tel.: +49-341-99-59-545; fax: +49-341-99-59-633. E-mail addresses: pwd@caltech.edu (P.W. Dondl), zimmer@mis.mpg.de (J. Zimmer). URL: http://www.mis.mpg.de/zimmer/ 0022-5096/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2004.03.001
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 invisible
2058 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 nontrivial 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 symmetric 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 feasibility 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 orthorhombic) 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
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 landscape
P.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 3niteelement 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
P W. Dondl, J. Zimmer/J. Mech. Phys. Solids 52(2004)2057-2077 Tetragonal Monoclinic Orthorhombic Fig. 1. Schematic phase diagram(see Fadda et al, 2002: Ondik and McMurdie, 1998). The triple point is near 1.8 GPa and 840K We remark that these ideas not only apply to multiphase crystals, but also to much more complicated situations, for example, energetic landscapes arising in molecular dynamics. Applications of our presented ideas in that context will be an area of future research. Zirconia is chosen as a prototype of a material with a triple point, due to its relevance for applications. Extraordinary mechanical properties like high corrosion resistance and a melting point at high temperature make Zirconia a potentially attractive material in engineering ceramics. Zirconia exhibits several solid-solid phase transitions that are responsible for the internal formation of microstructures. The phase changes are also the source of transformation toughening. That is considered a milestone in oughening agent for ceramics. Yet the high pressure and temperature at the triple point (see Fig. 1) render experimental investigations of the phase transformations difficult Theoretical modeling and numerical simulations, as presented here, can provide valuable Insights Zirconia also proves particularly challenging for the orbit space methods described in Section 3. The orthorhombic phases are much closer to the tetragonal phase than monoclinic ones. This scaling has to be resolved correctly The numerical simulations explore the pattern formation and nucleation in Zirconia We study a dynamic theory of phase transformations in a two-dimensional elastic lid, where the phenomenological energy for Zirconia, as developed before, is used The main purpose of the simulations is to show that, given the piecewise energy defined in Section 3, the three phases of Zirconia can be recovered correctly in a numerical setup. At the same time, a lowest-order polynomial energy fails to exhibit clearly distinguishable phases. That is due to the different heights of the energy barriers obtainable with this approach( Fig. 4 and the simulations in Section 4.4) A secondary theme of the simulation is to demonstrate the flexibility of a three-phase material, as opposed to a two-phase material. It is shown that the size of the boundary layer with high potential energy is significantly smaller for a three-phase material This indicates higher flexibility of such a material (regarding the accommodation of boundary conditions). Moreover, it suggests that the set of recoverable strains might be larger for a three-phase material than for a comparable two-phase material
2060 P.W. Dondl, J. Zimmer / J. Mech. Phys. Solids 52 (2004) 2057 – 2077 Fig. 1. Schematic phase diagram (see Fadda et al., 2002; Ondik and McMurdie, 1998). The triple point is near 1:8 GPa and 840K. We remark that these ideas not only apply to multiphase crystals, but also to much more complicated situations, for example, energetic landscapes arising in molecular dynamics. Applications of our presented ideas in that context will be an area of future research. Zirconia is chosen as a prototype of a material with a triple point, due to its relevance for applications. Extraordinary mechanical properties like high corrosion resistance and a melting point at high temperature make Zirconia a potentially attractive material in engineering ceramics. Zirconia exhibits several solid–solid phase transitions that are responsible for the internal formation of microstructures. The phase changes are also the source of transformation toughening. That is considered a milestone in achieving high strength ceramics of high toughness. Zirconia is the most important toughening agent for ceramics. Yet the high pressure and temperature at the triple point (see Fig. 1) render experimental investigations of the phase transformations diFcult. Theoretical modeling and numerical simulations, as presented here, can provide valuable insights. Zirconia also proves particularly challenging for the orbit space methods described in Section 3. The orthorhombic phases are much closer to the tetragonal phase than monoclinic ones. This scaling has to be resolved correctly. The numerical simulations explore the pattern formation and nucleation in Zirconia. We study a dynamic theory of phase transformations in a two-dimensional elastic solid, where the phenomenological energy for Zirconia, as developed before, is used. The main purpose of the simulations is to show that, given the piecewise energy de3ned in Section 3, the three phases of Zirconia can be recovered correctly in a numerical setup. At the same time, a lowest-order polynomial energy fails to exhibit clearly distinguishable phases. That is due to the diIerent heights of the energy barriers obtainable with this approach (Fig. 4 and the simulations in Section 4.4). A secondary theme of the simulation is to demonstrate the Hexibility of a three-phase material, as opposed to a two-phase material. It is shown that the size of the boundary layer with high potential energy is signi3cantly smaller for a three-phase material. This indicates higher Hexibility of such a material (regarding the accommodation of boundary conditions). Moreover, it suggests that the set of recoverable strains might be larger for a three-phase material than for a comparable two-phase material
P W. Dondl, J. Zimmer/J. Mech. Phys. Solids 52(2004)2057-2077 The article is further organized as follows: in Section 2, it is shown how the phase transition can be analyzed in a two-dimensional framework; and in Section 3, an energ. function is derived and fitted to the elastic moduli of the different phases. Numerical simulations using this energy are presented in Section 4. We close with a discussion in Section 5 2. Planar phase transformation We follow Truskinovsky and Zanzotto(2002)and consider a transformation path in Zirconia g the tetragonal phase and certain orthorhombic and monoclinic phases. We show that these phase transformations can be described as an in-plane tran formation, thus motivating the restriction to two space dimensions in the simulations in Section 4 As usual, we take the high symmetry phase as reference configuration. (This is jus- tified by the observation that one can define a so-called Ericksen-Pitteri neighborhood (Ericksen, 1980; Pitter, 1984 )of the lattice with the maximal symmetry in such a way that it comprises the lattices with a subgroup symmetry. For Zirconia, the tetragonal phase, denoted T3, is the high symmetry phase. To fix the notation, we list the elements of T3(Truskinovsky and Zanzotto(2002); the axes c1, c2, c3 are shown in Fig. 2; Ra tands for the rotation with angle a and axis a ) T3={1,R1,R2,R2,R1+e,R-,R2,R2 The orthorhombic subgroups of T3 are O,23:={1,RG,R2,R2} see Truskinovsky and Zanzotto (2002). Both orthorhombic groups form their own gacy class in T3 There are three conjugacy classes of monoclinic subgroups, from which we list one representative each M:={1,R,M1+2:={1,R+},M3:={1,R Of course, there is also the trivial triclinic subgroup (Id]. A schematic representation of the point groups is given by Truskinovsky and Zanzotto(2002, Fig 3) We assume that the symmetry breaking in ZrO2 occurs along the path This path, as we consider it, is different from the one usually studied for the tetragonal- monoclinic transformation(Fabris et al., 2000). Our path was first suggested by Truskinovsky and Zanzotto (2002 ); Fadda et al.(2002)based on experimental evi- dence collected there
P.W. Dondl, J. Zimmer / J. Mech. Phys. Solids 52 (2004) 2057 – 2077 2061 The article is further organized as follows: in Section 2, it is shown how the phase transition can be analyzed in a two-dimensional framework; and in Section 3, an energy function is derived and 3tted to the elastic moduli of the diIerent phases. Numerical simulations using this energy are presented in Section 4. We close with a discussion in Section 5. 2. Planar phase transformation We follow Truskinovsky and Zanzotto (2002) and consider a transformation path in Zirconia as joining the tetragonal phase and certain orthorhombic and monoclinic phases. We show that these phase transformations can be described as an in-plane transformation, thus motivating the restriction to two space dimensions in the simulations in Section 4. As usual, we take the high symmetry phase as reference con3guration. (This is justi3ed by the observation that one can de3ne a so-called Ericksen–Pitteri neighborhood (Ericksen, 1980; Pitteri, 1984) of the lattice with the maximal symmetry in such a way that it comprises the lattices with a subgroup symmetry.) For Zirconia, the tetragonal phase, denoted T3, is the high symmetry phase. To 3x the notation, we list the elements of T3 (Truskinovsky and Zanzotto (2002); the axes c1; c2; c3 are shown in Fig. 2; R a stands for the rotation with angle and axis a): T3 = {1; R c1 ; R c2 ; R c3 ; R c1+c2 ; R c1−c2 ; R =2 c3 ; R3 =2 c3 }: The orthorhombic subgroups of T3 are O1;2;3 := {1; R c1 ; R c2 ; R c3 } and O3;1±2 := {1; R c3 ; R c1+c2 ; R c1−c2 }; see Truskinovsky and Zanzotto (2002). Both orthorhombic groups form their own conjugacy class in T3. There are three conjugacy classes of monoclinic subgroups, from which we list one representative each M1 := {1; R c1 }; M1+2 := {1; R c1+c2 }; M3 := {1; R c3 }: Of course, there is also the trivial triclinic subgroup {Id}. A schematic representation of the point groups is given by Truskinovsky and Zanzotto (2002, Fig. 3). We assume that the symmetry breaking in ZrO2 occurs along the path T3 → O1;2;3 → M3: This path, as we consider it, is diIerent from the one usually studied for the tetragonalmonoclinic transformation (Fabris et al., 2000). Our path was 3rst suggested by Truskinovsky and Zanzotto (2002); Fadda et al. (2002) based on experimental evidence collected there
P W. Dondl, J. Zimmer/J. Mech. Phys. Solids 52(2004)2057-2077 Fig. 2. Tetragonal reference configuration. The axes c1, c2 and c3 of rotations in the tetragonal point group Fig. 3. Schematic of the three-dimensional P3=0 is shown. The bo tetragonal minimum is marked with a t. the orthorhombic is marked with an‘o’.Th ce in scaling of Pi and p2 in the real esh is too large to be accurately displayed here. The mor is not in the plan We study this phase transformation using a continuum theory by invoking the Cauchy-Born rule(Ericksen in Gurtin, 1984). Let 22 CR3 reference configura tion. The deformation of the crystal is given by y(r). The displacement is defined as u(x): =y(x)-x. The deformation gradient is F According to the Cauchy-Born rule, this deformation gradient serves as a measure of the deformation of the lattice It is well known that there are several variants of the low-symmetry phases, where the number of variants is given by the quotient of the order of the high symmet group and the low symmetry group(see, e.g., Bhattacharya, 2003, Section 4.3)
2062 P.W. Dondl, J. Zimmer / J. Mech. Phys. Solids 52 (2004) 2057 – 2077 Fig. 2. Tetragonal reference con3guration. The axes c1, c2 and c3 of rotations in the tetragonal point group are shown. Fig. 3. Schematic representation of the three-dimensional mesh used for the interpolation of the energy. A two-dimensional cut in the plane 3 = 0 is shown. The box with the tetragonal minimum is marked with a ‘t’, the orthorhombic minimum is marked with an ‘o’. The diIerence in scaling of 1 and 2 in the real mesh is too large to be accurately displayed here. The monoclinic minimum is not in the plane 3 = 0. We study this phase transformation using a continuum theory by invoking the Cauchy–Born rule (Ericksen in Gurtin, 1984). Let ⊂ R3 be the reference con3guration. The deformation of the crystal is given by y(x). The displacement is de3ned as u(x) := y(x) − x. The deformation gradient is Fij := @yj @xi : According to the Cauchy–Born rule, this deformation gradient serves as a measure of the deformation of the lattice. It is well known that there are several variants of the low-symmetry phases, where the number of variants is given by the quotient of the order of the high symmetry group and the low symmetry group (see, e.g., Bhattacharya, 2003, Section 4.3)
P w. Dond, J. Zimmer/J. Mech Phys. Solids 52(2004)2057-2077 For the readers convenience, the deformation gradients for the different variants ar listed below; see Truskinovsky and Zanzotto(2002). In particular, it can be seen that symmetry breaking takes place in the CIC2-plane shown in Fig. 2, to which we there fore devote our attention. Consequently, the third row and column of the deformation gradients are always given by(0, 0, 1 +u33)and will be suppressed from notation. For )123, there are two variants 1+l2 and F 1+ Similarly, for M3, there are four variants. It is easy to see that the corresponding deformation gradients F are given by the four matrices 1+uy ±12 1+u22±l 1+l2 Finally, deformation gradients preserving the tetragonal symmetry are of the form In the cIC2-plane depicted in Fig. 2, the tetragonal phase T3 is characterized by a C4 symmetry(the symmetry of a square ). This group is generated by an anti-clockwise rotation by 90, which will be denoted by o. The two orthorhombic phases have a planar C2 symmetry, since their restriction to he CIC2-plane is a rectangle. Finally, monoclinic variants reduce in the CIC2-plane to parallelograms, which also have C2 as the (orientation-preserving) planar point group. But for monoclinic phases, three-dimensional rotations by 180 along any axis in the cIC2-plane are no longer a self-mapping. Restricted to the cIC2-plane, this means that for monoclinic phases, reflections are no longer self-mappings. In sum- mary, our definition of the three phases (tetragonal, orthorhombic, monoclinic) is the standard one in a three-dimensional framework. There, phases can be defined by their orientation-preserving symmetry group. In equivalent terms, in a purely two-dimensional setting, we can define the phases by their symmetry subgroup in O(2), 1.e orientation preserving and orientation-reversing self-mappings. We think of the two-dimensiona framework studied here as a model reduction of three-dimensional phase transitions in Zirconia. Consequently, the groups operating on the phases will be the restrictions of he three-dimensional symmetry groups. Therefore, they are orientation-preserving 3. Derivation of a phenomenological free energy density The main input to the finite-el simulation will be a phenomenological en- ergy function modeling the phase ions in a two-dimensional setting. Fadda et al. (2002 ) Truskinovsky and Zanzotto(2002)have shown that, for the traditional approach based on invariant polynomials of lowest order, it is not possible to fit all available
P.W. Dondl, J. Zimmer / J. Mech. Phys. Solids 52 (2004) 2057 – 2077 2063 For the reader’s convenience, the deformation gradients for the diIerent variants are listed below; see Truskinovsky and Zanzotto (2002). In particular, it can be seen that symmetry breaking takes place in the c1c2-plane shown in Fig. 2, to which we therefore devote our attention. Consequently, the third row and column of the deformation gradients are always given by (0; 0; 1 + u33) and will be suppressed from notation. For O1;2;3, there are two variants, F = 1 + u11 1 + u22 and F = 1 + u22 1 + u11 : Similarly, for M3, there are four variants. It is easy to see that the corresponding deformation gradients F are given by the four matrices 1 + u11 ±u12 ±u12 1 + u22 and 1 + u22 ±u12 ±u12 1 + u11 : Finally, deformation gradients preserving the tetragonal symmetry are of the form F = 1 + u11 1 + u11 : In the c1c2-plane depicted in Fig. 2, the tetragonal phase T3 is characterized by a C4 symmetry (the symmetry of a square). This group is generated by an anti-clockwise rotation by 90◦, which will be denoted by . The two orthorhombic phases have a planar C2 symmetry, since their restriction to the c1c2-plane is a rectangle. Finally, monoclinic variants reduce in the c1c2-plane to parallelograms, which also have C2 as the (orientation-preserving) planar point group. But for monoclinic phases, three-dimensional rotations by 180◦ along any axis in the c1c2-plane are no longer a self-mapping. Restricted to the c1c2-plane, this means that for monoclinic phases, reHections are no longer self-mappings. In summary, our de3nition of the three phases (tetragonal, orthorhombic, monoclinic) is the standard one in a three-dimensional framework. There, phases can be de3ned by their orientation-preserving symmetry group. In equivalent terms, in a purely two-dimensional setting, we can de3ne the phases by their symmetry subgroup in O(2), i.e., orientationpreserving and orientation-reversing self-mappings. We think of the two-dimensional framework studied here as a model reduction of three-dimensional phase transitions in Zirconia. Consequently, the groups operating on the phases will be the restrictions of the three-dimensional symmetry groups. Therefore, they are orientation-preserving. 3. Derivation of a phenomenological free energy density The main input to the 3nite-element simulation will be a phenomenological energy function modeling the phase transitions in a two-dimensional setting. Fadda et al. (2002); Truskinovsky and Zanzotto (2002) have shown that, for the traditional approach based on invariant polynomials of lowest order, it is not possible to 3t all available
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 can
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, 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 polynomials 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
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)
P W. Dondl, J. Zimmer/J. Mech. Phys. Solids 52(2004)2057-2077 Table 2 Elastic Moduli (in GPa) Orthorhombic Monoclinic CCG 33.0 The values for the tetragonal and monoclinic phases are, except for rounding errors, the same as in Fadda et al.(2002, Tables ll(b), Iv(b))(orthorhombic data seem to be e). Note also the re-labeling of the indices in the monoclinic phase in Table IV in Fadda et alL. Here, the tetragonal phase labeling is always used. In the two-dimensional setting considered her trigonal and the orthorhombic configurations have the same independent moduli. However, C11- C22 holds only for the tetragonal phase, and not for the orthorhombic one Table Location of the breaks for the mesh on the 0.035 0.012 0.0000450.00002-0.0000050.000025 0.000225 0.007 0.05 0.0015 0.0005 0.0001 0.0125 0.06 This is a general property of orbit spaces, see, e.g., Zimmer(2004a). Table 1 lists the location of the other minima We turn towards the construction of a function on the orbit space such that p (p is a phenomenological energy function modeling the relevant mechanical properties of Zirconia at the t-o-m triple point. Since the Hilbert map identifies exactly the symmetry related variants as one point in the orbit space, can be an arbitrary function. It will be chosen to be a piecewise function. In this way, all available experimental and theoretical data of the elastic moduli can be fitted accurately. The values for the elastic moduli and the locations of the minima are taken from Fadda et al.(2002). No experimental data were available for the orthorhombic phase, so orthorhombic data of a similar agnitude as at the other phases were chosen before fitting the energy function. See Table 2 for the elastic moduli The definition of is done in two steps. First, a mesh on the orbit space is created The breaks are the locations where different pieces of the function will be joined. They form boxes in a natural way. The breaks are listed in Table 3. A comparison with the location of the minima on the orbit space in Table 1 shows that every minimum is
2066 P.W. Dondl, J. Zimmer / J. Mech. Phys. Solids 52 (2004) 2057 – 2077 Table 2 Elastic Moduli (in GPa) Tetragonal Orthorhombic Monoclinic C11 340 300 312 C22 340 350 350 C66 95.0 90.0 66.3 C12 33.0 33.0 35.2 C16 0 0 3.2 C26 0 0 4.3 The values for the tetragonal and monoclinic phases are, except for rounding errors, the same as in Fadda et al. (2002, Tables II(b), IV(b)) (orthorhombic data seem to be unavailable). Note also the re-labeling of the indices in the monoclinic phase in Table IV in Fadda et al. (2002). Here, the tetragonal phase’s labeling is always used. In the two-dimensional setting considered here, the tetragonal and the orthorhombic con3gurations have the same independent moduli. However, C11 = C22 holds only for the tetragonal phase, and not for the orthorhombic one. Table 3 Location of the breaks for the mesh on the orbit space 1 −0.05 −0.035 −0.02 −0.0025 0.0025 0.005 0.0075 0.0125 0.03 0.05 0.056 0.07 0.085 0.1 2 −0.0005 −0.000045 −0.00002 −0.000005 0.000025 0.00005 0.000075 0.000225 0.001 0.002 0.0026 0.004 0.0055 0.007 3 −0.05 −0.0015 −0.001 −0.0005 0.0001 0.0125 0.02 0.03 0.045 0.06 This is a general property of orbit spaces, see, e.g., Zimmer (2004a). Table 1 lists the location of the other minima. We turn towards the construction of a function on the orbit space such that () is a phenomenological energy function modeling the relevant mechanical properties of Zirconia at the t-o-m triple point. Since the Hilbert map identi3es exactly the symmetry related variants as one point in the orbit space, can be an arbitrary function. It will be chosen to be a piecewise function. In this way, all available experimental and theoretical data of the elastic moduli can be 3tted accurately. The values for the elastic moduli and the locations of the minima are taken from Fadda et al. (2002). No experimental data were available for the orthorhombic phase, so orthorhombic data of a similar magnitude as at the other phases were chosen before 3tting the energy function. See Table 2 for the elastic moduli. The de3nition of is done in two steps. First, a mesh on the orbit space is created. The breaks are the locations where diIerent pieces of the function will be joined. They form boxes in a natural way. The breaks are listed in Table 3. A comparison with the location of the minima on the orbit space in Table 1 shows that every minimum is