International Journal of Plasticity 60 (2014) 71–86

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International Journal of Plasticity

journal homepage: www.elsevier .com/locate / i jp las

Shape memory effect and pseudoelasticity behavior intetragonal zirconia polycrystals: A phase field study

http://dx.doi.org/10.1016/j.ijplas.2014.03.0180749-6419/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Center for Advanced Vehicular System, Mississippi State University, Starkville, MS 39762, United States. Tel.: +1 662 3E-mail address: mamivand@cavs.msstate.edu (M. Mamivand).

Mahmood Mamivand a,b,⇑, Mohsen Asle Zaeem c, Haitham El Kadiri a,b

a Center for Advanced Vehicular System, Mississippi State University, Starkville, MS 39762, United Statesb Department of Mechanical Engineering, Mississippi State University, Starkville, MS 39762, United Statesc Department of Materials Science and Engineering, Missouri University of Science and Technology, Rolla, MO 65409, United States

a r t i c l e i n f o

Article history:Received 30 October 2013Received in final revised form 10 March2014Available online 13 April 2014

Keywords:A. Tetragonal to monoclinic transformationB. Tetragonal zirconia polycrystalC. Phase field modelingA. Shape memory effectA. Pseudoelasticity

a b s t r a c t

Martensitic tetragonal-to-monoclinic transformation in zirconia is a ‘‘double-edgedsword’’, enabling transformation toughening or shape memory effects in favorable cases,but also cracks and phase degradation in undesirable scenarios. In stressed polycrystals,the transformation can burst from grain to grain, enabling stress field shielding and tough-ening in an autocatalysis fashion. This transformation strain can be recovered by an ade-quate thermal cycle at low temperatures (when monoclinic is stable) to provide a shapememory effect, or by unloading at higher temperatures (when tetragonal is stable) toprovide pseudoelasticity.

We capture the details of these processes by mining the associated microstructural evo-lutions through the phase field method. The model is both stress and temperature depen-dent, and incorporates inhomogeneous and anisotropic elasticity. Results of simulationsshow an ability to capture the effects of both forward (T ? M) and reverse (M ? T)transformation under certain boundary conditions.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Zirconia based ceramics are gaining a vast application in High-Tech industry. They are strong, hard, tough, smooth, andresistant to oxidation, consequently they naturally find integration in advanced cutting tools, gas sensors, refractories, andstructural opacifiers (Rashad and Baioumy, 2008). Owing to their biocompatibility, cutting-edge applications also comprisebiomedical and athletic industries (Chevalier et al., 2009; Thompson et al., 2007).

Zirconia exhibits three crystalline polymorphs: monoclinic, tetragonal and cubic. In pure zirconia, monoclinic is stablebelow 1170 �C, and above this temperature, the tetragonal structure takes place and persists up to 2370 �C. Then, transfor-mation to the cubic phase occurs and persists up to the melting temperature. During industrial or laboratory processing, thetetragonal phase transforms to monoclinic at temperatures which depend on the cooling rates and composition (970 �C atequilibrium). The tetragonal to monoclinic (T ? M) transformation proceeds by nucleation and migration of martensiticinterfaces (Hirth and Pond, 2011), which induce approximately 0.04 and 0.16 of dilatation and shear strain, respectively.Although the transformation strain enables a natural toughening mechanism in many applications (Hannink et al., 2000),it is generally regarded as a ‘‘double-edged sword’’. The inherent stress generated by the transformation can, in a limitingcase, develop detrimental triaxialities, which lead to crack formation and ultimately, material failure. Moreover, many

25 5566.

72 M. Mamivand et al. / International Journal of Plasticity 60 (2014) 71–86

applications require rather the tetragonal crystal structure at service operation so any phase transformation shall be pre-vented (Chevalier et al., 2009; Schmauder and Schubert, 1986).

A key to alloy and component design efforts is to understand how this transformation evolves in a boundary value prob-lem, and how internal and external stresses upset the driving forces. Countless examples in the thermal barrier coating (TBC)literature demonstrate where twin–twin interactions and transformation induced stress are associated with fracture initia-tion, and there are a number of micro-mechanism hypotheses to explain these phenomena.

As the T ? M transformation in zirconia is thermoelastic (Reyes-Morel et al., 1988; Wei et al., 1998), shape memory effect(SME), can be expected. In the last decades several experiments have shown SME and pseudoelasticity in zirconia ceramics(Lai et al., 2013; Rauchs et al., 2002; Reyes-Morel and Chen, 1988; Reyes-Morel et al., 1988).

In SME study, several constitutive models were constructed based on experimental observations. These constitutive mod-els are categorized into macro-phenomenological or micromechanical models. Macro-phenomenological models (Arghavaniet al., 2011; Auricchio et al., 1997; Brinson, 1993; Hartl et al., 2010; Lagoudas et al., 2012; Lexcellent et al., 2000; Peng et al.,2012; Reese and Christ, 2008) are constructed only from the macroscopic experimental data, but micromechanical models(e.g. crystal plasticity) (Gall and Sehitoglu, 1999; Manchiraju and Anderson, 2010; Patoor et al., 2006, 1996; Yu et al., 2013,2014) consider the microscopic physical nature of transformation.

Recently, the phase field (PF) method has become a powerful tool for simulating the microstructural evolution in a widevariety of material processes, such as solidification (Anderson et al., 2000; Beckermann et al., 1999; Boettinger et al., 2002;Kobayashi, 1993; Loginova et al., 2001; Provatas et al., 2005; Wheeler et al., 1992; Asle Zaeem et al., 2012), solid–state phasetransformations (Asle Zaeem et al., 2011a; Chen and Khachaturyan, 1991; Li et al., 2001; Onuki, 1989; Wang andKhachaturyan, 1997; Asle Zaeem and Mesarovic, 2011, 2010), precipitate growth and coarsening (Asle Zaeem et al., 2012;Diepers et al., 1999; Fan et al., 1998; Vaithyanathan and Chen, 2002), martensitic phase transformations (Fan and Chen,1995; Levitas and Preston, 2002a; Saxena et al., 1997; Wang and Khachaturyan, 1997) and grain growth (Asle Zaeemet al., 2011b; Chen and Yang, 1994; Krill Iii and Chen, 2002; Steinbach et al., 1996).

Although mesoscale by nature, the PF method has a phenomenological character, which enables an ease in the derivationof governing equations from classical thermodynamics and kinetics principles (Moelans et al., 2008; Asle Zaeem andMesarovic, 2010). Many theoretical studies showed the reliability of this method in predicting morphological and micro-structural evolution in several polycrystalline materials (Boettinger et al., 2002; Chen, 2002; Emmerich, 2003; Steinbach,2009).

For martensitic phase transformation (MPT) which gives rise to shape memory effect there are different phase fieldapproaches based on their order parameters, thermodynamic potentials, model formulation, and numerical methods.Recently, Mamivand et al. (2013b) reviewed and discussed extension of the PF modeling approach to MPT. Generally, thereare three different phase field modeling approaches to study MPT based on Ginzburg–Landau theory (Landau, 1965). In thefirst approach, primary order parameters or phase field variables are considered to be some components of the strain tensor,and consequently, in this approach the free energy density is a polynomial in terms of strain components (Ahluwalia et al.,2004, 2003; Barsch and Krumhansl, 1984; Cui et al., 2007; Dhote et al., 2012; Shchyglo et al., 2012). In the second approach,primary order parameters are related to atomic shuffles, and the free energy is a Landau polynomial in terms of primaryorder parameters plus a linear or quadratic term which couples order parameters and the strain tensor (Artemev et al.,2001, 2000; Jin et al., 2001; Malik et al., 2013a,b; Malik et al., 2012; Man et al., 2011; She et al., 2013; Wang andKhachaturyan, 1997; Yeddu et al., 2013c). The third approach uses the same order parameters as the second approach, how-ever, it couples the strain tensor components to the order parameter(s) through a 2–3–4 or higher order polynomial (Choet al., 2012, p. 20; Idesman et al., 2008; Levitas and Preston, 2002a, 2002b; Levitas, 2013; Levitas et al., 2010).

The approach we use primarily relies on a vision advanced by Khachaturyan, Chen and Wang (Artemev et al., 2000; Chen,2002; Wang and Khachaturyan, 1997). Wang and Khachaturyan (1997) presented the first 3D model for generic cubic totetragonal improper MPT which occurs mostly in ceramics, in a constrained single crystal; their model was able to predictthe major structural characteristics of martensite during the entire transformation including nucleation, growth and even-tually formation of internally twinned plates in thermoelastic equilibrium condition with the parent phase.

Wen et al. (2000, 1999) developed a phase field model to investigate the hexagonal to orthorhombic transformation; theystudied the effect of elastic interactions on the domain formation and evolution during nucleation, growth, and coarsening.They also investigated the effect of an applied strain field on the domain structure development and successfully predictedthe special patterns transformation, such as fan and star shape precipitates. The effect of external stress on MPT was studiedby Artemev et al. (2000); they presented a simulation of generic improper cubic to tetragonal MPT and they showed externalstresses increase the product variants having transformation strain aligned with the applied stresses. External stress alsochanges the morphology of martensite particles. Artemev et al. (2001) developed a phase field model for proper MPT andsimulated two different type of cubic to tetragonal transformation, with and without volumetric change in constrainedand unconstrained systems.

Jin et al. (2001) simulated the cubic to trigonal in AuCd for both single crystal and polycrystalline and studied the effect ofexternal load on transformation and domain boundary movement. Artemev et al. (2002) developed a phase field model todescribe a proper cubic-to-tetragonal MPT in a polycrystalline Fe-31at% Ni alloy under an applied stress. Koyama andOnodera (2003) studied another cubic to tetragonal transformation in Ni2MnGa under external stress and magnetic field.Wang et al. (2004) studied the effect of free surface in multi-variant proper MPT in polycrystals of Au-49.5at%Cd (cubic ?trigonal) and Fe-31wt%Ni (fcc ? bcc) alloys. They showed that martensite microstructures change in the surface layers

M. Mamivand et al. / International Journal of Plasticity 60 (2014) 71–86 73

because of formation of free surfaces, the concentration of martensite is high near free surfaces and free surfaces effect islocal. They also showed that the free surfaces have more effect in microstructures of the high-symmetry FeNi than that ofthe low-symmetry AuCd because the latter has more orientation variants and can achieve better stress accommodation.

Seol et al. (2003, 2002) developed a 3-D phase field model to study kinetics and morphologies of cubic to tetragonalproper MPT in thin film elastically constrained by a substrate. Yamanaka et al. (2008) introduced an elastoplastic phase fieldmodel based on Guo et al. (2005) to investigate cubic to tetragonal transformation, their model confirmed that plasticaccommodation largely reduces the elastic strain energy during the formation of the tetragonal phase because of both self-and plastic accommodations. Man et al. (2011) presented a phase field model to study forward and reverse proper MPT withcapability of treating continuously varying temperature. Recently, Yeddu and Malik have extensively studied the austenite tomartensite transformation in steel. They represent a 3D phase field model to capture the stress, strain and temperatureinduced MPT in steel (Malik et al., 2013a,b, 2012; Yeddu et al., 2013a–c; Yeddu et al., 2012a–c).

The above review shows that substantial research was recently performed to utilize the phase field approach in studyingcubic to tetragonal MPT (Artemev et al., 2002, 2001, 2000; Koyama and Onodera, 2003; Malik et al., 2012; Seol et al., 2003,2002; Wang and Khachaturyan, 1997; Wang et al., 1995; Yamanaka et al., 2008; Yeddu et al., 2013c). However, to the best ofour knowledge, there exists no phase field model for simulating the T ? M transformation in zirconia.

Recently, we reported a PF model for both thermally and stress induced T ? M transformation within a single crystal. Thecorresponding simulation results successfully agreed with classical observations. In fact, the model reproduced well impor-tant characteristics of the transformation, such twinning morphology and transformation toughening in zirconia (Mamivandet al., 2014, 2013a).

In this paper, we extend the above PF model for incorporating the process of T ? M transformation in an elastically aniso-tropic and inhomogeneous polycrystalline zirconia. We are concerned by reliably predicting the main microstructural, kinet-ics, and kinematics features that underlie autocatalytic transformation, shape memory effects, and pseudoelasticity.

2. Phase field model

The phase field method describes a multi domain microstructure through a set of phase field variables. In the case ofT ? M phase transformation, non-conserved phase field variables represent the possible variants of the monoclinic phase.Variants are all possible monoclinic unit cells which are crystalographically self-similar and obey colored symmetry pointgroup operations in a dichromatic complex between the two phases (Hirth et al., 2006). This is simplistically schematizedin Fig. 1.

The temporal and spatial evolutions of the non-conserved phase field variables are described by the phenomenologicaltime dependent Ginzburg–Landau kinetic equation (Landau, 1965):

@gpð r!; tÞ@t

¼ �LdF

dgpð r!; tÞp ¼ 1; . . . ; n; ð1Þ

where gp represent the pth variant of monoclinic, L is the kinetic coefficient, F is the total free energy of system, anddF=dgpð r!; tÞ is the thermodynamic deriving force for spatial and temporal evolution of gp. The value of gp varies from zeroto unity; where the limiting cases gp = 1 and gp = 0 indicate whether the variant pth of the monoclinic phase or either theparent phase or other variants exist respectively.

Fig. 1. Schematic illustration of possible variants of monoclinic phase during T ? M phase transformation in 2D.

74 M. Mamivand et al. / International Journal of Plasticity 60 (2014) 71–86

In the MPT problems, the total free energy can be written as the summation of chemical free energy and elastic strainenergy:

F ¼ Fch þ Fel: ð2Þ

2.1. Chemical free energy

The driving force of MPT, can be written as (Wang and Khachaturyan, 1997):

Fch ¼Z

Vf ðg1;g2; . . . ;gnÞ þ

12

Xn

p¼1

bijðpÞrigprjgp

" #dV n ¼ 1; . . . ;p; ð3Þ

where bij(p) is a positive gradient energy coefficient, r is the gradient operator, and f(g1,g2, . . . ,gn) is the local specific freeenergy density defining the basic bulk thermodynamic properties of the system. f(g1,g2, . . . ,gn) can be approximated by theLandau polynomial in terms of long-range order parameters gp. We selected the simplest sixth-order polynomial form for thelocal specific free energy which works for temperatures above and below equilibrium:

f ðg1;g2; . . . ;gnÞ ¼ DGa2

g21 þ g2

2 þ . . .þ g2n

� �� b

4g4

1 þ g42 þ . . .þ g4

n

� �þ c

6g2

1 þ g22 þ . . .þ g2

n

� �3� �

; ð4Þ

where DG is the chemical driving force representing the difference in the specific chemical free energy between the parentand the equilibrium martensitic phase, and a , b and c are the expansion coefficients.

Wang et al. (2006) calculated equilibrium temperature for T ? M phase transformation in pure zirconia, which they usedto assess Gibbs free energy of zirconia in different phases. According to their work (Wang et al., 2006), the equilibriumtemperature is a temperature at which the Gibbs free energy of both tetragonal and monoclinic phases are the same. Thistemperature for T ? M is 1367� 5 K, and Gibbs free energies for the monoclinic and tetragonal zirconia are:

GMZrO2¼ �1126163:5þ 424:8908T � 69:38751T ln T � 0:0037588T2 þ 683000T�1; ð5Þ

GTZrO2¼ 5468� 4T þ GM

ZrO2; ð6Þ

where Gibbs free energies and temperature (T) are expressed in the J/mole and Kelvin units, respectively.We assume that the positive gradient energy coefficient is isotropic (bij = bdij), therefore the chemical free energy can be

simplified as:

Fch ¼Z

Vf ðg1;g2; . . . ;gnÞ þ

12

Xn

p¼1

bðrigpÞ2

" #dV ; ð7Þ

2.2. Elastic strain energy

The elastic strain energy for a system under the external applied stress would be,

Fel ¼12

ZV

Cijkleelkle

elij dV �

ZVra

ije0ijdV ; ð8Þ

where the elastic strain eelij ð r!Þ is the difference between the total strain, etot

ij ð r!Þ, and the stress free strain, e0ijð r!Þ:

eelij ð~rÞ ¼ etot

ij ð~rÞ � e0ijð~rÞ ¼ etot

ij ð~rÞ �X

p

e00ij ðpÞg2

pð~rÞ: ð9Þ

To calculate the strain energy in the domain of study, the Ginzburg–Landau equations (Eq. (1)) must be coupled tomechanical equilibrium equations to give the displacements in the domain:

@rij

@rj¼ 0: ð10Þ

Following Mamivand et al. (2014), the external stresses were considered explicitly on the computational domain by add-ing them as boundary conditions to the mechanical equilibrium equations (Eq. (10)) (Malik et al., 2013a,b). In this method,we set ra = 0 in Eq. (8) and solve the equilibrium equations (Eq. (10)) with the following boundary condition:

@rij

@rj¼ 0; ta

i ¼ raijnj; ð11Þ

where raij is the applied stress tensor, ta

i is its traction vector and nj is the normal vector to the surface.

M. Mamivand et al. / International Journal of Plasticity 60 (2014) 71–86 75

2.3. Polycrystalline model

To describe the T ? M transformation in polycrystalline, we need to define a global coordinate to transfer all local ten-sorial quantities in different grains. Local coordinate in each grain lies on crystallographic axes of the parent phase (tetrag-onal). Since we treat a 2D problem, the at axis represents the local x-direction and ct axis represents the local y-direction. Toconsider elastic anisotropy, the tensorial quantities which transfer to global coordinate must comprise the transformationstrain (Eigen strain) and elastic stiffness.

eG00ij ¼ RikRjle00

kl ; ð12Þ

CGijkl ¼ RimRjnRkoRlpCmnop; ð13Þ

where the eG00ij is the transformation strain in global coordinate, Rij is the rotation tensor, e00

ij is the local transformation strainfor each grain and CG

ijkl and Cmnop are global and local elastic stiffness tensors, respectively. The 2D rotation tensor for a grainwhich makes an angle h with the global coordinates in a clockwise rotation is,

Rij ¼cosðhÞ sinðhÞ� sinðhÞ cosðhÞ

� �: ð14Þ

Another important physical aspect of MPT in polycrystals is the effect of grain boundaries (GBs). Several researchers haveshown that GBs act as an obstacle against the progress of MPT progress (Deville et al., 2004a; San Martín et al., 2008). Maliket al. (2013a,b) has incorporated a vision on the GB effect in phase field. He defined the GB as a rigid barrier opposing thetransmission of martensitic plates, so any martensitic domain in the next grain is assumed to form by a novel nucleationevent and not by propagation. Although not realistic for many other low symmetry materials (El Kadiri and Oppedal,2010; El Kadiri et al., 2013a, 2013b), we adopted the same idea in our model. Mathematically we assigned the kinetic coef-ficient at grain boundaries to be zero (L = 0), which stops the martensite lath as it reaches to the grain boundaries.

3. Model parameters

In the following, we envisage both thermal and stress induced T ? M phase transformation in a 2D polycrystalline zirco-nia. The domain dimensions are 2 � 2 lm and the initial value of g1 and g2 is zero (tetragonal) except for the single multi-variant monoclinic embryo where g1 = g2 = 0.5 . The initial value of displacement is zero in the whole domain, and theboundary condition for the order parameters is set by:

n � rgi ¼ 0; i ¼ 1; . . . ;p; ð15Þ

where n in the normal direction to the boundaries. A constant homogeneous temperature of T = 1170 K was considered.Three crystallographic correspondences or symmetry operations exist for the T ? M transformation. Each correspon-

dence shows which atom at the parent structure becomes which atom at the product phase. That is, one has ABC, BCAand CAB, indicating that at, bt, ct of the tetragonal phase change into am, bm, cm; bm, cm, am and cm, am, bm, respectively. Sinceat and bt are equal, two variants exist for each correspondence. Variants are crystallographically equivalent, but rotated 90�with respect to each other. For example in correspondence ABC one variant, ABC, emerges when at, bt, ct becomes am, bm, cm

and the other variant, BAC, rises when at, bt, ct becomes bm, am, cm. Inasmuch as the monoclinic axes are not perpendicular,two different orientations exist for each variant, depending which axis (am or cm) is parallel with their tetragonal counterpartaxis. For example the variant ABC-OR1 shows am parallel to at and ABC-OR2 shows cm parallel to ct. The more interestedreader may refer to (Deville et al., 2004b; Kelly and Francis Rose, 2002; Mamivand et al., 2013a) for more details on the crys-tallography of T ? M transformation.

The smallest transformation strains in zirconia for 2D T ? M belong to the correspondence ABC(ð001Þm � jjð001Þt ; ½100�m � jj½100�t), and are according to Mamivand et al. (2013b):

e00ij ð1Þ ¼

0:0049 0:07610:0761 0:0180

� �; e00

ij ð2Þ ¼0:0049 �0:0761�0:0761 0:0180

� �:

In 2D, the transformation strains for both ABC-OR1 and ABC-OR2 variants are identical, so the chance for having any ofthem is equal.

We considered inhomogeneous elasticity and defined a smooth transition from tetragonal to monoclinic elastic constantsthrough the following equation:

Cijkl ¼ PXn

i¼1

gi

!CM

ijkl þ 1� PXn

i¼1

gi

! !CT

ijkl; ð16Þ

where CMijkl and CT

ijkl are monoclinic and tetragonal elastic constants, respectively, n is the number of order parameters, and:

PðgÞ ¼ g3ð6g2 � 15gþ 10Þ: ð17Þ

Table 1Elastic constants for monoclinic zirconia (Gpa) (Chan et al., 1991; Zhao et al., 2011).

C11 C22 C33 C44 C55 C66 C12 C13 C16 C23 C26 C36 C45

361 408 258 100 81 126 142 55 �21 196 31 �18 �23

Table 2Elastic constants for tetragonal zirconia (Gpa) (Kisi and Howard, 1998; Zhao et al., 2011).

C11 C33 C44 C66 C12 C13

327 264 59 64 100 62

Table 3Numerical values used for calculation.

Temperature (K) 1170Chemical driving force (J mol�1) 788Gradient energy coefficient, b (J m�1) 1 � 10�8

Energy density coefficient, a 0.14Energy density coefficient, b 12.56Energy density coefficient, c 12.42Kinetic coefficient, L (m3 J�1 s�1) 2Domain size (nm � nm) 2000 � 2000

76 M. Mamivand et al. / International Journal of Plasticity 60 (2014) 71–86

We used COMSOL Multiphysics to solve the PDEs (COMSOL Multiphysics Users’ Guide, 2012). The input parameters of themodel are given in Tables 1–3. Chemical driving force was derived from Eqs. (5) and (6), gradient energy coefficient wasselected in a way to give a reasonable interface thickness (few nanometers) (Asle Zaeem et al., 2011a). a, b, and c are selectedsomehow to provide global minima at the parent phase and at all the product variants (Mamivand et al., 2013a). L, theGinzburg–Landau kinetic coefficient, was arbitrary selected since there is no report on the speed of the phase transformationin zirconia.

4. Simulation results and discussions

In order to study the effect of crystal lattice orientation with respect to the global coordinate on T ? M transformation,we first focus on the evolution of monoclinic in three single crystals with different lattice orientations. Fig. 2 shows thegeometry and dimensions of the single crystals model. The initial condition on the order parameter is a multi-variant singleembryo at the middle of crystal. For the displacement boundary condition, we assume the crystal embedded in an un-transformable tetragonal zirconia matrix.

The domain was discretized by four-noded quadratic rectangular elements with element size of 2 nm; for time integra-tion, an adaptive time-step algorithm was implemented. During early stages of the growth, time steps less than 1 � 10�10 swere used to guarantee the convergence of the solution, but at later stages of growth, time steps less than 6 � 10�10 s weresufficient.

Fig. 3 shows the final microstructural pattern of three single crystals with different lattice orientations. In 2D case, themonoclinic variants can happen in two ways: ABC-OR1 and ABC-OR2. In ABC-OR1 case, am remains parallel to at, and inABC-OR2 case, cm remains parallel to ct. ABC-OR1 and ABC-OR2 have the same transformation strain since they are relatedby a rigid body rotation. Therefore they have a same chance of formation in T ? M transformation. The twining planes forthese variants are 90� different and their formation depends on grain shape, nucleation site, external loadings, and boundaryconditions. System selects between ABC-OR1 and ABC-OR2 based on maximum accommodation and minimum strainenergy. The simulation results show that the h = 0� case leads to variant ABC-OR2 (with twinning plane of (100)m), whilefor h = 15� and h = 80�, the ABC-OR1 variant (with twinning plane of (001)m) takes place. When h = 0�, small tetragonalremains are visible next to the upper and lower boundary. However, for h = 15� and h = 80�, the entire domain transformsto the monoclinic phase.

The existence of (100)m and (001)m twining planes was confirmed in experimental observations of Bansal and Heuer(1974, 1972) and Buljan et al. (1976) respectively. Simha (1997) calculations also derived similar conclusions.

4.1. T ? M transformation in polycrystalline

We constructed a 2D tetragonal zirconia polycrystal (TZP) with 15 grains with average grain size of 600 nm. The domainsize was 2 � 2 lm, which we assumed constrained by an un-transformable tetragonal phase on the bottom, left, and

2μm

2μm

Multi-variant single embryo

80x80nm2

ct atθ

Fig. 2. Tetragonal single crystal with initial multi variant monoclinic embryo, embedded in un-transformable matrix.

o0=θ o15=θ o80=θVari. 1 Vari. 1 Vari. 1Vari. 2 Vari. 2 Vari. 2Tetra. Tetra. Tetra.

Fig. 3. Final microstructural patterns of T ? M transformation of embedded single crystals with different lattice rotation. (Plots represent g1 + 2g2.)

Fig. 4. The geometry and texture of polycrystal model, constrained by un-transformable tetragonal on three side and free from top.

M. Mamivand et al. / International Journal of Plasticity 60 (2014) 71–86 77

right -hand sides, but free for transformation on the top (Fig. 4). Different researchers (Deville et al., 2005; Gall et al., 2000;Kajiwara, 1986; Otsuka and Ren, 2005) independently showed that the GBs and other similar defects are favorable places for

σ11(Pa) σ11-σ22(Pa)Vari. 1 Vari. 2 Tetra.

(1)

(2)

(3)

(4)

(5)

Fig. 5. The temporal and spatial evolution of monoclinic embryo in tetragonal zirconia polycrystal. The left column shows the monoclinic variants evolution(plots represent g1 + 2g2). The second and third columns show the evolution of r11 and r11–r22. Rows (1)–(5) correspond to times 0 s, 2.4e�7 s, 2.8e�7 s,3.7e�7 s, and 5e�6 s, respectively.

78 M. Mamivand et al. / International Journal of Plasticity 60 (2014) 71–86

Fig. 6. Transformation of monoclinic variant from ABC-OR2 (a) to ABC-OR1 (c) due to internal stresses (b) developed by transformation strains inneighboring grains (deformation scale 5X).

Fig. 7. Domain geometry and its mechanical boundary and loading condition to study the stress–strain curve.

M. Mamivand et al. / International Journal of Plasticity 60 (2014) 71–86 79

martensitic nucleation. These conclusions have motivated us to put the multi-variant embryo at GBs. The polycrystal texturewas fairly random.

The initial condition for order parameters in the grain containing the seed is zero except at seed embryo whereg1 = g2 = 0.5. For the other grains, because the GBs constitute impenetrable barriers, we needed to assign a very small per-turbation (Normal distribution with mean = 0.01 and standard deviation = 0.002) to the order parameters for enabling phase

80 M. Mamivand et al. / International Journal of Plasticity 60 (2014) 71–86

transformation under any suitable local external stress imposed by interactions with neighboring grains. The initial condi-tion for displacement is zero in the whole domain, and boundary conditions for the ith order parameter is;

n � rgi ¼ 0; i ¼ 1; . . . ;p: ð18Þ

The boundary conditions for the mechanical equilibrium equation are defined in a fashion to satisfy the physical con-straints inherent to the geometry of Fig. 4. The input parameters are identical to those listed in Tables 1–3.

The temporal and special evolution of monoclinic variants (g1, g2), the stresses r11 and r11–r22 are shown in Fig. 5. Inpolycrystals which can experience the MPT, when the transforming region in one grain reaches the neighboring grains, localstress substantially deviating from the macroscopic stress arise. These local stresses may trigger stress-induced MPT in othergrains. This phenomenon is known as autocatalytic transformation and has been reported experimentally in TZP (Kelly andFrancis Rose, 2002; Reyes-Morel and Chen, 1988). Results reported in Fig. 5 demonstrate how the autocatalytic transforma-tion facilitates the MPT in tetragonal zirconia polycrystal.

At the first stages of MPT, the embryo grows in the parent grain, and so for the corresponding displacement field (Fig. 5).The displacement field reaches a critical magnitude for triggering a new the T ? M transformation in the neighboring grains.This nucleation cascade is well captured by the model. Row 2 shows the preference MPT nucleation at low angle boundariesin conformance with recent reports on deformation twinning nucleation in magnesium (El Kadiri et al., 2013a). BothGrain-35� and Grain-15� are in touch with parent grain (Grain-68�), but the T ? M was triggered first at Grain-35� sinceit forms a lower angle boundary with the parent grain. Row 3 shows the process of autocatalytic transformation. Whenthe monoclinic lath at Grain-35� reached the Grain-66�, the corresponding displacement field provided a sufficient externalstress level for triggering T ? M transformation in the Grain-66�. This process of autocatalytic transformation was respon-sible for the invasion of T ? M transformation in the whole simulated domain. Any monoclinic lath which reaches neighbor-ing tetragonal grain worked in triggering T ? M transformation in it. In row (4) it looks like the variants are crossing thegrain boundary, for e.g. Grain-66 and Grain-59, but it needs to be mentioned that the reason, it may looks like grain boundarypenetration is the autocatalysis transformation and low angle boundary between the grains (9�).

In Fig. 5, comparison between microstructures of Grain-35� (Fig. 4) at stages 4 and 5 shows how transformation strainsdeveloped in neighboring grains can change the monoclinic from one variant to another (Fig. 6). In Fig. 6(a), Grain-35� hasABC-OR2 with (100)m twining plane. As the neighboring grains transform to monoclinic, their deformations impose an inter-nal loading to Grain-35� (Fig. 6(b)). Since the existing twin cannot accommodate the imposed loading, it transforms to ABC-OR1 (Fig. 6(c)) which can easily accommodate it. Fig. 6 shows the process of variant reorientation due to developing internalstresses.

4.2. Stress–strain response of the TZP

In this section we study the stress–strain curve of tetragonal polycrystalline zirconia. As we did not consider the plasticstrain in our formulation (Eq. (9)) this model did not consider the transformation plasticity into account. Fig. 7 shows the

Fig. 8. Stress–strain curve for TZP under uniaxial tension and its corresponding microstructure evolution and deformed shapes.

M. Mamivand et al. / International Journal of Plasticity 60 (2014) 71–86 81

model we used for studying the stress–strain curve in TZP. The whole domain (ABLK) is 10 � 2 lm, and the size of the poly-crystal section (EFGH) is 2 � 2 lm.

The ABFE and GHLK are two un-transformable tetragonal domains that have been added to reduce the local effects ofboundary conditions. The texture of polycrystal domain (EFGH) was similar to Fig. 4. The lower horizontal edge (AB) wasfixed in y-direction, while the four short vertical edges on the top and bottom (AC, BD, IK, JL) were constrained in x-direction.A linear increasing external stress has been applied to the upper most horizontal edge (KL). The initial condition for orderparameters in all grains corresponded to a random normal distribution with mean = 0.01 and standard deviation = 0.002.The Ginzburg–Landau equations (Eq. (1)) were only assigned to the polycrystal section, but the equilibrium equation (Eq.(11)) governs the whole domain. The input parameters were identical to the previous simulations, except for the chemicaldriving force which was equal to 388 J/mole corresponding to T = 1270 K.

Fig. 8 shows the stress–strain curve for the TZP and corresponding domain microstructure at different loading conditions.Since the chemical driving force is not adequate, external mechanical energy is required to trigger the T ? M transformationas well. Three noticeable regimes exist in this curve. In the first regime, the material behaves elastically, and the microstruc-ture consists of a stable, homogeneous tetragonal. As stress increases, the tetragonal phase loses its stability and ultimately,

Fig. 9. Stress–strain curve for TZP under uniaxial tension at 1270 K with temperature–strain curve showing shape recovery on heating, correspondingmicrostructure evolution cycle and deformed shapes.

82 M. Mamivand et al. / International Journal of Plasticity 60 (2014) 71–86

the initiation of monoclinic phase at some grains occurs. During the second regime, the material shows a flat regime withconstant stress. At this stage the martensitic transformation propagates over an extended region without interruption(burst-like transformation). Indeed, this is the autocatalysis transformation which leads to a flat regime behavior of TZPs.Autocatalysis transformation terminates when the transformable matrix is partitioned into isolated pockets. The last stagecorresponds to a strain hardening regime with an increasing flow stress. This stage begins as autocatalysis terminates. Forthis regime, the microstructure is mainly monoclinic, except for some small portions of the tetragonal phase, which shrink asexternal load increases.

4.3. Shape memory effects

Since the T ? M transformation in zirconia is thermoelastic (Reyes-Morel et al., 1988; Wei et al., 1998), shape recovery, oralternatively termed shape memory effect (SME), can be expected. The shape memory effect in zirconia ceramics has beenreported in several experiments (Reyes-Morel et al., 1988; Swain, 1986; Zhang et al., 2002). Though zirconia ceramics arebrittle in comparison to other shape memory alloys, their chemical inertness, high strength and high operational tempera-ture make them an attractive shape memory ceramic (Jin, 2005). T ? M transformation occurs at characteristic tempera-tures, let say Ms (martensite start temperature). however, it can also be promoted by stress effects at temperatureshigher than Ms. In this case, if the specimen is reheated up to As (austenite start temperature), the M ? T soon occurs,and transformation strain can be recovered.

The same model described in preceding section (Fig. 7) was used to study the SME in TZP. The loading cycle started atconstant temperature (1270 K) with a linear increasing stress from zero to 45 Mpa in 2.1 ls, then it was kept constant for0.2 ls and again decreased to zero in 2.1 ls. After stress cycle, we increased the temperature from 1270 K to 1370 K, andthen cool it down again to 1270 K.

Fig. 9 illustrates the SME in TZP and the evolution of microstructure during the loading cycle. Reyes-Morel et al. (1988)also reported a similar SME cycle in their experimental studies on CeO2-srabilized tetragonal zirconia polycrystals (Ce-TZP).At the beginning, the chemical driving force is low, as tension stress increases the accumulation of chemical and mechanicalenergies reaches the critical value for triggering the T ? M (yield point in Fig. 9). Then, the specimen experiences a burst-like

Fig. 10. Stress–strain curve for TZP under uniaxial tension at 1380 K showing shape recovery on unloading (pseudoelasticity) and correspondingmicrostructure evolution.

M. Mamivand et al. / International Journal of Plasticity 60 (2014) 71–86 83

transformation due to the autocatalysis transformation. After complete unloading, 1.55% residual transformation strainremained in the material. Subsequent heating produced a gradual recovery of the axial strain due to the M ? T reversion.During the recovery phase, the material experienced another burst-like transformation which occurred when the tempera-ture increased to sufficient levels for enabling the formation of a stable tetragonal phase. The burst transformation in M ? Tis smaller than burst in T ? M, which is typical and has been reported in experiments as well (Reyes-Morel and Chen, 1988;Reyes-Morel et al., 1988).

4.4. Pseudoelasticity

Pseudoelasticity or shape recovery during unloading has been reported experimentally in zirconia based ceramics (Laiet al., 2013; Reyes-Morel et al., 1988). Pseudoelasticity happens at the temperatures higher than the equilibrium tempera-ture at which tetragonal is the stable phase. At high temperatures the direct and reverse phase transformations occur atstresses of the same sign, and transformation strains vanish at zero external stresses. At low temperatures, as we discussedin the prior section, there was residual transformation strain at zero external stresses, therefore a heat treatment was nec-essary to reverse the phase transformation, that behavior was referred as shape memory effect (pseudoplasticity).

The represented potential in Eq. (4) with the given parameters in Table 3 is suitable for temperatures lower than the equi-librium temperature; for temperatures higher than equilibrium temperature, we need to change a, b and c so that the localspecific free energy produces tetragonal as the stable phase and monoclinic as the metastable phase. a, b and c are the expan-sion coefficients at a fixed temperature, and although they are not dominant in microstructure evolution, they must beselected in a way that: (1) maintain the same value of the interfacial energy within the physical reasonable range, and(2) provide global minima at the parent phase (g1 = g2 = � � � = gn = 0) and at all the product variants. For pseudoelasticitymodel we assign a, b and c to be 40, 148 and 108, respectively. Fig. 10 demonstrates the mechanical behavior and microstruc-tural evolution of tetragonal zirconia polycrystal for a temperature higher than the equilibrium temperature (T = 1380 K).

Noticeable features in pseudoelasticity behavior of TZP are: (1) in this case there is no flat regime in stress–strain curve,instead, a stable regime of gradual strain hardening was found, (2) deformation proceeds even after unloading since phasetransformation continue at the first stages of unloading, and (3) in reverse strain section elastic modulus gradually decreasesbecause of reverse transformation until last portion of pseudoelastic loop which is elastic again with initial elastic modulus.Similar mechanical behavior was reported in experimental studies of Reyes-Morel et al. (1988).

5. Conclusion

A reliable phase field model was developed to study the tetragonal to monoclinic transformation in polycrystalline zir-conia, which incorporates inhomogeneous and anisotropic elasticity. The model was constructed to be both temperatureand stress sensitive, in an effort to capture the main physical and mechanical characteristics of autocatalytic transformation,shape memory effect and pseudoelasticity occurring in tetragonal polycrystals.

Autocatalysis transformation led to burst-like transformation and enhanced propagation of the T ? M transformationover an extended region without interruption. A homogeneous, metastable matrix and a constant stress or undercoolingwas required to trigger the burst. Burst-like transformation stems from internal stresses and finishes up when the tetragonaldomain become partitioned into isolated pockets. The small remained tetragonal regions shrunk as external stress increased.

Tetragonal zirconia polycrystal stress–strain curve showed a flat deformation regime because of autocatalysis transfor-mation. This regime terminated after burst-like transformation stopped, followed by a strain-hardening regime.

Shape memory effect in zirconia ceramics came on from the thermoelastic nature of reversible T ? M transformation.This property made the recovery of transformation strain possible under a thermal cycle, when the specimen is heated toabove As and cooled down to previous temperature.

Pseudoelasticity happens at higher temperatures where tetragonal is stable phase. At these temperatures martensitictransformation happens just because of external loadings. Due to stability of tetragonal, the forward and reverse transfor-mations occur at stresses of the same sign, and there is a complete shape recovery at zero stress.

The model further enabled monitoring the microstructure evolution at any loading condition, clearly showing howtwinned domains can be detwinned after heat treatment and unloading.

Acknowledgement

The authors appreciate the sponsorship of the Institute for Nuclear Energy Science and Technology Laboratory DirectedResearch and Development (INEST LDRD) and the Center for Advanced Vehicular Systems at Mississippi State University.

References

Ahluwalia, R., Lookman, T., Saxena, A., 2003. Elastic deformation of polycrystals. Phys. Rev. Lett. 91, 55501.Ahluwalia, R., Lookman, T., Saxena, A., Albers, R.C., 2004. Landau theory for shape memory polycrystals. Acta Mater. 52, 209–218.Anderson, D.M., McFadden, G.B., Wheeler, A.A., 2000. A phase-field model of solidification with convection. Phys. Nonlinear Phenom. 135, 175–194.Arghavani, J., Auricchio, F., Naghdabadi, R., 2011. A finite strain kinematic hardening constitutive model based on Hencky strain: general framework,

solution algorithm and application to shape memory alloys. Int. J. Plast. 27, 940–961.

84 M. Mamivand et al. / International Journal of Plasticity 60 (2014) 71–86

Artemev, A., Wang, Y., Khachaturyan, A., 2000. Three-dimensional phase field model and simulation of martensitic transformation in multilayer systemsunder applied stresses. Acta Mater. 48, 2503–2518.

Artemev, A., Jin, Y., Khachaturyan, A., 2001. Three-dimensional phase field model of proper martensitic transformation. Acta Mater. 49, 1165–1177.Artemev, A., Jin, Y., Khachaturyan, A., 2002. Three-dimensional phase field model and simulation of cubic ? tetragonal martensitic transformation in

polycrystals. Philos. Mag. A 82, 1249–1270.Asle Zaeem, M., Mesarovic, S.D., 2010. Finite element method for conserved phase fields: stress-mediated diffusional phase transformation. J. Comput. Phys.

229, 9135–9149.Asle Zaeem, M., Mesarovic, S.D., 2011. Morphological instabilities in thin films: evolution maps. Comput. Mater. Sci. 50, 1030–1036.Asle Zaeem, M., El Kadiri, H., Mesarovic, S.D., Horstemeyer, M.F., Wang, P.T., 2011a. Effect of the compositional strain on the diffusive interface thickness and

on the phase transformation in a phase-field model for binary alloys. J. Phase Equilib. Diffus. 32, 302–308.Asle Zaeem, M., El Kadiri, H., Wang, P.T., Horstemeyer, M.F., 2011b. Investigating the effects of grain boundary energy anisotropy and second-phase particles

on grain growth using a phase-field model. Comput. Mater. Sci. 50, 2488–2492.Asle Zaeem, M., Yin, H., Felicelli, S.D., 2012. Comparison of cellular automaton and phase field models to simulate dendrite growth in hexagonal crystals. J.

Mater. Sci. Technol. 28, 137–146.Asle Zaeem, M., El Kadiri, H., Horstemeyer, M.F., Khafizov, M., Utegulov, Z., 2012. Effects of internal stresses and intermediate phases on the coarsening of

coherent precipitates: a phase-field study. Curr. Appl. Phys. 12, 570–580.Auricchio, F., Taylor, R.L., Lubliner, J., 1997. Shape-memory alloys: macromodelling and numerical simulations of the superelastic behavior. Comput.

Methods Appl. Mech. Eng. 146, 281–312.Bansal, G.K., Heuer, A.H., 1972. On a martensitic phase transformation in zirconia (ZrO2)–I. Metallographic evidence. Acta Metall. 20, 1281–1289.Bansal, G.K., Heuer, A.H., 1974. On a martensitic phase transformation in zirconia (ZrO2)–II. Crystallographic aspects. Acta Metall. 22, 409–417.Barsch, G.R., Krumhansl, J.A., 1984. Twin boundaries in ferroelastic media without interface dislocations. Phys. Rev. Lett. 53, 1069–1072.Beckermann, C., Diepers, H.J., Steinbach, I., Karma, A., Tong, X., 1999. Modeling melt convection in phase-field simulations of solidification. J. Comput. Phys.

154, 468–496.Boettinger, W.J., Warren, J.A., Beckermann, C., Karma, A., 2002. Phase-field simulation of solidification. Annu. Rev. Mater. Res. 32, 163–194.Brinson, L.C., 1993. One-dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non-constant material functions

and redefined martensite internal variable. J. Intell. Mater. Syst. Struct. 4, 229–242.Buljan, S.T., McKinstry, H.A., Stubican, V.S., 1976. Optical and X-ray single crystal studies of the monoclinic � tetragonal transition in ZrO2. J. Am. Ceram.

Soc. 59, 351–354.Chan, S.K., Fang, Y., Grimsditch, M., Li, Z., Nevitt, M.V., Robertson, W.M., Zouboulis, E.S., 1991. Temperature dependence of the elastic moduli of monoclinic

zirconia. J. Am. Ceram. Soc. 74, 1742–1744.Chen, L.Q., 2002. Phase-field models for microstructure evolution. Annu. Rev. Mater. Res. 32, 113–140.Chen, L.Q., Khachaturyan, A., 1991. Computer simulation of structural transformations during precipitation of an ordered intermetallic phase. Acta Metall.

Mater. 39, 2533–2551.Chen, L.Q., Yang, W., 1994. Computer simulation of the domain dynamics of a quenched system with a large number of nonconserved order parameters: the

grain-growth kinetics. Phys. Rev. B 50, 15752.Chevalier, J., Gremillard, L., Virkar, A.V., Clarke, D.R., 2009. The tetragonal-monoclinic transformation in zirconia: lessons learned and future trends. J. Am.

Ceram. Soc. 92, 1901–1920.Cho, J.Y., Idesman, A.V., Levitas, V.I., Park, T., 2012. Finite element simulations of dynamics of multivariant martensitic phase transitions based on Ginzburg–

Landau theory. Int. J. Solids Struct. 49 (14), 1973–1992.COMSOL Multiphysics Users’ Guide, 2012. COMSOL Inc.Cui, Y.W., Koyama, T., Ohnuma, I., Oikawa, K., Kainuma, R., Ishida, K., 2007. Simulation of hexagonal–orthorhombic phase transformation in polycrystals.

Acta Mater. 55, 233–241.Deville, S., Guénin, G., Chevalier, J., 2004a. Martensitic transformation in zirconia: part II. Martensite growth. Acta Mater. 52, 5709–5721.Deville, S., Guénin, G., Chevalier, J., 2004b. Martensitic transformation in zirconia: part I. Nanometer scale prediction and measurement of transformation

induced relief. Acta Mater. 52, 5697–5707.Deville, S., Chevalier, J., El Attaoui, H., 2005. Atomic force microscopy study and qualitative analysis of martensite relief in zirconia. J. Am. Ceram. Soc. 88,

1261–1267.Dhote, R.P., Melnik, R.V.N., Zu, J., 2012. Dynamic thermo-mechanical coupling and size effects in finite shape memory alloy nanostructures. Comput. Mater.

Sci. 63, 105–117.Diepers, H.-J., Beckermann, C., Steinbach, I., 1999. Simulation of convection and ripening in a binary alloy mush using the phase-field method. Acta Mater.

47, 3663–3678.El Kadiri, H., Oppedal, A.L., 2010. A crystal plasticity theory for latent hardening by glide twinning through dislocation transmutation and twin

accommodation effects. J. Mech. Phys. Solids 58, 613–624.El Kadiri, H., Baird, J.C., Kapil, J., Oppedal, A.L., Cherkaoui, M., Vogel, S.C., 2013a. Flow asymmetry and nucleation stresses of twinning and non-basal slip in

magnesium. Int. J. Plast. 44, 111–120.El Kadiri, H., Kapil, J., Oppedal, A.L., Hector Jr., L.G., Agnew, S.R., Cherkaoui, M., Vogel, S.C., 2013b. The effect of twin–twin interactions on the nucleation and

propagation of twinning in magnesium. Acta Mater. 61, 3549–3563.Emmerich, H., 2003. The diffuse interface approach in materials science: thermodynamic concepts and applications of phase-field models, first ed. Springer.Fan, D., Chen, L.Q., 1995. Computer simulation of twin formation during the displacive c ? t0 phase transformation in the Zirconia–Yttria system. J. Am.

Ceram. Soc. 78, 769–773.Fan, D., Chen, L.Q., Chen, S.P., 1998. Numerical simulation of Zener pinning with growing second-phase particles. J. Am. Ceram. Soc. 81, 526–532.Gall, K., Sehitoglu, H., 1999. The role of texture in tension–compression asymmetry in polycrystalline NiTi. Int. J. Plast. 15, 69–92.Gall, K., Lim, T.J., McDowell, D.L., Sehitoglu, H., Chumlyakov, Y.I., 2000. The role of intergranular constraint on the stress-induced martensitic transformation

in textured polycrystalline NiTi. Int. J. Plast. 16, 1189–1214.Guo, X.H., Shi, S.Q., Ma, X.Q., 2005. Elastoplastic phase field model for microstructure evolution. Appl. Phys. Lett. 87, 221910.Hannink, R.H., Kelly, P.M., Muddle, B.C., 2000. Transformation toughening in zirconia-containing ceramics. J. Am. Ceram. Soc. 83, 461–487.Hartl, D.J., Chatzigeorgiou, G., Lagoudas, D.C., 2010. Three-dimensional modeling and numerical analysis of rate-dependent irrecoverable deformation in

shape memory alloys. Int. J. Plast. 26, 1485–1507.Hirth, J.P., Pond, R.C., 2011. Compatibility and accommodation in displacive phase transformations. Prog. Mater. Sci. 56, 586–636.Hirth, J.P., Pond, R.C., Lothe, J., 2006. Disconnections in tilt walls. Acta Mater. 54, 4237–4245.Idesman, A.V., Cho, J.Y., Levitas, V.I., 2008. Finite element modeling of dynamics of martensitic phase transitions. Appl. Phys. Lett. 93, 043102.Jin, X.-J., 2005. Martensitic transformation in zirconia containing ceramics and its applications. Curr. Opin. Solid State Mater. Sci. 9, 313–318.Jin, Y., Artemev, A., Khachaturyan, A., 2001. Three-dimensional phase field model of low-symmetry martensitic transformation in polycrystal: simulation of

f’2 martensite in AuCd alloys. Acta Mater. 49, 2309–2320.Kajiwara, S., 1986. Roles of dislocations and grain boundaries in martensite nucleation. Metall. Mater. Trans. A 17, 1693–1702.Kelly, P.M., Francis Rose, L., 2002. The martensitic transformation in ceramics–its role in transformation toughening. Prog. Mater. Sci. 47, 463–557.Kisi, E.H., Howard, C.J., 1998. Elastic constants of tetragonal zirconia measured by a new powder diffraction technique. J. Am. Ceram. Soc. 81, 1682–1684.Kobayashi, R., 1993. Modeling and numerical simulations of dendritic crystal growth. Phys. Nonlinear Phenom. 63, 410–423.

M. Mamivand et al. / International Journal of Plasticity 60 (2014) 71–86 85

Koyama, T., Onodera, H., 2003. Phase-field simulation of microstructure changes in Ni2MnGa ferromagnetic alloy under external stress and magnetic fields.Mater. Trans. 44, 2503–2508.

Krill Iii, C.E., Chen, L.Q., 2002. Computer simulation of 3-D grain growth using a phase-field model. Acta Mater. 50, 3059–3075.Lagoudas, D., Hartl, D., Chemisky, Y., Machado, L., Popov, P., 2012. Constitutive model for the numerical analysis of phase transformation in polycrystalline

shape memory alloys. Int. J. Plast. 32, 155–183.Lai, A., Du, Z., Gan, C.L., Schuh, C.A., 2013. Shape memory and superelastic ceramics at small scales. Science 341, 1505–1508.Landau, L.D., 1965. Collected papers of L.D. Landau. Pergamon Press.Levitas, V.I., 2013. Phase-field theory for martensitic phase transformations at large strains. Int. J. Plast. 49, 85–118.Levitas, V.I., Preston, D.L., 2002a. Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. I.

Austenite M martensite. Phys. Rev. B 66, 134206.Levitas, V.I., Preston, D.L., 2002b. Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. II. Multivariant

phase transformations and stress space analysis. Phys. Rev. B 66, 134207.Levitas, V.I., Lee, D.W., Preston, D.L., 2010. Interface propagation and microstructure evolution in phase field models of stress-induced martensitic phase

transformations. Int. J. Plast. 26, 395–422.Lexcellent, C., Leclercq, S., Gabry, B., Bourbon, G., 2000. The two way shape memory effect of shape memory alloys: an experimental study and a

phenomenological model. Int. J. Plast. 16, 1155–1168.Li, Y.L., Hu, S.Y., Liu, Z.K., Chen, L.Q., 2001. Phase-field model of domain structures in ferroelectric thin films. Appl. Phys. Lett. 78, 3878–3880.Loginova, I., Amberg, G., Ågren, J., 2001. Phase-field simulations of non-isothermal binary alloy solidification. Acta Mater. 49, 573–581.Malik, A., Yeddu, H.K., Amberg, G., Borgenstam, A., Ågren, J., 2012. Three dimensional elasto-plastic phase field simulation of martensitic transformation in

polycrystal. Mater. Sci. Eng. A 556, 221–232.Malik, Amer, Amberg, G., Borgenstam, A., Ågren, J., 2013a. Effect of external loading on the martensitic transformation – a phase field study. Acta Mater. 61,

7868–7880.Malik, A., Amberg, G., Borgenstam, A., Ågren, J., 2013b. Phase-field modelling of martensitic transformation: the effects of grain and twin boundaries. Model.

Simul. Mater. Sci. Eng. 21, 085003.Mamivand, M., Asle Zaeem, M., El Kadiri, H., 2014. Phase field modeling of stress induced T–M transformation in zirconia and its effect on transformation

toughening. Acta Mater. 64, 208–219.Mamivand, M., Asle Zaeem, M., El Kadiri, H., Chen, L.-Q., 2013a. Phase field modeling of the tetragonal-to-monoclinic phase transformation in zirconia. Acta

Mater. 61, 5223–5235.Mamivand, M., Asle Zaeem, M., El Kadiri, H., 2013b. A review on phase field modeling of martensitic phase transformation. Comput. Mater. Sci. 77, 304–311.Man, J., Zhang, J., Rong, Y., Zhou, N., 2011. Study of thermoelastic martensitic transformations using a phase-field model. Metall. Mater. Trans. A 42, 1154–

1164.Manchiraju, S., Anderson, P.M., 2010. Coupling between martensitic phase transformations and plasticity: a microstructure-based finite element model. Int.

J. Plast. 26, 1508–1526.Moelans, N., Blanpain, B., Wollants, P., 2008. An introduction to phase-field modeling of microstructure evolution. Calphad 32, 268–294.Onuki, A., 1989. Ginzburg–Landau approach to elastic effects in the phase separation of solids. J. Phys. Soc. Jpn. 58, 3065–3068.Otsuka, K., Ren, X., 2005. Physical metallurgy of Ti–Ni-based shape memory alloys. Prog. Mater. Sci. 50, 511–678.Patoor, E., Eberhardt, A., Berveiller, M., 1996. Micromechanical modelling of superelasticity in shape memory alloys. J. Phys. IV 6, C1-277.Patoor, E., Lagoudas, D.C., Entchev, P.B., Brinson, L.C., Gao, X., 2006. Shape memory alloys, part I: general properties and modeling of single crystals. Mech.

Mater. 38, 391–429.Peng, X., Chen, B., Chen, X., Wang, J., Wang, H., 2012. A constitutive model for transformation, reorientation and plastic deformation of shape memory alloys.

Acta Mech. Solida Sin. 25, 285–298.Provatas, N., Greenwood, M., Athreya, B., Goldenfeld, N., Dantzig, J., 2005. Multiscale modeling of solidification: phase-field methods to adaptive mesh

refinement. Int. J. Mod. Phys. B 19, 4525–4566.Rashad, M.M., Baioumy, H.M., 2008. Effect of thermal treatment on the crystal structure and morphology of zirconia nanopowders produced by three

different routes. J. Mater. Process. Technol. 195, 178–185.Rauchs, G., Fett, T., Munz, D., Oberacker, R., 2002. Tetragonal-to-monoclinic phase transformation in CeO2-stabilized zirconia under multiaxial loading. J.

Eur. Ceram. Soc. 22, 841–849.Reese, S., Christ, D., 2008. Finite deformation pseudo-elasticity of shape memory alloys–constitutive modelling and finite element implementation. Int. J.

Plast. 24, 455–482.Reyes-Morel, P.E., Chen, I.-W., 1988. Transformation plasticity of CeO2-stabilized tetragonal zirconia polycrystals: I, stress assistance and autocatalysis. J.

Am. Ceram. Soc. 71, 343–353.Reyes-Morel, P.E., Cherng, J.-S., Chen, I.-W., 1988. Transformation plasticity of CeO2-stabilized tetragonal zirconia polycrystals: II, pseudoelasticity and

shape memory effect. J. Am. Ceram. Soc. 71, 648–657.San Martín, D., Aarts, K.W.P., Rivera-Díaz-del-Castillo, P.E.J., van Dijk, N.H., Brück, E., van der Zwaag, S., 2008. Isothermal martensitic transformation in a

12Cr–9Ni–4Mo–2Cu stainless steel in applied magnetic fields. J. Magn. Magn. Mater. 320, 1722–1728.Saxena, A., Wu, Y., Lookman, T., Shenoy, S.R., Bishop, A.R., 1997. Hierarchical pattern formation in elastic materials. Phys. Stat. Mech. Appl. 239, 18–34.Schmauder, S., Schubert, H., 1986. Significance of internal stresses for the martensitic transformation in yttria-stabilized tetragonal zirconia polycrystals

during degradation. J. Am. Ceram. Soc. 69, 534–540.Seol, D.J., Hu, S.Y., Li, Y.L., Chen, L.Q., Oh, K.H., 2002. Computer simulation of martensitic transformation in constrained films. Mater. Sci. Forum 408–412,

1645–1650.Seol, D., Hu, S., Li, Y., Chen, L., Oh, K., 2003. Cubic to tetragonal martensitic transformation in a thin film elastically constrained by a substrate. Met. Mater.

Int. 9, 221–226.Shchyglo, O., Salman, U., Finel, A., 2012. Martensitic phase transformations in Ni–Ti-based shape memory alloys: the Landau theory. Acta Mater. 60, 6784–

6792.She, H., Liu, Y., Wang, B., 2013. Phase field simulation of heterogeneous cubic ? tetragonal martensite nucleation. Int. J. Solids Struct. 50, 1187–1191.Simha, N.K., 1997. Twin and habit plane microstructures due to the tetragonal to monoclinic transformation of zirconia. J. Mech. Phys. Solids 45, 261–292.Steinbach, I., 2009. Phase-field models in materials science. Model. Simul. Mater. Sci. Eng. 17, 073001.Steinbach, I., Pezzolla, F., Nestler, B., Seebelberg, M., Prieler, R., Schmitz, G.J., Rezende, J.L.L., 1996. A phase field concept for multiphase systems. Phys.

Nonlinear Phenom. 94, 135–147.Swain, M.V., 1986. Shape memory behaviour in partially stabilized zirconia ceramics. Nature 322, 234–236.Thompson, J.Y., Stoner, B.R., Piascik, J.R., 2007. Ceramics for restorative dentistry: critical aspects for fracture and fatigue resistance. Mater. Sci. Eng. C 27,

565–569.Vaithyanathan, V., Chen, L.Q., 2002. Coarsening of ordered intermetallic precipitates with coherency stress. Acta Mater. 50, 4061–4073.Wang, Y., Khachaturyan, A., 1997. Three-dimensional field model and computer modeling of martensitic transformations. Acta Mater. 45, 759–773.Wang, Y., Wang, H.Y., Chen, L.Q., Khachaturyan, A.G., 1995. Microstructural development of coherent tetragonal precipitates in magnesium-partially-

stabilized zirconia: a computer simulation. J. Am. Ceram. Soc. 78, 657–661.Wang, Y.U., Jin, Y.M., Khachaturyan, A.G., 2004. The effects of free surfaces on martensite microstructures: 3D phase field microelasticity simulation study.

Acta Mater. 52, 1039–1050.

86 M. Mamivand et al. / International Journal of Plasticity 60 (2014) 71–86

Wang, C., Zinkevich, M., Aldinger, F., 2006. The Zirconia–Hafnia system: DTA measurements and thermodynamic calculations. J. Am. Ceram. Soc. 89, 3751–3758.

Wei, Z.G., Sandstroröm, R., Miyazaki, S., 1998. Shape-memory materials and hybrid composites for smart systems: part I shape-memory materials. J. Mater.Sci. 33, 3743–3762.

Wen, Y., Wang, Y., Chen, L.Q., 2000. Phase-field simulation of domain structure evolution during a coherent hexagonal-to-orthorhombic transformation.Philos. Mag. A 80, 1967–1982.

Wheeler, A.A., Boettinger, W.J., McFadden, G.B., 1992. Phase-field model for isothermal phase transitions in binary alloys. Phys. Rev. A 45, 7424.Yamanaka, A., Takaki, T., Tomita, Y., 2008. Elastoplastic phase-field simulation of self-and plastic accommodations in cubic ? tetragonal martensitic

transformation. Mater. Sci. Eng. A 491, 378–384.Yeddu, H.K., Borgenstam, A., Hedström, P., Ågren, J., 2012a. A phase-field study of the physical concepts of martensitic transformations in steels. Mater. Sci.

Eng. A 538, 173–181.Yeddu, H.K., Malik, A., Ågren, J., Amberg, G., Borgenstam, A., 2012b. Three-dimensional phase-field modeling of martensitic microstructure evolution in

steels. Acta Mater. 60, 1538–1547.Yeddu, H.K., Razumovskiy, V.I., Borgenstam, A., Korzhavyi, P.A., Ruban, A.V., Ågren, J., 2012c. Multi-length scale modeling of martensitic transformations in

stainless steels. Acta Mater. 60, 6508–6517.Yeddu, H.K., Borgenstam, A., Ågren, J., 2013a. Stress-assisted martensitic transformations in steels: a 3-D phase-field study. Acta Mater. 61, 2595–2606.Yeddu, H.K., Borgenstam, A., Ågren, J., 2013b. Effect of martensite embryo potency on the martensitic transformations in steels—a 3D phase-field study. J.

Alloys Compd. 577 (suppl. 1), S141–S146.Yeddu, H.K., Lookman, T., Saxena, A., 2013c. Strain-induced martensitic transformation in stainless steels: a three-dimensional phase-field study. Acta

Mater. 61, 6972–6982.Yu, C., Kang, G., Kan, Q., Song, D., 2013. A micromechanical constitutive model based on crystal plasticity for thermo-mechanical cyclic deformation of NiTi

shape memory alloys. Int. J. Plast. 44, 161–191.Yu, C., Kang, G., Kan, Q., 2014. Crystal plasticity based constitutive model of NiTi shape memory alloy considering different mechanisms of inelastic

deformation. Int. J. Plast. 54, 132–162.Zhang, Y.F., Shi, J.L., Hsu, T.Y., Zhang, Y.L., Jin, X.J., 2002. Shape-memory effect in Ce-Y-TZP ceramics. Mater. Sci. Forum, 573–576.Zhao, X.S., Shang, S.L., Liu, Z.K., Shen, J.Y., 2011. Elastic properties of cubic, tetragonal and monoclinic ZrO2 from first-principles calculations. J. Nucl. Mater.

415 (1), 13–17.