Chapter 3 Constitutive modelling

Chapter 3 Constitutive modelling

35 Chapter 3 Constitutive Modelling 3.1 Introduction In this Chapter we will discuss three constitutive models of T T C materials. In contrast to ...

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Chapter 3

Constitutive Modelling 3.1


In this Chapter we will discuss three constitutive models of T T C materials. In contrast to conventional ceramic materials which exhibit linear elastic behaviour up to failure these materials exhibit a nonlinear stressstrain behaviour once a certain stress level is reached. It is believed that the occurrence of stress induced t --, rn transformation results in the formation of inelastic strains, which allow the material to redistribute stress and are an i m p o r t a n t feature in the crack growth resistance of TTC. To reduce the complexity of the transformation we assume that we can identify a material sample which is small compared to all macro-

F i g u r e 3.1: Schematic representation of the microstructure of PSZ ceramics, with transformed penny-shaped particles


Constitutive Modelling

scopic dimensions, but which is large enough that statistical averaging over all transformable particles is meaningful. For example, a. continuum element of PSZ ceramics may look like the schematic drawing in the left of Fig. 3.1. Such a material sample can then be treated as a continuum element for which all (macroscopic) quantities are averages over the sample. Phenomena on smaller scale are discarded. This means, for instance, that local stress and strain fields around individual particles are not considered, but only the macroscopic average of these fields over all particles in the sample is relevant. The differences in the three models to be discussed in w167 3.3, and 3.4 arise from the assumption of the influence of the shear component of the transformation. The first model is based on the assumption that the shear component may be neglected completely and includes only the dilatant part. The second model includes both the dilatant and the shear transformation strains but assumes that the strain deviator is parallel to the stress deviator. The third model does not make this assumption. In the first two models, the elastic constants E and u of matrix and inclusions are assumed to be identical. The third model relaxes this restrictive assumption. As the martensitic transformation proceeds with the speed of sound, we neglect any dynamical effects and assume that the transformation occurs instantaneously and is time independent.


C o n s t i t u t i v e M o d e l for Dilatant Transformation Behaviour

As described in Chapter 2 the transformation from tetragonal to monoclinic structure involves both dilatant and shear components. However, the first constitutive model developed by McMeeking and Evans (1982), and Budiansky et al. (1983) neglects the shear component, arguing that when the tetragonal particle is embedded in the surrounding matrix it transforms into a number of bands with the sense of the shear alternating from one band to the next. In this way the average shear component is much less than 16%, and may even play no role at all, while the dilatation remains about 4.5%. The effect of twinning on the residual shear is schematically demonstrated in Fig. 3.2. Of course, transformations which produce a single twin variant per particle will also interact with the shear strains, depending on the orientation of the particle. However, this is not considered in the modelling of dilatant transformation behaviour, for which we will follow closely the

3.2. Dilatant Transformation Behaviour


F i g u r e 3.2: Schematic representation of the influence of twinning on the transformation shear, demonstrating that twinning reduces the influence of the macroscopic shear component

exposition of Budiansky et al. (1983). Consider a special two component material comprising a linear elastic matrix material with embedded particles which undergo an irreversible inelastic dilatation. The assumed constitutive behaviour of the individual particles is not, in general, the same as that of a particle undergoing a true dilatant phase transformation. Nevertheless, the results for this idealized composite do lend insight to what can be expected for a more complicated system. Each component of the composite is assumed isotropic. The particle and matrix have identical linear shear behaviour so that in each

SiN = 2#eij


where 8ij is the stress deviator, eij is the strain deviator, and # is the shear modulus. The matrix material responds linearly under hydrostatic tension and compression with bulk modulus B according to 1

O'm -- -~O'pp- BCpp


where o'ij and Cij are the stress and strain tensors, am is the mean stress, and gpp is the total dilatation. The dilatant response of the particles is depicted in Fig. 3.3. Under monotonically increasing cpp, the particles satisfy (3.2) with the same bulk modulus B as the matrix material as long as



c is the critical mean stress associated with the start of transwhere a m formation. On the intermediate segment of the curve the incremental response is governed by B ~ according to


Constitutive Modelling















% Composite

F i g u r e 3.3: Dilatant stress-strain behaviour of particles and matrix material making up a two-phase composite. The shear behaviour is linear with the same shear modulus in both phases. The macroscopic behaviour of the composite is also shown

crm :


B t gpp

The inelastic, or transformed, part of the dilatation in the particle is denoted by Op and it is defined as the difference between the total dilatation and the linear elastic dilatation, i.e. Op = gpp -- O'm/B


The maximum transformed dilatation in each particle is 0T and thus (3.4) holds in the interval

B < epp < ---B--+ op



For larger values of Cpp the incremental response is again governed by the elastic modulus according to &m = B~pp


The volume fraction of the particles is denoted by c and no assumptions on their shapes need be made. The stress-strain behaviour to be shown for the composite is exact with the usual interpretation of average,

3.2. Dilatant Transformation Behaviour


or overall, stress and strain for a multi-phase material. The shear modulus of the composite is p at all strains and the bulk modulus B governs for cpv _< o ' C / B and again, incrementally, at sufficiently large Spy as shown in Fig. 3.3. Once cpv exceeds o'~n/B the response is incrementally linear with the macroscopic relation --


where B satisfies 1

+ 4p/3



B' + 4p/3



B + 4p/3


B is derived using Hill's (1963) method for the determination of the overall moduli of two-phase isotropic elastic composites both of whose phases have the same shear modulus. The macroscopic stress and strain of the composite are denoted by upper case letters. The dilatation is uniform in each particle and is the same in all particles. The overall transformed dilatation of the composite is defined by 0 -

Epp - E r n / B


On the intermediate segment of the curve 0 -

(1- B/B)(Epv - E~/B)


with the overall and local transformed dilatations related by (9 - c0v. The maximum, or complete, transformed dilatation of the composite is 0T




The strain range of the intermediate branch is

,~ < Ev v < 2 ~ B --B+





For Epv above the upper limit in (3.13) no further transformation occurs and (3.7) applies. If at any strain level beyond (r~/B the dilatation starts to decrease, the transformed dilatation in the particles is frozen and the incremental unloading response is governed by B. Equation (3.9) for B remains valid even for negative B', although this result is not necessarily unique when B' < - 4 # / 3 . If B / > - 4 # / 3 , the

Constitutive Modelling


equations governing the incremental behaviour of the particles are elliptic so that the stress and strain fields in the particles are necessarily smooth and unique. If B I < - 4 p / 3 the incremental equations are hyperbolic and certain discontinuities in the stress and strain fields in the particles become possible. Furthermore, if B I < - 4 # / 3 a particle in an infinite elastic matrix with moduli B and p can transform completely to Op T as soon as the critical mean stress is reached. From (3.9) it is seen that the transition from elliptic to hyperbolic behaviour in the particles at B ~ - - 4 # / 3 gives B - - 4 p / 3 , so that the corresponding transition for the composite coincides with that of the particles in this special system. In what follows we adopt the dilatant stress-strain behaviour for the composite shown in Fig. 3.3 and again in more detail in Fig. 3.4. It must be emphasized that we are not assuming that the particle response in an actual system is that specified above. Experimental observations suggest that partially transformed states of individual particles do not exist. A given particle is either untransformed or fully transformed. On the other hand, an actual composite system will usually have a distribution of particle sizes with an associated distribution of critical stresses. Thus, even though each particle transforms completely when its own critical stress is attained, a distribution of critical stresses may result in a composite whose incremental bulk modulus never drops below - 4 # / 3 . Indeed, if the distribution is sufficiently wide it is conceivable that the incremental bulk modulus of the composite might not even become negative. For the stress-strain behaviour in Fig. 3.4 with B > - 4 p / 3 , it can be shown that the spatial distribution of the macroscopic transformed di-


~m Loading//B c



B/ Unloading /0 ~ II



/ t)

/; ~Unloading /



d B=O


F i g u r e 3.4: Dilatant stress-strain behaviour of the composite

3.2. Dilatant Transformation Behaviour


latation 0 must be continuous. Thus, in general, there will exist regions in any nonhomogeneously deformed composite in which 0 < 0T, corresponding to partially transformed macroscopic states. If B < -4#/3, discontinuities in 0 can occur. The condition B < - 4 # / 3 has additional significance for the incremental behaviour of the composite. With # > 0, this is the condition for a real longitudinal wave speed. It is also the condition which excludes internal stretching necking modes that can, in principle, develop when ellipticity is lost. Not only is the physical response of the composite strongly dependent on whether B is greater or less than -4#/3, the mathematical techniques used to generate solutions to the crack problems posed in this Monograph also differ depending on whether B is above or below this transition. For this reason we introduce the following to distinguish the two ranges of behaviour m

B > -4#/3

<=> sub-critically transforming composite



critically transforming composite


super-critically transforming composite


B <-4#/3


The incremental form of the equations for the composite are analogous to those for an elastic-plastic solid. For any stress state not on the transforming branch






On the transforming branch there are two possible incremental responses which will be called loading and unloading, borrowing plasticity terminology. As shown in Fig. 3.4, loading occurs if Epp > 0 and




with O - ( 1 - B/B)/~pp


Unloading occurs if/~pp < 0 and then (3.15) applies. Upon unloading to zero mean stress the transformed, or permanent, dilatation is 0. If 0 is less than 0 T the material is said to be partially transformed, in the sense described above, while it is called fully transformed when 0 attains 0T. The full transformation 0 T c a n be identified with cOT. Given a transformed dilatation 0 the integrated stress-strain relations in three dimensions are

Constitutive Modelling


Eij = 2---~Ei~j+

E~Sij +-~05iy



r~ - 2,E~j + B(E~p - 0 ) ~


where superscript s denotes the deviatoric component. In plane strain (E33 = E13 = E23 = 0) the above reduce to



E ~ - 2# E ~ - -~Euu6~Z

) + B(Euu - 0)6~



E33 - - ~/.tEuu + B(Euu - o )


Em = -l -+-v- ~ E u . - ~E0


and E~



~..(2~ - uE..6~) + aft

l+u 06~ 3


where Greek subscripts range over 1 and 2 with a repeated Greek subscript indicating a sum over just 1 and 2. Here, u is Poisson's ratio of the elastic branch so that

# -

E 2(1+ u)'

B =

E 3(1- 2u)


where E is Young's modulus. In the sub-critical case, the transformation strain rate b0T can be calculated from R

(1-~ eel


_ -} < E ~ < r ~

B)Epp when E~ +









+ c m 0~~ 1-~-



whereas in the critical and super-critical cases, the transformation strain rate is undefined, but the transformation strains are simply given by

3.2. Shear and









-- CrnOT

Epp <






In the constitutive model under discussion, six material parameters determine the material behaviour" Poisson's ratio u, Young's modulus E, the bulk modulus on the intermediate segment B, a critical mean c the maximum volume fraction c m of transformable material stress ~rrn, and the unconstrained particle dilatation 0p. T Dimensional analysis and close examination of the governing equations reveal that the constitutive behaviour can be captured by three dimensionless variables" Poisson's ration u, the ratio B / # governing the slope of the intermediate stressstrain curve and the transformation strength parameter w, as defined by Amazigo & Budiansky (1988): i



[1+:] cr~



C o n s t i t u t i v e M o d e l for Shear and Dilatant Transformation Behaviour

Since the pioneering work of Budiansky et al. (1983) described above, much effort has gone into understanding the role of transformation shear strains and how to incorporate these in a constitutive description. Lambropoulos (1986) was the first to consider the twinning effect in a constitutive description in an approximate manner by ignoring interactions between transformable precipitates. As in the model described above, the martensitic phase transformation was assumed to be super-critical in the sense that it took place when a function of the macroscopic stress state attained a critical value. The resulting constitutive law was similar in structure to the incremental theories of metal plasticity in that it was characterized by a yield function, a loading criterion and a set of flow equations. It cannot however describe the material behaviour after initiation of transformation nor can it predict the real volume fraction of transformed material. Chen & Reyes-Morel (1986) also proposed a phenomenological transformation yield criterion which was pressure


Constitutive Modelling

sensitive to reflect the experimental data on transformation plasticity in compression. In the following we shall describe briefly the continuum model of Sun et al. (1991) which uses terminology and ideas from conventional plasticity theory. The exposition follows closely the work of Stam (1994).


Stress-Strain Relations during Transformation

The transformation plasticity model due to Sun et al. (1991) assumes the representative continuous element to consist of a large number of transformable inclusions (index I) coherently embedded in an elastic matrix (index M), as shown schematically in Fig. 3.1. Microscopic quantities (in the representative element) are denoted by lower case characters. The macroscopic quantities are found by taking the volume average ( ) of the microscopic quantities over the element. Thus the microscopic stress and strain tensors are denoted by O'ij and cij, respectively, for a given volume fraction of transformable metastable tetragonal inclusions c. The relation between microscopic and macroscopic stresses is

E0 -

(o'ij)v - -V

(rij dV - c(criJ)v, + (1 - c)(c~iJ>v.


where the volumes of the element, matrix and inclusions are given by V, VM, and VI respectively and c is the actual fraction of transformed material which is obviously less than or equal to cTM. The macroscopic strains are assumed to be small, and under isothermal conditions can be decomposed into an elastic part E~j and a "plastic" part E~j induced by the t --* m transformation in the inclusions

Eij -

Ei~ + EPj - MO.klrkl + c(cPj )VI

Here M ~ (Li~ -1, with Li~ inclusions and matrix

Lij kl -- 1 + u


being the elastic moduli of both

k~jl -'~ 6jk6il) + 1 - lJ 2u



The inelastic strain due to t ---, m transformation can in turn be written as a sum of dilatant and deviatoric parts distinguished by with superscripts d and s, respectively

3.3. Shear and Dilatant Transformation




E,? -



The rate of inelastic strain (designated by a superposed dot) during progressive transformation, 5 > 0, can be obtained by differentiating (3.30) or by averaging the transformation strain civj over the freshly transformed inclusions (per unit time) occupying the volume dVi, i.e. 9







= c(ciJ )dV, § c(CiJ )dV,


pd within each inclusion can be written in The dilatant part of strain Qj terms of the constant stress-free lattice dilatation gvd (_0V /3) which typically takes a value of 1.5% at room temperature, i.e. _


1 Pd~ij





The deviatoric part (gij)Vz is significantly less than the stress-free lattice shear strain of 16% because of twinning. Based on the earlier work of Reyes-Morel & Chen (1988), and Reyes-Morel et al. (1988), the rate of change b(Ci~s)dU~ is assumed to depend on the average deviatoric stress siM in the matrix according to

Here, A is a material function, which may be regarded as a measure of the constraint provided by the elastic matrix, and ~M is the von Mises stress in the matrix, which will be specified later. When ~M _ 0, A should be put equal to zero because there is no stress bias. However, experimental data of Chen & Reyes-Morel (1986, 1987), and Reyes-Morel Chen (1988) show that under proportional loading the value of A is almost constant during the transformation process. The macroscopic constitutive relationship (3.33) is assumed to apply to the ensemble of transformable particles in the continuum element. The deviatoric transformation strain over individual transformed particles will not depend on the local matrix stresses in such a simple manner because of twinning along well-defined directions on specific crystallographic planes and also

Constitutive Modelling


because the amount of twinning within a particle is dependent on its size (Evans & Cannon, 1986). Although much research has been devoted to nucleation and twinning in a single particle, these phenomena are still not well understood and need further attention. Some light will be shed on them when we describe the third constitutive model (w For the present model (3.33) is an acceptable approximation in the average sense over many grains with different orientations within d~/). From (3.31)-(3.33), the inelastic strain rate is


EPj - c(gPd~ij -]-(CiPS.)dVi )

The total macroscopic strain rate is obtained by adding (3.34) to the elastic strain rate E[j - M~

E~. _ E~j + Ef~



-- Mij]r162

~ ~- C(~Pd~ij ~- (~ij )dV I )


In an inverted form, we have

~ij -- 21-t(Eij -- Ern(~ij ) @ 3BErn~ij - c(3BgPd~ij + ~#@iPs}dV ) (3.36) where Em - Epp/3. For future reference, we note that under plane strain conditions E33 = E13 = E23 = 0, so that (3.28) and (3.36) reduce to 1

(3.37) and

1Et..5~Z) + BE.t.6~Z _ d(3BcPd6~z 2.k..

+ BE.. -~(aBc ~ + 2.(4;)..

Zm - B ( / ~ . . - 3de pd)

l+vE~ -


) E.


2~(g~)dVi )

3.3. Shear and Dilatant Transformation



T r a n s f o r m a t i o n Criterion and T r a n s f o r m e d Fraction of Material

The constitutive description is complete when the transformation condition and the evolution equation for ~ have been prescribed. This requires derivation of (forward and reverse) transformation yield conditions, for which we need to calculate the free energy of the continuum element by summing its elastic strain energy, the chemical free energy and surface energy (w We present without detail (Sun et al., 1991) the various energy components. The elastic strain energy W per unit volume of the continuum element is given by


-1~ i j MOklrkl - ~ 1


= -~r~,:~ M ~


- c

I v -o'T gijPdV 1 B~ A2 + ~B~(~d) 3 ~

[[email protected])vl(giPS)v 1 4- 3B2(gPd) 2]




-~T _ Cri7 + (-ffiJ )v M __ ~ i7 -c(cr~7)v,

is the transformation induced internal stress or eigenstress in the inclusion as defined by Mura (1987), and

0.i7 __ Li jOkl(Aklmn -- Iklmn )C~,~


is the Eshelby stress in an inclusion (Eshelby, 1957; 1961). The elements of the Eshelby tensor Akzm, for a spherical inclusion are Al111 -

A2222 -

A3333 =

7-5u 1 5 ( 1 - ~)



A3311 -


A1212- A2323- A3131 =

A2211 -

4-5~ 15(1 - u)

A3322 =

5~- 1 15(1 - u) (3.42)

Although the shape of the transforming particle influences the stress


Constitutive Modelling

field, the spherical shape has been assumed here for simplicity. M The deviatoric and mean stresses in the matrix siM and ~rm can be found using the averaging method of Mori & Tanaka (1973) for a body containing many transforming spherical particles


aiM _- .:-,,~v" - cB2c pd

where Sij = E i j -- Ern~ij and E,~ = Epp/3 are the deviatoric part and the mean stress of the macroscopic stress tensor Eij, and

B1 -

5u- 7 2# 15(1 - u)' -

B2 -

2u- 1 2B ~1 - u


These two parameters which resemble bulk moduli result from Mura's (1987) approach. The change in chemical free energy per unit volume is obtained by subtracting the chemical free energy in the martensitic phase from the chemical free energy in the tetragonal phase. This temperature (T) dependent contribution of the free energy is given by AGch~m(T)

= cAGt_m(T)


For equisized spherical particles, the total change in surface energy per unit volume is 67pc A~,r - c A o (3.46) a0

where a0 is the diameter of a particle, and 7p is the surface energy change per unit area during the t ---+rn transformation. The Helmoltz (or free) energy per unit volume, (I), can now be written by adding (3.45)and (3.46)to (3.39) O ( E i j , T, c, v ) -

W + As~,,. + AGch~,~

The complementary free energy is given by


3.3. Shear and Dilatant Transformation



1 A2 3 B +c 5B~ + ~ ~(c~) ~]

--lc22[[email protected])vi(EiSS)vx -Jr-3B2(cPd) 2] -cAo - cAGt_m


It is clear from (3.47) that the thermodynamic state of the representative continuum element is completely described by the variables Eij, T, c, and (ci~S)y~, in which c and (ci~S)u~ are the internal variables describing the microstructural changes in the material during transformation. Denoting the thermodynamic force conjugate to internal variables c and (ci~Sly~, by F c and Fief respectively it follows from the second law of thermodynamics that

,~v = Oq,



(3.49) where (vide (3.48))


1 A2 - -~B2(cPd) 3 Ao + AGt__.m- -~B1 2]

--C[/~1(ciPS)[email protected])v-t-3B2(cpd)2 1


and (3.51)

- c

Let us denote the total energy dissipation per unit volume by

W d = Do ccu



Constitutive Modelling

where Do is a material constant which can be determined by direct measurement or microstructural calculations, and the cumulative fraction of the transformed material during the whole deformation history is

f -




The energy dissipation rate W d is thus Wd -

Docc~, -



(forward transformation,

~ >_ 0)

(reverse transformation,

b <_ 0)


( -D0b

Since ~P must be equal to W d, (3.49) and (3.54) give for forward transformation Dob


Feb + Fi~ s (@iS)v, - -Doi:


FC~. + Fi]" (eiP])v, -

while for reverse transformation

Substitution of (3.50) and (3.51) into (3.55) and (3.56) and elimination of d by using (3.31) and (3.34) gives the yield condition in stress space for forward F+ and reverse F_ transformation F + ( ~ i j , C, T , (ciP.S}v1)

M c pd - Co(T, c) - 0 (3.57) Ao M + 3o m


F _ ( E i j , c, T, (ei~)v ) -

2-AaM + 3erM e pd - Co(T, c) - 0 (3.58) 3

where Co(T,c)-

Do + Ao + A G t - m ( T )

- ~3 B2(cpe)2 +


- -~BIA 2



and Co(T, c) -

Co(T, c) - 2D0


As the constitutive model is derived at the macroscopic level, the hard-

3.3. Shear and Dilatant Transformation


ening effect does not follow from the derivation itself, and Sun et al. (1991) introduced the last term in (3.59) on purely phenomenological grounds. The hardening response may be expected on a microstructural scale because of the following: 1. particle size distribution; it takes a higher stress level to transform smaller particles; 2. crystallographic orientation of particles; particles oriented along favourable planes transform first; and 3. the mutual interference of transformed particles. Introducing the parameters B0 and h0 Bo = 4 # ( l + v ) + P h ~ ( 2 8 - 2 0 v ) " 1_ v 5(1 _ v)


A h o - 3gPd

(361) "

the yield function in forward transformation (3.57) can be written as 2AcrM + 3cPda M + B o ( 1 - a)c(r

F+- 5

2- Co(T)


with an explicit linear hardening term, dependent on c. The yield condition (3.57) describes a "transformation surface" in stress space (Eij) within which transformation is excluded, similar to a yield surface in the theory of plasticity. When the transformation proceeds, the transformation surface expands (or contracts) as well as translates in stress space. Borrowing further from plasticity theory, the present material with transformation plasticity may be deemed to exhibit mixed hardening; isotropic expansion (or contraction) as a function of c is governed by Co(T, c), while kinematic hardening originates from the internal stresses appearing in the relations (3.43) between the matrix and macroscopic stresses. The incremental stress-strain relations for forward transformation may be obtained by the usual routine of internal variable constitutive theory (Rice, 1971) 9



+ c



Constitutive Modelling


= M~


+ ~ ePdsij --[- O"M


is determined by the consistency condition




O{eiS~)v' (ciJ )v, - 0


The solution of the above equation gives

OF+ O~-,ij

2BIA2 + (3B2 + o~Bo)(r 3



- 2B1A2 + (3B2 + c~Bo)(ePd)2 3

Expression (3.65) holds when the transformation is in progress, i.e. the current stress state satisfies the transformation condition (3.57) and there is still some transformable material left c < cm, and as long as no unloading takes place. We will call this the loading or transformation branch. If the response is elastic, because the criterion (3.57) is not satisfied, F+ < 0, or because elastic unloading occurs from a plastic state, F+ - 0 and ~ < 0 according to (3.65), ~ must be set equal to 0. To summarize,


Sij "4- 3gPd~m

-~B1A 2 + (3B2 + aBo)(r 0 when



(F+-Oand~_>O) (3.66)

(F+ < 0 ) o r ( F + - 0 a n d ~ < 0 )

Following along similar lines for reverse transformation, we have

2B1A2 + (3B2 + aBo)(cpd)2 0 when




(F_ < 0 ) or (F_ - 0 and ~ >_ 0)


3.3. Shear and Dilatant Transformation


From eqns (3.63) and (3.65)it follows that




i.e. the inelastic strain rate is normal to the yield surface (3.57) in stress space. The normality rule is thus identically satisfied and there is no need to assume it a priori, as is sometimes necessary in conventional phenomenological treatments. The constitutive equations may be rearranged into a form convenient for numerical analysis. For this we introduce the designations





OEij -- epdSij + A aM

g - -~B1 A2 + 3B2(cPd) 2 + e~Bo(gPd) 2 2


1 OF+ Ely


g ~ij and rewrite (3.63) as

E~ - M~

+ 1T~jT~,~, g

Next, we multiply (3.70) by (M~zij)- 1 1



Lk,i j o to give


LklijEij - Ekl + -LklijTijTmn~mn g


and again by Tkz to obtain


- (1 + 1TabL~ g



1 o T o _[~ij TmnEmn - (1 +-TabLabcd cd) -1 TmnLmnij g

Eliminating TktEkt from (3.71)and (3.73), we get


Constitutive Modelling



_ L O k l p q T p q T m n L mo n i j ~kl



LOklij -1


1 +-TabLabcdTcd g which may be rewritten in the following compact form ~ij


-- L i j k l E k l

The instantaneous moduli Lijkt in the above rate equation are --1L~ m n


Tm,~Tpq L p~q k l when F+ - 0 and ~ > 0

LOj kl -Lijkl


1 + 1Tab L~b~dTr g LiOkl




when F+ -r 0 or 5 _~ 0

Comparison Models





The continuum model of Sun et al. (1991) reduces to that of Budiansky et ps al 9 (1983) when the shear transformation strains are ignored, i.e. Qj - 0, for which we set h0 - 0. Noting that the dilatation 0pT in the first model corresponds to 3c pd in the second, it follows from (3.63) that

1L~ + 1 ~ , ~ + ~~,~ k,j = 2, 5~


which is identical to the rate expression of the integrated stress-strain equation (3.17), with the dilatational strain rate 9

E~ -



+ 3e~ ~


The corresponding b is given by (3.66)


B2r d 0


F+ - 0

and (~ > 0


F+ # 0 or b _< 0


3.3. Shear a n d D i l a t a n t T r a n s f o r m a t i o n


On the transformation branch, the volumetric inelastic strain rate is given by (vide (3.44)and (3.79)) 3F, pp B2

3ccPd --





(y+31 This result also follows from the model of Budiansky et al. (1983), when B - 4 p / 3 is substituted into (3.24). a > 0 corresponds to subcritical formulation and the relationship between a and B is now



3BB2 + 3B2B




BBo - BoB


B(B2 + 89



B2 + gc~Bo + 3B

The similarity between the two models extends to the onset of transformation when c - 0. In the model of Budiansky et al. (1983) this happens at the critical mean stress

r ~ - r~m


In the continuum model of Sun et al. (1991) the initial yield condition is obtained by putting c - 0 and (cPS)y~ - 0 in (3.57) F+ -

2hocPdEe 4- 3cPdEm -- C o ( T , O) - 0


where 2e

C o ( T , O) -


~ij ~ij,

~ij -- ~ i j -- ~ m ,

2 m -- -~ Epp

Do + Ao + A G t . . . . ~ ( T ) - 3Bl(hovpd) 2


3 )~ ~B~(c ~d (3.85)

If we denote by E C the critical stress


_ Co(T, 0) --



then (3.83) reduces to 2-hoEe + E m 3




This initial yield condition reduces to (3.82) when shear strains are ne-

Constitutive Modelling


glected, i.e. h0 = 0, and E c is appropriately reinterpreted.


Comparison with Experiment

To illustrate the capabilities of the continuum model of Sun et al. (1991), the predicted stress-strain relations are compared in Fig. 3.5 with experimental stress-strain curves for TZP and PSZ. The experimental data of TZP and PSZ were obtained under triaxial compression by Reyes-Morel & Chen (1988), and Chen & Reyes-Morel (1986), respectively. The broken lines show the theoretical response in the axial (xl) and radial (x2 and xa) directions. The strain in the axial direction is negative and the strains in the radial directions are positive. Obviously, inelastic strains of that nature cannot be predicted by the purely dilatant model. For TZP, the experimental data point towards a nearly perfect inelastic regime after the initiation of transformation followed by linear hardening. This bilinear transformation behaviour in TZP cannot be fully explained by the model of Sun et al. (1991). For PSZ, the transition from linear elasticity to transformation behaviour occurs more gradually, whereas in the theoretical model the transition is sharp. The experimental data may be used to estimate the various material parameters appearing in the constitutive model (v, w, h0 and a). For TZP tested by Reyes-Morel & Chen (1988) in hydrostatic compression the elastic properties are given as E=190 GPa and v=0.3. For the fully tetragonal material cm=l.0, and the constant lattice dilatation is always assumed to be cvd=0.015. The experimental stress-strain curve for TZP in Fig. 3.5(a) is used to derive the shear transformation strains in the Xl and x2 directions as a function of the transformed fraction. These strains are shown in Fig. 3.6. As the loading was almost proportional in the test, the shear transformation strains are given by

EiJ -- 3ch~



Using this relation the value of h0 can be estimated from the experimental curves of Fig. 3.6. The broken lines show the theoretical response for h0=l.4 in the axial (Xl) and radial (x2 and x3) directions. The critical transformation stress E c can be determined using the condition at the onset of transformation (3.87) 2 F+(Eij, (ci~)u) - -~hoEe +

Em -


- 0


3.3. Shear and Dilatant Transformation



x2 ~



x3 -Ell .............





4000 3500 ,'•


,:; ,;;,


i, ,,v'







.... ," " / /

2000 1500

El c! [MPa] 2500











1000 r/

1000 500 500 0


O r

0.00 0.01 0.02 0.03 0.04 0.05 Eli a)






0.0 0.005 0.01 0.015 0.02 El, b)


F i g u r e 3.5" Stress-strain curves for T Z P and PSZ obtained by triaxial compression at room temperature. The broken lines correspond to the predicted response

For a specimen loaded in compression E l l in addition to hydrostatic pressure P, Fig. 3.5(a), it follows from (3.89) that Ec _ 2h0- 1 - -----~--Ell

- P


From Fig. 3.5(a) the stress level at which the transformation is initiated is E~1=-1300 MPa, so that with P = 1 2 5 MPa, we find E c = 6 6 0 MPa. The value of the hardening parameter c~ is found by comparising the slope of predicted inelastic branch of the stress-strain curves of Fig. 3.5 with that of the experimental curves. In this manner the hardening parameter c~ is

Constitutive Modelling



Eij 0.05 [ p$

-E11 0.04






0.02 s s l l


s -


i I I s

0.00 ~ 0.0

, ..~











F i g u r e 3.6: Contribution of shear transformation to the transformed fraction for TZP. Broken lines show the predictions of the constitutive model estimated to be 1.16. As mentioned earlier, the experimentally observed bilinear regime of the transformation cannot be modelled, as the present model only takes into account linear hardening. For PSZ E=190 GPa, ~=0.3, cPd=0.015 and cm=0.35. The remaining parameters can be calculated in a similar way, as discussed above and are estimated to be: c~=1.2, h0=l.3, and E C ,,~490 MPa (based on E/11 - - 1 1 5 0 MPa and P = 1 2 0 MPa from Fig. 3.5(b)).


C o n s t i t u t i v e M o d e l for Z T C

We now present an incremental plasticity formulation due to Lam et al. (1995) which overcomes the two basic limitations of the model described in w namely that the transformable particles and the matrix have the same elastic properties and that the strain deviator is parallel to the

3.4. Constitutive Model for ZTC


F i g u r e 3.7: Configuration of the matrix-particle system stress deviator during transformation. This will allow the constitutive model to be used for all zirconia toughened ceramics, including dispersed zirconia ceramics. The incremental plasticity formulation for the transformation plasticity behaviour of zirconia-toughened ceramics is based on the micromechanics of heterogeneous media, which requires the solution of the inhomogenous inclusion problem. This will be accomplished via Eshelby's equivalent inclusion method. A single-particle model for the transformation yielding and twinning of an ellipsoidal zirconia particle, based on the end-point thermodynamics of martensitic phase transformation will be developed. The average eigenstrain inside the transformed particle is determined by the volume fractions of the four possible martensitic variants which minimize the total free energy of the matrix-particle system after transformation. The model can consider thermal expansion mismatch besides the elastic mismatch between the matrix and the particle, as well as thermoelastic anisotropy for both the matrix and the particle. First the overall fields of the particulate-toughened composites will be related to their local counterparts. An incremental plasticity model is then established by incorporating the single-particle model with the Mori-Tanaka (1973) estimate for the overall inelastic strain. 3.4.1

Equivalent Inclusion Method Inhomogeneity Problem


An ellipsoidal tetragonal zirconia particle (Fig. 3.7) is embedded in an

Constitutive Modelling


infinite elastic medium subjected to a load which produces a homoge0 in the matrix in the absence of the particle. The neous stress f i e l d ~-'ij homogeneous eigenstrains are EiT M and EiTP in the matrix and particle, respectively. Denote the elasticity tensors of the matrix and of the particle by cijMkt and ciPkz, respectively. They are not required to be the same and can be isotropic or anisotropic. The solution for the matrix-particle system is the superposition of the stress-free homogeneous eigenstrain E[~ M and the solution with a uniform eigenstrain (E T M - E TP) inside the particle. The latter is obtained by Eshelby's equivalent inclusion method for an inhomogeneity (Eshelby, 1957; Mura, 1987) which we shall briefly review below. Let E g be the stress perturbation caused by the inhomogeneity with eigenstrain E I - ( E T M - E TP) and let EiD be the corresponding perturbation of strains. The constitutive relations for the particle and the matrix are as follows 0

Eij + E D

- ciPk,(E~, + E D - EI,)


- cijM,(E ~ + E D)

+ E D

in 9




where E O


M (Cijk,)

- 1



is the homogeneous strain due to the external loading E~ in the absence of the ellipsoidal particle. According to the equivalent inclusion method, the inhomogeneity can be modelled by an inclusion (with the elastic properties of the matrix Cijkz) M in the homogeneous matrix with eigenstrain E/. plus an equivalent eigenstrain Ei) M Eij0 + ED _ Cijk,(E~ ' + E D _ E I,

Eijo +



(E ~ + E D)

* Eij)

in [2




Equations (3.91) and (3.93) are equivalent when the fictitious eigenstrain E'~j satisfies the following equation in the region f~ C iM j






P Cijk,(






For an ellipsoidal particle, the solution of (3.94) has been given by Eshelby (1957). The strain disturbance E D is uniform inside the particle and is given by

3.4. Constitutive Model for ZTC



A , ~ , ( E I, + E;,)



where Aijkt is the so-called Eshelby tensor that depends on the elasticity cijMkl and the geometry of the particle (The components of Aijkz for a spherical inclusion are given in (3.42)). Substituting (3.95) into (3.94) gives tensor

E i *j -

Pi;~, [(CMmn - C P m n ) E mon + C kPl m n E mIn ]

- EI


where Pijkl



P (Cijmn



(7.M M "-',;ran., ) A m n k l + C i j k l


Therefore, the total uniform strain and stress inside the inhomogeneity due to the external load Eij0 and the eigenstrains EiT M and EiTP are D


_ Eo + E , ? , -1 +AijklPklmn

M q [(Cmnp

- E-' ~ ~m. ~ , , ~~, 0 +




o - CijMklEkl0 + CijMkz(Akzmn- i k l m n ) p - 1mnpq

Cpqrs - c ~P, ~ , ) < , o + c ~P, ~ , ( < , T P -

<,T M



where Iijkl is the fourth-order identity tensor. Another expression for the average total strain E P can be obtained by substituting the second eqn (3.98) into (3.94)

= EiT M + Oijk,E~, M -1 Cpq~, P (E~TP - E~, TM ) +Q~ik,Ak,m,,(Cmnvq)


Q~jkz -

(I~ik~A~jm,~ ( c 2M~ ) - 1 M( c ~ - c ~ , )




Constitutive Modelling



Transformation Single Zirconia

Yielding Particle



of a

When the applied load ~ij0 increases to a critical level, the embedded zirconia particle experiences phase transformation from tetragonal to monoclinic form which is accompanied by dilatational and shear transformation straining. When only first-order effects are accounted for, the transformation strain can be assumed to be homogeneous inside the whole particle f~ and is denoted here by E n. According to Lam and Zhang (1992), for the transformation of zirconia crystal from tetragonal to monoclinic symmetry, there are only four variants for the right stretch tensor Uij, denoted by U1, U~j, U3 , and U4 , although there are many more variants of deformation gradients. The transformation of the monoclinic particle can be approximately considered as a uniform eigenstrain that is the volume average of the strains of the four martensitic variants 4

Ea -




fin EiN _ -2 E / 3 N ( u N u ~ - Iij)




where Iij is the unit tensor of the second order and /3N is the volume fraction of the martensitic variant UiN. The right-stretch tensors of all the four martensitic variants for the transformation of ZrO2 particles in Mg-PSZ (see Table 2.1 for lattice parameters) at room temperature are 1.0102 0 0.0826

0 1.0256 0

0.0824 0 1.0260

1.0102 0 -0.0826

0 1.0256 0

-0.0824 0 1.0260

1.0256 0 0

0 1.0102 0.0824

0 0.0824 1.0260

1.0256 0 0

0 0 1.0102 -0.0824 -0.0824 1.0260


3.4. Constitutive Model for ZTC


The average strain (3.101) is calculated in the following manner. Let F i j be the transformation gradient defined with respect to an orthonormal basis whose directions are consistent with three lattice vectors. Assume that (Fi 1, F~j ..... Fi~ ) are the deformation gradients of all the variants which may be twin-related to Fij. That is to say, for each deformation gradient F~ e {Fi~, F 2, ..., Fi')), there exist two orthogonal I N[) satisfying symmetry tensors RijI E O, H I E ~ and two vectors (ai, and jump conditions associated with twinning. Now, we will investigate which deformation gradients of the variants are twin-related to F / . In fact, if we replace (RiI , H I , ai, I N[) by (Rij' * H , / * , a i' * N , ~ * ) and Fij, by F / , we will easily find that the new equations of symmetry I and discontinuity have exactly n solutions (RilF]kHl,,...,RijF~kglt). This implies that there may be a variant whose deformation gradient is I J J I RijRjkFkIHlmHmn with respect to the given coordinate system. Similarly, we can obtain the variant with deformation gradient FIJ,...,K ij


I K K J I RikR~t...RzmFm,~Hnp...Hpqgqj


where I, J, K can be arbitrary integers among [1, 2, ..., hi. We now have a clear and complete picture of the twinned crystal. It IJ...K is built up of the homogeneous deformation regions Fij' ' which interface with each other only on a finite number of well-defined twin planes Fig. 3.8. As we are mainly interested in the macroscopic transformation strains of the twinned particle there are two approaches to averaging the deformations of the variants involved. One is based on the assumption that the averaged deformation gradient F~. of the particle is the volume average of the deformation gradient variants




F i g u r e 3.8: A schematic description of the twin structure of the transformed zirconia particle. A, B, C, D denote four possible martensitic variants

Constitutive Modelling





VIj .... ,KF/~' J ' ' K



where VI,j,...,K is the volume of the region with transformation deformation gradient F~ ' J ' ' K as defined by eqn (3.103), and V is the total volume of the particle considered. The strain tensor E~j of the particle is determined by 1

E,) - ~(r~F~% -



(3.105) I J . . . K 0t3...7

This averaging approach cannot be used directly for the calculation of transformation shear in the case of t ---, m phase transformation since there are too many gradient variants given by eqn (3.103). We shall employ an alternative approach which has been widely used in composite mechanics in which the macroscopic strains of a multi-phase material are considered as the volume average of the strains of the phases involved. For a given deformation gradient variant FiI J n the Green strain tensor E~ J K is

EI/...K__ -~(F~ 1 : ~ c F ; IJ...K s




....K _ I~j ) = - 2 ( u [ / ..K ~sl/k:


The right stretch tensor U / J n must take on one of the values Uff of (3.102). The average transformation strains are therefore expressed by z



v , , . . . ,< z


. . '< -

z IJ

(3 . o7)


When there are only four possible martensitic variants we obtain (3.101). The change in total potential energy of the matrix-particle system due to phase transformation is essential in determining the transformation yielding and twinning. To calculate this change we need to know the perturbation caused to the total potential energy by the inhomogeneity and the eigenstrain E~ relative to the homogenous stress field Eij0 and

3.4. Constitutive Model for ZTC


strain field E~ It can be expressed as follows (Mura, 1987) W D




)dV f~




r, u


Substituting (3.97) and (3.98) into (3.108) and taking into account the fact that the stress and strain fields inside the inhomogeneity are uniform, we have WD_

- - ~1V p { ~ i jo


[Pi;~, ((CMmn


P 0 + CijklEkl)] P I Cklmn)Emn

jk~(Aktmn Ikzmn)P mnpq -1 -

• ((CpqM~- cpP,.,)E~ + cpP~,E[~)] E/~ + E~ where Vp is the volume of the particle. The change in the total potential energy of the matrix-particle system during the phase transformation is the difference between the potential energy of the system after and before the transformation. Taking into account the fact that the eigenstrain inside the particle changes from E~TP to EiTP + Ei~ during the phase transformation it follows from (3.109) 1

AGM -- --~Vp[Ei~

+ E[jBijktZ~ + Ei~DijktZ~]


where -1 p Aij kl - Pijmn Cmnkl .jr.(Cijmn) M -1 (Cm~v~ M P -T ~ , + Iijkl - Cmnpq)Ppar,(hr, ,~ -- I,.~a, ) C ,M P -T M Bij kl -- Cijmn Pmnpq (Apqrs - Ipqrs )Crskl M

+Cijrnn (Amnpq - Imnpq )Pp-'qlrsCrPskl M Dijk, - Cijmn(Am,~vq - Im,~vq)Pp--ql,.,cPkt


According to the so-called end-point thermodynamics of phase transformation (Lange, 1982), the transformation can proceed only if the change in the free energy AG between the initial untransformed tetragonal state and its transformed monoclinic state is non-positive


Constitutive Modelling

AG = AGM + AGcH + AGs = AGcn < 0


where A G c n is a critical value that AG has to attain for the phase transformation to be thermodynamically favourable, AGM is the net change in the potential energy of the whole matrix-particle given by (3.110), A G s is the surface energy change both in the matrix-particle interface and in the interfaces between the twinned variants and A G c H is the change in chemical energy. When martensite is the low-temperature phase, A G c H is given approximately by AGcH Vp

_ g(1 -- t~ ~-)


with g being the enthalpy of the transformation of an infinitely large single crystal and ~, the stress-free equilibrium transformation temperature (Garvie & Swain, 1985). The expression for the interfacial energy change is quite complicated. Although its complete description requires full knowledge of the twin structure of the transformed particle, it can be reasonably approximated (Garvie & Swain, 1985) as follows AG______~s= ~--~(1 + ~2~(-)) vp r](n) ~

( n - 1)(n + 1) ll


~;1, ~2 > 0


where ~1 and ~2 are material constants, and n is the number of twins in the particle. Among the three components of the free energy change (3.112), only the change in potential energy AGM depends on the twinning combination (ill, ~2, ~3, and ~4), once we have accepted the approximation (3.114). Thermodynamically, the twinning combination (ill, fl~, 133, and f14) should be such that the whole matrix-particle system assumes a minimum free energy state after the transformation. If we know the critical stress E C at which the phase transformation takes place, the twinning combination is determined by minimizing the objective function AGM

ve subject to







E kt a



kz E a

+ EaDij

kz E a]

(3 9115)

3.4. Constitutive Model for ZTC

0 _< ~i _~ 1,


fll + f12 + f13 + f14 - 1


The minimum value of AGM/Vp should be (3.112)

AGM 1 = AGc - ~(AGcRVe vp

A G s - AGcH)


Conversely, if we know AGc, we can determine the yield stress E C and the twinning combination. Define the applied stress as

E~ - X ~ + ~rXij


where cr is a scalar loading parameter characterizing the loading level and is always taken as positive, X ~ is the initial stress tensor and Xij is a second-order symmetric tensor characterizing the loading direction and complexity. From (3.112) the loading parameter cr can be determined as follows

0"(~1, "", ~ 4 ) --

- 2 A G c - ( E [ j B i j k , + E~Dijk, +X~ Xij Aij kzEka


The critical loading parameter ~rcR is given by

acn - min{a(fll,fl2,~3,~4)}


subject to the constraint (3.116). It can be shown that subject to the condition (3.116) and XijAijktEt:~ > 0, the minimum of cr(fll, ~2, f13,/?4) also minimizes the potential energy change (3.115). Let the minimum of (3.120) be attained at (ill, /?2, /33, f14). For any other combination (fl~, fl~, ~ , ~ ) of the martensitic variants, satisfying Xij AijkiEk~' > 0, we always have ~cR

- ~(fl~, &, &, ~4) <

- 2 A G e - ( E I B i j k z + Eij Dijkt + X ~ Aijk,)Ek, XijAijkzE~t


where Eij is the average particle strain corresponding to the martensitic combination (fl~, fl~, fl~, fl~). From (3.121) it follows that the martensitic combination (ill, f12, f13, f14) minimizes the free energy change

Constitutive Modelling








(X~ +o'cRXij)AijklEkl +EIjBijklEij +Eij DijklEkl

+AGcH +AGs


in which AGcH and AGs are independent of the martensitic combination (13~,fl~, 13~,13~). The equality holds when (fl~, fl~, fl~, fl~) is equal to (~1, ~2, ~3, ~4)" The requirement X i j A i j k I E k ~ ~ > 0 can be justified thermodynamically. If the transformation takes place with a martensitic combination (~,~,~,~) satisfying XijAijkiEk~' <_ O, then it follows from (3.115) that the particle-matrix system will assume a lower or equal free energy state if the particle transforms with the same martensitic combination ( ~ , ~ , ~ , ~ ) before external stress is imposed on it. Therefore, the martensitic combination ( ~ , ~ , ~ , ~ ) could not be a twinning mode for a particle which has not transformed under the action of the original stress X ~ . The dependence of yield stress crcR on the particle size can be examined using (3.119). Let Vp = n~03, n > 0, so that



ACc -- vp~(/XacR- AGcH) - ~--(1 + ~271(n))


The first term at the right-hand side of (3.123) is found experimentally to be independent of particle size. I<1, g2 and g all are positive, and AGe decreases with decreasing particle size ~o. Therefore, smaller particles will transform at a higher critical loading parameter ~rcR, in agreement with test data. For given particle shape and loading complexity tensor X~ the twinning combination (~1, ~2,133, ~4) and the transformation strain E~ can be determined numerically by solving the constrained optimization problem (3.120). For the special case where the matrix and the particle share the same elastic stiffness tensors, the objective function (3.119) becomes

5r(~1,/~2, ~3, ~4) ----



{ - 2 A G c - E~


[(Aiyk,- Iijk,) Cklmn M

M C~jkl(hkzmn Ikzm,~)]

--EiajcijMk,(Ak,mn -- Ik,m~ ) E g -





3.4. Constitutive Model for ZTC


Extensive numerical analysis of the yielding behaviour of a single particle according to (3.124) has been given by Zhang & Lam (1994).


Overall Properties and Local Fields

To establish a proper constitutive relation between the overall macroscopic strain and stress fields based on the known thermostatic properties of the constituents we consider aligned ellipsoidal tetragonal particles embedded in an infinite matrix. The shape of the ellipsoids and the orientations of their tetragonal lattice relative to the principal axes of the respective ellipsoid are assumed to be the same for all particles. The particles are aligned along a given direction, but otherwise randomly distributed. Let us consider a representative volume V with surface S, the volume average of the mechanical response of the matrix and the included particles represents the macroscopic response of the ZTC. The matrix phase occupies a volume VM, whereas the particles occupy a total volume Vp so that

VM + V p - - V

)~M + )~P- 1,

,~M -- VM/V,

)~p - V p / V


Suppose that the constitutive relations of the matrix and the particle are known and are formally

~ ( ~ ) _ c,~, M (E~,(~) - El?' (x__)) 2~(~)



X_~ VM

x__E Vp


where NO(x_) and Eij(x_) are stress and strain tensors and EiTM(x) and EiTP(x) are eigenstrains in the matrix and particle, respectively. They need not be uniform. Define the overall stresses and strains as averages of the local quantities over the respective volumes

~,~ - ~ r~'7 + ~ r~ --Eij - ~ M EiM + ~ P E P


Constitutive Modelling

70 where M


E i M - - VM 1 / V ~ Eij(x__)dV, E P

1 l = ~pp



Substituting (3.126) into the first of (3.127) we get

(3.129) where

EiTM -- VM1 fY EijTM(x)dV_ M

EiTP - ~1/vpETP(x)d V_


are the average eigenstrains within the matrix and the particle, respectively. Now the problem is the computation of the average strains EiM and E P in the respective phases caused by uniform overall strain -Eij and by uniform eigenstrains both within the matrix and within the particles. The solution of this problem can be generally expressed in the form (Dvorak, 1990) w


z g - Ag

, + v ,'; z

+ v ,';f , z r2


where AijMkt and APkt are the mechanical concentration factor tensors MM MP PM PP and Dijkt , Dijkz , Dijkt , and Dijkz are the so-called eigenstrain concentration factor tensors. Once these concentration factor tensors are known, the overall elastic stiffness and eigenstrain tensors can be obtained by substituting (3.131) into (3.129) to give .





Eij -- Cijkl(Ekl- Ekl)



3.4. Constitutive Model for Z T C







,~MCijmn Amnkl + )tPCijmnAmn kl _-- cijMk l -[- /~p( Cijm, P ~

M ) Am,,kt P Cijm,~



--I M MM [~P r)PM C i j k l Z k , -- (,~MCijMk, __ ~MCijmn Dmnkl -- ,~p ,..,ijrnn~_,,mnkl ) E l y l--





PP mnDmnk )ETI P (3.134)

The concentration factor tensors are derived using the Mori-Tanaka (1973) method. Assume that the average strain E P in the interacting inhomogeneities in which there exists a uniform eigenstrain EiTP can be approximated by that of a single inhomogeneity with uniform eigenstrain EiT M embedded in an infinite matrix subjected to the uniform average matrix strain EiM. The solution of this problem can be easily derived from (3.98)

E P - Ei~ M "4- EiT M "4- Aij klPklmn-1 X

M [(Cmnpq - CmP


q)E q

+ C




TP q -




where EiT M is the average strain caused by stresses in the matrix which are to be determined. Substituting (3.135) into the second equation (3.127) the average strain Ei~ M is given by -1 P )ErsTM Ei~ M - I'(ijkzEkz - Kijk,(Iktrs - ,~pAkzmnPmnpqCpqrs -

)tKijkzAkzmn Pmnpq -1 Cpq,.sE~TP


where Kijkl -- {Iijkl + ~ p A i j m n Pmnpq(-1 Cpqk _ Cpqkl)P } -1

(3. 137)

Substituting (3.136)into (3.135) and comparing the latter with (3.131), we have AijMkl -- Kij kl APkz - Kijkl + Aijmn P~npq -1 (CpqrsM - Cpq,., P )Kr, kl


Constitutive MM



Iijkz - f f i j m n ( l m n k l


- ) ~ p A m n r s P~s~u ~Ct uPk l / ~


Dij kl -- - ~ P K i j m n Amnpq Pp-qlr, c P k , PM





Dij kz - Dij kz -- Aij m n Pmnpq {( Cpqrs - Cpqrs ) I~[rstu --1


x ( It~,k, - ,kpAt~,abPab~dC~dkt) + c P k , } PP MP - 1 M P D i j k l - D i j k l - A i j m n P~nnpq { A P ( C p q r s - Cpq,. s ) -1



X Kvstu AtuabPabcaVcakl -- Cpqkl

}(3.1 38)

It can be verified that the concentration factor tensors satisfy the following relations (Dvorak, 1992) / ~ M A iM j k l + ) ~ P A iPj k l - - I i j kl MM



Dij kl + Dij kz -- Iij kl -- Aij kz DijP Mkz + DijP Pkt - Iij kl -- A p kt

(3 139)

In the present formulation it is possible to include the eigenstrain due to thermal expansion mismatch between the particle and matrix besides the eigenstrain due to phase transformation. It is however assumed that the matrix is purely elastic. The eigenstrains in the particle E T P and matrix EiT M are

(3.140) where m p and m iM are the thermal expansion coefficients of the particle and the matrix, respectively, OF is the fabrication temperature and E P is the average transformation strain in the zirconia particles. Substituting (3.140) into (3.133) gives the constitutive relation for ZTC .





Eij -- C i j k t ( E k l - -mkl(v ~ -- ~ F ) -- E k l ) ~I


E i j - m--~j(t9 -- I~F) + E i j



3.4. Constitutive Model for Z T C

-m- i j -- -Ci~ 1k l { ( ~ M C M p q


) i M C M m n D mMnM p q _ lPCPmn

--P Eij


PM )m M Dmnpq

Dmnpq - I P C k l m n D m n p q ) m p q

1 (A,, P Cklp q - ) i u C M m n Dm.~p M P q -- ApCPmnD...~pq)Epq PP P kl

(3 9142)


m--~j is

the overall thermal expansion coefficient tensor and Eij the overall transformation inelastic strain. As pointed out by Hill (1967), the -=_P

inelastic strain Eij is usually not equal to the volume average A p E P. Only for the special case where the matrix and particles share the same elastic stiffness t e n s o r s ( C i M j k l -- C iPj k l ) , we have AijMk --

-- Iij

MM D i j k l -- ) i F A i j kl MP D i j kl -- --.'~p A i j kl PM D i j k I -- --,'~M A i j kl PP D i j kl -- ,'~M A i j kl -mij - ) i M m i M + ~ P m P ..=p


Eij - )iRE P




and Twinning

of TTC

To complete the description of the transformation plasticity of TTC we need a yield criterion e,




and a flow rule --P dEij

_ O P (Eij, v~, Q)d0


The former determines the critical overall stress level at which the zirconia particles with characteristic radius 0 transform from tetragonal to

Constitutive Modelling


monoclinic phase, and the latter the increment to the average inelastic --P

strains Eij due to the phase transformation of particles with radii between ~ - d~0 and ~. We already considered this problem in w where we introduced both the transformation yield condition and the flow rule. If we assume that under the action of overall external load Eij, all zirconia particles with radii larger than ~ have transformed to monoclinic --p phase resulting in an overall inelastic transformation strain Eij , then the overall strain is . . . . I - ---:P E i j - M i j k l ~ k l + Eij -- M i j k l ~ k l + m i j ( v q -- ~ F ) + E i j


Mijkl--Cijkl --p

with m~j and Eij defined by (3.142). Substituting (3.146) into (3.136) we obtain the average matrix stress ~.,.S.M ~j ~ S M _ y~M K,SM ij ~ i j kl-~kl

M{ Kklm.--Mmnpq~pq

---- Cijkl

+ [Kklmnmm.


M Kkzm,~mm.

-1 CPtu(mtu M - mPu)] (v~- OF) +ApKklmnAmnpqPpqr, -J"Kklmn'-E --mnP

- ApKkzmnAmnpq pparsCPtu-1



Next we impose the Mori-Tanaka (1973) assumption again. We assume that under an increment AEij - do'-Xij(O) to the overall stress Eij, an untransformed tetragonal zirconia particle of radius between Q and 0 - d o will transform to monoclinic phase with the same martensitic combination (/~l,/32,/~3, ~4) as when it is embedded alone in an infinite matrix with uniform external loading E~ M + AE~ M -

EiTM + dcrCijM,Iik,,~.-M~,~pq-Kpq(~)


In accordance with the minimization problem (3.115)-(3.116) the transformation plastic strain of particles transformed under increasing external loading is determined by the minimization of the objective function AGM



3.4. Constitutive Model for ZTC


+ E I Bijk,E~ + EiUjDijk,E~]


subject to the constraint (3.116) with E~ depending on the twinning combination (/31, t32,/33,/34), as in (3.101). The increment in the average particle deformation is 1 --p

dEij _ Mij kl( )~pCPpq _ ,,~MCMmn Dmnp --

PC m. Dmnpq PP )dE


where f(co)dco is the total volume of the zirconia particles with radii between co- dco and co, satisfying

fo ~ f(co)dco - Vp


The relation between dco and dcr is determined by the function AGe. It is related to the minimum of the objective function as follows


> AGc(co);


- AGc(CO);

neutral loading

< AGc(e);



In the first two cases, no particles transform to monoclinic phase, i.e. dco = 0, whereas in the third case

CO- dco - AGcl(min(AGM/Vp))


where A G c 1 is the inverse of the monotonically decreasing function

AGe. Equivalently, the relation between do and d(r can be determined by minimizing the objective function

d ~ r - min

1 M


- -

- -

[Ci i ktIi~klm,~MmnpqXpq( O)]Aij~, Ea.

Constitutive Modelling


x [-2AGc(~o - dQ) -(E[jBiik, + E~Dijkt + EijSM Aijk,)


Transformation Loading






Among the geometric and physical parameters required in the preceding constitutive description, the temperatures t~, OF, ~),, the thermal expansion coefficients miM and m P, the elastic stiffness tensors cijMt and P and the particle size distribution f(~o) are easily measured and have Cijkt been reported separately for different zirconia toughened ceramics. The energy parameter AGc that is essential to the calculation of yield stress and plastic flow rate, involves the energy barrier AGcR that depends on the size and volume fraction of transformable precipitates, the chemical energy change AGcH and the interfacial energy change AGs. The last two terms can be approximately estimated in terms of the temperature t9 and particle radius ~0using (3.113) and (3.114), while the first term is assumed to be zero in most of the previous calculations (Lange, 1982; Garvie & Swain, 1985). To discuss how the parameter AGc can be established by experiment on ceramics toughened with aligned zirconia particles, we investigate the transformation yielding of ceramics with aligned zirconia particles of equal size under uniaxial loading. Theoretically, all these particles will transform simultaneously, so that ~ij



o'~UA, -


f(o) - Vp6(O- Oc) --P



- 0

where 5(L0- Oc) is the Dirac delta function,


nlFt2 Y/lYt31 (3.156)

[n3nl n3n2 n3n3_] and (nl, n2, n 3 ) T z (cos/~ Cos ct, COS/3 sin a, sin/3) T is the loading direction relative to the principal axes of the ellipsoidal particle. Substituting (3.155) into (3.149) gives




[(E~ M~ + o-cRCij~tKktmn'-Mmr~pqEpq )Aijmr~



Constitutive Model for ZTC


(3.157) where ~SM0

M M -- Cij kl { I'(klrnn m m n -- Iiklrnn m m n





is the average thermal residual stress in the matrix. Once the yield stress crcR is known from uniaxial tension tests, the critical parameter AGc can be calculated. The accuracy of the above constitutive model for the transformation yield stress and flow rate is best demonstrated by comparing the predicted values with experimental data. For this let us consider the pressure sensitivity of the uniaxial tensile yield stress of ceramics toughened with zirconia spheres at the initiation of transformation under uniaxial tensile loading q ~ u a superposed on given confining pressure -plij, -




Both the matrix and the particles are assumed to be elastically isotropic with elastic stiffness tensor

C i j k l -- #

~ik~jl + 6il6jk "t-

1 "2lz2u 6ij6kl)


where # is the shear modulus, v is the Poisson ratio and 6ij is the Kronecker delta, p = 8 3 . 2 G P a and v=0.30 will be used for both the matrix and the particle (Garvie & Swain, 1985). Figure 3.9 shows plots of the uniaxial tensile yield stress Yc against the hydrostatic pressure p for prolate spheroidal particles. The aspect ratio is r = c/a = c/b = 0.2, with a = b > c being the three principal radii of the ellipsoid. It is assumed that there is no thermal expansion mismatch between the matrix and the particles. The volume fraction of the tetragonal zirconia particles is taken as 0.35. For the five loading directions considered, the uniaxial tensile yield stress Yc is approximately linearly dependent on the hydrostatic pressure p. The pressure sensitivity coefficients r are between 1.118 and 1.318 except for the case a = 0,/3 = 7r/2. The tensile yield stress at zero hydrostatic pressure is found to be strongly dependent on the critical energy parameter AGc. In the numerical calculations, A G c = --0.22AGcH/Vp,

Constitutive Modelling

78 1700 Experimental // [ - Mg-PSZ ( M S ) / ( [ ,, Mg-PSZ (TS) / 1500

1300 r cD r~ tD

9~ 1100

900 ~ / 7








zx 0 rd2 o ~6 rd6 v ~3 ~/3 700 , I , l 0 100 200 300 400 500 Confining Pressure, p (MPa)

F i g u r e 3.9" Uniaxial tensile yield stress versus hydrostatic pressure" theoretical and experimental results

with AGcu/~/~ = - 0 . 2 8 5 ( 1 - 10/1448)GPa (Garvie & Swain, 1985). If it is assumed that AGcn = 0 and Garvie & Swain's (1985) estimate for the total interfacial energy change AGs/Vp = 5.41~o-ljm -2 is used, the particle size corresponding to the critical energy parameter AGe is found to be Q=24.51nm. It can also be seen that for ceramics toughened with aligned zirconia particles the transformation yield surface is anisotropic. The tensile yield stresses are higher along loading directions in the tetragonal lattice planes (/3 = 0 or /3 = 7r/2) than along other loading directions. The anisotropy is quite similar to that for a single zirconia particle, embedded in an infinite matrix (Zhang & Lam, 1994). Experimental results on the pressure sensitivity coefficients for two grades of Mg-PSZ, MS (maximum stress) and TS (thermal shock resistance), have been reported by Reyes-Morel (1986), and C h e n & ReyesMorel (1986). The theoretical values of the tensile transformation yield stress for such ceramics should be understood as orientation averages of the results for ceramics with aligned particles because the lenticular

3.4. Constitutive Model for ZTC


tetragonal zirconia precipitates inside the tested ceramics are randomly orientated. The experimental points are also shown in Fig. 3.9. The pressure sensitivity coefficients according to these experimental data are 1.26 for Mg-PSZ MS and 1.21 for Mg-PSZ TS (Reyes-Morel, 1986) - in good agreement with the theoretical predictions.