Chapter 11 Toughening in DZC

343

Chapter 11

Toughening in DZC 11.1

Introduction

In this Chapter we shall consider ceramics which contain dispersed zirconia precipitates in various proportions. An example of such dispersed zirconia ceramics (DZC) is the commonly used zirconia toughened alumina (ZTA). The toughening in DZC can arise from two complementary mechanisms depending on the content of t-ZrO2. Recent in situ transmission electron microscopic studies by Riihle et al. (1986) on various ZTA compositions containing a fixed total content of ZrO2 (15 vol%) but a variable proportion of t-ZrO2 (between 0.23 and 0.86) have demonstrated the complementary nature of phase transformation and microcrack mechanisms in the toughening of these ZTA. They found that at low volume fractions of t-ZrO2 there was no stress-induced phase transformation, so that the toughness increment was primarily due to microcrack-induced dilatation around thermally formed m-ZrO2 precipitates (see the low t-ZrO2 end of Fig. 11.1). With an increase in the volume fraction of t-ZrO2, the proportion of t ---, m transformation due to the high stress at a sharp crack tip was seen to increase. In fact, at the largest volume fraction of t-ZrO2 studied (86% of the total ZrO2 content) the stress induced t ---. m transformation toughening mechanism would seem completely to dominate over the microcrack mechanism (see the high t-ZrO2 end of Fig 11.1). This is because the stress reduction generated by the t ---, m dilatation would not permit stress-induced microcrack initiation from m-ZrO2 created by the phase transformation. In this Chapter we shall first (w consider the extreme situation when the ZTA composition contains mostly t-ZrO2 precipitates, so that

344

Toughening in DZC

the toughening is a result of phase transformation alone. We shall then (w consider the other extreme situation when the ZTA composition contains mostly m-ZrO2 precipitates, so that the toughening is primarily induced by microcracking. We note en passant that the contribution of microcracking to the toughening of PSZ or TZP is believed to be only minimal. But even in these materials slight mismatch in the elastic constants of t-ZrO2 and m-ZrO2 can have a significant effect upon the toughening process. We shall study the effect of small moduli differences upon the toughening of TTC in Section 11.4. When the differences in the elastic moduli are large, as in all DZC, the perturbation approach taken in w11.4 is no longer applicable. In these cases we shall introduce in w an approach based on the concept of effective transformation strain.

11.2

Contribution of P h a s e Transformation to the Toughening of D Z C

We shall calculate the toughness increment resulting from the stressinduced dilatational component of the t ---. m transformation in a DZC on the example of a zirconia-toughened-alumina (ZTA) composition containing a high proportion of tetragonal zirconia precipitates and show that it agrees very well with the experimental value. The good agreement is made possible by allowing for the mismatch in the elastic constants between the zirconia particles and the alumina matrix, and for the observed variation in the size of the transformable tetragonal particles with the height of the transformation zone. The actual variation is estimated from experimental data (Riihle et al., 1986) which indicate that large particles (_>0.18#m) are more prone to stress-induced transformation than are the small ones. As far as the mismatch in the elastic constants is concerned, it is taken into account by calculating the two-dimensional dilatation appropriate to the composite of ZrO2 and A1203 (McMeeking, 1986; Rose, 1987a; see w No attempt will be made to estimate the influence of the shear component of the phase transformation or of a stress-induced transformation criterion other than the critical mean stress criterion. The exposition will follow closely the paper by Karihaloo (1991).

345

11.2. Phase Transformation and Toughening of DZC

1300 7

1200

t

5 4 3 exo

=

2

1100 ~ 1000 ~

900

~

800

r~

7OO

1

600

0

I

0

20

I

I

I

100 40 60 80 Tetragonal ZrO2 [%]

F i g u r e 11.1" Bend strength and fracture toughness of ZTA (total ZrO2 content - 15 vol%) as function of t-ZrO2

11.2.1

Experimental

Evidence

For future use and completeness of presentation, it is convenient to summarize briefly the experimental evidence on mechanical properties, transformation characteristics, and microcrack density (Riihle et al., 1986; Evans, 1989). Figure 11.1 shows the variation of four-point-bend strength and ISB (indentation strength in bending) fracture toughness with increasing tZrO2 content. All ZTA compositions containing a fixed total content (15 vol%) of ZrO2 have much higher toughness than pure A1203 (~3.5 to 4 MPax/~). The ZrO2 size distributions in the compositions containing the highest (86% t-ZrO2) and lowest (23% t-ZrO2) fractions of t-ZrO2 have been studied stereologically in a TEM. These studies have shown that the mean particle size of ZTA with 86% t-ZrO2 is 0.4pm and that a critical particle size of 0.6#m exists for spontaneous t ---. m transformation on cooling. From in situ straining experiments in TEM it was found that ZTA with 86% t-ZrO2 had a well-defined transformation zone and that larger particles (>0.18#m) were more prone to stress-induced transformation

346

Toughening in DZC

Nm Win+N,

5

-10

-5

~_4_ 4 4

~4

5_5 .

.5_

L

676__6

6--

3_ -.37 7

7__

s ssr 0 5 10 Distance from crack plane [gm]

F i g u r e 11.2: Variation of ZrO2 particle size in the transformation zone of ZTA containing 86% t-ZrO2. Numbers refer to size range groups of Table 11.1. Nm and Nt are the fractions of m- and t-ZrO2

Size group

Size range log scale (gm)

4 5 6 7 8

0.18-0.24 0.24-0.33 0.33-0.44 0.44-0.59 0.59-0.79

9 N f ( o ) - N2~N, Fig.ll.2 0.140 0.225 0.349 0.186 0.070

VI 0.100 0.200 0.325 0.250 0.125

Weighted NF(O) = Nf(O)Vf 0.014 0.051 0.106 0.046 0.009

T a b l e 11.1: Calculation of f(0)

than were the smaller particles. The size distribution of particles was found to vary along the height of the transformation zone, as can be seen from Fig. 11.2. The various size groups noted on this figure are defined in Table 11.1. In ZTA with only 23% t-ZrO2, on the other hand, no transformation zone was observed. However, from thin foils of known thickness TEM studies showed radial matrix microcracks. All such radial microcracks occurred along grain boundaries in A1203. Moreover, the interface between the A1203 and ZrO2 was usually debonded at the origin of the microcracks. We shall study the microcracking mechanism below in w11.3.

11.2. Phase Transformation and Toughening of DZC

347

E

x

a) A1203 = 85.00 % t-ZrO 2 = 3.45 %

0.i5 " "Microcrack density variation

Crack \

m-ZrO 2 = 11.55 %

/Transformation process zone / /Microcrack process zone 9 y

E

X

0.15 "

b) A1203 = 85.00 %

Crack \

t-ZrO 2 = 12.9 % m-ZrO 2 = 2.1%

F i g u r e 11.3: Steady-state t ~ m transformation (hatched) and microcrack (cross-hatched) process zone around a macrocrack in two ZTA compositions

11.2.2

D i l a t a t i o n a l C o n t r i b u t i o n to the T o u g h e n i n g of ZTA

The aforementioned experimental evidence is graphically illustrated in Fig. 11.3, which shows the steady-state t ~ m transformation (hatched) and microcrack (cross-hatched) process zones around a macrocrack in the two ZTA compositions. Also shown are the distribution of microcrack density parameter and, where applicable, the size distribution of transformed particles, f(y) along the height of the process zone. As mentioned above, the toughness increment in the ZTA composition containing only 23% t-ZrO2 is due mainly to microcracking around the thermally formed m-ZrO2 particles (see Fig. l l.3a). This increment will be calculated in w

Toughening in DZC

348

Here we calculate the toughness increment in the ZTA composition containing 86% t-ZrO2 (i.e. 12.9 vol% of t-ZrO2 out of the total 15 vol% of ZrO2). It is evident from Fig. l l.3b that the toughness increment for this composition must result from both the microcracking around the thermally formed m-ZrO2 particles and the t ~ m transformation of the particles. Note, however, that since t --~ m dilatation results in a reduction in the hydrostatic stress in the transformation zone, no microcracking can be expected from the m-ZrO2 particles formed by stress-induced t ~ m transformation. From the measured heights of process zones due to t ---, m transformation and microcracking, it is clear that the latter zone is completely enveloped by the former. It is therefore reasonable to ignore the minor contribution of microcrack mechanism to the toughening of this compositi.on and to assume that the toughness increment is due almost exclusively to the dilatation resulting from the t ---. m transformation. A rough estimate of the contribution from microcracking and from the deflection of the microcracks may be obtained from the following relation (Li & Huang, 1990) F =

V/1 § 0.S7V! ~/1

-

:'~ Y/(1-

(11.1) ~,~)

where F is the ratio of the effective fracture toughness of the composite in the presence of microcracks and crack deflection to the fracture toughness of the matrix. Vl is the volume fraction of ZrO2 (=0.15) and um (=0.2) is Poisson's ratio of A1203. For the composition under study, the toughening ratio is just 1.07. Neglecting the small contribution from microcracking, we can formally write an expression for the toughness increment from the dilatational component of the phase transformation under steady-state plane strain conditions (w AI~"'iP

--

-ilia

(1 - P)

D(y)

o 2V~

cos

(3r162

(112)

where Ac refers to the area of transformation zone above the crack plane, and P are the effective shear modulus and Poisson's ratio of the ZTA composite, and D(y) is the two-dimensional plane strain dilatation corresponding to the lattice dilatation eT (~0.04) due to t ---. m transformation of a t-ZrO2 particle (Fig. 11.4; see (7.1)). Since the elastic constants of the transforming ZrO2 inclusion (#i 78GPa, ui ~ 0.31) differ much from those of the nontransforming A1203

11.2. Phase Transformation and Toughening of DZC

/

Transformationprocess zone

~Y

349

r

L 0) J r-

v

~)

w!

F i g u r e 11.4: Steady-state t --, m transformation zone in ZTA containing 86% t-ZrO2, showing the coordinate system and the approximate size distribution of transformable t-ZrO2 particles

matrix (#.~ ,,~ 169GPa, ~m ~ 0.2), the effective elastic moduli are appropriate for relating D(y) to eT. The effective moduli of the two-phase composite ~ and ~ can be estimated by using Hill's (1963) self-consistent approach which requires the solution of the following two nonlinear simultaneous equations Vm

t~

N-B----~. + ~ - B m Vm - #~

t~ [

--

-]2

-- ~m

3

- 3B+4~ 6 (B + 2~) 5~ (3B + 4~) __

(11.3)

where B, Bi and Bm refer to the effective bulk modulus, the bulk modulus of ZrO2 (~ 180GPa), and the bulk modulus of A1203 (,~ 226GPa), respectively, and Vm (=0.85) and ~ (-0.15) are the volume fractions of the matrix and the transforming phase. The solution of eqns (11.3) with the indicated values of Bi, t~, Brn, Vm is ~=151GPa, B=218GPa. The relation between the two-dimensional plane strain dilatation D(y) and the lattice dilatation, eT can also be estimated in the spirit of self-consistent theory by considering the deformation of a single, quasispherical transforming ZrO2 particle within a homogeneous matrix that has the elastic constants of the composite (Rose, 1987a)

D(y)-

2(1---ff)f(y)eT 1 + 4-fi/3Bi

(11.4)

where f(y) is the local value of the volume fraction of transformed ma-

Toughening in DZC

350

terial. Experiments show (Rfihle et al., 1986) (see Fig. 11.2) that follows a bell shaped curve along the height of the transformation (H ~ :t:10ttm). However, to simplify calculations, we assume that diminishes linearly along the height of the transformation zone, that in the upper half of the zone (y >_ 0)

s(.) - s(0)(,-

Y

f(y) zone

f(y) such

(11.5)

where the volume fraction of transformed material adjacent to the crack faces, f(0), is estimated from the experimental data as follows. The fraction of various size groups that have transformed within a height of lpm on either side of the crack plane are averaged (Fig. 11.2) and weighted by the corresponding volume fraction of the size group. This is explained in Table 11.1. The sum of these weighted fractions gives f(0) = ENf (0)Vf = 0.226. Substituting eqn (11.5) into eqn (11.4) with this value of f(0), and referring to Fig. 11.4, the steady-state toughness increment (11.1) from t --* m transformation may be rewritten as

AK tip-A

/~'/3J0Br176162 L" r-l/2

.so

L hlsin(r

+A

r -1/2

1 (

1

/3

where H

H rsinr

cos ( 3- r-

cos

H

drdr

(3~) - - drdr (11 6) "

10pm,B - 8H/(ax/3) (see (7.18)), and 'fif (O)eT A-

(1 + 4~/3Bi)

= 0.00341

(11.7)

The first of the two integrals in eqn (11.6) that gives the contribution from the zone in front of the crack tip bounded within the fan between r - 0 and r - 7r/3 can be evaluated analytically and is equal to 1.344Av/-ff. The second integral that gives the contribution from the wake of transformation zone (7r/3 < r < 7r) has been evaluated numerically and is equal to -2.149Ax/~. Substituting these two contributions together with the constant A from eqn (11.7) into eqn (11.6) gives AKtip=-I.31MPax/~, the negative sign indicating the shielding effect of transformation on the crack tip. The fracture toughness of ZTA containing 85 vol% A1203, 12.9 vol% t-ZrO2, and 2.1 vol% m-ZrO2 is therefore approximately equal to K~41umina- AKtip=4.81 to 5.31 MPavfm. This

11.3. Contribution of Microcracking to the Toughening of DZC

351

value is in close agreement with the measured ISB fracture toughness (5.25 5= 0.35 MPav/m-) for this composition (Fig. 11.1). It is interesting to note that were f(y) assumed to remain constant and equal to f(0) over 0 _< y < H as is customarily done (see w the resulting fracture toughness of the composition would work out to be 6.12 to 6.62 MPax/~, which would overestimate the measured value.

11.3 11.3.1

Contribution of Microcracking to the Toughening of DZC Introduction

As mentioned above, there is growing evidence (Riihle et al. 1987) that microcracking in regions of high stress concentration or at the tip of a macroscopic crack may postpone the onset of unstable macroscopic crack propagation in brittle solids such as DZC. For this mechanism to operate it is essential that the microcracks arrest at grain boundaries or particle interfaces and be highly stable in the arrested configuration. Ultimately the macroscopic crack advances by interaction and coalescence of the microcracks. But the microcrack zone can also have a shielding effect on the macroscopic crack tip, redistributing and reducing the average near-tip stresses. There are two sources of the redistribution of stresses in the near-tip stress field of the macroscopic crack. One is due to the reduction in the effective elastic moduli resulting from microcracking. The other is the strain arising from the release of residual stresses when microcracks are formed. The residual stresses in question develop in the fabrication of polycrystalline or multi-phase materials due to thermal mismatches between phases or thermal anisotropies of the single crystals. The spatial variation of these stresses is set by the grain size or by the scale of second phase particles. These residual stresses play an important role in determining the onset and extent of microcracking. Moreover, the microcracks partly relieve the residual stresses producing strains which are manifest on the macroscopic scale as inelastic strains. A continuum approach developed by Hutchinson (1987) will be described below in which it is assumed that a typical material element contains a cloud of microcracks. The stress-strain behaviour of the element is obtained as an average over many microcracks. A characteristic tensile stress-strain curve is shown in Fig. 11.5. The Young modulus E of the uncracked material governs for stresses below ~rc where microcracking

Toughening in DZC

352

~s

~c E ,,,,l

L..

,,

E

T

F i g u r e 11.5: Characteristic tensile stress-strain curve

first sets in. It will be assumed that microcracking ceases, or saturates, above some stress ~r~. The assumption of the existence of a saturated state of microcracking is fairly essential to the analysis described below, as will become evident later. It does seem reasonable to expect that the sites for nucleation of microcracks will tend to become exhausted above some applied stress level when local residual stresses are playing a central role in the microcracking process. Thus, it is tacitly assumed that there exists a zone of nominally constant reduced moduli surrounding an even smaller fracture process zone within which the microcracks ultimately link up. A reduced modulus E8 governs incremental behaviour for stresses above cry. The offset of this branch of the stress-strain curve with the strain axis, ~T, is the contribution from microcracking due to release of the local residual stresses. It can be thought of as a transformation strain. Two of the most important assumptions involved in the formulation of the constitutive law deal with the distribution of the orientations of the microcracks, whether the reduced moduli are isotropic or anisotropic for example, and the stress conditions for the nucleation of the microcracks. Recent microscopic observations of a zirconia toughened alumina (Rfihle et al. (1986) suggest that the microcracks which form in this material have a more-or-less random orientation with no preferred orientation relative to the applied stress. This would be consistent with the random nature of the residual stresses expected for this system. Nevertheless, there is not yet nearly enough observational information or theoretical

11.3. Contribution of Microcracking to the Toughening of DZC

353

insight to justify any one constitutive assumption. The approach taken below is to consider a number of reasonable options, so that the results discussed here will serve to bracket actual behaviour and give some indication of which uncertainties are most crucial to further development.

Y

E,v

F i g u r e 11.6: Geometry of the microcracked zone surrounding the tip of a semi-infinite crack From the point of view of mechanics, we consider the problem shown in Fig. 11.6. A microcracked region, Ac, surrounding the crack tip has reduced moduli which are uniform and isotropic. In analogy with phase transformation, a uniform dilatation ~T is also present associated with the release of residual stress. The crack is semi-infinite with a remote stress field specified by the applied stress intensity factor K appt, modelling a finite length crack under small scale microcracking conditions. The near-tip fields have the same classical form but their stress intensity factor, K tip, is different. It is the toughening ratio KaPPZ/Ktip which is sought as a function of the moduli differences, ~T and the shape of the zone. The knowledge of this ratio is not sufficient to predict the toughening increment due to shielding, because microcracking reduces the intrinsic toughness Kc of the matrix. The knowledge of KaPPZ/Ktip for different situations, such as stationary or growing cracks, can be used to make comparative assessments of macrocracking behaviour and to gain insight into phenomena such as stable crack growth.

11.3.2

Reduction Stress

in Moduli

and Release

of Residual

The following two examples are chosen to illustrate the way microcracking can reduce the moduli of a brittle material and give rise to inelastic strain by release of residual stress.

354

T o u g h e n i n g in D Z C

P e n n y - s h a p e d m i c r o c r a c k s in a p r e s t r e s s e d s p h e r i c a l p a r t i c l e

E,v

a) ,,

2b E,v

b) r

-I

F i g u r e 11.7: Two prototypical microcrack geometries" (a) pennyshaped microcrack in a spherical particle, (b) annular microcrack outside a spherical particle Consider the configuration of Fig. l l . 7 a which shows an isolated spherical particle or grain of radius b embedded in an infinite matrix. Both particle and matrix are assumed isotropic and with common Young's modulus E and Poisson's ratio u. Suppose the particle sustains a uniform residual stress prior to cracking. Let ~rR denote the normal component (assumed positive) acting across the plane where the microcrack will form. There is zero tangential traction on this plane. Now suppose a penny-shaped microcrack is nucleated which arrests at the interface of the particle and the matrix as shown in Fig. 11.7a. The volume of the opened microcrack is 16b3 aR AV - T ( 1 - u 2) E (11.8) The release of the residual stress creates an inelastic strain contribution. If the microcrack forms within a material element of volume V and if interaction with other microcracks is ignored, the inelastic strain contribution is Aeij

= A V V ninj -

16 b3 O'R 3 V (1 - u 2 ) - - f f n i n j

(11.9)

where ni is the unit normal to the plane of the microcrack. This is a

11.3. Contribution of Microcracking to the Toughening of DZC

355

uniaxial strain contribution with dilatational component Ackk =

16b a an (1 - u2)__~ 15 3 V

(11.10)

The formulation of the microcrack also increases the compliance of the material element (Budiansky & O'Connell, 1976). If crij is the macroscopic stress experienced by the material element, the increase in strain due to a component of stress acting normal to the plane of the microcrack (i.e. crnn = O ' i j n i n j ) is b3 finn (11 11) A~nn = -3- ( 1 - u 2) V E Any component of stress acting tangential to the plane of the microcrack (i.e. crnt - ~rijnitj, where ti is parallel to the crack face) gives rise to an increase in the corresponding strain component which is 16 (1 - g2)b 3 ~ t Ac~, = y ( 2 _ u ) V E

(11.12)

These contributions to the strain are also based on the assumption that interaction between the microcrack and its neighbours can be ignored. If the microcracks have random orientation with no preferred alignment, the microcracked material will be elastically isotropic on the macroscale and the strain due to the release of the residual stresses will be a pure dilatation. Suppose there are N microcracks per unit volume and let 0 be the measure of the microcrack density, where g is the average of Nb 3. With E and P denoting Young's modulus and Poisson's ratio of the microcracked material, the total strain following microcracking is obtained by averaging the contributions (11.9)-(11.12) over all orientations with the result (cf. (3.17))

+ v O'ij -- ~~ gij -- 1 __ -~O'kk~ij Jr- -310T~i j

(11 13)

0T _ ~ ( 1 - - U 2 ) 0 E

(11.14)

E

where

The notation here is deliberately chosen to be the same as that for a dilatational phase transformation since at the macroscopic level the dilatation due to release of the residual stress is indistinguishable from that due to phase transformation. The modulus E and Poisson's ratio

Toughening in DZC

356

of the microcracked material can be obtained from 3 2 ( 1 - v ) ( 5 - v) P = 1 + -~0 45 (2-v)

(11.15)

and B --

B

16 (1 - v 2) -

1 + -

9 (1 - 2u)

~

(11.16)

where # and B are the shear and bulk moduli of the uncracked material and ~ and B are the corresponding moduli for the microcracked material. These estimates of the moduli, which ignore microcrack interaction, agree with the dilute limit of estimates which approximate interaction (Budiansky &: O'Connell, 1976). They are reasonably accurate for values of Q less that about 0.2 and 0.3, and it is expected that the residual stress contribution in (11.13) will be accurate within this range as well. Annular

microcrack

around

a prestressed

spherical particle

Now, consider a spherical particle which has a residual compressive stress due, for example, to transformation or developed during processing as a result of thermal mismatch between particle and matrix. Referring to Fig. 11.7b, we suppose that the particle nucleates an axisymmetric microcrack at its equator with the outer edge of the crack arrested by some feature of the microstructure. Usually such a microcrack runs along a grain boundary and arrests at a boundary junction. We model the situation by taking the particle to be under a residual uniform hydrostatic compression O'ij "-- --O'R6ij prior to cracking. If, for example, this residual stress arises as a result of a dilatational transformation strain in the particle of r l~T~ij, then

crn =

2EOT 9 ( 1 - v)

(11.17)

The moduli of the matrix and particle are again taken to be the same. The residual normal traction in the matrix acting across the plane of the potential crack is a-

a0

(11.18)

where ~0 - crn/2 is the tensile circumferential stress in the matrix just outside the particle, and r is the distance from the centre of the particle.

11.3.

Contribution of Microcracking to the Toughening of DZC

357

The volume of the annular crack due to the partial release of the residual stress (11.18) is given approximately by AV -- 7r2(1-

v2)ab2 ( 1 _ ab ) 2 (r0 E

(11.19)

Once the microcrack is nucleated it gives rise to an additional strain contribution (in a material element of volume V) in the direction normal to the crack plane

bc2-A~,~n - ~r2(1-v2)-~

( 1 + ~2c) -ann ~ F(c/b)

(11.20)

where ann is again the macroscopic stress component normal to the plane of the crack, and c = a - b. The function E(c/b)is 1 when c / b - 0 and monotonically decreases to 0.81 when c/b --+ co; it is very close to 1 for c/b _< 1. (The formula (11.20) can be derived from results given in the handbook by Tada et al. (1985). The counterpart to (11.20) for the shear strain contribution Ac,~t is not available). With N noninteracting annular, randomly oriented microcracks per unit volume, the strain is still given in terms of the macroscopic stress by (11.13) where now from (11.19)

0T -

NTr2(1-

~,Z)ab2 1 -

-~-

(11.21 /

The result (11.20) is not sufficient to determine estimates for E and since A~nt is also needed. However, if one assumes that the ratio of ent/trnt to ~nn/trnn is the same, or at least approximately the same, for the annular crack as for the penny-shaped crack, then E and F can still be obtained from (11.15) and (11.16). Now, however, by comparing (11.11) and (11.20), one sees that the crack density parameter must be taken as

37r2Nbc2 1 + ~-

F(c/b)

(11.22)

16

This formulation provides the density of annular microcracks measured in an equivalent density of penny-shaped cracks for the purpose of determining the reduction in moduli. The parameter proposed for arbitrarily shaped microcracks, Q = 2NA2/(TrP) where A and P are the area and perimeter (inner plus outer) of the crack, provides an excellent simple

358

T o u g h e n i n g in D Z C

approximation to (11.22). Riihle et al. (1987) found in ZTA containing a low volume fraction of t-ZrO2 that each ZrO2 particle is circumvented by a radial microcrack, consistent with the symmetry of the residual strain field around each particle. They also found that the microcrack density diminished with distance from the crack plane; the maximum density ~0c adjacent to the crack faces suggests a saturation value determined by m-ZrO2 content (Fig. 11.3a).

11.3.3

Ktip/K ~ppt for Arbitrarily Shaped Regions Containing a Dilute Distribution of Randomly Oriented Microcracks

Uniformly distributed microcracks Some general results for the plane deformation problem depicted in Fig 11.6 will now be presented. A semi-infinite crack lies on the negative x-axis. Within the microcracked region r _< R(0) , the material is governed by (11.13) where 0T can be thought of as a stress-free dilatational transformation strain. Within Ac, E, P and 0T are taken to be uniform. Outside this region the material is governed by s

l+v

v

E

O'ij -- --~O'kk~ij

--

(11.23)

The region Ac is restricted to be symmetric with respect to the x-axis. In analogy with phase transformation, when E - E and P - v, K tip is given by (7.14). When E and V differ from E and v, numerical work is generally required to obtain the relation K tip and K ~ppz. However, this relation can be obtained in closed form to lowest order in the differences between the moduli governing behaviour within and without A~. Moreover, to lowest order in these differences the contributions to K tip from 0 T and from the reduction in moduli within Ar can be superimposed. We shall see later in w that the superposition assumption is only partially valid. We proceed by considering the case 0 T - O, when one can conclude from dimensional analysis alone that l~[ t i p

Kapp z -

--~ F(---~, u,-if)

(11.24)

where F also depends on the shape of Ac, but not on its size. However, it is known that this relationship can be reduced to dependence on just two special combinations of the moduli (the so-called Dunders' parameters).

11.3.

Contribution of Microcracking to the Toughening of DZC

359

For present purposes the most convenient choice of moduli parameters is 1

61--i; v

1

62 - 1 ; / /

u~ - u

]

(11 26)

which both vanish in the absence of any discontinuity across the boundary of Ac. These parameters emerge naturally in the analysis which we shall here omit. Interested readers may consult the paper by Hutchinson (1987). With this choice Ktip I~appl --

f(61,~2,shape of

(11.27)

Ac)

The following result is exact to lowest order in 61 and 62 K tip i~appl

3 ---- 1 + (kl - ~5)61 + (k2 + )52

(11.28)

where kl -

k2 =

~

lf0~

(11 cos 0 + 8 cos 20 - 3 cos 30)

ln[R(O)]dO

27rl~0?i"(cosO+cos20)ln[R(O)]dO

(11.29)

(11.30)

The integrals defining kl and k2 also appear in a different context, as we shall see in the next section (w11.4). Since the collection of terms in each integrand multiplying ln[R(0)] integrates to zero, kl and k2 are unchanged when R(O) is replaced by AR(O) and are thus dependent on the shape, but not on the size, of Ac. If Ac is a circular region centred at the tip, kl = k2 = 0. 11.3.4

KtiP/K~PPt f o r tionary

and

two Nucleation Steadily-Growing

Criteria

for Sta-

Cracks

The results of the previous Section are now specialized to specific zone shapes dictated by two possible microcrack nucleation criteria. The first is based on the mean stress; the second is based on the maximum normal stress. In each case, it will be assumed that there is no preferred orientation of microcracks so that the reduced moduli are isotropic. Results

Toughening in DZC

360

for both stationary cracks and cracks which have achieved steady-state growth conditions will be given so as to assess the potential for crack growth resistance following initiation. In every example, the zone shape and size are determined using the unperturbed elastic stress field (7.3) since this is consistent with our limited aim of obtaining just lowest order contribution to K tip. The perturbation of the size of the zone is likely to be relatively unimportant for the effect of the reduced moduli even for non-dilute crack distribution since the lowest order results for Ktip/K appl are independent of zone size, as discussed in the previous Section.

S t a t i o n a r y Crack w i t h N u c l e a t i o n at a Critical M e a n S t r e s s

Zm

Saturated state

$

Zm

Saturated state

C

Zm

Zm

C

Zm

Simplified criterion

,

a)

N

b)

N

F i g u r e 11.8: Variation of microcrack density N with mean stress With Em = ~rkk/3 as the mean stress, suppose microcracks begin nucleating at E ~ and the nucleation is complete at E~ with a variation in microcrack density N as indicated in Fig. 11.8a. To lowest order the elastic stress distribution (7.3) can be used to determine the zone shape and the distribution of the microcrack density within the zone. The distribution of the density and the relation of the inner region of uniform muduli to the full microcrack region fits precisely into the situation discussed in the preceding Section. Thus, the change in K tip due to the moduli reduction is the same, to lowest order, as when the microcracks are uniformly distributed throughout the zone. We will therefore restrict attention to the simplified nucleation criterion indicated in Fig. l l.8b and take

11.3. Contribution of Microcracking to the Toughening of DZC

~-0

361

for (Y]m)max < ~Crn (11.31)

= N for ( ~ m ) m a x ~ ~ c

The 0T-contribution to K tip does depend on the distribution of the microcrack density, but this can be evaluated fairly simply using (7.10) if desired. Here only the results for the simplified nucleation criterion (11.31) will be given. There will be a transition region just within the boundary to Ac in which the microcrack density varies from zero to the saturated value, but in the limit corresponding to the lowest order problem the transition region shrinks to zero. Imposing E m - E~ on the elastic field (7.3) one finds

R(O) -

2

~--~(1 + u) 2

(Kappt)20

cos 2 2

E~

(11.32)

which is identical in form to the transformation zone boundary (7.12), except that the critical mean stresses for transformation and microcrack nucleation can be significantly different. The boundary of the microcracked zone is shown in Fig. 11.9a. Then, evaluating kl and k2 in (11.29) and (11.30), one obtains ]r -- 3/16 and k2 = - 1 / 4 . The 0T-contribution is found to be identically zero (as for a stationary crack under phase transformation (w so that the combined effect is given by just (11.28)

K tip

K.pp t = 1 -

_~

1

61 + ~62

(11.33)

To specialize the result even further we will use the results (11.15) and (11.16) for the reduced moduli ~ and B in terms of the crack density parameter ~0 which in turn is given by the average of Nb 3, or by (11.22), or by any other appropriate choice depending on the nature of microcracking. To lowest order in ~ one can show that

-

u-

163 (3 - v)(1 - v 2) 1--5( 2 - u) 0

and

32(5-

(~1 -- ~

_ U) ~0 -- 1.0990

1

(11.34)

Toughening in DZC

362

2 appl

(l+v) (K

c 2

/ Era)

1.0 Boundary of wake for steady-state problem

/

0.5

a)

Boundary for stationary roblem

0

0.0

I

'

I

0.5

-

1.0

x

c 2 (1 +v) 2(g appl/ ]~m)

(KPPl[ ]E1c 2)

0.4

H

b) 0.0

0.4

0.2

(Fppl

c

2

/Z I ) F i g u r e 11.9: Zones of microcracked material for stationary and steadily growing cracks for two nucleation criteria. (a) Critical mean stress criterion, (b) critical m a x i m u m principal stress criterion

62 = 16u(1 - 8u + 3u 2) 4 5 ( 2 - u) ~-

-0.095~o

(u - 1 ~)

(11.35)

and thus Ktip

= 1 - 2 ( 3 5 - l l u + 32u 2 - 12u 3)~, Kappl 4 5 ( 2 - u) - 1-0.9196

(u-

1

~)

(11.36)

11.3. Contribution of Microcracking to the Toughening of DZC

363

S t e a d i l y - G r o w i n g crack w i t h N u c l e a t i o n at a C r i t i c a l M e a n Stress A crack which has extended at constant K appl has a wake of microcracks as indicated in Fig. 11.9a. With the nucleation criterion (11.31) in effect, the leading edge of the microcracked zone is given by (11.32) for [01 < 600 , and the half-height of the zone is given by H -

x / ~ ( l + u ) 2 (KaVVt) 127r \ ~

(11.37)

The values of kl and k2, which have been computed by numerical integration, are kl - 0 . 0 1 6 6

and k2 = - 0 . 0 4 3 3

(11.38)

The combined effect of moduli reduction and microcrack dilatation is given by (11.28) and (7.14) 1

K tip

is

-

-

1 - 0.60861 4- 0.70762

(1 4- u) EO T

47rx/~ (1 - u) E~

= 1 - 0.60861 4- 0.70762 - 2(1 - u)Kavv z

(11.39)

where the 0T-contribution is the same as that for the corresponding transformation problem (7.21). Equations (11.35) for 51 and 52 still pertain and for tt = 1/3 Ktip EOT v/-H Kapp I -- 1 - 1.2780-0.3215 i~app-"-'-------------[-

(11.40)

By comparing (11.36) and (11.40), one notes that the shielding contribution due to moduli reduction is about 40% larger for the growing crack than for the stationary crack. This will add to crack growth resistance but the major source of resistance is likely to come from the release of residual stress (i.e. from 0w). Even without growth, however, moduli reduction provides some shielding according to (11.36) although how much extra toughness this generates cannot be predicted without knowledge of the toughness of the microcracked material within Ac, as already emphasized.

Toughening in DZC

364

Stationary C r a c k w i t h N u c l e a t i o n at a C r i t i c a l M a x i m u m N o r mal Stress Now suppose that the microcracks are still nucleated with no preferred orientation so that within Ac the stress-strain relation is still (11.13), but suppose that nucleation occurs when a maximum principal stress O"I reaches a critical value E~, i.e. ~0- 0 for (~I)max < ~ (11.41)

= N for (~I)max _> ~

where as before, ( )max signifies the maximum value attained over the history. The boundary Ac as determined by (7.3) is now (of. (10.6)) 1 (

R(O) - ~

0 1 )2(I.~vPz)2 cos ~ + ~sin [0[ El

(11.42)

and this is shown in Fig. 11.9b. The value of kl and k2 have been obtained by numerical integration of (11.29) and (11.30) with the result kl

--

0.0779,

k2 - -0.0756

(11.43)

The combined effect of moduli reduction and microcrack dilatation is given by (11.28) and (7.14)

Ktip EoT Kapp I - 1 - 0.54761 + 0.67462- 6 r ( 1 - v)E~ = 1 - 0.54761 + 0.67462 - O.1060EOT (1 - l])]~ appl (11.44) where the half-height of Ar from (11.42) is obtained at 0 is

Hfor v -

1/3 and with

61

0.2504(KappZ) 2 E~

(11.45)

and 62 given (11.35), (11.44)reduces to

Ktip i~appl

74.840 and

EOT v ~

---- 1 -

1.1530-0.159 K,~pp-----------7

(11.46)

11.3. Contribution of Microcracking to the Toughening of D Z C

365

S t e a d i l y - g r o w i n g c r a c k with n u c l e a t i o n at a c r i t i c a l m a x i m u m normal stress Now the zone Ac is specified by (11.42) for IOl < 74.840 and by H for 101 > 74.840 where H is given by (11.45). Evaluating the integrals in (11.29), (11.30) and (7.14) numerically, one finds

KtiV

EO T

K apvt

= 1 - 0.6736a + 0.82262 - 0.1329

= 1 - 0.67361 + 0.82262 - 0.2656 which for u -

( 1 - v) E~I

EOr~/g (1 - u)Kapp i

(11.47)

1/3 and 61 and 62 given by (11.36) becomes

Ktip EOT v/-~ Kapp I = 1 - 1 . 4 1 7 0 - 0 . 3 9 8 Kapp-----------~

(11.48)

The predictions for this case are not very different from those based on a critical mean stress. The shielding due to reduction in moduli is larger in each case by about the same amount for the steadily-growing crack compared to the stationary crack. For nucleation at a critical maximum principal stress there is some shielding even for the stationary problems due to 0T. This is not the case for nucleation at a critical mean stress. Effect of z o n e s h a p e o n shielding For cases, such as those discussed above, in which the moduli of the microcracked material are isotropic and the release of residual stress gives a pure dilatation 0T, the general result can be used to gain qualitative insight into the effect of zone shape on shielding. For the 0T-contribution, it follows immediately that, because R1/2(0) is modulated by cos(30/2)in (7.14), decreases in R in the range 101 < 600 and increases in the range 101 > 600 will increase shielding. The trend is similar for shielding due to the reduction in moduli. Note from (11.36) that 61 is generally much larger in magnitude than 62 and will be the dominant of the two parameters in determining K tip. Therefore, the influence of shape on K tip comes about mainly through kl. By (11.29), the integral for kl involves ln[R(0)] modulated by

f(O) -

1 327r (11 cosO + 8cos20 - 3cos30)

(11.49)

Toughening in DZC

366

0.2 -

f ( 0 ) = ( l l c o s 0 + 8cos 2 0 - 3cos 30)/(32x)

0.1 -

~

i

Increasing R(0) increases shielding

0.0 ..................... '------~---

-0.1 Decreasing R(0) increases shielding -0.2 0~

~ 90 ~

l 180 ~

F i g u r e 11.10: Plot of the function f(O) appearing in the expression for kl and its implication for change in shielding stemming from changes in shape of microcracked zone

This function is plotted in Fig. 11.10, and is seen to be positive for 101 < 70.50 and negative for 101 > 70.5 ~ Thus, with a circular shape of reference (kl = k2 = 0), shape changes involving decreases in R for 101 < 70.50 and increases for 101 > 70.50 will increase shielding. However, the influence of shape change is not nearly as strong as in the case of the 0T-contribution. The examples worked out above suggest that k l and k2 are generally quite small so that shielding will not be markedly different than that afforded by a circular zone centred at the tip. Even the addition of the wake in the steady-state problems only increases the shielding by 30-40% over the circular zone. To summarize, the increase in shielding of the growing crack over the stationary crack due to the reduction in moduli (the Q--contribution) is between 30 and 40%. Values of ~ of about 0.3 near the crack tip have been observed by Riihle et al. (1987), corresponding to about a 40% reduction in K tip due to this effect. The shielding contribution due to release in residual stress (the OT-contribution) is exactly the same as in the corresponding transformation problem, and the shielding is significantly greater for the steadily growing crack than for the stationary crack. It would appear that strong resistance curve behaviour would stem mainly from the release of the residual stresses.

11.3. Small Moduli Differences and Toughening of TTC

11.4

Contribution of Small M o d u l i Differences to the T o u g h e n i n g of TTC

11.4.1

Introduction

367

In all chapters dealing with toughening induced by phase transformation it was assumed that the elastic constants of the transformed particles are identical to those of the untransformed matrix material and in the case of toughened zirconia this is very nearly true. However even in this material there is a small difference between the elastic constants of the tetragonal and monoclinic phases (Green et al., 1989). In this Section a perturbation expansion, that exploits the smallness of this difference in elastic constants, will be used to make an approximate estimation of the effect of this difference on the fracture toughness of the material. The exposition will follow closely the paper by Huang et al. (1993). As expected the influence of lowest order moduli differences is negligible, but the perturbation technique reveals two rather unexpected features of the solution. First, it shows that, even to the lowest order, the fracture toughness cannot be assumed to be a simple superposition of contributions from dilatation and moduli mismatch considered in isolation from each other. Secondly, it shows that the joint effect of the two is qualitatively different from the prediction based on the concept of effective dilatational strain (see w below). In view of the well-known similarity between the crack tip shielding by transformation and microcrack induced dilatation, the above features are likely to carry over to the microcracking problem that we considered in w We will again consider the plane strain model for steady-state crack growth that we investigated in w It will be assumed that the elasticity tensors Ca~76 of the transformed material (t-ZrO2) and Ca~7~ of the composite material (t-ZrO2 + m-ZrO2) consisting of particles that have undergone a mean stress-induced dilatant transformation embedded in a matrix of untransformed material in the neighbourhood of the crack are isotropic. In common with the study in w the shear strains induced by the phase transformation will not be included in the analysis. In order to make any progress it will be further assumed that Cap76 and Ca~-y6 are proportional so that (11.50)

368

T o u g h e n i n g in D Z C

As will be seen below this last assumption entails that Poisson's ratios v, ~ of the untransformed precipitates and the composite material be equal. The plane strain elasticity tensors are

C ~ . y 6 - 2~

-

-

C ~ ~.y 6 - 2-fi

{ {-

1

}

1 - 2v 6~6"y6 + ~(6,~.y6z6 + 6~.y6~6) v 6~.y6 1 - 20

+

1(6~.y6~6 + ~.y~,~ ) )

-2

6

(11.51)

with ~-(1

+e)p;

V- v

and the Greek subscripts range over x and y. The effective shear modulus ~ for the composite material in the neighbourhood of the crack can be calculated from the moduli p and pt of the untransformed and transformed materials respectively by Hill's selfconsistent method. For this we need to solve the two equations (11.3). In the case of toughened zirconia with volume fractions of the matrix Vm and transforming phase Vt equal to 0.7 and 0.3, shear moduli # 78.93 G P a and #t _ 96.29 GPa, bulk moduli B - 143.06 G P a and B t - 174.54 GPa and Poisson's ratio v ~ v t - 0.267 (Green et al., 1989) the moduli of the composite are ~ - 84.08 GPa and B - 151.73 GPa. This leads to the value : - 0.065 for the small parameter.

11.4.2

Mathematical

Formulation

Let the x, y plane D contain a semi-infinite crack coincident with the negative x-axis y - 0, x _< 0 subject to a remote mode I loading that would induce a stress intensity factor K appl at the crack tip in the absence of transformation. Let ~ be the region of steady-state transformed material surrounding the crack consisting of a parallel sided wake region of width 2H behind a small region of transformation ahead of the crack tip bounded by a smooth curve C. The rest of the D-plane exterior to ~ will be designated D - f~ (Fig. 11.11). It will be assumed that the concentration c of transformed particles is constant throughout ~. The plane strain c~Z T due to transformation is related to the stress free dilatation 0T that would occur in an unconstrained particle by equation

:.~ - g ( l +

(11.52)

11.4. Small Moduli Differences and Toughening of T T C

369

where cOT is the volumetric transformation strain. The concept of effective transformation strain introduced by McMeeking (1986) (see w below) for transforming composites in which the elastic properties of the transforming particles differ from those of the matrix would require that c in (11.52) be replaced by an effective coefficient ~. For purely dilatant transformation strains, E = { B t ( B B)}/{-B(B t - B)}. For the zirconia composition under consideration, the effective dilatational strain would be ~0T, with ~ - 0.3168. In analogy with the three-dimensional Eshelby formalism used in Chapter 9 (w if c~z is the total strain then the stress is given by -

/C~z~6

in D - f~ (11.53)

t

--

T)inf ]

The equilibrium equations are

T )] , Z _ 0 in

(11.54)

and continuity of surface traction across the boundary 0f2 of the transformed region gives C

OUT IN T ~.y~%~ n~ - -C~-y6(e~6 - c.y6 )n~

(11.55)

where nz is the outward normal to the boundary of the transformed region, assumed positive if pointing from the inside (IN) to the outside (OUT) of this region. The perturbation scheme is straight forward; all the dependent variables are expanded in power series in the small parameter c 0 1 (ra~ -- ~ra~ + cera~ + ...

(11.56)

and so on. When these expansions are substituted into the above equations and the coefficients of like powers of c on both sides of the resulting equations are equated the following hierarchy of perturbation equations is obtained: The O(1) equations are

370

Toughening in D Z C

appl

g, v

(D-~)

Transformation zone

~Y

Crac~k':,,,,,,,,,,,~Cx

I Kapp I

Figure 11.11" Steady-state transformation zone surrounding a semiinfinite crack, showing the coordinate system

o Co~~.r6c.r 6,~

~0 -

in D-f~

T ( Co~76E76,~

-

(11.57) in f~

with OUT

C~Z~6 [r176

T

nz -- -C~z.r6c.r6n ~

(11.58)

on the boundary OQ. The O(c) equations are

1 _{0 C~z'r6 c~6'Z

in D - f ~ o

(11.59)

_OIN )nil nZ -- -Cafl'r6(c'r6T - 6-'r6

(11.60)

with Cafl76 [516]~IN

on the boundary 0f~. From (11.57) it can be seen that the O(1) strain r

is due to a

11.4. S m a l l M o d u l i Differences and Toughening o f T T C

371

T inf2(cf. (9.6)), and surdistribution of body force F ~ - -Ca~6~76,Z face traction T ~ - Co, Z.y6c.~Tnz on the surface 0f~ (cf. (9.7)) plus the strain due to the external load. In Chapter 9 we showed how the weight function ha can be used to calculate the change in stress intensity factor due to the transformation. The same is applicable to two-dimensional problems, so that we may write

K~176176

We have retained the symbol ha for the two-dimensional weight function. It will be defined later. When the expressions for F ~ and T~ are substituted into this expression and the divergence theorem is used to reduce the second integral above the final result is

K~ f f

(11.61)

Equation (11.61) is the two-dimensional mode I counterpart of the three-dimensional eqn (9.9). The O(~) change in the stress intensity factor can be calculated by the same method from (11.59). The result is K1 - f i n

C,~ z'y ~( ayT - ~-y 6 ,~ d A o )ha

= K~- /

(11.62)

fr~ Caz.y6r o ha ,~ d A

and the stress intensity factor at the crack tip is K tip = K avpl + K ~ + c K 1 "b O(c 2)

(11.63)

In order to calculate the above integrals we introduce the independent complex variable z = x + iy (-5 = x - iy), the complex displacement w~ -

u~

(11.64)

+ iu ~

and the complex plane strain weight function 1

{-X

1

4~--x/~

(11.65)

where X = 3 - 4~. The two-dimensional mode I weight function was

372

T o u g h e n i n g in D Z C

derived by Bueckner (1972) and Rice (1972) before the corresponding three-dimensional weight functions (w 1987). It is however instructive to derive it from the three-dimensional mode I counterpart (9.15). This requires an integration of eqns (9.20) from -cx~ to oc with respect to the coordinate z (which we here donote s to avoid confusion with the complex variable z), and so giving F ~ P ( ~ , y, ~)d~ - - 2(1 _ 1~ , ) , ~ I m

P -

f -

~ Q ( x , y, s ) d s -

f

1/

= -~

1 2(1 - u ) v ~ Re

{ 1~} -

- a , ~ (11.66)

{1}

- G,, (11.67)

~

f,zaZ+f

1/ (d,. -d,~ )d~

( d , . + d , ~ )dz + -~

1

- 4(1 - u)v/~

(/1

-~dz +

_ 1 x/~) - 2(1- u)v/~ (V~ +

(11.68)

In common with the three-dimensional weight functions which were all expressed in terms of the derivatives of G, the two-dimensional mode I weight function can be expressed in terms of the derivatives of (~ (cf. (9.15)) hlx=-(1

- 2 u ) C Rez -1/2 -

CYlmz-3/2 2

hly=-(2-

2 u ) C I m z -1/2 --

CYRez-3/2 2

(11.69)

where C = [2(1 - u ) ~ ] -1 . We are now in a position to construct the complex weight function

h - nix "t- ihly

(11.70)

As we shall only consider mode I loading in this Section, hr, and hIu will be simply denoted as hi and h2 (11.65). The two-dimensional weight

373

11.4. Small Moduli Differences and Toughening of TTC function may finally be written as h=-(1- 2v)-~-

ilm~

2

1 ( -~z~z 1 2x/~(1 - u) + x/~

1)

Vff

z+:) 4:a/2

(11.71)

Substituting (11.51) and (11.52)into (11.61) gives g 0

2/~(1 + u)cOT 3(1- 2u)

_-

Next,.we show that h~,~ - 2Re

f/,h.,odA

(11.72)

{oh}

(11.73)

The following identities are true by definition h,x - h,z + h ~ - h:,~ + ih2,~ -ih,u - h,z - h y - -ih:,u + h2,u.

whence we obtain (

1

( X

1)} 4~/2

ho~O~- 2Re v/~( ~_ u) 4~/2 X-1 { 1 1 } 4Vc~(1_ u)z--3~ + z~2 (1-2u) { 1 1} 4x/~(1 - u) ~ + ~

(11.74)

Substitution of (11.74)into (11.72) gives (cf. (7.10)) K~ = 6v~-~(1- u)

~

+~

1}

dg

(11.75)

Integration with respect to x further reduces this result to a contour integral around the curved boundary C of the transformed region ahead of the crack tip (Fig. 11.11)

374

Toughening in D Z C

/fa

{ 1

1

c

1

z-~ + ~-nT~} dA - /_oo d~ / dY { z-~ 1 + ~--~}

1 } - - 2 / C { z -1~ + -vv dv + lim ~0 0 ~0 2~"2 cos(O/2) cos(O)dOv/Tdr

0---*0

(11.76)

The last term results from the exclusion of a small (singular) circular region from the area integral. It vanishes as ~ tends to zero, so that KO -

P(I+u)cO T /c{

- 3x/2"~(1 - u)

1

1 }

- ~ + -~z dy

(11.77)

If the effect of the transformation on the location of the boundary C is neglected, i.e. only the far-field stresses due to K ~ppz are taken into account, such that the mean stress is (7.11) then the shape of C is given by (7.12)

1 }

( l + u)Kavpt { 1

3,/~~

~ +

7r < a r g ( z ) < 7r 3 _ ~ (11.78)

--1,

where ~m c is the critical mean stress that induces the tetragonal to monoclinic phase transformation. As shown on several occasions, on this boundary the integral on the right of eqn (11.77) can be calculated exactly

T c #cOv ~m [~" - - (1 7-~-~Tvv, jy dy

KO _

(11.79)

l

where from

( 11.78) y~, - - y , - rl~ sin(Tr/2) - 3-~23 ((1 + u)I'(app') 3v/~a~

so that ooK app l K~= - ~

4,/~

(11.80)

where r (3.26) is the parameter introduced by Amazigo & Budiansky (1988) which is a measure of the strength of transformation in region f~.

11.4. S m a l l M o d u l i D i f f e r e n c e s a n d T o u g h e n i n g o f T T C

375

In the method proposed by McMeeking (1986) for binary transforming composites which we shall describe in w below, the expression for K 0 retains the form of (11.80), but in the definition ofw (3.26) the matrix elastic constants and c must be replaced with the composite elastic constants and ~, respectively, The corresponding strength of transformation will be designated ~-. When expression (11.51) is inserted into the integral in (11.62) it becomes I -

C~6e~6

- 2p

j/o{.

0

1 - 2v ~`~`~h~'z + ~ o' ~ h ' ~ ' z

}

dA

(11.81)

In order to calculate this integral it is necessary to calculate c~ from the complex displacement w ~ (11.64). Either the weight function method of Rice (1985a) or the method of Rose (1987a) can be used to do this. 11.4.3

Calculation

of Displacement

Field

The complex displacement field w ~ consists of three parts: the displacement field w T due to transformation in the uncracked body; w L due to the external load K appl and w tip due to transformation in the cracked body. First we derive w T. The Muskhelishvili complex potentials (I)(z) and ~(z) for a centre of dilatation of unit strength, lying at any point z0 in an uncracked infinite body are given in (4.28) and (4.25). The corresponding displacements in plane strain are then obtained from (4.24), with w T = u , + iu u. For constant dilatational strain inside the transformation zone, D in (4.28)is given by (7.9). Thus, for a single centre of dilatation at z0 wT =

(1 + ~,)cOT 6~r(1 -- ~,)(z- z0)

(11.82)

Next, the plane strain displacement due to external load K appt can be easily found out UL

K~VVti r

Toughening in DZC

376

vL - K appl i ' r

- 4#

2--~r[(2X +

1) sin(C/2) - sin(3r

The complex displacement field is W L -- UL + iv L - ~

e-

(

e-

z+:)

(11.83)

Finally, we adopt the method of Rice (1985b) to derive the complex displacement field due to transformation strain in a cracked body ~. From the invariance property of weight functions, if the crack tip is moved from x = l to the neighbouring position x = l + 51, the body is subjected to O(1) change in the stress intensity factor (i.e. the change in stress intensity factor due to transformation strain without moduli difference) denoted K ~ given by

Ow tip Ol =

2(1 -

v 2)

E

K~

l)

(11.84)

where w tip = u tip + iv tip is the part of the displacement field due to the crack growing from l = -cxD to l = 0 into the region Ft

2(1--v2)#(1+V)c0T //~{

cgwtiP 0l

-

E

6 x / ~ ( 1 - v)

1

2 ~ / ~ ( 1 - ,,)

-X

.. ~

2,/2 - t

1

1

(zo - 0 3/2 § (:o - l ) 3/2

1 ~/:-t

z+-2-21 4(:- l)~/:-

} l

dAo (11.85)

Integration with respect to l gives

wtip

( 1 + v)cO~ = 247r(1- v ) / / a +

X { - ,/~(~%- + 48)

X

2

+

v ~ ( ~ / 5 + v~) z+:

+ v~) 2

}

+

1

]

377

11.4. S m a l l M o d u l i Differences a n d T o u g h e n i n g o f T T C

1

1

}

(v/_~_ 4- v/~) 2 - ( v f ~ 4- x/~) 2

dAo

(11.86)

The total displacement is the sum of(11.82), (11.83) and (11.86) W 0

{

---

(i4-U)cSpT//~ { 4 4-

247r(1 - v)

.5- .50

4- r

~, zo) + r

~, To) } d A o ( 1 1 . 8 7 )

where

r

zo) -

v~(~/~ + v~)

~ ( ~ / ~ + vq)

1

z4-.5

( v ~ + v~) ~ In terms of w ~

'-2#//~

2~ - ~ ( v ~

+ ~)~

and h(z,.5) the integral (11.81)is ~ R2 e { ( 1 _ 2u)

{ Oh

4 - 2 R e { O - h O w05~

(11.88)

where the identities resulting from the definition h - hi + ih2 introduced above have been used, as well as similar identities for w ~ - w~ + iw ~ w o,X - w o,z + w ~,Z ~ wO,x 4- iw~ _iw~ , - w o, z _ w ~, z

_ i w o , y 4- w 2,y ~

All the derivatives with respect to x, y may now be transferred to z,.5"

hl,y - -Im{h,z - h,~-}, h2,x - Im{h,z + h,~-}

w ~ - Re{w~ 4- w ~

- Re{w?z - w,~

w~ - -Im{w~ - w,~-}, w~ - Im{ w~ 4- w~

(11.89)

Toughening in DZC

378 As _

_

(11.90)

+

or

~~2 - C~ - Imw~

(11.91)

it follows that e~

- 2[Reh ~Rew ~ + Reh ~-Rew ~ + Imh ~-Imw ~ = 2[Re{h~-w~

+

RehzRew~ ]

(11.92)

Some of the integrals in (11.88) can be calculated analytically, while the rest have to be evaluated numerically. In analytical integration care has to be exercised to isolate any non-integrable singularity at the crack tip. This is done by surrounding the latter with a circular core with the matrix moduli as suggested by Hutchinson (1987). (This procedure was used by Hutchinson in arriving at eqns (11.28)-(11.30)). It is of course now essential to realize that the corresponding value of the integral is not a contribution to the desired K tip, but to the singular fields within this inner circular core. To obtain the correct contribution to K tip, the procedure proposed by Hutchinson (1987) for the corresponding microcrack shielding problem is adopted here. Of course, this procedure is only approximate for our purposes as it ignores the interaction effects, but to the lowest order differences in moduli the error is expected to be negligible.

11.4.4

Evaluation

of Some

Integrals

The calculation of the integrals in (11.88) will be given in some detail. For the first term, the first step is to simplify the derivative of the displacement: 2

Re{0W ~

Kavpl { 1 +

1 }

(I+v)cOT f /a {g(z, zo) + 247r(1 -- u)

zo)

+g(z,-2o) + g(-s T0)} dAo

(11.93)

11.4.

Small Moduli Differences and Toughening of TTC

where

379

1

g(z, zo) =

zv~(v ~ + ~-)~

Integration with respect to x0 reduces the integral in the above equation to a contour integral around the boundary C 2

Is

Re(0W ~

( 1

(1- 2u------) -~-z } - 2#v/~ ~ + 2(1 + u)cOT / c 24~(1 - u)

1} {G(z, zo) + G(-5,zo)

+C(z, ~0) + G(~, ~0)} duo

(11.94)

where

C(z, zo) = ~(~/~-~ + This term is multiplied by

Re{0h

(1-2u)

z} -

8V~-~(1-v)

4#

/f,~

( 1

1 1

z - - a ~ + z~-]-~

The first term

(1 - 2u) ~ , ,

Oh

Ow~

Re{ ~zz } R e { - ~ z

}dA

(11.95)

in (11.88) can now be calculated (after isolating any non-integrable singularities). Some of the integrals can be evaluated exactly analytically for the approximate transformation zone boundary (11.78), the rest can be reduced to contour integrals over C which then have to be evaluated numerically. Examples are given below. The leading term, after isolation of the non-integrable singularity at z - 0, is

//o(1

+ ~

1)(1 1) ~

+ ~

dA - - 3 ~ +

2_~ (11.96)

where the underlined contribution is from Izl = ~ which is independent of ~. The area integral (11.96) can be reduced by integrating with respect to x to a contour integral over the front of the transformation zone C and a circular region C e isolating z = 0

Toughening in DZC

380

+ __/c(1

+ 1 )

dA (

1 )

cos 8dO =e

= - 3 v / 3 + 27r

(11.97)

There are four terms in (11.94) of the type

H(z, z o ) - / L

z-~ 1 Iv G(z, zo)dyodxdy

- / c d Y / c dy~

Zo-~ In

x/~+~/~

21}

z0v/~

zV ~

(11.98)

and when these four terms are combined the non-logarithmic part of the integral can be evaluated exactly (again after isolating the point z = 0 with Izl = 8), and the remaining part evaluated numerically. The result is

H(z, zo) + H(-5,zo) + H(z,-fo) + H(-5,-5o)= 3

- (5V/37r + g

)

(11.99)

where B = {2(1 + v)KaPP'}/{3V'~a'i}. The numerical factor AI(= -1.1188) is the contribution from the logarithmic terms in the integrand which have been evaluated numerically, whereas the underlined term is again the contribution from ]z I = ~owhich is independent of 8. The four terms remaining in the integral (11.95) have the form H1 (z, z0)=

--

~

G( z, zo)dyodxdy

/cd v /c{v " g (2( z V~- ~)(z0 z - +~) z + ~)

2 (zo - z + ~)3/2

11.4.

Small Moduli Differences and Toughening of TTC

381

When these four terms are combined the apparent singularity on the x-axis when (z - 2) - 0 is removed. This is easy to see when the sum of the first terms in the four integrals H (z, z0), H (2, zo), H (z, 2o), H(2, 20) is evaluated giving

~/~

+

(zo + 2iy)(iy)[z[

4~ z

(-go - 2iy)(-iy)[z[

v~

v~ z

+ (20 + 2iy)(iy)lz I + (zo - 2iy)(-iy)lz[ X/2~ox(-4y0)

ff~-'ozox (-4yo)

[zlir.].r 2_ +

(-v~o

+ (-4~o

Izlir~r 2_

- v ~ z o ) - 2 ( 4 ~ - ~ + v~Tz)

- 4~7zo) - 2 ( v ~ izlr 2

+ v~)

(11.1Ol)

where ~= - ~o~ + (so 9 2u) ~

The non-vanishing parts of the integrals are now integrated numerically to give

Hl(z, zo) + Hl(2, zo) + Hx(z,2o) + H1(2,2o) _- 2(1 +

v)h'~'VP'A2

(11.102)

where the numerical factor A2 - - 1 . 6 9 1 5 . The second term in (11.88)

2u / /a 2Re ( O-fiOw~ Oz 02 ) d A will now be evaluated. As above, the derivative bw~ calculated from (11.87) can be simplified by integration with respect to xo. The result is

Toughening in DZC

382

~w ~

,,.~

O-e

8~~

(z_:) -e3/

[ 4 12~r(1 - u)

-5 - -50

§

z--5 /

~-~(4~ + 4~) ~

2

1

~~o

1

~ ~~o

1

z4~(4~ + 4~)~

}] dy0(11 103)

This term is to be multiplied by

.~Oz m

--_

, 16x/2~(1- u)

( z 5/2 ) Z---5

Some of the integrals can again be evaluated analytically for the zone boundary (11.78) while the rest can be reduced to contour integrals over C which can then be evaluated numerically. After an integration with respect to x, the leading term reduces to

G1 - 2Re

//~

(z - -5)2 dA

~/2z~/2

:_,e/,(z_'~)~~,/~-~(-)"'z 1 -5

} dy

(11.104)

The next term must vanish

z-~)

-5o -~/ 2

d yo - 0

(11.105)

because a non-zero value would imply a contribution to the stress intensity factor from phase transformation in the absence of the crack which is clearly absurd. It is easily verified that this integral does indeed vanish. The contour integral re(-5--5o)-1 dyo can be easily calculated. The indefinite integral is

11.4. Small Moduli Differences and Toughening of T T C

dyo yo Y = - arctan -f - -20 xo - x

383

i In Iz - z012 2

(11.106)

Substitution of (11.106) into the left hand side of (11.105) gives -16Re

+i

~ ~ cos ~9 In Iz IL"{(-' '

ycos2

Yo - Y) zol' + ysin 5t~ 2 arctan X 0 --

5 In Iz - zo arctan Yoo -- Y x _t_y ~ sin ~e

[2)t

X

(11.107)

If Zo E Ce, zo ---, 0, the above integral may be rewritten as -16Re

~

- ~ ~os ~O In Izl ~ + ~sin ~e a,~tan X

The integrand is an odd function of 9, whereas gt is symmetric with respect to 9, so that the above integral must vanish for z0 E Ce. For z0 E C, it is found that the integrand F(9, 9o) in (11.107) satisfies

F (0, 00) + F (0, -00 ) + F (-0, 00) + F (-0, -00 ) - 0 so that the integral over the contour C which is also symmetric with respect to 0 again vanishes, thereby confirming the validity of (11.105). The next two integrals together, may be written as

/cG1(~o-2

+-~ol )dyo - i

3

o-~

~

(11108).

where G1 is given by (11.104) The penultimate term

"~

1

( z - -2)2

d yo

2

z~/~v~(v~+ v ~ ) ~

(11.109)

is to be evaluated numerically and is equal to 2(1 + v)K appt

3v57~,

A3

where the numerical factor A3 = 8.7426. The last term

(11.110)

Toughening in DZC

384

( z - -5)2 dyo 1 v~ 2 z~/~(,/7+

,e/l

V~) ~

(11.111)

must also be evaluated numerically, but is equal to the previous integral (11.110). This completes the evaluation of all integrals appearing in (11.88). 11.4.5

Correction

for Moduli

Differences

To summarise, 2# /

2 ~Re{~} Jn ( 1 - 2v ) (1 -

16~r(1

Re{ -N-z 0w~}dA-

2v) -

v) Kapp' (3v/~- 2~)

2887r2(1 - v) K"PPz -1.1188 + 6v/'3

7r3

4 ) - 1.6915] -(5V~r + 57r2 2#s

ljo {o o o} 2Re

Oz 0-5

8~r(1 - u)

+

02

1927r2(I - v)

(1!.112)

dA-

- 6

KaPP z

2v/3

- 6

- 8.7426 (11.113)

Finally from (11.80), (11.112)-(11.113)the O(c) change in stress intensity factor (for v = 0.267) can be evaluated: /~-1 = K 0 _ 0.0070K,Ppz _ 0.0022wKappz

(11.114)

where K ~ is given by (11.80). The total stress intensity factor at the "crack-tip" is (e = 0.065)

11.4. S m a l l M o d u l i Differences and T o u g h e n i n g o f T T C

Primary problem A

385

~Y

I Li -

", ,,

~,V

3

. X

~1, V

Auxiliary problem B

y IH

].t,V

]a,V

//'

.~ x

~, V

F i g u r e 11.12: Primary problem A (Fig. 11.11) and the definition of auxiliary problem B

~.tiv _ KappZ + e(_O.O481wKavvt _ O.O070Kappt) _ O.0459wKavpZ ~,tip _ K,ppl _ O.O031~zKappt _ O.O005Kappt _ O.0459~Kappt (11.115)

As previously noted, the above f(tip is not the desired value (hence the use of distinguishing tilde). It has to be corrected (albeit approximately) by the procedure adopted by Hutchinson (1987), as follows. It is worth recalling that our primary task is the determination of K t i p / K "pvz for the geometry of Fig. 11.11, which is designated as primary problem A in Fig. 11.12. We have so far actually solved the auxiliary problem B in which the composite material with elasticity tensor C~Z76 covers only the region ~ - Fte where 12e is a disk centred at ori-

Toughening in D Z C

386

gin with radius ~ and boundary Ce. Thus, the integrations in integral I (11.88) were carried over ~ - ~e only. When ~ ---+ 0, a logarithmic singularity will appear in I, so that we have to limit 8 to be positive, albeit infinitesimally small. A tilde over [~[tip and /~-i indicates that they correspond to Problem B. In dimensionless form, the solution to Problem B looks like ktip

Ko

k I

K aPPz = I + K aPvz + ~ K aPv---7 '

whereas that of Problem A would look like Ktip Ko K1 K aPvz = 1 + K avvz + e K aPv---------i

The contribution from transformation alone to Problem A is exactly the same as to Problem B, and is equal to K ~ avvz. We need only consider the correction to the remaining part (1 + K 1 / K avvz) in which K 1 is due to the moduli mismatch within f~ and its interaction with transformation strain 0~. /-<1, which we have already calculated is due to the reduced moduli within f~ - f~e and its interaction with transformation strain opT in f~. To adopt the technique used by Hutchinson (1987), we will neglect the contribution to K 1 and K 1 from the interaction of transformation over f~ and moduli mismatch. This is equivalent to the assumption that the ratio K appl + K 1 Kapp I + K 1

is invariant with respect to a change in opT. The technique of Hutchinson (1987), on the other hand, is equivalent to the assumption that this ratio is invariant with respect to a change in the geometry of fl, such that if a solution to a special geometry of f~ to primary problem A and auxiliary problem B were obtained then the solution to the problem of any geometry of fl can be factored through the following identity KaPVt + K 1

K'~r'Pz+ K 1

K appt

K"Pvl + ~"1

K"r'vz

K appl .q_ ~"1

Applying this identity to our problem, and using the already known solution of Problem B, the solution of problem A may be obtained through

11.4. S m a l l M o d u l i Differences and T o u g h e n i n g o f T T C

K "VPl + K 1 K"vvz

387

K "r'Pz + K 1 K "VVz + f f 1 -

K"vvt + r~"1

K"vvz

The problems A and B have been solved by Hutchinson (1987) for the special geometry of an infinite strip to obtain K appl + ~.1

5

3

K appl -t- K 1 = 1 + ~ 1 - ~ 2

(11.116)

which is not limited to the lowest order moduli differences. Here ~1 = [ ( # / ~ ) - 1 ] / ( 1 - u)and 52 - U~l. Inserting (11.80)into (11.114) we have ~.1 _ _O.O070KaVpz _ O.04814wKavpZ

where the last term is the contribution from the interaction between transformation and moduli mismatch over D - D e. This, plus the contribution from the external load gives K"vPt + K i Kapp I

K"PPt+ ~. 1 K x + K appZ --

K"vvz

/~- I + K"Pvl

1 - 0.0070- 0.04814w l + g 561 - ~ 3~ 2

(11.117)

Comparison of (11.115) and (11.117) shows that the net result is that the first three terms of (11.115) get divided by (11.116). The last term is the (correct) contribution from dilatation alone (11.80). For the zirconia composition under study, 61 = -0.0835, ~2 = -0.0223, so that for the lowest order differences in moduli we have from (11.115) and (11.117) K tip -

1 . 0 3 6 2 K appt - 0 . 0 0 3 2 ~ K appl'- 0 . 0 4 5 9 ~ K appt (11.118)

The first term in the right hand side of (11.118) is the contrubution of moduli changes alone and this contribution would appear to be deleterious to the overall toughening of the material under study. The second term is the contribution from the joint effect of the phase transformation and the lowest order moduli changes induced by this transformation. It is seen that this effect is synergistic. The last term is the contribution of dilatation alone. It is instructive to compare (11.118) with the corresponding expression for microcrack induced shielding obtained by adding (11.80) to (11.28) (w

388

Toughening in DZC

Ktip

= l + ( k l + g l 651 + ( k 2 +

K ,,ppl

3)52 - 0.0459w

(11.119)

where w is still defined by (3.26) but with cOT reinterpreted as the dilatation due to the formation of microcracks, and kl and k2 are given by (11.29) and (11.30), respectively, k 1 and k2 can be shown to correspond to the leading terms in (11.112) and (11.113), i.e. the terms independent of w. To within constant multipliers involving # and u the integrals giving these terms are (11.96) and (11.104). From (11.96), we have

//n(

1

-

fo

-

4

1

os(3O/2)cos(0/2) 4

/o"

In R ( 0 / ( ~ o s 2O + cos 0td0

From (11.104), we have

Gl(z)=Re

f/(~

-~)~

~ d A _53/2z5/2

- - 1 6 fo '~ In R(0) sin 2 0 cos 0dO When w

-

0, we have therefore from (11.62), (11.112) and (11.113) K 1

Kapp I

:--C

(i

-

47r(1 8a'(1

-

29) u)

In R(0)(cos 20 + cos O)dO

3 u) f0~In R(O) sin 2 0 cos OdO -

(1 - ~,)

g ( 1 1 ~os 0 + 8 ~os 2o - a cos a0)

-2u(cos 0 + cos 20) } In R(O)dO = ki61 + k262

(ll.12o)

11.4. Small Moduli Differences and Toughening of TTC

11.4.6

389

R e s u l t s and D i s c u s s i o n

The contribution from first order moduli differences considered in isolation from phase transformation is given by (11.120). For the steadilygrowing crack (Fig. 11.11), we calculated kl = -0.0166, k2 = -0.0433 in w (see (11.38)). If we now assume, as we did in w that the contributions from moduli differences and phase transformation can be superimposed we will get from (11.120) and (11.80) Ktip I~ app i

= 1.0350 - 0.0459w

(11.121)

Comparison of (11.121) and (11.118) shows that, even to the lowest order differences in moduli, the fracture toughness is not a simple superposition of individual contributions from the dilatation and the moduli mismatch, but that there is a coupling between the two. Equation (11.118) also shows that, the net shielding effect in a binary transforming composite cannot be calculated by a simple replacement of w with ~ in the definition of K ~ (11.80). To see this, let us calculate ~from w (3.26) after replaceing p, u and c with/~, v and ~, respectively. For the zirconia composition under study this gives ~- = 1.1246w, so that the toughness ratio according to the effective dilatational strain approach is given by (7.22) Ktip

Kapp z = 1 - 0 . 0 4 5 9 ~ - 1 - 0.0516w

(11.122)

whereas the perturbation technique gives (11.118) Ktip I~ app I

= 1.0362 - 0.0491w

(11.123)

The effective transformation strain technique only calculates the coupling effect of dilatation and moduli mismatch. It does not take into account the effect of moduli mismatch alone. A comparison of (11.122) and (11.123) shows that, already for lowest order moduli mismatch, this can make not only a small quantitative, but also an important qualitative difference to the predicted shielding effect. Thus, for example if w = 5, (11.122) and (11.123) give KtiV/K appt = 0.742 and 0.7907, respectively. Comparison with the shielding effect of dilatation alone (Ktip/K appz - 1 - 0.0459w - 0.7705)shows that whereas the present perturbation technique predicts a reduction of about 3%, the effective transformation strain technique predicts an increase of nearly 4%. This

Toughening in DZC

390

needs to be borne in mind when studying the crack tip sheilding in composites, such as ZTA which have large differences in the elastic properties of the transforming and matrix phases.

11.5

Effective Transformation Strain in Binary Composites

11.5.1

Introduction

Notwithstanding the cautionary remark at the end of Section 11.4, we shall introduce the effective transformation strain concept for binary composites containing one transformable phase. This is the only way of treating binary composites whose phases have very different elastic moduli, although the treatment is approximate and based on analogy with thermoelastic properties of binary isotropic composites (Budiansky, 1970; Laws, 1973). It will be recalled that for composites with homogeneous elastic properties, an effective volumetric transformation strain ~T was defined by (3.12) in terms of the volume fraction c of transformed particles and of the volumetric transformation strain of a particle 0T unconstrained by the matrix. The quantity 0 T served as the effective transformation strain in regions of the continuum in which many particles were undergoing a stress-induced martensitic phase transformation. In this Section, the relation (3.12) will be generalized to transforming composites in which the elastic properties of the transforming phase differ significantly from those of the matrix phase. The ZTA composite that we studied in w is a good example. There we replaced relation (3.12) with the effective dilatation (11.4), dependent not only on the particle transformation strain but also on the composite elastic properties. We shall exploit the notion that thermal expansion and transformation strains in binary elastic composites are equivalent and restate the results found in the literature on thermal expansion strains in a convenient form for the transformation strains. This will in particular allow us to state bounds and estimates on effective transformation strains when the composite contains dilute concentrations of transformable phase. We shall find that the effective dilatant transformation is less than cOT, if the nontransforming matrix is stiffer than the transforming particle, as in ZTA. The exposition follows closely the paper by McMeeking (1986).

11.5. Effective Transformation 11.5.2

Effective

Strain in Binary C o m p o s i t e s

Transformation

391

Strains

Laws (1973) investigated the effective thermal expansion coefficient of a binary anisotropic composite. The interpretation of Laws' result for dilatant transformation is as follows. The elasticity law for the nontransforming phase is (rij -

Ci'~klCkz

(11.124)

where o'ij is the stress, cij the strain and Ci'~k z is the tensor of elastic moduli. The elasticity law for the transforming phase is o'ij -

C~ktckz

(11.125)

before transformation and ~ij

--

C~kz(CkZ -

Tp

(11.126)

CkZ )

after the phase change has taken place, where cTp is the homogeneous particle transformation strain. The superscripts m and p denote matrix and particles, respectively, but the two phases can also be intermingled in any way that retains the binary feature. The behaviour of the composite in a macroscopic sense is determined by O'ij -- CijklCki

(11.127)

-

(11.128)

before transformation and ~i

C~ikt(ckz- c~z)

after transformation. In (11.127) and (11.128) the stresses and strains are macroscopic averages in the sense of Hill (1963) and thus c T. is the effective transformation strain of the composite. The exact result obtained by Rosen & Hashin (1970) and Laws (1973) can now be restated as gT

-

~ Ci~l(Cklmn -- Cklmn)(

c mP. ~ - c ~ . ~ )

-~

T~ (11 . 129) C;qrsCrs

Hence, when each phase and the mixture are isotropic but the deviatoric part of the transformation strain in every particle is uniformly aligned 1 oT + eT eT _ ~8,i

(11.130/

Toughening in DZC

392

oT _ B p ( B - Bin) T -- B(Bp

-

T

B m ) Op -

-- p ( i t p _ l z m ) e i j

-ddOp

-- c s e i j

(11.131)

(11.132)

Tp where tt and B are the shear and bulk moduli, respectively, and eij is the deviatoric part of the particle transformation strain. The terms Cd and cs have been introduced to represent the effective coefficients. The alignment of the shear part of the transformation among all the particles Tp obviously precludes a locally random orientation of eij in which case eT would be zero. For purely dilatant transformations, eqn (3.12) should be replaced by (11.131) and inserted into (7 21) to give the toughening effect. In (7.21), Young's modulus and Poisson's ratio for the composite should be used, as we did in arriving at (11.122). When the shear moduli are the same for both phases, the limit of (11.132) can be found by first bounding through the use of the Reuss and Voigt estimates (Hill, 1963) of it, and then allowing ttp to approach ttm. Both bounds approach c and so

Bp ( B - Bm ) 15 T ceTP sT = B ( B p - B m ) - 3 'jOp +

(11.133)

which can be simplified by use of the result (Hill, 1963) B - Bp +

1

1--C

c

(11.134)

+ Bm - B,

B, +

which leads to

!cS'J

cT. _

( 1 - c)(Bm - Bp)(34-p) 1+

(11.135)

Bp (Bin + g

The effective dilatant transformation strain is thus increased above c8T if Bp > Bin. In addition, it can be seen clearly that when the two phases have identical elastic properties ~T _ cQTP confirming generally the expression (3.12).

(11.136)

11.5. Effective Transformation Strain in Binary Composites

11.5.3

General

Bounds

and

Dilute

393

Estimates

In the case of an arbitrary isotropic mixture, there are no general results for B and #; hence, measurements, bounds, or estimates must be used. Hill (1963) gives bounds on the bulk modulus for an isotropic mixture of two phases with the configuration otherwise arbitrary. The bounds are

c

< B - Bm <

(Bp-Bm)(1-c)

1"~-

- Bp-Bm

C

(Bp

-

(Bm..~_ 4#m)

Bin)(1

-

1-Jr-

-

c) (11.137)

4

when the signs of Bp - Bm and #v - P m are the same (the inequalities are reversed when the signs are different). As an example, consider two phases with the same Poisson's ratio, but with B,~/Bp - 2, pertinent to ZTA. The bounds in (11.137) become 9c

< 5d _<

13 -- 4c --

7c 9 -- 2c

(11.138)

Notice the satisfactory results when c - 0 and c - 1. For the shear modulus, the Hashin and Shtrikman (1963) bounds can be used. These are (ttm PV +

--

# p ) ( 1

--

c) ~ _< P _< #m + 1

)

(#v #,~)c l + J 3 m ( 1 - c ) ( 2z~ -1).,,, -

(11.139)

where #m - # p and B m - Bp are both positive (the inequalities are reversed when both are negative) and ~ = [ 2 ( 4 - 5 u ) ] / [ 1 5 ( 1 - u ) ] . Taking the case Izm/l~p = 2, as before, and u = 0.25 45c 67c < 5, _< 6 8 - 23c 9 0 - 23c

(11.140)

Alternatively, estimates for B and/~ (and subsequently for 5d and 5s) can be obtained using the self-consistent method of Hill (1965), and Budiansky (1965). However, the results must be calculated numerically for each composition and particle shape. Instead, for simplicity, estimates for 5d and 58 which are valid when the concentration of the transforming phase is dilute, are based on the average of the Voigt Bv and Reuss BR estimates

Bv - (1 - c)Bm + cBp

(11.141)

394

Toughening in D Z C

1

1-c

Bn =

Bm

c (11.142)

~ Bv

The resulting average leads to an estimate of ~d in a series expansion in c truncated at 2 terms _ Cd-

(Bp + B.~) c 2Bin

1-

c(B 2BpBm

(11 143) "

Similarly for cs ~-

c(PV#+P'~) ~ ) [] 1 -2 c(#~ t 2ttp#.~ t -m

(11.144)

Note that, when B m / B p = l*m/PV = 2, the estimates for Cd and Cs lie near the upper limits in (11.138) and (11.139). As a result, the expressions (11.143) and (11.144)should be taken with caution.