Chapter 8 R-curve analysis

187

Chapter 8

R-Curve Analysis In Chapter 7 we studied the toughness increment induced by phase transformation under steady-state conditions. In this Chapter, we will present a complete R-curve analysis of the quasi-static growth of a crack, taking into account the development of transformation zone. We begin with the analysis of a semi-infinite crack in an infinite medium, followed by that of a finite internal crack and an array of collinear internal cracks. We conclude the Chapter with the analysis of a single and of a periodic array of edge cracks. We shall assume throughout that the phase transformation only induces dilatation and that super-critical transformation takes place according to the critical mean stress criterion (Chapter 3).

8.1

Semi-Infinite

Cracks y

T H

ro

F i g u r e 8.1: Semi-infinite crack

R-Curve Analysis

188

The model considered in the present Section consists of a semi-infinite crack C in an infinite plane body subjected to a load which induces an applied stress intensity factor K appl, as depicted in Fig. 8.1. The exposition follows that of Stump & Budiansky (1989a). Muskhelishvili's theory of plane elasticity is applied (Muskhelishvili, 1954), and the notations are in accordance those introduced in Chapters 4 and 5. In order to solve the problem depicted in Fig. 8.1 the traction induced on the crack-line by the transformation zone has to be cancelled. Due T (X), which is to symmetry the traction is given by crack-line stress cry~ obtained from (4.21) and (4.52)

T %y(x)

-

EeTJs 4~(1-u)

x-z0

- ~ - ~1 ) x-~0

dyo

(8.1)

This equation holds irrespective of whether the crack tip is inside or outside the transformation zone. Thus the constant terms in the stress potentials of (4.51) do not enter the expression for the crack-line stress. There is however a jump in this stress across the zone boundary, which is embedded in the integral (8.1). The potentials for a crack with crack-line stresses balancing out the stresses from transformation are determined by the standard method devised by Muskhelishvili (1954). The image potential Oi(z) arising from the cancellation of the crack-line stress is determined from (5.56) by inserting the stress (8.1). The image potential O~(z) becomes

r

8 (1-

+

)VZ

+ (vz+

1

) dy0

(8.2)

The potential (~avvZ(z)from the applied loads is obtained from standard results (Muskhelishvili, 1954) Kapp l

2 /Y z

(8.3)

The potentials Oi(z) and ~aPPt(z) associated with the O-potentials of (8.2) and (8.3) which are necessary for the complete determination of the elastic fields are obtainable from the respective q~-potentials, but for the present analysis these potentials are not needed. The complete O-potentials are obtained by superposition of the image potential due to transformation (8.2) and the potential due to the applied load (8.3). In the following these results are used to obtain

8.1. Semi-Intinite Cracks

189

equations determining the transformation zone shape and crack growth behaviour. The mean stress cr~pt from the applied load is obtained from (8.3) and (7.23)

a~n-

g-----~Re

(8.4)

where the characteristic length L (7.28) has been used. The mean stress due to the image potential (hi(z) is obtained from (8.2) and (7.23) o'm_

1

w

1

( v/'~ + V/-~ ) v/-~ + ( V/-~ + ~ o ) v/-~ d y o (8.5)

~r~ - - lS-'-~Re

where the transformation parameter w (3.26) has been used. From (4.52) and (4.21)it is seen, that the mean stress due to the transformation zone itself vanishes outside the zone and takes on a constant value inside it. Thus the mean stress from transformation itself does not enter the equations needed here. The applied stress intensity factor K appt in (8.4) has to be tied to the intrinsic toughness Kc through a crack growth criterion in order to obtain a complete system of equations for the present problem. The stress intensity at the crack tip K tip can be viewed as consisting of the applied stress intensity factor K ~ppt determined by the far field load and an additional contribution from transformation A K tip

K tip = KappI+ A K tip

(8.6)

A K tip is the stress intensity factor from the image stress and is given by

AKtip_

Re{2O/(z) 2x/~-xrz}~__.0+

(8.7)

Introducing (8.2)into (8.7)gives

Kr

= 18r

+

dyo/L

(8.8)

To determine the transformation zone boundary prior to crack growth in accordance with the critical mean stress criterion for transformation,

190

R-Curve Analysis

the mean stress must approach the critical mean stress as the zone is approached from the outside. At the onset of crack growth the stress intensity factor at the crack tip K tip equals the intrinsic toughness Kc of the material. Adding the image mean stress from (8.5) to the applied mean stress from (8.4) and equating the sum to the critical mean stress results in the following system of equations for determining the transformation zone boundary at the onset of crack growth

K~

1

1 + (x/T + e~-5) e 7

- 1S----~Re

) zES

(s.9)

Kc = K tip = K appt + A K tip

The unknowns in the two equations (8.9) are the zone shape S and the applied stress intensity factor K appz. At the onset of crack growth the stress intensity factor at the crack tip K tip must equal the intrinsic toughness of the material Kc. This acts as a side condition to the nonlinear integral equation (8.9) determining the zone shape S. For a growing crack it is assumed that a wake of transformed material will develop behind the crack tip due to nonreversible transformation. To model the growing crack, the crack length is incremented. The crack tip therefore moves into the transformed zone. This will result in an increase in the mean stress ahead of the crack tip, while the mean stress behind it will decline. If the zone shape behind the crack tip is fixed and the stress intensity factor at the crack tip K tip is maintained at Kc, a new zone front and the applied stress intensity factor K appz necessary for quasi-static crack growth can be determined. For a small increment of the crack Aa the equations determining the transformation zone boundary can be written as

~ R e Kr

lSr

Re

(

1

(V/_~. + v/~)x/,_~

K~ = K tip = K "ppz + A K tip

+

1

(e7 +

)e7 )

z f: Sfron t

8.1. Semi-Infinite Cracks

191 (8.10)

Swake(a -4- Aa) = Swake(a)

where the wake Swake(a) is obtained by a reverse translation of the transformation zone by the distance Aa. The unknowns in eqn (8.10) are the zone front Sf,-ont and the applied stress intensity factor K appz. These are determined incrementally using the solution of (8.9) as the initial zone shape. As the crack is incremented, the integral equation (8.10) determines the new transformation zone front. The corresponding applied stress intensity factor t~ appz equalling the apparent toughness of the material is obtained from the side condition of (8.10) with the aid of (8.8). For sufficiently small crack increments Aa the transformation zone wake S,~ak~ joins smoothly with the front SI,.ont. These increments are however too small for an efficient computational scheme. Larger increments can be handled by imposing a smoothness condition on the zone shape. By joining the wake and the front by common tangents (Fig. 8.2) much larger increments of crack lengths can be handled without causing numerical difficulties.

F i g u r e 8.2: Transformation zone of a growing crack joined by common tangent to the wake

8.1.1

Stationary

and

Growing

Semi-Infinite

Crack

The transformation zone shapes for various values of w ranging from 0 to 30 are shown in Fig. 8.3. The transformation zone size is seen to increase with the transformation parameter w. (The transformation zone for w = 0 is the cardioid, described in w The transformation zone dissociates from the crack tip for non-zero values of w and approaches

R-Curve Analysis

192

1.2 I--

~

0.8

o~=30 25 20 15 10

0.6

5

1.0

0.4 0.2 0.0 -0.2

J

0.0 0.2 0.4 0.6 0.8

1.0 1.2 1.4 1.6 1.8x/L

F i g u r e 8.3: Initial transformation zone shapes for a semi-infinite crack

the crack face at right angles, in contrast to the situation for w = 0 where it terminates smoothly at the crack tip. The toughness increment for a stationary crack is negligible, less than 0.5% for 0 < w < 30. There is no increment when w = 0, (w Once the initial zone shape for a stationary crack has been found, it is possible to determine the growth around an advancing crack tip from (8.10). As a growing crack tip moves into the zone, material in its vicinity attains the critical mean stress and transforms, while due to the irreversibility of the transformation a wake of transformed material is left behind. Along a frontal portion of the transformed zone boundary the mean stress criterion is satisfied, while on the wake portion of the c . As we have seen boundary the mean stress will have dropped below ~rm (w the transformed region behind the radial lines running through the tip at +7r/3 reduces the stress intensity at the tip. To continue driving the crack forward, the applied stress intensity must be adjusted. Consequently to solve the growing crack problem, both the stress intensity and the zone shape must be determined as functions of the crack extension. Zone shapes for growing cracks for w = 5 and 10 are shown in Fig. 8.4. The development of toughening ratio Kappt/Kc or the R-curves corresponding to these growing cracks are shown in Fig. 8.5. The zone half-height H/L and the toughening ratio Kappt/Kc both overshoot the steady-state levels (broken lines; w for finite amounts of crack growth before approaching them asymptotically from above. The half-height of transformation zone at initiation of crack growth is

8.1. Semi-Infinite Cracks

193

y/L 1.O 0.8 0.6

i :

i i

,, i

'

i

i

1

0.4 0.2 0 a)

0.0

8

4 ~

~

4

~

I

I

8

12 I

I

12

Aa= 20

16 I

I

I

16

20

x/L

y/L 2.0

1.5 ,

1.O

,

,

0.5 b)

0.0

J 10

10 J

J 20

20 J

J 30

30 I

, 40

40 ,

Aa = 50 I 50

x/L

F i g u r e 8.4: Transformation zone shapes for a growing semi-infinite crack, ( a ) w - 5, (b) w - 10

shown in Fig. 8.6, together with the half-height at peak toughening and under steady-state conditions. The value at peak toughening diverges as the transformation parameter w attains the value of approximately 20.2. This is accompanied by unlimited increase in toughness, i.e. by lock-up. Under steady-state conditions the lock-up value of w is approximately 30.0, as described in Chapter 7. Reciprocal peak toughening is shown in Fig. 8.7 and compared with the reciprocal peak toughening under steady-state conditions, described

194

R-Curve

Analysis

appl

K

IK c

1.30 1.25 1.20

1.15 1.10 1.05

1.O0 a)

0.95

i 5

0

t 10

t 15

J 20

AalL

I 50

Aa/L

appl _

K

IK c

1.8 1.6 1.4 1.2 1.0 b)

0.8 0

J 10

I 20

I 30

I 40

P i g u r e 8.5: Development of a p p a r e n t toughness for a growing semiinfinite crack, (a) w - 5, (b) w - 10

in C h a p t e r 7. Surprisingly, the linear e s t i m a t e of s t e a d y - s t a t e toughening (see Fig. 7.6) agrees very well with the peak toughness values up to w~18.

195

8.1. Semi-Intinite Cracks

H/L 5

_ ~~../Steady-state Initial 0

0

!

I

I

I

I

I

5

10

15

20

25

30

F i g u r e 8.6: Half-height of transformation zone at initiation of crack growth, at peak-toughening, and under steady-state conditions

gc/K appl 10

L!near 0.8 0.6 0.4 0.2 0.0

0

5

10

15

20

25

30 o)

F i g u r e 8.7: Reciprocal toughening at peak transformation and under steady-state conditions

R-Curve Analysis

196

8.2

Single Internal Cracks

In this Section we shall analyse the R-curve behaviour of T T C containing short internal cracks whose size can be commensurate with flaws that inevitably form in such materials. We shall consider first a single short internal crack and then an array of internal cracks.

F i g u r e 8.8" Internal crack with transformation zones The micromechanical model consists of a central crack C of length 2a in an infinite plane body subjected to a remote transverse stress ~r~ as shown in Fig. 8.8. The transformation zones of dilated material are bounded by S + at the right hand tip of the crack and by S - at the left tip. (For later use, a T-stress parallel to the crack is also included in the model). To solve the problem depicted in Fig. 8.8 the normal stress induced on the crack-line by the two transformation zones is to be cancelled. This stress is given by (4.21) and (4.52)

T (X)

%v

--

dyo

+

47r(1-v)

+

x-zo

x-:o

+

+ -

1 47r(1 - u)

+

x -

X

- -

dyo gg

- -

ZO

1

-} zo

Z0

x -

1 :o

x +

zo

1 )dyo

x+:o

(8.

11)

8.2. Single Internal Cracks

197

Here, E is Young's modulus, u Poisson's ratio, e T the plane dilatational strain, and z the complex coordinate z = x + iy. The boundary S + (and S - ) is determined by the critical mean stress criterion. In arriving at (8.11) the double symmetry of the problem has been exploited. The image potential (I)i(z) arising from the cancellation of the crackline stress (8.11) is determined from eqn (5.56)

ePi(z)- 2 (EeT 1 - u) fS + (A(z, z0) +A(z,T0) - A ( z , - z 0 ) - A(z,-To)) dyo

(8.12)

where A(z, A) -

x/A2 - a2 - x/z2 - a2 4~'(z - A)x/z 2 - a 2

(8.13)

The potential (~avVl(z) corresponding to the applied loads is /

r

) _

( z \ 2~/z 2 - a 2

l2

T )\ /

(814)

The mean stress (r,~pt from the applied loads is obtained from (8.14) and (7.23)

cr,~v' o'~ -

KaVV' i2La K---~

{ Re

z

l-T}

(815)

x / z 2 _ a 2 -- ~

The applied stress intensity factor is given by K appt = cr~ V/-~. The mean stress due to the image potential epi(z) is obtained from (8.12) and (7.23) O'm - C

O'rn

Re

{A(z, zo) + A(z,-Zo) +

- A ( z , - z 0 ) - A(z,-T0)} dyo

(8.16)

The toughening increment A K tip for the internal crack is given by (8.17) whence, vide (8.12)

R-Curve Analysis

198

A K tip

-w

K~

+ a .... z0 - a

zo

367r

+

-i

+

a -+ zo

-5o + a - a

l a - - z+~ ~o )

(8.18)

At the onset of crack growth the stress intensity factor at the crack tip K tip must equal the intrinsic toughness K~ of the material. Adding the image mean stress from (8.16) to the applied mean stress from (8.15) and equating the sum to the critical mean stress results in the following system of equations for determining the transformation zone boundary at the onset of crack growth K~Pi~ K~

+--~--Re

{ Re

z ~/z 2 - a 2

+ (A(z, z0) +

h(z, ~0)

l-T} 2

- A ( z , - z 0 ) - h ( z , - ~ 0 ) ) d~0

zES+

K~ = K tip -- Kappl -t- A K tip

(8.19)

As in w for a small increment of the crack Aa the equations determining the transformation zone boundary can be written as

_

_

Kc

+ --~-Re

a + Aa

X//z2 - (a + Aa) 2

+ (h(z, z0) + h(z, ~0) - A(z,

2

-zo) - h(z, -~o)) dUO[z ES;ron t

Kc = K tip

= Kappl w

~wake(a zt" Aa) --

A K tip

~wake(a)

(8.20)

The procedure for the solution of (8.20) is the same as for the system (8.10). We shall therefore omit the details and present only the results.

8.2. Single Internal Cracks

8.2.1

199

Stationary and Growing Internal Crack

Some results obtained by solving eqns (8.19) and (8.20) for imminent crack growth and for growing cracks, respectively are given in the following. The shape of the transformed zone at the onset of crack growth depends on the crack length and the transformation parameter w. The shapes of transformation zone at the onset of crack growth obtained from (8.19) are shown in Fig. 8.9 for ao/L = 5 and 10 and several values of w. The intermediate values of transformation parameter w and of initial crack length are of principal interest. This is because when the cracks are very short the transformation zone will diverge before crack growth appears, as the mean stress induced by applied load will exceed the

y/t, 0.7 0.6 0.5

~ / ~

0.4 0.3 0.2

0)=30 /25 20 15 10 5 0

0.1

a)

0.0

-0.2

J

0.0

0.2

0.4

0.6

0.8

1.0

1.2 x/L

y/L 0.7 0.6 0.5

/

0)=30 25 2O 15 10 5

0.4 0.3

b)

0.2 0.1 0.0 -0.2

i I

0.0

0.2

0.4

0.6

0.8

1.0

1.2 x/L

F i g u r e 8.9: Initial transformation zone shapes under uniaxial load (T = 0), ( a ) a o / L - 5, ( b ) a o / L - 10

200

R - C u r v e Analysis

rdL 5.0 4.0 3.0

aolL=500 50 10

2.0 1.0

0.0

0

5

10

15

:

_-

i

-

20

25

30

35

.

0.5

40 o3

F i g u r e 8.10" Frontal zone intercept r0 for uniaxial load. The transformation zone size diverges for ao/L = 0.5, as w ---* 0

critical mean stress before the stress intensity factor at their tips attains the critical value Kc. The critical crack length at which this occurs for uniaxially loaded cracks is easily obtained from eqn (8.19) by letting w = 0 and K appz = Kc; the critical crack length is ao/L = 0.5. For very large values of transformation parameter w self-cracking will occur (Stump & Budiansky, 1989b), but these values of w are well beyond those of practical interest. Two measures of transformation zone size are of interest. In analyzing steady-state toughening, the more practical size measure from an experimental point of view is the transformation zone height. For growing cracks, the height is not easily measured, but the distance between the crack tip and the transformation zone boundary intercept with the crack-line ahead of it is uniquely defined. Under steady-state conditions the zone height and the frontal intercept are proportional to each other and thus easily interchanged (see (7.17) and (7.28)). The frontal zone intercept r0 for several initial crack lengths is shown in Fig. 8.10. For very small cracks the transformation zone size diverges as the transformation parameter w tends to zero. The critical crack length for this to occur is ao/L = 0.5, as previously mentioned. Due to a toughness decrement at the onset of crack growth, finite values of the transformation parameter w give nondiverging transformation zone sizes for this critical crack length.

8.2. Single Internal Cracks

201

appl

K

/Kc

1.0

adL=~

0.9

10

50

0.8

3

0.7 1

0.5 0.5

0.5 . . . .

0

I

I

I

1

I

I

I

I

5

10

15

20

25

30

35

40

F i g u r e 8.11: Stress intensity factor at the onset of crack growth. The broken lines indicate first order linear estimates The apparent toughness at the onset of crack growth is shown in Fig. 8.11 for various values of crack length and transformation parameter w. Prior to crack growth, transformation reduces the apparent toughness, and the more so, the smaller the crack. For very long cracks however a small increment in toughness of approximately 0.5% at w = 40 appears. One of the few linear approximations for the present theory as w ---. 0 can be readily obtained by first solving (8.19) with w = 0 and K appz = Kc for the transformation zone shape and then using this result in (8.18) to obtain the toughness decrement at the onset of crack growth. The linear approximations are shown in Fig. 8.11 by broken lines. For growing cracks, R-curves for several initial crack lengths obtained by solving eqns (8.20) are shown in Fig. 8.12 for two values of the transformation parameter w. The apparent toughness goes through a peak before reaching the steady-state value from below. The peak gets shallower as the initial crack length gets smaller. For short cracks, the peak value is less than the steady-state value, whereas for long cracks the peak overshoots it. There is experimental evidence (Swain & Hannink, 1984) in support of this. The applied stress cr~176 necessary for quasi-static crack growth corresponding to the R-curve results of Fig. 8.12 can be calculated from the following relation between toughness, stress, and crack length cr~176 cro

_ Kappt(Aa)/ ao Kc V a o + Aa

(8.21)

202

R-Curve

Analysis

appl g

/K c

1.30 1.25 .,

1.20

1.15 1.10 1.05

1.00 a)

0.95

l 5

0

t 15

10

i 20

Aa/L

appl K

IK c

1.6 1.4 1.2 1.0 I b)

0.8 0

I 10

l 20

I 30

,1 40

I 50 AalL

F i g u r e 8.12: R-curves under uniaxial load, (a) w = 5, (b) w : 10

The normalizing stress or0 is the stress necessary for initiating the growth of a crack of length a0 in the absence of transformation toughening, i.e. the inherent strength of the material. The stress needed for quasi-static crack growth is shown in Fig. 8.13. Due to the R-curve behaviour the applied stress has to be raised in order to maintain crack growth. After a finite amount of crack growth the stress curves go through a peak as the effect of the increased crack length becomes the dominating factor. The ultimate strength of the transforming ceramic

8.2. Single Internal Cracks

203

o~176 1.4

F

.a_~L~_-_=

1.2

5--~-"-~

1.0 0.8-

5

0.6 0.4 0.2 a)

0.0

0

i

i

J

i

5

10

15

20

Aa/L

o**/co 1.8

_

1.6 2~

a0{__L~_~

-

I4tL.

---

1.2 1.0 0.8

b)

0"6 0.4 f 0.2 0.0 0

5 l 10

t 20

~ I 30

J 40

J 50 Aa/L

F i g u r e 8.13: Applied uniaxial stress necessary for crack growth, (a) -5,(b)~-10

is given by the peak value ap of the applied stress cr~176 The peaks in the curves shown in Fig. 8.13 indicate that a certain amount of stable crack growth can be sustained before a transformation toughening ceramic will fail catastrophically. Assuming that a certain population of small cracks is present in the ceramic, the stable initial growth of the most critical crack among this population can cause other

R-Curve Analysis

204

y/L

ao/L=5 10 50

4

o,o

0 -2 -4 -5

tl -30

I

I

I

I

I

I

-25

-20

-15

-10

-5

0

x/L

F i g u r e 8.14: Transformation zone boundaries under uniaxial load, w 10

K c/K appl 1.0 0.8 0.6 0.4 0.2 0.0

I

0

5

10

15

20

~

"'t

25

30

0)

F i g u r e 8.15: Reciprocal peak toughening under uniaxial load

smaller or less critical cracks to develop transformation zones around them and even to start growing before the critical crack itself eventually becomes unstable. If the population of inherent flaws or cracks has a sufficiently narrow size distribution a large number of such cracks may become active before catastrophic failure. This would give rise to a certain deviation from linearity in the stress-strain behaviour for these ceramics apart from the nonlinearity induced by the transformation itself.

205

8.2. Single Internal Cracks

~o/t~ p 1.0 0.8 0.6 0.4 0.2 0.0

0

5

10

15

20

F i g u r e 8.16" Reciprocal peak strengthening ratio under uniaxial load The peaks in the R and applied stress curves are reflected in the transformation zone shapes (Fig. 8.14) through a certain zone widening before steady-state conditions are reached. Reciprocal peak toughening ratio is depicted in Fig. 8.15. For comparison the steady-state toughness estimate consistent with the present theory is shown by the broken curve (Amazigo & Budiansky, 1988). The reciprocal peak strengthening ratio is shown in Fig. 8.16; the strengthening ratio decreases with decreasing internal crack size. Thus initially strong materials are less susceptible to strengthening by transformation toughening than are the initially weak materials. This effect can reduce the scatter in strengths, and thereby increase the Weibull modulus of these materials (Shetty & Wang, 1989). 8.2.2

Relation Between Strengthening

Toughening

and

The reciprocal peak strengthening ratio (Fig. 8.16) can be correlated with the steady-state toughening ratio (broken curve in Fig. 8.15). The result is shown in Fig. 8.17. Three microstructural parameters enter this correlation, namely the initial crack length a0, the characteristic length L (7.28), and the transformation strength parameter w (3.26). The last two parameters are defined with the critical mean stress ~r~ in the denominator. Besides,

206

R-Curve Analysis

the transformation parameter w is proportional the transformation density cOT . In correlating the peak strength and steady-state toughness data, as in Fig. 8.17 it is expedient to associate each curve with a specific microstructure. For variable transformation density cOT , the curves in Fig. 8.17 can be related to a microstructure with a specific initial c For this microstructure, crack length a0 and critical mean stress (rm. the strength increases monotonically with toughness, so that even without changing the critical mean stress, additional transformable particles improve the strength, as well as the toughness. oP/%

3.5 3.0 2.5 2.0 1.5-

~'

I 1.0

l

1.0

1.2

1.4

1.6

$$

1.8

2.0

2.2

2.4

K IK c

F i g u r e 8.17: Peak strengthening versus steady-state toughening for increasing transformation density cOT An alternative way of increasing the amount of transformation accompanying the crack growth is by lowering the critical mean stress a~n" c is decreased, both the characteristic length L and the transformaIf (rm tion parameter w are increased. From Fig. 8.16 it can be seen that this produces two opposite effects in peak strengthening. An increase in L reduces the reciprocal peak strengthening ratio, whereas an increase in w increases it. Both effects can be captured through the parameter

/3 -

w2ao/L - ~

K~(1-u)

(8.22)

8.2.

Single

Internal

207

Cracks

The peak strength and steady-state toughness relation is shown in Fig. 8.18 for fixed values of transformation density cO T and initial crack length a0, i.e. ~ is fixed and corresponding values of w and L are obtained through (8.22). The peak strengthening ratio reaches a maximum at finite amounts of toughening. In an ageing process, where coarsening dominates over precipitation, it can be expected that the density of transformable precipitates is constant, whereas due to the coarsening the critical mean stress for transformation decreases. In that case the result in Fig. 8.18 can be interpreted as leading to peak strengthening before peak toughening is reached. Eventually, toughening ceases as lowering the critical mean stress leads to spontaneous transformation during cooling. The results shown in Fig. 8.18 are in qualitative agreement with the results reported by Swain (1986), and Swain & Rose (1986). The present theory offers the possibility of predicting the effect of alloying, precipitation, ageing and other treatments of transformation toughening ceramics on the strength-toughness relationship, if.the influence of the c and c~T specific treatment upon the microstructural parameters a0, a m, is known. Alternatively, a theoretical estimate for the optimal critical mean stress necessary for maximum strengthening can be obtained for a given microstructure with minimized initial flaw sizes and maximized density of transformable particles.

oP/oo 1.4

-

1.3

-

1.2 1.1 1.0 9

C

Decreasing ~m -~ 0.9

1.0

I

l

l

!

l

I

I

1.2

1.4

1.6

1.8

2.0

2.2

2.4

ss K /K c

Figure 8.18: Strengthening versus steady-state toughening for various values of/3

208

8.2.3

R - C u r v e Analysis

Biaxially Loaded Internal Crack

Some results obtained by solving eqns (8.19) and (8.20) for imminent crack growth and for growing cracks under equal biaxial tension (T = 1) are given in the following. The shape of the transformed zone at the onset of crack growth depends on the applied load, the crack length and the transformation parameter w. Some examples of transformation zone at the onset of crack growth obtained from eqn (8.19) are shown in Fig. 8.19 for two initial crack lengths ao/L. The corresponding frontal zone intercept r0 for sev-

y/t, 1.0

{o=30

25 20 15 10 5 0

0.8 0.6 0.4 0.2

a)

0.0

i

0.0

0.4

0.8

1.2

1.6

x/L

y/L 1.0

=30 25 20 15 10

0.8 0.6 0.4 0.2

b)

0.0 0.0

0.4

0.8

1.2

1.6

x/L

F i g u r e 8.19: Initial transformation zone shapes under equal biaxial tension (T = 1), ( a ) a o / L = 5, (b) ao/L = 10

8.2.

Single Internal Cracks

209

rdL 5.0 4.0 3.0

~

2.0

ao/L=500

~ ~ ~

--~~" 1.O 0.0 0

i

i

i

i

i

i

i

i

5

10

15

20

25

30

35

40

5 3 2

co

F i g u r e 8 . 2 0 : Frontal zone intercept r0 under equal biaxial tension. T h e t r a n s f o r m a t i o n zone size diverges for ao/L - 2, as w ---+ 0

r

appl

IKc ao/L=oo

1.0

50 0.9 0.8

10

0.7

5

0.6

3 2

0.5 0

i

i

i

I

J

I

!

J

5

10

15

20

25

30

35

40

to

F i g u r e 8.21" T o u g h e n i n g ratio at the onset of crack growth u n d e r equal biaxial tension. T h e broken lines indicate first order linear e s t i m a t e s

210

R-Curve Analysis

appl /K c

K

1.3 1.2 1.1 1.0 0.9 a)

0.8

0

K

t

i

1

J

5

10

15

20

Aa/L

appl /K c

2.0 1.8 1.6 1.4 1.2 1.O

b)

0.8

Figure

8.22:

0

I.

l

10

20

.d

30

..

l

I

40

50

R-curves under equal biaxial tension, ( a ) w

Aa/L

-

5, (b)

w-lO eral initial crack lengths is shown in Fig. 8.20. For very small cracks the transformation zone diverges as the transformation parameter r tends to zero. The critical crack length at which this occurs is a o / L = 2. However, due to a toughness decrement at the onset of crack growth, finite values of the transformation parameter w give nondiverging transformation zones for this critical crack length. The apparent toughness at the onset of crack growth is shown in

8.2. Single Internal Cracks

211

0~1760

15t

1.4

aolL=**

1.2 1.0 0~

-

0.6

0.4 0.2 0.0

a)

" 0

t

I

I

I

5

10

15

20

Aa/L

0~1760

2.0 1.8 1.6 1.4 1.2

adL=** 5O

1.0 0.8 0.6 0.4 0.2 b)

0.0

0

1

i

i

i

10

20

30

40

i

50 Aa/L

F i g u r e 8.23: Applied equal biaxial tension necessary for crack growth,

(~) ~

-

5, (b)

~

-

~0

Fig. 8.21 for various values of initial crack length and transformation parameter w. For growing cracks the R-curves for several initial crack lengths obtained by solving eqn (8.20) are shown in Fig. 8.22 for two values of the transformation parameter w. The apparent toughness goes through a

212

R - C u r v e Analysis

y/L 5 4

5

2 0 -2 -4 -5

) -30

-25

- 0

-15

-10

-

0

x/L

F i g u r e 8.24: Transformation zone boundaries for equal biaxial tension, w-10

i~

,,, a p p l c/l{

1.0

Steady-state

0.8 0.6 0.4

I

0.2 \ \

0.0

0

5

10

15

20

I

i,

25

30

F i g u r e 8.25" Reciprocal peak toughening ratio under equal biaxial tension

peak before reaching the steady-state value from above. The peak values are the larger the shorter the initial crack length, which is just the opposite of that observed under uniaxial tension (Fig. 8.12). Moreover, the peak value is always above the steady-state value (Andreasen, 1990; Andreasen & Karihaloo 1993a). Another noticeable dissimilarity in R-curves between the equal bi-

8.2. Single Internal Cracks

213

axial load and the uniaxial load is in the crack advance needed before the steady-state conditions are reached. Under equal biaxial tension, the crack advance necessary for attaining the peak toughness increases with diminishing initial crack length, while under uniaxial tension it reduces. The applied equal biaxial tension necessary for quasi-static crack growth corresponding to the R-curves of Fig. 8.22 can be calculated from (8.21). The results are shown in Fig. 8.23. As under uniaxial tension, so also under equal biaxial tension, the peaks in the R and stress curves are reflected in the transformation zone shapes through a zone widening before the steady-state conditions are reached. Under equal biaxial tension, as opposed to uniaxial tension, the zone widening is more pronounced for shorter initial cracks (cf. Figs. 8.14 and 8.24). The reciprocal peak toughening ratio under equal biaxial tension is shown in Fig. 8.25. For comparison the steady-state toughness estimate consistent with the present theory is shown by the broken curve (Amazigo & Budiansky, 1988). The appearance of peaks lead to diverging toughening or "lock-up" for values of transformation strength w lower than that expected from steady-state analysis, as reported by Rose (1987a). For finite crack lengths, the lock-up values of w are above the lock-up value w = 20.2 for semi-infinite cracks (Stump & Budiansky, 1989a) and less than the lock-up value of w = 30.0 for steady-state

1.O 0.8 0.6

-

10

50

0.4 0.2 0.0

0

5

10

15

20

F i g u r e 8.26: Reciprocal peak strengthening ratio under equal biaxial tension

214

R-Curve Analysis

conditions (Amazigo & Budiansky, 1988). The reciprocal peak strengthening ratio is shown in Fig. 8.26. It diminishes with diminishing initial crack length, in much the same manner as under uniaxial tension (Fig. 8.16).

8.3

Array of Internal Cracks

We will extend the discussion of Section 8.2 to a collinear array of internal cracks. The method used is based on dislocation formalism and complex potentials and it is similar to the method used in the previous Section. A collinear array of equally spaced cracks is illustrated in Fig. 8.27. The spacing is denoted d, and c is half the length of each crack designated C. For simplicity, one of the cracks in the array is assumed to be situated with its centre at the origin. This crack is referred to as the central crack. The plane is loaded at infinity by an external stress a ~ , normal to the cracks resulting in pure opening mode I. At the tip of each crack a transformation zone with the boundary, S develops when the plane is loaded and the criterion for transformation is satisfied, i.e. c It is also assumed that the the mean stress reaches a critical value, am. transformation is accompanied by a purely dilatational strain inside the transformation zone. 8.3.1

Mathematical

Formulation

Two governing equations are derived from the following conditions 1. Traction-free crack faces, ayy(z) = 0, z E C. C 2. Critical mean stress on S, am - am, z E S.

In a detailed form these conditions are

T ( z ) + ayy ~(z) - O,zzC ayy + ayy cr ~m + ~ T( z )

+ ~ D( z )

-

~ c ,z ~ s

(8.23)

where superscripts oc, T and D denote stress contributions from the remote applied stress, transformation and dislocations, respectively. The crack-line stress from the applied load a r can be written as

[~[appl ay~ = B 0 x f ~

(8.24)

8.3. Array of InternM Cracks

215

F i g u r e 8.27: Collinear array of internal plane cracks in a transforming ceramic where B0 a geometry factor given by

B0 -

I

d 7rc ~cc tan(--~-)

(8.25)

The crack-line stress due to the transformation can be written as r

T (z) -

EcT [ r 27r(1- u) Js [GuT(x, Zo)- Guu(x,-~0)] dyo

(8.26) x6.C

where GuT(z, zo) are given by (4.60), and symmetry has been exploited to reduce the integration along the zone boundaries to that along the right hand zone for the central crack. The crack-line stress due to the dislocations is given by (YyDy (z)

--

/0cD" (t) [HuD'u(x, t) -- HyyD'u(x , --t)] dt I

(8.27)

xEC

D,y where Hyy (z, zl) are given by (6.25). Again, symmetry has been exploited to reduce the integration to only the right hand half of the crack C. The mean stress along S from the dislocations can be written as

~(z) -

fc

(8.28)

D* (t) [H~D~u (z, t) - H~Dg~y (z, --t)] dt z6_S

216

R-Curve Analysis

where HaDg~y (z, zo) are given by (6.25). Introducing the dislocation density function D*(t)

D*(t) -

E/a~n

127r(1 - u)

D(t)

(8.29)

the condition of imminent crack growth (K tip = Kc) can be written as / lim 27r~/C- XD,(x) 9~ c v L

1

(8.30)

The system of equations determining the transformation zone shape at the onset of crack growth can now be written as

O~

K appl

w J~s D* (t) [Hy u

Kappl + 1 - BoK~

iC

, z0) -

- Huy

dy0

at xEC

D* (t) [ HD'Y(Z D'Y(Z, --t)] dt ~ , t) - H,~

(8.31)

z6.S

The effect of the interaction of cracks on the strength and toughness of a transformation toughened ceramic can be studied in two ways. First, the initial crack length, co is kept constant and the distance between the cracks varied. This procedure is useful for understanding the effect of crack separation. It is also the physically most comprehensible way, especially for comparison with the results of a single internal crack. Secondly, the initial length of the unbroken material between the cracks, = ( d - 2 c 0 ) is kept constant by varying the distance between the cracks and the initial crack length. This procedure is relevant to a ceramic with a non-transformable matrix toughened by transformable particles, as shown in Fig. 8.28. Following this procedure the interaction is studied for different particle sizes and area fractions, Aj = ( d - 2co)2/d 2 of transformable particles. If the cracks in the array are very close to one another either initially or after growth, neighbouring transformations zones would merge together. This situation however will not be treated in the present analysis.

8.3. Array of Internal Cracks

217

F i g u r e 8.28: Cross-section of a particulate transformation toughened ceramic The shape and size of the transformation zone forming at the tip of each crack depend on the load, the crack length, the distance between the cracks and the transformation strength parameter, w. Due to the symmetry of the transformation zones only the upper right half of a zone will be shown in the following. The lower limit on the length of a single internal crack liable to grow was shown to be co/L = 0.5 (w For shorter cracks the applied far field mean stress necessary to initiate crack growth would exceed the critical mean stress for transformation leading to spontaneous transformation of the whole material. This limit will also be used here, even though the limit for an array of cracks would be lower than that for a single crack due to the interaction of the cracks in the array and the resulting reduction in the applied load necessary for ~=rack initiation.

8.3.2

Onset

of crack

growth

The transformation zones at the onset of crack growth are shown in Figs. 8.29-8.32 for ~o = 10 and four initial crack lengths, co/L = 0.5, 1, 5, 50. In each figure the distance, d / L is varied to show the effect of interaction between the cracks and their transformation zones. Generally, the transformation zones increase in height and length, as the cracks are brought closer. The length of the zone is characterized by the distance from the crack tip to the frontal intercept of the zone boundary with the z-axis. The intercept behind the crack tip, xc is virtually unchanged. For the longer cracks, co/L = 5 and co/L = 50, an increase in both transformation zone height and frontal intercept is observed. The same behaviour is observed in Fig. 8.33 for w = 30 and co/L = 5, but, as expected with larger zones than the corresponding zones for w = 10. For the very short cracks (e.g. co/L = 0.5) both the frontal intercept

R-Curve Analysis

218

y/L 0.5 0.4

.... dlL=10 i --

f

. . . . .

. . . . . . . . .

3.8 .....

0.3

.

.

.

.

.

0.2 O.1 0.0 0.4

0.6

I

I

0.8

1.O

I

1.2

1.4 x/L

F i g u r e 8.29: Transformation zones at the onset of crack growth for co/L - 0.5, w - 10 and d/L - oc, 10, 5, 4, 3.8, 3.5

y/t,

~f 0.4

=

.

5

0.3 0.2 0.1 0.0 1.0

J

J

i

1.2

1.4

1.6

1.8

x/L

F i g u r e 8.30" Transformation zones at the onset of crack growth for co/L - 1, w - 10 and d/L - oc, 10, 5, 4.5, 4.25 and the zone height decrease for d/L = 10 and d/L = 5, compared to the single crack. As the cracks are brought even closer the height keeps decreasing, while the frontal intercept increases. For coiL = 1 the long crack behaviour described above prevails, except when the cracks are very close, e.g. d/L = 4.25, where a reduction in height is observed. Figures 8.34 and 8.35 show the frontal intercept, r0 and the zone height, H respectively, as a function of log d-2c~ for different initial crack lengths. L

8.3.

Array of Internal Cracks

219

y~

~f

0.5 0.4

13.5

0.3 0.2 0.1 0.0 5.0

5.2

5.4

5.6

5.8

6.0

x/L

F i g u r e 8.31: Transformation zones at the onset of crack growth for coiL = 5, w = 10 and d/L = co, 50, 20, 15, 14, 13.5

y/L

~ F 0.7 0.6 0.5 0.4 0.3

0.0

.7

t

50.0

I

t

I

-'i ''=

I

50.2 50.4 50.6 50.8 51.0 51.2 51.4 x/L

F i g u r e 8.32: Transformation zones at the onset of crack growth for coiL = 50, w = 10 and d/L = co, 200,150,125,110,107,105,104.7

From Fig. 8.34 it can be concluded that the frontal intercept progressively increases as the distance between the cracks decreases, with the exception of the very short cracks when a weak decrease is observed at relatively large values of d/L. This behaviour could possibly be also observed for the longer cracks, at similar long separation distances. Consistency of the general behaviour might also have been observed for the height, i f a larger range of (d-2co)/L was included in the analysis.

R - C u r v e Analysis

220

y/t, 0.6 0.5 0.4 0.3 0.2 0.1 0.0 4.8

5.0

5.2

5.4

5.6

5.8

6.0

x/L

F i g u r e 8 . 3 3 : T r a n s f o r m a t i o n zones at the onset of crack g r o w t h for co/L - 5, w - 30 a n d d / L - oc, 50, 20, 18, 17

rdt, 1.4 1.3 -

o)=10

/

1.1 1.0 0.9 0.8 0.7 2.0

! 1.8

i 1.6

I 1.4

i ----r1.2 1.0

I 0.8

i 0.6

I 0.4

i d_2c ~ 0.2 log L

F i g u r e 8 . 3 4 : F r o n t a l t r a n s f o r m a t i o n zone i n t e r c e p t as a f u n c t i o n of l o g ( d - ~ c ~ at the onset of crack g r o w t h for w - 10, co/L - 0.5, 1,5, 50 a n d for w - 30, co/L - 5

8.3. Array of Internal Cracks

221

H/L

0.75

-

0.70

r

~

0.65

co/L=50

0.60 0.55 0.50 0.45 0.40 0.35

2.0

l 1.8

i 1.6

1 1.4

i 1.2

.---------1 ---~-- ~ 0.5 i L J I J d_2c 0 1.0 0.8 0.6 0.4 0.2 log L

F i g u r e 8.35: Transformation zone height as a function of log(d-2c~ L ) at the onset of crack growth for w - 10, c o / L - 0.5, 1, 5, 50 and for w - 30,

coiL - 5 For the height, the general trend is an increase up to a peak value, as the cracks are brought closer, and a subsequent decrease when the cracks are very close. Figures 8.36-8.38 show the zone shapes for constant length of unbroken material between the cracks, )~ = ( d - 2 c 0 ) . From the different values of )~/L, different ratios of broken to unbroken material can be calculated in order to study the effect of the area fraction of transformable material on the mechanical properties of a particulate transformation toughened ceramic. In these figures each value of c o / L corresponds to a certain area fraction. It is seen that the height and length decrease as c o / L decreases (i.e. area fraction increases) except for very short cracks ( c o / L = 0.5) where the opposite happens. This pattern was also observed in the single crack analysis when c o / L is varied with the other parameters kept constant. The differences in zone shapes in Figs. 8.36-8.38 must be seen in the light of the change in the applied load as d / L is varied in order to m a i n t a i n K tip = Kr The toughening ratio KapVz/Kc at the onset of crack growth is plotted in Fig. 8.39 and 8.40 for different values of c o / L and A / L as a function of log d-2co and A/ respectively. The curves L also represent the normalized strength, cr~176 where a0 - Kc/~v/'ff-~ is the strength in the absence of transformation, i.e. KavVz/Kc = ~r~176

R-Curve Analysis

222

y/L 0.7 0.6

~

co/L=10

0.5 0.4 0.3 0.2 0.1 0.0 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

i

1.2 (x-q,)/L

F i g u r e 8.36: Transformation zones at the onset of crack growth for $/L = 4, w = 10 and co = 0.5, 1 , 2 , 3 , 4 , 5 , 10

y/t, 0.8 0.7

-f

cdL=50

0.6 0.50.4 0.3 0.2

J

0.1 0.0 -0.2

I

0.0

0.2

0.4

0.6

0.8

1.0

1.2 (x-q~)lL

F i g u r e 8.37: Transformation zones at the onset of crack growth for )~/L = 10, w = 10 and co = 0.5, 1,2, 3, 4, 5, 7, 10, 15, 50

at the onset of crack growth. When the initial crack length is kept constant and d/L is varied, it is seen from Fig. 8.39 that the curves decay monotonically, as the length of unbroken material is reduced. The initially long cracks start at a high level of applied stress intensity at long distances but decrease

8.3.

Array of Internal Cracks

223

y/t, 0.8 0.7 0.6

,~,

0.5

~

0.4

\

co/L=200 AF--'0.040

~,

0.3 0.2

0.1

A 0.98

0.0

~,xl

-0.2

0.0

I

I

I

0.2

0.4

0.6

II

I

0.8

1.0

1.2

(x-q))/L

F i g u r e 8 . 3 8 : T r a n s f o r m a t i o n zones at the onset of crack g r o w t h for ~ / L - 100, w - 10 and co - 0.5, 1, 5, 10, 20, 50, 70,100, 150,200

Kappt/Kc =_ 0"~/00

1.00 ~

5

0

oo=lO

0.95

.......

0.90 0.85

f

0.70 / 0.65 2.0

i 1.8

i 1.6

I 1.4

"

I 1.2

I 1.0

I 0.8

I

J 0.6

0.4

J d-2c o 0.2 log

F i g u r e 8 . 3 9 : N o r m a l i z e d applied stress intensity factor or applied stress as a f u n c t i o n of log d-2cQ at the onset of crack g r o w t h for w - 10, L c o / L = 0.5, 1, 5, 50 a n d for w = 30, c o / L = 5

R-Curve Analysis

224

K

appl

/Kc = 0"*/~o

1.O0 ML=IO0 0.95 0.90 0.85 0.80 0.75 t 0.0 0.1

t I I t I t l I J 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 A t

F i g u r e 8.40: Normalized applied stress intensity factor or applied stress as a function of area fraction at the onset of crack growth for w - 10 and A/L = 4, 10,100

more rapidly to a lower level than the shorter cracks. To conclude, the normalized strength and toughness are seen to decrease, the Closer the cracks are for a constant initial crack length. In other words, the detrimental effect of the transformation at the onset of crack growth is amplified by the interaction of the crack tips and transformation zones, compared to a single crack. By comparing the curves for w = 10 and w - 30 in Fig. 8.39 it is again seen that a higher value of w has a detrimental effect on the normalized toughness and strength (KaPPt/Kc and ~r~176 are always below unity). This means that the transformation weakens the material at the onset of crack growth. This result is in agreement with previously obtained results for a single crack. In Fig. 8.40 the amount of unbroken material between the cracks is kept constant by an appropriate variation of co/L and d/L. For moderate )~/L(= 4, 10) it is seen that the toughness at the onset of crack growth goes through a peak whose location depends on the value of )~/L. For a particulate toughened material it is concluded that at the onset of crack growth an o p t i m u m area fraction of particles exists, and that large particles are favourable with respect to the toughness and the strength, relative to a non-toughened material with a similar crack configuration.

8.3.

A r r a y of I n t e r n a l C r a c k s

225

o~.,E Kc co/t~o.5

0.7

{o=10 ~

0.6

{0=30 ........

0.5 0.4 0.3 5

0.2 O.1 --------I I 0.0 2.0 1.8 1.6

50 I

I

I

7---

I

1.4

1.2

1.0

0.8

0.6

i 0.4

i d_2c ~ 0.2 log L

F i g u r e 8 . 4 1 : Actual normalized applied far field stress as a function of log d-2c0 at the onset of crack growth for w - 10 , c o / L - 0.5 , 1 , 5 , 50 L and for ~ - 30, c o / L 5

0.7 0.6 0.5

-

l0

/

0.4 0.3 0.2 0.1 0.0 0.0

I 0.1

I

0.2

I

l

0.3 0.4

1

I

I

i

I

0.5

0.6

0.7

0.8

0.9

I

1.0 At

F i g u r e 8 . 4 2 : Actual normalized applied far field stress as a function of area fraction at the onset of crack growth for w - 10 and A/L - 4, 10,100

R-Curve Analysis

226

Figures 8.41 and 8.42 show the actual normalized applied far field stress corresponding to Figs. 8.39 and 8.40. Figure 8.42 shows that the actual normalized strength increases monotonically with increasing fraction of transformable particles.

8.3.3

Growing cracks

Each individual crack in the array is grown quasi-statically by the same amount by adjusting the applied load, so that K tip = Kc, as described in Section 8.2. The toughening ratio is plotted in Figs. 8.43 and 8.44 as a function of the crack advance for co/L = 5 and co/L = 50, respectively and different values of normalized distance, d/L. The curves correspond to the R-curves for an array of cracks. For large separation distances, e.g. d/L=lO0 in Fig. 8.43, the curve follows the single crack R-curve relatively closely until the cracks have grown sufficiently to interact, whereafter the curve decreases rapidly. For the smaller values of d/L the curves peak earlier as the neighboring crack tips and the transformation zones interact at an earlier stage of crack advance. Two factors are responsible for this behaviour. First, the initial de-

Kapp I

Kc 1.6

dlL= .~ 1.5 I00

1.4 1.3 1.2

~20

1.1 l.O 0.9

0

I

I

I

I

I

I

I

.J

5

10

15

20

25

30

35

40

C-C O

L

F i g u r e 8.43: Normalized applied stress intensity factor as a function of crack advance for w - 10, c o / L - 5 and d/L - oo, 100, 50, 30, 20

8.3.

A r r a y of Internal Cracks

227

Kapp I

Kc 1.7 1.6

d/L

-

oo

1.5 1.4 1.3 1.2 1.1

112

1.0 0.9 0

i 5

i 10

i 15

i 20

I

I

I

I

25

30

35

40

c-c o

L

F i g u r e 8 . 4 4 : N o r m a l i z e d a p p l i e d stress i n t e n s i t y factor as a f u n c t i o n of crack a d v a n c e for w - 10, c o / L - 50 a n d d / L - oo, 1 2 5 , 1 1 2

~'/~o 1.3 F 1.2 1.1 1.0 0.9 0.8 -

30 30"

~~-...._

0.7 _ 0.6 0

Figure for w -

5 I 2

I 4

I 6

t 8

0 t 10

d/L = ,,,, ~ I 12

100 I 14

C.Co L

8 . 4 5 : N o r m a l i z e d a p p l i e d stress as a f u n c t i o n of crack a d v a n c e 10, co/L - 5 a n d d/L - oo, 100, 50, 30, 20

228

R-Curve

Analysis

~~176 1.6 1

.5 ~ - d / L

1.4

5

= oo

~

1.3 1.2 1.1

~112

1.0 0.9 0

I 2

I 4

I 6

J 8

I 10

I 12

t 14

c-c o

L

F i g u r e 8 . 4 6 : N o r m a l i z e d a p p l i e d stress as a f u n c t i o n of crack a d v a n c e for w - 10, c o / L - 50 a n d d / L - ~ , 1 2 5 , 1 1 2

~/(~0 0.90

co/L = 2

0.88

3

1 ------------4

0.86 0.84 0.82 0.80 0

Figure for w -

i 0.02

i

i

i

0.04

0.06

0.08

c-c O

L

8 . 4 7 : N o r m a l i z e d a p p l i e d stress as a f u n c t i o n of crack a d v a n c e 10, A / L - 4 a n d c o / L - 1, 2, 3, 4, 10

8.3.

Array o f Internal Cracks

229

o*"/o 0 1.15 1.10

15

-

co~L= 10

_4

5

1.05

1.O0 0.95 1

0.90 0.85

F---0.5

0.80 0

Figure for w -

I 0.2

I 0.4

I 0.6

i 0.8

I 1.0

I c-~ 1.2 L

8 . 4 8 : N o r m a l i z e d a p p l i e d s t r e s s as a f u n c t i o n of c r a c k a d v a n c e 10, A / L - 10 a n d c o / L - 0.5, 1, 2, 3, 4, 5, 10, 15, 50

~**/ao 1.7 [1.6 1.5 ].4

co~L= 200

t

2-0

1.3 1.2 1.1 1.0 0.9 I 1 0.8 0

Figure for w -

I

I

1

2

2 3

I 4

I 5

I 6

I 7

I

8

c-c o

L

8 . 4 9 : N o r m a l i z e d a p p l i e d s t r e s s as a f u n c t i o n of c r a c k a d v a n c e 10, A / L - 100 a n d c o / L - 1,5, 10, 20, 5 0 , 1 0 0 , 2 0 0

230

R-Curve Analysis

velopment of transformation zone wakes behind the crack tips reduces K tip, and consequently K appz must be increased to maintain K tip - Kc. Secondly, the interaction of the crack tips and transformation zones in the array of cracks increases K tip as the cracks come closer, so that K appz must be reduced to maintain K tip = Kc. The interaction of the crack tips is determined by the geometrical factor B0 used to calculate K "ppt (8.25). For a single crack, these two factors will result in the R-curve going through a peak before reaching a steady-state level. For both c o / L = 5 and c o / L = 50 it is evident that the interaction of the crack tips and transformation zones drastically reduces the peak toughness of the material. Figures 8.45 and 8.46 show the normalized applied stress against the crack advance. It is seen that the stress curves peak earlier than the corresponding R-curves, but still allow for some subcritical crack growth, if loaded by a monotonically increasing far field stress. The applied stress, normalized with respect to ~0, is shown as a function of crack advance in Figs. 8 . 4 7 - 8.49 for the three values of )~/L = 4, 10,100, respectively. The peak stresses from these figures are then plotted in Fig. 8.50. For the small particle sizes ~ / L = 4, 10, the strength peaks at a finite area fraction but for )~/L = 4 it is below the

cP/c0 1.7 1.5 1.5 1.4 1.3 1.2 1.1 1.0 0.9 J 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

A]'

F i g u r e 8.50: Normalized applied peak stress as a function of area fraction, w - 10 and )~/L - 4, 10,100

8.4. S u r f a c e C r a c k s

231

o"4Z Kc 0.7 0.6 0.5

k/L = 4

0.4

10

0.3 0.2 0.1 0.0

I

I

I

I

I

I

I

1

i

I

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Af

F i g u r e 8.51: Actual normalized applied peak stress as a function of area fraction, w = 10 and )~/L = 4, 10,100

strength of a non-toughened material. From this it can be concluded that the transformation toughening is only efficient for large particle sizes for which an optimum particle fraction for the relative strength exists, but as shown in Fig. 8.51 the actual normalized strength is highest for small particles in high concentration.

8.4

Surface Cracks

In this Section a model for a single surface crack in transformation toughening ceramics is described with a view to examining the development of transformation zone, toughening, and strengthening along the lines of analysis presented for an internal crack in Section 8.2 (Andreasen & Karihaloo, 1994). Surface damage is a fundamental issue in the analysis of transformation toughened ceramics. The model described below is expected to form a good basis for analyzing thermal shock, fatigue, wear and other phenomena in which the presence of surface cracks plays a fundamental role. Some of these phenomena will be further examined in Part III of this Monograph.

R-Curve Analysis

232

8.4.1

Model

Description

and

Theory

The problem of surface cracks in the absence of transformation has been solved by a number of investigators, see e.g. Nemat-Nasser et al. (1978), Keer et al. (1979), and Nemat-Nasser et al. (1980). In these references the stability of thermally induced surface crack growth was studied.

F i g u r e 8.52: Model configuration for a surface crack The model for a surface crack in T T C is shown in Fig. 8.52. A surface crack C of length c is situated in the half plane and loaded at infinity by a constant transverse stress (r~ . At the tip of the crack a zone of transformed material bounded by the contour S develops as the load is applied. The transformation strains are assumed to be purely dilatational and constant inside the zone. The transformation is assumed to be induced by a critical mean stress crc In the analysis to follow the free-surface problem is solved analytically for a dislocation and for a homogeneous inclusion of arbitrary shape by means of Muskhelishvili's theory of plane elasticity (Chapter 4). The crack is modelled by a pile-up of dislocations (Chapter 6). The density of dislocations in the pile-up is adjusted to meet the traction-free crack condition. The transformation zone boundary is determined by the critical mean stress criterion. The traction-free condition and the critical mean stress criterion lead to two coupled singular integral equations which are solved numerically. The pile-up of dislocations can be described through a dislocation density function D(yo), such that D(yo)dyo is the Burgers vector of the dislocations between y0 and Yo + dyo. The stress due to this pile-up on the crack-line (r~xc gives an integral equation from which the dislocation density function can be determined such that the stress across the crack

8.4. Surface Cracks

233

C vanishes as described in Chapter 6. Formally, the crack-line stress (Fig. 8.52)can be written as ~(z)

-

0 -

~oo + ~ (~~ )

+

~(~) 1 ~

(8.32)

In terms of the applied stress intensity factor K appz the load r ~176 can be written as ~oo =

Kappl

(8.33)

B0x/~

For a single crack B0 is approximately 1.1215, (see e.g. Tada, 1985 or Murakami et al., 1987). The crack-line stress r from the transformation is calculated from

T gxx

o'xx

(ir, zo)dxo]

(8.34) rEC" zoE S

where the weight function gT~(z, zo) is given by (4.56), with z - 0-4- iv. The crack-line stress from the dislocations is calculated from

E

crD~(z)- 47r(1- t~2) / c D(yo )hxD~x (it, is)ds

(8.35) r,sEC

where the weight function h~DJ(r, s)is given by (6.19) (with z - 0 + iv and z0 = 0 + is). The transformation zone boundary is determined from the critical mean stress criterion, which can be written as crr m 1 + t, (~r~ + a~,(z)) eo + ~r,,(z) T D I m --

3

(8.36)

zES

The mean stress from the applied load is ~r~a~176 _ ~oo and it is given by (8.33) in terms of the applied stress intensity factor. The mean stress from transformation can be written as

l3q~ l,, ~T( z )

T - ~w / s g~(z, zold~o I~,~o~S

(8.37)

where g~a T ( z , )z0 is given by (4.56). Finally, the mean stress from the dislocations can be written as

R-Curve Analysis

234

l+u

D _

3(r~ aa~

E(l+u) 127r(1- v2)a~

/cD(yo)hD~(z, is)dsl

zes. ,ec

(8.38)

where the weight function h~D3~(z,s) is given by (6.19). The two equations for zero traction across the crack and for the transformation zone boundary can be rewritten in similar forms. For this we introduce the dislocation density function D*(s) via

D*(s) -

E/cr~

127r(1 - v)

D(s)

(8.39)

and substitute it into (8.35) and (8.38). From (8.32)-(8.35), and (8.36)(8.38), we finally get

O~

rEC Kappi~~c Tx (z, zo)dxo + /C D* (s)ha~; Dx (z, is)ds [(8.40) BoKc + ~w ~s ga'd

IzES

The dislocation density function D*(s) and the transformation zone boundary S can be obtained from (8.40) for a given load K appt and transformation parameter w. In order to analyse initial toughening and R-curve behaviour, the stress intensity factor at the crack tip K tip is required so as to impose a steady-state crack growth condition. The dislocation density function D* (s) has a square root singularity as the crack tip is approached and the stress intensity factor K tip is given through the following limit

Ktip Kc

=

lim 27rD*(s) i ~--~+

c+s L

(8.41)

At the onset of crack growth, as well as for quasi-static crack growth, the stress intensity factor at the crack tip K tip must equal the intrinsic toughness Kc, i.e. J

1-

lim 8--~-- C+

r

27rD*(s),/c~ s v

(8.42)

L

The steady-state growth condition supplements the two eqns (8.40), thereby allowing K appl - or the R-curve - to be determined.

8.4.

Surface Cracks

235

The two integral equations (8.40) contain a number of singularities. As already mentioned, the dislocation density function D* (y) has a square root singularity at the crack tip. The crack-line stress imposed by the transformation zone contains a discontinuity as the transformation zone boundary is crossed. This leads to a logarithmic singularity in the dislocation density function. The weight functions contain singularities of the ordinary Cauchy type, as well as weak singularities at the surface and at the transformation zone boundary intersection by the crack. In order to obtain accurate numerical solutions, it is imperative to have good control over these singularities. Thus, the singularities are isolated and treated analytically as far as possible, in order to ensure that only regular functions are numerically integrated. Solutions to (8.40) are obtained by improving a guess for the transformation zone shape through a number of perturbations. Inverting the first integral equation (8.40) for each perturbed shape, an improved zone shape is obtained by Newton-Raphson's method. For a growing crack the two coupled integral equations (8.40) can be restated in an incremental form as follows:

O

g appl

/

BoKr

V2(c + Ac)

L

gTx( ir' zo)dxo + Iv D*(s)hxDf~x(ir'is)ds rEC K appl /

BoKr +~

L

V2(c + Ac)

gag (z,

lim 27rD* (s) s~-(c+ac)+ S(c + ac)wok, =

+

D*

D,~

is)ds zeS/,'ont

/Cv + Ac + s L (8.43)

The assumption of no reverse transformation is imposed by the last side condition in (8.43). The transformation zone shape is only changed at the front Sf,-o,t of the crack tip, where the mean stress is rising, while a wake S,oake of transformed material is allowed to develop behind the

R-Curve Analysis

236

tip of the growing crack, where the mean stress is declining. The procedure for solving (8.43) is based on a guess for the transformation zone front SIront and the solution of the first integral equation (8.43) using this guess. The resulting dislocation density function D*(yo) is substituted into the second integral equation (8.43), whose solution gives an improved estimate of Si,,o,~t. This procedure is applied repeatedly until convergence criteria are met. The two side conditions in (8.43) are met by adjusting K appz iteratively and by joining the transformation zone wake S~ake and front S],-ont by common tangents until sufficient accuracy is attained (see w

8.4.2

Single Surface Cracks

In the following initial transformation zone shapes, initial toughening, and R-curve behaviour are discussed. Some general results on peak toughness and strengthening are also presented. The discussion follows the same lines as for internal cracks (w167 and where appropriate comparisons between surface cracks and internal cracks are made to emphasize relevant differences or similarities. Transformation zone shapes at the onset of crack growth obtained by solving (8.40) are shown in Fig. 8.53. The detachment of the trans-

y//.,

/{o=30

1.0

//is -/10

0.8 0.6 0.4 0.2 0.0

I

0.0

0.4

0.8

1.2

1.4

x/L

F i g u r e 8.53: Initial transformation zone shapes for single surface crack,

co/L- 10

8.4.

Surface Cracks

237

gappl/g c 1.0

co/L=500 50

0.9

lO

0.8

_

5

0.7 0.6 0

I 5

t 10

t 15

l 20

I 25

J 30

F i g u r e 8.54: Toughening ratio at the onset of crack growth for single surface crack formation zone wake from the crack tip is characteristic of the model being studied. The transformation zone size increases monotonically with the transformation parameter ~o when the initial crack is not too small. The critical initial crack length co/L at which the transformation zone diverges for vanishing r because the mean stress from the applied load cr~176 exceeds the critical value a ~ before crack growth is initiated is approximately co/L -- 0.3975 for a single surface crack, as opposed to co/L = 0.5 for a single internal crack (w However, as with internal cracks, the toughness decreases for w :/: 0, so that the transformation zone is bounded, even for an initial crack length equal to the critical value. The apparent toughness (i.e. the toughening ratio) at the onset of crack growth for several initial crack lengths is shown in Fig. 8.54. In general, the ratio decreases before crack growth, except for very long cracks when a slight increase is observed. The toughening ratio for a semi-infinite crack is within 0.5% of the toughening ratio for the surface crack of length c0/L = 500. The R-curves determined from the solution of (8.43) are shown in Fig. 8.55 for several initial crack lengths and for two values of the transformation parameter w. The limiting case of an infinite crack, shown with a broken line was first solved by Stump & Budiansky (1989a). The toughening ratio peaks before the reaching steady-state level. The pres-

238

R - C u r v e Analysis

K

appl /K c

1.3

m

1.2 1.1 oo

1.O 0.9 a)

0.8 0 K

I

I

I

5

l0

15

I

20 Ac/L

appl /K c

1.8 1.6 1.4 1.2

b)

,~ I 0.8

0

9

t

t

l

I

J

10

20

30

40

50

Ac/L

F i g u r e 8.55" R-curves for single surface crack, (a) w - 5, (b) w - 10

ence of a free surface causes the peak value to drop for short initial cracks; the peak value may even drop below the steady-state level. In Fig. 8.55b this is evident for the crack of length co/L = 5. Monotonically rising R-curves are also observed (e.g. for co/L = 5 in Fig. 8.55a), but this behaviour is the exception rather than the rule. The appearance of peaks in the toughening ratio prior to the attainment of the steady-state level seems to be an inherent feature of models based on the critical mean stress transformation criterion. They ap-

8.4. Surface Cracks

239

~**/% 1.4

co/L=500

1.2

50

1.0 0.8 0.6 0.4 0.2 a)

0.0

0

'

'

'

'

5

10

15

20

Ac/L

t~**/% 1.8 1.6

co~L=500

1.4

50

1.2 1.0 0.8

10

0.6

5

0.4 0.2 b)

0.0

0

I

I

I

I

10

20

30

40

I

50 Ac/L

F i g u r e 8.56: Strengthening ratio for single surface crack, (a) w - 5, (b) w - 10

pear in R-curves for semi-infinite cracks (w for internal cracks under uniaxial tension (w for internal cracks under equal biaxial tension (w and now for surface cracks. The peaks get shallower as the initial crack gets shorter. This trend is the same as we observed for internal cracks under uniaxial tension (w but it is contrary to that seen for internal cracks under equal biaxial tension (w The toughening ratio for a semi-infinite crack shown in Fig. 8.55 has converged to within a

R-Curve Analysis

240

fraction of a percent of the steady-state toughening value. In comparison with semi-infinite cracks, the convergence to steady-state values of the R-curves for finite initial crack lengths is seen to be quite slow. The applied stress ~r~176 at infinity necessary for maintaining quasistatic crack growth is shown in Fig. 8.56 for several initial crack lengths and for two values of the transformation parameter w. The strengthening ratio in Fig. 8.56 corresponds to the toughening ratio of Fig. 8.55. The strengthening ratio is obtained from the following relation between stress and toughness

~176176 _- K avptKr i co+c~Ac

(8.44)

where rr0 is the stress necessary to induce crack growth in the absence of transformation. Comparison of Figs. 8.55 and 8.56 shows that the peak strengthening ratio that determines the ultimate strength is attained at a shorter crack advance than is necessary to obtain peak toughness. Thus the peak toughness is not fully available for strengthening of the material, except for very long initial cracks. The peaks in the R-curve behaviour have their origin in the widening of the transformation zone (Fig. 8.57). The reciprocal peak toughening is shown in Fig. 8.58 for values of

y/t, 1.6 1.2 0.8 0.4 0.0

-30

-25

-20

-15

-10

I

I

-5

0

x/L

F i g u r e 8.57: Transformation zone shapes for single surface cracks, w 10

8.4. Surface Cracks

241

Kc/Kpeak 1.o 0.8

cd~5 lO 50 500

0.6 0.4 0.2 0.0 0

i

I

t

I

I

i

i

J

2

4

6

8

10

12

14

16

F i g u r e 8.58: Reciprocal peak toughening for single surface cracks

1.0 0.8 J

0.6 50 0.4 0.2 0.0 0

I

I

I

5

10

15

20

F i g u r e 8.59: Reciprocal peak strengthening

R-Curve Analysis

242

the transformation parameter w up to 16. Lock-up values of the transformation parameter w at which the transformation zone diverges and the peak toughness tends to infinity are not known for the geometry of Fig. 8.52. For initial cracks of length co/L = 5 and 10, the lockup values are less than the lock-up value w = 20.2 for a semi-infinite crack (Stump & Budiansky 1989a), whereas the lock-up values for initial cracks of length co/L = 50 and 500 are expected to lie between this value for a semi-infinite crack and the value under steady-state conditions (w = 30.0). The peak toughening ratio reduces with decreasing initial crack length for moderate values of w. For w larger than about 12, this trend is reversed (Fig. 8.58). The reciprocal peak strengthening is shown in Fig. 8.59 for values of the transformation parameter up to 20. These results are consistent with the results for internal cracks in that initially weak materials with long inherent cracks are more susceptible to strengthening than initially strong materials. The qualitative similarity of the strengthening results for single surface cracks with those for single internal cracks presented in w suggests similarity in their peak strengthening correlation with toughness (see Figs. 8.17 and 8.18).

8.5

Array of Surface Cracks

We shall now extend the discussion of the previous Section to an array of surface cracks. Surface damage is a fundamental issue in the application of ceramic materials. As a first step towards modelling this damage in transformation toughening ceramics, an array of surface cracks is introduced and analysed in a manner similar to that for a single surface crack. The configuration for an array of surface cracks is shown in Fig. 8.60. An infinite array of equally spaced (spacing d) surface cracks C of length c is situated in the half plane and loaded at infinity by a constant transverse stress aoo. At the tip of each crack a zone of transformed material bounded by the contour S develops as the load is applied. Effects of elastic mismatch between matrix and transforming particles are neglected, and reverse transformation is assumed not to take place. The applied stress can be expressed in terms of the applied stress intensity factor K avvt via a~ =

Kappl

Bo~/~

(8.45)

8.5. Array

of Surface Cracks

243

F i g u r e 8.60: An array of surface cracks

where B0 varies with the crack spacing (see e.g. Tada, 1985 or Murakami et al., 1987). The governing equations determining the transformation zone shape and the dislocation density function at the onset of crack growth are obtained from (8.40) by replacing the weight functions for a single surface crack with those for an array of surface cracks

O m

EC

BoKc

+~

G~

, )dxo +

D*

is)ds zES

1-

lim 2 r D * ( s ) ~/ / c + s u---- c+ V L

(8.46)

zo) T The weight functions due to transformation GTx(z, zo) and a~.(z, are given by (4.64), and the weight functions due to dislocations D x D,x g,~, (z, z0)and H,~, (z, zo) by (6.36). For a growing crack (8.46) can be restated in an incremental form

/ L BoKc V2(c + Ac)

K appl O

__

+ ~

fs G T~ (ir' z~176 + /c D* (s)H~Da;x(ir' is)ds[ rEC

R-Curve Analysis

244

K appl/ L BoK~ V2(c + Ac)

__

+ ~W ~ s C~a T,x (z, zo)dxo+/cD. (s)H~ D,x(z, is)ds zESfront 1 -

27rD*(s) ,/c

lim

~-.-(c+Ac)+

S(c + Ac)wake

=

+ Ac + s

V

L

S(c)wake

(8.47)

where the assumption of no reverse transformations is imposed by the last condition. Examples of initial transformation zone shape for an array of surface cracks obtained by solving (8.46) are shown in Fig. 8.61 for several crack spacings. The initial toughening accompanying the zones varies from a decrement of approximately 3% for infinite crack spacing to an increment of approximately 8% for crack spacing equal to the crack length (cold = 1; d/L = 10). The increment in apparent toughness at the onset of crack growth for crack spacing d/L less than approximately 40 is in contrast to the results for a single crack on initial toughening where only a slight increment in toughness appears for very long cracks and relatively high values of the

x/L

1.6f 1.4 1.2

1.0 0.8 0.6 0.4 0.2 0.0 -0.5

30 40

0.0

I

I!

0.5

1.0

II!

I

I

I

1.5

2.0

I

I

2.5

y/L

F i g u r e 8.61" Initial transformation zone shapes for arrays of surface cracks, w = 10 and c0/L = 10

8.5. Array o f Surface Cracks

245

appl

g

/K c

2.0 1.8 1.6 1.4 1.2 oo

1.0 0.8

0

Figure

co/L-

8.62:

J

'

'

'

10

20

30

40

'

50 Ac/L

R-curves for an array of surface cracks, w -

10 and

10

transformation parameter w. R-curves for various initial crack lengths obtained by solving the equations of (8.47) are shown in Fig. 8.62. The peaks in the apparent toughness induced by the transformation are the steeper, the smaller the crack spacing. The strengthening ratio ~r~176 corresponding to the toughening ratio of Fig 8.62 is shown in Fig. 8.63. This ratio is obtained from the relation 0"~

_

Cro -

K appl B o [

Kr

co

B Vco+Ac

(8.48)

Note that the geometry factor B depends on the crack length and therefore on the crack growth increment Ac. The peak value of the strengthening ratio determines the ultimate strength of the ceramic. As the cracks grow, the effect of the free surface diminishes and the stress necessary for continued crack growth depends more on the crack spacing, rather than on the crack length. In the limiting case where the steady-state conditions prevail, the stress needed to give quasi-static crack growth becomes constant and vanishes for a single surface crack (d/L = cxz). In contrast to the single surface crack model previously presented, the strengthening effect of transformation is enhanced by the presence

246

R-Curve Analysis

of multiple surface cracks, so that initially strong materials with closely spaced cracks are more amenable to strengthening that initially weak materials with widely spaced cracks. The above model for interacting surface cracks is expected to be a good first approximation for the analysis of surface damage. However, it is important to consider stability of growing cracks. The R-curve behaviour induced by transformation ensures a certain degree of stability in the crack growth. In the absence of transformation, small variations in crack length in ideally brittle materials will cause only the longest crack to grow. In the presence of transformation however, the R-curve behaviour counteracts this tendency and a large number of surface cracks can be expected to grow together before failure eventually is caused by the growth of the longest crack. Another important factor to be considered is the crack path stability. Small variations in crack length or crack spacing will cause the crack paths to depart from the initial crack plane implied in the present model. This can cause cracks to coalesce or provide additional shielding of the smaller cracks, whereby the plasticity effects are reduced by lowering the number of growing cracks.

o

/o o

2.0 1.6 1.2 5O 1~

0.8 0.4 0.0 0

a

J

I

t

10

2O

3O

4O

I

5O Ac/L

F i g u r e 8.63: Strengthening ratio for arrays of surface cracks, w - 10 and co/L = 10

8.6. Steady-State Analysis of an Array of Semi-Infinite Cracks

8.6

247

Steady-State Analysis of an Array of Semi-Infinite Cracks

In this Section an analysis of an array of semi-infinite edge cracks in transformation toughening ceramics under steady-state conditions is presented. It transpires that the transformation zones between the cracks cannot coalesce, but that for transformation densities above a critical value two transformation zone solutions are possible. One solution pertains to quasi-static crack growth and the other to pretransformed materials. The latter can cause excessive transformation to appear during loading before crack growth is initiated. The multiplicity of solutions is a consequence of the semi-infinite crack length. As was shown above, an array of finite surface cracks very effectively shields the crack tips in comparison with single surface cracks. For crack spacing less than about 5 times the crack length the stress intensity factor for an array of finite surface cracks is within 2% of the stress intensity factor for a similar array of semi-infinite surface cracks (Tada, 1985). The multiplicity of solutions that emerges from the study of steady-

F i g u r e 8.64: Model configuration for an array of semi-infinite cracks

248

R-Curve Analysis

state growth of semi-infinite edge cracks was not found in the similar study of finite surface cracks (w Surface grinding of transformation toughening ceramics can induce a certain strengthening of the component if the grinding gives rise to transformation. The grinding-induced transformation can be the result of at least two mechanisms. First, as the contact stresses between the grinding agent and the ceramic are locally very large, and possibly singular if the grinding agent consists of irregular particles, transformation is likely to take place in the vicinity of contact. A second, less direct mechanism is that the grinding just induces small cracks in the surface, but the transformation is brought about by subsequent loading of the ceramic during either the grinding process or service. The latter mechanism can be expected to give rise to transformation in a thicker surface layer in comparison with the former mechanism. Limited crack growth can be sustained by an array of cracks, where R-curve behaviour induced by transformation prevents the instability of this configuration that would otherwise occur. In the following, this mechanism where the transformation is a result of crack growth will be considered. The model configuration of an array of semi-infinite edge cracks is depicted in Fig. 8.64. An infinite array of equally spaced parallel cracks C (spacing d) is loaded at infinity by a constant normal stress cr~ . Each crack is bounded by a zone S of transformation formed during loading and crack growth. The zones are assumed to continue along the crack faces to infinity along the negative x-axis, such that steady-state conditions prevail. The transformation strains are assumed to be constant and purely dilatational in the zones in accordance with the super-critical transformation assumption. The transformation zone boundary ahead of a crack is determined by the critical mean stress criterion. A similar model for a single semi-infinite crack was described in Chapter 7. The transformation toughening behaviour at the onset of crack growth, and during stable crack growth, for an array of finite surface cracks was presented in the previous Section. The analysis in this Section gives the results appropriate for the limiting case of semi-infinite edge cracks in steady-state conditions. Multiple solutions, which did not emerge in the growth of finite cracks, now seem possible. These solutions suggest the possibility of having crack systems in transformation toughening brittle materials, whose growth is preceded by excessive transformation, thus leading to inelastic behaviour before failure. In the theoretical analysis to follow the cracks are modelled by a pile-up of appropriate dislocations. The density of the dislocations in the pile-up is adjusted to meet the traction free crack condition. The

8.6. Steady-State Analysis of an Array of Semi-Infinite Cracks

249

transformation zone is modelled as a homogeneous inclusion, and the transformation zone boundary ahead of the crack is determined by the critical mean stress criterion. The two conditions mentioned are expressed through two coupled integral equations which are solved numerically. The first condition, that of zero traction will be considered in some detail. Each crack is modelled by a pile-up of dislocations described through a dislocation density function D(t) such that Burgers vector b between t and t + dt is b = D(t)dt. Summing the stresses from each dislocation in the array of cracks and integrating the pile-up along the central crack, the following crack-line stress due to these dislocations is obtained

r (X)

_

E

47r(1 - ~2) j_oo D(t)H D'u (x, t)dt

(8.49)

where HuD,Y(z, zo ) is given by (6.31). x and t are along the central crack C. Taking advantage of the fact that y - y0 - 0 in uyD'Y(z, zo)in (6.31) the weight function Hy~,y(x, t) reduces to Hyy (x,t) -

~

2coth(Tr

d

)- ~(x-

t)cosech(~r

d

) (8.50)

The crack-line stress from the array of transformation zones is similarly obtained by summation as T (.)

r

_

r(* , z o ) d y o

ayy

27r(1 - v)

(8.51)

where Gyu(x, T zo)is given by (4.62). The integration along the transformation zone boundary S in the crack-line stress in eqn (8.51) can be reduced to a line integral over the transformation zone front by exploiting the fact that the integrand of this equation reduces to a constant at infinity along the negative direction of the x-axis, and so giving

-

r

271"(1 -- P)

H

, z0) -

dy0

(8.52)

The bounds of the integral + H in eqn (8.52) are the half-height of the transformation zone (see Fig. 8.64). It is determined from the transformation zone front as the point where dyo/dxo vanishes along the zone boundary. It will be shown below that H is limited to a quarter

R-Curve Analysis

250

of the crack spacing, so that coalescence of neighbouring transformation zones cannot occur. The transformation zone wakes along the crack faces are a result of the assumption of no reverse transformation. The term -Tr/d is due to the closed part of the transformation boundary S at infinity along the negative z-axis. If the boundary S is assumed not to close at - c ~ , a non-zero far-field stress appears in the x-direction. Elimination of this stress at infinity yields the same governing equation, as (8.52) above. The condition of no traction across the cracks can now be obtained by adding the crack-line stresses from (8.49) and (8.52) and a constant stress dryy 0 needed to ensure the stress conditions at infinity

0 -

0 + 2~'(1 - u) o'yy

"

Z( g

E

r Guy

-~ dyo

)

/~__o~ D(t)HyyD,y (z, t)dt

+ 4 r ( 1 - u2)

(8.53)

Due to symmetry the crack-line shear stress automatically vanishes. The constant stress ~ryy 0 introduced in eqn (8.53) is determined by considering the stresses at infinity in the y-direction. The stress from the dislocations is obtained by letting x tend to infinity in eqn (8.49). The stress from the transformation zones given by eqn (8.52) vanishes, so the resulting stress ~r~ at infinity is given by

cr~

-

o +

~r~y

2d(1

-

u 2)

oo

D(t)dt

(8.54)

The stress at infinity in the z-direction automatically vanishes. Eliminating the constant stress ~ryy 0 from eqn (8.53), by introducing the more practical stress at infinity ~r~176 corresponding to the far field loading (see Fig. 8.64) from eqn (8.54), finally gives the following condition of vanishing crack-line stress

0

-

or~17627r(1 - u)

+ 4 r ( 1 - u2 )

H

~ D(t)

zo)

-

-j

HyD'y(x,t) - 2-~ dt

(8.55)

The transformation zone boundary is determined from the critical

8.6. Steady-State Analysis of an Array of Semi-Infinite Cracks

251

mean stress criterion. Following the same method as for the crack-line stresses, the mean stresses are obtained through the summation for all cracks in the array and integration along the central crack. The resulting c the mean stress gives through the critical mean stress criterion am _ am following equation for the transformation zone boundary 1_

l+v(aco 3r

E f ( 7r) ) + 27r(1 - v2 ) co D(t) u D ~ ( z , t) -- 2-~ dt (8.56)

where H,D&~ (z, Zo) is given by (6.31). The transformation itself does not contribute directly to the mean stress outside the transformation zone (see eqn (4.62)), so the additional mean stress from the transformation appears only indirectly as a result of the change imposed on the dislocation density function D(t) from eqn (8.55). The two integral equations for zero traction across each crack in the array (8.55) and for the transformation zone boundary (8.56) can be rewritten in similar forms. For this we introduce the transformation parameter w given in (3.26) and the length measure L given in (7.28). The stress at infinity is conveniently normalized by the critical applied stress or0 which would induce transformation in an uncracked specimen, namely 3 r = ~a~ (8.57) l+u Introducing a new dislocation density function D0(t) through

E/ L

Do(t)-

(8.58)

1 2 ( 1 _ v) D(t)

finally gives the integral equations for the traction-free crack (8.55) and the transformation zone boundary (8.56) 0 - -- + if0 ~ +-

1/

7f"

1 - -frO

Guy (x, zo) -

dyo

S yD'y (x, t) - 2

dt

H

Do(t)

(

co

+ 7r

Do(t) co

H~D~(z, t) -- 2

dt

R-Curve Analysis

252

From (8.59) the dislocation density function Do(t) and the transformation zone boundary S can be obtained for a given load ac~ and a value of the transformation parameter w. In order to obtain the specific solution for quasi-static crack growth at steady-state conditions the stress intensity factor at the crack tip K tip is needed to impose a crack growth criterion. The dislocation density function Do(t) has a square root singularity as the crack tip is approached and the stress intensity factor K tip is given through the following limit

Ktiv Is

lim 2D0(x) ~--,o-

(8.60)

L

For quasi-static crack growth the stress intensity factor at the crack tip K tip equals the intrinsic toughness Kc

K tip -- Kc

(8.61)

Combining eqns (8.60) and (8.61) and the condition for determining the transformation zone height H gives the following supplementary conditions for the solution of the system of equations (8.59)

1 - z-.0-1im2 D o ( X ) I L x

0 - odU~

y0)

S

(8.62)

Solutions to eqns (8.59) and (8.62) for specific values of the transformation parameter w are obtained by initially guessing a transformation zone boundary and then iteratively solving the first eqn (8.59) to obtain the dislocation density function Do(t). This is used to obtain an improved guess for the transformation zone boundary S from the second eqn (8.59) and the second side condition (8.62). The side condition of imminent crack growth expressed through the first eqn (8.62) is met by adjusting the applied stress ~r~176 at each iteration. The roles of the transformation parameter w and the applied stress ~r~176 can be interchanged such that the applied stress is fixed and the transformation parameter acts as the unknown to be obtained from eqns (8.59) and (8.62). The first eqn (8.59) is Cauchy singular, as can be seen fi'om eqn

8.6. Steady-State Analysis of an Array of Semi-Infinite Cracks

253

(8.50), and the integral is to be evaluated in its principal value sense. The dislocation density function Do(t) has a square root singularity appropriate for the singular stress field at the crack tip but is otherwise well behaved, that is it is continuous and differentiable except at the crack tip. Consequently the inversion of this equation can be performed by standard Gauss-quadrature techniques (Erdogan et al., 1972) without difficulty and with good accuracy. 8.6.1

Results

and Discussion

On the basis of the model described above for an array of parallel semiinfinite edge cracks some results relating to the strengthening of ceramics with damaged surfaces and transformation induced by crack growth are presented in the following. Before presenting general results, it is interesting to consider the limiting case of diverging transformation zones, to delineate the conditions under which such solutions exist. An upper limit on the transformation zone height H can be obtained by considering the second eqn (8.59) for the critical mean stress criterion. As the applied stress goo is less than the critical applied stress ~r0, the D due to the dislocations given by the integral in the second mean stress grn eqn (8.59) and by (6.31) must give a positive contribution to the mean stress, so that the following inequality must hold //

0 <--

(

oo Do(t)

sinh(27r~~d ) c o s h ( 2 7 r ~ ) - cos(27r~) - 1 dt

(8.63)

Numerical studies show that the dislocation density function Do(t) is always positive. For y0 > d/4 the integrand increases monotonically for fixed t, and tends to zero from below as x0 is allowed to increase. Therefore the integral is negative for y0 > d/4 and the inequality (8.63) is violateit. The limiting value of the transformation zone height is therefore H = d/4, and coalescence of neighbouring transformation zones cannot take place. At this limit for H, the zone front diverges, i.e. x0 ---* exp. When x0 ---* cxz and g ---. d/4 eqns (8.59) reduce to (with x ~ -cx~) 0 "c~

0-

o'o

r 4/~ dyo Do(t)dt 9d J-d/4 "d oo

(Too

(8.64)

1 --

0"0

254

R-Curve

Analysis

The second of the two eqns (8.64) gives the value of the remote load ~r~176 at which divergent transformation zones are possible. The integral of the dislocation density function D o ( t ) is related to the crack opening displacement, which in turn is related to the stress intensity factor. Considering the fundamental solution to the crack problem in the absence of transformation, the stress intensity factor is given by K - c r ~ 1 7 6 (Tada 1985), and by elementary analysis the opening of the crack is equal to the displacement of a plane strain strip of height d in the direction of the applied stress aoo, i.e. v+ - v- = aoo d(1 - u 2 ) / E . Imposing the side condition of imminent crack growth (8.61) and expressing the crack opening in terms of the dislocation density function D ( t ) gives v + - v-

-

o

D(t)dt -

X/~(1 -- ~2)

oo

Rewriting this in terms of D o ( t ) using (8.58) gives

-~

Do(t)dt

-

(8.65)

oo

Substituting (8.65) into (8.64) then gives the critical value for the transformation parameter wc for which a diverged transformation zone is the solution

F i g u r e 8.65: Transformation zone for a single semi-infinite crack used for analyzing possible divergence

8.6. Steady-State Analysis of an Array of Semi-Infinite Cracks

w~-

18 ( 1 -

~/-~)

255

(8.66)

For the limiting case when the crack spacing d/L tends to infinity, i.e. for a single semi-infinite crack, the critical value for the transformation parameter is wc = 18. This result was obtained by initially assuming that the transformation zone has diverged. If a finite transformation zone at a single semi-infinite crack were initially assumed and the limit of diverging transformation zones were obtained by gradually increasing the transformation parameter w in eqns (8.59) and (8.62), a lock-up value for the transformation parameter of approximately 30.0 would be obtained (Amazigo & Budiansky, 1988). The result (8.66) indicates that a diverged zone can be a solution for a single semi-infinite crack for w = 18. It is worthwhile considering this point in a little more detail. The closing stresses on a single semi-infinite crack can be obtained from (4.52) and (4.21) as T(Z) _ R e ( Ec T ~ 1 dyo} ~r~y 27r(1 - v) x - z0

x<0

The rectangular zone depicted in Fig. 8.65 is the limiting convex shape giving the maximum closing stress. Integrating along this zone front gives (ryy

- v--------~ 2(1 c~

x < 0

H

ira -

arctan

Y0 x

-- r 0

I

_< 1

(8.67)

-H

The equality sign holds when the zone height H is first allowed to diverge, thereby rendering the size of the frontal zone intercept r0 irrelevant. If on the other hand, r0 tends to infinity with H but the ratio r 0 / H remains fixed, the inequality sign holds, and a will be less than one. Rewriting the inequality in terms of the transformation parameter gives l+v T > E~ T 1 + v = _ _ _w 3~r#~ (ryy _ 6a~n 1 - v 18 In order to annul the crack-line stress, it is necessary to apply a load cr~ equal and opposite to ~u~ at infinity. The applied stress must be

R-Curve Analysis

256

sufficiently large to yield a mean stress at infinity equal to the critical mean stress in order that the diverged zone is a solution. Thus ~ ( 1 + ~ ) / 3 - ~r~, and the inequality finally becomes

>

(8.68)

In the above line of reasoning the limiting behaviour was obtained directly by considering a single semi-infinite crack, and the limit is in agreement with that obtained in (8.66) by considering the limiting behaviour of an array of parallel semi-infinite cracks with increasing spacing. The limiting value actually obtained depends on how the limiting process is performed, i.e. the actual value of a in (8.67). As already mentioned, the value obtained by increasing the transformation parameter w for a single semi-infinite crack is approximately 30.0 (Amazigo Budiansky 1988). On the other hand, the limiting value obtained by increasing w and the crack spacing d successively is approximately 36.6. For sufficiently close spacing of the cracks in the array the applied load for crack growth in the absence of transformation crc~ - Kc/v/-d-/2 (Tada, 1985) is sufficient to induce transformation by exceeding the critical mean stress ~rm. Thus the transformation zones will diverge and cover the entire plane ahead of the cracks. From (8.66) this critical crack spacing is obtained as dr 7r.

(3"1q30

1.0 0.8

6

)

0.6

8

12 2O

0.4

0"2 f 0.0

0

l

~

i

i

i

i

5

10

15

20

25

30

F i g u r e 8.66" Strengthening ratio for array of semi-infinite cracks

8.6. Steady-State Analysis of an Array of Semi-Infinite Cracks

257

CO

40-

30

20 (Omax

l0 0

0.0

0.2

0.4

0.6

0.8

~.0 x--E

F i g u r e 8.67: Critical and maximum values for the transformation parameter

For nondiverging transformation zones, solutions to eqns (8.59) and (8.62) are obtained numerically. The strengthening ratio for various crack spacings d/L is depicted in Fig. 8.66. It is seen that solutions to eqns (8.59) and (8.62) can be obtained for the transformation parameter equal to the critical transformation parameter wc of eqn (8.66) but with lower strengthening than the critical strengthening given by eqn (8.63). For these solutions, the transformation zones do not diverge, and for the transformation parameter w greater than the critical value wc but less than a certain maximum Wmaz, two finite transformation zone solutions to eqn (8.59) are possible. These limits are shown in Fig. 8.67. The result for the crack spacing d/L = 50 shown in Fig. 8.66 is redrawn in a slightly more explicit form in Fig. 8.68a. The stable region is now above the curve, the unstable region below it, and the curve itself pertains to quasi-static crack growth. For w = 22, the line A-D is indicated in the diagram. This line is followed from A to D as the applied load cr~176 is increased. The part from A to B is in the stable region, and as the load is increased from the point A no crack growth appears. When point B is reached quasi-static crack growth is possible. A further increase in the load will lead to unstable crack growth, as indicated by the broken line between B and C. The derivative of the crack tip stress intensity factor K tip with respect to the applied load a ~176 is positive at the point B, as indicated in Fig. 8.68b. Therefore it is not possible to go

R-Curve Analysis

258

20 15

lI a)

0 ~ 0

dg*/K,

I 0.2

I 0.4

0.6

i 0.8

I 0.2

I 0.4

I 0.6

J 0.8

0.2

0.4

0.6

0.8

?---X

1-~

IJo

~

oo

o Io o

-5

-lO b)

-15 0

{y ~0

xclL, YclL 25 20 15 C lO

c)

0

0

F i g u r e 8.68: C h a r a c t e r i s t i c r e s u l t s for

d/L-

1-~oo 50

8.6. Steady-State Analysis of an Array of Semi-Intinite Cracks

259

from B to C just by increasing the load on the specimen. If however the situation pertaining to point C is brought about by some other means, quasi-static crack growth is possible at a higher load at C compared to the load at B. Increasing the load from point C towards point D leads to a decrease in the crack tip stress intensity factor, as indicated by the negative derivative in Fig. 8.68b at point C. Therefore a new stable region is reached and the point C is a "superstable" point at which an increase in the load stops crack growth by enhancing the transformation, i.e. the toughening effect of the transformation grows more rapidly than the increase in applied stress intensity factor. Under these circumstances failure will initiate first by divergence of the transformation zones as the applied load ~ approaches the critical load ~0 (see eqn (8.64)) and thereafter by crack growth as the surrounding matrix material loses its ability to enclose the transformation zone. Due to the assumption of no reverse transformation the configuration of larger transformation zones pertaining to the left branch cannot revert to the right branch simply by lowering the applied load, as the derivative of the crack tip stress intensity factor K tip with respect to the applied load ~r~ is positive for fixed transformation zone shapes, as indicated by the dotted line in Fig. 8.68b. The transformation zone boundary intercept with the crack line extension zc and the height of the transformation zone y~ - H associated with the quasi-static solutions of Fig. 8.68a are shown in Fig. 8.68c, with the points B and C indicating the load cases just described. Transformation zone shapes for crack spacing d/L = 50 and various loading ratios ~r are depicted in Fig. 8.69. The toughening ratio Kappz/Ke corresponding to the strengthening ratio of Fig. 8.69 is depicted in Fig. 8.70. In terms of the strength values shown in Fig. 8.66 the toughening ratio is

Kappl K~

O.c~ I

-

~/ (to v ~

,.

d

(8

69)

The broken curve is the limiting result for a single semi-infinite crack obtained in w 7.3 and the dotted line pertains to the critical value of the transformation parameter wc given by (8.65). From the above analysis it is evident that diverging transformation zones can exist for transformation strengths less than those expected from conventional lock-up analyses, and crack configurations in transforming ceramics can exist which induce excessive transformation for quite low transformation strengths before crack growth is initiated.

R-Curve Analysis

260

y/L 45 I

o'0/o0= 0.9

|

o.8

3 0.7 2

0"5"6

i

0 0

.4 1

2

3

4

5

6

7

8

x/L

F i g u r e 8.69: Transformation zone shapes for various strengthening ratios, and one crack spacing d/L = 50

Kc/Kappl 1.0 0.8 0.6 0.4 0.2 0.0

0

5

10

15

20

25

30

F i g u r e 8.70: Toughening ratio for array of semi-infinite cracks

8.7. Solution Strategies for Interacting Cracks and Inclusions

261

These circumstances cannot however be brought about simply by loading a precracked transformation toughening ceramic without an initial transformation zone. The latter must be induced by some other means, such as surface grinding or thermal chock.

8.7

Solution Strategies for Interacting Cracks and Inclusions

A numerical method for the integration of the singular integral equation resulting from the interaction of a surface crack with a subsurface inclusion is presented (Andreasen & Karihaloo, 1993b). This examplities the solution method applied in w167 The dislocation density function is partitioned into three parts: A singular term due to the load discontinuity imposed by the inclusion, a square-root singular term from the crack tip, and a bounded and continuous residual term. By integrating the singular terms explicitly the well-behaved residual dislocation density function only has to be determined numerically, together with the intensity of the square-root singular term. The method is applied to the determination of the stress intensity factor for a surface crack growing towards, and through, a circular inclusion. The objective is to provide an accurate numerical solution method for this problem in order to develop solution strategies applicable to the determination of transformation zones of arbitrary shape. In the latter problem, it is imperative to have good control over the singularities contained in the mathematical formulation in order to be able accurately to determine the boundary of the transformed region. The integral equation for determining the dislocation density function contains a number of noticeable features. At the crack tip the solution is square-root singular; at the intersection of the crack-line by the inclusion boundary the solution has a logarithmic singularity, and at the free surface the otherwise Cauchy singular kernel must vanish. All of these features have to be taken into account, if accurate numerical solutions are to be obtained. A widely used and very effective numerical solution method for integral equations with Cauchy singular kernels was given by Erdogan et al. (1972). By means of certain Gauss-quadrature formulas which explicitly take possible singular endpoints into account the integral equations are transformed into a set of linear algebraic equations. The quadrature formulas can be applied directly to the singular integral provided that the collocation points are appropriately chosen. Due to these features

R-Curve Analysis

262

the method is simple to implement and has gained widespread acceptance. A disadvantage of the method is that little freedom is left for choosing collocation and integration points. In the problem at hand the residual dislocation density function may vary rapidly in the vicinity of the point of intersection of the crack-line by the transformation zone boundary, thus control over the position of collocation and integration points is important in order to obtain sufficient numerical accuracy. Another drawback in relation to the present problem is that the common quadrature formulas are not readily applicable to surface crack problems. This can be overcome by symmetric continuation of the singular integral across the free surface (Gupta & Erdogan, 1974), but in general the dislocation density function cannot be continued in a smooth manner, and the numerical accuracy suffers. In the solution method to be described below the accuracy of the solution is of prime concern. Accordingly, the singularities of the problem are isolated and handled analytically in order to avoid any numerical difficulties. As we have seen above, the problem of interaction between a transformation zone and a surface crack reduces to the solution of two coupled singular integral equations, one ensuring zero crack-line stress and the other determining the transformation zone boundary by a critical mean stress criterion. The interaction of crackline by the transformation boundary introduces a discontinuity in crack-line stress. To simplify the discussion, the transformation zone boundary is fixed a priori, and the coupling between the equations is thereby avoided. The crack-line stress due to an arbitrary inclusion can be written as T

O'xx

_

~rT/s ( 3(y + y0) +

+2Y x~176

+

Y - Yo ) - x g + ( y - y0) 2

+

dxo

+

(8.70)

where S is the boundary of the inclusion, see Fig. 8.71. The singular term induces a discontinuity in the crack-line stress imposed by the inclusion. This discontinuity is fixed at 0 " T irrespective of the shape of the inclusion. Therefore without limiting the generality of the analysis to follow, the shape of the inclusion is fixed to be circular, so that the above crack-line stress can be analytically integrated. For a circular region, (8.70) becomes (Mura, 1987)

8.7. Solution Strategies for Interacting Cracks and Inclusions

263

F i g u r e 8.71: Model configuration

T _ ~rT ( ~==

T

r2 4r2y + (~~+ h)~ z r (u-h)~

3r2

(u--h) ~

;)

(s.71)

-lzE

where r is the radius of the circular inclusion and h is the distance from the surface to its centre, h - a + r, and R is the region occupied by the inclusion. The uniform dilatational transformation strain in the inclusion is described through the parameter a T which is given by E• T

~T =

(8.72)

3(1 - u)

where 0T is the dilatation, E is Young's modulus and u Poisson's ratio. The parameter a T was introduced by Rose (1987a). a T (8.72) equals the crack-line stress discontinuity appearing from (8.71), when it is crossed by the boundary of the inclusion. The crack-line stress from a dislocation can be written as D

Eb

~r~ -

D,x

47r(1 - u 2)

h~x (y, y0)

(8.73)

where the weight function h=nj(y, yo) is given by (6.19). Taking advantage of the central position of the crack (x - x0 - 0), it reduces to

h=D~=(Y, Yo) -

1 +u0

+

6y (u+u0)~

-

4y 2 (~-u0)~

1 u-u0

(s.74)

R-Curve Analysis

264

It should be noted that (8.74) is Cauchy singular and vanishes at the free surface (y = 0). A dislocation density function D(yo) can be introduced such that D(yo)dyo is proportional to the Burgers vector b between y0 and yo +dyo

D(yo)dyo =

Eb 47r(1

-

v 2)

(8.75)

The integral equation determining the dislocation density function

D(yo) for a surface crack which annuls crack-line stress due to the inclusion can now be written from (8.71)-(8.75)

0 - ~r** T+ l

D(yo)g(y, yo)dyo

(8.76)

C

where c is the crack length, and g(y, yo) is given by (8.74). Before proceeding with the numerical inversion of the integral equation (8.76), the singular nature of the dislocation density function D(yo) is discussed in some detail. The displacement jump across the crack faces v(s) near the crack tip can be expanded as v(s) = A181/2 +0(8 3/2) (Barenblatt, 1962), s being the positive distance ahead of the crack tip. The dislocation density function can be obtained by differentiation of the crack face displacement to within a multiplying constant; thus the expansion of the dislocation density function near the crack tip can be written as D(s) - A2s -1/2 + 0(sl/2). A1 and A2 are proportional to the stress intensity factor KI. It should be noted that apart from the inverse square-root singularity, the near-tip expansion implies that the dislocation density function vanishes at the crack tip. At the crack load discontinuity induced by the inclusion, the crack face displacement contains a term proportional to s in Isl (Bilby et al., 1963), which leads to a logarithmic singularity in the dislocation density function with the expansion D(s) = A3 In Isl+O(s~ with s now being the distance from the crack load discontinuity. A3 is proportional to the crack load discontinuity a T . Bearing in mind the singular behaviour, the dislocation density function D(yo) is conveniently written as a sum of three parts, as follows

D(yo ) -

KtiP / -yo 7r 2x/"2~ c + Yo

aTi:+YOln

7r2

+ Yc

Yc - Y0 + Do(yo) c + Y0

(8.77)

The first part gives the square-root singularity pertinent to the stress

8.7. Solution Strategies for Interacting Cracks and Inclusions

265

intensity factor at the crack tip K tip. That this term indeed gives the singularity consistent with the stress intensity factor K tip is seen by expanding the stress O'xxD(8.73)-(8.75) ahead of the crack through the following limit K tip - limr--.0 crD~2X/~-~, where r is the distance on a straight extension of the crack-line. The second part in (8.77) gives a logarithmic singularity at y~, which leads to a crack-line stress discontinuity equal to O"T at yr without violating the near-tip expansion for the dislocation density function, as discussed above. The logarithmic term has a very simple integral formulation which will be exploited later. The last term Do(Yo) is a nonsingular and continuous function. In order not to violate the near-tip expansion the condition D o ( - c ) = 0 is imposed. This condition also ensures that no part of the crack tip singularity in the dislocation density function D(yo) is captured in the residual dislocation density function Do(yo). A more common way of representing a dislocation density function in terms of singular and regular functions is by products rather than sums (Erdogan et al., 1972). The representation chosen here offers some advantages over a product representation in the analytical integrations performed below, and simplifies the transformation of the integral equation (8.76) into an ordinary integral equation with a continuous crack-line load. Introducing the dislocation density function (8.77) into the integral equation (8.76) gives

0 -- O'xxC+ / ~ J ( ~" 2~-~~c-Y~

Yo + D0(Y0))g(y, yo)dyo

(8.78)

with c axx -

T axx

~2

c

+ Y0 log Yc -- Yo g(y, yo)dyo + Yc c + Yo

(8.79)

The unknowns of the singular integral equation (8.78) are the stress intensity factor K tip and the residual dislocation density function Do(yo). The modified crack-line stress r c is bounded and continuous, as it will be demonstrated below. The integral term in (8.79) is discontinuous due to the singularities of the integrand. As is demonstrated below, integration of the logarithmic term together with the Cauchy singularity of (8.79) creates a discontinuity which cancels out the discontinuity induced by the inclusion, thus rendering a~xc continuous over the entire crack. For a better understanding of the subsequent calculations, the crackline stress induced by the dislocations is written as the following limit

R-Curve Analysis

266 along any line z not coinciding with the crack-line C

~r~,:c(y) - ~,--.olimJc D(Y~ (g'~ (Y' Y~ + Re z --iiyo } ) dyo (8.80) where the Cauchy singular term of the weight function g(y, yo) (8.74) has been separated out, such that gn,(y, yo) is nonsingular, i.e. g(y, yo)-

gn,(y, Yo) = 1/(y - Yo). The logarithmic part of the dislocation density function (8.77) is conveniently rewritten in an integral form as

f (Yo ) log Yc - Yo -- f(Yo) c+yo

Yp _ y--------~dyp

O"T ~/C + Y0

f(Yo)- ---~

(8.81)

c + Yc

where the function f(yo) is finite and differentiable. Introducing (8.81) into (8.80) and for the moment disregarding the nonsingular part of the weight function gns(y, y0), the crack-line stress can be written as the limit of a double integral lim

-i

x ~ o

c

c

Yp

-

Yo

z

-

i yo

Changing the order of integration, the integral (8.82) can be rewritten

as lim Re x~o

/v__.~jo__ f(yo) 1 ~dyodyp c Y p - yo iz + yo

= lim Re { J j ~

~-o +

1

~ iz + yp

vc 1 c iz+yp

f(yp)

c

yp - yo

cYp-Yo

+ f(y)

f(Y_O)-zz+ Yof(Y)) dyodyp}

ciz+yo

dyp

(8.83)

f(y) is the value at y for z tending to zero along any path z. Provided that f(y) is bounded and continuous the first double integral is real and nonsingular. To see this, consider the integrands of the inner integral. These integrands are continuous and differentiable by virtue of the properties of f(y). Formally integrating the inner integral shows by the same reasoning that the integrand of the outer integral is continuous as

8.7. Solution Strategies for Interacting Cracks and Inclusions

267

well. From the fundamental theory of Cauchy integrals (Gakhov, 1966) it follows that in this case the limiting process and the integrations can be interchanged, provided that only the real part is needed, as in the present case. Adding and subtracting the Cauchy principal value integral for x = 0 of the last double integral in the above equality, gives

v__~j: f(Yo) ~ d 1y o d y p c c Yp - Yo iz + Yo

lim Re

x~o f~

-1

+jffr r

(f(Yo)_-f(Yp)f(Yo)_-f(Y))

f -1

J;

f

+f(y) lim Re f ~ ~o

9__ dyo ) c Y - Yo dyp

dyo

vo - I ( Y )

yp -

l

f ( 1

~ i z + yp

~

1

+ i z + yo

Y-

) dypdyo (8.84) Yo

By similar reasoning as applied above, the second integral is seen to be nonsingular. The discontinuity induced by the logarithmic part of the dislocation density function can now be obtained by carrying out the integrations of the singular double integral and taking the limit as follows fc 1 j_~ 1 1 ) f(y) lim Re + dyodyp

~o

~ iz + yp

= f(y) lim Re f ~ 9 ~o

= f(y) lim Re x---.o

1 ~ iz+yp

~ iz + yo

( log ~ iz

-iz+c

Y - Yo

+ i~r - log - y ) y+c

dyp

f c l ( iz -y) log - log dyp c iz + yp -iz + c y+c

+ f (Y) ~-,olimRe { ilr l~ iz +- yrc

=f(v)

-~r 2

-c<_y
- 2/2 0

v-v yc
(8.85)

268

R - C u r v e Analysis

which through (8.81) gives a crack-line stress discontinuity opposite to that induced by the inclusion. The discontinuity can be obtained simply by integrating (8.82) not in the principal value sense but by retaining first the imaginary terms and then taking the real part only. This gives the correct discontinuity but the value at y = yr is not obtained. This value can be important if the transformation zone boundary is to be determined as a part of the problem. The full expression for the crack-line stress (8.79) induced by the logarithmic singularity can be written as c

~

T _

-- ~

y)

~r2x/c + yc

+[1 + 6y

r

+ 4y2~y2]A(c, yp, - y )

whereA= 1 fory
dyp - mv/c + y (8.86)

1/2 f o r y = y r

1 ( x/c + 7/log

A(c, 7/, ~) - 7/- ~

(ssT)

-V/c + ~ log

The integral has been transformed from one in y0 from - c to 0 with the logarithmic singular integrand (8.79) to a regular integral in yp from - c to yc (8.86). In order to invert the integral equation (8.78) the residual dislocation density function Do(yo) is linearly interpolated so that it is transformed into a system of linear algebraic equations i~ti p 0 __ O.xxC(Yi) +

~~-~F(yi)r

n + E Do(Yk)Vk(Yi)

k=l

(8.88)

The functions F(y) and Vk(y) are obtained by analytical integrations as given below. The system (8.88) consists of n + 1 linear equations with unknowns Do(yi), i = 1...n and K tip. The inversion of the integral equation (8.78) can now be performed simply by Gaussian elimination of the system of linear equations (8.88). The function r(y) in (8.88) is obtained by analytical integration of

8.7.

269

S o l u t i o n S t r a t e g i e s for I n t e r a c t i n g Cracks a n d Inclusions

square-root singular part of the dislocation density function (8.78) and the weight function (8.74) as 1 f~

F(y) -

./-Yo

dyo-

u)

y(2y2-6yc+7c

-~ j _ ~ g ( v , vo ) V c + vo

2(c - v) ~ v / - v ( c

- v)

(S.89)

T h e functions Vk(y) in (8.88) are obtained by linear interpolation of the residual dislocation density function and analytical integration of the weight function (8.74) as follows vk(v) -

yk vo - v k - ~ vk-~ Yk - Yk-1

g(Y, yo )dyo

+

Y - Yk-1 log Yk - Y Yk - Yk- 1 Yk-1 - y 7y+yk_llog Yk -- Yk-1

jfyyk+l Yo Y k + l g(y, yo )dyo k Yk - Yk+l

Y - Yk+l log Yk - Y Yk - Yk+l Yk+l -- Y

Yk+Y _ 7y+yk+llog Yk-1 + y Yk -- Yk+l

Yk+Y Yk+l + Y

yk+l - yk-1

(s.90)

__,.,y2O (Yk-1 + Y)(Yk + Y)(Yk+I -~- Y)

T h e integration and collocation points are n u m b e r e d from the crack tip towards the free surface. Defining yk = - c for k = 0 and yk = 0 for k = n + 1 formula (8.88) holds for k = 1 and the condition D o ( - c ) = 0 is a u t o m a t i c a l l y satisfied, but for k = n - 1, n, i.e. at the surface, additional t e r m s appear if the residual dislocation density function is linearly e x t r a p o l a t e d from the last integration point to the surface. T h e additional terms A Vn-1 and A Vn are to be added to the corresponding functions from (8.90)

A Yrt-1 --

-1 Yn -- Y n -

A V,~ -

(vo - v,~ )g(v, vo )dvo 1

1 Y,~ - Y,~- 1

(Yo - Y,~ )g(y, yo )dyo - (y,~ - y ) l o g

,., 0

Yn Y -- Yn Y

(Y0 - Y,~)g(Y, yo)dyo

270

R-Curve Analysis

- ( 7 y + yn)log y + y" + 6y,~ y

2y~ (8.91) Y+ Y.

This concludes the problem of inverting the integral equation (8.78) for a surface crack with a continuous crack load. If collocation points are placed at the midpoints of the integration points used in (8.90), the numerical implementation is somewhat simplified because the integration points do not coincide with the collocation points, the special points at the surface, the point at which the transformation zone crosses the crack-line or the crack tip where otherwise limiting processes may be needed for determining function values.