Phase-transformation zones and shielding stress intensity factors for cracks in finite bodies

Engineering Fracture Mechanics Vol. 59, No. 1, pp. 47-55, 1998

Pergamon

© 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain

PII: S0013-7944(97)00117-3

PHASE-TRANSFORMATION STRESS INTENSITY

ZONES AND

FACTORS

oo13-7944/98 $19.oo+ o.00

SHIELDING

FOR CRACKS

IN FINITE

BODIES T. FETT Forschungszentrum Karlsruhe, Institut fiir Materialforschung II, Postfach 3640, D-76021, Karlsruhe, Germany Abstract--The paper deals with the determination of phase-transformation zones and residual stress intensity factors in transformation toughened ceramics. The influence of finite specimen dimensions is discussed in detail. It results that in case of non-propagating cracks the residual stress intensity factor does not disappear, in contrast to computations made for small-scale transformations. Additionally, it can be shown that for propagating cracks the residual (shielding) stress intensity factor depends on the type of loading (tension, bending), the relative crack length, and on the ratio of zone size to specimen size. © 1997 Elsevier Science Ltd Keywords---phase-transformations, shielding stress intensity factor, R-curve.

1. INTRODUCTION IF A cracked body made of transformation toughened ceramics undergoes growing external loading, a phase transformation zone extends at the crack tip. For the special case of a crack in an infnite body McMeeking and Evans[l] and Budiansky et al.[2] computed the transformation zone ahead of a crack tip. Besides the condition of small-scale phase transformation behaviour (zone size<
2. BASIC RELATIONS 2.1. Transformation zone

A crack of length a in a body of width W (see Fig. 1) is considered. Due to the externally applied load a phase transformation from the tetragonal phase to the monoclinic phase occurs near the crack tip. For transformations triggered by the hydrostatic stress ~hyd the zone extends 47

48

T. FETT

I y o.::

r 2L

×-..~ •

I[

W

Fig. 1. Geometric data of an edge-cracked plate.

to that distance from the crack tip for which the condition O'hyd = O'~yd

(1)

is fulfilled. The total stress state in a body, and especially the hydrostatic stresses, are known if the Airy stress function for this state is available. In case of a crack with length a in an externally loaded component the series representation of the Airy stress function according to Williams [6] leads to the hydrostatic stress (see Appendix A) O'hyd ~ - 4 1 4- v

tr0

3

n__~o[/A, [/ =r] \n-1/2(n 4- 3/2)(n 4- 1/2) cos(n - 1/2)~b 4- A*(r'~n(n + 1)cosn~b] n ~ W ,] n - 1/2 k ~w/ d

(2)

where W is the width of the component and tro is a characteristic stress value of the applied load (e.g. the outer fibre stress in bending). The first coefficient Ao is related to the applied stress intensity factor by Kl = croF,~'~-d, F = A o v / - ~ I ~ ,

~z = a l W .

(3)

If we use a three-term representation, taking into account the coefficients Ao, Ao and A~, we obtain for the location where the critical hydrostatic stress is reached r(~b)= 2 , 21 (. a_~yd 2 ,) W 22 4- 10~o 4- 2 2 / 1 + 20 AAo~_~; A0 cos ~b/2 \2(1 4- v)tro + 3 A° "

(4)

If only the first term Ao, is used one obtains (1 4- I))2 8 r = -----=co cos2(~/2), co -- - (K1/O~yd) 2 ,

3,/3

4v%

(5)

as derived by McMeeking and Evans [1]. The quantity co is the zone height in the asymptotic case of a semi-infinite crack in an infinite body, i.e. the zone height for small-scale conditions. For higher-order approximations the radius r(~b) has to be determined numerically using a zeroroutine. The first coefficients of the stress function were determined in ref. [7] for the tension loading case~

A0(1 - ~)3/2/~/~ _-- 0.26434 - 0.39652~ + 1.58060fl - 2.8451~ 3 + 2.50550c4 - 0.844450c5,

(6)

A1(1 - ~)5/2q,~ = -0.022795 + 0.041073~ + 0.032314afl + 0.24705~ 3 - 0.32412~ 4 + 0.13585~ 5,

(7)

Cracks in finite bodies A2(1

-

49

- 0 . 0 0 6 7 9 6 + 0.01721 l~t + 0.031207~ 2 - 0.055757ct 3 + 0.10248ct 4 - 0.0302a 5,

0 0 7 / 2 ~ 3/2 •

(8) A~(1 - ct)2 = 0.13149 - 0.16024a - 0.051233a 2 - 0.18874a 3 + 0.19936a 4 - 0.04915ct 5, A~(1 - ~)3~ = 0.048125 - 0.1062ct - 0.08187ct 2 + 0.32757ct 3 - 0.40916ct 4 + 0.15114a 5.

(9) (10)

F o r bending the corresponding coefficients are reported in ref. [7] in a tabulated form, appropriate for interpolations by cubic splines.

2.2.

Stress intensity factors and near-tip weightfunction The residual stress intensity factor caused by a phase transformation zone is given as[l] Klres = p fv n- hdS,

(11)

where h = (hx, hy, 0) -r is the vector o f the weight function, n is the vector normal to the b o u n d ary line F o f the transformation zone, and p is given by

~'rfE P-

(12)

3(1 - 2v)"

The residual stress intensity factor is the stress intensity factor in the absence o f an external load which only results f r o m the residual stress field occurring after a phase transformation. Since all residual stress intensity factors c o m p u t e d here are m o d e I stress intensity factors, the subscript " I " will be d r o p p e d subsequently. In eq. (12) ea" is the unconstrained volumetric phase transformation strain, f is the volume fracture o f the transformed material, E is the Y o u n g ' s modulus, and v is Poisson's ratio. U n d e r conditions o f small-scale transformations (co<
1

1

hy'v/W = D ~ f o y ~ +

/)-

+ D

/-7

--

/~-

r

r

Do)~y+ DLf,y r ~ + ~Cly-~+Dzfzy(w) 3/2,

(14)

with the angular functions f~x = [(n + 4v - 7/2)cos((n - 3/2)~b) - (n - 3/2)cos((n + 1/2)~b)]cos q5 + [(n - 4v + 5/2)sin((n - 3/2)4)) - (n - 3/2)sin((n + 1/2)~b)]sin ~b,

(15)

f~y = [(n + 4v - 7/2)cos((n - 3/2)q~) - ((n - 3/2)q~)cos((n + 1/2)~b)]sin ~b - [(n - 4v + 5/2)sin((n - 3/2)q~) - (n - 3/2)sin((n + 1/2)~b)]cos qS,

(16)

~

f~x =[(n + 4v - 3)cos((n - 1)~b) - (n + 1)cos((n + 1)~b)]cos q~ + [(n - 4v + 3)sin((n - 1)q~) - (n + 1)sin((n + 1)~b)]sin ~b,

(17)

fny =[(n + 4v - 3)cos((n - 1)q~) - (n + 1)cos((n + 1)q~)]sin ~b -

[(n - 4v + 3)sin((n - 1)q~) - (n + 1)sin((n + 1)q~)]cos q~,

(18)

T. F E T T

50

and

the first coefficients are Do

D1

----

-

1 ~/~(1 - v)'

(19)

- 0 . 1 5 0 3 7 + 0.4555a - 8.06442~ 2 + 21.842~ 3 - 32.310a 4 + 23.643a 5 - 6.7556a 6 C ' ~ ( 1 - v ) ~ ( l - ~)5/2A0 ,

D2 --

0.57978 - 2.0378a + 2.8263a 2 + 21.45a 3 - 37.619~ 4 + 29.121a 5 - 8.151a 6 18q/~-~(1 _ v ) g 3 / 2 ( l - g)7/2Ao

D0 =

3~-~(1

(20)

(21)

(22)

- v)Ao'

/ ) l - - 0 . 0 9 6 2 5 + 0.31514a - 0.09897a 2 - 1.2214a 3 + 1.8045a 4

--

0.94675a 5 + 0.14745a 6

(23)

3~¢/~(1 - v)a(1 - a)3Ao

3. T R A N S F O R M A T I O N

ZONES AND RESIDUAL STRESS INTENSITY FACTORS

I n the w e a k - t r a n s f o r m a t i o n limit the zone d i m e n s i o n s at the onset o f c r a c k extension are c o m p u t e d for the c o n d i t i o n t h a t the a p p l i e d stress intensity f a c t o r reaches a critical value Klo, the so-called c r a c k - t i p toughness, i.e.

KIo.

gappl =

(24)

T h e externally a p p l i e d l o a d at this state is, a c c o r d i n g to eq. (3)

glo ao - ~ A o "

(25)

h/r.,o 1.00

: ~

0.9G

ih

~

g

0.80 .00

fension ',~ ,

.20 I

,

.40 I

,

.60

a/w

I

,

.80 I

i

1.00 I

Fig. 2. Zone height h for ¢7~yd = 100 MPa, normalized to the zone height co for small-scale conditions.

Cracks in finite bodies

h/U

51

bending 300 250 200

.oo

.20

I

.40

.60

1.oo

.80

Fig. 3. Zone height h for different e&d, normalized to o (bending).

Most of the numerical computations data-set K/Q= 4 MPa&i, leading to w = 115 pm (o/W=

were performed with five coefficients for eq. (2) using the $&, = 100 MPa, v = 0.25, W = 4.5 mm,

l/40) according to eq. (5).

weight

O.lOO-

function

---

5 terms

-

3 terms

0.050 -

I

20

I

I

I

J

.40

.60

.80

1 .oo

a/W Fig. 4. Residual stress intensity factor for non-propagated cracks. Solid curve: computed with three terms in the weight function and three coefficients in the hydrostatic stress solution. Dashed curve: computed with five terms in the weight function and five coefficients for the hydrostatic stresses (o/W = l/ 40).

(26)

52

T. FETT

Kres/(p031/2)

bending

0.100

Oh%d(MPo)

.

/

100

~

0.050

~

1 50

_

'

0.00(

'

'

.20

'

.40

'

'

'

.60

'

200 250 300 40O

'

.80

'

1.00

a/W Fig. 5. Residual stress intensity factors for different values of O'~yd.

3.1. Results for non-propagated cracks The height h of the phase-transformation zone is shown in Fig. 2 as a function of the relative crack length for bending and tension. The zone height becomes maximum near a/W = 0.50.6. Figure 3 illustrates the influence of the critical transformation stress on the zone height. In the limit case O'~yd---~oo the ratio h/o) approaches 1. The residual stress intensity factor was calculated for the data given in eq. (26). The results for bending and tension are represented in Fig. 4. It is of interest that the residual stress intensity factor does not disappear, in contrast to the well-known behaviour in the small-scale transformation case. Moreover, an influence of the loading type is visible. In order to give an impression of the accuracy of the computations, two degrees of approximation were used. Whilst the solid curves were computed with a three-terms representation of the hydrostatic stresses and a three-term weight function, the dashed curves were computed with a five-term stress relation and a five-term weight function. The differences are sufficiently small so that the threeterm computations may be used in first estimations. It should be mentioned in this context that three-term calculations of zone size can be performed analytically by application of eq. (5) which reduces drastically the numerical effort. Figure 5 illustrates the influence of the critical

-/X Ksh/p 031/2

number

0.50

of

terms 2

0.40

~

1

0.30

~

/

5

0.20

o lot 0.0

%

..''...__./ '

1'

'

2'

'

3' ' 4' Ao/03

'

ao:O5

5'

~

-~

Fig. 6. Change of shielding stress intensity factor A Ksh = Ksh(a)- AKsh(d0) with crack propagation. Parameter: number of terms used for the hydrostatic stresses and for the near-tip weight function (e)/ W = 1/40).

Cracks in finite bodies

53

-/XKsh/PEO 1/2 limit

case

0.40

0.30

nsion

0.20

bending ~o =0.5

0"10 I

0.006

I

I

I

I

I

I

I

I

I

1

2

.3

4

5

6

7

8

9

I0

,

1

I

11

Z a/oJ Fig. 7. Influence of the loading type on the shielding stress intensity factor (gkol]/W = 1/40). Thin line: limit-case for a semi-infinite crack in an infinite body[l].

transformation stress on Kres. Especially for low critical stresses the residual stress intensity factor is not negligible. For large values of the critical transformation stress (resulting in very small zone sizes) the residual stress intensity factor disappears. It should be noted that in this case small-scale transformation is approached. It can be seen from Fig. 3 that the deviations from the limit case h ~ 0 (i.e. h/o9--~ 1) decrease with increasing critical s t r e s s tr~yd. It should be noted that a small overshooting effect (h/~o > 1) occurs which has also been shown in ref. [9].

-fKsh/PCO 1/2 0.40

0.30

0.20 0.10

0.0. )0

I

.20

,

I

I

I

.40

.60

.80

,

I

1.00

a/W Fig. 8. Influence of the initial crack length ~ = ao/W on the shielding stress intensity factor (bending) ( ~ / W = 1/40).

54

T. FETT

3.2. Results for propagated cracks The shielding stress intensity factor for propagated cracks has been computed with that phase-transformation zone which is given by the envelope of all zones belonging to intermediate crack lengths a' within ao<<.a'<<,a (see insert in Fig. 6). The result is plotted in Fig. 6 for an increasing number of coefficients used to calculate the hydrostatic stress as well as in the computation of the weight function. The value -AKsh denotes the difference between the residual stress intensity factor after a certain crack extension Aa and the initial value for Aa = 0, i.e. -AKsh = AKre~= Kres(a)- AKrcs(a0). The convergence is very fast for at least three terms. The influence exerted by the loading type becomes obvious from Fig. 7. Only small deviations can be detected between the tension and the bending loading cases. Computations of several initial crack sizes show that the resulting shielding stress intensity factors depend on a0. The maximum shielding stress intensity factors decrease with increasing initial crack lengths, Fig. 8. Finally, the influence of the relative zone size, co/W, is shown in Fig. 9. The variation in co/W may be obtained for a fixed ratio a~yd/Kl0 by variation of the specimen size W. For the limit case co/ W--, 0 the shielding stress intensity factor approaches the value -AKsh/(p~/'~) = 0.44

(27)

(with v = 0.25), which is in agreement with the result given by McMeeking and Evans[l]. Experimental R-curves for transformation toughened zirconia obtained by the compliance method are less steep at the beginning of stable crack propagation compared with the computations. One reason may be the fact that the development of the initial transformation zone with its plastic deformation will result in a deviation from the initial straight-line behaviour. The interpretation of all deviations from the initial straight line as an increase in compliance by crack propagation must falsely lead to an early apparent crack extension and to a moderately increasing R-curve. In cases where the crack-length measurements are performed with a microscope steeper curves have to be expected.

4. CONCLUSIONS Phase-transformation zones in edge-cracked bars made of transformation toughened ceramics have been investigated numerically. The size and shape of the transformation zones were

-AKsh/PE0 I/2 0.50 0.40

bending

O3/W

1/64-o

"

1/160 0.30 0.20

I/40 1/20 S o =0.5

0.10 0.00

,

.50

I

.60

,

I

.70

G/W

i

I

I

.80

.90

Fig. 9. Shape of R-curves for different values of co/W.

Cracks in finite bodies

55

computed, using the Airy stress function, for cracks with the first five coefficients on the assumption of a critical hydrostatic triggering stress. The related residual stress intensity factors were computed with a five-term near-tip weight function. The main results are: • In the case of non-propagating cracks the residual stress intensity factor does not disappear, in contrast to small-scale behaviour; and • for both non-propagating and propagating cracks the residual (shielding) stress intensity factor depends on the type of loading (tension, bending), the relative crack length, and on the ratio of zone size to specimen size.

REFERENCES 1. McMeeking, R. M. and Evans, A. G. Mechanics of transformation toughening in brittle materials. Journal of the American Ceramic Society, 1982, 65, 242-246. 2. Budiansky, B., Hutchinson, J. W. and Lambropoulos, J. C. Continuum theory of dilatant transformation toughening in ceramics. International Journal of Solids and Structures, 1983, 19, 337-355. 3. Stump, D. M. and Budiansky, B. Crack-growth resistance in transformation-toughened ceramics. International Journal of Solids and Structures, 1989, 25, 635-646. 4. Amazigo, J. C. and Budiansky, B. Steady-state crack growth in supercritically transforming materials. International Journal of Solids and Structures, 1988, 24, 751-755. 5. Stump, D. M. and Budiansky, B. Finite cracks in transformation-toughened ceramics. Acta Metallurgica, 1989, 37, 3297-3304. 6. Williams, M. L.. Journal of Applied Mechanics, 1957, 24, 109-114. 7. Fett, T. A near-tip weight function for the edge-cracked rectangular plate. International Journal of Fracture, Reports of Current Research 1996, 76, RI l-R17. 8. Bueckner, H . . ZAMM, 1970, 50, 529-546. 9. Fett, T., Contributions to the R-curve behaviour of ceramic materials. KfK-Report 5291, Kernforschungszentrum Karlsruhe, 1994. 10. Fett, T. and Munz, D., Stress intensity factors and weight function solutions. KfK-Report 5290. Kernforschungszentrum Karlsruhe, 1994.

(Received 22 January 1997, accepted 8 July 1997)

APPENDIX The transformation zone can be calculated from the Airy stress function ~, which satisfies the bi-potential equation AA* = O. (Al) If this function is known for a given crack in a component, the stresses result as lOt~ 1 02~ O"r = r ' ~ - r -I rE 0~b2,

(A2)

02~ a~o- Ota .

(A3)

The symmetric part of the stress function--the only part of interest here--reads[6]

• d(eoW2) =~_,(r/Wy+3/2a. cos(n+ 3/2)q~- ~

cos(n- 1/2)q~

n--O

+ E(r/W)"+2A*~Icos(n + 2)~ - cos n~].

(A4)

n=0

For plane strain conditions the hydrostatic stress is l+v.

O'hyd = - - - ~ ( 0 " r + O'~b),

(A5)

and consequently ahyd .l+v ao -4~.

~ n--'O

. /rx,-I/2(n+3/2)(n+l/2)cos(n_l/2)dp+A, " .1~) n - 1/2

(A6)