a dugdale model for crack opening and crack growth under creep conditions

Materials Science and Engineering, 30 (1977) 187 - 196 © Elsevier Sequoia S.A., Lausanne -- Printed in the Netherlands

187

A Dugdale Model for Crack Opening and Crack Growth under Creep Conditions

H. RIEDEL

Max-Planck-Institut fiir Eisenforschung GmbH, Max-Planck-Str. I, 4000 Diisseldorf (F.R.G.)

(Received April 1, 1977)

SUMMARY

Creep deformation ahead of stationary and propagating crack tips has been studied by means of a time dependent Dugdale model. Analytical expressions for the stresses, strains, and for the crack tip opening displacement (COD), are derived from a non- linear partial integro-differential equation. Combined with a crack extension criterion, e.g., the critical COD criterion, the stress analysis allows for the calculation of crack growth rates, ~, and of crack growth initiation t i m e s , ti, both as functions of the applied stress intensity factor, K~. The prerequisites for a unique correlation between ~ and KI are steady state crack extension and small scale yielding conditions. These are investigated in the present paper.

I. INTRODUCTION

The present theoretical study is aimed towards an understanding of the behaviour of pre,existing cracks in components operat- ing under creep conditions. A number of pre- mature failures in power generating plants have been attributed to creep crack propagation [1]. In the past, the lack of precise criteria, as to whether cracks are harmless or dangerous, has lead to over-conservative and non-realistic inspection standards and accep- tance limits. For instance the ASME Boiler and Pressure Vessel Code presently does not permit any crack-like defects in ferritic steels that are employed under creep conditions.

One of the prerequisites for less restrictive standards is a time-dependent stress analysis of pre-cracked bodies under creep conditions.

McEvily and Wells [2] have proposed to treat a Dugdale type crack model, as a first step. Vitek [3] has analysed a Dugdale model by means of a computer simulation method. Taira and Ohtani [4] employed the finite element method to study creep deformation at a notch and the subsequent crack initiation and propagation.

It also appears reasonable to develop analytical solutions for creep problems in cracked bodies, which will reveal general correlations more clearly than the numerical methods. Therefore, in the present paper, a semi-analytical solution of the Dugdale model under creep conditions is presented. The case of steady state crack extension is considered, as well as the opening of a stationary crack by creep deformation. The results of the stress analyses may be written in a concise analytical form (eqns. (19}- (21)). Assuming any specific crack extension criterion, e.g., the crack tip opening displacement {COD) criterion, the analysis allows for the calculation of crack growth initiation times, ti, and time rates of crack growth, ~. In particular, a relation of the form ~ ~ K~ (KI = stress intensity factor} will be derived which has often been reported as an empirical relationship [1, 5- 14].

In recent literature, the question has been discussed whether the stress intensity factor is descriptive of the creep crack growth behaviour (cf. refs. 9, 15, 16). From a theoretical point of view, a unique correlation between KI and ~ may only be expected if both the "small scale yielding" and the "steady state" approximations are valid. The limitations of these approximations will be investigated quantitatively as a function of material properties, of the specimen size, and of the loading conditions.

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2. ANALYSIS OF A DUGDALE MODEL UNDER CREEP CONDITIONS

The relation among strain rate, ~, strain, e, and stress, o, obtained in uniaxial creep tests may be represented approximately by [ 17]

= B.(o -- Oo)" " e (1) (B, Oo, n, p are materials parameters). The stress exponent, n, is frequently n ~- 4, but values between 1 and 20 have been reported. If p = 0, then steady state creep is prevalent, while strain softening is characterized by --1 < pn < 0; p > 0 implies strain hardening; in particular, Andrade's primary creep law is described when pn = 2. The friction stress, o0, may often be neglected. However, in the case of precipitation hardened alloys, a non- zero friction stress, o0, is physically meaning- ful [18] . If the stress is not time dependent, eqn. (1) may readily be integrated with respect to the time:

e = B'(o - - a o ) " / (1 +P")t 11(1 +P"). (2)

At crack tips, however, the stresses are space and time dependent: in the first instant after loading an elastic stress distribution with a singularity at the crack tip builds up. The creep deformation subsequently relaxes the stress concentration and distributes the stresses across the section of the specimen. Crack propagation is initiated once a critical amount of damage (which is often given by COD or by the creep strain) has been accumulated at the crack tip [3, 13] .

Figure 1 shows a two- dimensional semi- infinite crack, the crack tip being either at rest or propagating with a constant velocity, ~. In this analysis it is assumed that the material behaves elastically except in a thin creep zone ahead of the crack tip (y = 0, x > 0). In the creep zone, the creep law stated in eqn. (1) is assumed. The creep strains are expressed as e = u/h, where u is the relative displacement in the y<lirection between the material imme- diately above and below the creep zone and h is the height of the creep zone in the y<lirection.

In addition, the present theory is confined to the small-scale yielding range because of mathematical difficulties. In the small-scale yielding approximation, the influence of the external loading system on the creep zone appears only in the singular stress term

! crack

y

~ : B . ( a - ~ ) ' . ~ -p" L I I t . . a

o r r r r 7 r

creep zone

Fig. 1. Crack with co-planar creep zone.

Ki/(2rrx) 112. The creep zone is modelled by a continu-

ous array of edge dislocations. The dislocation density D(x, t) is correlated with the displacement u and the creep strain e through

e(x, t) = u(x, t)/h = f D(x: t)dx'/h. (3) X

The stresses OD, due to a given one<limen- sional dislocation density in a cracked body, have been derived in previous papers, e.g., refs. 19, 20. In the small-scale yielding limit, these stresses, OD, reduce to the particularly simple form

? ;:z OD(X, t) = A x,)/dx'.t o (4)

The first term of eqn. (4) describes the direct stresses of the dislocations, whereas the second term describes the image stresses. For plane strain: A = E/[4•(1 -- v2)], E = Young's modulus, and v = Poisson's ratio. Substituting the strains and stresses, eqns. (3), (4) into eqn. (1), an equation for the unknown dislocation density, D(x, t), results

a at f D(x: t)d ' = Bh I+pn •

l " x' 112 D(x: t) KI ~" ao,x]([--I ~ d x ' + - - Oo/ " x - - x ' x /2 ,x

~ p n

lf I (5)

The non-linear partial integro-differential eqn. (5) will be solved for two cases:

(A) Beginning from the time t = 0, a cons- rant stress intensity factor, KI, acts on a sta- t ionary crack. Then eqn. (5) has to be solved

together with the initial condi t ion D(x, t = O ) =0.

(B) Under the action of a constant stress intensity factor, the crack tip propagates with a constant velocity, ~. Then a steady state solution of eqn. (5) is required.

These two cases are treated in the following three subsections (a) - (c) since, within case (B), it is convenient to distinguish between o o = 0 (subsection (b)), and o0> 0 (subsection (c)).

(a) Stationary crack. To solve this problem, the friction stress, o0, had to be neglected (a 0 = 0). Then the procedure to solve eqn. (5) is: in t roduce dimensionless space and t ime coor- dinates ~ and r according to

r = t/to (6)

= X/Xo(r). (7)

Insert the factorized form of the dislocation densi ty

D(x, t) = f(r)' A(~) (8)

into eqn. (5). Equat ion (8) satisfies eqn. (5) if (KI/A )~I + p,O

to = (1 + n + p n ) A " B h 1 +P" ;

X 0 = • T2J(l+n+pn); f = T--1/(l+n+pn)

(9) After rearranging, the remaining integral equa- t ion for the normalized dislocation density, A(~), is

112 A(,~ ) , t - A(~')d~" o ,

I/ 1 A(~')d~' = - - ( 2 ~ ) -1/2. (10)

(b) Propagating crack, o0 = 0. If the crack velocity, ~, is constant , the t ime is easily eli- minated f rom eqn. (5) by the subst i tut ion

x - ~ t l~ - - - ( 1 1 )

Xo

With

A~/xo A(~) - D(x, t) (12)

gl

and

1 8 9

IA"Bh x+p" - A l + p n - n 2](pn+n-1)

x ° = ; a (KII) f (13)

eqn. (5) is writ ten in the dimensionless form

1 f A(~')d~'l p = --(2~) -x/2 . (14)

The boundary condit ion, A(oo) = 0, implies that steady state solutions are only possible if n > 2 .

(c) Propagating crack (Oo > 0). If Oo > 0, the creep zone has a finite length, L, along the x- axis, and a different procedure has to be employed. With the dimensionless coordinate, ~, velocity, v, and dislocation density, A

= (x - - ~ t ) / L (15)

v = ~ o~Bh1+p~ (16)

~(~) = A . D ( x , t)/oo, (17)

eqn. (5) becomes

; ( ~ ' ) x'2 A ( ~ ' ) ~ - ~ _ ~ , d~ ' - -v l / "A(~) li".

I I p (2~) -112 = I. (18)

1 g, f ~(~')d~' +

o04L i

The dimensionless velocity, v, as well as the exponents n and p, are parameters of eqn. (18), whereas the normalized dislocation density, A, and the combination Ki/(oox/L) are the unknown quantities of eqn. (18). The non- linear singular integral equation (18) has to be solved together with the boundary condition A(1) = 0 [20].

The ordinary non-linear integral eqns. (I0), (14), (18) have been solved by iterative proce- dures which will not be described here in detail. In each iterative step a linear singular integral equat ion must be solved. This is done by converting the integrals into sums based on k appropriately spaced discrete points ~i. The tail of the funct ion A(~) for ~ < ~ < oo may be included analytically since the asymp- tot ic behaviour, A(~ -~ oo), may be derived

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easily from eqns. (10) and (14). Then the resulting set of k linear algebraic equations is solved by one of the standard computation- al routines. The numerical costs are small: each iteration step with k = 40 points needs less than four seconds computing time on a DEC PDP-10 computer, and three to eight iterative steps have always been sufficient.

It should be noted that the linear (n = 1, p = 0) eqn. (18) has been solved by Glennie [21] and Riedel [22], who treated dynamic crack propagation in strain rate sensitive materials.

3. R E S U L T S

3.1 Stresses, strains and COD Figure 2(a) - (c) shows the solutions A(~)

of eqns. (10), (14), {18). Once the normalized dislocation density, A(~), is known, the creep strain, e, the tensile stress, o, in the creep zone, and the COD-value, ~, may be determined by numerical integration according to the following eqns. (19) - (21). Formulae labelled with (a) are valid for stationary cracks, formulae with (b) refer to steady state crack growth and vanishing friction stress (o0 = 0), formulae with (c) refer to steady state crack extension and non-zero friction stress, oo.

_1 (KiI 2Tll<l+,+p,) / A(~')d~'. (19a}*

f A(~')d~'. (19b)

Lo ° 1 = f A(~')d~'. (19c)

a = A.r-ll(l+n+Pn).12~A(~) + / A(~')d~'l lln.

l ( ~ ) 2pn d__.. +p- 11/~"+"-1) o = A" A n a l "

(20a)*

(20b)

l/ I" O = 0 0 + (70vlInA(~) TM A(~')d~' (20c)

--IKll 2rll(l+n+Pn) / o

(21a)*

/ A(~)d~ 0

(21b)

K2(1 _ ~2) = ~(n, p, v) (21c)

ooE The definite integrals appearing in eqn. (21) are shown in Fig. 3(a) - (c). The dimensionless factor, f3, (eqn. (21c)) depends on the parameters v, n and p, but is independent of KI, 0o, E and v. Its functional form will be given later (Fig. 6).

Figure 4 demonstrates how the initial elastic stress concentration at the tip of a stationary crack is relaxed by creep deformation (eqn. (20a)). In Fig. 4, the temporal sequence is characterized by the dimensionless COD-value ~/(KI/A) 2. The corresponding times may be obtained by inverting eqn. (21a). Figure 4 also shows that the stress relaxation does not depend strongly on the exponents n and p if the dimensionless COD is chosen as the temporal parameter instead of the time.

Figure 5 shows the rapid increase of the strains (eqn. (19a)) at the beginning of the

1"

*If in eqn, (1) B is a func t ion of the time (explicit t ime hardening), replace T by y B(t)dt/B. o

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~ ( a ) 2 ( b ) .2 . . . . . . . . . . . . .

1 "<1

n = 2 i

0 t 2 0 1

i

v= t...Xk,,, i n=2 "%',edO i

2 0 5 1

~ ,norm. d i s t a n c e f r o m c r a c k t ip ~ = ~ =

Fig. 2. Normalized dislocation density, A(}). (a) Solution of eqn. (10): stationary crack. (b) Solution of eqn. (14): steady state crack growth and o 0 = 0. (c) Solution of eqn. (18): steady state crack growth and o 0 > 0.

2 4 6 8 lO 2 ,; 6 8 tO

s t ress exponent , n = n -

3. Definite integrals ~A(~)d~ vs. stress exponent Fig. V

n. Cases (a) and (b) as in Fig. 2.

deformation process and the subsequent dece- leration due to the stress relaxation.

3.2 Crack growth initiation time, growth rate and t ime to failure

In accord with Vitek's conclusions [3], we assume that crack growth and growth initia- t ion are controlled by a critical COD-value, 5 c. This is analogous to the critical strain failure criterion which is approximately valid in uniaxial creep tests [23]. Other crack extension criteria can be combined with the preceding stress analysis as well.

During the initiation time, ti, the COD value increases from zero to the critical value, 8 c. Inverting eqns. (21a, b) one obtains the initiation time, ti, and the velocity, ~, of steady state crack growth:

0

1 + n + pn

(22a)

A" 1

(1 + n + pn)Bhl+pn Ki ~

I n +p n - -1

0

(22b)

A 2-" Bhl+pnK 2(n-l) (if o0 = 0).

Note that eqn. (22b) has the form ~ K~ t " - l ) , and therefore the power law, ~ K~', which has been found in many experiments [1, 5 - 14] is in agreement with the present theory. For non-zero friction stress, %, the crack growth rate cannot be written in closed form as a function of Kx. Inverting eqn. (21c) numerically, one obtains the dimensionless crack growth rate, v (defined in eqn. (16)), as a function of KI. The relation v vs. KI, which is shown in Fig. 6, is characterized by a threshold value Ki,m below which no crack extension occurs. If KI approaches its threshold value, the curves in Fig. 6 exhibit considerable deviations from the power law

In order to obtain the time of crack growth from initation to final instability, we assume that eqn. (22b) is approximately valid also for accelerating cracks, although eqn. (22b) has been derived for a constant crack velocity only. This assumption will be discussed in Section 4.2. For any given relation between

1 9 2

T .02

.0!

.02

.01

I ~ l r i [ I I r l I I

p--O ~ p--O

12

n---4 . . ~ 0 n=lO p--I~12 p= 0

0 2 ,~ 6 8 (xlO 3) 0 2 ~ 6 8 (xlO 3) tO

norm. d/s tance f r o m crack t ip ,# ,= ~ =

Fig. 4. Tensile stress, o, in the creep zone ahead o f a stationary crack vs. normalized distance from the crack tip = x/(Ki/A)2. Parameter is the normalized t ime-dependent COD-value ~ = ~ / (KI/A )2.

10

1

5

"6

o c I

/ /

0 l 5 I0 15

normalized t ime "E

Fig. 5. Normalized strain ~ = e.h/(Ki/A)2 in the creep zone ahead of a stationary crack us. normalized t ime r (eqn. (6)) , at various normalized distances from the crack tip ~ = x/(KI/A)2, n = 4, p = 0.

K[ and the crack length, a, eqn. (22b) may be integrated with respect to the time. For instance, KI (a) may have the form

Kt(a) = Kx'(a/ai) 1/2 (23)

(KI = stress intensity factor associated with the initial crack length, ai). In this case, the time, tg, during which the crack grows from its initial length, ai, to the final length, af is

A , - Z a i a ~ - 2 _ ai n - 2 tg -" (n - - 2 ) B h I +P" K 2 t ( n - 1 ) . a ~ - 2

f 2

.5

.2 2 3 4 5 I0

Fig. 6. Log- log plot o f normalized crack growth rate, v (eqn. (16)) , vs. normalized stress intensity factor KI = KI((1 --P2)/OoE~c) 1/2 ffi ~ 3r-1/2. Note the threshold value against crack propagation l~ i , th --- 1.

n = ] ~

15c //A(~)d~l -I+"÷p" (24) 0

The initiation time, ti, contributes a considerable fraction to the total life time, tf = ti + tg, if the applied stress intensity K I is small and if the material is ductile (large critical COD-value, $c).

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The crack ~ o w t h rate, ~, and crack opening rate, 5, at the onset of crack growth are correlated through eqns. (21a) and (22b). After eliminating the stress intensity factor, KI, a relation results which is similar to an empirical relation found by Haigh [15]

~ ~(,-1)/ , . (25)

4. DISCUSSION

The preceding results have been derived on the basis of the small-scale yielding approximation and of the steady state solution. As a consequence, the creep crack growth characteristics were univocal functions of the stress intensity factor. In the following two subsections, 4.1 and 4.2, the limits of these approximations will be discussed. Beyond these limits, no unique description of creep crack growth by the stress intensity factor may be expected.

4.1. Small scale yielding approximation. Creep brittleness.

In the fundamental integral eqn. (5), the applied stress field has been represented by the stress intensity factor, KI, alone. This is reasonable if an appreciable fraction of the creep strain is concentrated in a region, R, near the crack tip where the singular stress term Ki/(2~x) 1/2 predominates. In most specimens, the extent of this region, R, is approximately given by half the smallest relevant geometrical dimension, d, of the specimen [24], i.e., d = crack length or load carrying ligament width. To be definite we consider the small scale yielding approximation to be sufficiently accurate if more than 90 per cent. of the dislocations (eqn. (8)) are concentrated in a region 0 ,~ x < d/lO. Evaluating this numerically, the condition for small scale yielding becomes:

5 < 0.4x/d'KI/A, (26)

or, for crack growth initiation under small scale yielding conditions:

d > 4"(5~E/K~) 2. (27)

In eqns. (26), (27), the numerical factors are almost independent (within 20 per c e n t . ) o f the exponents n and p, if 3 ~ n ~ 20.

Crack extension in the small scale yielding range is strongly affected by the initial stress and strain concentration at the crack tip. The material then behaves "creep brit t le" or flaw sensitive. Creep ductile behaviour, on the other hand, is characterized by crack extension after the stresses and strains have been distributed across the section of the specimen. Therefore, the net section stress, rather than the stress intensity factor, will govern the crack extension under creep ductile conditions. Relation (27) demonstrates that creep brittleness is not only determined by material properties, represented by 5¢ and E. Increas- ing the geometrical dimensions, d, and the stress intensity factor, KI, favours creep brittleness. The exponents n and p have only a small influence on the numerical factors in eqns. (26}, (27) if n > 3. Williams and Price [25] arrived at the opposite conclusion: according to their agruments, the exponent n alone should decide whether a material behaves creep brittle or ductile, irrespective of the applied stress, the specimen geometry, and material properties such as critical strain.

It should be noted that the analogous small scale yielding condition in static plasticity (ASTM E-399)

d > 2.5 (g~¢/oy) 2 (28)

depends on the yield stress, oy, and on the fracture toughness, Kxc. The applied stress intensity factor, K~, cannot be chosen inde- pendently. In creep tests, however, the specimen size, required for brittle behaviour of a given material, may be reduced by increasing the stress intensity factor (see relation (27)).

4.2 Steady state approximation In most laboratory tests, the crack velocity

increases during the crack extension. The steady state solution that has been derived in the present paper will be approximately valid only if the crack acceleration, ei', is sufficiently small. Otherwise the crack growth rate will depend not only on the instantaneous value of K~, but also on the prior history of the stress intensity factor. To be definite we consider the steady state solution to be accepta- ble if the crack velocity increases by less than 10 per cent. while the crack tip traverses the region in which 90 per cent. of the dislocations are concentrated. Together with eqn. (22b), this definition yields the following

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approximate numerical result: the steady state solution is applicable at each instant of crack propagation, if

1 ~K, x / ( n - 2 ) ( K I 12 - -~<0.3 n - - 1 \~-~-/ ( n > 3 ) . (29)

Taking into account that in most laboratory test specimens under constant load conditions the stress intensity factor approximately obeys the relation ~Ki/3a = Ki/2d, one obtains from eqn. (29):

n - - 1 d > 13/(n-~--~)(~c'E/K~)2; (n > 3). (30)

A comparison of eqns. (30) and (27) shows that the steady state condition, eqn. (30}, imposes a more severe requirement on the specimen size, d, than the small scale yielding condition, eqn. (27). The condition eqn. (30) may be relieved if crack growth tests are con- ducted under a constant stress intensity factor rather than under constant load.

4. 3. Dugdale model It is an intrinsic weakness of the Dugdale

model that the height, h, of the plastic or creep zone is an undefined parameter. For plane stress conditions, it has been suggested to equate h to the plate thickness. For plane strain, h has been put equal to the ratio between critical COD and critical strain [21]. This uncertain value of h renders the predic- tion of lifetimes uncertain through the factor h I+p" in eqns. (22) and {24). This is a serious limitation for predicting absolute lifetime values from the Dugdale model. COD values, however, depend less markedly on the uncertain quantity, h, (eqn. (21)) if the exponent n is large.

Another deficiency of the Dugdale model becomes apparent if a linear creep law (n = 1, p = 0; Nabarro-Herring creep) is considered. For'a stationary crack, the exact solution of the linear creep problem may be inferred from the theory of linear visco-elasticity [26]. The exact solution predicts that no internal stresses develop during visco-elastic flow and that the initial elastic stress distribution remains unchanged. In the Dugdale model, on the other hand, internal stresses due to the inhomogeneous dislocation density, D(x, t), are present, and the elastic stress singularity at the crack tip is relaxed as shown in Fig. 4.

The author expects that for increasing non- linearity, n, the discrepancy between the present Dugdale model and the prospective exact solution will decrease.

4.4. J-integral In creep problems, the attainment of a cri-

tical J-integral [27] is unlikely to be the condition for the onset of crack growth. For small scale yielding, namely, the J-integral has the time independent value

J = K~2/(4~A). (31)

Therefore, the critical value, Jc, would either be attained instantaneously when the load is applied or it would never be attained. This obviously does not describe crack propagation under creep conditions, where crack growth starts when a finite time, tl, has elapsed after applying the load.

In the numerical calculations, the J-integral can serve as a convenient, self-consistency parameter, since the solutions D(x, t) of eqn. (5) must satisfy the condition

J = / a(x, t) D(x, t) dx = K~/(4rA). o

(32)

This relationship is fulfilled by the present solutions better than within 2 per cent.

4.5. Comparison with data from the literature The present results qualitatively confirm

the results which Vitek [3] obtained from a computer simulation of a stationary Dugdale crack under creep conditions. However, the numerical COD-values of the present study are about 20 per cent. smaller than the values given by Vitek [3]. This discrepancy can pro- bably be attributed to numerical difficulties during the first time increment in the computer simulation method [3] (personal com- munication}. In addition, the stress maximum ahead of the crack tip (Fig. 4) has not been found in ref. 3. This stress maximum is not a numerical artefact as might be suspected but an analytical consequence of eqn. (10). Vitek's principal conclusions, however, remain unaf- fected by these differences: the results following from the Dugdale model, combined with the COD crack extension criterion, are consistent with the measurements of Haigh [15] and of Batte [28].

According to the present theory, the exponent, m, which appears in the relation ~ ~ KI ~, eqn. {22b), is given by m = 2(n -- 1). Kenyon et al. [10] have suspected that m is approximately equal to the stress exponent, n. Part of the experimental results are in agreement with the present theory [6] but, unfortunate- ly, uniaxial creep data and creep crack growth data are rarely reported for the very same material. The discrepancies which do occur between theory and experiment may at present not be clarified univocally. They might be attributed to experimental difficulties [8] , to environmental effects, to the deficiencies of the Dugdale model, to a critical COD which might depend on the crack growth rate, 5c{a), to the threshold gi,th associated with the friction stress, o0 (Fig. 6), to large scale yielding, and to the constant load conditions applied in most experiments, which often violate the steady state condition stated in eqns. (29), (30).

5. CONCLUSIONS

In the preceding discussion it has been de- monstrated that it is by no means clear, a priori, that a Dugdale model will yield realistic results under creep conditions. The measurements discussed in the preceding Section lend some support to the validity of the Dugdale model and the COD crack extension criterion. For instance, the empirical crack growth law,

= K~, is in agreement with the theoretical result. Also, the calculated growth initiation times, ti, are consistent with a limited number of measurements. But the experimental evi- dence is not yet sufficient to prove or disprove the usefulness of a Dugdale model under creep conditions.

Forthcoming experimental studies should compile the data obtained from uniaxial creep tests (B, n, p, o0) , as well as the creep crack growth data. A rational laboratory test design will take into account the requirements of small scale yielding, eqn. (27) and of steady state crack growth, eqns. (29, 30). If these requirements are fulfilled, a unique correlation between KI and the crack growth rate may be expected.

Quantities that can be compared between theory and experiment are mainly the t ime dependent COD during the initiation period,

195

eqn. (21a), the crack growth initiation time, ti, eqn. (22a), and the rate of crack growth

(eqn. (22b) and Fig. 6), all as functions of KI. Finally, an experimental assessment of the COD crack extension criterion is neces- sary which has been employed in the present theory. It should be emphasized, however, that the stress analysis of Section 3.1 remains valid irrespective of the crack extension criterion.

ACKNOWLEDGEMENT

Financial support from the European Coal and Steel Community is gratefully acknow- ledged.

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28 A. D. Batte, Parsons Tech. Mem. No. MET 75- 128, 1975.

a dugdale model for crack opening and crack growth under creep conditions

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