Crack propagation under creep and fatigue

Nuclear Engineering and Design 83 (1984) 303-323 North-Holland, Amsterdam

CRACK PROPAGATION

UNDER

303

CREEP AND FATIGUE

K. S A D A N A N D A Material Science and Technology Division, Naval Research Laboratory, Washington, DC 20375, USA

Received April 1984

Crack growth under cyclic, static and combined loads in several high temperature alloys is presented from both material and fracture mechanics aspects. Parametric representation of high temperature crack growth in terms of linear and non-linear elastic fracture mechanics is discussed along with the experimental determination of these parameters for several specimen geometries. Crack growth is characterized as cycle-dependent, time-dependent or combined cycle- and time-dependent processes depending on material, temperature, load frequency and environment. While the cycledependent process is due to fatigue damage, the time-dependent process could be due to creep or environmental effects or both. It is shown that the applicability of a particular fracture mechanics parameter to characterize high temperature crack growth depends on micromechanics of the growth.

1. Introduction Components in structures and machinery are seldom subjected to uniform stresses. Pre-existing flaws or defects produced during processing or welding contribute to stress concentrated regions. Non-uniform cross sections or eccentric loading will also give rise to stress gradients. Because of this, fatigue as well as creep damage becomes highly localized and nonuniform. As the defects grow into the damage region, the stresses in the remaining ligament get accentuated giving rise to accelerated crack growth. Furthermore, since crack growth often proceeds from surface defects, particularly under fatigue, component life can be further affected by the presence of an aggressive environment. In many instances the crack nucleation stage is either circumvented or minimized and in such cases component life is limited to the crack propagation stage. Life prediction techniques based on uniformly stressed specimens may be, therefore, inaccurate or inadequate in predicting component life in service. In fact, many alloys that have very high fatigue or creep strengths show very poor resistance to crack growth. This emphasizes the fact that knowledge and understanding of crack growth behavior in the creep-fatigue range are important in the characterization of alloys for high temperature applications as well as in predicting service life of components. Crack growth under creep-fatigue conditions is

affected by several variables, namely specimen and load geometry, type and magnitude of load, temperature, frequency, environment and microstructure as well as the material flow properties. Investigations generally have been directed towards identifying a single controlling parameter for crack growth that is independent of specimen and load geometry, crack length and load. The presumption that a single field parameter can uniquely characterize the crack tip stress fields has led to the use of fracture mechanics techniques to characterize high temperature growth. If, for example, a linear elastic parameter, the stress intensity factor K is the controlling parameter for crack growth, then the growth rates represented in terms of K will be independent of the specimen and load geometry. In such a case, prediction of the extent of crack growth in a structure is possible, using crack growth laws (relations representing the correlation of crack growth rates in terms of the controlling parameter K) developed on the basis of laboratory generated data, provided proper K solutions for the structure of interest are available or can be determined using stress analysis techniques. The success of linear elastic fracture mechanics in providing a single field parameter to characterize crack growth under fatigue at low temperature and thus to predict life of components has stimulated efforts to extend the techniques to high temperature fatigue and creep crack growth. With increasing

0029-5493/84/$03.00 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

304

K. Sadananda / Crack propagation under creep and fatigue

temperature, plasticity effects become prominent particularly because of the decrease in yield strength of the material with temperature and because of the incipience of non-linear time dependent effects. Therefore, the use of elastic-plastic fracture mechanics parameters such as AJ, J and J* integrals becomes important. In addition these parameters become more pertinent, particularly when the microstructure becomes unstable due to time-temperature dependent transformations occurring during prolonged service. Definitions and applications as well as the methods of determination of both linear z:nd non-linear elastic fracture parameters will be discussed with reference to crack growth in high temperature reactor materials such as Inconel 600, Inconel 625, Incoloy 800, austenitic stainless steels and low alloy steels. In the following sections a general description of high temperature crack growth in terms of micromechanisms of growth involving cycle and time dependent processes under various types of loading conditions, advances in the fracture mechanics methodology in correlating the crack growth and finally the behavior of different high temperature structural alloys under cyclic, static and combined loads are presented.

2. Crack growth processes Crack growth rate in a material depends, in addition to specimen and load geometry, on the type of load, that is, cyclic, static or combined loads. Fig. 1 shows several load sequences used for characterization of high temperature crack growth. These represent the basic types generally encountered in service. For convenience the figures are represented in terms of initial stress intensity rather than load. Crack growth under cyclic load (fig. la) depends on the magnitude of stress intensity range AK, temperature, environment, frequency, wave shape and the ratio of minimum to maximum load R. The effects of strain rate or loading rate and mean stress could be resolvable in terms of the above variables. Fig. l b represents the initial stress intensity K as a function of time for a static load test. Crack growth occurs continuously under static load if the applied K exceeds K, hc, the threshold stress intensity for time-dependent crack growth. For K less Kt,~, crack growth may not occur or may occur at a decreasing rate leading to crack arrest. However, creep deformation could still

K

T ZXK

AI ~--t---I ..... TIME K ~Kth (c)

TIME

l

I"

KMAX

Kthc

(b) K

TIME Kth~

c (d)

TIME NI

"I'

th

I

NI

(e)K I~tl'-~-th~l TIME

(f)

TIME

Fig. 1. Schematic representation of several types of loads (a) continuous cycling; (b) static load; (c) hold time at peak load; (d) hold time at minimum load; (e) hold time defined keeping total frequency constant; (f) periodic cycling and static loads.

occur for K < Kth~and result in relaxation of crack tip stresses. Service components in general are subjected to neither pure cyclic nor pure static loads but to a combination of these. Figs. lc and ld show two limiting cases where the two types of loads are combined in each cycle by imposing a hold period at maximum load or at minimum load. Clearly, under such combined loads, the contribution from cyclic load occurs if AK is greater than AKthand from static load if K at the hold period is greater than K, hc. In addition to each contribution, synergistic effects could be superimposed which could either accelerate or retard the growth. Note that in figs. lc and ld the effects of superimposed hold time can be easily separated from the effects of continuous cycling by keeping the loading and unloading rates, 1/t~ and 1/t=, the same as in continuous cycling. Note also that at high temperature-time-dependent contributions can be significant even during loading and unloading such that combined loading effects may manifest even without a hold period. James [1] used a different type

305

K. Sadananda / Crack propagation under creep and fatigue

of cycle to define hold time effects which is represented in fig. le. In this cycle the total frequency v is kept constant. Finally, fig. If shows a different combination of cyclic and static loads where the two types of loads are applied sequentially. The effects under such loads may be treated under mechanical history effects. Instead of characterizing crack growth on the basis of type of load, it is more advantageous, particularly in the creep range, to do so on the basis of the micromechanics of the growth processes. For example, at very high temperatures and low frequencies, the creep process may become predominant even during cycling to the extent that crack growth under cyclic, static and combined loads may be essentially the same implying that the controlling parameter would also be the same for all of the loading conditions. Crack growth processes may be broadly classified as cycle- and time-dependent. Time-dependent processes include both creep and environmental effects which are generally thermally activated and thus cause a large temperature dependence in crack growth rates. Clearly, crack growth under static load (fig. lb) is only time-dependent. On the other hand, crack growth under cyclic load or under combined loads could be either cycle-dependent or timedependent or a combination of both depending upon the material, temperature, frequency, environment and stress amplitude. The temperature and frequency dependence of crack growth of an alloy under cyclic load is schematically shown in fig. 2, where d a / d N is the crack growth increment per cycle, T is temperature and u is frequency. High temperature and low frequencies favor the time-dependent process and low temperatures and high frequencies favor the cycledependent process. At intermediate temperatures and frequencies, a combination of the two processes could occur depending on the stress amplitude. Since many service conditions fall in the range where crack

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growth is neither pure cycle-dependent nor pure timedependent, but a combination of the two, a number of studies have been aimed at understanding crack growth behavior under these complex loading conditions. Attempts have been made to predict the crack growth under these loading conditions by simple linear addition of the growth rates caused by each individual process. In many cases, however, synergistic effects occur which give rise to either enhanced crack growth rates or decreased growth rates in relation to the simple linear addition. These synergistic effects are termed creep-fatigue-environmental interactions. Fig. 3 shows the temperature dependence of crack growth rates under cyclic load for many high temperature alloys [2] for A K = 3 3 M P a m 1/2 and for A K / E = I . 6 x l O 4m'/2 where E is the Young's modulus. If change in crack growth rates with temperature is due to change in elastic stiffness then normalization with respect to Young's modulus would eliminate any temperature or even material dependence. Fig. 3 shows that temperature dependence persists even after normalization and could be the result of different thermally activated processes at different temperature regimes. Since the prediction of crack propagation life squarely depends on the validity of characterization of crack growth rates using

v

Fig. 2. Schematic illustration showing crack growth rate as a function of (a) temperature and (b) frequency.

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Fig. 3. Temperature dependence of crack growth rate in many high temperature alloys (from ref. [2]).

306

K. Sadananda / Crack propagation under creep and fatigue

fracture mechanics methods, it is important to discuss what parameters characterize various crack growth processes and how to determine these experimentally.

3. Fracture mechanics parameters 3.1. Linear elastic parameters

Fracture mechanics techniques are based on a fundamental concept that the stress field ahead of the leading edge of a sharp crack can be characterized by a single parameter K, the stress intensity factor, which is proportional to both nominal stress o- and crack length a. For a mode I crack, that is, for a tensile crack where local displacements are symmetric with respect to parallel and perpendicular directions to the crack plane, K can be determined from K = Ytrx/a,

(l)

where Y is the compliance relation for a given specimen and load geometry and is readily available in the literature for various fracture mechanics type specimens. For time-dependent crack growth, K,,,x or K is the controlling parameter if non-linear effects are confined close to the crack tip. On the other hand, if crack growth is cycle-dependent, the growth is controlled predominantly by the cyclic stress amplitude rather than by the peak stress. A linear elastic stress intensity factor range AK is defined to characterize the growth and is given by AK = Kmax(1 - R ) ,

(2)

where R is the ratio of minimum load to maximum load. The stress intensity factor range AK is affected, particularly at low R values, by the plasticity induced compressive residual stresses at the crack tip and the tip may remain closed during portion of the loading part of the cycle, thereby effectively reducing AK at the crack tip. In addition, closure of the crack tip can also occur by oxidation products produced by environmental interactions and by surface ridges produced during crack growth. An effective AK which can be determined from the knowledge of closure loads provides the controlling parameter for cycle-dependent crack growth. At high K values or with increasing temperature the plasticity effects become important to the extent that linear elastic concepts may not be valid and one has to resort to non-linear parameters such as AJ and J* and these parameters are discussed below.

3.2. Non-linear elastic parameters AJ and J* integrals

The concept of the J integral was originally introduced by Rice [3] and its applicability as a fracture criterion was proposed by Begley and Landes [4]. The value of the J integral can be experimentally determined for a cracked body from the load displacement curve using a simple equation: ldU 13~a ~'

J-

(3)

where B is the specimen thickness and the differential corresponds to the change in strain energy U due to an infinitesimal change in the crack length d a at a given crack displacement 8. The J-integral technique has been applied to room temperature fatigue by Dowling and Begley [5] and to high temperatures by Sadananda and Shahinian [6]. Experimental determination of the J integral for fatigue involves the following procedure. For displacement-controlled fatigue, determination of J is rather straightforward and follows directly from eq. (3). This is illustrated schematically in fig. 4. In particular, the rising part of the load-displacement curves in fig. 4a for crack lengths aj and az are displaced to a common

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t LOAD

L1 I / /

6--

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(a)

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6~ (b)

MAX LOAD

MAX LOAD -

LOAD .~ .~" 61 62

MIN LOAD

61, d2

MIN LOAD

6~

d~

(c)

(d)

Fig. 4. Determination of the J-integral for displacement-

controlled (a, b) and load controlled (b, c) fatigue. (a) and (c) hysteresis loops for crack lengths a~ and a2; (b) and (d) translation of loading parts of the loops to a common reference for integration.

K. Sadananda / Crack propagation under creep and fatigue origin as in fig. 4b. The J integral is then given by the hatched area [5]. For load-controlled fatigue, there is an ambiguity in terms of defining the proper limits of integration. It has been shown [6] that the following procedure ensures that the J integral is compatible with the ~lK value and it involves selecting minimum load as a reference point for the integral. In particular, the rising parts of the load-displacement curves in fig. 4c are displaced to a common origin as in fig. 4d and the J integral is determined from the hatched area. Thus in load-controlled fatigue the cumulative cyclic strain such as 8~ and 32 (fig. 4c) is assumed to have no effect on the subsequent fatigue crack growth. In effect it is similar to the AK concept. Depending on the minimum and maximum loads, a proper J ( d J to be precise) can be determined which reflects R effects or mean stress effects, just as in d a / d N versus AK plots. While the above procedure requires the use of at least two load-displacement curves, one for each crack length, it is possible to estimate d J from a single load-displacement curve using an approximation procedure. Thus for the compact tension specimen geometry AJ can be determined from J = 2 (cq U + ct2PSm) Bb

(4)

where a, and a2 are the coefficients developed by Merkle and Corten [7] and are given in graphical form elsewhere [6], U is the total area under the load-displacement curve, b is the uncracked ligament of the specimen and t~m is the displacement at maximum load. It has been shown [6] with reference to Udiment 700 fatigue crack growth data that the estimation procedure gives AJ values very close to that determined following the procedure in fig. 4. At high temperatures where creep becomes important one should consider J* parameter which is an energy rate integral that characterizes crack tip stress-strain rate fields. Experimental determination of J* involves measurement of crack length and displacement along the loading line as a function of time and then following a step-by-step data reduction scheme [8]. Since the majority of creep tests involve application of a constant load, a scheme for the determination of J and J* for such tests is shown in fig. 5. A similar procedure for displacement-controlled tests was developed earlier by Landes and Begley [9]. There have also been several approximate methods developed to determine J* using displacement and crack length data as a function of load. All of them assume that the material ahead of the crack

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Fig. 5. Data reduction scheme for the determination of J (left column) and J* (right column) integrals. In this figure, a, a, 8, ~, U, /[/and L respectively are crack length, crack growth rate, displacement, displacement rate, strain energy, strain energy rate and load (from ref. [8]).

tip creeps at a rate given by a power law of the form = Ao'",

(5)

where A and n are constants, which is generally valid for uniformly stressed specimens creeping at steady rate (stage II creep). In eq. (5), the constants A and n depend on material, temperature and stress range. Table 1 gives expressions used by various authors for the estimation of J* for different fracture mechanics type specimens. Harper and Ellison[10] have shown for low alloy steels and Sadananda and Shahinian [11] for type 316 stainless steel that J* determined from

308

K. Sadananda / Crack propagation under creep and fatigue

Table 1 Expressions for J* for several types of fracture mechanics specimens ~) Specimen

./*

Ref.

Center-notched (CN)

_ (( nn +- l 1) ) Cr"et8. ~ P8 for n ~ 1

[31] [30]

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-

[12]

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[30]

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[32] [30]

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a) A and n are from eq. (5); gl{2a/W, n} and hl{a/W, n} are Hutchinson's functions represented graphically in ref. [12]; = displacement rate; r = radius; B,, - thickness within side grooves; B = specimen thickness; b = uncracked ligament size; P = applied load; W = width and a = crack length. the data reduction scheme and the estimation procedure are nearly the same while Saxena [12] found for type 304 stainless steel that the two differ anywhere from 8 to 4 0 % .

3.3. Stress field ahead of creep crack It is important to know the degree of modification of the stress field ahead of a crack by creep deformation in a given material so that one can determine what load parameter is required to characterize creep or creep--fatigue crack growth. If a creep zone is defined in analogy with a low temperature plastic zone, then a small scale creeping condition can be defined under which the elastic parameter K may be valid for creep crack growth. Riedel and Rice [13] have analyzed the asymptotic solutions for stresses in the creep zone for a material deforming by power law creep [eq. (5)] and these are shown to be of the order of r I/("-') singularity in comparison to the inverse square root singularity for an elastic crack. Clearly for n-< 3 the stress singularity corresponds to that of an elastic crack implying that K may be valid for such materials. From the analysis, the creep zone is given by

r c = ~ Ti ~

((n ', +~ - ~

.j

F.(o),

(6)

where K~ is the initial stress intensity factor, E is Young's modulus, A and n are defined in eq. (5), aT, is approximately equal to unity. For(O) is a function of angle 0 measured from the plane of the crack in an anticlockwise direction and is given in graphical form in ref. [13]. T h e creep zone boundary corresponds to the position where creep strains are equal to the elastic strains. Note that the boundary is a function of initial stress intensity K i and time under load. In analogy with small scale yielding, small scale creep is defined when rc is less than the crack size and the appropriate specimen dimensions, Riedel and Rice have also defined a characteristic time for the transition from a small scale creep to extensive creep of the whole specimen given by K2(1 - v2) t~ -

E(n + 1 ) J * '

(7)

where J * can be determined experimentally or analytically using the expressions given in table 1, and v is Poisson's ratio. For plane stress, (1 - u 2) is replaced by 1. T o determine if K is valid for J * is a valid

K. Sadananda / Crack propagation under creep and fatigue

parameter, incubation time for creep crack growth, for example, can be compared with the characteristic time, and the small scale creep condition can be presumed to exist if incubation time is less than the characteristic time. While the above analysis pertains to stationary cracks, attempts are being made to extend the analysis to moving cracks. Some aspects of the dynamic crack have been analyzed by Hart [14] and Hui and Riedel [15]. Assuming a general functional relation for velocity of the creep crack, Hart showed that inverse square root singularity exists at a moving crack and a stress intensity factor can be defined which is the sum of elastic, Ke, and plastic, Kp, contributions. Thus, total stress intensity is given by KT = K~ + K p ,

(8)

where Kp is generally negative. Under a "steady state" condition, i.e. when stress field becomes independent of time (dynamic equilibrium), Kp is proportional to Ke such that crack growth rates under such steady state conditions are K controlled. Whether K is valid or not may still depend on whether the proportionality constant is independent of specimen and load geometry. If it does depend on specimen geometry, then the knowledge of the functional dependence of the proportionality factor is required before K can be used. Also it was noted by Hart that if stress intensity is less than some minimum, steady state condition cannot be reached and crack growth if any occurs at a decreasing rate ultimately leading to crack arrest. We may note that all of the above conclusions are phenomenological and are not based on the micromechanism of crack growth. Hui and Riedel [15] also concluded that for a dynamic crack for n < 3, inverse square root singularity exists, while for n > 3, stressstrain singularity is of the form r -urn-t) and depends only on crack growth rate and not on applied load. The implication is that for crack growth rates greater than some minimum value the rates are proportional to K" for small scale creep and for growth rates less than the minimum, no stable steady state growth is possible. The latter condition, perhaps, defines a threshold (depending on the crack growth criteria used) and is similar to Hart's conclusion. It is clear from the above discussion that the choice of the load parameter that controls creep crack growth rates is intimately related to material creep properties particularly to the value of the creep exponent n for materials deforming by power law creep. Eqs. (6) and (7) provide some guidelines in

309

terms of determining the conditions for which a linear or non-linear elastic parameter is rate controlling. It should be noted, however, that the above analyses are based on homogeneous continuum models whereas internal inhomogeneities such as grain boundaries or inclusions are known to play a major role in creep damage. Inhomogeneous deformation on a local scale such as grain boundary sliding could affect the characterizing parameter. Another factor that could influence the crack growth kinetics is the environmental sensitivity of the material. For example, if a material is highly sensitive to environment, crack extension can occur before crack tip stresses are relaxed due to creep. Finally, stress-induced microstructural instabilities would also affect the crack growth process. Therefore, it should be noted that creep properties alone do not uniquely define the crack growth behavior under static load.

3.4. Other parameters

In addition to the above fracture mechanics parameters, other parameters have been suggested to characterize creep crack growth. If creep can occur rapidly, the stress concentration is relaxed at a rate faster than the increase of stress due to crack extension such that stresses become essentially homogeneous except in cases where severe bending moments exist. The specimen under such conditions behaves similar to a creep rupture specimen. Crack growth and creep life then could be predicted using net section stress, or more accurately, the reference stress [16] which takes into consideration the presence of a crack in the specimen. A reference stress [17,18] is defined for a component as that stress which when applied to an uniaxial specimen will give the same displacement rate. In engineering terms a single stress can be specified for a component under a stress gradient which will predict its deformation behavior relative to uniaxial tests. Williams and Price [19] have shown that the reference stress in a cracked body is equal to the stress at a skeletal point which is a position within the structure where stress is approximately constant for all values of creep index n. The reference stress for the fracture mechanics specimens under plane strain is defined as [16] O~ref

~g_ - -

m

P mBW

,

(9)

where % is the gross section stress, P is the applied

310

K. Sadananda / Crack propagation under creep and fatigue

Table 2 Reference stress for fracture mechanics specimens [19] Specimen Center-notched (CN)

Reference stress, Oref P = O'net = B b

Compact tension

= - P (2.02 ( l + a l W ) ~

Single-edge-notch tension (SENT)

_

(CT)

t~W

J

1.004 ( P ) (1 - a / W)2 ~ for a / W < 0.5

=2.oo7 ~

ree- o,n, bend

~

l-a/

(0V ) BW

-~

for a/W>0.5

, (1 - a/W-~

load, /3 and W respectively are specimen thickness and width and m is the ratio of the load to produce general yield in a cracked body to the load to yield an uncracked body of the same overall dimensions. Values of rn for the standard test piece configurations are given by Haigh and Richards [16]. While eq. (9) refers to the time-independent stress (yield stress), its applicability to creep could be extended without much error for n > 5. Williams and Price [19] have obtained solutions for the reference stress for the different specimen geometries considering time dependency of stress and these are listed in table 2 for easy reference. The reference stress concept thus provides a way of predicting the component life of a structure containing a flaw from conventional creep rupture data. While generally only the creep life could be predicted, the concept can be extended to correlate the creep crack growth rates since the stress depends on crack length, applied load and specimen geometry. In summary, the characterizing parameter for crack growth in a given material essentially depends on the micromechanism of crack growth which is dictated by the plastic flow properties of the material, its environmental sensitivity (for aggressive environments) in addition to the type of load, temperature and frequency. For a purely cycle-dependent process AK or its non-linear counterpart AJ and for purely time-dependent processes K . . . . . . "]max, J* or ~rr~ could be used to correlate the crack growth rate data. Validity of any particular parameter is assessed by

showing the independence of crack growth rates on specimen geometry. When both processes occur simultaneously, the choice of the parameter may depend on which of the two is predominant. In general, representation of crack growth in terms of cycle ( d a / d N ) and time (da/dt) would help to identify the synergistic effects that occur when processes are present simultaneously. With the above background on the various parameters and their limitations, we can now examine the crack growth behavior of different high temperature structural alloys under cyclic, static and combined loads.

4. Cycle-dependent crack growth 4.1. Effect of temperature As noted in the last section, crack growth under cyclic load is essentially cycle dependent at low temperatures, high frequencies and inert environment. Cycle dependent crack growth or fatigue crack growth has been studied extensively since the early success of the application of fracture mechanics techniques to characterize the growth. Since there have been several reviews in this field, for brevity we shall limit our discussion to effects of temperature on crack growth and the recent developments in the method of predicting growth rate.

K. Sadananda / Crack propagation under creep and fatigue 10 2

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Fig. 6. Normalization of fatigue crack growth data in terms of elastic modulus. Speidel curve corresponds to room temperature behavior of several metals and alloys (from ref. [21]). If the creep or environmental contribution to crack growth is not significant, then the extent to which temperature and microstructure affect fatigue crack growth depends on whether the growth occurs by the plastic blunting process or by cumulative damage mechanisms. The plastic blunting process results in the formation of ductile striations on the fracture surfaces while the cumulative damage mechanisms could result in formation of facets or voids. In the ductile striation mode the net plastic crack tip displacement in each cycle determines the crack extension in that cycle and the effect temperature and

311

microstructure on the growth materials through their effect on the crack tip displacement. Since for a given stress intensity, the displacement is related to the elastic modulus of the material, any variation of the modulus with temperature affects crack growth rates. Speidel [20] has shown in terms of several metals and alloys that crack growth rates in vacuum at room temperature fall essentially on a single curve if AK is normalized by their modulus. Fig. 6 shows crack growth rate data at different temperatures in vacuum for several alloys [21] along with Speidel's curve. With the exception of Udimet 700 wherein crack growth occurs by a faceted mode, the data for each alloy at different temperatures fall essentially on a single curve, although the curve for each alloy differs. Effect of temperature on crack growth rates in air however differs among alloys depending on their environmental sensitivity. Figs. 7 to 10 show the growth rates represented as a function of temperature for different austenitic stainless steels, Incoloy 800, Iconel 600 and Inconel 625, respectively [2,22]. In some cases there will be in situ microstructural changes involving time, temperature-dependent transformations which may affect crack growth rates. Fig. 11 shows crack growth rates in cold worked type 316 stainless steel at 593°C where the material undergoes slow recovery at the test temperature [23]. In terms of AK, the data at low load (long time tests) do not fall along those for high load. The implication is that one cannot use the laboratory generated data for long time applications. On the other hand, fig. 11 indicates that data correlates very well in terms of AJ, where /U is determined using the load-displacement curves wherein the changes in the material flow properties are reflected. The results indicate that for long term high temperature service applications, involving slow and gradual changes in the microstructure, the use of elastic-plastic parameter AJ is more appropriate in the life prediction of structural components. 4.2. Prediction of crack growth rates A number of theoretical models have been proposed in the literature which explain the second or the fourth power dependence of fatigue crack growth rate on AK. On a microscale fatigue crack growth occurs from mechanisms operating at the crack tip region such as that represented by Laird's plastic blunting model [24] which gives rise to ductile striations or mechanisms operating ahead of the crack tip by a cumulative damage process involving microcrack

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t

348 10-6 I I I 1 I I I I I0 20 40 60 80 I00 STRESS INTENSITY FACTOR RANGE - ( KSI -'¢"~'. )

Fig. 7. Effect of temperature on fatigue crack growth rate in various austenitic stainless steels (from ref. [22]).

10-6 I0 20 40 60 80 I00 STRESS INTENSITY FACTOR RANGE - (KSI '~fi'N.)

Fig. 8. Fatigue crack growth rates in Incoloy 800 as a function of stress intensity factor range (from ref. [2]).

313

K. Sadananda I Crack propagation under creep and [atigue

20

5x I0 -4

(MPo ¢"~) 40

60

80 I00

20

(MPo ¢'~ ) 40 60

4xlO -4

INCOI~

10-2

80

I00

10-2

INCONEL 625

tO-4 10-4 hi .--I 0 >rJ

.J

704* C (1500*F)

Z

10-3

Z v

I

i

hi I-
on~

10-3

taJ I--

hi --I (.} )-0

-rI.--

E

.J

0

)-

(.9

(.9

E

v o

24* C (75"F)

10-5 (E (~

10-5

427"C (80CPF)

5 (, (llO

593°C (IIO0°F) 10-4 10- 4

10-6

J I , J J I I I I0 20 40 60 80 I00 STRESS INTENSITY FACTOR RANGE (KSI I,,/~.)

IO-S

Fig. 9. Fatigue crack growth rates in Inconel 600 as a function of stress intensity factor range (from ref. [2]).

I ~ I ~ I ~ I ,I I0 20 40 60 80 I00 STRESS INTENSITY FACTOR RANGE (KSI I¢~.)

Fig. 10. Fatigue crack growth rates in Inconel 625 as a function of stress intensity range (from ref. [2]).

314

K. Sadananda / Crack propagation under creep and fatigue

[

10 2

[

I

I

10-2

T Y P E 316 20% COLD WORKED 593 ° C

TYPE 316 20% COLD WORKED 593 °C

tj

-$

E E < tr "r

v

E E

uJ

10-3

tr "r

10 3

o .,v."

0

(..9 .v (J <

v < n,-

.,v,. {J

LOAD (kN)

L O A D (kN)

,~ 8

A 8

8.9

[] 8.9

0 13.3 10 4

O 13.3

10 4

4x10

510

20 AK

40

60

80 100

5x10-5

(MPa ~)

10

I

l 20 ~AJ

i •E

i 40

~

1 60

t

80

100

(MPa k / ~ )

Fig. 11. Crack growth data for 20% cold worked stainless steel in terms of (a) stress intensity range and (b) J integral parameter at the temperature where the material undergoes slow recovery (from ref. [23]).

nucleation or void coalescence. The plastic blunting process is essentially related to the crack tip opening displacement which from continuum mechanics is related to the square of AK. A natural consequence of this is the second power relation in the crack growth rate - A K equation. On the other hand, all of the cumulative damage processes which involve accumulated plastic displacement or balance of total energy provide apparently a fourth power relation in

the crack growth rate equation. Liu et al. [25] have analyzed the plastic blunting process involving alternate shear which they term "unzipping model" using finite element analysis and deduced an empirical equation of the form da / d N = 0.17 AK2 /( Etrc) ,

(10)

where E is the Young's modulus and oc is the cyclic yield stress. On the other hand, for a cumulative

K. Sadananda / Crack propagation under creep and fatigue

1°°/

I

I

l

I

I

I

I

5. Time dependent crack growth

I I

EXPERIMENTAL PREDICTED

5.1. Crack growth in alloys

MA 956

/ 1000°C

10-1~ ///

INCONEL600

>-

EE 10-2 "1L9

10-3

650°C//~ /"/ / /

(_)

~ / /

--

/--.~ MA956 25°C

_

60

200

10-4

/ 10-5 10

20

40

80 100

~K (MPa,~)

Fig. 12. Comparison of theoretically predicted and experimentally observed fatigue crack growth rates in several alloys.

Study of the time-dependent crack growth behavior in several alloys indicates that the growth occurs either by a brittle mode or by a ductile mode. In the brittle mode, the growth occurs in essentially a continuous manner along grain boundaries. Grain boundary sliding provides a major driving force for this type of growth which generally occurs in high strength materials. While an aggressive environment generally accentuates this process, creep deformation of the matrix may relax the crack tip stress field and retard the growth. Since crack growth occurs rapidly before any extensive creep deformation and thus crack tip stress relaxation, linear elastic fracture mechanics can adequately describe this growth. In contrast to this, the ductile mode of crack growth occurs by the nucleation of microcracks ahead of the main crack and subsequent growth and coalescence with the main crack. Growth of these microcracks occurs predominantly by deformation and results in considerably slower crack growth rates requiring higher stress intensities than for the brittle mode. Applicability of linear elastic fracture mechanics becomes questionable and therefore non-linear parameters may be required. In the following we shall review the time-dependent crack growth in several 101

damage process the crack growth rates are given by [26] da AK 4 d N - g a'rr2 bl,o-~ U '

315

(11)

where U is the effective surface energy for fatigue crack growth and /.t the shear modulus. Eq. (11) is purely phenomenological and can be deduced based on simple energy balance. The effective surface energy U depends on the specific micromechanism of damage and can be determined experimentally by measuring the hysteresis energy during the crack increment. By measuring the hysteresis energy in each cycle using the load-displacement loop such as shown in fig. 4c, crack growth rates have been calculated and are compared with the experimental data for some selected alloys in fig. 12. Except in the cases where crack growth occurs by the ductile striation process, the experimental and predicted growth rates agree reasonably well within a factor of 3.

I

1

Cr-Mo-VSTEELS 550° 10o

DATA FROMCHRISTIANet al

(z: i E

uJ 10

0102

PILKINGTONet al ~

DATABAND 1 Cr-Mo-V

10-3 ....

DATA

10 -4

10-3

I

10-2

I

BAND

0.5 Cr-Mo-V

:

I

10-1 100 J* (KJOULES/m2HR)

I

101

102

Fig. 13. Creep crack growth rates in low alloy steels in terms of J* parameter (from refs. [27,28]).

316

K. Sadananda / Crack propagation under creep and [atigue

X

200 260 240220 200•

z

160 180

X

0

~

X

i

140

120 ,I{

TyPE

g

SENT

ioo O A w J

V 80

25 40 40 40 40 40 40 40 40 65

&

5

04

5

0 ii25

15

0

I112S

IS ~5 i5

0 0 2 03

25 20 25

0 0 0 0

IS

•

CT WOL

C) x

CCP 50 S PLAIN BAR DATA

ll~

•

_ •

~ G

•

~.

0

__

04

I 3 54 i

~

v

~

C~

e

V _~

60

IO°

,~,,

,b2

,b3

6'

FA,LORE T,ME ,,, 300"

250 -

O

~

? E 200-

O

X

~-'~

W(mm) c~ uJ 150-

z

/W

as

2s

o-,

2S 40

25 ~ S

.O.4 0 " I I 2 .~

s

0.3

50 50 50

25 25 25 e

O" I 0 t 04

SO

SO e

04

so SO

s

so"

~

e~

~

v

CeNT

~

v--

~

04

~

"e

x~

= B I--

03

04 O I IOO 25 m 0 4 140 S 0"4 140 251 04 26 I 3e 03 40 20 ~ 0 3 40 S 0'4 PLAIN BAR DATA IOOlo 5~DE GROOVED

IO0

/

~'x-~

0.2

s

/

®

~&E)

a

o.,

40

•

" ~

s

40

I00-

B(mm)

2s ,~o

g

PLAIN BAR RUPTURE LINE

"~. o

Z :E

0

~ --

x ~

\

io0"

5

"l ) CT

FAILURE T I M E , It

Fig. 14. Correlation of creep life in notched and u n n o t c h e d specimens in terms of equivalent stress (a) in an untempered brittle condition (b) in tempered ductile condition (from ref. [29]).

K. Sadananda / Crack propagation under creep and fatigue structural alloys in terms of the applicability of fracture mechanics concepts to characterize growth rate.

5.2. Crack growth in low alloy steels There has been a large amount of study on the creep crack growth behavior of low alloy C r - M o - V steels for different heat treated conditions which result in either ductile or brittle behavior. Extensive analysis [27,28] involving several different specimen geometries indicates that crack growth rates are best correlated by J* rather than by K or O'ref. The total spread in the data extends by a factor of 50 for a given J* and this factor is even more in terms of K or trrcf. In fig. 13, 0.5 C r - M o - V steel was considerably more brittle with a creep ductility in the range of 0.3 to 4% than the 1 C r - M o - V steel which had ductility of 5 to 20% over the same stress range [27]. Note the spread in the data is much larger in the brittle steel. But in both cases life is determined by the growth of a single macroscopic crack. If the ductility is much larger, microcracks may form throughout the minimum gauge section and life may be dictated by the reference or equivalent stress. Fig. 14 shows, for 0.5 C r - M o - V steel in two different heat treated conditions initial equivalent stress versus creep life for notched specimens along with a uniformly stressed bar specimen [29]. In an untempered bainitic structure, the extent of crack tip stress relaxation is limited and fig. 14a shows that creep life of the notched specimens cannot be predicted from that of plain bar specimens. In contrast, in the tempered state (fig. 14b), the steel is more ductile and stresses relax rapidly such that life of notched specimens can be predicted reasonably well from that of plain bar specimens. Equivalent stress thus provides a convenient parameter to define notch weakening or strengthening of materials under creep. The method of determining reference stress or equivalent stress has already been discussed earlier (table 2).

5.3. Crack growth in austenitic stainless steels Creep crack growth behavior in austenitic steels has been studied by several investigators and again stress intensity, net section stress, reference stress and J* parameters have been used to correlate the data. A limited correlation using a particular geometry was obtained in terms of stress intensity or net section stress. More extensive work involving different specimen geometries indicates that J* indeed correlates the data much better than other parameters [11-

317

22,3-33]. Fig. 15 shows the crack growth data in types 304 and 316 stainless steels at 593 and 650°C. Again the total spread in the data at each temperature is about a factor of 20. This is considerably less compared to the spread in terms of the other two parameters. In reality most of the spread in crack growth rates in terms of J* parameters arises during the initial crack growth period in each specimen wherein the crack length increment occurs as the creep damage is still forming far ahead of the crack tip. In other words, steady state growth condition were yet to be established during this initial crack growth period resulting in large spread in the data in terms of J* parameter. Depending on temperature and material creep properties, this initial period could be quite extensive to the extent that considerable crack length increment may occur before the steady state condition is established. The degree of spread in terms of J* could be considerable but this does not necessarily imply that K or O'ref are any better. There may be alternate parameters based on incremental plasticity theory but until these are developed, J* may be the best available parameter for creep crack growth.

5.4. Crack growth in nickel base alloys Creep crack growth behavior of a number of nickel base alloys has been studied but most of the work involves characterization of the growth behavior in terms of linear elastic parameters. Many of the alloys are high strength creep resistant alloys and several of them are very susceptible to environmental damage at the crack tip. Crack growth rates in terms of J* is shown for several alloys [34] in fig. 16 along with the data band for type 304 stainless steel at 650°C. Most of the data for the alloys pertain to one specimen geometry. If type 304 stainless steel behavior at 650°C is considered as representative of the purely creep deformation controlled process with crack growth at nearly steady state stress conditions, then higher growth rates in the other alloys can represent the various degrees of environmental contributions in each of the alloys. In fact, it was shown that crack growth rates in Alloy 718 and Inconel X-750 are more than two orders of magnitude lower in vacuum [35] with the result that when the growth is purely creep controlled the growth rates may fall within the data band. There was an attempt to measure creep crack growth rates in Inconel 600 and Hastelloy X at 540 and 650°C using compact tension specimens. At both

318

K. Sadananda / Crack propagation under creep and fatigue 102

I

I

I

TYPE 304 A N D 316 S T A I N L E S S STEELS

D A T A BAND AT 593°C ( S A D A N A N D A AND SHAHINIAN, SAXENA) ..3 .

101

100

E <

0¢v~9

DATA BAND AT 650°C (Ohji et al, Taira et al, Koterazawa and Mori)

< rr"

10-2

10-3

10-41 10 -1

1 1

I 10

1 102

I 103

104

J* (kJoules/m 2 hr) Fig. 15. Comparison of crack growth data for type 316 and 304 stainless steels at 593°C with those of type 304 stainless steel at

650°C (from ref. [11]). temperatures the specimens continued to deform significantly resulting in the formation of blunted cracks with very little crack growth [34]. The materials are too ductile to the extent that crack tip stress fields are significantly reduced by creep. For these cases reference stress could serve as a convenient parameter to predict life.

6. Combined cycle- and time-dependent crack growth With decreasing frequency and with superimposed

hold time, time dependent contributions will be superimposed on the cycle-dependent process and depending on material and test conditions synergistic interactions can occur which can either increase or decrease crack growth rates. Fig. 17 shows crack growth rates in C r - M o - V steel at 550°C in air and vacuum for different frequencies [36]. At higher AK values, crack growth rates in air increase with decrease in frequency. But with decreasing zlK, the frequency dependence appears to reverse. This is related to the fact that threshold stress intensity increases with decrease in frequency. These effects are absent in vacuum. The increase of threshold was

K. Sadananda / Crack propagation under creep and fatigue 102 -

~INCONEL X-750

/75o,c

/

/"\'9"" /~,~o~ _ / , , /

~\

/

/

/ ~s~o--- /:4.,,~1~.~0o~ ~,

= ~o-1

--"/

.'~ o \ ~ ? / / + ~ ' ~ /

i

",?7 /

/

~x/ /

I/ 10-41 10-2

/ 10-I

/

- - D A T A B A N D FOR 5 0 4 STAINLESS S T E E L

"-,e~--~

I/

.-

(65o0c) I I I 10 0 101 102 J* (KILOJOULES/m 2 hr)

103

Fig. 16. Creep crack growth rates in several nickel base alloys in relation to growth rates in type 304 stainless steel in terms of J* parameter (from ref. [34]). 102

I

I

I ' I '1

C r - Mo - V STEEL 550°C 10

(FROM SKELTON AND HAIGH) AIR

/

VACUUM - - - -

7

E

E 10

"I1-

O105 59 <

(10 2

10 Hz) /

10 6 1Hz

10 7

I 2

i

J

4

,

1

6

,

,11

8 10

240

40

_~K (MPa Vm)

Fig. 17. Effect of frequency on high temperature crack growth behavior in a low alloy steel in air and vacuum (from ref. [36]).

319

attributed to oxide bridging which decreases the COD and contributes to lower growth rates. Ellison and Walton [37] studied the crack growth behavior of the same alloy with hold time at peak load in addition to that under cyclic and static loads. With increase in hold time from 3 to 30 min, the crack growth rates change from cycle-dependent to time-dependent behavior, but the rates at a given hold period were higher than the linear summation under individual loads. Crack growth behavior under creep--fatigue conditions has been studied in Types 304 and 316 stainless steels in annealed, aged, cold-worked, and coldworked and aged conditions in the temperature range 540--650°C. While most of the studies were done in air, some were conducted in vacuum or liquefied sodium environment. James [1] has shown that crack growth rates increase with decreasing frequency at 540°C if frequency is less than 7 Hz. In contrast, he shows no hold time effect or, in fact, decreasing growth rates with hold time. The apparent inconsistency is because of the way he has defined hold time effect. Shahinian [38] and subsequently Michel and Smith [39] have shown that hold time increases the crack growth rates at 593°C. Cold work increases the hold time effect substantially while aging decreases the effect. There has been a question concerning whether these effects are due to environmental interactions or due to superimposed creep. James [40] based on his work at 540°C in vacuum at 3-7 Hz, concluded that temperature- and time-dependent contributions come predominantly from environmental effects. Fig. 18 shows that in cold worked stainless steel at 593°C, crack growth rates are the same in air and vacuum [41]. It is possible that environmental effects become important at low AK values and low growth rates. Realization that the time-dependent contribution for crack growth under creep-fatigue conditions in stainless steel comes predominantly from creep, particularly at high temperatures, and since K appears not to be a valid parameter for creep crack growth in stainless steels (at least in the annealed condition), attempts [42,43] have been made to express crack growth rates in terms of the J* integral. Fig. 19 shows recent results of Koterazawa and Mori showing crack growth rates at 550 and 650°C at 0.(133 and 0.0033 Hz using center-notched specimens but with variable thickness, width and initial crack lengths in terms of AK and J* parameters where J* was computed using load-displacement rate curves. Crack growth rates at high temperature and low frequency,

320

K. Sadananda / Crack propagation under creep and fatigue

1.0

1

'

I

'

I

'

I

3oc 1 / 10-1

EE

LOAD 13.3kN

p ,3o0 //,

nr "r" 10-2

VACUUM - AIR

0

r~ U

ZEROH O J 6 1

//o J]/!

10-3

_

(JAMES)

where the growth is time-dependent, fall on a narrow band when represented in terms of J* and do not in terms of AK. The rest of the data do not fall in the band due to the presence of the superimposed cycledependent process. Taira et al. [42] represented crack growth in Type 304 stainless steel at 650°C under cyclic load at low and high frequencies in terms of the J integral and showed that cycle- and time-dependent regimes are in two separate bands. It is expected that at intermediate frequencies the data should fall between these two bands although no such data have been presented. Fig. 20 shows the crack growth rates in Hastelloy X at 800°C in air as a function of AK and J* parameters [44]. Crack growth appears to be essentially time dependent and the growth rates correlate well with J* value. The extent of data available in other high temperature alloys in terms of different crack growth parameters is very limited. Analysis of crack growth behavior in Inconel 600 at 650°C indicates that while no crack growth was observed under static load due to excessive creep of the specimen, crack growth under continuous cycling and 1 min hold essentially is cycle dependent and correlate with both AK and ,~J parameters [45]. There are several instances wherein hold times instead of accelerating crack growth retarded growth and even contributed to complete crack arrest [46,47]. In all these cases, the stress intensity during hold is generally less than the threshold stress intensity for creep crack growth. Even if it does not result in crack growth, creep still occurs near the crack tip, contributing to stress relaxation sometimes to the extent that it affects the crack growth process during the cyclic load. Additional work involving different specimen geometries and temperatures for different high temperature materials is still required before it would be possible to state conditions under which fracture mechanics methodology can be used to predict crack growth rates under complex creep-fatigue conditions.

250(2

7.

Summary

and

conclusions

A brief review of fracture mechanics methodology as applied to crack growth under creep and fatigue has been made with an emphasis on high temperature

/ / /

10 4 10

2O

,

I

AK (MPa

40

V'-~)

I

60

,

1

80 100

Fig. 18. Hold time effects in air and vacuum for cold worked type 316 stainless steel. Note the absence of environmental contribution (from ref. [41]).

2x10

l

[

TYPE 304 STAINLESS STEEL 650°C 650~C 10

./ E F

l

/

/ /

/

/

/

? /

10 °

o.oo33~-o.o33.~

/ 550°C

:/J i

< n"1I-

o~ lo L9 <

/

~1

g 0 (650°C 0.033

BAND FOR

/~'-~(650cC 0.0033 Hz)

\ °c

10 10

_

1

12

I

I

15 20 ,~K (MPa \ m)

lO

I

25

30

35

2

1

lO 1

10

I 0

102

101

10

J* (kJ/m2hr)

Fig. 19. Crack growth rates in type 304 stainless steel in terms of AK and J* under creep-fatigue conditions. Different symbols correspond to different specimen sizes and loads (from ref. [42]).

- Haste.'lloy X 800°C in air - Triangular strain LCF _---t~in(%) "~in(%/min) 2.0 0.2 1.0 2 1.0 0.1 1.0 0.2 0.5 1

._.R E --~ cJ >

E E +~ "O

E z"

--~ ~= 10 -1

10-

"O m

O

tO

~

-

0.2

E

0.1

T

I

to

0.4

~ 10-2

_-

/:/

/

#

/

0.2/

:

/.~, o ¢~

-

£

10-:

-

,

•

Q

10-1

1

Creep J-integral

10 J, kgf/mm-min

10-"

20

50

100

20

50

100

Fig. 20. Crack propagation rate versus stress intensity factor range (left), and propagation rate versus creep J integral (right) in cyclic creep of the Hastelloy X at 800°C (creep J integral is the same as J* integral) (from ref. [43]).

200

Stress intensity f a c t o r range AK, k g f / m m 3/2

321

322

K. Sadananda / Crack propagation under creep and fatigue

structural materials. Crack growth is characterized as cycle-dependent, time-dependent or combinations of both depending on material, test temperature and loading conditions. Correlation of crack growth rates in terms of linear elastic and elastic-plastic fracture mechanics parameters was discussed along with the methods of measuring these parameters. It is shown that elastic-plastic parameter A./ integral is particularly useful in correlating crack growth data not only when plasticity effects b e c o m e important but also for conditions where slow t i m e - t e m p e r a t u r e dependent transformations occur that alter the material flow properties during service. Similarly, the J * integral is useful to characterize creep crack growth where the growth occurs under steady state stress conditions. U n d e r combined loading conditions AJ or ./* could be used depending on whether fatigue or creep contributes most to crack growth. When plasticity effects are localized or when environmental effects are dominating the linear elastic fracture mechanics parameters J K and K for cyclic and static loads respectively can be used safely to correlate crack growth rates.

Acknowledgements It is a pleasure to acknowledge Dr P. Shahinian for his stimulating discussions on high temperature crack growth behavior and for his critical review of the manuscript. T h e present research effort was supported by the Office of Naval Research.

References [1] L.A. James, Atomic Energy Review 14 (1976) 37-85. [2] P. Shahinian, Metals Technology 5 (1978) 372-80. [3] J.R. Rice, in Fracture: An Advanced Treatise, vol. ll, ed. H. Liebowitz (Academic Press, New York, 1968) pp. 192-314. [4] J.A. Begley and J.D. Landes, Fracture Toughness, ASTM-STP 514, (American Society for Testing and Materials, Philadelphia, 1972) pp. 1-2(I. [5] N.E. Dowling and J.A. Begley, Mechanics of Crack Growth, ASTM STP 590 (American Society for Testing and Materials, Philadelphia, 1976) pp. 19-32. [6] K. Sadananda and P. Shafiinian, Engrg. Fracture Mechanics 11 (1979) 73-86. [7] J.G. Merkle and H.T. Corten, J. Pressure Vessel Technology, Trans. ASME 96 (1976) 286-292. [8] K. Sadananda and P. Shahinian, Met. Trans. A, 8A (1977) 439-449.

[9] J.D. Landes and J.A. Begley, Mechanics of Crack Growth, ASTM STP 590 (American Society for Testing and Materials, Philadelphia, 1976) pp. 128-48. [10] M.P. Harper and E.G. Ellison, J. Strain Analysis 12 (1977) 167-179. [11] K. Sadananda and P. Shahinian, Fracture Mechanics: Fourteenth Symp. Vol. II: Testing and Applications, ASTM STP 791 (American Society for Testing and Materials, Philadelphia, 1981) pp. II 182-196. [12] A. Saxena, Fracture Mechanics: Twelfth Conf., ASTM STP 700 (American Society for Testing and Materials, Philadelphia, 1980) pp. 131-151. [13] H. Riedel and J.R. Rice, Fracture Mechanics: Twelfth Conf., ASTM STP 700 (American Society for Testing and Materials, Philadelphia, 1980) pp. 112-130. [14] E.W. Hart, Stability of Crack Extension Rates in Ductile Materials in Micro and Macro Mechanics of Crack Growth, ed. K. Sadananda et al. (The Metallurgical Society of A1ME, Warrendale) pp. 97-1l)6. [15] C.Y. Hui and H. Riedel, Int. J. Fracture 17 (1981)) 4(19-425. [16] J.R. Haigh and C.E. Richards, in: Creep and Fatigue in Elevated Temperature Applications (American Society of Mechanical Engineers, New York, 1973) paper 159/73. [17] A.C. MacKenzie, lnt. J. Mech. Sci. 10 (1968) 441-453. [18] R.G. Anderson, L.R.T. Gardner and W.R. Hodgkins, J. Mech. Eng. Sci. 5 (1962) 238-244. [19] J.A. Williams and A.T. Price, J. Eng. Mat. Technol., Trans. ASME 97 (1975) 214-221. [20] M.O. Speidel, in: High Temperature Materials in Gas Turbines, eds. P.R. Sahm and M.O. Speidel (Elsevier Scientific, Amsterdam, 1974) pp. 207-52. [21] K. Sadananda and P. Shahinian, Met. Trans. A, 12A (1981) 343-351. [22] P. Shahinian, H.E. Smith and H.E. Watson, in: Fatigue at Elevated Temperatures, ASTM STP 520 (American Society for Testing and Materials, Philadelphia, 1973) pp, 387-398. [23] K. Sadananda and P. Shahinian, Int. J. Fracture 15 (1979) R81-83. [24] C. Laird, in: Fatigue Crack Propagation, ASTM STP 415 (American Society for Testing and Materials, Philadelphia, 1967) p. 131. [25] H.W. Liu, C.Y. Young and A.S. Kuo in: Fracture Mechanics, eds. N. Perrone et al. (University Press of Virginia, Charlottesville, 1978) p. 629. [26] J. Weertman, in: Fatigue and Microstructure (American Society for Metals, Columbus, OH, 1978) p. 279. [27] E.M. Christian, D.J. Smith, G.A. Webster and E.G. EIlison, in: Advances in Fracture Research, Vol. 3, ed. D. Francois, 5th Intern. Conf. on Fracture (Pergamon Press, Oxford, 1981) p. 1295. [28] R. Pilkington, D.A, Miller and D. Worswick, Met. Trans. A, 12A (1981) 173-181. [29] C.J. Neate, Mat. Sci. Eng. 33 (1978) 165-173.

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