Astm stp 995 v1 1989

Creep Crack Growth The papers on creep crack growth deal with the issues of crack growth under small-scale creep conditions, the usefulness of the recently proposed C, parameter, the ap

Nonlinear fracture mechanics/A Saxena, J D Landes, and J L Bassani, editors (STP ;995)

Papers presented at the Third International Symposium on Nonlinear Fracture Mechanics, held 6-8 Oct 1986 in Knoxville, Tenn., and sponsored by ASTM Committee E-24 on Fracture Testing

"ASTM publication code number (PCN) 04-995001-30."

Includes bibliographies and indexes

Contents: v 1 Time-dependent fracture

ISBN 0-8031-1174-6

1 Fracture mechanics Congresses I Saxena, A (Ashok) II Landes, J D (John D.) III Bassani, J L (John L.) IV International Symposium on Nonlinear Fracture Mechanics (3rd : 1986 : Knoxville, Tenn.) V ASTM Committee E-24 on Fracture Testing

VI Series: ASTM special technical publication ; 995

Peer Review Policy

Each paper published in this volume was evaluated by three peer reviewers The authors addressed all of the reviewers' comments to the satisfaction of both the technical editor(s) and the ASTM Committee on Publications

The quality of the papers in this publication reflects not only the obvious efforts of the authors and the technical editor(s), but also the work of these peer reviewers The ASTM Committee on Publications acknowledges with appreciation their dedication and contribution

of time and effort on behalf of ASTM

Printed in Ann Arbor, MI February 1989

Foreword

This publication, Nonlinear Fracture Mechanics: Volume I Time-Dependent Fracture,

contains papers presented at the Third International Symposium on Nonlinear Fracture

Mechanics, which was held 6-8 Oct 1986 in Knoxville Tennessee ASTM Committee E-24

on Fracture Testing sponsored the event The cochairmen for the symposium section on

Time-Dependent Fracture were A Saxena, Georgia Institute of Technology, and J L

Bassani, University of Pennsylvania Both men, along with J D Landes, University of

Tennessee, served as editors of this publication

ROLAND PIQUES, AND ANDRE PINEAU

A Numerical Study of Non-Steady-State Creep at Stationary Crack Tips

CHUN-POK LEUNG, DAVID L MCDOWELL, AND ASHOK SAXENA

Crack Growth in Small-Scale C r e e p - - J O H N L BASSANI, DONALD E HAWK, AND

FWU-HWEI WU

Growth of Macroscopic Cracks by Void Coalescence Under Extensive Creeping

C o n d i t i o n s - - C H U N G - Y U E N HUI AND KUANG-CHONG WU

Creep Crack Growth of Alloy 800H in Controlled-Impurity Helium

JUDE R FOULDS

Low-Alloy Steels: An Assessment of the Effects of Residual Impurity

Elements and Postweld Heat Treatment Condition on Creep Ductility and

Crack Growth sH[NJI KONOSU AND KEIKICH[ MAEDA

Influence of Aging on High-Temperature Creep Crack Growth in Type 304H

Stainless Steel G M BUCHHEIM, C BECHT, K M NIKBIN, V DIMOPOLOS,

G A WEBSTER~ AND D ft SMITH

An Anisotropic, Damage-Coupled Viscoplastic Model for Creep-Dominated Cyclic Loading DAVID L MCDOWELL, KWANG-IL HO, AND JAMES STALLEY

Experimental Determination of the High-Temperature Crack Growth Behavior of lncoloy 8 0 0 H - - T H O M A S HOLLSTEIN AND BERT VOSS

Three-Dimensional Transient Analysis of a Dynamically Loaded

An Experimental Study of the Validity of a Delta J Criterion for Fatigue

Crack Growth DAVID A JABLONSKI

Combined-Mode Low-Cycle Fatigue Crack Growth Under Torsional Loading

R O Y A W I L L I A M S A N D W E L D O N W W I L K E N I N G

Fatigue Crack-Tip Mechanics in 7075-T6 Aluminum Alloy from High-Sensitivity

Dislocation-Free Zone Model of Fracture Under Reverse Loading

Author Index

Subject Index

475

477

STP995-EB/Jan 1988

OVERVIEW

Elastic-Plastic Fracture Mechanics (EPFM) had its birth in the late 1960s and early 1970s

In nearly two decades of growing effort, the field has seen a maturing trend as well as a change in emphasis EPFM developed in response to a real technology need: the parent technology, linear elastic fracture mechanics (LEFM), did not apply to many of the engi- neering materials used in modern structures New and better materials were developed to attain more ductility and higher fracture toughness Where LEFM could no longer be used for analyzing failures in these materials, EPFM provided the solution To organize and document the results of the growing research effort in the field, ASTM Committee E-24 on Fracture Testing sponsored the First International Elastic-Plastic Fracture Symposium in Atlanta, Georgia, in 1977 The bulk of this symposium, as peer-reviewed papers, is published

on this subject was held in Philadelphia in 1981, which resulted in the two-volume A S T M STP 803, Elastic-Plastic Fracture: Second Symposium

The 1980s saw a rise in more general interest in nonlinear fracture mechanics topics, particularly time-dependent fracture mechanics Therefore, the title for the next symposium was modified to include this emerging field As a result, the Third International Symposium

on Nonlinear Fracture Mechanics was held in Knoxville, Tennessee, in 1986 This sympos- ium, sponsored by ASTM Committee E-24 and its Subcommittee E24.08 on Elastic-Plastic and Fully Plastic Fracture Mechanics Technology, featured both time-dependent and elastic- plastic topics in fracture mechanics The time-dependent fracture mechanics papers are published in Volume I (this volume) of this Special Technical Publication (ASTM STP 995)

Volume II features the elastic-plastic contributions to the symposium

In the mid-1970s, when consensus in the approaches to elastic-plastic fracture was emerg- ing, the attention of some researchers shifted to elevated-temperature crack growth behavior The motivation for this work came primarily from projects active at the time, and was directed toward building commercial advanced nuclear reactors, improving energy conver- sion efficiencies of conventional power plants and jet engines, exploring the feasibility of alternate energy sources such as coal gasification, and understanding failures in major equip- ment, such as Tennessee Valley Authority's Gallatin steam turbine rotor New concepts which could adequately account for the presence of time-dependent creep strains in cracked body analysis were needed for integrity assessment and prevention of failures in these components A creep analog to the J-integral called C* was proposed in 1974, which over time has proven to be the first major breakthrough in the development of time-dependent fracture mechanics (TDFM) In its range of applicability, C* is now a well-accepted crack- tip parameter

At the time of the second elastic-plastic fracture symposium in 1981, it was becoming clear that the application of C* is limited to cracked bodies undergoing dominantly sec-

ondary-stage creep Researchers were engaged in understanding the limitations of C* and

also in extending the concept into the small-scale creep (SSC) regime, where a good portion

of the practical problems lie Only single session was devoted to papers on this subject at

the second symposium In the third symposium, TDFM was one of the prominent themes

and several sessions were organized on the subject The papers from these sessions are

included in the first section of this volume

Creep Crack Growth

The papers on creep crack growth deal with the issues of crack growth under small-scale

creep conditions, the usefulness of the recently proposed C, parameter, the applicability of

damage mechanics concepts in understanding micromechanics and micromechanisms of

creep crack growth, embrittlement due to aging in service and its influence on creep crack

growth behavior, and experimental methods While significant progress has occurred since

the last symposium on this topic, a lot more remains to be done Some issues not addressed

in the papers at the symposium include the influence of cyclic loading and inclusion of creep

deformation other than that represented by power-law creep These are areas of current

research Also, further evaluation of the C, parameter is likely to continue until a consensus

can be reached, and stable and unstable crack growth and fracture at elevated temperature

should be addressed Therefore, a good number of problems still remain unresolved in this

area Although some of the original reasons for developing T D F M are no longer the primary

driving force, the field has found considerable use in remaining life assessment of fossil

power-plant components and will be useful in the development of advanced aircraft Hence,

this area is expected to be represented in future symposia on nonlinear fracture mechanics

Dynamic Fracture

The second section of this volume is devoted to dynamic fracture This is also one of the

newer areas of research in fracture mechanics The papers in this section deal with the issue

of calculating the crack driving force, with proper emphasis on inertial effects and the

measurement of fracture toughness under conditions of high rate loading This area continues

to be of significant interest to the nuclear power industry and the U.S Navy

Cyclic Loading

The papers in the section on cyclic loading are concerned with experimental evaluation

of AJ for characterizing fatigue crack growth behavior under gross plasticity conditions and

with cracking under mixed-mode loading Crack-tip mechanics under cyclic loading was

studied by measuring displacements, using optical interferometry Damage accumulation in

the form of dislocation motion at the crack-tip field was modeled in another paper

Fracture of Nonmetals

The final section of the book is devoted to papers based on exploratory work in the area

of fracture in nonmetallic materials, such as polymers and ceramics This is an emerging

field in which there is considerable need for new ideas

On glancing over the Third International Symposium on Nonlinear Fracture Mechanics

and Volume I of this Special Technical Publication, it is c/ear that very significant progress

has occurred in the field of TDFM The field is not very far along in its readiness for

OVERVIEW 3

applications when we compare its recent progress to the status of its parent technologies,

L E F M and EPFM The concepts are based on sound principles which should ensure their

widespread acceptance and usage in the future The same is true for the status of dynamic

fracture mechanics In the area of cyclic loading, the h J p a r a m e t e r has survived ten years

of criticism, and it appears that the theory behind its success in correlating experimental

data is becoming increasingly understood Fracture mechanics of new materials such as

polymers, ceramics, and composites are fields in which considerable interest is expected in

the near future

A Saxena

Georgia Institute of Technology, Atlanta, GA 30332; symposium cochairman and editor

J D Landes

University of Tennessee, Knoxville, TN 37996;

symposium cochairman and editor

J L Bassani

University of Pennsylvania, Philadelphia, PA 19104; symposium cochairman and editor

Creep Crack Growth

John L Bassani,1 Donald E Hawk, 2 and A s h o k Saxena 3

Characterizing Creep Crack Growth Rate the Transient Regime

in

Mechanics: Volume l Time-Dependent Fracture, ASTM STP 995, A Saxena, J D Landes,

and J L Bassani, Eds., American Society for Testing and Materials, Philadelphia, 1989, pp 7-26

characterizing the creep crack growth behavior under small-scale-creep (transient) to extensive creep conditions, is conducted The evaluation includes a thorough examination of the as- sumptions made in the definition of C,, an experimental evaluation, and a finite-element analysis of a compact specimen used in an actual creep crack growth experiment

The experimental correlations between da/dt and C, are extremely encouraging C, is in-

terpreted as the stress-power (or energy rate) release rate during crack extension Under extensive creep conditions, C, therefore equals C*, which is given in terms of a path-inde- pendent integral as well as the stress-power release rate Under small-scale creep, C, is shown

to characterize the rate of growth of the creep zone More finite-element analyses and ex- periments under highly transient conditions are recommended for completing the understand- ing of the creep crack growth behavior in this regime

KEY WORDS: creep, crack, transient regime C~ parameter, Cr-Mo-V steel, fracture me- chanics, nonlinear fracture mechanics

Creep crack growth is an important concern in the design of steam turbines, boilers, and steam pipes for power plants, advanced nuclear reactor components, and aircraft engine components Most elevated temperature components in these machines are large and are designed to resist extensive (or widespread) creep deformation However, significant lo- calized creep deformation that is constrained by surrounding elastic material tends to occur

in the vicinity of stress raisers such as preexisting or service-initiated cracks (or crack-like defects) This constrained crack-tip creep gives rise to a strong time-dependence in the crack-

tip fields and, therefore, in the crack growth behavior [1-6] Even though the creep defor-

mation is small scale, it must be properly taken into account in a sound methodology for predicting creep crack growth

On the other hand, laboratory specimens for measuring creep crack growth rates are small

1 Associate professor, Department of Mechanical Engineering and Applied Mechanics, University

and generally undergo extensive creep during most of the test Therefore, to simulate exactly

the conditions experienced by the components in service, it will be necessary to test full-

scale models for test durations equaling the service time (40 years for electrical power

generation equipment) Since such a test is impractical, there is a real need for identifying

a crack-tip parameter that can characterize creep crack growth behavior under a wide range

of creep conditions so that, based on the data developed in the laboratory under extensive

creep conditions, the behavior of the components can be predicted Furthermore, test

specimens often do not lie entirely in the extensive creep regime, and such data needs to

be meaningfully represented

Recently, a parameter C, has been proposed for characterizing the creep crack growth

behavior under wide-range creep conditions including the transient condition of small-scale

creep [1] Under extensive creep conditions, the C*-integral [2, 4, 7-9] is now widely accepted

as the crack-tip parameter for characterizing creep crack growth, and it can be shown that

C, reduces to C* in this regime

In this paper, we will concentrate on the methodology for characterizing creep crack

growth in the small-scale-creep regime and in the transition regime between small-scale and

extensive creep conditions We first carefully examine the assumptions made in the definition

of C, in an effort to understand better its mechanics basis and also its limitations In the

small-scale-creep regime it is shown that C, is proportional to the rate of growth of the crack-

tip creep zone, which is not uniquely related to an instantaneous measure of the intensity

of the crack-tip stress or strain fields Subsequently, we present some experimental results

showing the correlations between creep crack growth rate and C, We also present detailed

finite-element analyses results for both stationary and growing cracks in a compact specimen

At present, the methodology based upon C, has been developed only for power-law sec-

ondary creep

In the next section, the crack-tip fields for a stationary crack in an elastic, power-law

creeping material are reviewed

Crack-Tip Stress Fields

For brevity we will consider only sharp, Mode I cracks in solids that deform both elastically

and by power-law (secondary) creep Results analogous to those summarized in this section

for stationary cracks that include the effects of primary creep (strain hardening) and recovery

have been given by Riedel [10] This investigation was based upon extensions of the power-

law constitutive relations

Total strains are taken as the sum of the elastic and creep strains: 9 = 9 + 9 For

isotropic linear elasticity, the strains are given in the usual form in terms of the stresses,

Young's modulus, E, and Poisson's ratio, v The isotropic, incompressible power-law creep

relation is given in terms of the constant material parameters ~o, G0, and n as

\o'o/

where s~ i = crj~ - ~ q ( ~ k k / 3 ) is the stress deviator and ~ = [(3s~is~j)12] "2 is the Mises effective

stress With g = [(24ij4ij)/3] 1/2, Eq 1 implies that ~ = G0r/cr0)" In the power-law form G

and % are not parameters to be chosen independently; often A - G/%" is specified instead

With ~0 and or0 kept separate, dimensional consistency is more easily seen in the results that

follow

As depicted in Fig 1, the crack is assumed to be two-dimensional, lying in the x z = 0

plane with r and 0 denoting the crack-tip polar coordinates, and 0 = 0 ahead of the crack

BASSANI ET AL ON THE C, PARAMETER P,V

C r e o n e

FIG 1 Mode I crack

along the x 2 = 0 plane Mode I or tensile loading conditions are emphasized in the discussion

below

Stationary Cracks

The plane-strain Mode I crack under constant applied load has been investigated by Riedel

and Rice [2], Bassani and McClintock [4], and Ehlers and Riedel [11] Ohji, Ogura, and

Kubo [3] have considered cracks under load variations of the form P(t) = Pot% where t

denotes time, and P0 and "y are constants Recently, Riedel [12] has also considered cyclic

loading

For instantaneously applied loads (or sudden load changes), the material responds elast-

ically, since creep deformation takes time, and the crack-tip stress field (r ~ 0) is the well-

known elastic one [2,4]

gl

V A~r

where KI, the elastic stress-intensity factor, and f0 are known nondimensional functions of

0 For most metals undergoing power-law creep n > 3 in Eq 1, and this nonlinear stress

dependence causes the creep strain rates to dominate the elastic ones after any sudden load

changes in the high-stress crack-tip region The form of the crack-tip stresses then resembles

the Hutchinson, Rice, and Rosengren (HRR) fields in elastic-plastic fracture mechanics [11-

15], in terms of known nondimensional functions 6ij(0;n), which are normalized with the

maximum over 0 of (3/2)6%Gj = 1, and I,, which is in the range 3.8 to 6.3 for plane strain,

specimen geometry, crack length, material properties, and time The size of the region

around the crack tip where this field dominates is discussed below

If a suddenly applied load is thereafter held constant, then the crack-tip stresses relax as

creep strains build up a r o u n d t h e crack tip As long as small-scale creep (SSC) conditions

persist (that is, the region where creep strains are greater than elastic strains is very small

compared to crack length), the amplitude factor is approximately

( 1 - v g K i ~

This defines the short-time behavior For long times under constant applied load when

extensive creep occurs everywhere in the specimen, C approaches the constant, steady-state

value C* This steady-state value is given in terms of a path-independent integral [7] The

transition time tr between small-scale creep and extensive creep is approximated by setting

Eq 4 equal to C* [2,4]

tr (n + 1)EC*

The values of K~ and C* for various creep exponent n and crack sizes are tabulated for

several specimen geometries and loadings in Refs 17 and 18, respectively Riedel [11] has

suggested an interpolation formula for C(t) between SSC and extensive creep

In SSC the creep zone is defined as the region around the crack tip where the creep strains

exceed the elastic strains [2] A measure of the creep zone size is denoted re, as depicted

in Fig 1 The creep zone tends to grow in time, and its radial extent around the crack tip

as a function of KI, t, and 0 is

= 1 (K,~z F (n + 1)~0l, t ~:'(" ')L(O) (7)

2 0%

where L is a nondimensi0nal function that can be approximated in terms of the flj(0) and

6-0(0;n ) functions in Eqs 2 and 3, as plotted in Ref2 Otherwise, r~(O,t) must be determined

numerically In what follows, for plane-strain conditions, since r~ is roughly maximum at 0

= 90 ~ we will refer to r,.(t) as re(0 = 90~ At 0 = 90 ~ i,.(0) = 0.2 to 0.5 depending on

n; from our finite-element calculation for n = 10 t;c(0) = 0.4

Fracture Parameters Associated with Crack-Tip Stress Fields

Under predominately elastic conditions that is, when the size of the creep zone is smaller

than the fracture process z o n e - - t h e stress-intensity factor KI can be used to correlate crack

growth This correlation, which is consistent with Eq 2, has been observed in laboratory

tests on nickel-base alloys that are susceptible to oxidation [19]

At the other extreme, in behavior under extensive creep conditions (t >> &), a large body

of experimental data (see, for example, Refs 1,5, 7, 8, and 9) supports a correlation between

BASSANI ET AL ON THE C, PARAMETER 1 1

the crack velocity and C* The crack-tip field summarized above is consistent with this

correlation since the amplitudes of the crack-tip stress, strain rate, and strain fields are given

in terms of C* = C(t >> tr) in Eq 3 In the extensive creep regime, C* can be either (a)

directly measured using load and deflection rate measurements as in Ref 1

t'9

= ~]BW

where P and 9 are the load and load-line displacement rate (see Fig 1), B and W are the

specimen thickness and width, and -q is a geometry factor; or (b) calculated from the Electric

Power Research Institute (EPRI) handbook [18] from measured load and geometry data

where

Since 17 ~ P" under extensive creep conditions, Eqs 8a and 8b are consistent For the

compact type specimen, -q in Eq 8a is defined later in Eq 23, and the full form of Eq 8b is

given in Eq 26

Under small-scale creep conditions (t < tr) and in the transition regime (t ~- tr), it is not

obvious what the fracture parameter should be (Recall that Eqs 8a and 8b only are defined

for the extensive creep regime where ~c >> ~e everywhere in the specimen.) For example,

the amplitude of the crack-tip stress and strain-rate fields in the SSC regime are powers of

K2/t ~ C(t), so that these fields relax with time under constant load Correlations with C(t)

have been explored but have been unsuccessful In what follows, the C, parameter is intro-

duced to correlate crack growth in the regime where t ~< tT In this regime C, ~ K~i'c, where

the size of the creep zone rc is given in Eq 7 The complete definition of C, reduces to C*

in the limit as t >> tr, otherwise Ct/> C*, which reflects relaxation effects for t < tr An

expression for C, that is analogous to Eq 8a is also given

C, Parameter

The definition of C, for t ~< tr is based upon a partitioning of the load-line deflection, V,

into a purely instantaneous part, V~, and a part due to the growth of the crack-tip creep

zone, Vc It will be shown that C, is an extension of the C*-integral into the transient creep

regime via its stress-power (or energy rate) release rate interpretation [1] The growth of

the creep zone, and therefore Vc, is intimately connected with the elastic strain rates that

arise due to crack-tip stress relaxation as well as the creep strain rates Since Ct is defined

with the assumption that Eq 1 describes the creep behavior, its use at the present time is

restricted to those materials in which creep deformation is dominated by power-law sec-

ondary creep

To define the partitioning of load-line displacements in small-scale creep, imagine a spec-

imen (Fig 1) with a crack of length, a, subjected to load P The total load-line deflection

is expressed as

V = L + V ~ where Ve(P,a) is the total deflection of an identical specimen that undergoes only the

instantaneous elastic deformation, and V~ is the remaining deflection which accumulates with

time for the actual specimen undergoing both elastic and creep deformation The latter elastic

deformation is caused by the redistribution of stresses at the crack tip It is important to

note that Ve and Ve are not compatible with the elastic and creep strains, respectively, of

the specimen In fact, since under SSC, Vc is due to the growth of the crack-tip creep zone,

it is intimately connected with the crack-tip stress relaxation and the associated elastic strain

rates as well It may be recalled that in the time-independent elastic-plastic case, Edmonds

and Willis [20] have demonstrated that this partitioning is asymptotically exact (see Hutch-

inson [21] for an analogous discussion based upon the Dugdale solution) Also, note that

for a stationary crack under constant load, 1? = f'c

Analogous to the strain energy release rate interpretation of J in the elastic-plastic regime

release rate) Following Saxena [1], consider several identical pairs of cracked specimens

For each pair, one specimen has a crack length a and the other has crack length a + Aa

The specimens of each pair are loaded to various constant load levels PI, P2, 9 9 , Pi at

elevated temperature It is assumed that no crack extension occurs in any of the specimens,

and the instantaneous response is linear elastic with load-line deflection V~, as in the problem

formulated by Riedel and Rice [2], Bassani and McClintock [4], and Ohji et al [3] As

creep deformation progresses in the various specimens, additional deflection accumulates

at the load-line, Vc, due to the growth of the creep zone, as shown in Fig 2a Part of this

time-dependent deflection can be attributed to secondary creep strain, e~, and part to the

change in elastic strain, e~, due to stress relaxation within the body In any case, the rate

at which the creep zone grows,/'c, within the relaxing crack-tip stress field should be directly

related to the rate of deflection at the load-line, Vc This is the basis for the definition of

the Ct parameter

In Fig 2b the instantaneous value of 1?~ is plotted at time, t, as a function of load, P, for

specimens with crack length a and for specimens with crack length a + Aa For other times,

similar plots of P versus Vc can be constructed The area between the curves corresponding

to crack lengths a and a + Aa is designated AU,* The subscript t denotes the value for a

fixed time C, is then defined as

where B is the specimen thickness Under extensive creep conditions, C, =- C* by definition

because A U* assumes a steady-state value h U*

Since the creep zone is a relatively soft region surrounding the crack tip, the additional

deflection due to the presence of the creep zone can be estimated using Irwin's concept of

effective crack length, ar [21]

where a is the physical crack length, rc is the creep zone size given in Eq 7, and 13 is a scaling

parameter With rc =- r~(0 = 90~ the value of 13 = 1/3 has been determined by finite-

element analysis as discussed in a later section

The additional deflection due to the creep zone for a constant load is given by

d C

In the small-scale creep where C = V e / P is the elastic compliance In the small-scale-creep

regime, C,, as defined in Eq 9, can be expressed in terms of Vc by making use of the

BASSANI ET AL ON THE C, PARAMETER 13

FIG 2 (a) Load-line deflection V~ is a function of time for bodies of crack lengths a and

a + Aa at various load levels (b) The definiton of the Ct parameter

stationary crack-tip fields as [1]

where the geometric factors F and F ' are defined as

Although Eq 8a for C* and Eq 12 for (C,)~r are similar, we note that the geometry factor

~q is determined from extensive creep (nonlinear) behavior, while F ' / F is determined from

purely elastic behavior As Saxena noted [1], -q and F ' / F c a n be roughly the same magnitude

for compact type specimens, so that measured values of (Ct)~sc and C* may not be too

different in these specimens The important distinction between the two parameters arises when they are predicted using handbook values of K~ and C*

Another expression for (C,)s~c can be obtained by substituting Eq 11 into Eq 12 and using the well-known result dC/da = (1 - vz)(2B/E)(K/P) z to give

between the experiments and analysis most likely results from primary creep deformation which was not considered in the analysis

In SSC, C, is directly related to the rate of growth of the crack-tip creep zone as it evolves with time (Eq 14) Furthermore, note from Eq 15 that

Recall that setting C(t) in Eq 4 equal to C* yields an estimate for the transition time tr

given in Eq 5 Similarly, C, in Eq 14 can be equated to C* to give another estimate of the transition time t, which must closely approximate tr Therefore, with i substituted for tr,

an interpolation formula for C, from small-scale to extensive creep that is similar to Eq 6

(18)

BASSANI ET AL ON THE C~ PARAMETER 15

which is in good agreement with the finite-element results presented later in Figs 9 and 10 Recall that the short-time behavior in Eq 18, or more exactly Eq 15, is based on the short- time growth of the crack-fip creep zone that is predicted by the Riedel-Rice analysis [2] From the experimental results discussed below, C, is interpolated between the small-scale creep form of Eq 12 and the extensive creep limit C* using Eq 8a as [23]

Experimental Verification of C,

Recently, creep crack growth experiments on ASTM Grade A470 Class 8 (1Cr-lMo- 0.25V) steam turbine rotor steels were performed under the auspices of the ASTM coop- erative test program on creep crack growth testing [24] The pertinent details of the test procedure are described elsewhere [1, 7,24] The steady-state creep deformation behavior

is shown in Fig 3 at 593~ (]100~ and 538~ (1000~ the two test temperatures used in this study Data are presented from six tests conducted at 593~ (ll00~ using 1T compact- type specimens Four specimens were 6.3 mm (0.25 in.) thick and did not have side grooves

The other two specimens were nominally 25.4 mm (1.0 in.) thick and had side grooves which were approximately 3.1 mm deep on each face The following expressions were used with

Eq 19 for estimating C, in compact-type specimens [24]

Recall that C* is defined only for extensive creep (or steady-state) conditions when t ,> tr,

and also that C* ~ PI)'~ is given in Eq 8a with -q for the compact specimen given in Eq 23 Under a wide range of conditions, C, is given in terms of Eqs 18 and 19, both of which reduce to C* for t ,> tr Experimental values of C, are most readily given by Eq 19 (or Eq

12 in SSC), while predicted values are most readily given by Eq 18 with Eq 5 (or Eq 15 in SSC)

Figure 4 shows a plot of da/dt versus the plastically corrected value of K ( = V/-E-J) for the six creep crack growth tests The correlation between da/dt and K is not good in these tests Next, da/dt is plotted in Fig 5 as a function C* calculated from Eq 26 Similar lack of correlation between da/dt and C* is observed In this correlation, C* is calculated from Eq

26 since it cannot be assured that steady-state conditions existed throughout the test, and the ~ measured is in fact the steady-state value In Fig 6, the same data are plotted with

C, calculated from Eq 19, and all the data from the six tests collapse in a single trend There may be a tendency for the crack growth rates from the side-grooved specimens to be marginally higher than the 6.3-mm (0.25-in.)-thick specimens without side grooves This could be related to the differences in the states of stress between the two specimen types Further testing should be done to resolve this difference

The above data clearly support the validity of the C, parameter This correlation is also

BASSANI ET AL ON THE C, PARAMETER 17

independent of the extent of creep deformation because Fig 6 includes data obtained under

essentially steady-state conditions and also in the transient region, as discussed in the next

paragraph Similar correlation between da/dt and C, was also shown previously in separate

studies [1,24]

In order to estimate the extent of SSC or transient conditions which exsited during the

various tests, the following parameter, ~, was calculated

C*

when steady-state conditions are reached, C, = C*, and 9 approaches 1.0 When highly

transient conditions exist, r ~ 1 The value of T at the beginning and end of the tests are

listed in Table 1 The Riedel-Rice transition time, tr, from Eq 5 and the test duration, to,

is also listed for comparison Comparing the values of r and toltr in Table 1 can lead to

somewhat contradicting results The values of T indicate that the data presented in Figs 5

and 6 include several points which were generated under transient conditions, as well as

those which were generated under steady-state conditions The values of to/tr indicated that

all tests were under extensive creep conditions If the latter were true, da/dt should correlate

with C*, which was not observed in Fig 5 Therefore, r appears to be a better parameter

than tr for growing crack tests for determining the extent of creep deformation in the test

specimens It was also observed that the tests began under more extensive creep conditions,

which is in agreement with the short transition times, but as the tests progressed, transient

+ v

v

g

% + V O 0

i 0 -6

i0-"

.0-1

conditions developed in the specimens

Since C, is a stationary crack parameter, it is of interest to know to what extent the crack growth affects the determination of C, Because C, is linked with the growth of the creep zone, the pertinent question should be how the growing crack affects the creep zone size

as given by E q 7 When deflection rates are measured, such as in a test, the influence of the growing crack on the extent of the creep zone is reflected in the measurement Therefore,

TABLE 1 - - T h e values o f T and transition time during the various creep crack growth tests

BASSANI ET AL ON THE Ct PARAMETER 19

v = CT-21 10- z + = CT-22

the growing crack effects are already included in the calculation of C, during experiments

This is, h o w e v e r , n o t t h e case w h e n E q s 15 or 18 with E q 5 are used in applications This

will be discussed further in the section on finite-element analysis

F i n i t e - E l e m e n t A n a l y s i s

In this section, the results of a f i n i t e - e l e m e n t analysis of an actual c o m p a c t specimen used

in a c r e e p crack growth test are described T h e c o m p a c t specimen analyzed is a 25.4-ram-

thick and 50.8-mm-wide s p e c i m e n f r o m a ll/4Cr-V2Mo boiler pressure vessel steel T h e test

t e m p e r a t u r e was 538~ (1000~ T h e deflection versus time data f r o m the test are shown

in Fig 7 T h e applied l o a d was 15.35 k N (3450 lb) and the material p r o p e r t i e s used in t h e

analysis are as follows [23]:

w h e r e (r0 is the 0 2 % yield strength of the material, and t0 is t h e creep strain rate at u0

T h e specimen is m o d e l e d by the f i n i t e - e l e m e n t m e s h with 1158 e l e m e n t s and 1218 nodal

FIG 7 Comparison between the experimental and the finite-elements estimate of the de-

flection as a function of time

points shown in Fig 8 The mesh is shown in two regions that fit into each other, as indicated

by the arrows Cross-triangle elements are used to avoid problems with incompressibility The bottom region, made up of square elements, is the region where crack growth occurs The size of the elements in this region is 0.254 mm (0.01 in.) The initial crack tip is two elements to the left of the center node of the rectangular region The crack is allowed to extend through 16 elements for a total crack extension of 4.064 mm (0.16 in.)

The finite-element program models crack extension by releasing nodal points based on experimentally measured crack velocities The 0 implicit (tangent stiffness) time-integration method is incorporated The details of the finite-element procedure are given in a paper by

Hawk and Bassani [25]

For the stationary crack with a crack length equal to 23.9 mm (0.941 in.), K = 23.67

MPa V m (21.52 ksi ~ ) , and C*, from the E P R I expression [18], is 77.12 J/m2/h (0.412

in 9 lb/in.2/h) This yields an initial transition time of 4.6 h from Eq 5 The predicted displacement for the growing crack from finite-element analysis is also plotted in Fig 7 for comparison with the experimental values It is apparent that the finite-element estimates of deflection are considerably below the measured estimates There are a number of possible reasons for this difference First, the finite-element analysis did not account for fast primary creep or instantaneous plasticity, which was significant in this specimen due to its low yield strength Second, in the analysis it was assumed that plane-strain conditions prevailed in the specimen as opposed to a mixed-mode condition, which may be the case in reality Not accounting for each of these factors is expected to result in a lower predicted displacement, leading to a significant cumulative discrepancy Therefore, in this preliminary evaluation of the C, parameter, no effort is made to compare the finite-element results directly with the experimental results Much more work remains for the future in reconciling these differences Nevertheless, the finite-element results in themselves are interesting and can be utilized for evaluating C, as a crack-tip parameter and also the methods for obtaining C,

BASSANI ET AL ON THE C, PARAMETER 21

!

\ L Y \ I /1":'1

-f \ -(

C ( t ) = fFr~o [ n ~(rij'i~nl - (rnOU~] 'J J Ox J ds (29) where nj is the unit outward normal to F As can be seen from the definition, C-integral becomes C* for extensive creep, and it becomes path-independent throughout the specimen

To begin with, finite-element results for the stationary crack will be discussed Figure 9

is a plot of both the amplitude of the R R field and the C, parameter Values of C(t) calculated near the crack tip from the integral in Eq 29 (see Refs 4, 12, and 25) agree welt with the approximate interpolation formula C(t) = C*(tr/t + 1), where C* is calculated from the handbook value [18] Note that extensive creep conditions are reached, to a good approx- imation, when t > lOtr Similarly, excellent agreement is found between C,(Pf'), which is

3.o

-

0 ,-.q

FIG 9 Results of the stationary crack analysis of the compact type specimen

An interesting result is shown in Fig 10, where it is apparent that C, from Eq 18 and a calculated value of the C-integral (Eq 29) along a path that follows the boundary of the

is only path independent as r ~ 0 and is path dependent outside of the crack-tip region This result does not appear to be general In a parallel study on center-cracked specimens,

For our model of the actual crack growth experiment, the C-integral was calculated for

C-integral appears to be path independent for up to a fairly large path radius (Here, and

respective values corresponding to the initial crack length, a0.) This is indicative of extensive creep conditions in the specimen early during the test However, the very early portion of

FIG lO -Comparison of the analytical estimate of C, with various finite-element estimates

of the C-integral for stationary cracks

BASSANt ET AL ON THE C~ PARAMETER 23

0.50

0.25 0.00

FIG l l Comparison of the estimate of C, from various expressions' for a growing crack

the test did lie in the transition regime between SSC and the extensive creep regimes, as

indicated by path-dependent values of the C-integral for short times

In Fig 11 the values of C, from Eq 18 and C,(PIT) from Eq 19 are plotted There is a

maximum discrepancy of about 25% in the transition creep regime during the initial portions

of the test Some of this discrepancy may be due to inaccuracies in estimating displacement

rates in the begining of the test The major cause for this discrepancy is due to crack growth

1.75

~o

c)

1.50 1.25 1.00 0.75 0.50 0.25 0.00

-0.25 0.5

FIG 12 Comparison of the estimates of Ct with the magnitudes of the C-integral near and

far from the crack tip for a growing crack

itself, that is, the crack tip extending away from the creep zone This tends to reduce the

extent of the Riedel-Rice [25] zone on which the expression for Ctin Eq 15 is based However,

Eq 15 is expected to yield conservative estimates of C,, which is important for applications

Figure 12 shows comparisons between C, from Eq 18, which was also approximately equal

to r/W = 0 and the other on the boundary of the specimen r/W = 0.4 The close agreement

between C(r/W = 0.4) and C, from Eq 18 that is found for the stationary crack does not

exist for the growing crack essentially because of the effects of crack growth, as mentioned

above However, except for the very initial region, close agreement is obtained between

small-scale-creep regime

Summary and Conclusions

An in-depth evaluation of the C, parameter proposed for characterizing the creep crack

growth behavior under transient conditions has been carried out The evaluation was made

by including a thorough examination of the assumptions made in the definition of C,, in an

in-depth experimental evaluation and finite-element modeling of an actual compact specimen

used in a creep crack growth experiment The following conclusions can be drawn from this

study:

1 The experimental correlation between creep crack growth rates, da/dt, in the transient

regime and C, is extremely encouraging This correlation appears to hold irrespective of the

extent of creep deformation in the specimen The correlation between da/dt and other

parameters such as K and C* is not good in the transient regime

2 For stationary cracks, C, characterizes the rate of growth of the creep zone size under

small-scale creep conditions, and it is also the stress-power release rate Under steady-state

conditions, C, is identical to C* In this latter regime, C, retains its interpretation as the

stress-power release rate and, in addition, also characterizes the amplitude of the H R R

field These may be the reasons why C, correlates with the creep crack growth behavior

under a wide range of conditions

3 In small-scale creep, C, is equal to the value of the C-integral at the boundary of the

specimen Since the C-integral is path dependent and its value at the boundary of the

specimen does not bear a unique relationship with the crack-tip stresses, C, cannot be

interpreted as the instantaneous amplitude of the H R R field in SSC

4 For growing cracks, there are some differences (25%) between the C~ values obtained

from the product of the load and deflection rates C, ( P f ' ) , and those obtained from the

estimates of the rate of growth of the creep zone size, given in Eq 18 However, the latter

are expected to be higher, which will give conservative results if used in a life prediction

analysis There seems to be a correlation between C, ( P f ' ) and the C-integral along the

specimen boundary

5 More experimental work and analysis should be performed using specimen sizes, geo-

metries and materials that will promote transient conditions for longer durations Primary

creep should be incorporated into the analyses

6 In life prediction, it is important to account for the transient effects due to creep in

the small-scale and transition regimes Not accounting for these transient effects can lead

to significantly nonconservative life predictions

BASSANI ET AL ON THE Ct PARAMETER 25

Acknowledgments

The authors wish to acknowledge the stimulating discussions with Professors D M Parks

of the Massachusetts Institute of Technology, C F Shih of Brown University, and C Y Hui of Cornell University Acknowledgment is also due to Drs C E Jaske of Battele Columbus Laboratories, T Hollstein of Fraunhoffer Institute in West Germany, and H Kino of Mitsubishi Heavy Industries in Japan for contributing creep crack growth data to the ASTM program used in this paper Dr J H a n of the Georgia Institute of Technology assisted in analyzing the data

The financial support for the work by D E Hawk and J L Bassani was provided by the N S F / M R L program at the University of Pennsylvania under Grant No 8216718, and the financial support for the work by A Saxena was provided by the Electric Power Research Institute (Contract RP 2253-10)

[3] Ohji, K., Ogura, K., and Dubo, S., Transactions of the Japan Society of Mechanical Engineers,

No 790-13, 1979, pp 18-20 (in Japanese)

[4] Bassani, J L and McClintock, E A., International Journal of Solids and Structures, Vol 7, 1981,

pp 479-492

[5] Saxena, A and Landes, J D in Proceedings of the Sixth International Conference on Fracture,

ICF-6, Pergamon Press, New York, 1984, pp 3977-3988

[6] Hui, C Y and Riedel, H., International Journal of Fracture, Vol 17, No 4, 1981, pp 409-425 [7] Landes, J D and Begley, J A in Mechanics of Crack Growth, ASTM STP 590, American Society for Testing and Materials, Philadelphia, 1979, pp 128-148

[8] Nikbin, K M., Webster, G A., and Turner, C E in Cracks and Fracture (Ninth Conference), ASTM STP 601, American Society for Testing and Materials, Philadelphia, 1976, pp 47-62 [9] Saxena, A in Fracture Mechanics: Twelfth Conference, ASTM STP 700, American Society for Testing and Materials, Philadelphia, 1980, pp 131-151

[10] Riedel, H., Journal of Mechanics and Physics of Solids, 1981, Vol 29, pp 35-49

[11] Ehlers, R and Riedel, H in Advances in Fracture Research, ICF-5, Vol 2, Pergamon Press, New York, 1981, pp 691-698

[12] Riedel, H in Elastic-Plastic Fracture: Second Symposium, Vol I, Inelastic Crack Analyses ASTM STP 803, 1983, pp 1-505-1-521

[13] Hutchinson, J W., Journal of the Mechanics and Physics of Solids, Vol 16, 1968, pp 13-31

[14] Rice, J R and Rosengren, G F., Journal of the Mechanics and Physics of Solids, Vol 16, 1968,

[19] Sandananda, K and Shahinian, P., Metallurgical Transactions A, Vol 9A, January 1978, 79-84

[20] Edmunds, J and Willis, W., Journal of the Mechanics and Physics of Solids, 1976, Vol 24, pp 205-225

[21] Hutchinson, J W., "Nonlinear Fracture Mechanics," Technical University of Denmark Report, University of Denmark, Lyngby, Denmark, 1979

[22] Rice, J R in Fracture, Vol 2, H Liebowitz, Ed., Academic Press, New York, 1971, pp 191-

311

[23] Saxena, A and Liaw, P K., "Remaining-Life Estimation of Boiler Pressure Parts-Crack Growth

Studies," EPRI-CS-4688, Electric Power Research Institute, Palo Alto, CA, July 1986

[24] Saxena, A and Han, J., "Evaluation of Crack-Tip Parameters for Characterizing Crack Growth

Behavior in Creeping Materials," Proceedings, ASTM Workshop on Creep Crack Growth, Knox-

Philippe Bensussan, 1 R o l a n d Piques, 2 a n d A n d r e Pineau 2

A Critical Assessment of Global Mechanical Approaches to Creep Crack Initiation and Creep Crack Growth in 316L Steel

Mechanical Approaches to Creep Crack Initiation and Creep Crack Growth in 316L Steel,"

ena, J D Landes, and J L Bassani, Eds., American Society for Testing and Materials, Philadelphia, 1989, pp 27-54

studied in 316L stainless steel at 575 to 650~ This alloy was found to be creep ductile, which can be justified in terms of fracture mechanics concepts applied to creeping solids Fracture mechanics is yet unable to provide unique correlations with global load-geometry parameters, such as K, J, or C*, for all the stages of both creep crack initiation and growth Unique correlations nevertheless exist between C* and the time to initiation, and between C* and the Stage I to Stage II crack growth transition time Time versus C* diagrams based on these latter correlations, in which creep crack initiation, slow creep crack growth, and fast creep crack growth regimes can be delimited, are the only global approach data which can be safely used for life prediction purposes in engineering applications

viscoplasticity, viseoplastic fracture mechanics, finite-element analysis, creep damage, 316L stainless steel, nonlinear fracture mechanics

Macroscopic cracks can initiate and propagate in metallic parts during high-temperature service under the combined influence of creep and fatigue damages The need to reach higher service temperatures and stresses has led to lower safety margins Applications in- volved in the conception of nuclear reactor plants and jet engines, for example, have mo- tivated a large research effort in the field of high-temperature crack behavior In particular, creep cracking, that is, the initiation and propagation of a single macroscopic crack under

a sustained or slow-varying load at temperatures well within the creep regime, is of interest for components under steady output service conditions

Creep cracking has been extensively studied in structural alloys such as stainless steels, low-alloy steels, nickel-base superalloys, and aluminum alloys (for a review see, for cxample, Refs 1 to 3) Most of the published work has been devoted to creep crack growth (CCG) only and not to creep crack initiation (CCI), defined here as the time necessary for an existing defect to start growing or to grow by a predetermined increment under creep conditions, yet this latter stage often lasts for a large part of the rupture life of a cracked component

Senior scientist, Etablissement Technique Central de l'Armement, 94114 Arcueil Cedex, France

2 Graduate student and professor, respectively, Ecole Nafionale Suprrieure des Mines de Paris Centre des Matrriaux, 91003 Evry Cedex, France

Most CCG studies have focused on the identification of unique correlations between the

CCG rates, da/dt, and the load-geometry parameters Materials susceptible to creep cracking

can be said to be either creep brittle or creep ductile Creep brittle and creep ductile materials

fail by creep cracking under, respectively, small- and large-scale creep conditions Based

on the Riedel and Rice extension of non-linear elastic fracture mechanics (NLEFM) to

creeping solids [4,5], the elastic stress-intensity factor, K, and the creep C*-integral, which

theoretically characterize the time-dependent crack-tip stress-strain fields for short and long

times, have often been proposed as correlating parameters with da/dt in creep brittle

[6-9] and creep ductile [8-14] materials, respectively

Recent work performed on a creep brittle 2219-T851 aluminum alloy at 175~ [1,3,6,7,15]

has shown that unique correlations actually exist between K and the time to initiation, t~,

and K and da/dt during the Stage II quasi-steady state crack growth regime only

The present paper deals with CCI and C C G in a creep ductile 316L stainless steel at 575

to 650~ Previous results obtained on the same heat of material [2,15-17], some of which are discussed below along with new results, have shown that a unique t~-C* correlation can

be established Apparent approximately linear correlations between Stage II da/dt and C*

calculated from load and total load-point displacement rates have still been found to reduce

to trivial correlations between da/dt and da/dt Given the fact that similar da/dt-C* apparent

correlations have very often been reported in the literature for creep ductile as well as creep

brittle materials [3,11-14], their validity and limitations, of vital importance for engineering

applications, are critically discussed below

In the present paper, viscoplastic fracture mechanics concepts are first reviewed Theo-

retical crack-tip stress-strain fields are then compared with finite-element numerical results

obtained in finite-size specimens Major limitations of the application to 316L stainless steel

of a global mechanical approach to creep cracking, consisting of the identification of unique

correlations between t~ or da/dt and global load-geometry parameters, are then pointed out and discussed

Fracture Mechanics of Creeping Solids

The stress and strain fields around the tip of a stationary sharp crack in an infinite creeping solid can be calculated in an approximate way as a function of time and distance from the crack tip [4,5] In a creeping solid, the total strains can be assumed to be equal to the sum

of time-independent instantaneous elastic and plastic strains and of time-dependent creep strains For such idealized conditions, asymptotic stresses can be determined close to the

crack tip The domains of validity of these purely mathematical stresses may not exist around

cracks in real materials as crack tip blunting and specimen finite size, for example, cannot always be neglected

Hutchinson-Rice-Rosengren ( H R R ) Initial Plastic Stress Fields

Upon loading at t = 0, the stresses in the instantaneous plastic zone are then given for power-law hardening materials for r ~ 0 by the well-known Hutchinson-Rice-Rosengren ( H R R ) singularities [18-20]

BENSUSSAN ET AL ON CREEP CRACK INITIATION AND GROWTH 29

is the plastic constitutive equation law;

J = th e J-integral [21];

(rij(O,np) = the geometrical tabulated functions [18-20];

I,p -~ 7r; and

n p = plastic work hardening coefficient

As time increases, creep zones, in which significant stress relaxation takes place, develop

and grow around the crack tip into the instantaneous plastic zone

Riedel and Rice Creep Stress Fields

For an ideal material which creeps by secondary power-law creep only and for the as-

sumptions listed above, the stresses in the secondary creep zone are given for r ~ 0 by the

time-dependent Riedel and Rice (RR) singularities [4,5]

Under these conditions, the size of the creep zone, Rop, in which the creep strains are

dominant, increases for a short time as [4,5]

t < t,,, Rue ~(J)(BzE"2t) 2/("2-~) (5d)

The creep zone grows from dimensions that are negligible with respect to characteristic

specimen dimensions for short times (t < t,r) to large dimensions for long times (t >~ ttr)

For the sake of clarity, primary creep strains were not considered up to this point For a

material such as 316L stainless steel that deforms both by primary and secondary creep, a

primary creep zone first develops and grows around the crack tip As time increases, the

primary creep zone overgrows the instantaneous plastic zone, and the stresses relax toward

time-independent distributions Under such primary creep conditions, Riedel [4] has shown

that the crack-tip stresses in the primary creep zone are given for r ~ 0 by

where J(t) is the J-integral calculated from the stresses and time-dependent primary creep

strains around the crack tip, and

n'l + 1 \ c : /

is a transition time between small-scale and large-scale primary creep conditions Here,

C~ is actually time independent for time-independent stress distributions if the initial tran-

sient stress redistribution regime is neglected (for example, Refs 4, 16, and 17), since Eq 7

is equivalent, for constant ~r, to

The stresses in the secondary creep zone, which grows inside the primary creep zone for

short times and eventually overgrows it for long times, have also been calculated as a function

of J, C~', and C* [4] However, these expressions can be largely simplified by assuming

that crack-tip stress relaxation by primary creep is very fast and that large-scale primary

creep conditions prevail at t ~ 0 The crack-tip stresses are then equal to the RR singularities

given by Eq 3 where [4]

nzp'l + 1~ C;~

BENSUSSAN ET AL ON CREEP CRACK INITIATION AND GROWTH 31

where t2 is a transition time between small-scale and large-scale secondary creep conditions,

defined by

Characteristic Load-Geometry Parameters

By assuming fast stress relaxation by primary creep and, in particular, that t~ < t2, the

stress, strain, and strain rate crack-tip fields are thus theoretically fully characterized for

r ~ 0 by [4,5]:

(a) J in the small-scale primary creep zone and the surrounding instantaneous plastic zone

for short times (t < h), which reduces to K for small-scale yielding loading conditions for

which J ~ KZ/E [21];

(b) C~" in the small-scale secondary creep zone and the surrounding primary creep zone

for short times (h < t < h) and loading conditions leading to large-scale primary creep at

t = 0; and

(c) C* in the large-scale secondary creep zone for long times (t >> h)

Slightly different conclusions would have been obtained by neglecting primary creep

Under such an assumption, and according to Eqs 3 and 5, only the J and C* dominated

regimes remain

The case of a growing crack has been studied by Hui and Riedel [23] The zone over

which the corresponding stress singularities are dominant has been shown to be of negligible

dimensions in most cases [23-25] These singularities do not need to be considered here

According to Eqs 1 through 10, J, or K in the case of small-scale yielding loading con-

ditions, appear to be logical load-geometry parameters to correlate with t~ or da/dt in creep

brittle materials in which tl is expected to be larger than the test characteristic times On

the other hand, C* and C* would be respectively better suited in creep ductile materials in

which t2 or tl would be shorter than the test characteristic times

On the basis of the material data and plasticity and creep constitutive laws presented in

the following sections, transition times tx and t2 have been estimated for 316L stainless steel

at 600~ in compact tension (CT) specimens with w = 40 mm, b = 10 mm, and a 15.16

mm (a/w = 0.379), subjected to an applied load of 9189 N For such loading conditions,

the values of J, C~', and C* calculated by the numerical methods described in the Appendix

t2 ~ 6000 h

which is, as a matter of fact, much larger than tl

TABLE 1 Chemical composition in weight percent of Type 316L stainless steel

of the alloy at the test temperatures are given in Table 2 [2,15-17] The stress and strain exponents, as well as the constants of the uniaxial plastic and creep strain power laws for these temperatures, are listed in Table 3 [2,15-17]

Test Procedures

Creep cracking tests were performed on CT specimens and on axisymmetrically cracked specimens The specimen geometries and dimensions are shown in Fig 1 CT specimens were machined in the T-L orientation, for which the crack propagates along the rolling direction, and the load was applied along the long-transverse direction; the axisymmetrically cracked specimens were machined in the T direction, for which the axial load is applied parallel to the long-transverse direction Tests had also been previously performed on double- edge-notched (DEN) specimens; those results have been described elsewhere [2,15-17]

Creep tests were carried out at 575,600, and 650~ on either dead-weight level arm or servohydraulic machines All specimens were carefully fatigue precracked at room temper- ature in tension at load levels so that the maximum stress-intensity factor remained below

15 MPa ~mm, which is much lower than the high-temperature test initial load levels Straight

TABLE 2 Conventional ~nsi~ properties of Type 316L sminless steel

BENSUSSAN ET AL ON CREEP CRACK INITIATION AND GROWTH 33

TABLE 3 Plasticity and creep constitutive laws at 575, 600, and 650~

and circular crack fronts w e r e always o b t a i n e d in C T and axisymmetrical specimens, re-

spectively T h e initial crack lengths w e r e such that 0.3 ~< a / w <~ 0.6 in C T specimens, and

a / b ~ 0.45 in axisymmetrically cracked specimens

Crack lengths a(t) w e r e m e a s u r e d by the d-c potential d r o p t e c h n i q u e [26] L o a d - l i n e

displacements, g(t), w e r e o b t a i n e d by m e a n s of e x t e n s o m e t e r s attached to the specimens, with a gage length of L0 = 25 m m for the axisymmetrically cracked specimens (Fig 1) Typical crack length and load-line displacement versus time e x p e r i m e n t a l data f r o m typical tests p e r f o r m e d on axisymmetrical specimens are given in Figs 2 and 3 B o t h the t i m e to initiation, that is the t i m e necessary for the crack to grow f r o m the initial fatigue precrack front o v e r a critical distance Xc = 50 Ixm, and crack growth rates can be d e d u c e d f r o m such data

L o a d - g e o m e t r y p a r a m e t e r s , such as K, J, C*, and Cff w e r e e s t i m a t e d according to the

m e t h o d s described in the A p p e n d i x T h e estimation of J, Cff, and C* w e r e derived f r o m

FIG 1 Dimensions of the specimens used in this study: (a) CT specimens and (b) axisym-

metrically cracked specimens

FIG 2 Load-line displacement, ~, versus time Axisymmetrically cracked Specimen S Q F A

FIG 3 Crack length increment, Aa, as a function o f time Specimen S Q F A 15 was tested

BENSUSSAN ET AL ON CREEP CRACK INITIATION AND GROWTH 3 5

limit-load analysis of the specimens [27] It is worth noting that they required knowledge

of the instantaneous crack length, the applied load, and the load-line displacement for J

and C~ or the load-line displacement rate for C* Both 8 and 8 (dS/dt) can be obtained

either experimentally or numerically

Finite-Element Calculations

Numerical Methods

Two elastic-plastic-viscoplastic two-dimensional finite-element analyses of a 316L stainless

steel cracked CT specimen at 600~ were performed The first numerical simulation was

done with the finite-element code Z E B U L O N , developed at the Centre des Mat6riaux of

the Ecole des Mines de Paris, and the second numerical simulation was done with the

N O V N L code developed at Novatome Both codes, rather similar as far as material behavior

is concerned, have been described elsewhere [28,29]

Stress-strain fields obtained by both numerical simulations were shown to agree with each

other very well, although a coarser crack-tip mesh size of 180 ~m was used with the

Z E B U L O N code, compared with 30 p~m for the N O V N L code In both cases, the instan-

taneous initial loading of the specimen was simulated by elastic-plastic calculations F o r the

simulation of the hold time under a constant applied load, total strains were assumed to be

the sum of the elastic, plastic, and creep strain No coupling between plastic and creep strain

was considered Both primary and secondary creep strains were taken into account since

the transition between both regimes was assumed to occur when the equivalent primary and

secondary creep strain rates became equal

Stress-strain fields were computed in a 10-mm-thick, 40-mm-wide CT specimen subjected

to a 9189-N applied load with a 15.16-mm-long stationary crack (a/w = 0.379) under plane-

stress conditions This latter assumption has been justified elsewhere for 10-ram-thick 316L

CT specimens [2,15-17] The values of the corresponding J, C~, and C* are given in the

section on the fracture mechanics of creeping solids (see the Appendix)

Material data and constitutive laws given in Tables 2 and 3 were used

Numerical Results

The variations with time and distance from the crack tip of the "numerical" stress-strain

fields obtained by finite-element analysis were compared with the "theoretical" ones deduced

from the application of fracture mechanics

According to the review of the application of fracture mechanics to creeping solids pre-

sented in a previous section, the singular asymptotic stresses in the plastic, small- and large-

scale primary and secondary creep zones around the crack tip can be written as

where n and B, are, respectively, equal to np and Bp, n'~ and B'~, or nz and B2; and A ,

independent of r and t, is proportional to J, C;~, or C*

Theoretical stress fields were evaluated according to Eq 11 The expressions of A , p, and

-q issued from the application of fracture mechanics (Eqs 1 through 10) are given in Table

4 for each type of deformation regime listed above in the case where tl < tz (compare with

the section on the fracture mechanics of creeping solids) The material constitutive laws are