Experimental evaluation of the j or c parameter for a range of cracked geometries

Nikbin 1Experimental Evaluation of the J or C* Parameter for a Range of Cracked Geometries ABSTRACT: In the current ASTM Standard Test Method for Measurement of Creep Crack Growth Rates

Imperial College London

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C M Davies, 1 M Kourmpetis, 1 N P O’Dowd, 1 and K M Nikbin 1

Experimental Evaluation of the J or C* Parameter for a

Range of Cracked Geometries

ABSTRACT: In the current ASTM Standard Test Method for Measurement of Creep Crack Growth Rates in

Metals共E 1457兲 the experimental C* parameter is related to the load and creep load line displacement rate

through the geometry related ␩ factor In this work ␩ factors for a range of geometries are presented The geometries examined are the compact tension specimen, C 共T兲, single edge notch specimen in tension, SEN 共T兲, and bending, SEN共B兲, double edge notch specimen in tension, DEN共T兲, middle crack specimen in tension, M 共T兲 and the C-shaped specimen in tension CS共T兲 Calculations have been performed for a linear elastic-power law hardening material but the resulting ␩ factors are applicable to either power law plastic or power law creeping materials Values for ␩LLDand ␩CMOD, based on the load line displacement and crack mouth opening displacement, respectively, have been determined A wide range of crack depths, 0.1艋a/W艋0.7, where a is crack length and W is specimen width, and hardening exponents, 3 艋N 艋10,

under plane stress and plane strain conditions have been examined using the finite element method The influence of specimen length, crack length, material properties and out of plane stress state on the ␩ factor has also been considered It has been found that for shallow cracks the value of ␩ depends quite strongly

on the exponent, N in the material power law, regardless of whether␩ is defined based on the load line displacement or crack mouth opening displacement The ␩LLD

factor has also been found to be strongly

sensitive to plane stress/strain conditions imposed, a/W and specimen length, whereas ␩CMOD

KEYWORDS: eta factor, creep crack growth, finite element, specimen geometry, C*, J, load line

displacement, crack mouth opening displacement

Nomenclature

B , B n ⫽ Specimen thickness and 共net兲 thickness between side grooves

Manuscript received April 28, 2005; accepted for publication January 30, 2006; published online March 2006 Presented at ASTM Symposium on Fatigue and Fracture Mechanics: 35th, on 18–20 May 2005 in Reno, NV;

R E Link and K M Nikbin, Guest Editors.

1 Department of Mechanical Engineering, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK.

Paper ID JAI13221 Available online at www.astm.org

Copyright © 2006 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959.

J e , J p ⫽ Elastic and plastic components of J

␣ ⫽ Constant in Ramberg-Osgood law

␧ ⫽ Strain

␯ ⫽ Poisson’s ratio

␴ ⫽ Stress

Introduction

to the CCG rate of the material, a wider range of test geometries is needed in order to apply the resultsfrom experimental laboratory test specimens to the range of conditions experienced by engineering com-ponents under service conditions This trend towards CCG testing using nonstandard specimens has drawn

共sometimes referred to as center crack tension, CC共T兲兲 These geometries are shown schematically in Fig

obtained from the literature and additional values have been generated using finite element analysis

to a nonlinear power law plastic material is provided and the generalization to a power law creepingmaterial is then considered The effect of material properties, specimen geometry and out-of-plane stress

examined This will allow a user to determine values of C* for a wide range of cracked geometries.

where the creep stress exponent, n, and the constant C, are temperature dependent material properties.

Estimation of J and C* from Load-Displacement History

The nonlinear elastic crack tip parameter, J, may be evaluated from a line integral along a contour

rate of a cracked body,

FIG 1—Specimen geometry definition.

J = − ⳵U

共3兲

For a nonlinear material, in order to obtain the value of J directly from a load-displacement history, it

also depend on material hardening exponent, N, normalized length, L / W, and specimen thickness B.

The linear elastic component of the displacement of a cracked geometry is directly related to the applied

curve, and (b) plastic secant area A sec , under the load-plastic displacement curve.

N − 1

can be represented by a power law as in the Ramberg-Osgood material law, Eq 1 Replacing the area A in

J p= P⌬p

general, H will depend on N and specimen geometry During testing, the displacement of the specimen is

under three point loading, 2L is the spacing between the support points.

6 The superscripts LLD and CMOD are used with H to indicate that they are associated with the load line and crack mouth opening displacement, respectively The values of H in Table 1 are appropriate for a

from

J p= A

TABLE 1—Definitions of H, H⬘and A for each specimen.

The steady-state creep crack tip characterizing parameter, C*, defined under widespread共steady-state兲

test using the relations,

C * = P ⌬˙ c

that for a power law plastic material with exponent, N = n.

Review of ␩ Factors in the Literature

the loading pin, displacement or load controlled analysis, etc.兲 and in the precise method used to calculate

␩ Also, the displacement measuring positions and the material properties used vary among the analyses

Numerical Analysis

elastic-plastic analyses The values obtained are found to be in good agreement with those available in the

HP⌬p

共16兲Two-dimensional, plane stress, and plane strain analyses have been performed Each specimen hasbeen modeled with a focused mesh and multiple nodes at the crack tip to capture the crack tip singularity.Small geometry change conditions are used Symmetry conditions are fully employed For all specimens

2000 elements and nodes and the J integral and K values have been averaged over 41 contours.

the plastic strains are sufficiently high that power law material behavior prevails throughout the specimen

Results

and SEN共B兲 a summary highlighting the effects of normalized length, L/W on the ␩ factor are also

SEN 共B兲-3PB SEN 共T兲

关24兴

SEN 共T兲 DEN 共T兲 SEN 共B兲-3PB

High strength structural

Single Edge Notched Tension Specimen, SEN(T)

with uniform remote stress and thus bending is allowed in the specimen, which can be significant for deep

cracked specimen, a / W = 0.7 The results shown are for hardening exponent N = 10 under plane stress and plane strain conditions The trends given for N = 3 are similar and thus not shown here It may be seen that

insensitive to L / W This sensitivity for a shallow crack is related to the significant amount of remote

plasticity which increases with increasing specimen length It may be seen in Fig 3 that the value of

fit” is an approximate fit to the mean value of the dataset at each value of a / W, and the upper bound and

lower bound line have been estimated to cover between one and two standard deviations of the dataset

TABLE 3—Summary of ␩ and J estimation studies in literature.

Author

Specimen Type

关18兴

Analysis Kim and

Schwalbe

关28兴

M 共T兲

C 共T兲 SEN 共B兲-3PB

LLD CMOD

solutions Kim and

of Ref 关20兴 Sumpter

关31兴

limited range of data.

Wu et al.

关32兴

3PB

CC 共T兲 SEN 共T兲 DEN 共T兲

LLD 0.125 – 0.875 L / W = 2 Slip Line Field

& EPRI Solutions.

aP ␴⬅ Plane stress, P␧⬅ Plane strain.

various conditions; no attempt has been made to describe the individual dependence on material or

given in Appendix A

variability is approximately 5 % of the mean value for the cases examined The recommended solutions

Middle Cracked Tension Specimen, M(T)

a / W = 0.1, being strongly sensitive to specimen length, while␩CMODis unaffected by the specimen length

almost coincide This indicates that for deep cracks the majority of the deformation is occurring in the

shallow and (b) deeply cracked SEN(T) specimens, N = 10.

range of crack lengths for a SEN(T) specimen.

In Fig 7 the sensitivity to N and to plane stress or plane strain conditions is examined It is seen, that

as for the SEN共T兲 specimen the influence of out of plane constraint is insignificant compared to the

The full set of results, including those taken from the literature, is shown in Fig 8 Good agreement is

seen that some of the data for a / W = 0.1 and 0.3 lie above the upper bound line These data correspond to

particularly for shallow cracks It is therefore recommended that, where practicable, J or C* be estimated

Double Edge Notched Tension Specimen, DEN(T)

Figures 9 and 10 illustrate the sensitivity to specimen length, out of plane constraint and strain hardening

on strain hardening, N, is somewhat greater for deep cracks and out-of-plane constraint effects on the value

sources.

deeply cracked M(T) specimens.

the mean fit over the range of crack lengths For␩LLDthe variability is between approximately 75 % for

variability and strong dependence on a / W, care should be used when determining J or C* from this

Single Edge Notched Bend Specimen, SEN(B)

The specimen has been analyzed under three point bending Results under plane strain conditions are

a / W⬍0.5, for ␩LLD and a / W⬍0.3 for ␩CMOD For deeper cracks 共a/W⬎0.5兲, ␩ LLD is essentially pendent of out-of-plane stress state and strain hardening and is close to 2, as expected for a deeply cracked

and strain hardening and has a near linear variation with a / W.

a / W; for a / W⬎0.3 the variability is lower and ␩LLD is more weakly dependent on a / W On average, the

considered The relatively small variations at a given a / W are mainly due to the dependence on N Over

conditions, over a range of crack lengths for a M(T) specimen.

sources.

the range 0.1艋a/W艋0.7 the value of ␩ CMOD deviates from the mean value by about 10 % for the cases

Compact Tension Specimen, C(T)

measured at the point of load application This assumption has been confirmed by the analyses carried out

have been compared, and these are almost identical Note that due to differences between the specimen andfinite element model geometry for the cases examined here, the position of the node used to determine

the results presented here the pin hole indicated in Fig 1 was not modeled explicitly

关9兴 and ASTM E 1457 关4兴 is ␩ solution for the C共T兲 specimen is widely used and is given by

deeply cracked DEN(T) specimens.

conditions, over a range of crack lengths for a DEN(T) specimen.

assumption that the displacement is measured at the loading point, though in Refs.关12,15兴 J values found

the load point and at the crack mouth

line, provides an upper bound to the data at a / W = 0.45 and a lower bound at a / W = 0.7 Thus an adequate

共␩=2.2兲 The upper and lower bounds are less than 5 % of the mean fit value in the range of crack lengthsconsidered

C-shaped Specimen, CS(T)

The variation in␩LLDfrom the plane stress and plane strain analyses is relatively small compared with

numerical data for comparison have been found hence only the values determined from the currentanalysis are shown in Fig 18 As before, mean, upper, and lower bound lines are provided, though the data

a / W = 0.54 Values of ␩LLD of approximately 2.24 and 2.44 were obtained from 2-D and 3-D analyses,respectively These results are consistent with the values shown in Fig 18共a兲 An analytical estimate for

with the upper bound for a / W = 0.4 and the mean fit at a / W = 0.55 However, the solution is only relevant

for deep cracks and there is a large discrepancy between this solution and the data for shallower cracks

approximately 5 % of the mean value This variability is mainly due to the dependence on N for shallow

The limited number of crack depths examined here makes it difficult to provide definitive

conditions, over a range of crack lengths for a SEN(B) specimen.

sources.

Due to the limited data available for this specimen and the relatively large variability for shallow

Discussion

Analyses have been reported which allow the fracture mechanics parameters J and C* to be evaluated

H used to evaluate J or C* have been tabulated and␩ values plotted for a range of crack geometries usingeither the load line displacement or crack mouth opening displacement The influence of material response共power law exponent兲, out-of-plane stress state 共plane stress/plane strain兲, and specimen length on the

Most of the results presented here are from a 2-D analysis There may also be an influence from

specimen thickness, B / W, and specimen side-grooving which has not been fully explored It is expected

speci-mens Some 3-D calculations and discussion on thickness and side-grooving effects can be found in Refs.关3,10,17,19兴

material properties are being determined the lower bound will provide a conservative measure of the

uncertainty, evaluated as a percentage of the mean, should be quantified in each case Since C* is linearly

range of crack lengths for a C(T) specimen.

related to␩ 共see Eq 15兲 then, for example, a 20 % uncertainty in ␩ would lead to a 20 % uncertainty in

C*, in addition to other experimental uncertainties In some experiments both ⌬LLD and ⌬CMOD may be

available The values of J or C* obtained from the two measurements should agree within the quoted

from FE analysis or otherwise The analysis carried out here indicates that for most specimens the

that within the FE analysis certain assumptions have been made, e.g., in most cases the rigid pin has not

to the specimen at the specimen boundaries In an experiment a minimum gage length is required for a pinloaded structure to ensure a relatively uniform stress distribution to be achieved in the region of the crackplane

crack mouth opening displacements and the geometric functions presented here

Conclusions

respectively, have been presented for a range of geometries The geometries examined are the compact

conditions, over a range of crack lengths for a CS(T) specimen.

specimen tension CS共T兲 Calculations have been performed for a Ramberg-Osgood 共power law hardening兲

plane stress and plane strain conditions have been examined using the finite element method Specimen

and crack length These results are generally consistent with those in the literature Mean linear fits, that

state, and other effects, are accommodated by providing an upper and lower bound to the data Generally,

more confidence is expected in J or C* values derived from crack mouth opening displacement

measure-ments

Acknowledgments

discussions with Prof C E Turner are gratefully acknowledged

APPENDIX A—Equations for the Mean Fit to ␩ Including a Measure of Uncertainty

crack length, x = a / W, are given below, together with an approximate absolute measure of the uncertainty

in the data, thus providing an upper and lower bound estimate The restrictions on specimen geometry and

hardening exponent, N, are also specified.

TABLE 4—Details of the numerical models for each specimen.