neuber method for fatigue

8/10/2019 Neuber method for fatigue

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EN G I N E E R I N G S T R U C T U R A L IN T E G R I T Y A S S E S S M E N T

EMAS 2004 1

APPLICATION OF THE NEUBER APPROACH TO PREDICT CRACKINITIATION IN COMPONENTS

C. M. Davies*, N. P. ODowd

*, K. M. Nikbin

*, F. Biglari

, G. A. Webster

*

The Neuber method is commonly used to estimate the stress field ahead ofa crack tip in an elastic-plastic material and has been incorporated into anumber of failure assessment procedures. In this work a compact tension

specimen is examined under plane strain conditions using theRamberg-Osgood power law plasticity model. It is found that existing

procedures provide a reasonable estimate of the equivalent von Misesstress over a wide range of load levels, but non-conservative estimates ofthe maximum principal stress. A method is proposed to determine the

maximum principal stress from the equivalent stress determined by theNeuber method. Through the use of the proposed techniques, accurate

estimates of the von Mises equivalent and maximum principal stressahead of a sharp crack tip can be obtained. The implications of the result

in terms of the lifetime prediction of cracked components are discussed.

INTRODUCTION

In component lifetime assessments a procedure, known as the sigma-d method, has

been proposed to estimate the crack initiation period from pre-existing defects, i.e. the

time required for the onset of crack growth due to creep and/or fatigue. The method,developed by Moulin et al. [1], predicts that initiation occurs under creep conditionswhen the stress at a distance dahead of the crack tip is equal to the stress which causes

rupture under uniaxial conditions.

The sigma-d method has recently been introduced into the British Energy defect

assessment procedure, R5 [2]. The method is currently based on the elastic-plasticstress field generated ahead of the crack tip and does not consider the relaxation of

stress due to creep. This stress field may be estimated by a number of methods

including procedures based on Neubers rule [3].

In Davies et al. [4] a comparison of the crack tip stress field predictions obtained

from a number of methods, including Neubers method, with full field finite element

solutions has been performed, for a CT specimen under plane strain conditions. Some

of the key results of [4] are reviewed here. In addition, the sensitivity of the sigma-d

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

Department of Mechanical Engineering, Amirbkabir University of Technology, Hefez Avenue, Tehran, Iran

Contact author email address: [email protected]


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method, and hence lifetime predictions, to the stress estimate will be considered.

STRESS ESTIMATES FROM NEUBERS METHOD

Neuber [3] proposed that, for a material which obeys a non-linear deformation law thatis linear at vanishingly small stresses, the maximum stress in the vicinity of a sharply

curved notch is related to the nominal (remote) stress through a linear elastic stress

concentration factor. Neuber [3] suggests that the method may be generalised to any

arbitrary loading state through the use of the equivalent stress.

Different interpretations of the Neuber method are found in the literature, which

predict either the stress normal to the crack plane (which is also the maximum principalstress in the plane of the crack under Mode I conditions) or the equivalent von Mises

stress. A brief description of four Neuber methods detailed in [4] follows and asummary of the key equations are given in Table 1.

TABLE 1 Summary of Neuber methods

Method

Name

Stress

Predicted

Equation

NeuberEquivalent,

d0 0

0 0

N N

refd d de ded de

refE E E E

+ = +

(1)

Neuber-R5Max.

Principal,

d

0 0

0 0

N N

refd d de

d deE E E E

+ = +

(2)

Neuber-

R5(A)

Equivalent,d

0 0

0 0

N N

refd d ded de

E E E E

+ = +

(3)

Neuber(M)Max.

Principal,

d

0 0

0 0

N N

refd d d dede

M ME E M E E

+ = +

(4)

The derivation of Eqs. (1)(4), shown in Table 1, is given in [4]. In Table 1 d andd refer to, respectively, the equivalent von Mises stress and the maximum principalstress in the plane of the crack (or stress normal to the crack plane) at a distance d

ahead of the crack tip. Likewise, d eandderefer to the equivalent von Mises linearelastic stress and the maximum principal linear elastic stress in the plane of the crack, at

a distance dahead of the crack tip. All the equations in Table 1 contain terms of the


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form stress strain. In the equations the stress-strain relation is described by aRamberg-Osgood type equation whereNis the strain hardening power law dependence,

E is the Youngs Modulus, 0 is the normalising stress and is a constant. The

definition of equivalent strain used in Eqs. (1) and (3) is based on the relationship

proposed in Harkegaard and Mann [6] where the total equivalent strain is given by the

sum of the von Mises elastic and plastic deviatoric strains, giving rise to an effective

elastic modulus 3 (2(1 ))E E v= + where is the Poissons ratio. An alternativedefinition of equivalent strain (see e.g. Chen et al. [7]) may be used, in which case

E= E. Also in Eqs (1)(4) the nominal stress is taken to be the reference stress, ref(see e.g. Webster and Ainsworth [5]).

For a sharp crack the linear elastic equivalent (von Mises) stress in the crack plane

at a distance ddirectly ahead of the crack tip, d e,is determined from the Kfield,

2de

K

d

= where = {

1 2 for plane strain

1 for plane stress.

v (5)

In Eq. (5) K is the linear elastic stress intensity factor. Similarly the linear elastic

maximum principal stress at d, de, is given by

2de

K

d

=

(6)

under both plane stress and plane strain conditions.

The equivalent (von Mises) elastic-plastic stress at a distance dahead of the crack

tip, d

, may be obtained by the solution of the Neuber method (Eq. 1). In the R5

sigma-d procedure, the maximum principal elastic-plastic stress at a distance ddirectly

ahead of the crack tip, d, is given by the solution of Eq. (2), here denoted the

Neuber-R5 method. (A value of dequal to 50 m is recommended in the R5 procedurefor austenitic stainless steels).

An alternative form of the R5 equation is proposed in [4] to predict the equivalent

stress by replacing the maximum principal stresses d and de in Eq. (2) by their

equivalent stress values, d and d e, respectively. The resultant equation, Eq. (3) inTable 1, is denoted the adapted R5 equation or the Neuber-R5(A) method and differs

from Eq. (1) since the second term in the bracket on the right hand side (the equivalent

plastic strain at reference stress) is not amplified by the linear elastic stress

concentration factor (d e/ref). Hence the Neuber-R5(A) method does not precisely

follow the original formulation of Neuber given by Eq. (1).

In the defect assessment procedures R5 [3] and A16 [8] the maximum principal

stress (stress normal to the crack plane), d, is used to predict the time to crack

initiation. An alternative approach to determining dhas been proposed in [4]. There dwas obtained by scaling the corresponding equivalent stress, d , obtained from the


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Neuber method. For a given d the value of the ratio d/d is dependent on load,specimen geometry and strain hardening power law dependence, N. Based on finite

element studies on a CT specimen a representative value of d/d , denotedM, has been

used to determinedfrom the equivalent Mises stress obtained from theNeuber-R5(A)

method. The value ofMhas the following dependency on load [4]

0

0.22 2.96refd

d

M

= = +

. (7)

The functionMshown in Eq. (7) has been chosen to be conservative over the relevant

crack tip distances, forNbetween 5 and 10, and to give optimal agreement with the FE

solutions at d= 50 m. The maximum principal stress, d, may be estimated by solvingEq. (4) using theMvalue calculated from Eq. (7).

ANALYSIS METHODS AND MODELS

Specimen Geometry, Material and Finite Element Model

A compact tension (CT) specimen has been examined with a crack length to specimenwidth ratio, a/W, equal to 0.5. The Ramberg-Osgood material model is used with

= 0.1, E/0= 912, representative of austenitic stainless steel at around 600oC. Two

values ofNhave been used,N= 5 and 10. Plane strain conditions have been examined

and small displacement theory used (i.e. effects of crack blunting have been ignored).

Finite element calculations have been performed using the commercial software

package ABAQUS [9]. Only one half of each specimen has been modelled due to

symmetry and a sharp crack has been represented with a focused mesh at the crack tip.

The mesh for the CT specimen consists of 2005 nodes and 1895 plane strain, linearhybrid elements. Further details of the finite element analysis are provided in [4].

Stress Estimates and Initiation Time Predictions

The four Neuber equations, shown in Table 1, have been solved iteratively to give the

stress d or d . The accuracy of the Neuber stress estimates has been assessed by

comparison with finite element solutions. The sensitivity of the initiation time

predictions given by sigma-d method to the accuracy of the various stress estimates has

also been examined. The sensitivity is assessed by calculating the ratio between the

initiation time predicted from the Neuber stress estimate to that obtained from the finite

element stress solution. For this purpose, a stress to rupture time relation has been

assumed such that the initiation time,ti, is estimated from (see e.g.[5])

pit B

= . (8)


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where p and B are the rupture stress exponent and constant, respectively and is therelevant stress estimate (i.e. d or d ). A typical value of p equal to 10 has been

assumed. The results are expressed here in normalised form and are independent ofB.

RESULTS

Maximum Principal Stress PredictionsNeuber-R5 Method

A comparison of the maximum principal stress distributions, d, over a range of crack

tip distances d, obtained from the Neuber-R5 method (Eq. 2) and the finite element

(FE) results is presented in Fig. 1 forN= 10. All stress values are normalised by 0 and

the distance, d, is normalised by crack length, a. The results for ref /0= 0.1 and 1.8

are shown in Fig. 1(a) and (b), which correspond to small scale and large scale yielding

conditions, respectively. At ref/0= 0.1 the stress predicted by the Neuber-R5 method

falls below the FE solution at d/a< 0.004. Very poor agreement between the

Neuber-R5 estimate and FE solution is seen in Fig. 1(b) for ref/0= 1.8.

Based on a standard CT specimen (W = 50 mm), the ratio between the d stress

from the Neuber-R5 method and the FE solution at d= 50 m is shown in Fig. 2(a) forN = 5 and 10 (A distance d= 50 m corresponds to d/a = 0.002 for a standard CTspecimen). The Neuber stress estimate and the finite element solution are represented in

the figure by the subscripts N and FE, respectively. It is seen in Fig 2(a) that a

non-conservative prediction of d(indicated by a stress ratio less than 1) is obtained by

the Neuber-R5 method over the entire load range. The ratio of the respective initiation

time predictions from Eq. (8) is shown in Fig. 2(b). It is seen that the values of

initiation time, ti, predicted from the Neuber-R5 stress estimate for ref /0 > 0.2 are

non-conservative and up to four orders of magnitude greater than the predictions basedon the FE stress solution. Note that the implicit assumption here is that initiation is

controlled by the maximum principal stress and hence in Eq. (8) is equal to d.

Equivalent von Mises Stress PredictionsNeuber and Neuber-R5(A)

A comparison of the d stress from the Neuber method, Eq. (1), the Neuber-R5(A)

method, Eq. (3), and the FE solution is presented in Fig. 3 for N= 10. For ref/0= 0.1

(Fig. 3a) the solutions are almost indistinguishable from each other. At ref/0= 1.8

(Fig. 3b) it can be seen that both methods provide a conservative prediction of the

equivalent von Mises stress distribution, relative to the FE distribution, over the

distances of interest.

In Fig. 4(a), the ratios between the d stress from the Neuber and Neuber-R5(A)

methods and the FE solution are shown over the load range considered, ford= 50 m(based on a standard CT specimen). For ref /0< 1.0 results from both solutions are

virtually indistinguishable, and are slightly non-conservative for ref /0 < 0.2. At

higher loads a conservative prediction is achieved by both methods. It is also seen in


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the figure that the prediction from the Neuber-R5(A) method (the solid line)consistently falls below that of the Neuber method (the dash line) and is significantly

less conservative, though more accurate, at high loads (ref /0 > 0.8). Figure 4(b)

shows the ratios of the predicted initiation times, assuming that initiation is controlled

by the equivalent stress. It is seen that the Neuber stress estimate (Eq. 1), in conjunction

with Eq. (8), leads to a prediction of ti that is more conservative than that from the

Neuber-R5(A) stress estimate and is in fact on the order of four times more

conservative forN= 10 at high loads.

Maximum Principal Stress Predictions Modified Neuber Method

A comparison of the d stress determined by the modified Neuber method (Neuber(M))

is shown in Fig. 5 for the CT specimen with N= 10. At ref /0 = 0.1 (Fig. 5a), the

Neuber(M) stress estimate is almost indistinguishable from the FE solution. At thehigher load of ref /0 = 1.8 (Fig. 5b) the scaling factor used provides conservative

results for d/a > 0.002. In this region the Neuber(M) stress is approximately 20%

higher than the finite element solution.

The ratio between the maximum principal stress in the plane of the crack, d,

estimated by the modified Neuber method and that obtained from the FE solution is

shown in Fig. 6(a) forN= 5 and 10. Results are in all cases conservative, indicated by a

ratio greater than 1. Consequently, conservative initiation time predictions are obtainedby the Neuber(M) stress estimate in relation to the FE solution, indicated by a ratio less

than 1 in Fig. 6(b). It is seen in this figure that initiation time predictions from

Neuber(M) are very close to those predicted from the FE stress at intermediate loads,

and within an order of magnitude of the FE prediction over the load range.

Comparison of Neuber R5 Maximum Principal Stress Estimates to the FE von MisesStress Solution

The ratio between the maximum principal stress estimate at d= 50 m from theNeuber-R5 method, (d)N, and the FE solution for the equivalent Mises stress, (d ) FE

is shown in Fig. 7(a). If it is assumed that initiation time is controlled by the equivalent

stress d , as ine.g.[5], and not by the maximum principal stress, d, as proposed in the

sigma-d method, Fig. 7(a) indicates that the use of the Neuber- R5 method (giving d)

will provide a conservative prediction for the cases considered, under plane strain

conditions. Within the load range (0.2 ref/0 1.4) for N= 10, the ratio

(d )N/ (d ) FE is generally constant at around 1.3 (see Fig. 7a). Therefore using Eq. (8)

and assuming that crack initiation under creep conditions is controlled by the equivalent

stress; the resultant tipredicted from the Neuber-R5 method would be conservative

(underpredict the initiation time) by a factor of around fourteen for this load range,under plane strain conditions, as shown in Fig 7(b) ( i.e. considerably more conservativethan the Neuber(M) method).


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EMAS 2004 7

DISCUSSION AND CONCLUSIONS

Crack tip stress distributions have been predicted by the Neuber method and the full

field solution has been obtained from finite element calculations on a CT specimen.

Under plane strain conditions non-conservative predictions of the maximum principal

stress in the plane of the crack, d, over a range of distances d, have been obtained from

the Neuber method, as proposed in the current R5 procedure [2]. An adapted form of

the R5 Neuber method (Neuber-R5(A)) is proposed, which replaces the maximumprincipal stress with the equivalent von Mises stress. This method provides

conservative estimates of the equivalent von Mises stress d . The method, here

designated Neuber-R5(A), used is not precisely consistent with the original form of

Neubers rule, but it is found that, under plane strain conditions and for the material

properties chosen, more accurate (though less conservative) estimates of d are

obtained using this equation than are obtained using an equation consistent withNeubers original formulation. Encouraging results are also obtained from the

Neuber(M) method, which has been proposed to obtain an estimate of d.

The stress estimates at a distance d= 50 m ahead of the crack tip (based on astandard CT specimen, a= 25 mm) have subsequently been used in the sigma-d methodto predict the time to crack initiation under creep conditions and compared to

predictions based on the corresponding FE stress solution. Only the Neuber(M) stress

estimate leads to conservative initiation time predictions, compared to predictions

based on the FE stress solutions, for ref/0 > 0.2. If failure is controlled by the

maximum principal stress non-conservative estimates of initiation time may be

obtained using the current versions of the R5 sigma-d procedure. However, if initiationis controlled by the equivalent stress, conservative initiation time predictions are

expected from the procedure since the value of the Neuber-R5 stress estimate is found

to be greater than the value of the respective finite element equivalent von Mises stress.The discussion here has been limited to the case when crack initiation is controlled

either by the maximum principal stress or the equivalent stress. However, more

generally, initiation may be controlled by some combination of the maximum principal

and equivalent stress values. For this situation conservative results are expected to beobtained through the combined use of the Neuber-R5(A) and Neuber(M) equations.

ACKNOWLEDGEMENTS

The authors would like gratefully to acknowledge helpful discussions with Dr Manus

ODonnell of British Energy Generation Ltd.

LIST OF SYMBOLS

a, W Crack length and width of specimen, respectivelyB,p Creep rupture constant and exponent, respectively

d Distance directly ahead of crack tip in the plane of the crack


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E, E Youngs modulus, Effective elastic modulusK Linear elastic crack tip stress intensity factor

, 0,N Constants in Ramberg-Osgood plasticity law

d Equivalent (von Mises) stress at a distance dahead of crack tip

d e Equivalent linear elastic stress at a distance d ahead of the crack tip

d Stress normal to crack plane (Maximum principal stress in the crack

plane) at a distance dahead of the crack tip

de Linear elastic dstressref Reference stress

v Poissons ratio

REFERENCE LIST

(1)

Moulin, D., Drubay, B., Acker, D., and Laiarinandrasana, L., A Practical MethodBased on Stress Evaluation (d Criterion) to Predict Initiation of Crack Under

Creep and Creep-Fatigue Conditions,J. Press. Vess-T ASME, 1995, Vol. 117, pp.

16.

(2) British Energy Generation Ltd., R5: Assessment Procedure for the High

Temperature Response of Structures.BEGL., Issue 3, 2003.

(3) Neuber, H., Theory of Stress Concentration for Shear Strained Prismatic Bodies

with Arbitrary Non-Linear Stress Strain Law. J. App. Mech-T ASME, 1961, pp.

544550.

(4) Davies, C. M., ODowd, N. P., Nikbin, K. M., Webster, G. A., Biglari, F.,

Comparison of Methods for Obtaining Crack Tip Stress Distributions in an

Elastic-Plastic Material, Submitted for publication, 2004.

(5)

Webster, G. A. and Ainsworth, R. A., High Temperature Component LifeAssessment.1st ed. 1994, Chapman and Hall: London.

(6) Harkegaard, G. and Mann, T., Neuber Prediction of Elastic-Plastic StrainConcentration in Notched Tensile Specimens Under Large-Scale Yielding. J.

Strain Anal. Eng, 2003, Vol. 38(1), pp. 7994.

(7) Chen, G. X., Wang, C. H., and Rose, L. R. F., A Perturbation Solution for a Crack

in a Power Law Material Under Gross Yielding. Fatigue Frac. Eng. M, 2002,Vol. 25, pp. 231242.

(8) RCC-MR, Guide for Leak Before Break Analysis and Defect Assessment,Appendix A16, 2002, AFCEN.

(9) ABAQUS Users Manual, Version 6.2, Hibbit, Karlsson & Sorenson Inc., 2001.


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FIGURES

d

/0

0.0

1.0

2.0

3.0

4.0

5.0

0.00 0.01 0.02 0.03 0.04

d/a

FE

Neuber - R5

ref /0 = 0.1

(a)

N= 10

0.0

2.0

4.0

6.0

8.0

10.0

0.00 0.01 0.02 0.03 0.04

d/a

FE

Neuber - R5

ref /0 = 1.8

(b)

N= 10

d

/0

Figure 1 Comparison of normalised dstress predictions from Neuber-R5 and FE for

(a) ref/0= 0.1 and (b) ref/0= 1.8.

(d

)N

/(d

)FE

ti(d

)N

/ti(d

)FE

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8ref /0

N= 10 N= 5

(b)

d = 50m

a = 25mm

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8ref/0

N= 10

N= 5

(a)

d = 50m

a = 25mm

Figure 2 Ratio between (a) dstress and (b) tiprediction from Neuber-R5 and FE.


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d/

0

0.0

0.5

1.0

1.5

2.0

2.5

0.00 0.01 0.02 0.03 0.04d/a

FE

Neuber

Neuber - R5(A)

ref /0 =0.1

(a)

N = 10

0.0

0.5

1.0

1.5

2.0

2.5

0.00 0.01 0.02 0.03 0.04d/a

Neuber

Neuber - R5(A)

FE

ref /0 =1.8

(b)

N = 10

d/

0

Figure 3 Comparison of normalised d stress predictions Neuber-R5(A) and FE for

(a) ref/0= 0.1 and (b) ref/0= 1.8.

ti(d

)

N

/ti(d

)

FE

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8ref /0

Neuber

Neuber R5(A)

N= 10N= 5

(b)

d = 50m

a = 25mm

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8ref/0

N = 10

N= 5

Neuber

Neuber - R5(A)

(a)

d = 50 m

a = 25 mm

(d

)

N

/(d

)

FE

Figure 4 Ratio between the (a) von Mises stress and (b) initiation time prediction

from the Neuber method, the adapted R5 Neuber method and FE.

d

/0

0.0

1.0

2.0

3.0

4.0

5.0

0.00 0.01 0.02 0.03 0.04

d/a

Neuber (M)

FE

(a)

ref /0 = 0.1

N = 10

0.0

2.0

4.0

6.0

8.0

10.0

0.00 0.01 0.02 0.03 0.04

d/a

Neuber (M)

FE

ref /0 = 1.8

(b)

N = 10

d

/0

Figure 5 Comparison of normalised d stress distributions from Neuber(M) and FE

for (a) ref/0= 0.1 and (b) ref/0= 1.8.


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(d

)N

/(d

)FE

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8ref /0

N = 5N = 10

(a)

d = 50 m

a = 25 mm

ti(d

)N

/ti(d

)FE

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8ref /0

N= 10 N= 5

(b)

d =50m

a =25mm

Figure 6 Ratio between (a) dstress and (b) t

iprediction from Neuber(M) and FE.

(d

)N

/(d

)

FE

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8ref/0

N= 10

N= 5

d = 50 m

a = 25mm

(a)

ti(d

)N

/ti(d

)

FE

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8ref /0

N= 10

N= 5(b)

d = 50m

a = 25mm

Figure 7 Ratio between (a) dfrom Neuber-R5 and d from FE and (b) theirrespective initiation time predictions.

neuber method for fatigue

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