Failure assessment diagram (FAD) for I–II mixed-mode crack structures under biaxial loading

Im. J. Pm. Ves. & Piping 69 (tY96) 53-58 Copyright 0 1996 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0308-0161/96/$15.00

0308-0161(95)00076-3

Failure assessment diagram (FAD) for I-II mixed-mode crack structures under biaxial loading Weng-Long Huang & Jin-Zhu Tan Nanjing

Irwtitute

of Chemical

Technology,

Nanjing,

Jiangsu, 210009, People’s

Republic

of China

(Received 4 August 1995: accepted14 August 1995)

In this paper the results of fracture experiments under biaxial loading and theoretical analysisare used to study the fracture behaviour of a cruciform specimen,made of Chineseindustrial standard 16MnR steel, with a central penetrated I-II mixed-mode crack under biaxial loading. The failure assessment diagram (FAD) for evaluating the safety margin of structurescontaining I-II mixed-mode cracks under biaxial loading is explored. The fracture parameter J-integral of I-II mixed-mode cracks is calculated by the elastoplastic finite-element method and the instantaneousloads for the initiation of I-II mixed-mode cracks are detected by DC electric potential measurement. Modified methodsfor the failure assessment diagramof CEGB R6 and EPRI engineeringapproach for biaxial loading and inclined crack angle have been proposed in the present work. The experimental results reveal that the J-integralsbasedon the measuredinstantaneousload of initiation of growth of I-II mixed-mode crack, 4, are almost equal under different ratio of biaxial load. The instantaneousload of initiation predicted by mixed mode J-integral fracture criterion showsreasonableagreementwith the experimentalresults.It is however found that the modified failure assessment diagramproposedEPRI method for biaxial loading is more in agreementwith the experimental results. Copyright 0 1996Elsevier ScienceLtd.

NOMENCLATURE

semi-crack length (mm) initial semi-crack length (mm) increment of crack length (mm) stress in the x and y direction (MPa) ultimate stress, yield stress and effective stress (MPa) limit stress and limit load (MPa, kN) strain hardening parameters initiating load (kN) effective stress (MPa) effective strain ratio of biaxial loads, u.~/u-” ratio of biaxial stress inclined crack angle from the direction of the maximum principal stress to the crack plane (deg.) crack initiating angle (deg.) J resistance curve (kJ/m2) elastic component of J (kJ/m2)

plastic component of J (kJ/m’) fracture toughness (MPa(m)“‘) stress intensity factor relative to fracture toughness load (or stress) relative to limit load (or limit stress) K,, Ku, K,,, mode I, II, III linear elastic stress intensity factor, respectively K rff effective linear elastic stress intensity factor in mixed mode loading 1 INTRODUCTION

In recent years, several methods have been proposed to assessthe safety of certain structures which were made of medium or low strength and high toughness materials commonly used in the chemical and nuclear industries. Two important methods are the CEGB R6’ and the EPRI JFAD.2 The basic procedure recognizes both

54

Weng-Long

Huang, Jin-Zhu

Tan

brittle fracture and plastic collapse of the structure, using the format of a failure assessment diagram, to assess the integrity of structures containing defects. In this paper the results of the fracture experiment under biaxial loading and theoretical analysis are used to study the fracture behaviour of a cruciform specimen made of a Chinese industrial standard 16 MnR steel for pressure vessels, with a central penetrated inclined crack, and the failure assessment diagram (FAD) for evaluating the safety margin of structures containing I-II mixed-mode cracks under biaxial loading is explored. .

320------/ Fig. 1. Test specimen.

2 EXPERIMENT 2.1 Material

and test specimen

The mechanical properties and the chemical composition of the 16MnR steel obtained by tests are represented in Table 1. The specimen is of a cruciform type, containing a centre notch which is inclined at an angle, p, to the vertical direction, and is machined through the thickness of the plate. The details of the test specimen are shown in Fig. 1.

In fracture tests, three different crack lengths and inclined angles were selected, see Table 2. The fracture tests were carried out at a ratio of biaxial load k = 0.5, to study the effect of the second stress and inclined angle of crack on initiation of the crack. In the test, the initiating load of the crack and the initiating angle of the crack were measured. The crack opening displacement, COD, was also measured by clip-gauges during loading.

2.2 The fracture loading

2.3 Determination

experiment

under

biaxial

An experiment was performed on a 2.5ton biaxial fatigue test machine. The loads in the X, Y axes were imposed by ramp wave, and controlled by a function generator. The loading speed was 139Nlsec. The test machine and equipment are shown in Fig. 2. Table

1. Mechanical

properties and chemical of 16 MnR steel

of crack initiating

load

The specimen’s thickness is 5 mm, so the specimen is taken as under plane stress condition. Local DC potential measurement method was used to detect the initiating load. The initiating loads are tabulated in Table 3.

composition

Mechanical properties CY (M?a)

(&a)

(M?‘a)

g,

&,

386.6

571.9

2.1E5

25.0

56.2

0.98

n 8.24

Chemical composition (G,

(C,

(yif)

0.15

0.51

1.4

(4, 0.009

(Gil, 0.02

Fig. 2. The testing machineand equipment.

Failure assessment Table

2. Crack lengths and inclined

diagram for biaxial loading

55

Table 4. Comparison

angles

Test specimen

A

B

C

Crack length 2a (mm) Inclined angle p (“) Specimen number

58 30 5

44 45 9

36 60 5

Specimen DC potential Multisection Error (“da)

of crack initiation

methods

no.

Bl

B3

B4

B5

B6

.I, (kJ/mZ) .I, (kJ/m’)

52.0 51-2 l-5

53-5 51-2 4.49

53.0 51.2 3.5

51.5 51.2 0.6

50.0 51-2 -2.3

By comparing these results with the data of multisection method, it was found that the initiating J-integral is Ji = 51.2 KJ/m’; maximum error was 4.49% (2a = 44 mm, p = 45”). The results are in good agreement. It shows the reliability of the local DC potential method. Comparison between DC potential and multisection methods is shown in Table 4.

I-mode crack structure’s FAD and the material properties Klc, cr,, incorporating the effects of II-, III-modes into the effective stress intensity factor KeK and the plastic yield load. The failure assessment curve of the I-II mixed-mode crack structure is based on

2.4 Determination

where

of the crack initiating angle

The initial angle of I-II mixed-mode cracks was measured under X50 optical microscope. In figure 3, OA is the crack growth length, 8, is the crack initiating angle, n is the open angle of original crack, y is the open angle of expanded crack. The crack initiating angles are tabulated in Table 5. 3 FAILURE ASSESSMENT DIAGRAM FOR I-II MIXED MODE CRACK STRUCTURE UNDER BIAXIAL LOADING

The effective stress intensity factor Keff is defined bY Kefl = [K:(a) + ti,(a)]“’ (3) Considering the plate under biaxial loading shown in Fig. 4, gaPp can be replaced by the effective (T, u, = [$[(U.” - u*)2 + a: + a;]]“” (4) By substituting k = o-,/u, into eqn (4), the expression for (T, is obtained when k = 0: u, = cry = u when k = 0.5: v3 ue=----u~=-u 2

The appendix VII of CEGB (Rev. 3)3 proposes a failure assessment analysis diagram based on the 3. Initiating

Specimen no.

k

Al A2 A3 A4 B2 Bl B3 B4 B5 B6 C4 Cl c2 c3

0 o-5 0.5 0.5 0 o-5 0.5 o-5 0.5 0.5 0 0.5 0.5 0.5

loads and initiating

di.) 30 30 30 30 4.5 45 45 45 45 2 60 60 60

s,> syaxJ

K, = 0

3.1 The CEGB failure assessment method

Table

s,spax (1)

J-integral

2a (mm>

(k&

(k.JTm’)

58-23 58.31 58.41 58.20 44.50 44.13 44.29 44.83 44.44 44.40 36.27 36.51 36.315 36.44

224 220 204 207 203 206.8 211.7 211.2 206 203 194 206.25 209 203.75

51.0 x,3 52.3 52.1 53.0 52.6 53.5 53.0 51.5 50.0 53.0 52.0 51.5 51.2

ti 2

The limit stress u0 of the square plate containing cracks in eqn (2) is obtained by the elasto-plastic finite-element method. The modified FAD can be derived from eqn (1) when k = 0, 0.5 and p = 30”, 45”, W, see Figs 5-7.

By substituting experimental data into the related equations, K, and S, can be evaluated. By comparing the modified failure assessment curve

Fig. 3. The crack initiating

angle.

56

Weng-Long Huang, Jin-Zhu Tan Table 5. The crack initiating

angle q k=O.SEXP

Specimen no.

A3 B4

Cl Al I32 c4

k 0.5 0.5 0.5

0 0 0

(dfg.) 30 45 60 30 45 60

O,-left crack tip; B,right

(dfk.)

(d:;.)

55 44 40 66 45.5 43.5

47 39 38.5 65 50.5 47.5

8” (mean) (deg.1 51 41.5 39-2 65.5 48 45

I

3.2 EPRI failure assessment method For a material obeying the Ramberg-Osgood law, He Ming Yuan4 suggested the value of J in a structure containing an inclined crack as (5)

0.2

IIIIIh 0.4

0.6

Ill 0.8

1.0

1.2

1.4

1.6

1.8

2.0

Sr 5. Comparison between the modified FAD experimental resultsat p = 30”.

Fig.

with the test data assessment points (K,, Sr) it is found that the test data points are far away from the curve. The tests above reveal that the modified failure assessment curve is the lower limit curve and gives a conservative prediction.

me)

0.4

0

crack tip.

J = aa;e;h(n,

0.6 2

and

which was introduced by Shich,’ is used here. The definition of M” is

(7) h(n, M”) = ---!-U,E,LY According to this definition, M’ ranges from 0 to 1, with Me = 0 for mode II case and M” = 1 for mode I. For a square plate, the expression of a& is (see Fig. 4): u;&l = 0) = z” [(l + k) + (k - 1)cos 2p1

(8)

(6)

and the expression of aTB is k-l u sin 2p a;,(8 = 0) = (9) 2 By substituting &, aTo into eqn (7), M” can be written as (k + 1) + (k - 1)cos 2p (lo) Me = 2 arctan ?c (k - 1)sin 2p

where S, and (T, (= $S’ijS;j)“’ are the stress stress, respectively. and effective deviator Coefficient (Y and exponent yz are material constants. In mixed mode crack the parameter M”,

The values of H(n, M”) at k = 0, 0.5; /3 = 30”, 45”, 60” are tabulated in Table 6. It can be seen from the above discussion that an engineering estimation scheme for J can be used to evaluate the elasto-plastic J-integral for infinite plate containing inclined cracks. According

where a:, E; are the remote effective stress and effective strain, respectively. By considering a material in the multi-axial stress state, the Ramberg-Osgood law may be described by 5-3 - - a(UJU,)n-‘SjjlUS Es 2

1

0.8 0.6 2

0.4 0.2 0

Fig. Fig. 4. The squarethin plate.

Sr 6. Comparison between the modified FAD experimental resultsat p = 45”.

and

Failure assessment

diagram for biaxial Loading

57

1 .o 0 k=0.5EXP 0.8 0.6

0.8

-

0.6

-

0.2

-

---k=OEST

d

2

A

0.4 0.2 0

0.4

0.2

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0

2.0

k=OFEM k=OSFEM

q

I

I

I

I

I

I

I

I

I

l

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

sr

Sr

Fig. 7. Comparison between the modified experimental results at p = 60”.

FAD

and

Fig. 9. Comparison between the modified .I FAD

and

elasto-plasticfinite-element resultsat p = 30”.

to eqn (5) the elasto-plastic J-integral, JeP, is as follows. Jep= J, + Jp = a!a,e,h(l, M’)a(a,/aJ + ~~Oa,h(n, Me)u(ae/a,)“+’

(11) 0

0.2

0.4

0.6

0.8

The normalized coordinates are defined by

Then the J-integral controlling the failure assessment curve may be summarized as

According curve can load and 8-13. By

1.2

1.4

1.6

1.8

2.0

sr

(12)

K = (Je$E$‘”

1.0

=f(Sr)

(13)

to eqn (13), the failure assessment be plotted for different ratios of biaxiaI crack inclined angles, shown in Figs comparing the test data with results

Fig. 10. Comparison between the modified J FAD and

experimented resultsat p = 45”.

calculated by the elasto-plastic finite-element method, it is found that the J FAD results agree more with the experimental data whereas the results detected by the finite-element method give agreement only near the crack initiating margin; errors appear outside the range. The main reason is that the constants (Y and n of Ramberg-Osgood law were applied near the crack initiating point Ji. Another reason is that 1.0

Table

6. Values

0.8

of h(n, M’)

0.6

n

k = 0 for p (deg.)

1 8.24

k = 0.5 for p (deg.)

30

45

60

30

45

60

0.986 2,38

1.68 6.53

2.14 11.25

2.38 17.6

2.42 17.9

2.84 22.2

b-i-

0.4

A _

0.2 0

k=OFEM k-0.5 FEM

q

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Fig. 11. Comparison between the modified J FAD

and

elasto-plasticfinite-element resultsat p = 45”. 1.0 0.8 0.6 SC*

- - -k=OEST A 0

0.2 0

0.6 SC*

0.4

k=OEXP k=OJEXP

0.4 0.2

I 0.2

I 0.4

I 0.6

I 0.8

I 1.0

I 1.2

I 1.4

I 1.6

I 1.8

] 2.0

experimented resultsat /3 = 30”.

_

I 0

sr Fig. 8. Comparison between the modified J FAD

-

---k=OEST -k = 0.5 EST A k=OEXP 0 k=0.5EXP

I

I

I

I

I

I

I

I

I

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1 2.0

%

and

Fig. 12. Comparison between the modified J FAD and

experimentedresultsat p = 60”.

58

Weng-Long

Huang, Jin-Zhu

0.8 0.6 ti-

---k=OEST -k=OSEST 0.4 A k=OFEM n k=O.SFEM 0.2 _ I I I I I I I I I I 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 s*

Fig. 13. Comparison between the modified J FAD and elasto-plasticfinite-element resultsat j3 = 60”.

for finite square plate the free edge effect on the stress field of the crack tip was neglected. It is necessary to modify the results with a coefficient f(2a/b) where f(2alb) > 1; further work is required in this aspect. Results obtained with an infinite plate give conservative results for finite width plates. 4 CONCLUSION

AND DISCUSSION

It should be emphasized that all the experimental and theoretical analyses are based on biaxial cruciform specimens with a centre-penetrated I-II mixed-mode crack structure. For a reliable assessment of the integrity of pressure vessels containing defects, the effective coefficient of expansion of the inclined crack has to be considered.

Tan

The experimental results reveal that (1) The method of using DC electric potential to measure the initial load under biaxial loading is acceptable. (2) The CEGBs modified FAD gives a conservative curve; the investigation of this paper reveals that the CEGBs modified FAD for evaluating the safety margin of I-II mixed-mode crack structure is acceptable. (3) By comparison, it is found that the modified failure assessment diagram of the EPRI method proposed for I-II mixedmode crack structure under biaxial loading agrees better with the experimental results than the other approach.

REFERENCES 1. Harrison, R. P., Loosemore,K., Milne, I. & Dowling, A. R., Assessmentof the integrity of structure containing defects, CEGB R/H/R-Rev 2, 1980. i. Bloom, J. M. & Malik, S. M., Procedure for the assessment of integrity of nuclear pressurevesselsand piping containing defects, Research project 1237-2, EPRI, 1982. 3. Milne, I., Ainsworth, R. A., Dowling, A. R. & Stewart, A. T., Assessmentof the integrity of structure containing defects, CEGB RIHIR6-Rev 3, 1986. 4. He Ming Yuan, Fully plastic analysis of mixed mode plane strain crack problem, JWASTM STP, 803 (1983). 5. Shich, C. F., Fracture Analysis, ASTM STP 5601974, 187-210.