Elastic-plastic defect assessment based on ductile fracture process

Elastic-plastic defect assessment based on ductile fracture process

Nuclear Engineering and Design 142 (1993) 27-41 North-Holland 27 Elastic-plastic defect a s s e s s m e n t b a s e d on ductile fracture process * ...

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Nuclear Engineering and Design 142 (1993) 27-41 North-Holland


Elastic-plastic defect a s s e s s m e n t b a s e d on ductile fracture process * Y.M. Dong, W. Y a n g and K.C. H w a n g

Department of Engineering Mechanics, Tsinghua University, Beijing 100084, People's Republic of China Received 26 November 1992

A systematical scheme effective to the defect assessment of reactor pressure vessels and pipes is proposed in the present paper. Theoretical fracture resistance curves for power law strain hardening materials are proposed, which are characterized by two fracture parameters, namely the fracture toughness Jlc and the initial tearing modulus Tc. The defect assessment method constructed upon the theoretical resistance curves is then referred to as the "Two Parameter Assessment Scheme" for ductile fracture processes, in which the tedious measurement on resistance curves is replaced by the tabulated data of Jlc and Tc. A blunting correction is advanced which enables the extension of the present methodology to pressure vessel steels of high toughness. The versatility of the two parameter assessment method is demonstrated by its four different representations, and its applicability is verified by measured burst pressure data of 6 model vessels. Approximate but analytical expressions for J integral pertinent to a variety of crack geometries are provided, whose accuracy satisfies the requirement of engineering assessment.

I. Introduction T h e d o m i n a n t failure m o d e for various engineering structures is characterized by ductile fracture processes. Consequently, the predictions about ductile fracture instability consist of a critical link in the structure integrity assessment of reactor pressure vessels and pipes. Considerations on ductile crack growth procedure, featuring the material strain hardening and the safety margin offered by stable crack growth, have b e e n i n c o r p o r a t e d in the newly revised editions of various influential assessment schemes in the world [1,2,3]. T h e recent research trend clearly shows the convergence of the C O D design curve m e t h o d [2], the C E G B failure assessment diagram m e t h o d [2] and the E P R I engineering estimate m e t h o d [3]. A systematic investigation on the elastic-plastic defect assessment m e t h o d for pressurized reactor vessels and pipes is u n d e r t a k e n by the authors u n d e r the e n t r u s t m e n t of the National Nuclear Safety Administration of China. T h e aim of this p a p e r is to p r o p o s e a new m e t h o d o l o g y which combines the basic scientific p r o c e d u r e of E P R I fracture instability assessment and the engineering appealing of C O D design curve and C E G B failure assessment diagram. T h e m e t h o d is based u p o n the theoretical fracture resistance curves for power law strain hardening materials [4] to simplify the assessment procedure. T h e theoretical resistance curve is characterized by two fracture parameters, namely the fracture toughness JIc for crack initiation and the initial tearing m o d u l u s Tc. T h e m e t h o d so developed is t e r m e d as the " T w o P a r a m e t e r Assessment S c h e m e " for ductile fracture process. T h e advantage of this m e t h o d o l o g y lies in the r e p l a c e m e n t of the tedious m e a s u r e m e n t of resistance curves by easily attainable values of Jic and T c. By taking into account the correction on crack tip blunting e m b e d d e d in the J testing procedure, we are able to extend the theoretical resistance curves to the pressure vessel steels ranging from low to high toughness. T h e versatility of the two p a r a m e t e r assessment scheme is * Sponsored by the National Nuclear Safety Administration of China under the research project "Defect Assessment of Pressurized Reactor Vessels and Pipes". 0 0 2 9 - 5 4 9 3 / 9 3 / $ 0 6 . 0 0 © 1993 - Elsevier Science Publishers B.V. All rights reserved


Y.M. Dong et a L / Elastic-plastic defect assessment

illustrated by its four variations, namely the intersection method of driving and resistance curves, the design curve method, the failure assessment diagram and the method of admissible stress curve. The four alternative forms of the present methodology unify the guidelines of several influential defect assessment schemes. The theoretical predictions under this approach are checked with the measured burst pressure data of 6 model vessels, where satisfactory agreement is found between theory and experiment. Approximate but analytical expressions for J integral pertinent to a variety of crack geometries are proposed, and an estimate based on the experimentally recorded failure stress ~rf and the largest attainable crack growth Aafis formulated. Under this estimate, the predicted burst pressures of six axially cracked cylinders agree well with the experimental measurements. The following sections will be devoted to the discussions on theoretical resistance curve, two parameter assessment scheme and J integral estimate.

2. Theoretical resistance curves

2.1. Theoretical resistance curves for power law hardening materials The structural steels used in reactor pressure vessels and pipes can be characterized by the following Ramberg-Osgood constitutive law between the strain tensor 6ij and the stress tensor ~j:



(1 + v) °riJ . .




3ij +




where e 0 and ~r0 denote the reference yield strain and yield stress of the material, with ~o = Eeo. Tensor indices have the range from 1 to 3, and summation is implied for repeated indices. E and u are Young's modulus and Poisson's ratio of the material, sij, a 0 and n in turn represent the stress deviator, hardening coefficient and hardening exponent. The effective stress o"e is defined in the J2 sense. If an infinite strip set along x 2 direction is loaded by remote uniform tensile stress ~r22 under plane strain condition, its response can be described by (2)

e = o- + oto .n,

where e--



2(1 - u 2) 60 ,




3% 4 ( 1 - v 2)


From an asymptotic investigation on a growing crack-tip field, the crack opening profile for unsteady crack growth in a Ramberg-Osgood material could relate to the J integral and its derivative with respect to the cracklength a. As represented by Luo, Yang and Hwang [4] (1987) and by Hwang and Yang [5] (1989),

6 = Br

(~-~r)-l/(n+1)1-~o--dadJ+ __/3~r°rEIn (2"718AEJ) "/(n-',~r2r


/3 and A in (4) are given in table 1 for selected values of a and n. The B

2n (n + 1)I,




blunting factor B in (4) is given by (5)

Y.M. Dong et al. / Elastic-plastic defect assessment


Table 1 Parameters to describe theoretical resistance curves


fl(a = 0.1)

/3(a = 1.0)

5 9 12 oo

-4.08 -

2.65 3.52 3.61 5.30

h(a = 0.1)

)t(ct = 1.0)




0.25 -

0.43 0.29 0.27 0.16

1.08 0.95 0.91 0.79

5.02 4.60 4.44 3.72

2.368 2.063 1.944 1.480

where U2(~r, n) and I n are the known functions of n, and they are furnished by the H R R solution for a stationary crack. Their typical values for different n are included in table 1. Following the argument by Rice and Sorensen [6] (1978), the theoretical J resistance curve for elastic-plastic materials could be formulated by a constant crack profile criterion, = 6 c at r = r c








where 6 c and r c are the material parameters characterizing ductile fracture, and the dimensionless constant/~ ranges from 0.5 to 1.5. The quantity d was calculated from H R R solution as

where D n is also listed in table 1 for different n. It was shown by Yang [7] (1987), Yang et al. [8] (1988), and Hwang and Yang [5] (1989) that equations (4) and (6) can be cast into the following two parameter form:

I 'l]

- d3 = 1 ~


f~c - fxl(n+l)[(ln JR)n/(n-1)_ (In JIc)

where various dimensionless quantities are: ^ JR fR=JIcj--~c'

2.718A ~e0-~'


2~rB (2.718A)l/(n+DTc 7~c=



~ =

flo" o



The subscript " C " represents crack initiation. In (9), the initiation fracture toughness Jic and the normalized initial tearing modulus Tc

27r%2 ~



are the two fracture parameters, signifying the material resistance to crack initiation and growth, to define the theoretical resistance curve (8).

2.2. Blunting correction of materials of high toughness It is recognized that the basic expression (4) for crack opening displacement stands for a simple superposition of the stationary and the steady state crack solutions. As remarked by Dong, Yang and


Y.M. Dong et al. / Elastic-plastic defect assessment

Hwang [9] (1990), the theoretical resistance curve (8) is applicable only if the following restriction is satisfied

¢c <

"(n -+ n- 1

f,c) 'j(°-l', IC



which ensures the shape of theoretical resistance curve as convex upward. Some structural steels, such as 18CrNiWA, satisfy the condition (11), henceforth the experimental measurements for J resistance curve agree well with the theoretical prediction (8), e.g., Hwang and Yang [5] (1989). However, the condition (11) fails to observe for steels of high toughness. For instance, our experimental JR curve for A533B-1 disagrees substantially with the theoretical prediction (8) after small amount of crack growth. This discrepancy could attribute to two reasons: (1) the account on blunting implied in the superposition (4) is inconsistent with the testing procedure of J integral; and (2) the breakdown of J-controlled crack growth assumption. The second source was explored recently by Wang, Yang and Hwang [10] (1992), and whose result alone could not explain the substantial difference between the steels of low or moderate toughness (and high yield strength) such as 18CrNiWA, and the steels of high toughness (and low yield strength) such as A533B-1 and 18MnNiMoNb. A plausible way to modify the theoretical resistance curve is to scale down the initial blunting effect which should diminish after considerable amount of crack growth. As indicated in equation (4), the initial blunting effect is proportional to the blunting factor B. The reduction of B would abate Tc and consequently lead to the satisfaction of (11). Furthermore, the asymptote of the J resistance curve, Js~, would also decline dramatically by the reduction of B, as exemplified by the extreme case of perfect plasticity. Then equation (8) and the third expression of (9) would yield 7~c = In


2 ~-B - -Tc. Jic ]3


The above discussion suggests an effective way for the blunting correction of the theoretical resistance curve (8). That is, the blunting factor B defined in (5) only suits for low toughness materials. For high toughness material, B should be reduced to subdue the relative contribution from the stationary crack solution. Accordingly, B should maintain at the value defined in (5) for low toughness materials (/~c << 7~c) and be reduced by a factor q < 1 for high toughness materials (Tc >> 7~c), in order to satisfy the condition (11). We are now in the position to generalize the theoretical resistance curve (8) previously valid for low toughness materials to almost all structural steels, by modifying Tc as: 7~o d

27rfB (2.718A),/(,+l)Tc,



where the reduction factor f is roughly interpolated from unity for extremely low toughness steels to about q for extremely high toughness materials,


l 2 ql+q l-q -


+tanh A


)1) .


The parameters A and/~ characterize the speed and the location for the f(Tc/7~c) curve to change from q to 1, respectively, and they are determined by correlating the experimental resistance curve data from a spectrum of structural steels.

Y.M. Dong et al. / Elastic-plastic defect assessment


Table 2 Material characterization of four structural steels (the units for ~r0 and Jlc are M N / m e and MN/m, respectively, the others are dimensionless) Materials







A533B-1 18MnNiMoNb 18CrNiWA 38CrMoAI

10 17 22.2 6.33

425 572 1030 850

0.2 0.2 0.1 0.093

54.1 42.6 3.5 8.34

78.74 52.10 3.94 26.50

0.35 0.57 8.34 1.18




7 ~ °d

0.136 0.186 0.030 0.035

0.164 0.247 1.000 0.493

12.90 12.87 3.94 13.10

The ratio q in (14) relies on the properties of tough materials. A simple estimate on q is provided in the following. For extremely tough materials, the resistance curve (8) is strongly influenced by the first term which offers the blunting contribution to 6 from the stationary crack solution. This blunting effect, however, is subtracted in the data processing on the required testing procedure of J resistance curve measurement. To correct this difference between the theoretical model and the testing standard, the factor q in (14) should stands for the ratio between the slope of the blunting line (namely J i c / A a c ) and the slope of the experimental J resistance curve: q

=w( Aa c

dJ R

The apparent crack growth amount of a fully blunted crack is, Schwalbe, Neale and Ingham [11] (1988), Jic

Aa c = 0 . 4 d - -


The combination of the above two expressions furnishes the final expression of q, q = 0.8~e0dTc


and obviously, different materials will have different q values. 2.3. Experimental verification After the determination of factor q, we are able to correlate the parameters A and /7 from the experimental resistance curves for the structural steels listed in table 2, A=2,




1.0 0.8 0.6 0.4





0.2 0.0







Fig. 1. Reduction factor f versus toughness ratio/~c / 7~c- (1) 18MnNiMoNb; (2) A533B-1; (3) 38CrMoAI; (4) 18CrNiWA.


Y.M. Dong et al. / Elastic-plastic defect assessment

2.5 2.0 1.5 1.0 0.5












°'°o ~a(mrn) 5 i0 Fig. 2. Comparison between the theoretical J resistance curves (dashed lines) and the experimental measurements (solid lines). (1) 18MnNiMoNb; (2)A533B-1; (3)38CrMoAI; (4) 18CrNiWA.

From them one can proceed to calculate the reduction factor f in (14). The f versus T c / T c curve is plotted in fig. 1 for four different structural steels. Using the modified values of the dimensionless initial tearing modulus given in (13), we are able to compute the theoretical J resistance curves for four structural steels listed in first column of table 2. As shown in fig. 2, the prediction of equation (8) agrees satisfactorily with the experimental data, even under large amount of crack growth. Various materials parameters involved in the definition of theoretical resistance curves are listed in table 2. Consequently, the theoretical resistance curve (after blunting correction) may be used in the defect assessment of structural components made of various steels with confidence.

3. Two parameter assessment scheme

As described in the previous section, a theoretical resistance curve can be constructed analytically according to (8) for structural steels, provided two fracture parameters, namely the fracture toughness Jtc and the initial tearing modulus Tc, are known. Based on the theoretical resistance curve, defect assessment of a ductile fracture process becomes defect assessment by two fracture parameters. The assessment methodology of this feature is termed the two parameter assessment scheme or J - T assessment scheme. Four variations of this scheme will be described below.

3.1. Intersection of driving and resistance curves (intersection method) The EPRI defect assessment methodology [3] formulates the crack growth instability by a common tangent criterion,

J(a,cr) =JR(a--ao)


T= T R,


where a 0 represents the initial crack length, and the driving and resistance tearing moduli T and T R are defined by

E [ l[aJ~, E dJ R T= 27ro'------~ -~a ],, TR = 27r~ da


Y.M. Dong et al. / Elastic-plastic defect assessment


The determination of crack instability relies on successive searching of the common tangent of crack driving and resistance curves. Sizable error might be caused by the graphical common tangent construction. An alternative to the tangent method, featuring an intersection construction of the driving and resistance curves, was advanced by Yang [7] (1987). The basic reference configuration for fracture assessment is chosen as the infinite CCT plate with half crack length a~, where the subscript o0 is used to label the quantities for this configuration. From dimensionality, the J integral for the reference configuration is a linear homogeneous function of a~. The dimensionless driving forces qb and T~ are defined by @=

E J~ 2~-o-2 a= -F=(troo),


E dJ~ 2zro"ff daoo"


Obviously, ~= = T=, as suggested by the linearity of Joo with respect to the crack length. For the actual structure components, the dimensionless J integral and the tearing modulus T are denoted by E


2rrtr z a - F ( t r , g ) ,

E dJ T = 2~.tr2 d a '


where g represents a collection of dimensionless geometric parameters. According to the J equivalence principle for crack growth, Yang [7] (1987), Dong, Yang and Hwang [9] (1990), J = Joo when tr = tr=. When converted to the basic reference configuration, the equivalent crack size of a is given by am

F(cr,g) F=( tr-----~a ,


and the tearing moduli of the reference and the actual configurations are linked through T = T~( -~-a da= ]]~"


The driving curves q~ and T are plotted in fig. 3 versus the normalized strain e defined in (3), combined with the resistance curves (denoted by q~R and TR) established in the previous section. The characteristic points e i and ef in fig. 3 signify the crack initiation and failure instability strains, respectively. The fracture instability point, as phrased in (19) by EPRI common tangent method, is now redefined by the intersection of T and TR curves, as delineated in fig. 3. An example of crack growth instability

~,T 7"c

oc1 1 0





Fig. 3. Defectassessmentdiagramfor the basic referenceconfiguration(curveintersectionapproach).

Y.M. Dong et al. / Elastic-plastic defect assessment


T , TR aO 50 40 30 20

0.8~ /

i0 I



= 1.16

o'f :













Fig. 4. T e a r i n g instability o f a c t u a l c r a c k e d s p e c i m e n ( S E C P , p l a n e strain). L e f t curves: at) = 0.1 m, o-f = 0.88; r i g h t curve: a 0 = 0.05 m, O'f = 1.16.

assessment by the present curve intersection method is performed in fig. 4 for SECP specimen. The material selected is A533B-1 with the following material properties: a = 1.14, n = 10, tr 0 = 425MPa, e o = % / E = 0.002, Jic = 0 . 2 M N / m , Tc = 57.3. The SECP specimen has a total width of 0.4 m and two initial crack lengths of 0.05 m and 0.1 m. The solid curve in fig. 4 represents the dimensionless resistance curve T R of the specified material, whereas the dashed curve denotes the driving force T for the actual configuration. Both of them are plotted with respect to the normalized stress or, so their appearances are different from the conventional resistance and driving curves plotted against the crack growth a a . The result of fig. 4 indicates that the curve intersection approach is highly accurate, with a deviation from the exact E P R I result of less than 2%. 3.2. Design curve approach

Based on the theoretical J resistance curve (8), with derived from the criterion of J-controlled crack growth curve so derived is comparable to the conventional C O D Referred to fig. 3, the initial values of ( ~ a and T R a r e

Tc modified by (13), a Design Curve can be instability. The form of the defect assessment design curve. denoted by q~c and To respectively. The ratio (25)

y = Tc/CI2 c

is termed the initial tearing toughness ratio which depends also on the initial crack length a 0. According to the value of Y, all ductile fracture processes can be classified into two categories. (1) Y < R. R = rar d is a revision factor, r a is the transformation ratio between the equivalent crack length a~ and the actual crack length a

/ ra

\ da

1,~ =af

=£ .... f


and is introduced to describe the effect of geometric change on crack instability at equivalent failure crack length af. All quantities with subscripts f describe their values at failure. The last equality in (26) also comes from the J-equivalence principle of crack growth. The second factor r d was introduced by Yang et al. [8] (1988) to model the effect of high strain gradient near the defect nucleation region. In the case of Y < R, q' = qbc as the remote normalized strain e (the abscissa in fig. 3) reaches e i, resulting in crack initiation. Because the driving force T has long surpassed T R, instability and catastrophic failure occur immediately after initiation.


KM. Dong et al. / Elastic-plastic defect assessment

(2) Y > R. This is exactly the case delineated in fig. 3, and encompasses most ductile fracture problems. (/) = q)c when e = el, crack is initiated but cannot propagate, due to the lack of driving force shown by the gap in fig. 3 between T and T R. As the applied strain e gradually increases, q)R increases continuously along the q) curve and T R decreases according to the theoretical resistance curve (8) under stable crack growth. When e reaches el, the value of instability strain, T R and T curves intersect, and instability occurs. The second case suggests a growth induced toughening factor y defined by Y = q)f/q)c.


From the theoretical resistance curve (8), an equation governing the crack growth instability can be derived as Y-RY



(In J1C)


for small amount of crack growth. Equation (28) possesses a positive real root only if Y > R. The design curve can be derived by

c~ = qg( ef ) / y < q~c


and a computational scheme for q~(e) is outlined as follows. First, if Y < R, qfi(e) becomes the the reference curve q~(e) itself. If Y is greater than R, the ordinate of q~(e) curve can be determined through equation (28) as 4=

(In JIcY) ~" , . / ( . - 1 ) -- (In "~ , . / ( . - 1 ) J,c) 27rB ( ~ cod ) l/(, + 1) Y -- RF n/(n + 1) /3


for any y > 1. The abscissa e in the q~(e) curve can then be obtained via the following implicit equation (/)(e) = y 4


for the same value of y. Two families of design curves are plotted in fig. 5 to illustrate the influence of Y and R on the defect assessment curves. The material parameters are chosen to be: a = 1, n = 12 and /.t•0(aE0 ) l / n = 0.003.



Y < i /-PD6493



/// Y=2 ~ /JWES2085K 3 / /" / /

5 I0 2





/ CVDA-1984















Fig. 5. Computed design curves (without safety margin). Left: R = 1; right: Y = 5.



Y.M. Dong et al. / Elastic-plastic defect assessment

The existing design curves, including those stated by PD6493, JWES-2805K [12] and CVDA-1984 [13] codes, are also shown in fig. 5 as the dashed curves for comparison. According to our understanding, these dashed curves signify conservative and empirical envelops (with a safety factor of two with respect to the crack length) on the experimental data, while the effect of strain hardening, stable crack growth and nonuniform strain distribution are either neglected or treated in an engineering average sense. 3.3. Failure assessment diagram

Based on the same development which connects the E P R I approach to the Failure Assessment (FAD), the present two p a r a m e t e r assessment scheme can be phrased in terms of the C E G B / R 6 formalism. The crack growth instability condition J(trf) = Jf can be written as



F(~r r, g ) -

2~.a~r 2


in view of (22), (27) and the assumption on small crack growth amount. The above equation can be arranged in the form of F A D as follows:

P/eo'~ + Hno-fn+l

=J~ = K 2,








- - - - ,

9 l + tr2



"-4 ( 1 - v 2) a K


and a K and ao~ are the converted crack lengths in the infinite CCT configuration calculated from the K-equivalence and J-equivalence principles, respectively. Typical failure assessment diagram is shown in fig. 6 for the special case of infinite CCT configuration (ao~ = aK). The material parameters are selected such that v = 0.3 and a 0 = 1.

/X~r 1.0 0.8




0.4 0.2


I .2

! .4

I ,6

I .8




Fig. 6. Failure assessment diagram, infinite CCT, v = 0.3, a o = 1.

Y.M. Dong et al. / Elastic-plastic defect assessment


3.4. Admissible stress curve The stress-crack growth relation tr(a) admissible by the material J resistance curve is referred to as the admissible stress curve. Accordingly, the function

A(~r, a) - J ( a , or(a)) --JR(a--ao)


should be identically zero to maintain a stable crack growth. Differentiating (35) with respect to the crack length a and setting the derivative as zero, one find (Dong, Yang and Hwang [9], 1990) da(o', a) da

OJ do" - -

80" da

8J +


dJ R - -





Therefore, the tearing instability T I~ = TR will be imminent provided dtr(a) d----~ = 0,


namely the peak of the admissible stress curve represents the failure stress at tearing instability and the largest attainable stable crack growth. We exemplify this approach by pressure vessels made of low-carbon seamless steel with axial cracks along the vessel exteriors. The material characterization for the vessel material is: Or0 = 2 5 6 M P a ,

E = 2 . 0 8 X 1 0 5 MPa,



a JR=1.903 ~


)0.414 MN/m. (38)

The stress admissible curves for six different geometries of cracked vessels are plotted in fig. 7. These curves enable us to predict the burst pressures of those vessels, as shown in the 4th row of table 3. Those theoretical predictions agree well with the experimental measurements taken on the same cracked vessels by Zhang and Wang [14] (1987).


6 0.15 0.14 0.I~ 0.I: 0.ii




3 1

0.09 0.08



.2 .3







Fig. 7. A d m i s s i b l e stress curves for six p r e s s u r e vessels. T h e d i m e n s i o n l e s s p e a k p r e s s u r e s (p/~r o) for various curves are listed as follows, curve 1: 0.100; curve 2: 0.086;; curve 3: 0.113; curve 4: 0.131; curve 5: 0.128; curve 6: 0.156.

Y.M. Donget aL/ Elastic-plasticdefect assessment


Table 3 Defect assessment by admissible stress method for six precracked pressure vessels No.

R i/R o


Burst pressure (predicted) (MPa)

Burst pressure (measured) (MPa)


Stable crack growth (ram)

1 2 3 4 5 6

59.5/68.65 59.5/68.40 59.5/68.65 59.5/71.4 59.5/71.4 59.5/71.4

0.595 0.663 0.524 0.597 0.609 0.489

25.6 22.0 28.9 33.5 32.8 39.9

23.6 20.1 31.0 34.8 33.7 37.9

+ 8.8% + 9.4% - 6.8% - 3.7% - 2.7% + 5.3%

0.30 0.30 0.35 0.40 0.40 0.50

4. Estimates on fully plastic J-integrals Attention is now focused on the determination of the crack driving forces. It is established in the E P R I engineering estimates [3,15] that the J integral for the general elastic-plastic case can be approximated by the additive decomposition of an elastic and a plastic J integral, namely J = Je + Jp. The latter is computed under a pure power-law hardening material and is tabulated for selected crack geometries through extensive finite element calculations. To facilitate the engineering applications, we intent to derive approximate but analytical expressions of Jp in this section.

4.1. The approximate scheme Suppose that the material under consideration is characterized by the Ramberg-Osgood law, namely equation (1). The fully plastic J integral under a remote load P may be expressed as: [

p ]n+l (39)

Jp=°lo°'o%~aHl[-~o )

from the E P R I engineering approach. In (39), sc is a known scale factor relating to the crack geometry; P0 represents the limit load of the crack containing structure; and Hi(a/b, n), where b stands for the ligament length, is the dimensionless solution of the fully plastic J integral by finite element method. The pure elastic J, on the other hand, can be expressed in terms of the energy release rate G:


K2(1 _p2) E

- O - o E o ( l - v 2 ) S 2 7 r a ¢ ~ 2,



where ~r= is the remote stress acting perpendicularly to the crack, and S is the elastic shape factor of the cracked specimen. Combining (39) and (40), we are able to recast the fully plastic J integral as, Dong and Shen [16] (1992): (41)

Jp = aoG&n-lD( a / b , n ) , where D has the following expression p


D ( a / b , n) =~ Po---~)

H1 ~-(1 - v 2 ) S 2


Y.M. Donget al. / Elastic-plasticdefectassessment


and is termed the plastic growing factor. It is known from the asymptotic behavior at either purely elastic or fully plastic extremes that the function D should be a monotone increase function and has the following properties:

D ( a / b , 1) = 1; D ' ( a / b , 1) = 0 ;

D ( a / b , n) > 1.


The above conditions may be regarded as the basic restrictions for the construction of a fully plastic estimate. We postulate an expression of D satisfying these restrictions as

D(a/b, n) = c o s h ( ~ - ~ b ) ,


where the geometry dependent phase angle ~b is approximated by s

with two parameters to and s determined by the experimental data of trf and af of a scaled down model test. Consequently, an approximate estimate for Jp is given by (41), (44) and (45), and which enables us to obtain the fully plastic J integral for a class of actual crack geometries from one scaled down model test. Combining the elastic and plastic J, we obtain

J = G [ l +ao cosh(~n ldp)~n-11.


According to the methodology mentioned above, the authors present analytical Jp estimates for two classes of defect geometries: (a) Internal Axial Crack - A533B - 1, a 0 = 1.14, n = 10, 0.45 < a/b < 0.60, to = 4.03



S = 0.2118.


External Axial Crack - Seamless tube made by low carbon steel, a 0 = 1, n = 6.582,


to = 5.01 X 101°,

s = 0.1149.


More elaborated expression of D is given by Dong and Shen [16] (1992) where s is regarded as a function of the relative crack growth amount. The burst pressures for vessels with external axial surface cracks are listed in table 4, where the values of to and s in (48) are adopted. It is shown that the theoretical predictions by the present fully plastic estimates agree well with the experimental measurements.

Table 4 Comparison between the estimate (46) and the experiments R i/R o


Pf/tro (experiment)

Pf/~ro (prediction)

65/75 65/75 65/75 59.5/68.7 59.5/68.7 59.5/68.7

0.523 0.549 0.540 0.595 0.663 0.524

0.1188 0.1073 O.1155 0.0922 0.0785 0.1211

0.1199 0.1092 O.1122 0.0945 0.0763 0.1194

Error 0.9% 1.8% 2.9% 2.5% 2.8% 1.4%

Y.M. Dong et al. / Elastic-plastic deject assessment


Table 5 Estimate on H 1 (a / b ~ O, CCP, plane stress)











4.2. Estimate formula for short cracks

The evaluation of plastic J integral of short cracks ( a / b < 0.125) has been a difficult problem in numerical calculation. The present estimate scheme (41) and (44), however, can be utilized to arrive simple and analytical formula of H l function under the short crack limit. The phase angle 4) in the expression of the plastic growing factor can be more accurately correlated by 4) = M e ~a/~,


where the two parameters M and .(2 can be extrapolated by two known fully plastic solutions H~(a~/b) and H l ( a z / b ) for any a 1 > a 2 > 0.125b, I2 =

b In(Az/A1) a 2 -



M = A 1 e a,~/b,

n Ai =


n -- 1

^ -n+l ~i(P/°'Poi)



n,( a,/b,n),

i=1,2. (50)

For the case of a / b tends to zero and n is large, satisfactory predictions are acquired by the above approximate estimate. Table 5 listed the values of H 1 function under infinitesimal a / b , for the geometry of center cracked plate under plane stress condition (in which the factor 1 - u 2 should be removed from the relevant formulae). The data shown in table 5 agree quite well with the predictions by Shih and Hutchinson [17] (1976) for the same configuration. Thus, the approximate estimate on fully plastic J integral described here might provide insight for the engineering calculations of short crack problems.

5. Conclusions The following conclusions are made from the present investigation: (1) The theoretical fracture resistance curves, suitable for materials ranging from brittle to very ductile structure steels, are established for the plane problems. This development might bypass the cumbersome testing works on experimental J resistance curves, and will provide an analytical characterization of the defect assessment procedure. (2) A defect assessment methodology based on the two fracture parameters, namely J~c and T c, has been proposed. The four variants of this assessment methodology, i.e. the curve intersection approach, the design curve approach, the failure assessment diagram and the admissible stress curve method, are elucidated. These representations encompass all known ductile fracture assessment approaches. (3) The theoretical predictions under the present methodology have been verified by the experimental testing data of several precracked pressure vessel models. (4) Approximate but analytical estimates of the fully plastic J integrals can be obtained from the instability parameters ~rf and af recorded in the scaled-down model test of the same steel. These estimates are shown to have sufficient accuracy for engineering applications, and may be utilized to replace vast amount of numerical data previously supplied by finite element calculations.

Y.M. Dong et al. / Elastic-plastic defect assessment


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