The virtual crack extension method for evaluation of J - and Ĵ-integrals

0013-7944l8’ 53.00 + Ilo S 198? Pergamon Press Ltd

THEVIRTUALCRACKEXTENSIONMETHODFOR EVALUATIONOF J- AND&INTEGRALS Department

N. MIYAZAKI of Chemical Engineering, Kyushu University, Hakozaki, Higashi-ku, Fukuokashi, Fukuoka-ken, Japan

T. WATANABE Century Research Center Corporation . 2.3-chome. Hon-cho, Nihonbashi. Japan

Chuo-ku, Tokyo.

and

Department

of Nuclear Engineering,

G. YAGAWA University of Tokyo, Bunkyo-ku, Tokyo, Japan

Abstract-The

equation for evaluating the nonlinear fracture mechanics parameters J- and jintegrals are derived using the virtual crack extension method. The validity of the equations derived here are checked by solving several numerical examples, that is, the J-integral analyses of compact tension specimen and three-point bend specimen, and the j-integral analysis of centrally cracked plate. Reasonably good agreement is found between the virtual crack extension method and the line integral method.

1. INTRODUCTION THE FRACTUREmechanics should be applied to evaluate the integrity of a cracked structure. The path-independent J-integral proposed by Rice[l, 23has been used widely as a fracture mechanics parameter which can be applicable to fracture phenomena under a large scale yielding condition. The line integral method is often used to calculate the J-integral. The alternative to calculating the J-integral is the virtual crack extension method which has been originally proposed by Parks[3] and Hellen[4] to calculate the potential energy decrease per unit crack advance in elastic body. Thereafter Parks[S, 61 has generalized the virtual crack extension method to evaluate the J-integral for nonlinear elastic material. Recently deLorenzi[7] has proposed independently another technique of the virtual crack extension method to the energy release rate for the nonlinear elastic material. In the presence of body force, thermal strain and irreversible plastic strain, the J-integral loses the property of path-independence and cannot be interpreted as the energy-release rate. Kishimoto et a1.[8] has proposed the j-integral, which is a generalization of Rice’s J-integral and takes account of the effects of body force, thermal strain and irreversible plastic strain. In the present paper, the equation for the virtual crack extension method is derived to evaluate the J-integral from a different viewpoint from the method proposed by Parks[S, 63, who used the total potential energy defined for a nonlinear elastic body. The equation proposed here for evaluating the J-integral is derived without using the total potential energy. Then the procedure for the J-integral is extended to calculate the j-integral, which includes the effects of plastic strain based on the incremental plastic theory, so that the total potential energy cannot be defined. Several numerical examples are presented to show the validity of the equations derived in the present paper.

J- OR&INTEGRALUSING VIRTUALCRACKEXTENSIONMETHOD

2. CALCULATIONPROCEDUREF~R J-Integral

Let us consider two frames O-X,X2 and o-x1x2 as shown in Fig. 1, where the frame OX,X2 is fixed in the space and the other, 0-x1x2, moves with the crack advance. The relation 975

976

N. MIYAZAKI er al.

between the two frames is given by XI = x1 - a,

x2 = X2,

(1)

where Q denotes the crack length. The following expression, obtained from Ref. [I], is for the J-integral of no&near elastic material: J=

T,

dlri

jTi;;d” -

Here I’ is a integral path su~~unding the crack tip, A the area enclosed by I’, Ti the traction force defined according to the outward normal nj along r and W the strain-energy density defined by (3) where aijand Eddenote the stress tensor and strain tensor, respectively. The material derivatives of ui and W with respect to crack length a in eqn (2) can be expressed by ddxl,

~2% al

da

=

hhl,

-

X2t

dW(xr, xzr a) = aWx,,

da

-

4

+

aa

*

axi -a

a&w,,

ax, aa x2,

aff

a)

+

81%’ 8x1 --s

ax,da

x2,

a)

aa awx,

1 x2,

aa

’ 4

f4f (5)

Using eqns (4) and (S), the J-integral expressed by eqn (2) can be rewritten in terms of fixed frame O-XIX:! as follows:

(6) Let us assume that the crack is extended by 6a under the conditions that the traction force Tj on the contour f is kept constant and that I’ is fixed in the space. Then, the following equation is obtained from eqn (6): Jtia =

$ j-_

W dAGa,

(71

Evaluation of J- and .bintegrals

977

The above equation shows that the virtual work of the crack extension force Jtia is equal to the difference between a virtual increment of the external work and the internal work. Now let us consider the virtual crack extension 6a as shown in Fig. 2. During the crack extension the contour I is assumed to be fixed and the contour I1 is assumed to be rigidly translated by 6a. In the figure, the area enclosed between Ii and F, and that enclosed by Ii are represented by A1 and A*, respectively. The superscript “0” of Al, AZ, r, and I denotes the value before the crack extension. According to eqn (7), the following equation can be obtained: Ji5a =

$ Jr

W dAGa.

When only the area enclosed by Ii is assumed to be rigidly translated by 6a under the condition that the displacement ui and the traction force Ti on the contour I are kept constant, the eqn (8) becomes

(9) In deriving the above equation, i

the following relations are used:

(Tiui) = 0,

aw aa=

0,

a’ dr/@ I =

o

aa

a I CWCM aa

I = o

r,

(10)

in AZ.

(11)

on

In eqns (9), (10) and (II), 1 dAl!dA’: ( et al. are the absolute values of the determinant of the Jacobian matrices. Equation (9) represents the equation for the evaluation of the J-integral value using the virtual crack extension method. It is the same as the method of Parks[S, 61, which was derived from the relation J = -adaa (n: potential energy).

(a)

Fig. 2. Virtual crack extension for evaluation of J-integral.

978

N. MIYAZAKI et al.

Q-Zntegral

Kishimoto er a/.[81 have proposed the path-independent j-integral as the energy-release rate due to crack extension. Let us consider a virtual crack extension from the point 0 to o as shown in Fig. 3, where the frame O-XIX2 is fixed in the space and other one, u-x1x2, moves with crack advance. The following equation can be obtained using the energy balance in the region A [8]:

(12) where Fi is a body force. The strain Eij is a sum of the elastic strain et, the thermal strain ~ffi and the plastic strain e$ as follows: Eij

where EF denotes the nonelastic the fixed frame as follows:

=

E; +

E; + Q = E; + EF’,

(13)

strain. By using eqn (4), eqn (12) can be written in terms of

(14) The expression for the virtual crack extension method can be derived in the same manner as in the case of the J-integral. Let us consider a virtual crack extension 6a as shown in Fig. 4, during which Ti, Fi and eij are assumed to be kept constant. Then, eqn (14) becomes jfj,

=

a T,u; dl%u aa II’+I-,,+I‘,~

+ a ikl

I AI+Az

Fiu, dAGa - ’ da

I AI+A?

UijEi,dAGa.

(15)

Considering that during the virtual crack extension the energy variation on r and r,, are equal to zero, eqn (15) is written as

(16)

Fig. 3. Coordinate systems for evaluation of j-integral.

Evaluation of J- and hntegrals

979

b)

Fig. 4. Virtual crack extension for evaluation of j-integral.

where A = A, + AZ. In deriving the above equation, the following relations are used:

substituting eqn (13) into (16) yields

(20) where We and W”” denote the elastic and nonelastic components of the strain-energy density expressed by

N. MIYAZAKI et al.

980

W”’ corresponds to the plastic strain-energy density Wp given by Hi11[9], if E$ is equal to zero. Equation (20) represents the equation for the evaluation of the j-integral value using the virtual crack extension method.

3. SELECTION OF VIRTUAL CRACK EXTENSION VALUE In the virtual crack extension method, the J- or j-integral value is evaluated from the difference of energy caused by a virtual crack extension. Thus, the virtual crack extension value &a should be selected so as not to deteriorate the accuracy of the results. A stress intensity factor of a centrally cracked plate under uniform tension shown in Fig. 5 were calculated by varying the virtual crack extension value 6a to find out its appropriate value. The curve of nondimensional stress intensity factor vs virtual crack extension value are shown in Fig. 6 where 1 denotes the side length of the element measured in the direction of crack extension. b

Fig. 5. Finite element mesh for centrally cracked plate.

1.1

0.8

I a ’ o I n ’ ’ 100 1o-2 lo-4 lo+ 6W Fig. 6. (F,),/(F,), vs &ah curve.

’ 10-t

981

Evaluation of J- and &integrals

FI is the nondimensional

stress intensity factor defined by FI = Kh,~(na)

and the subscripts “e” and “a” represent the values obtained by the present method and Isida[lO], respectively. The figure shows that the stress intensity factor converges to Isida’s when Wl is smaller than the value of 10e2. The accuracy of solution is deteriorated sharply when hall is larger than the value of 10p2. On the other hand, the accuracy of solution is also thought to deteriorate due to the round off error when gall is relatively small compared with the number of digits which the computer can represent. In the present analysis, the deterioration of accuracy cannot be found even at aall of lo-’ because the CDC6600 machine used in the numerical calculation has 60bits per one word. Thus, the value of &z/l ranging from IO-* to 1O-3 was chosen in the following numerical examples presented in the paper. 4. NUMERICAL

EXAMPLES AND DISCUSSION

The J- or j-integral values of a compact tension specimen, a three-point bend specimen and a centrally cracked plate was obtained from the virtual crack extension method (VCE) and the line integral method (L.INT.). All the numerical examples presented here were analyzed under the plane strain condition. Compact tension specimen

The first numerical example is a compact tension specimen shown in Fig. 7(a) which was dealt with in the analytical round robin conducted by JSME [ 111. The stress-strain curve of

B : Thickness

26 -+--24

60mm

f

P

65 -

P : Load

Zorle No. Crack tip Fig. 7. Compact tension specimen.

(a) Geometry.

(b) Finite element mesh.

N. MIYAZAKI et al.

982

the material for a nuclear pressure vessel was approximated

by the following equation: for u < a,,

CSIE

E

=

l

= u/E + {(u/B)” - (u,/B)“}

where the following material properties

for u 2 a,,

were used in the analysis:

E(Young’s modulus)

= 2.059 x lo5 MPa,

v(Poisson’s ratio) = 0.3, u,(yield stress) = 549.2 MPa, n = 3.0,

B = 1.5 x IO3 MPa.

Figure 7(b) shows the finite element mesh of a half of the specimen and the J-integral paths or zones. Four solutions PI through Ps were obtained in the present analysis. The basic information for these solutions are shown in Table 1 together with that for the reference solutions Al, C and F qu_oted from Ref. [ 11J. The J vs load curves are shown in Fig. 8, where 5 is the average of the J-integral values obtained from the line integral or the virtual crack extension. The figure shows that the present solutions PI through P4 agree well with other solutions A I, C and F. Fairly good agreement is found between the virtual crack extension method and the line integral method. Three-point

bend specimen

The results of the J-integral analysis are presented here for a three-point bend specimen which was treated in the analytical round robin undertaken as a task under ASTM Committee E24.01.09[12]. Figure 9(a) shows the geometry and loading condition for the three-point bend specimen. The finite element mesh of a half of the specimen and the J-integral paths or zones are shown in Fig. 9(b). The stress-strain curve of A533B steel was approximated by the equation

Table 1. Basic information for solutions of compact tension specimen Model

Incremental method

PI

Marcal’s method

p2

Marcal’s method

p3

Marcal’s method

p.l

Marcal’s method

Ai

Yamada’s method

C

Yamada’s method

F

Yamada’s method

Element type 8-node isoparametric element 8-zz;ei;parametric 4-node isoparametric element I-t-$;ei;parametric 3-node triangular element &node isoparametric element. singular element at crack tip S-node isoparametric element

Calculation of J

No. of nodes No. of element

Integral along Gaussian points (LINT.) Virtual crack extension method WCE) Integral along Gaussian points (LINT.) Virtual crack extension method (VCE) u = (a, + u2)/2 (LINT.) Integral along Gaussian points (L.INT.)

259 78 259 78 91 78 91 78 1.54 248 145 38

Integral along Gaussian points (L.INT.)

204 54

Nores. a: Crack length, oi ,uz: Stress of the element.

element 2

(Size of element at crack tip)/a I .2126 I .2/26 1.2126 I .?I26 4126 21’26

2126

Evfuation

0 0.0

t.0

983

of J- and j-integrals

2.0 3.0 4.0 5.0 Pi3 ~kN~~rn~

Fig. 8. _?vs load curves for compact tension specimen.

where the material constants B. and n were equal to 827.4 MPa and 10, respectively, and Young’s modulus E was assumed to be 2.18 x 105MFa. The value of Poisson’s ratio was 0.3. In the present analysis four solutions PI through P4 were obtained and compared with other solutions A, F and J quoted from Ref. 1121.Table 2 shows the basic information for the above solutions. The solutions of ASTM round robin range from the solutions J to F. Figure 10 shows the f vs load curves. The figure shows that the present solutions P, and PZ which use the &node isoparametric elements agree relatively well with the solutions A and F. It is also found from the figure that the solutions PJ and P4 are much smaller than other solutions, because the 4-node isoparametric element cannot represent accuratefy the bending P/2 W

2H

TT P

a

P/2~

a/w = 0.5 H/W= 2.0 W= 25.4 mm P : Load B : Thickness=25.4mm

Kg.9.Three-point bend specimen.

Zone No. Geometry.

(b) Finite element mesh.

984

N. MIYAZAKI et al. Table 2. Basic information for solutions of three-point

Model

Plasticity method

PI

Tangent modulus (Increment1 theory) Tangent modulus (Incremental theory) Tangent modulus (Incremental theory) Tangent modulus (Incremental theory) Secant modulus (Incremental theory) Tangent modulus (Incremental theory) Tangent modulus (Incremental theory)

P2 P3 p4

A F J

element type g-node isoparametric element I-node isoparametric element 4-node isoparametric element 4-node isoparametric element 3-node triangular element g-node isoparametric element g-node isoparametric element, singular element at crack tip

bend specimen

Calculation of J

No. of nodes No. of elements

(Size of element at crack tip)/a

Integral along Gaussian points (LINT.) Virtual crack extension method (VCE) Integral along Gaussian points (L.INT.) Virtual crack extension method (VCE) Line integral method (L.INT.) Line integral method (LINT.) Line integral method (L.INT.)

I89 62 I89 62 64 62 64 62 218 369 803 252 725 222

0.126 0.126 0.126 0.126 0.0016 0.0625 0.0200

Note. a: Crack length

deformation of the specimen. Therefore, to obtain accurate results.

in these cases many more elements may be necessary

Centrally cracked plate

The j-integral value was calculated using the virtual crack extension method and the line integral method for a centrally cracked plate shown in Fig. 1l(a) whose solution has been already obtained by Aoki et aZ.[13]. Figure 1l(b) shows the finite element mesh of a quarter of the plate and the .&integral paths or zones. The following material properties were assumed in the analysis: &Young’s

modulus)

= 2.06 x 10’ MPa,

v(Poisson’s ratio) = 0.3, a,(yield

stress) = 480 MPa,

H’(strain hardening modulus)

= E/100.

The loading history used in the analysis is shown in Fig. 12.

200-

P2: V.C.E.

8-Node Element b-Node Element

J

5 1-3 loo-

P (kN) Fig. 10. 1 vs load curves for three-point

bend specimen.

985

Evaluation of J- and j-integrals

2H

I.

2a

MC

2w

c

a =lOmm W=ZOmm H =50mm

Zone No. Fig. 11. Centrally cracked plate. (a) Geometry.

(b) Finite element mesh.

N. MIYAZAKI

986

er al.

Fig. 12. Loading history for centrally cracked plate.

I

0

0.2

0.4

0.6

C

WG Fig. 13. J vs oOiu~ curves for centrally cracked plate.

The j vs uoluy curves are presented in Fig. 13, where the curve obtained by Aoki et al. is also depicted. The figure shows that the virtual crack extension method gives relatively good result, compared with the line integral method and that of Aoki er al. 5. CONCLUDING REMARKS In the present paper, the equations for evaluating the nonlinear fracture mechanics parameters J- and j-integral were derived using the virtual crack extension method. The virtual crack extension method presented here is more general than that of Parks, because it can be applied to calculating the j-integral which is an extension of the J-integral proposed by Rice. The validity of the equations derived here were checked by solving several numerical examples. Acknowledgemenrs-The authors wish to thank Associate Professor K. Watanabe of the Institute of Industrial Science, the University of Tokyo and Dr K. Kishimoto of the Tokyo Institute of Technology for their fruitful comments.

REFERENCES 111 J. R. Rice, Mathematical analysis of mechanics of fracture, Fracture Academic Press, New York (1968).

2 (Edited by H. Liebowitz),

pp. 192-311.

Evaluation of J- and J-integrals [2] J. R. Rice, A path independent J. Appl. Mech.

Trans. ASME

integral and the approximate analysis of strain concentration Ser. E 35, 379-386

987 by notches and cracks.

(1968).

[3] D. M. Parks, A stiffness derivative finite element technique for determination of crack tip stress intensity factor. ht. J. Fracture 10, 487-502 (1974). [4] T. K. Hellen, On the method of virtual crack extensions. ht. J. Numer. Meth. Engng 9, 187-207 (1975). [5] D. M. Parks, The virtual crack extension method for nonlinear material behavior. Comp. Meth. Appl. Mech.

Engng 12, 353-364 (1977). 161 D. M. Parks. Virtual crack extension: L

.

A general finite element technique for J-integral evaluation. Proc. 1st Inr. pp. 265-478 (1978). On the energy release rate and the J-integral for 3-D crack configurations. Int. J. Fracture 19,

Conf. on Nuke. Meth. Fract.

Mech..

[7] H. G. deLorenzi, 183-193 (1982). [8] K. Kishimoto, S. Aoki and M. Sakata, On the path independent

[9] [IO] [I I] [12] [13]

integral-J. Engng Fracture Mech. 13, 845-850 (1980). R. Hill, The Mathematical Theory of Plasticity, pp. 14-49. Clarendon Press, London (1954). M. Isida, Effect of width and length on stress intensity factors of internally cracked plates under various boundary conditions. Int. J. Fracture Mech. 7, 301-316 (1971). M. Shiratori and T. Miyoshi, A comparison of finite element solutions for J-integral analysis of compact specimen (A round robin test in Japan), Proc. 2nd Int. Conf. on Num. Meth. Fract. Mech., pp. 417-431 (1981). W. K. Wilson and J. R. Osias, A comparison of finite element solutions for an elastic-plastic crack problem. Int. J. Fracture 4, R95-RI08 (1978). S. Aoki, K. Kishimoto, M. Nabeta and M. Sakata, Elastic-plastic fracture mechanics parameter for preloaded specimen. J. Sot. Materio/ Sci. Japan 31, 370-375 (1982). (Rcceil,ed

9 Nolvmber

1984)