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Evaluation of the J-integral for surface cracks using the line-spring model
Engineering Fromrc Mechanics Vol. 49. No. 2. pp. 251-263. 1994 CopyrightC 1994Elswier Science Ltd Printedin Great Britain. All rights reserved 0013-7944/94 s7.00+ 0.00
Pergmon
EVALUATION OF THE J-INTEGRAL FOR SURFACE CRACKS USING THE LINE-SPRING MODEL Z. Z. HUANG and Q. GAO Institute of Applied Mechanics, Southwest Jiaotong University, Chengdu, Sichuan 61fIf)31,P.R.C. Abstract-The line-spring model for the analysis of surface cracks is discussed in this paper. The complementary energy method [J. Press. Vess. Technol. 108. 305 (1986)] is extended to derive the fully plastic solutions of a single edge cracked plate in combinations of tension and bending. A simple method is proposed to specify the undetermined factors in complementary energy by making use of Shih’s [J. uppl. Mech. 51.48 (1984)] finite element results. The obtained fully plastic solutions are in a quasi-analytic form, and suitable for arbitrary combinations of tension and bending. Finally, the present line-spring model is incorporated into a plateeshell finite element code, and the semi-elliptical surface cracked plates in tension are analyzed. The estimated stress intensity factors and J-integral are in keeping with those of threedimensional finite element analysis.
INTRODUCTION
THE FRACTURE ANALYSIS and safety assessment for surface cracked plates or shells play an important role in various engineering fields. The fracture analysis of a surface crack is a complicated three-dimensional (3D) problem. From the viewpoint of application, a reasonably accurate and economical estimate method is needed. The line-spring model (LSM) proposed by Rice and Levy [ 11,which reduces the 3D surface crack problem to a shell problem, has been shown to be a suitable approximate method and has received considerable interest in the past decade. The stress intensity factors (SIF) estimated by the LSM are in good agreement with those by 3D finite element analysis, especially at the deepest point of a surface crack [2]. The LSM has been further developed to analyze surface cracks in the elastic-plastic range. The application of the engineering approach of elastic-plastic fracture mechanics [3] to the LSM was first reported by Kumar et al. [4]. According to the engineering approach, the elastic-plastic J-integral of a surface crack can be obtained by superimposing the modified elastic J-integral on the fully plastic J-integral. Thus, the fully plastic analysis is needed, which is much easier than the elastic-plastic analysis. The main idea of the LSM is illustrated in Fig. 1. The key point of the LSM is the introduction of a line-spring, which is equivalent to a plane strain single edge cracked plate (SECP) under combined tension and bending. The solutions of SECP, such as the relations of displacements due to the existence of crack with forces at load point (compliance) and J-integral, play an important role in the LSM. Some tabulated fully plastic solutions of SECP have been obtained by Kumar et al. [3,4] and Shih et al. [5] through incompressible deformation finite element analysis. It is obvious that solutions of sufficient scope and accuracy are needed. Efforts have been made by Miyoshi et al. [6] and Shawki er al. [7] trying to give an explicit solution of SECP. A complementary energy method (CEM) was proposed by Miyoshi et al. [6], by which the compliances and J-integral of SECP can be derived from the complementary energy of SECP. There are three undetermined factors in the suggested complementary energy expression. A method was proposed to determine these factors by calculating the complementary energy of SECP for a given material and different crack geometries through elastic-plastic finite element analysis. In their paper, the factors were obtained only for one material because of the difficulties of the calculations. In this work, the CEM proposed by Miyoshi et al. [6] is extended to estimate the fully plastic solutions of SECP. Then, the obtained fully plastic LSM with a plate-shell element is applied for the analysis of semi-elliptical surface cracked plates in tension. 251
252
Z. Z. HUANG and Q. GAO
t
t
t
t
t
Y
I
Fig. I. The concept of the line-spring model.
THE FULLY PLASTIC
COMPLEMENTARY
ENERGY OF SECP
The material considered in the following analysis is assumed to follow the Rambergagood power hardening relation. The stress-strain relation of fully plastic material in simple tension is (1)
c/E0 = a(a/ccJ
where 4 and o. are some reference strain and stress, n is the hardening exponent, and a is a material constant. Generalizing this uniaxial relation by the deformation theory of plasticity leads to Q/EO= (3/2)a(c,/c,)+
WJ~,
(2a)
where sg and cl/ are deviatoric stress and plastic strain deviator, respectively. The equivalent stress is defined as Q, = ,/(3S,S,/Z).
(2b)
Let’s consider the SECP under the state of plane strain subjected to combined axial force N and bending moment M (Fig. 1). Following Miyoshi et al. [6], the fully plastic complentary energy of the SECP is assumed as
where f=
AN2 + 2BNMIt + CM2/t2 ’ ogr - C)2
(3b)
where the subscript c denotes that the quantity is due to the existence of a crack, and A, B and C are undetermined factors. The displacement and rotation at the load point of SECP are given as 4 =s
= T;;* (”,;; (AN + BM/t) 0 -
(4)
0 -an, ‘-dM
/OF)‘2 ao(t-o2
(5)
(BN + CM/f).
253
Analysis of surface cracks using the LSM
Equations (4) and (5) can be called the generalized fully plastic constitutive relations of SECP. In the elastic range (n = l), eqs (4) and (5) are equivalent to those given by Rice and Levy [I]. The fully plastic J-integral of SECP can be evaluated as
where [ is the crack length, and the F is defined as F = AN2 + 2BNMJt + CM2/t2. DETERMINATION
OF FACTORS
(W
A, B AND C
Let’s firstly consider the following fully plastic displacements et al. [4]
8, = ~~~t~/t,
of SECP given by Kumar
(8)
n, ~)tN/N~~,
where N,, is the reference load, tZis defined as 1 = M/(Nr), and h3 and h, are dimensionless functions or calibration functions of t/t, n and 1. The reference load is defined as
where
fo=‘l-----2 [-]2n+5/tI+J(l
-5/t)2+(21+r/f)29 J(3) whereq=l forA#O,andq=1.26for1=0. Considering the definitions off and 2, eqs (4) and (5) are as
t9b)
A, = % t’(A + 232 + C12)(“- ‘)‘2(A+ BJ)jV” o;l(t - ()“+I
(10)
e, = cqz(A + 281 + CJ2)‘“- ‘)‘2(B+ CJ)N”
(11)
a;f(t - r)n+’
By comparing eqs (7) with (10) and (8) with (1 I), respectively, it can be recognized that the relations from the CEM give explicitly the approximate forms of hJ and hs, which assist us to obtain the factors A, B and C by making use of the results from the previous authors. From eqs (8)-(1 l), we obtain (A + 2BR + C,i2)1n-1112(A+ BA) = a;(t - ~~+i~t-2N~“~~(~/r,~, (A + 2BJ. + C;i2)t”-“‘2(B + C;c) = o;r(r - t)“+‘t -‘N,“h,({/t, It can be seen from the above equations that A, B and C, as the specified if two values of I and correlated h functions are given. In tension (A = 0) and pure bending (d = co) are considered as the factors. The h functions of Shih et al. [5] are employed, because consistency checks [8]. For pure tension (L = 0), the A and B can be obtained from A = (1 - (/t)2[t/t
B = (1 - e,f)2[C/t h:(r/t,
A)
(12)
n, 11).
(13)
functions of t/t and n, can be the following analysis the pure conditions to determine these they have been examined by eqs (12) and (13). That is
h:((/z, n)fJ2'("+')
(14)
n)f(y”]” -nMn+‘)h:(e/t, n)&
(1%
where the superscript T denotes the pure tension. For pure bending (N = 0), the relation given by Shih et al. is written as
End 4w-o
254
Z. Z. HUANG and Q. GAO
where the superscript B denotes pure bending, and (16b) The rotation obtained from eq. (5) by N = 0 is as follows
The factor C is then obtained as
It can be seen from the above equations that A, B and C are functions of relative crack length t/r and power hardening exponent n. The polynomial fitting for A, B and C comes across some difficulties because of the steep gradient in the scope of r/t. The alternative factors of A,, B, and C, are defined and fitted as A, =
B, =
,io u,(r/z)’
[h:(~/t,n)]2’(“+‘)= 1
h:((/f, n)/((/r)[h:([/r,
n)]” -“)‘(“+ ‘) = i
(19)
b,({/r)’
(20)
i=O
c, =[h:(
i Ci(5/f)i.
(21)
i-l
The polynomial coefficients ui, bi and ci for A,, B, and C, are given in Table 1. Some of A,, 8, and C, against t/r are illustrated in Figs 24, respectively. The A, B and C then be rewritten as: Table 1. The polynomial cuefficients for factors A,, B, and C, n
A,
0.064
2 3 5 7 10 13 16 20
0.346 0.710 0.951 1.229 1.526 1.583 1.546 1.367
n I
B,
an
1
2 3 5 7 10 13 ::
bo
CI
2 3 5 7 IO 13 16 20
4
0.06 - 10.60
-3.46 - 72.79 - 54.04 -37.15 -39.31 -70.14 -99.01
b,
-3.12 2.03 - 13.94 40.27 - 106.62 415.02 255.18 -632.98 137.72 - 164.49 104.40 - 128.39 136.35 -230.84 350.51 - 986.44 502.82 - 1447.18
b,
h
a7
0.00
0.00
-2.18 - 740.47 1123.09 95.02 73.85 224.54 1722.99 2547.43
-42.64 752.08 - 1254.03 -21.33 - 15.97 - 122.97 - 1816.02 - 2678.70
b,
b,
0.015 0.329 - 0.972 5.190 10.454 7.043 8.352
-0.099 -4.043 49.545 -39.185 - 140.778 -73.688 - 118.494
23.48 178.25 -360.10 149.00 933.13 351.03 792.72
-81.8 - 1328.4 1217.9 -341.0 -3616.1 -907.4 -3027.1
121.1 4538.0 -2267.0 514.9 8865.4 1376.2 7164.5
- 72.9 -8515.1 2385.7 -519.3 - 13634.8 - 1240.7 - 10703.2
19.3 9061.1 - 1333.2 313.0 127%. I 620.6 9795.3
7.197 8.666
- 139.214 117.310
1029.13 879.13
- 3657.4 -4254.2
10704.3 9227.0
- 16745. 14443.8 I
13677.3 15866.8
G
G
0.565 5.828 10.728 27.947 34.218 40.633 44.514 47.113 49.449
28.572 18.156 - 3.078 -235.120 -314.318 -409.255 -473.670 - 520.783 - 565.807
- 55.37 - 109.91 - 119.63 1203.57 1629.03 2191.06 2601.34 2917.12 3229.92
- 15.5 - 308.4 -616.7 6039.7 8266.8 11324.3 13705.4 15610.3 17556.3
0.0
n I
aA
0,
3.152 6.392 5.630 9.986 6.893 2.751 1.966 3.671 6.672
G 46.0
267.5 445.0 - 3555.6 -4832.3 - 6586.0 - 7922.2 - 8978.9 - 10047.1
C, 142.9 313.9 - 5420.7 -7502.1 - 10306.6 - 12510.2 - 14277.0 - 16090.3
0.0 0.0 0.0
2007.4 2812.9 3870.6 4702.3 5366.9 6049.5
0.00
23.84 -429.32 761.46 0.00 0.00 29.68 1047.67 1535.30
4
0.0 0.0
107.8 - 189.8 0.0 0.0 0.0 -252.9 - 367.8
4
b,
0.0 -5136.5 308.2 -82.8 -6671.4 - 133.2 - 5000.3
0.0 1206.6 0.0 0.0 1478.I 0.0 1088.5
-7161.5 -8321.4
1851.2 1589.5
255
Analysis of surface cracks using the LSM
2.0
1.5
4 1.0
0.5
0.0 -
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 2. Factor A, vs C/r.
A = (1 - 5/1)2[r/rf~“]2”“+‘)A,
(22)
B = (1 - 5/1)2[r/rf,-“]2’(“+“B,
(23)
c = 0.364-2”“” + “C,.
(24)
As the factors A, B and C have been expressed in a polynomial form, the fully plastic J-integral and compliance of SECP given by eqs (4)-(6) are then in a quasi-analytic form. ‘Ihe main advantages of the present fully plastic solutions of SECP are continuous c/t and arbitrary combinations of tension and bending are available due to the explicit t/r and 1.
r
5.0
_._
0.0
0.2
0.6
0.4 w
Fig. 3. Factor H, vs
0.8
1.0
2%
2. Z. HUANG and Q. GAO
0.0 0.0
0.2
0.4
0.6
. 0.8
1.0
S/t
Fig. 4. Factor C, vs C/t.
COMPARING THE CEM RESULTS
WITH FINITE ELEMENT RESULTS
A method named “consistency checks” was proposed by Parks et al. [9] to estimate the accuracy of finite element solutions of fully plastic crack problems. Afterwards, it was employed by Shih and Needelman [8] for fully plastic solutions of SECP subjected to pure tension and bending. According to the consistency checks, the fully plastic solutions should follow a “consistency relation”, which is a linear relation between h, , h3 and ah,/& It is a severe check, as the errors in h, will be greatly amplified in ah,/@. In the present analysis, the complementary energy of eq. (3), in fact, depends on the fully displacements (h, and h,), so the fully plastic J-integral of eq. (6) is related to the displacements and their partial derivatives. Comparing the J-integral of eq. (6) with those of finite element analysis is also a severe check. The fully plastic J-integral by calibration function is [4,5]
By comparing eqs (25) with (6), the corresponding L # co is given as n+ I
h,(r;r,
A, n) =
(Ii
calibration
functions h, from the CEM for
I)/2
f” G XMl -
Kf - t) aG/‘%+ =I,
where G = A + 2Bk + CA’, and for I = co is as h,({/t, n) = 0.364”+‘(1 - e/f~-‘c@-‘@[(t
-t;) &T/a& + 2C].
(26b)
The h, vs c/t in pure tension and bending obtained from eqs (26) and Shih‘s [S] finite element analysis are shown in Figs 5 and 6, respectively. Good agreement is found, and the errors are within 2% in all the range of r/t. In the case of combined tension and bending, only the EPRI’s [4] finite element results are available. Figures 7 and 8 show the h, vs 5/r in the case of I = l/8 and a = l/16, respectively, and the EPRI’s finite element results are also given for comparison.
Analysis of surface cracks using the LSM 7.0
257
Present CEM 0 0 0 Shill [S]
6.0 Pure toolloll
5.0
4.0
.ci3.0
2.0
1.0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Fig. 5. h, vs
It can be seen from Fig. 7 that the present CEM results are in good agreement with finite element results in the range of c/t > 0.25. The CEM results of n = 3 and n = 5 are greater than those of EPRI when r/t < 0.25. Figure 8 shows the h, vs r/t in the case of 2, = - l/16. The CEM results agree well for t/t > 0.3. But adverse variations are found for c/t < 0.3, and the difference increases as {fr decreases. Further study will be made by the authors about the disagreement between the results by the present CEM and EPRI’s finite element analysis, and the consistency checks for the EPRI’s results are needed. 6.0
3.0
4.0
fi-
P
-I -
t
-
Present CEM
000
Shih [S]
Pure bending .n=20
3.0
2.0
1.0
Fig. 6. h, vs c/r in pun bending.
Z. Z. HUANG and Q. GAO
258 6.0
-PrwentCEM -O- Knmu[4]
r 5.0
4.0
i
3.0 I
nn _._ 0.0
0.2
0.4
0.6
0.8
1.0
s/t Fig.7. h,vs
EVALUATIONS
OF THE J-INTEGRAL FOR SURFACE IN TENSION
CRACKED
PLATES
As an application of the present fully plastic LSM, semi-elliptical surface cracked plates in tension are analyzed. A four-noded isoparametric plate-shell element is employed, in which the effect of shearing deformation has been taken into account. To be compatible with the plate-shell element, the isoparametric line-spring element with bilinear shape functions is derived. The formulation for the line-spring element is analogous to that in [lo]. The finite element mesh for
--O-.
_.0.0
0.2
0.6
0.4
Present CEM Kumar [4]
0.8
5/t Fig. 8. h, vs C/r in the case of 1 = -I/16.
1.0
259
Analysis of surface cracks using the LSM
t
t
t
t
t
t
t
t
t”
no-spring elamalt
Fig. 9. The finite element mesh for a quarter of the surface cracked plate in tension.
2.5
-a-- Prosent LSM -Newman [ll]
ale = 0.2 2.0
C %
1.5
% 5 b 2
1.0
0.5
0.0 0.0
I 0.2
I
I 0.6
0.4
I 0.8
I 1.0
2+/x Fig. 10. The normalized SIF vs the crack angle for surface cracked plates in tension (a/c 10.2, Q = 1.104).
Z. Z. HUANG and 0. GAO
260 2.0
rlc
= 0.4
-o-
PrMent LSM
-
Newman (111
r/t = 0.8. 1.5
C “h 1
1.0
b 2
0.5
0.0 __
I
I
I
I
I
0.2
0.4
0.6
0.8
1.0
__
0.u
20/r Fig. 11. The normalized SIF vs the crack angle for surface cracked plates in tension (a/c = 0.4, Q = 1.324).
a quarter of the plate is shown in Fig. 9, containing 81 plateshell elements and 16 line spring elements. For all of the crack geometries the ratios of L/W and W/c are fixed as 2 and 4, respectively. The computations are executed on a VAX-l l/780 computer. In the elastic range (n = l), the SIF is obtained by means of the J-K relation. Figures lo-12 show the normalized SIF along the crack front by the present analysis and by the 3D finite analysis of Raju and Newman [l 11. At the deepest points of the surface cracks, the two solutions have a
2.0 aft
= 0.6
-o-
Prssent LSM
-
Newman [II)
o.“O.o
1.0
291% Fig. 12. The normalized SIF vs the crack angle for surface cracked plates Q = 1.629).
in tension (a/c =
0.6.
261
Analysis of surface cracks using the LSM 5.0
a - 5.0 et= 1.0 r/c = 0.2
Fig.
13. The f-integral
r/t
I 0.6 P
^
vs the far field tension stress for surface cracked (o/c = 0.2, n = 5, a = I, v = 0.3).
plates in tension
difference of less than 6%. The LSM results are much lower near the comer points. The reason should be that at the deepest point the state of stress is plane strain, which coincides with the definition of line-spring, while that at the comer point is plane stress. The present results basically agree with those reported by German et al., and support the conclusion that the LSM for shallow surface cracks is better than for deep cracks [2]. In the fuhy plastic analysis, the hardening exponents n = 3,5,7 and material constant a = 1, are considered. The finite element analysis of the fully plastic problem with the deformation theory
ait = 0.6 P
0.0
0.0
Fig.
14. The J-integral
0.2
0.4
0.6
0.8
vs the far field tension stress for surfaa (a/c = 0.4, n = 5, a = 1, v = 0.3).
1.0
cracked
plates in tension
262
2. Z. HUANG and Q. GAO 5.0 alt
= 0.6
4.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
alao Fig. IS. The J-integral vs the far field tension stress for surface cracked plates in tension (u/c = 0.6, n = 5,u = 1, v = 0.3).
of plasticity is equal to that of a non-linear elastic one. In an iteration sequence, the Newton-Raphson and the Modified Newton-Raphson method are employed alternately. In accordance with the engineering approach, the J-integral is the sum of the modified elastic and the fully plastic one. Figures 13-15 show the normalized J-integral vs the norma&& remote tension stress for n = 5 and a = 1. The J-integral results of 3D finite element analysis by Parks et al. [12] are given in Fig. 16 to be compared with the LSM results. The comparison shows that the proposed fully plastic LSM is efficient with a reasonable accuracy.
* Present LSM ----Parks 1121
n = 5.0 cl=
1.0
Y f 0.3
Na IS.0
B ii 10.0
5.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
ala, Fig. 16. The J-integral obtained by present LSM and by 3D finite element method vs the far field tension stress in tension (n/r ==0.6,n =$a = t,v -0.3).
263
Analysis of surface cracks using the LSM
CONCLUSIONS
The LSM discussed in this paper is mainly based on the engineering approach of elastic-plastic fracture mechanics and the complementary energy method. A simple method is suggested to specify the undetermined factors in the complementary energy expression of SECP. The obtained fully plastic solutions of SECP are in a quasi-analytic form and are suitable for arbitrary values of C/t and 1. The surface cracked plates in tension are analyzed by this model with a plate-shell finite element. The estimated J-integral results are in keeping with those obtained by 3D finite element analysis. REFERENCES J. R. Rice and N. Levy, The part through surface crack in an elastic plate. J. 185 (1972). M. D. German, V. Kumar and H. G. Delorenzi, Analysis of surface cracks in plates and shells using line-spring model 1;; and ADINA. Cornpar. Srructures 17, 881 (1983). appl. Mech. 39,
131V. Kumar, M.D. German and C. F. Shih, An engineering approach for elastic-plastic fracture analysis. EPRI NP-193 I, Topical Rep., Res. Project 1237-1, General Electric Company, Schenectady, NY (1981). 141V. Kumar er al., Further development in the engineering approach for elastic-plastic fracture analysis. EPRI NP-3607, Final Rep., Res. Project 1237-1, General Electric Company, Schenectady, NY (1983). C. F. Shih and A. Needelman, Fully plastic crack problems. J. appl. Mech. 51, 48 (1984). 1:; T. Miyoshi, M. Shiratori and Y. Yoshida, Analysis of J-integral and crack growth for surface cracks by line spring method. J. Press. Vess. Technol. 108, 305 (1986). [7l T. G. Shawki, T. Nakamura and D. M. Parks, Line-spring analysis of surface flawed plates and shells using deformation theory. Inc. J. Frucrure 41, 23 (1989). 181 D. M. Parks, V. Kumar and C. F. Shih, Consistency checks for power-law calibration functions. Second Symposium on Elasric-Plasric Fracrure, Vol. 1, Inelasric Crack Analysis (Edited by C. F. Shih and J. P. Gudas) ASTM STP 803, 1370 (1983).
[91 C. F. Shih and A. Needelman. Fully plastic crack problems. J. uppl. Mech. 51, 51 (1984). 1101M. Shiratori and T. Miyoshi, Evaluation of J-integral for surface cracks. Second Symposium on Elasric-Plastic Fracrure, Vol. 1 Inelusric Crack Analysis (Edited by C. F. Shih and J. P. Gudas), ASTM STP 803, 1410 (1983). 1111J. C. Newman, Jr and I. S. Raju, Stress-intensity factors for a wide range of semi-elliptical surface cracks in finitethickness plates. Engng Fracture Mech. 11, 817 (1979). 1121 D. M. Parks and Y. Y. Wang, Elastic analysis of part-through surface cracks, in Analyrical, Numerical, and Experimental Aspecrs of Three Dimensional Fracture Processes (Edited by A. J. Rosakis, K. Ravi-Chander and Y. Rajapakse), New York (1988). (Received 19 July 1993)
Pergmon
EVALUATION OF THE J-INTEGRAL FOR SURFACE CRACKS USING THE LINE-SPRING MODEL Z. Z. HUANG and Q. GAO Institute of Applied Mechanics, Southwest Jiaotong University, Chengdu, Sichuan 61fIf)31,P.R.C. Abstract-The line-spring model for the analysis of surface cracks is discussed in this paper. The complementary energy method [J. Press. Vess. Technol. 108. 305 (1986)] is extended to derive the fully plastic solutions of a single edge cracked plate in combinations of tension and bending. A simple method is proposed to specify the undetermined factors in complementary energy by making use of Shih’s [J. uppl. Mech. 51.48 (1984)] finite element results. The obtained fully plastic solutions are in a quasi-analytic form, and suitable for arbitrary combinations of tension and bending. Finally, the present line-spring model is incorporated into a plateeshell finite element code, and the semi-elliptical surface cracked plates in tension are analyzed. The estimated stress intensity factors and J-integral are in keeping with those of threedimensional finite element analysis.
INTRODUCTION
THE FRACTURE ANALYSIS and safety assessment for surface cracked plates or shells play an important role in various engineering fields. The fracture analysis of a surface crack is a complicated three-dimensional (3D) problem. From the viewpoint of application, a reasonably accurate and economical estimate method is needed. The line-spring model (LSM) proposed by Rice and Levy [ 11,which reduces the 3D surface crack problem to a shell problem, has been shown to be a suitable approximate method and has received considerable interest in the past decade. The stress intensity factors (SIF) estimated by the LSM are in good agreement with those by 3D finite element analysis, especially at the deepest point of a surface crack [2]. The LSM has been further developed to analyze surface cracks in the elastic-plastic range. The application of the engineering approach of elastic-plastic fracture mechanics [3] to the LSM was first reported by Kumar et al. [4]. According to the engineering approach, the elastic-plastic J-integral of a surface crack can be obtained by superimposing the modified elastic J-integral on the fully plastic J-integral. Thus, the fully plastic analysis is needed, which is much easier than the elastic-plastic analysis. The main idea of the LSM is illustrated in Fig. 1. The key point of the LSM is the introduction of a line-spring, which is equivalent to a plane strain single edge cracked plate (SECP) under combined tension and bending. The solutions of SECP, such as the relations of displacements due to the existence of crack with forces at load point (compliance) and J-integral, play an important role in the LSM. Some tabulated fully plastic solutions of SECP have been obtained by Kumar et al. [3,4] and Shih et al. [5] through incompressible deformation finite element analysis. It is obvious that solutions of sufficient scope and accuracy are needed. Efforts have been made by Miyoshi et al. [6] and Shawki er al. [7] trying to give an explicit solution of SECP. A complementary energy method (CEM) was proposed by Miyoshi et al. [6], by which the compliances and J-integral of SECP can be derived from the complementary energy of SECP. There are three undetermined factors in the suggested complementary energy expression. A method was proposed to determine these factors by calculating the complementary energy of SECP for a given material and different crack geometries through elastic-plastic finite element analysis. In their paper, the factors were obtained only for one material because of the difficulties of the calculations. In this work, the CEM proposed by Miyoshi et al. [6] is extended to estimate the fully plastic solutions of SECP. Then, the obtained fully plastic LSM with a plate-shell element is applied for the analysis of semi-elliptical surface cracked plates in tension. 251
252
Z. Z. HUANG and Q. GAO
t
t
t
t
t
Y
I
Fig. I. The concept of the line-spring model.
THE FULLY PLASTIC
COMPLEMENTARY
ENERGY OF SECP
The material considered in the following analysis is assumed to follow the Rambergagood power hardening relation. The stress-strain relation of fully plastic material in simple tension is (1)
c/E0 = a(a/ccJ
where 4 and o. are some reference strain and stress, n is the hardening exponent, and a is a material constant. Generalizing this uniaxial relation by the deformation theory of plasticity leads to Q/EO= (3/2)a(c,/c,)+
WJ~,
(2a)
where sg and cl/ are deviatoric stress and plastic strain deviator, respectively. The equivalent stress is defined as Q, = ,/(3S,S,/Z).
(2b)
Let’s consider the SECP under the state of plane strain subjected to combined axial force N and bending moment M (Fig. 1). Following Miyoshi et al. [6], the fully plastic complentary energy of the SECP is assumed as
where f=
AN2 + 2BNMIt + CM2/t2 ’ ogr - C)2
(3b)
where the subscript c denotes that the quantity is due to the existence of a crack, and A, B and C are undetermined factors. The displacement and rotation at the load point of SECP are given as 4 =s
= T;;* (”,;; (AN + BM/t) 0 -
(4)
0 -an, ‘-dM
/OF)‘2 ao(t-o2
(5)
(BN + CM/f).
253
Analysis of surface cracks using the LSM
Equations (4) and (5) can be called the generalized fully plastic constitutive relations of SECP. In the elastic range (n = l), eqs (4) and (5) are equivalent to those given by Rice and Levy [I]. The fully plastic J-integral of SECP can be evaluated as
where [ is the crack length, and the F is defined as F = AN2 + 2BNMJt + CM2/t2. DETERMINATION
OF FACTORS
(W
A, B AND C
Let’s firstly consider the following fully plastic displacements et al. [4]
8, = ~~~t~/t,
of SECP given by Kumar
(8)
n, ~)tN/N~~,
where N,, is the reference load, tZis defined as 1 = M/(Nr), and h3 and h, are dimensionless functions or calibration functions of t/t, n and 1. The reference load is defined as
where
fo=‘l-----2 [-]2n+5/tI+J(l
-5/t)2+(21+r/f)29 J(3) whereq=l forA#O,andq=1.26for1=0. Considering the definitions off and 2, eqs (4) and (5) are as
t9b)
A, = % t’(A + 232 + C12)(“- ‘)‘2(A+ BJ)jV” o;l(t - ()“+I
(10)
e, = cqz(A + 281 + CJ2)‘“- ‘)‘2(B+ CJ)N”
(11)
a;f(t - r)n+’
By comparing eqs (7) with (10) and (8) with (1 I), respectively, it can be recognized that the relations from the CEM give explicitly the approximate forms of hJ and hs, which assist us to obtain the factors A, B and C by making use of the results from the previous authors. From eqs (8)-(1 l), we obtain (A + 2BR + C,i2)1n-1112(A+ BA) = a;(t - ~~+i~t-2N~“~~(~/r,~, (A + 2BJ. + C;i2)t”-“‘2(B + C;c) = o;r(r - t)“+‘t -‘N,“h,({/t, It can be seen from the above equations that A, B and C, as the specified if two values of I and correlated h functions are given. In tension (A = 0) and pure bending (d = co) are considered as the factors. The h functions of Shih et al. [5] are employed, because consistency checks [8]. For pure tension (L = 0), the A and B can be obtained from A = (1 - (/t)2[t/t
B = (1 - e,f)2[C/t h:(r/t,
A)
(12)
n, 11).
(13)
functions of t/t and n, can be the following analysis the pure conditions to determine these they have been examined by eqs (12) and (13). That is
h:((/z, n)fJ2'("+')
(14)
n)f(y”]” -nMn+‘)h:(e/t, n)&
(1%
where the superscript T denotes the pure tension. For pure bending (N = 0), the relation given by Shih et al. is written as
End 4w-o
254
Z. Z. HUANG and Q. GAO
where the superscript B denotes pure bending, and (16b) The rotation obtained from eq. (5) by N = 0 is as follows
The factor C is then obtained as
It can be seen from the above equations that A, B and C are functions of relative crack length t/r and power hardening exponent n. The polynomial fitting for A, B and C comes across some difficulties because of the steep gradient in the scope of r/t. The alternative factors of A,, B, and C, are defined and fitted as A, =
B, =
,io u,(r/z)’
[h:(~/t,n)]2’(“+‘)= 1
h:((/f, n)/((/r)[h:([/r,
n)]” -“)‘(“+ ‘) = i
(19)
b,({/r)’
(20)
i=O
c, =[h:(
i Ci(5/f)i.
(21)
i-l
The polynomial coefficients ui, bi and ci for A,, B, and C, are given in Table 1. Some of A,, 8, and C, against t/r are illustrated in Figs 24, respectively. The A, B and C then be rewritten as: Table 1. The polynomial cuefficients for factors A,, B, and C, n
A,
0.064
2 3 5 7 10 13 16 20
0.346 0.710 0.951 1.229 1.526 1.583 1.546 1.367
n I
B,
an
1
2 3 5 7 10 13 ::
bo
CI
2 3 5 7 IO 13 16 20
4
0.06 - 10.60
-3.46 - 72.79 - 54.04 -37.15 -39.31 -70.14 -99.01
b,
-3.12 2.03 - 13.94 40.27 - 106.62 415.02 255.18 -632.98 137.72 - 164.49 104.40 - 128.39 136.35 -230.84 350.51 - 986.44 502.82 - 1447.18
b,
h
a7
0.00
0.00
-2.18 - 740.47 1123.09 95.02 73.85 224.54 1722.99 2547.43
-42.64 752.08 - 1254.03 -21.33 - 15.97 - 122.97 - 1816.02 - 2678.70
b,
b,
0.015 0.329 - 0.972 5.190 10.454 7.043 8.352
-0.099 -4.043 49.545 -39.185 - 140.778 -73.688 - 118.494
23.48 178.25 -360.10 149.00 933.13 351.03 792.72
-81.8 - 1328.4 1217.9 -341.0 -3616.1 -907.4 -3027.1
121.1 4538.0 -2267.0 514.9 8865.4 1376.2 7164.5
- 72.9 -8515.1 2385.7 -519.3 - 13634.8 - 1240.7 - 10703.2
19.3 9061.1 - 1333.2 313.0 127%. I 620.6 9795.3
7.197 8.666
- 139.214 117.310
1029.13 879.13
- 3657.4 -4254.2
10704.3 9227.0
- 16745. 14443.8 I
13677.3 15866.8
G
G
0.565 5.828 10.728 27.947 34.218 40.633 44.514 47.113 49.449
28.572 18.156 - 3.078 -235.120 -314.318 -409.255 -473.670 - 520.783 - 565.807
- 55.37 - 109.91 - 119.63 1203.57 1629.03 2191.06 2601.34 2917.12 3229.92
- 15.5 - 308.4 -616.7 6039.7 8266.8 11324.3 13705.4 15610.3 17556.3
0.0
n I
aA
0,
3.152 6.392 5.630 9.986 6.893 2.751 1.966 3.671 6.672
G 46.0
267.5 445.0 - 3555.6 -4832.3 - 6586.0 - 7922.2 - 8978.9 - 10047.1
C, 142.9 313.9 - 5420.7 -7502.1 - 10306.6 - 12510.2 - 14277.0 - 16090.3
0.0 0.0 0.0
2007.4 2812.9 3870.6 4702.3 5366.9 6049.5
0.00
23.84 -429.32 761.46 0.00 0.00 29.68 1047.67 1535.30
4
0.0 0.0
107.8 - 189.8 0.0 0.0 0.0 -252.9 - 367.8
4
b,
0.0 -5136.5 308.2 -82.8 -6671.4 - 133.2 - 5000.3
0.0 1206.6 0.0 0.0 1478.I 0.0 1088.5
-7161.5 -8321.4
1851.2 1589.5
255
Analysis of surface cracks using the LSM
2.0
1.5
4 1.0
0.5
0.0 -
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 2. Factor A, vs C/r.
A = (1 - 5/1)2[r/rf~“]2”“+‘)A,
(22)
B = (1 - 5/1)2[r/rf,-“]2’(“+“B,
(23)
c = 0.364-2”“” + “C,.
(24)
As the factors A, B and C have been expressed in a polynomial form, the fully plastic J-integral and compliance of SECP given by eqs (4)-(6) are then in a quasi-analytic form. ‘Ihe main advantages of the present fully plastic solutions of SECP are continuous c/t and arbitrary combinations of tension and bending are available due to the explicit t/r and 1.
r
5.0
_._
0.0
0.2
0.6
0.4 w
Fig. 3. Factor H, vs
0.8
1.0
2%
2. Z. HUANG and Q. GAO
0.0 0.0
0.2
0.4
0.6
. 0.8
1.0
S/t
Fig. 4. Factor C, vs C/t.
COMPARING THE CEM RESULTS
WITH FINITE ELEMENT RESULTS
A method named “consistency checks” was proposed by Parks et al. [9] to estimate the accuracy of finite element solutions of fully plastic crack problems. Afterwards, it was employed by Shih and Needelman [8] for fully plastic solutions of SECP subjected to pure tension and bending. According to the consistency checks, the fully plastic solutions should follow a “consistency relation”, which is a linear relation between h, , h3 and ah,/& It is a severe check, as the errors in h, will be greatly amplified in ah,/@. In the present analysis, the complementary energy of eq. (3), in fact, depends on the fully displacements (h, and h,), so the fully plastic J-integral of eq. (6) is related to the displacements and their partial derivatives. Comparing the J-integral of eq. (6) with those of finite element analysis is also a severe check. The fully plastic J-integral by calibration function is [4,5]
By comparing eqs (25) with (6), the corresponding L # co is given as n+ I
h,(r;r,
A, n) =
(Ii
calibration
functions h, from the CEM for
I)/2
f” G XMl -
Kf - t) aG/‘%+ =I,
where G = A + 2Bk + CA’, and for I = co is as h,({/t, n) = 0.364”+‘(1 - e/f~-‘c@-‘@[(t
-t;) &T/a& + 2C].
(26b)
The h, vs c/t in pure tension and bending obtained from eqs (26) and Shih‘s [S] finite element analysis are shown in Figs 5 and 6, respectively. Good agreement is found, and the errors are within 2% in all the range of r/t. In the case of combined tension and bending, only the EPRI’s [4] finite element results are available. Figures 7 and 8 show the h, vs 5/r in the case of I = l/8 and a = l/16, respectively, and the EPRI’s finite element results are also given for comparison.
Analysis of surface cracks using the LSM 7.0
257
Present CEM 0 0 0 Shill [S]
6.0 Pure toolloll
5.0
4.0
.ci3.0
2.0
1.0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Fig. 5. h, vs
It can be seen from Fig. 7 that the present CEM results are in good agreement with finite element results in the range of c/t > 0.25. The CEM results of n = 3 and n = 5 are greater than those of EPRI when r/t < 0.25. Figure 8 shows the h, vs r/t in the case of 2, = - l/16. The CEM results agree well for t/t > 0.3. But adverse variations are found for c/t < 0.3, and the difference increases as {fr decreases. Further study will be made by the authors about the disagreement between the results by the present CEM and EPRI’s finite element analysis, and the consistency checks for the EPRI’s results are needed. 6.0
3.0
4.0
fi-
P
-I -
t
-
Present CEM
000
Shih [S]
Pure bending .n=20
3.0
2.0
1.0
Fig. 6. h, vs c/r in pun bending.
Z. Z. HUANG and Q. GAO
258 6.0
-PrwentCEM -O- Knmu[4]
r 5.0
4.0
i
3.0 I
nn _._ 0.0
0.2
0.4
0.6
0.8
1.0
s/t Fig.7. h,vs
EVALUATIONS
OF THE J-INTEGRAL FOR SURFACE IN TENSION
CRACKED
PLATES
As an application of the present fully plastic LSM, semi-elliptical surface cracked plates in tension are analyzed. A four-noded isoparametric plate-shell element is employed, in which the effect of shearing deformation has been taken into account. To be compatible with the plate-shell element, the isoparametric line-spring element with bilinear shape functions is derived. The formulation for the line-spring element is analogous to that in [lo]. The finite element mesh for
--O-.
_.0.0
0.2
0.6
0.4
Present CEM Kumar [4]
0.8
5/t Fig. 8. h, vs C/r in the case of 1 = -I/16.
1.0
259
Analysis of surface cracks using the LSM
t
t
t
t
t
t
t
t
t”
no-spring elamalt
Fig. 9. The finite element mesh for a quarter of the surface cracked plate in tension.
2.5
-a-- Prosent LSM -Newman [ll]
ale = 0.2 2.0
C %
1.5
% 5 b 2
1.0
0.5
0.0 0.0
I 0.2
I
I 0.6
0.4
I 0.8
I 1.0
2+/x Fig. 10. The normalized SIF vs the crack angle for surface cracked plates in tension (a/c 10.2, Q = 1.104).
Z. Z. HUANG and 0. GAO
260 2.0
rlc
= 0.4
-o-
PrMent LSM
-
Newman (111
r/t = 0.8. 1.5
C “h 1
1.0
b 2
0.5
0.0 __
I
I
I
I
I
0.2
0.4
0.6
0.8
1.0
__
0.u
20/r Fig. 11. The normalized SIF vs the crack angle for surface cracked plates in tension (a/c = 0.4, Q = 1.324).
a quarter of the plate is shown in Fig. 9, containing 81 plateshell elements and 16 line spring elements. For all of the crack geometries the ratios of L/W and W/c are fixed as 2 and 4, respectively. The computations are executed on a VAX-l l/780 computer. In the elastic range (n = l), the SIF is obtained by means of the J-K relation. Figures lo-12 show the normalized SIF along the crack front by the present analysis and by the 3D finite analysis of Raju and Newman [l 11. At the deepest points of the surface cracks, the two solutions have a
2.0 aft
= 0.6
-o-
Prssent LSM
-
Newman [II)
o.“O.o
1.0
291% Fig. 12. The normalized SIF vs the crack angle for surface cracked plates Q = 1.629).
in tension (a/c =
0.6.
261
Analysis of surface cracks using the LSM 5.0
a - 5.0 et= 1.0 r/c = 0.2
Fig.
13. The f-integral
r/t
I 0.6 P
^
vs the far field tension stress for surface cracked (o/c = 0.2, n = 5, a = I, v = 0.3).
plates in tension
difference of less than 6%. The LSM results are much lower near the comer points. The reason should be that at the deepest point the state of stress is plane strain, which coincides with the definition of line-spring, while that at the comer point is plane stress. The present results basically agree with those reported by German et al., and support the conclusion that the LSM for shallow surface cracks is better than for deep cracks [2]. In the fuhy plastic analysis, the hardening exponents n = 3,5,7 and material constant a = 1, are considered. The finite element analysis of the fully plastic problem with the deformation theory
ait = 0.6 P
0.0
0.0
Fig.
14. The J-integral
0.2
0.4
0.6
0.8
vs the far field tension stress for surfaa (a/c = 0.4, n = 5, a = 1, v = 0.3).
1.0
cracked
plates in tension
262
2. Z. HUANG and Q. GAO 5.0 alt
= 0.6
4.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
alao Fig. IS. The J-integral vs the far field tension stress for surface cracked plates in tension (u/c = 0.6, n = 5,u = 1, v = 0.3).
of plasticity is equal to that of a non-linear elastic one. In an iteration sequence, the Newton-Raphson and the Modified Newton-Raphson method are employed alternately. In accordance with the engineering approach, the J-integral is the sum of the modified elastic and the fully plastic one. Figures 13-15 show the normalized J-integral vs the norma&& remote tension stress for n = 5 and a = 1. The J-integral results of 3D finite element analysis by Parks et al. [12] are given in Fig. 16 to be compared with the LSM results. The comparison shows that the proposed fully plastic LSM is efficient with a reasonable accuracy.
* Present LSM ----Parks 1121
n = 5.0 cl=
1.0
Y f 0.3
Na IS.0
B ii 10.0
5.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
ala, Fig. 16. The J-integral obtained by present LSM and by 3D finite element method vs the far field tension stress in tension (n/r ==0.6,n =$a = t,v -0.3).
263
Analysis of surface cracks using the LSM
CONCLUSIONS
The LSM discussed in this paper is mainly based on the engineering approach of elastic-plastic fracture mechanics and the complementary energy method. A simple method is suggested to specify the undetermined factors in the complementary energy expression of SECP. The obtained fully plastic solutions of SECP are in a quasi-analytic form and are suitable for arbitrary values of C/t and 1. The surface cracked plates in tension are analyzed by this model with a plate-shell finite element. The estimated J-integral results are in keeping with those obtained by 3D finite element analysis. REFERENCES J. R. Rice and N. Levy, The part through surface crack in an elastic plate. J. 185 (1972). M. D. German, V. Kumar and H. G. Delorenzi, Analysis of surface cracks in plates and shells using line-spring model 1;; and ADINA. Cornpar. Srructures 17, 881 (1983). appl. Mech. 39,
131V. Kumar, M.D. German and C. F. Shih, An engineering approach for elastic-plastic fracture analysis. EPRI NP-193 I, Topical Rep., Res. Project 1237-1, General Electric Company, Schenectady, NY (1981). 141V. Kumar er al., Further development in the engineering approach for elastic-plastic fracture analysis. EPRI NP-3607, Final Rep., Res. Project 1237-1, General Electric Company, Schenectady, NY (1983). C. F. Shih and A. Needelman, Fully plastic crack problems. J. appl. Mech. 51, 48 (1984). 1:; T. Miyoshi, M. Shiratori and Y. Yoshida, Analysis of J-integral and crack growth for surface cracks by line spring method. J. Press. Vess. Technol. 108, 305 (1986). [7l T. G. Shawki, T. Nakamura and D. M. Parks, Line-spring analysis of surface flawed plates and shells using deformation theory. Inc. J. Frucrure 41, 23 (1989). 181 D. M. Parks, V. Kumar and C. F. Shih, Consistency checks for power-law calibration functions. Second Symposium on Elasric-Plasric Fracrure, Vol. 1, Inelasric Crack Analysis (Edited by C. F. Shih and J. P. Gudas) ASTM STP 803, 1370 (1983).
[91 C. F. Shih and A. Needelman. Fully plastic crack problems. J. uppl. Mech. 51, 51 (1984). 1101M. Shiratori and T. Miyoshi, Evaluation of J-integral for surface cracks. Second Symposium on Elasric-Plastic Fracrure, Vol. 1 Inelusric Crack Analysis (Edited by C. F. Shih and J. P. Gudas), ASTM STP 803, 1410 (1983). 1111J. C. Newman, Jr and I. S. Raju, Stress-intensity factors for a wide range of semi-elliptical surface cracks in finitethickness plates. Engng Fracture Mech. 11, 817 (1979). 1121 D. M. Parks and Y. Y. Wang, Elastic analysis of part-through surface cracks, in Analyrical, Numerical, and Experimental Aspecrs of Three Dimensional Fracture Processes (Edited by A. J. Rosakis, K. Ravi-Chander and Y. Rajapakse), New York (1988). (Received 19 July 1993)