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Elastic-plastic fracture analysis of a thick-walled cylinder
hr. J. Pres. Printed ELSEVIER
0308-0161(94)00048-4
Ves. & Piping 63 (1995) 165-168 0 1995 Elsevier Science Limited in Northern Ireland. All rights reserved 030%0161/9S,‘$O9.50
Elastic-plastic fracture analysis of a thick-walled cylinder Wang Zhiqun Department
of Applied
Mechanics,
Nanjing
University
of Science and Technology,
Nanjing,
People’s
Republic
of China
(Received 20 May 1994;accepted20 June 1994)
In this paper the elastic-plastic fracture parameter-J-integral-for thickwalled cylinder autofrettage is calculated using a nonlinear finite element method basedon the flow theory of plasticity. The &-resistance curve of the material used is obtained experimentally. According to the J-based fracture criteria, crack initiation and crack instability in a thick-walled cylinder under autofrettage pressuresare analyzed. The relationship between the initial crack sizesand o&strains is determined.
1 INTRODUCTION Autofrettage has been developed to increase the strength in relation to bore pressure of a thick-walled cylinder. Large scale yielding occurs in the autofrettaged thick-walled cylinder wall. The theory of elastic-plastic fracture mechanics is used for this aspect of fracture analysis; the J-integral is one of the most valuable parameters for characterizing crack initiation, stable growth and instability. In this paper the J-integral for a thick-walled cylinder, under conditions of large plastic deformations when subjected to autofrettage pressures, is calculated using a nonlinear finite element method based on the flow theory of plasticity. The &-resistance curve of the material is obtained through experiment. According to the J-based fracture criteria, crack initiation, stable growth and crack instability of a thick-walled cylinder under autofrettage pressures are analyzed. The relationship between initial crack sizes and overstrains is determined. 2 J-INTEGRAL IN LARGE SCALE YIELDING OF THICK-WALLED CYLINDER WALL
employing deformation theory of plasticity or using engineering meth0ds.l In this paper, for a thick-walled cylinder, the J-integral in large scale yielding is calculated using the nonlinear finite element method and employing the flow theory of plasticity. 2.1 Constitutive equation of flow theory
W = (PI” - Pl%W where [D]’ is the elastic matrix and
in which and f(a,, material obtained material.
k is a function of the plastic work W, ~5, k) = 0 expresses yielding surface of hardening. The parameter A can be from results of axial tension of the
A = { $}T[D]e{
J-integral solutions are available for relatively thin-walled cylinders. For a thin-walled cylinder its solutions are based on the finite element and
(1)
5)
+ $$‘{rr}‘{
E}
(4)
The material used is a Cr-Ni-MO-V steel. The axial tension U---I curve can be simplified by
166
Wang Zhiqun
6 1
E
Fig, 1. Stress-strain
curve.
visualizing the process as one of double-linear hardening, as shown in Fig. 1:
q - EET
(5)
E-E, E, = tg8,
E = tge,
I-
‘J2
Fig. 3. Finite element mesh and J-integral paths for the zone A.
J-integral is given by
(6)
2.2 Finite element mesh With strain hardening of the material close to perfect plasticity, the strain field near the crack tip has approximately Y-’ singularity. Thus, we choose f midpoint isoparameter element around the crack tip in which the strain has r-’ singularity. The other meshes are the standard isoparameter elements as shown in Fig. 2.
The integral paths pass the Gauss integral points of isoparameters. For the convenience of comparison, five integral paths are chosen around the crack tip as shown in Fig. 3. IS, the J-integral value of the fifth path, is distinctly different from that of the others. This is due to the tolerance caused by the fifth integral passing a seven-node isoparameter.
2.3 Loading and integral paths Applying the flow theory of plasticity, the loading is divided into 20 steps. The method of changing stiffness is adopted. The formula for the
FLjFFig. 2. Finite element mesh for a cylinder.
Y-
2.4 Example The outside and inside radius, R2 and RI, of the thick-walled cylinder are taken to be 125 mm and 60 mm, respectively. The U--E curve of the material is shown in Fig. 1 and is modeled by a double-linear representation with the following parameters: a, = 1280 MPa, E = 200 GPa, and E, = 3630 MPa. As the cylinder possesses symmetry, only half of the thick-walled cylinder is considered and 64 elements and 186 nodes are taken. The results of calculation are listed in Table 1 and 2. Table 1 shows that the values of the J-integral calculated by using flow theory of plasticity are conservational. The values of the J-integral in Table 2 are the mean values of &, J3 and J4. The curve giving the driving force for crack
Elastic-plastic Table
1. J-integral
values of several integral
300 400 500 600 700 800 900
Jz
J3
J4
29.32 53.40 82.35 125.99 179.26 245.63 312.48
29-23 53.28 85.35 125.68 183.00 254.11 315.12
29.31 53.40 85.52 128.71 180.00 258.69 316.44
2. J-integral
values for several pressures
75-11 11.40 13.54 5.7 141.2 89-65 668.71
depths
JR =
---
a=1.5
a=3
8.27 14.83 24.04 39.6 57.26 74.27 92.64 116.05 136.85
16.36 29.28 46.66 71.80 102.90 132-67 164.00 210.05 305.00
~~
a=6
and
a = 14
a = 10
29.29 53.36 84.41 126.79 180.70 252.81 314.68 448.65 665.50
47.60 85.77 137,85 202.50 29760 420.10 525.10 810.50 2 073.00
65.72 118.30 190.80 285.50 424.20 655.60 995.80 1995.00 4 892.00
In accordance with the National Standard GB203880, the J-integral resistance JR corresponding to the crack growth ha for the Cr-Ni-MO-V steel are measured as shown in Table 3. The yielding stress of the material us = 1280 MPa, and the ultimate stress ub = 1385 MPa. Processing the first five data in Table 3 with a linear approximation, the JR - Aa equation is obtained:
0.0346 58.33
INITIATION
First, the internal pressure corresponding to the overstrain of the thick-walled cylinder is calculated and then, in terms of Table 2, the crack driving force equation with load held fixed, expressed by the J-integral J(a), can be obtained. Using the crack initiation condition: = J,
(10)
J(a), = JR aJ(a> U aa .=da
(11)
[ 1
critical crack length a, for crack instability and the initial crack length a, for its stable growth under the corresponding internal pressure are found. The value of a, divided by a factor of safety gives the allowable crack length. In our example, when the autofrettage internal pressure is 700 MPa, under the condition a d 10 the crack driving force equation is approximately linear, i.e.: J(a), = 3.145a (12) Making use of eqn (lo), we find the critical crack length for crack initiation ai = 1.79 mm. Letting Aa = a -a, in eqn (9), the JR resistance equation is rewritten as: J,(Aa) = J,(a - a,) = 15.8 + 6*831g(a - a,)
(8)
(13
3. JR test results for Cr-Ni-MO-V 0.168 80.8
(9
the critical crack length ai for crack initiation can be found. According to the crack instability condition:
3 ELASTIC-PLASTIC FRACTURE BEHAVIOUR OF MATERIAL
JR = 5.35 -t- 1964 Aa
4 ANALYSIS OF CRACK AND INSTABILITY
J(a),
growth as denoted by the J-integral with the applied load held fixed can be obtained from the results in Table 2.
Table
15.8 + 6.831g Aa
which is the JR resistance equation.
J (kN m-‘)
P Q$W
300 400 500 600 700 800 900 1000 1 100
The equation for the blunting line is given by JR = 399.75 Aa, from which we find the crack initiation toughness J = 56.3 kN m-‘. Processing the data in Table 3 with a logarithmic approximation, we obtain:
paths
JS
crack
167
cylinder
analysis of a thick-walled
J(kNm-‘)
P NW
Table
fracture
0.209 96,4
0.373 141.6
0.492 149.5
steel 1.35 166.1
3.21
192
4.35
226
Wang Zhiqun
168
Substituting eqns (12) and (13) into eqn (ll), we find a, = 5.76 mm and a, = 3.59 mm.
- 80 - 70 - 60
5 RELATIONSHIP BETWEEN THE CRITICAL CRACK LENGTH AND OVERSTRAIN
- 50-
s .5
- 40 g
- 30
The material used for the thick-walled cylinder in this paper is assumed to be in conformity with double-linear hardening and the relationship between the applied autofrettage internal pressure p and the plastic region radius p is given by Ref. 2. For the closed end: 2u, ‘=T5
1 -In:+; i l+a,
(
,
I--$ 2
I-kp E Ei-
H=
WI
1-s (
1
2
3
4
5
6
Crack lengtha(mm) Fig. 4. Variation of a,, a0 and a, with the pressurep.
from eqn (14). Then, the critical crack length a;, a, and a, are found from eqns (10) and (11). Figure 4 shows the relationship between the overstrain, autofrettage internal pressure and ai, a, and a, of the thick-walled cylinder. From Fig. 4, the critical crack length ai for crack initiation, initial crack length a,, for crack stable growth and critical crack length a, for crack instability corresponding to the overstrain can be found. Figure 4 shows that af and a, decrease rapidly with the increase of overstrain and then decrease slowly after attaining some 40% of overstrain.
(14)
+ a, = $H
1
0.5
1
The material constants for Cr-Ni-MO-V steel are taken to be E = 200 MPa and p = 0.27. The value of k in eqn (15) depends on the end condition. With a free end, k = 0 and a, = O-0246, so 20-,/V? in eqn (14) is modified and changed to l*llu,. The radius p is determined by the overstrain internal pressure p and the autofrettage corresponding to the overstrain can be obtained
REFERENCES 1. Kumar, V., German, M. D. & Shih, C. F., An Engineering Approach for Elastic Plastic Fracture Analysis. Topical Report No. EPRI NP-1931, General Company Schenectady,NY, July 1981. 2. Wuxue, Z., & Zichu, Z., An elastic-plastic analysis of autofrettage thick-walled cylinders. In Proc. of the Znt. Conf: on Non-Linear Mechanics, Shanghai, 198.5,pp. 663-7, in Chinese.
0308-0161(94)00048-4
Ves. & Piping 63 (1995) 165-168 0 1995 Elsevier Science Limited in Northern Ireland. All rights reserved 030%0161/9S,‘$O9.50
Elastic-plastic fracture analysis of a thick-walled cylinder Wang Zhiqun Department
of Applied
Mechanics,
Nanjing
University
of Science and Technology,
Nanjing,
People’s
Republic
of China
(Received 20 May 1994;accepted20 June 1994)
In this paper the elastic-plastic fracture parameter-J-integral-for thickwalled cylinder autofrettage is calculated using a nonlinear finite element method basedon the flow theory of plasticity. The &-resistance curve of the material used is obtained experimentally. According to the J-based fracture criteria, crack initiation and crack instability in a thick-walled cylinder under autofrettage pressuresare analyzed. The relationship between the initial crack sizesand o&strains is determined.
1 INTRODUCTION Autofrettage has been developed to increase the strength in relation to bore pressure of a thick-walled cylinder. Large scale yielding occurs in the autofrettaged thick-walled cylinder wall. The theory of elastic-plastic fracture mechanics is used for this aspect of fracture analysis; the J-integral is one of the most valuable parameters for characterizing crack initiation, stable growth and instability. In this paper the J-integral for a thick-walled cylinder, under conditions of large plastic deformations when subjected to autofrettage pressures, is calculated using a nonlinear finite element method based on the flow theory of plasticity. The &-resistance curve of the material is obtained through experiment. According to the J-based fracture criteria, crack initiation, stable growth and crack instability of a thick-walled cylinder under autofrettage pressures are analyzed. The relationship between initial crack sizes and overstrains is determined. 2 J-INTEGRAL IN LARGE SCALE YIELDING OF THICK-WALLED CYLINDER WALL
employing deformation theory of plasticity or using engineering meth0ds.l In this paper, for a thick-walled cylinder, the J-integral in large scale yielding is calculated using the nonlinear finite element method and employing the flow theory of plasticity. 2.1 Constitutive equation of flow theory
W = (PI” - Pl%W where [D]’ is the elastic matrix and
in which and f(a,, material obtained material.
k is a function of the plastic work W, ~5, k) = 0 expresses yielding surface of hardening. The parameter A can be from results of axial tension of the
A = { $}T[D]e{
J-integral solutions are available for relatively thin-walled cylinders. For a thin-walled cylinder its solutions are based on the finite element and
(1)
5)
+ $$‘{rr}‘{
E}
(4)
The material used is a Cr-Ni-MO-V steel. The axial tension U---I curve can be simplified by
166
Wang Zhiqun
6 1
E
Fig, 1. Stress-strain
curve.
visualizing the process as one of double-linear hardening, as shown in Fig. 1:
q - EET
(5)
E-E, E, = tg8,
E = tge,
I-
‘J2
Fig. 3. Finite element mesh and J-integral paths for the zone A.
J-integral is given by
(6)
2.2 Finite element mesh With strain hardening of the material close to perfect plasticity, the strain field near the crack tip has approximately Y-’ singularity. Thus, we choose f midpoint isoparameter element around the crack tip in which the strain has r-’ singularity. The other meshes are the standard isoparameter elements as shown in Fig. 2.
The integral paths pass the Gauss integral points of isoparameters. For the convenience of comparison, five integral paths are chosen around the crack tip as shown in Fig. 3. IS, the J-integral value of the fifth path, is distinctly different from that of the others. This is due to the tolerance caused by the fifth integral passing a seven-node isoparameter.
2.3 Loading and integral paths Applying the flow theory of plasticity, the loading is divided into 20 steps. The method of changing stiffness is adopted. The formula for the
FLjFFig. 2. Finite element mesh for a cylinder.
Y-
2.4 Example The outside and inside radius, R2 and RI, of the thick-walled cylinder are taken to be 125 mm and 60 mm, respectively. The U--E curve of the material is shown in Fig. 1 and is modeled by a double-linear representation with the following parameters: a, = 1280 MPa, E = 200 GPa, and E, = 3630 MPa. As the cylinder possesses symmetry, only half of the thick-walled cylinder is considered and 64 elements and 186 nodes are taken. The results of calculation are listed in Table 1 and 2. Table 1 shows that the values of the J-integral calculated by using flow theory of plasticity are conservational. The values of the J-integral in Table 2 are the mean values of &, J3 and J4. The curve giving the driving force for crack
Elastic-plastic Table
1. J-integral
values of several integral
300 400 500 600 700 800 900
Jz
J3
J4
29.32 53.40 82.35 125.99 179.26 245.63 312.48
29-23 53.28 85.35 125.68 183.00 254.11 315.12
29.31 53.40 85.52 128.71 180.00 258.69 316.44
2. J-integral
values for several pressures
75-11 11.40 13.54 5.7 141.2 89-65 668.71
depths
JR =
---
a=1.5
a=3
8.27 14.83 24.04 39.6 57.26 74.27 92.64 116.05 136.85
16.36 29.28 46.66 71.80 102.90 132-67 164.00 210.05 305.00
~~
a=6
and
a = 14
a = 10
29.29 53.36 84.41 126.79 180.70 252.81 314.68 448.65 665.50
47.60 85.77 137,85 202.50 29760 420.10 525.10 810.50 2 073.00
65.72 118.30 190.80 285.50 424.20 655.60 995.80 1995.00 4 892.00
In accordance with the National Standard GB203880, the J-integral resistance JR corresponding to the crack growth ha for the Cr-Ni-MO-V steel are measured as shown in Table 3. The yielding stress of the material us = 1280 MPa, and the ultimate stress ub = 1385 MPa. Processing the first five data in Table 3 with a linear approximation, the JR - Aa equation is obtained:
0.0346 58.33
INITIATION
First, the internal pressure corresponding to the overstrain of the thick-walled cylinder is calculated and then, in terms of Table 2, the crack driving force equation with load held fixed, expressed by the J-integral J(a), can be obtained. Using the crack initiation condition: = J,
(10)
J(a), = JR aJ(a> U aa .=da
(11)
[ 1
critical crack length a, for crack instability and the initial crack length a, for its stable growth under the corresponding internal pressure are found. The value of a, divided by a factor of safety gives the allowable crack length. In our example, when the autofrettage internal pressure is 700 MPa, under the condition a d 10 the crack driving force equation is approximately linear, i.e.: J(a), = 3.145a (12) Making use of eqn (lo), we find the critical crack length for crack initiation ai = 1.79 mm. Letting Aa = a -a, in eqn (9), the JR resistance equation is rewritten as: J,(Aa) = J,(a - a,) = 15.8 + 6*831g(a - a,)
(8)
(13
3. JR test results for Cr-Ni-MO-V 0.168 80.8
(9
the critical crack length ai for crack initiation can be found. According to the crack instability condition:
3 ELASTIC-PLASTIC FRACTURE BEHAVIOUR OF MATERIAL
JR = 5.35 -t- 1964 Aa
4 ANALYSIS OF CRACK AND INSTABILITY
J(a),
growth as denoted by the J-integral with the applied load held fixed can be obtained from the results in Table 2.
Table
15.8 + 6.831g Aa
which is the JR resistance equation.
J (kN m-‘)
P Q$W
300 400 500 600 700 800 900 1000 1 100
The equation for the blunting line is given by JR = 399.75 Aa, from which we find the crack initiation toughness J = 56.3 kN m-‘. Processing the data in Table 3 with a logarithmic approximation, we obtain:
paths
JS
crack
167
cylinder
analysis of a thick-walled
J(kNm-‘)
P NW
Table
fracture
0.209 96,4
0.373 141.6
0.492 149.5
steel 1.35 166.1
3.21
192
4.35
226
Wang Zhiqun
168
Substituting eqns (12) and (13) into eqn (ll), we find a, = 5.76 mm and a, = 3.59 mm.
- 80 - 70 - 60
5 RELATIONSHIP BETWEEN THE CRITICAL CRACK LENGTH AND OVERSTRAIN
- 50-
s .5
- 40 g
- 30
The material used for the thick-walled cylinder in this paper is assumed to be in conformity with double-linear hardening and the relationship between the applied autofrettage internal pressure p and the plastic region radius p is given by Ref. 2. For the closed end: 2u, ‘=T5
1 -In:+; i l+a,
(
,
I--$ 2
I-kp E Ei-
H=
WI
1-s (
1
2
3
4
5
6
Crack lengtha(mm) Fig. 4. Variation of a,, a0 and a, with the pressurep.
from eqn (14). Then, the critical crack length a;, a, and a, are found from eqns (10) and (11). Figure 4 shows the relationship between the overstrain, autofrettage internal pressure and ai, a, and a, of the thick-walled cylinder. From Fig. 4, the critical crack length ai for crack initiation, initial crack length a,, for crack stable growth and critical crack length a, for crack instability corresponding to the overstrain can be found. Figure 4 shows that af and a, decrease rapidly with the increase of overstrain and then decrease slowly after attaining some 40% of overstrain.
(14)
+ a, = $H
1
0.5
1
The material constants for Cr-Ni-MO-V steel are taken to be E = 200 MPa and p = 0.27. The value of k in eqn (15) depends on the end condition. With a free end, k = 0 and a, = O-0246, so 20-,/V? in eqn (14) is modified and changed to l*llu,. The radius p is determined by the overstrain internal pressure p and the autofrettage corresponding to the overstrain can be obtained
REFERENCES 1. Kumar, V., German, M. D. & Shih, C. F., An Engineering Approach for Elastic Plastic Fracture Analysis. Topical Report No. EPRI NP-1931, General Company Schenectady,NY, July 1981. 2. Wuxue, Z., & Zichu, Z., An elastic-plastic analysis of autofrettage thick-walled cylinders. In Proc. of the Znt. Conf: on Non-Linear Mechanics, Shanghai, 198.5,pp. 663-7, in Chinese.