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A parametric study on J-integral under pressurized thermal shock by using statically indeterminate fracture mechanics
Nuclear Engineering and Design 196 (2000) 153 – 159 www.elsevier.com/locate/nucengdes
A parametric study on J-integral under pressurized thermal shock by using statically indeterminate fracture mechanics M. Matsubara * Faculty of Engineering, Gunma Uni6ersity, Kiryu-shi, Gunma 376 -8515, Japan Received 16 September 1999; accepted 15 October 1999
Abstract The method of statically indeterminate fracture mechanics (SIFM) is application of elastic-plastic fracture mechanics to statically indeterminate problems. Application of SIFM has been developed for axially cracked cylinder problems under axisymmetric pressurized thermal shock, PTS loading. This method allows us to evaluate the J-integral in an explicit form and is efficient in clarifying the mechanical characteristics of the PTS event. This paper describes a parametric study of the J-integral under PTS loading by using SIFM. © 2000 Elsevier Science S.A. All rights reserved.
1. Introduction The J-integral is valid for elastic-plastic fracture mechanics analysis, especially in ductile fracture analysis. But the original J-integral proposed (Rice, 1968) is not used for fracture analysis under thermal loading, because the J-integral cannot keep its path independence in the case. Special path independent integrals have been proposed for the purpose (Kishimoto et al., 1980). The special path independent integrals can only be evaluated by using a numerical analysis method,
* Tel.: +81-277-30-1536; fax: + 81-277-30-1599. E-mail address: [email protected] (M. Matsubara)
finite element method and so on. They are not generally familiar to engineers because of their difficulty and complexity. The author has proposed applying statically indeterminate fracture mechanics (SIFM) to a single-edge cracked specimen subjected to combined tension and bending, which is typical for structures in fracture mechanics analysis (Matsubara, 1990). SIFM enables us to evaluate the J-integral value of such a cracked member under displacement controlled condition. In previous works, SIFM was used for obtaining the J-integral for a cylinder containing a long axial crack under pressurized thermal shock, PTS (Matsubara, 1991; Matsubara and Soneda, 1992). This paper describes a parametric study of the J-integral under PTS based on the application of SIFM to a axially cracked cylinder problem.
0029-5493/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 2 9 - 5 4 9 3 ( 9 9 ) 0 0 2 9 0 - 3
M. Matsubara / Nuclear Engineering and Design 196 (2000) 153–159
154
2. Analysis model Fig. 1 shows the model of the cylinder of inner radius Ri, outer radius Ro and wall thickness W= Ro − Ri,. It is assumed that the cylinder had a uniform initial temperature equal to TW over thickness of the wall and that beginning of the cooling phase starts at t = 0 (the start of the thermal transient), with a medium temperature of TC. Temperature profiles are assumed to be sym-
metrical about the axis and independent of the axial coordinate, z. The cylinder is internally pressurized. The type of flaw is a two-dimensional inner-surface axial crack with depth a. The internal pressure p and coolant temperature TC are assumed to be constant through the transient. Table 1 shows the condition considered in the analysis. Parameters which are assumed to apply are thermal conductivity, heat transfer coefficient and stress–strain relationship.
3. Thermal analysis In the cylindrical coordinate system, the dependent variable, temperature T, is related to independent variables, time t and distance r from the center in the conducting body, by means of a heat balance differential equation derived from Fourier’s law of heat conduction. The partial differential equation is
( 2T(r) 1 (T(r) (T(r) =k + (t (r 2 r (r
rc
Fig. 1. PTS model considered in analysis. Table 1 Conditions considered in analysis Model cylinder Mean radius (m) Wall thickness (m) Young’s modulus (GPa) Poisson’s ratio Ramberg–Osgood’s law Work-hardening constant Work-hardening exponent Reference stress (MPa) Reference strain Coefficient of thermal expansion (1/K) Specific weight (kg/m3) Specific heat (kJ/(kg K)) Thermal conductivitty (W/m K) Thermo-hydraulic condition Internal pressure (MPa) Initial wall temperature (K) Coolant temperature (K) Heat transfer coefficient (W/(m2 K)) Cooling rate (1/min)
(1)
Here r is the density, c the specific heat and k the thermal conductivity. The boundary condition is continuity of inside surface (at r= Ri) flux. (T =h(T− TC ) (r
(2)
k 2.0 0.18–0.22 193–206 0.3 0.0208–1.12 9.71–11.5 375–414 2.37×10−3 1.5×10−5 7800 0.5 30–50 15 560 300 2500–7500
Here h is the heat transfer coefficient. The outside surface (at r= Ro) of the cylinder may be thermally insulated. (T =0 (r
(3)
The initial boundary condition on t= 0 is (4)
T(r)= Tw
The hoop stress component, su (r= Ri + x), over thickness of the wall at any time during pressurized thermal transient may be written by su =
&
aE r 2 + R 2i 2 (1− 6)r R 2o − R 2i
Ro
Ri
Trdr +
&
r
Ri
Trdr − Tr 2
(5)
Here E is Young’s modulus, a the coefficient of thermal expansion and 6 Poisson’s ratio. su is
M. Matsubara / Nuclear Engineering and Design 196 (2000) 153–159
155
4. Statically indeterminate fracture mechanics
4.1. Application of SIFM
Fig. 2. The hoop stress through wall thickness under PTS.
The hoop stress in the model analyzed is induced by internal pressure and thermal shock. Prior to the application of SIFM, the hoop stress is divided into three components, membrane stress, bending stress and stress varying non-linearly through the wall thickness sn (r), as shown in Fig. 2. SIFM is applied to evaluation of the J-integral due to membrane stress and bending stress which vary linearly. A small axial strip containing the crack is conceptually separated from the cylindrical shell as shown in Fig. 3. The tensile load, P, and moment, M, per unit axial length of the cylinder are the changes in axial load and moment due to introduction of the crack. P0 and M0 are, respectively, the initial distributed axial force and moment per unit length of the cylinder on the ends of the slit cylinder. The displacement, u, and the rotation, u, are those at the ends of the slit cylinder. Du and Du are those arising from the presence of the crack. By continuity of displacement and rotation, u and u must, respectively, equal Du and Du. That is, u= Du u= Du
Fig. 3. Statically indeterminate fracture mechanics model.
necessary for evaluation of the J-integral under PTS. Fig. 2 shows the hoop stress components at a particular time 400 s after start of thermal shock. The hoop stress under PTS has three components, membrane tensile stress, bending stress and stress varying non-linearly through the wall thickness. The non-linear stress component is balanced in the direction of the wall thickness. The bending stress by constraint has almost reached the maximum value after 400 s.
"
(6)
P and M can be obtained by solving the simultaneous equation (Eq. (6)), because u, u, Du and Du are considered to be functions of P and M. The P and M values obtained can provide the J-integral as described below. The displacement and rotation of the split cylinder due to P and M may be written as
!"
u CP = u CPM
CPM CM
n!
P0 − P M0 − M
"
(7)
Here CP is the extension compliance of a split cylinder due to the surface hoop forces on crack faces, CM the angular compliance of the cylinder due to the surface moments and CPM the extension compliance of the cylinder due to the moments or the angular compliance of the cylinder due to the hoop forces.
M. Matsubara / Nuclear Engineering and Design 196 (2000) 153–159
156
CP, CPM and CM are given as 3pRm CP = E%W
! "
1 1 1+ − k 3 2p 1 CPM = − 1+ E%W k C CM = − PM Rm
Â Ã Ã Ã Ì Ã Ã Ã Å
(8)
Here  à E E%= à (1− 6) à W3 à I= à 12 Ì 1 1 +l à k= − 1 + log 2l 1 −l à à W à l= 2Rm à Å
(9)
where Rm is the mean radius of the cylinder. The displacement and rotation of the cracked strip element, Du and Du, may be written as
! "
Du DCP = DCPM Du
DCPM DCM
n! " P M
(10)
Here DC eP, DC ePM and DC eM are the extension and angular compliance increments due to the presence of crack of depth a, for a flat plate in plane strain tension and bending conditions. They may be expressed as the sum of the elastic components, DC eP, DC ePM and DC eM, and the plastic components, DC pP, DC pPM and DC pM, i.e. DCP =DC eP +DC pP Â Ã DCPM =DC ePM Ì Ã DCM = DC eM +DC pM Å
(11)
If the length of the cracked strip element is small enough (Cheng et al., 1984), the elastic components of the compliance can be obtained from
& & &
 KP 2 à da P à 0 a à 2 K K P M da Ì DC ePM = M E% 0 P à a 2 à KM 2 e da à DC M = E% 0 M Å DC eP =
2 E
a
(12)
Here KP is the stress intensity factor of a singleedge cracked specimen subjected to tension and KM is that subjected to bending. Based on the similar assumption to the elastic components of DCP, DCPM and DCM and the complimentary energy concept, the plastic ones are derived by Parks et al. (unpublished) as n−1 Â PM M2 2 Ã 2 AP + 2B +C 2 W W A Ã 2 p DC P = aRo0s0W P 2C Ã P 2C Ã n−1 2 Ã 2 PM M AP 2 + 2B +C 2 Ì W W B Ã DC pPM = aRo0s0W P 2C Ã P 2C n−1 Ã PM M2 2 Ã AP 2 + 2B +C 2 W W C Ã p DC M = aRo0s0 P 2C Å P 2C
:
: :
;
; ;
(13) The constitutive relation is completed by specifying the Ramberg–Osgood’s law (Ramberg and Osgood, 1943), i.e. o s s n = + aR (14) o0 s0 s0
Here o0 is the reference strain, s0 the reference stress, aR the work-hardening constant, n the work-hardening exponent, o the true strain and s the true stress. A, B and C in Eq. (13) are functions of a and n. They can be derived from fully plastic solutions and expressed as 2  h uPj(1− j) n + 1 à A= à (1.445h)n à 2 à h uPAn + 1(1−j) Ì (15) B= à (1.445h)n 2 à n+1 h uM à C= 0.364n(1−j)n − 1 à Å
!
!
"
"
M. Matsubara / Nuclear Engineering and Design 196 (2000) 153–159
Here h uP and h uP are the fully plastic solutions corresponding to the displacement and rotation of the single-edge cracked specimen subjected to tension, and h uM that corresponding to the rotation of the specimen subjected to bending given by Shih and Needleman (1984) j and h are
h=
a W
Â Ã Ì 2 j 1 à 1+ − 1−j 1−jÅ j=
'
(16)
PC is the reference load by per unit axial distance as
157
PC = s0(W−a)
(17)
Substitution of Eq. (7) and Eq. (10) into the continuity conditions (Eq. (6)) leads to the following simultaneous equations. To obtain P and M on crack faces, P and M must be combined with P0 and M0 on the ends of the split cylinder.
CP + DC eP + DC pP DC ePM + DC pPM
=
CP CPM
CPM CM
DC ePM + DC pPM C + DC eM + DC pM
n! " P0 M0
n! " P M
(18)
Solving the non-linear simultaneous equations, P and M can be obtained.
4.2. J-integral e6aluation The J-integral value of a single-edge cracked specimen subjected to P and M can be evaluated by complementary energy method (Miyoshi et al., 1986) as  à 1 Je = (KP + KM + Kn )2 à E% n−1 à PM M2 2 à AP 2 + 2B +C 2 à W W 1 JP = ao0s0W 2 Ì 2 2 PC à 1 (A 2 (B PM (C M 2 2 (PCà · P +2 + − 3 à (a W (a W 2 P C (a à P 2C (a à PM M2 AP 2 + 2B +C 2 Å W W (19)
:
Fig. 4. The effect of differences in thermal conductivity on J-integral under PTS.
J=Je + Jp
!
"
;
Where Je and Jp are the elastic and plastic components of the J-integral and Kn is the contribution of the stress varying non-linearly through the wall thickness under PTS. Kn may be written as
&
'
s (x)G
a n
x ,j a
(20) x 2
pa·(1−j) 1− a Here G is the correction factor for the stress intensity factor of a single-edge cracked plate by surface tractions on the crack faces by Tada et al. (1985).
Kn =
Fig. 5. The effect of differences in heat transfer coefficient on J-integral under PTS.
2
3 2
0
158
M. Matsubara / Nuclear Engineering and Design 196 (2000) 153–159
Fig. 6. The effect of differences in constants of the Ramgerg – Osgood’s law on J-integral under PTS.
ductivity increase, heat transfer coefficient increase would make the J-integral slightly increase for each crack depth. The relations between J-integral and a/W of Figs. 4 and 5 are similar in configuration. Fig. 6 explains the change in J-integral — a/W relationship profile with variation in a and n of the Ramberg–Osgood’s law for the elapsed time of 400 s since the start of thermal shock. The Ramberg–Osgood’s law is an identical stress– strain characteristic for evaluating the J-integral and the compliances DC pP, DC ePM and DC eM. The contours of J-integral — a/W curve change with the constants a and n of the Ramberg–Osgood’s law. The contour would tend to be in the form of the parabola opening upwards with the increase in n.
The above methodology is essentially valid for a thin-walled cylinder structure. 6. Conclusion 5. Results and discussion J-integral values are calculated at 0.01 intervals of non-dimensional crack depth in Figs. 4–6. Fig. 4 shows the effect of thermal conductivity on the J-integral under PTS for the elapsed time of 400 s since the start of thermal transient. The J-integral would almost become largest for each crack depth, because the bending stress by constraint has almost reached the maximum value at that time. The thermal conductivity exerted only a minimal influence on the J-integral over three thermal conductivities considered. Thermal conductivity increase would increase slightly the J-integral for each crack depth. The J-integral shows the first peak at about a/W=0.25. The first peak arises from positive bending, membrane and nonlinear stresses (see Fig. 2). The interaction of bending and membrane stresses causes the second peak at about a/W = 0.5. The J-integral under PTS almost tends as crack grows except for crack depth of about a/W = 0.25 and 0.5. Fig. 5 shows the J-integral under PTS for the elapsed time of 400 s as a function of crack depth a/W under the three conditions of heat transfer coefficient. The heat transfer coefficient has little effect on the J-integral. Like to the thermal con-
The study in this paper has demonstrated the parametric effect on the J-integral for the simplified model of an axially-cracked cylinder without cladding under PTS. The parameters considered are thermal conductivity, heat transfer coefficient and stress-strain relation. Both the thermal conductivity and the heat transfer coefficient do not noticeably affect the J-integral. The profiles of J-integral — a/W curve depend on the constants a and n of the Ramberg–Osgood’s law.
References Cheng, W., Stevick, G., Finnie, I., 1984. Predictions of the stress intensity factor for an internal circumferential crack at a butt-weld between cylinders using the plane strain solution, Transactions of the ASME. J. Eng. Mater. Technol. 106, 21 – 24. Kishimoto, K., Aoki, S., Sakata, M., 1980. On the path independent integral-J. . Eng. Fract. Mech. 13, 841 – 850. Matsubara, M., 1990. Application of statically indeterminate elastic-plastic fracture mechanics to a single-edge cracked member subjected to combined tension and bending. Trans. JSME 56 (526), 1396 – 1399. Matsubara, M., 1991. Application of statically indeterminate elastic-plastic fracture mechanics to pressurized thermal shock problem, pressure vessel integrity. PVP 213, 269– 273.
M. Matsubara / Nuclear Engineering and Design 196 (2000) 153–159 Matsubara, M., Soneda, N., 1992. Application of statically indeterminate elastic-plastic fracture mechanics to PROJECT FALSIRE problem, Pressure Vessel Technology, vol. 1, Design, Analysis, Materials (1), vol. 1, pp. 543 – 553. Miyoshi, T., Shiratori, M., Yoshida, Y., 1986. Analysis of J-integral and crack growth for surface cracks by line spring method, Transactions of the ASME. J. Pressure Vessel Tech. 108, 305–311. Parks, D.M., Rodin, G.J., Locket, R.R. The line-spring model for J-analysis of surface-cracked plates and shells: calibration of the power law spring, unpublished manuscript. Ramberg, W., Osgood, W.R., 1943. Description of stress –
.
159
strain curves by three parameters. NASA Technical Notes 902. Rice, J.R., 1968. A path independent integral and the approximate analysis of stress concentration of notches and cracks. Transactions of the ASME. J. Appl. Mech., 379– 386. Shih, C.F., Needleman, A., 1984. Fully plastic crack problems part 1: solutions by a penalty method, Transactions of the ASME. J. Appl. Mech. 51, 48 – 56. Tada, H., Paris, P.C., Irwin, G.R., 1985. The Stress Analysis of Crack Handbook, second ed. Paris Productions Incorporated, pp. 2 – 27.
A parametric study on J-integral under pressurized thermal shock by using statically indeterminate fracture mechanics M. Matsubara * Faculty of Engineering, Gunma Uni6ersity, Kiryu-shi, Gunma 376 -8515, Japan Received 16 September 1999; accepted 15 October 1999
Abstract The method of statically indeterminate fracture mechanics (SIFM) is application of elastic-plastic fracture mechanics to statically indeterminate problems. Application of SIFM has been developed for axially cracked cylinder problems under axisymmetric pressurized thermal shock, PTS loading. This method allows us to evaluate the J-integral in an explicit form and is efficient in clarifying the mechanical characteristics of the PTS event. This paper describes a parametric study of the J-integral under PTS loading by using SIFM. © 2000 Elsevier Science S.A. All rights reserved.
1. Introduction The J-integral is valid for elastic-plastic fracture mechanics analysis, especially in ductile fracture analysis. But the original J-integral proposed (Rice, 1968) is not used for fracture analysis under thermal loading, because the J-integral cannot keep its path independence in the case. Special path independent integrals have been proposed for the purpose (Kishimoto et al., 1980). The special path independent integrals can only be evaluated by using a numerical analysis method,
* Tel.: +81-277-30-1536; fax: + 81-277-30-1599. E-mail address: [email protected] (M. Matsubara)
finite element method and so on. They are not generally familiar to engineers because of their difficulty and complexity. The author has proposed applying statically indeterminate fracture mechanics (SIFM) to a single-edge cracked specimen subjected to combined tension and bending, which is typical for structures in fracture mechanics analysis (Matsubara, 1990). SIFM enables us to evaluate the J-integral value of such a cracked member under displacement controlled condition. In previous works, SIFM was used for obtaining the J-integral for a cylinder containing a long axial crack under pressurized thermal shock, PTS (Matsubara, 1991; Matsubara and Soneda, 1992). This paper describes a parametric study of the J-integral under PTS based on the application of SIFM to a axially cracked cylinder problem.
0029-5493/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 2 9 - 5 4 9 3 ( 9 9 ) 0 0 2 9 0 - 3
M. Matsubara / Nuclear Engineering and Design 196 (2000) 153–159
154
2. Analysis model Fig. 1 shows the model of the cylinder of inner radius Ri, outer radius Ro and wall thickness W= Ro − Ri,. It is assumed that the cylinder had a uniform initial temperature equal to TW over thickness of the wall and that beginning of the cooling phase starts at t = 0 (the start of the thermal transient), with a medium temperature of TC. Temperature profiles are assumed to be sym-
metrical about the axis and independent of the axial coordinate, z. The cylinder is internally pressurized. The type of flaw is a two-dimensional inner-surface axial crack with depth a. The internal pressure p and coolant temperature TC are assumed to be constant through the transient. Table 1 shows the condition considered in the analysis. Parameters which are assumed to apply are thermal conductivity, heat transfer coefficient and stress–strain relationship.
3. Thermal analysis In the cylindrical coordinate system, the dependent variable, temperature T, is related to independent variables, time t and distance r from the center in the conducting body, by means of a heat balance differential equation derived from Fourier’s law of heat conduction. The partial differential equation is
( 2T(r) 1 (T(r) (T(r) =k + (t (r 2 r (r
rc
Fig. 1. PTS model considered in analysis. Table 1 Conditions considered in analysis Model cylinder Mean radius (m) Wall thickness (m) Young’s modulus (GPa) Poisson’s ratio Ramberg–Osgood’s law Work-hardening constant Work-hardening exponent Reference stress (MPa) Reference strain Coefficient of thermal expansion (1/K) Specific weight (kg/m3) Specific heat (kJ/(kg K)) Thermal conductivitty (W/m K) Thermo-hydraulic condition Internal pressure (MPa) Initial wall temperature (K) Coolant temperature (K) Heat transfer coefficient (W/(m2 K)) Cooling rate (1/min)
(1)
Here r is the density, c the specific heat and k the thermal conductivity. The boundary condition is continuity of inside surface (at r= Ri) flux. (T =h(T− TC ) (r
(2)
k 2.0 0.18–0.22 193–206 0.3 0.0208–1.12 9.71–11.5 375–414 2.37×10−3 1.5×10−5 7800 0.5 30–50 15 560 300 2500–7500
Here h is the heat transfer coefficient. The outside surface (at r= Ro) of the cylinder may be thermally insulated. (T =0 (r
(3)
The initial boundary condition on t= 0 is (4)
T(r)= Tw
The hoop stress component, su (r= Ri + x), over thickness of the wall at any time during pressurized thermal transient may be written by su =
&
aE r 2 + R 2i 2 (1− 6)r R 2o − R 2i
Ro
Ri
Trdr +
&
r
Ri
Trdr − Tr 2
(5)
Here E is Young’s modulus, a the coefficient of thermal expansion and 6 Poisson’s ratio. su is
M. Matsubara / Nuclear Engineering and Design 196 (2000) 153–159
155
4. Statically indeterminate fracture mechanics
4.1. Application of SIFM
Fig. 2. The hoop stress through wall thickness under PTS.
The hoop stress in the model analyzed is induced by internal pressure and thermal shock. Prior to the application of SIFM, the hoop stress is divided into three components, membrane stress, bending stress and stress varying non-linearly through the wall thickness sn (r), as shown in Fig. 2. SIFM is applied to evaluation of the J-integral due to membrane stress and bending stress which vary linearly. A small axial strip containing the crack is conceptually separated from the cylindrical shell as shown in Fig. 3. The tensile load, P, and moment, M, per unit axial length of the cylinder are the changes in axial load and moment due to introduction of the crack. P0 and M0 are, respectively, the initial distributed axial force and moment per unit length of the cylinder on the ends of the slit cylinder. The displacement, u, and the rotation, u, are those at the ends of the slit cylinder. Du and Du are those arising from the presence of the crack. By continuity of displacement and rotation, u and u must, respectively, equal Du and Du. That is, u= Du u= Du
Fig. 3. Statically indeterminate fracture mechanics model.
necessary for evaluation of the J-integral under PTS. Fig. 2 shows the hoop stress components at a particular time 400 s after start of thermal shock. The hoop stress under PTS has three components, membrane tensile stress, bending stress and stress varying non-linearly through the wall thickness. The non-linear stress component is balanced in the direction of the wall thickness. The bending stress by constraint has almost reached the maximum value after 400 s.
"
(6)
P and M can be obtained by solving the simultaneous equation (Eq. (6)), because u, u, Du and Du are considered to be functions of P and M. The P and M values obtained can provide the J-integral as described below. The displacement and rotation of the split cylinder due to P and M may be written as
!"
u CP = u CPM
CPM CM
n!
P0 − P M0 − M
"
(7)
Here CP is the extension compliance of a split cylinder due to the surface hoop forces on crack faces, CM the angular compliance of the cylinder due to the surface moments and CPM the extension compliance of the cylinder due to the moments or the angular compliance of the cylinder due to the hoop forces.
M. Matsubara / Nuclear Engineering and Design 196 (2000) 153–159
156
CP, CPM and CM are given as 3pRm CP = E%W
! "
1 1 1+ − k 3 2p 1 CPM = − 1+ E%W k C CM = − PM Rm
Â Ã Ã Ã Ì Ã Ã Ã Å
(8)
Here  à E E%= à (1− 6) à W3 à I= à 12 Ì 1 1 +l à k= − 1 + log 2l 1 −l à à W à l= 2Rm à Å
(9)
where Rm is the mean radius of the cylinder. The displacement and rotation of the cracked strip element, Du and Du, may be written as
! "
Du DCP = DCPM Du
DCPM DCM
n! " P M
(10)
Here DC eP, DC ePM and DC eM are the extension and angular compliance increments due to the presence of crack of depth a, for a flat plate in plane strain tension and bending conditions. They may be expressed as the sum of the elastic components, DC eP, DC ePM and DC eM, and the plastic components, DC pP, DC pPM and DC pM, i.e. DCP =DC eP +DC pP Â Ã DCPM =DC ePM Ì Ã DCM = DC eM +DC pM Å
(11)
If the length of the cracked strip element is small enough (Cheng et al., 1984), the elastic components of the compliance can be obtained from
& & &
 KP 2 à da P à 0 a à 2 K K P M da Ì DC ePM = M E% 0 P à a 2 à KM 2 e da à DC M = E% 0 M Å DC eP =
2 E
a
(12)
Here KP is the stress intensity factor of a singleedge cracked specimen subjected to tension and KM is that subjected to bending. Based on the similar assumption to the elastic components of DCP, DCPM and DCM and the complimentary energy concept, the plastic ones are derived by Parks et al. (unpublished) as n−1 Â PM M2 2 Ã 2 AP + 2B +C 2 W W A Ã 2 p DC P = aRo0s0W P 2C Ã P 2C Ã n−1 2 Ã 2 PM M AP 2 + 2B +C 2 Ì W W B Ã DC pPM = aRo0s0W P 2C Ã P 2C n−1 Ã PM M2 2 Ã AP 2 + 2B +C 2 W W C Ã p DC M = aRo0s0 P 2C Å P 2C
:
: :
;
; ;
(13) The constitutive relation is completed by specifying the Ramberg–Osgood’s law (Ramberg and Osgood, 1943), i.e. o s s n = + aR (14) o0 s0 s0
Here o0 is the reference strain, s0 the reference stress, aR the work-hardening constant, n the work-hardening exponent, o the true strain and s the true stress. A, B and C in Eq. (13) are functions of a and n. They can be derived from fully plastic solutions and expressed as 2  h uPj(1− j) n + 1 à A= à (1.445h)n à 2 à h uPAn + 1(1−j) Ì (15) B= à (1.445h)n 2 à n+1 h uM à C= 0.364n(1−j)n − 1 à Å
!
!
"
"
M. Matsubara / Nuclear Engineering and Design 196 (2000) 153–159
Here h uP and h uP are the fully plastic solutions corresponding to the displacement and rotation of the single-edge cracked specimen subjected to tension, and h uM that corresponding to the rotation of the specimen subjected to bending given by Shih and Needleman (1984) j and h are
h=
a W
Â Ã Ì 2 j 1 à 1+ − 1−j 1−jÅ j=
'
(16)
PC is the reference load by per unit axial distance as
157
PC = s0(W−a)
(17)
Substitution of Eq. (7) and Eq. (10) into the continuity conditions (Eq. (6)) leads to the following simultaneous equations. To obtain P and M on crack faces, P and M must be combined with P0 and M0 on the ends of the split cylinder.
CP + DC eP + DC pP DC ePM + DC pPM
=
CP CPM
CPM CM
DC ePM + DC pPM C + DC eM + DC pM
n! " P0 M0
n! " P M
(18)
Solving the non-linear simultaneous equations, P and M can be obtained.
4.2. J-integral e6aluation The J-integral value of a single-edge cracked specimen subjected to P and M can be evaluated by complementary energy method (Miyoshi et al., 1986) as  à 1 Je = (KP + KM + Kn )2 à E% n−1 à PM M2 2 à AP 2 + 2B +C 2 à W W 1 JP = ao0s0W 2 Ì 2 2 PC à 1 (A 2 (B PM (C M 2 2 (PCà · P +2 + − 3 à (a W (a W 2 P C (a à P 2C (a à PM M2 AP 2 + 2B +C 2 Å W W (19)
:
Fig. 4. The effect of differences in thermal conductivity on J-integral under PTS.
J=Je + Jp
!
"
;
Where Je and Jp are the elastic and plastic components of the J-integral and Kn is the contribution of the stress varying non-linearly through the wall thickness under PTS. Kn may be written as
&
'
s (x)G
a n
x ,j a
(20) x 2
pa·(1−j) 1− a Here G is the correction factor for the stress intensity factor of a single-edge cracked plate by surface tractions on the crack faces by Tada et al. (1985).
Kn =
Fig. 5. The effect of differences in heat transfer coefficient on J-integral under PTS.
2
3 2
0
158
M. Matsubara / Nuclear Engineering and Design 196 (2000) 153–159
Fig. 6. The effect of differences in constants of the Ramgerg – Osgood’s law on J-integral under PTS.
ductivity increase, heat transfer coefficient increase would make the J-integral slightly increase for each crack depth. The relations between J-integral and a/W of Figs. 4 and 5 are similar in configuration. Fig. 6 explains the change in J-integral — a/W relationship profile with variation in a and n of the Ramberg–Osgood’s law for the elapsed time of 400 s since the start of thermal shock. The Ramberg–Osgood’s law is an identical stress– strain characteristic for evaluating the J-integral and the compliances DC pP, DC ePM and DC eM. The contours of J-integral — a/W curve change with the constants a and n of the Ramberg–Osgood’s law. The contour would tend to be in the form of the parabola opening upwards with the increase in n.
The above methodology is essentially valid for a thin-walled cylinder structure. 6. Conclusion 5. Results and discussion J-integral values are calculated at 0.01 intervals of non-dimensional crack depth in Figs. 4–6. Fig. 4 shows the effect of thermal conductivity on the J-integral under PTS for the elapsed time of 400 s since the start of thermal transient. The J-integral would almost become largest for each crack depth, because the bending stress by constraint has almost reached the maximum value at that time. The thermal conductivity exerted only a minimal influence on the J-integral over three thermal conductivities considered. Thermal conductivity increase would increase slightly the J-integral for each crack depth. The J-integral shows the first peak at about a/W=0.25. The first peak arises from positive bending, membrane and nonlinear stresses (see Fig. 2). The interaction of bending and membrane stresses causes the second peak at about a/W = 0.5. The J-integral under PTS almost tends as crack grows except for crack depth of about a/W = 0.25 and 0.5. Fig. 5 shows the J-integral under PTS for the elapsed time of 400 s as a function of crack depth a/W under the three conditions of heat transfer coefficient. The heat transfer coefficient has little effect on the J-integral. Like to the thermal con-
The study in this paper has demonstrated the parametric effect on the J-integral for the simplified model of an axially-cracked cylinder without cladding under PTS. The parameters considered are thermal conductivity, heat transfer coefficient and stress-strain relation. Both the thermal conductivity and the heat transfer coefficient do not noticeably affect the J-integral. The profiles of J-integral — a/W curve depend on the constants a and n of the Ramberg–Osgood’s law.
References Cheng, W., Stevick, G., Finnie, I., 1984. Predictions of the stress intensity factor for an internal circumferential crack at a butt-weld between cylinders using the plane strain solution, Transactions of the ASME. J. Eng. Mater. Technol. 106, 21 – 24. Kishimoto, K., Aoki, S., Sakata, M., 1980. On the path independent integral-J. . Eng. Fract. Mech. 13, 841 – 850. Matsubara, M., 1990. Application of statically indeterminate elastic-plastic fracture mechanics to a single-edge cracked member subjected to combined tension and bending. Trans. JSME 56 (526), 1396 – 1399. Matsubara, M., 1991. Application of statically indeterminate elastic-plastic fracture mechanics to pressurized thermal shock problem, pressure vessel integrity. PVP 213, 269– 273.
M. Matsubara / Nuclear Engineering and Design 196 (2000) 153–159 Matsubara, M., Soneda, N., 1992. Application of statically indeterminate elastic-plastic fracture mechanics to PROJECT FALSIRE problem, Pressure Vessel Technology, vol. 1, Design, Analysis, Materials (1), vol. 1, pp. 543 – 553. Miyoshi, T., Shiratori, M., Yoshida, Y., 1986. Analysis of J-integral and crack growth for surface cracks by line spring method, Transactions of the ASME. J. Pressure Vessel Tech. 108, 305–311. Parks, D.M., Rodin, G.J., Locket, R.R. The line-spring model for J-analysis of surface-cracked plates and shells: calibration of the power law spring, unpublished manuscript. Ramberg, W., Osgood, W.R., 1943. Description of stress –
.
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strain curves by three parameters. NASA Technical Notes 902. Rice, J.R., 1968. A path independent integral and the approximate analysis of stress concentration of notches and cracks. Transactions of the ASME. J. Appl. Mech., 379– 386. Shih, C.F., Needleman, A., 1984. Fully plastic crack problems part 1: solutions by a penalty method, Transactions of the ASME. J. Appl. Mech. 51, 48 – 56. Tada, H., Paris, P.C., Irwin, G.R., 1985. The Stress Analysis of Crack Handbook, second ed. Paris Productions Incorporated, pp. 2 – 27.