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Three-dimensional thermoelastic analysis of a cylindrical pipe with an internal surface crack under convection cooling
Nuclear Engineering and Design 132 (1991) 143-151 North-Holland
143
Three-dimensional thermoelastic analysis of a cylindrical pipe with an internal surface crack under convection cooling Wen-Hwa Chen I and Chin-Cheng Huang 2 Department of Power Mechanical Engineering, National Tsing Hua Unit,ersity, Hsinchu, Taiwan 30043, ROC Received 24 August 1990; revised version 9 April 1991
This paper presents a three-dimensional thermoelastic fracture analysis for a cylindrical pipe containing an internal surface crack subjected to convection cooling on its inner surface. To predict its thermoelastic fracture behaviors, the path-independent integral fl, which is physically the energy release rate per unit area of crack extension along the direction of crack growth, is computed by an accurate three-dimensional finite element model which provides both heat transfer and thermal stress analysis. The influence of realistic convection cooling on the inner surface of the cylindrical pipe on the computation of the temperature and the thermal stress intensity factor is evaluated. The variation of the thermal stress intensity factor along the crack front versus various configuration parameters and Blot numbers is also presented. This work is helpful to the safety evaluation of cylindrical pipes subjected to convection cooling.
I. Introduction
The failure of cylindrical pipes serviced in severe temperature environments, e.g. chemical conduits, pressure vessels and nuclear reactors, etc., has been traced to unavoidable surface cracks or crack like defects arising during the manufacturing process of such structures. The local thermal stresses at the regions near the imperfections are elevated by an intensified temperature gradient and may initiate crack growth or breakdown of the structure. To assure the safety of the structure, the necessity of three-dimensional thermoelastic fracture analysis is obvious. Kobayashi et al. [1] examined a hollow cylinder with a surface crack under internal pressure and thermal shock loadings by using the S c h w a r z - N e u m a n n alternating method and the curvature correction factors obtained from the analogue of edge-cracked strips. The influence of cylinder configuration parameters on the computation of thermal stress intensity factors was studied. However, the realistic three-dimensional effects and the influence of surface heat convection were not considered. Schmitt et al. [2] computed the thermal stress intensity factors using a three-dimensional finite
x Professor. z Graduate student. 0029-5493/91/$03.50
element model via the concept of energy release rates for a hollow cylinder with an axial semi-elliptical surface crack under given temperature or pressure loadings. The temperature field was prescribed in advance and the influence of crack geometry on thermoelastic fracture behaviors was not investigated. In addition, G r e b n e r et al. [3] studied a cylindrical pipe with an internal semi-elliptical surface crack subjected to thermal loadings by weight function method. The stress intensity factors were obtained only at the surface and the deepest point along the crack front. Recently, Kikuchi et al. [4-6] analyzed a pressure vessel with an internal semi-elliptical surface crack under thermal transient loadings by the J-integral introduced by Miyamoto et al. [7]. The temperature distribution in the problem was simulated by a given simple profile. It is mentioned that none of the above studies [1-7] were contributed to the study of convection cooling occurred on the inner surface of the cylindrical pipe. As is known, an infinite heat transfer coefficient is generally not attainable in a practical pipeline cooling system. Hence, the evaluation of convection cooling on thermoelastic fracture mechanics analysis for realistic cylindrical pipes is required. This work is thus to explore the three-dimensional thermoelastic fracture behaviors of a cylindrical pipe with an internal semi-circular or semi-elliptical surface crack subjected to convection cooling on its inner sur-
© 1991 - E l s e v i e r S c i e n c e P u b l i s h e r s B.V. A l l r i g h t s r e s e r v e d
144
IV..-H. Chen, C.-C. Huang / Analysis of a cylindrical pipe with an internal surface crack
face. The influence of various cylinder configuration parameters (e.g. crack depth, crack aspect ratio and thickness of the cylindrical pipe) and the heat transfer coefficient on the computation of the temperature distribution and the thermal stress intensity factor is then discussed. The finite element model for thermal convection and fracture mechanics analysis is extended from the formulation of the principal author's work [8] for two-dimensional problems. To simulate the singularities of the temperature gradient and the thermal stress around the crack front effectively, the collapsed quarter-point singular elements [9] instead of complicated hybrid singular elements [8] are adopted. An accurate path-independent integral fl which is physically the energy release rate per unit area of the crack extension along the direction of crack growth as derived in the authors' earlier work [10] is employed. The intensification of hoop thermal stress near the crack regions and the variation of the thermal stress intensity factor along the crack front are also drawn.
2. Description of the problem An internal surface crack can be induced in a cylindrical pipe during the manufacturing or operating of the structure, such as cooling equipment. The circulation of coolant flowing on the inner surface of the pipe may induce thermal stresses due to the variation of temperature across the thickness of the pipe. Since the heat is assumed to flow in radial direction, without loss of generality, only a mode-I fracture problem is considered here. To study the realistic pipeline system, various convection conditions on the inner surface of the cylindrical pipe are also simulated. Figure 1 displays the problem of a cylindrical pipe with an internal semi-elliptical surface crack subjected to convection cooling. 0 i and 0o denote the temperature of the coolant and the pipe respectively. The temperature Oo is selected as the reference temperature where a stress-free state is assumed (#i < 0o). The ends of the cylindrical pipe are constrained in the z
IY front
t
Z
L
X L
j
'
I~<0.
Fig. l. Finite elemental model for a cylindrical pipe with an internal semi-elliptical surface crack.
145
W.-H. Chen, C.-C. Huang /Analysis of a cylindrical pipe with an internal surface crack
direction and the temperature of the outer surface is prescribed as 0o. For simplicity, 0o can be taken as zero here. Due to convection cooling applied to the inner surface, the shrinkage of the surface may cause the damage of the structure. The effects of different configuration parameters R o / R i , a / b and a / ( R o - R i) are investigated. R o and R i are the outer and inner radius of the pipe (R i is kept constant in all analyses), a and b are the half length of the minor and major axes of the elliptical crack and L is the half length of the pipe respectively. As L > 5Ri, numerical experiments show that the variation of the thermal stress intensity factor due to the change of the length of pipe can be neglected.
derivative with respect to X 1. The relation between E 1 and the path-independent fl-integral, which is the energy release rate per unit area of the crack extension along the X 1 direction in the volume surrounding the crack front increment, can be found as
where ds is the arc length of the crack front increment between s 1 and s 2 (see fig. 2) in the fracture region. After appropriate derivations, the ./l-integral at the midpoint s o of the crack front increment can then be written as [10] 1
Y, ( S o )
[ . OWc
= s2 -
Ou i
[L
dv - fA
3. Calculation of the thermal stress intensity factors One of the three-dimensional path-independent integrals derived by the authors [10] is employed to determine the mode-I thermal stress intensity factor herein. As shown in fig. 2, a slice of cracked body is considered. The rate of energy change E 1 per unit length of crack extension along the X 1 direction in the fracture region can be expressed as Ou i --jAfTi~l .
E l =
dA,
(i = 1, 2, 3),
(1)
where Af is the bounded surface enclosing the fracture region. T/ is the traction acting on the surface and u i is the displacement. O ( . . . ) / O X l denotes the partial
A A.
Fig. 2. Integration domains of the path-independent integral
sl.
c
+ jvO'U-~l d V
],
+Ag
Ti
(i, j = 1, e, 3)
dA
(3)
where We is the elastic strain energy density, ~ij is the stress tensor, * = a t ~ i j A O is the thermal strain tensor, a is the thermal expansion coefficient, 8u is the Kronecker delta and A0 is the temperature variation. V is the volume enclosed by A, A s and Af. A denotes the entire surface enclosing the slice of the cracked body selected except the area of crack surface A s and fracture region Af. The detailed formulation of the Jl-integral has been documented in ref. [10] and, for want of space, is not repeated here. The variation of fl across the crack front increment is assumed to be linear. Using the Gaussian quadrature in the finite element analysis model, the path-independent Jl-integral as stated in eq. (3) can be calculated through simple surface and volume integrations. Based on the relation between the integral and the thermal stress intensity factor [10],
•ij
K 1 = ( E J l / / ( 1 - v 2 )) "'1/2 ,
(4)
the thermal stress intensity factor K 1 along the crack front is determined indirectly. E is Young's modulus and v is Poisson's ratio. For convenience, the same one finite element mesh is employed to compute both the temperature and the thermal stress of the problem. Due to the symmetry of the problem, a quadrant of the problem as seen in fig. 1 is modeled. There are 180 elements (including 24 collapsed quarter-point singular elements and 156 conventional twenty-node isoparametric brick elements) and 1083 nodes adopted in the model. The collapsed quarter-point singular elements [9] are selected to sim-
146
HK-H. Chen, C.-C. Huang
/Analysis o f
a cylindrical pipe with an internal surface crack
0.0
ulate the singularities of temperature gradient and thermal stress around the crack front. Conventional brick elements are taken in the domain far away from the crack region. To calculate the path-independent fl-integral as stated in eq. (3), the integration domains as shown by four shaded regions surrounding the crack front increment perpendicular to the minor axis of the semi-elliptical crack are also displayed in fig. 1.
°~'e'°'~"
°//'/~
- o . 2 - ~"
/ ' /
..# 8*
,.///
A( ' . I
4. Numerical results and discussions
,mj
B~= I00 (" r.~,.,.
B~= ac, )
....
,;/ ,/
B, = =
- ' -
1
"~ J
O0
"
i0
.:: Bt =
--1.0
I 0.2
0.0
.~
A
/
/ // me
-0.0
B, = 1
m
/V Based on the aspect of uncoupled thermoelasticity, the present thermoelastic fracture problem is also solved by two distinct steps [11]: (i) determine the temperature field due to convection cooling and (ii) evaluate the thermal stress based on the computed temperature variations. Since the Biot number denotes the strength of convection cooling, the significance of the magnitude of the Biot number on the computation of the temperature and the thermal stress intensity factor is also discussed.
=1.2
R/R,
,)s/~o/
I
l
[
0.4
0.8
0.8
1.0
r - R, R,-
R,
Fig. 3. Normalized temperature distribution across the wall thickness of a thin pipe (R o / R i = 1.2) for various Biot numbers.
4.1. T h e r m a l a n a l y s i s
The exact solution of the temperature field 0 across the wall thickness of the cylindrical pipe with an internal surface crack subjected to convection cooling on its inner surface can be derived by solving the heat equation and satisfying the corresponding thermal boundary conditions on inner and outer surfaces as follows 0i - 0°
0 = 0o +
In R° , 1
In
because of its bigger thermal resistance. The temperature distribution for the cylindrical pipe with an internal surface crack is thus calculated accurately by the present finite element analysis model. 0.0
.,¢lr'" ~ ,.¢y°°clr"°'O'"I j
(5)
-0.2-
r
/o /
+ B--~ / o°
where B i = h R J k is the Blot number, h and k represent the heat transfer coefficient on the inner surface and the thermal conductivity of the pipe respectively, r is the radial distance from the axis of the pipe. Since the heat flows in radial direction only, the crack surface can be thus considered as insulated. The normalized temperature distributions 0 " ( = ( 0 - 0 o ) / ( 0 o0i)) computed by the present finite element model across the wall thickness of thin ( R o / R i = 1.2) and thick ( R o / R i = 2) pipes are displayed in figs. 3 and 4 respectively. Excellent agreements between the finite element solution and exact solution as shown in eq. (5) are noted. As would be expected, a more profound convection is found for larger Biot numbers. A better convection efficiency between the inner surface of the pipe and the coolant is also observed for a thick pipe
..~ ~ ' * ~ " o °
/
t,•
.m'/
/°
,
-0.4-
/
/"
e"
R./Rj =2
S
. P .~"
-0.6 -
,./ /
/
/'~/ -0.8
•'~"
/
-
o
B, =
[] a
_s,=too
~" ....
Ri=
O0
J
B, =
I
--
J
--
-.--
/
-I.0 0.0
I 0.2
I 0.4 r -
I
.
)
B,= io
~, Zxact
B~= co
i
I 0.6
I 0.8
1.0
R,
R.- R,
Fig. 4. Normalized temperature distribution across the wall thickness of a thick pipe (R o / R i = 2) for various Biot numbers.
t~-H. Chen, C-C, Huang / Analysis of a cylindrical pipe with an internal surface crack
147
3.0 e ..,
2,0
a/(R,-l~)=0.2~ Present a/(R,-RO=O.4S(with crack)
- - - Boley et at. [12] ~ Without
1.0
) crack
--m-- Present
(7"* 0.0
-1.0
-2.0
.....
~........
0.0
~
0.2
014
0.8
0;8 r-
1.0
R~ Rt
R,-
Fig. 5. Variation of the normalized hoop thermal stress across the wall thickness of a thick pipe (R o/R~ = 2).
4.2. Thermoelastic fracture analysis
Based on the temperature distribution computed above, the thermoelastic fracture behaviors can be further investigated. To compute the thermal stress intensity factor K~, the path-independent fFintegral as stated in eq. (3) needs to be first calculated and K~ can
be obtained by the relation as mentioned in eq, (4). Figure 5 shows the variation of the normalized hoop thermal stress ~r*(= ¢ r h / [ E / ( 1 - v)]a(0o - 0i)) across the wall thickness (along x axis) of the thick pipe ( R o / R i = 2) with an internal semi-circular crack of different crack lengths. ~rh is the hoop thermal stress
o.2s T
0.2-
-,,
a/(R.-R,) = o.2/../b,
-
a/(R°-R~)
•
-~ =
0.4
a/(R,-R,) = 0 . 2 ~ / b , a/(R, -R~ )
=
1
J
,n - ~
0,45
t/3
al(Ro-Ri) = 0.2 I a/b = I a/(R,-R~ ) 0.4
0.~,J
A/(R,~-RI )
0.2
a / ( R , , - R ~ ) = 0.4 )
0.15 1
F~ o.1-
0.15 0.05
0.0
t 0.0
0,25
0,5 2 Y'/'~
O,'t5
1.0
Fig. 6. Variation of the normalized thermal stress intensity factor F 1 along the crack front in a thin pipe ( R e / R i = 1.2).
0.0
~
0.0
0,25
l
0.5 2P/'~
~
0,75
........... 1,0
Fig. 7. Variation of the normalized thermal stress intensity factor Ft along the crack front in a thick pipe (R o / R i = 2).
148
W.-H. Chen, C.-C. Huang / Analysis of a cylindrical pipe with an internal surface crack
O'e~l
- ¢ ,8
i
i(l!° 0.4
'L
I
f~
(1)Deformation
(2)Temperature (a)Ro/R,
-
(3)'rhermal s t ~ s s
=t.2
Undeformed
~
- - -
Deformed
,,
J
,
(1)Deformation !
I
e'= I-). !
-o.8
-0.8
-0.4
-0.2
O.
I
(2)Temperature
0 o. !
o
ot
(3)~ermal etreee (b)R./R, =2 Fig. 8. Deformation and distribution of the normalized temperature and the hoop thermal stress near a semi-circular crack for different wall thicknesses.
W.-H. Chen, C.-C. Huang /Analysis of a cylindrical pipe with an internal surface crack
149
o"
L
!)Deformation
(2)Temperature
(3)~nermAI stress
( a ) R . / R , =1.2
Undeformed ---
Deformed
(1)DeformaUon
~.
-0.0
-0.6
-0.4
-0.2
O.
(2)Temperature
0":
Lt
i
O.l 01
-0.1
-0.2
(3)Thermal stress
( b ) R . / ~ =2 Fig. 9. Deformation and distribution of the normalized temperature and the hoop thermal stress near a semi-elliptical crack for different wall thicknesses.
W. -H. Chen, C.-C. Huang / Analysis of a cylindrical pipe with an internal surface crack
150
of the pipe computed for B i = oc. It depicts that ~ * approaches to infinity near the crack tip for both a / ( R o - R i) = 0.2 and 0.4. Further, to display the effect of the existence of a semi-circular crack (a/b = 1), the exact solution of the hoop thermal stress o-h for the pipe without crack under same thermal conditions [12]
0.5
0.45-
i
a/b
=
l
m
a/b
=
I/3
Ea(Oo Oi) -
O"h
=
2(1
Ro p) I n - -
0.3Ro/R~ =2
Ri 0.15
0,0
and the present computed finite element solution are also shown in fig. 5. Tensile and compressive thermal stresses occur at the regions near inner and outer surfaces of the cylindrical pipe with initial traction free boundaries respectively. Little difference of ~ * between the cases with and without crack is found at the region near the outer surface. Figures 6 and 7 display the variation of the normalized thermal stress intensity factor FI(= K I / [ E ( 1 - v)--la(Oo- Oi)(Ri )1/2] along the semi-circular and semi-elliptical surface crack front (a/b = l / 3 ) with a/(Ro - R i) = 0.2 and 0.4 and B i = zc for Ro/R i = 1.2 and 2 respectively. Maximum variation of the J : i n t e g r a l calculated from four selected integration domains for semi-elliptical surface crack front is about 8.3%. In general, good path-independence of the integral is found. These two figures depict that the maximum F L occurs at the intersection of the crack front and the inner surface of the pipe i.e. d~ = 0. F~ then decreases as d~ increases and has a minimum at the deepest point of the crack front i.e. & = ~-/2. It is seen that the mentioned tendency is not true for the shallower crack a / ( R o - R i) = 0 . 2 as a/b = 1 / 3 for which maximum and minimum F~ make less difference. It is worthwhile to mention that, since the boundary layer near the free surface is very small in these problems (d~ = ~-/32) [13], the boundary layer effect is neglected here. Figures 6 and 7 can be used to predict the most probable location along the crack front where crack will propagate. In addition, F I is larger at 4~ = 0 and smaller at d~ = ~ ' / 2 as the aspect ratio a/b increases. That is, the crack region at d~ = 0 for a semicircular crack is more severe and should receive more attention. As the normalized crack depth a / ( R o - R i) increases, FI becomes larger at d, = 0 and smaller at ~h = 7r/2 for both a/b = 1 and 1/3. The unexpected
I
0.0
0.25
I
0,5
I
0.75
1.0
210/~ Fig. 10. Influence of the wall thickness on the normalized thermal stress intensity factor F I along a crack front. results for F~ at d~ = ~ ' / 2 can be explained by the higher tensile thermal stresses found ahead of the crack tip for shallower crack as shown in fig. 5. To further investigate the influence of the wall thickness of the pipe on the evaluation of thermoelastic fracture behaviors (the crack surface of semi-circular or semi-elliptical crack is kept constant here), the deformed shape and the distributions of 0* and o-* on the plane with the semi-circular or semi-elliptical surface crack are displayed in figs. 8 and 9 for different wall thicknesses respectively. The crack regions arc shaded and the undeformed plane is shown by a solid line. Shrinkage of the inner and outer surface of the pipe occurs in the radial direction due to the constraints in the z direction. Larger deformation and a lower 0* distribution accompanied by a larger ~r* near the crack front are also seen for thick pipe (Ro/R i = 2). This explains why the F I value of a thick pipe (Ro/R i = 2) is larger than that of a thin pipe (Ro/R i = 1.2) as shown in fig. 10. That means the safety evaluation of the surface crack existing in the pipe becomes more important as the wall thickness of the pipe increases. Figure 11 shows the variation of F I along the semielliptical crack front versus various Biot numbers. The influence of convection cooling with various surface heat transfer coefficients on the computation of F~ is depicted. Since there is better convection efficiency occurred on the inner surface of the pipe as B i increases, the value of F~ at d~ = 0 thus becomes larger as a lower temperature is induced. The variation of F~
W.-H. Chen, C.-C. Huang / Analysis of a cylindrical pipe with an internal surface crack 0.6
the pipe increases. Since t h e r e is b e t t e r convection efficiency o c c u r r e d as the Biot n u m b e r increases, a larger t h e r m a l stress intensity factor along the crack front is also observed. T h e p r e s e n t analysis should be of help in evaluating the structural integrity of a cylindrical pipe containing internal surface cracks.
Ro/Rt =2
a/b =1/3 0.45 -
1~
a/(R,-R,)
=
0.4
o ®
Bt= Bi =
l 10
= A
Bi = Bl=
t00 OO
151
0.3References
1 Q
0.15
0.0
I
0.0
0.25
I
0.5
I
0.75
t.0
2W~t Fig. If. Influence of the Biot number on the normalized thermal stress intensity factor F t along a crack front.
along the crack front is nearly the same as B i varies. In addition, it is f o u n d that the convection efficiency as B i is over 100 almost achieves the ideal situation with a discrepancy less t h a n 2%. Similar results are also obt a i n e d for a pipe with different configuration p a r a m e ters or crack aspect ratios.
5. Conclusions
A cylindrical pipe with an internal semi-circular or semi-elliptical surface crack subjected to convection cooling on its i n n e r surface has b e e n studied. Based on a t h r e e - d i m e n s i o n a l finite e l e m e n t analysis model and the p a t h - i n d e p e n d e n t f~-integral, the t e m p e r a t u r e distribution a n d t h e t h e r m a l stress intensity factor can be calculated accurately. T h e influence of crack depth, crack aspect ratio, wall thickness of the pipe a n d Blot n u m b e r o n the c o m p u t a t i o n of the t h e r m a l stress intensity factor are thus investigated thoroughly. T h e t h e r m a l stress intensity factor has a m a x i m u m at ~h = 0 a n d a m i n i m u m at ~h = 7 r / 2 except for a shallower crack with aspect ratio a / b = 1 / 3 for which m a x i m u m and m i n i m u m t h e r m a l stress intensity factors m a k e less difference. A s the crack d e p t h or the crack aspect ratio increases, t h e t h e r m a l stress intensity factor b e c o m e s larger at 4, = 0 a n d smaller at ~b = 7r/2. In addition, as the crack surface is kept constant, the t h e r m a l stress intensity factor b e c o m e s larger as the wall thickness of
[1] A.S. Kobayashi, A.F. Emery, N. Polvanich and W.J. Love, Surface flaw in a pressurized and thermally shocked hollow cylinder, Int. J. Press. Vessels and Piping 5 (1977)103-122. [2] W. Schmitt and E. Keim, Linear elastic analysis of semielliptical axial surface cracks in hollow cylinder, Int. J. Press. Vessels and Piping 7 (1979) 105-118. [3] H. Grebner and U. Strathmeier, Stress intensity factors for circumferential semi-elliptical surface cracks in a pipe under thermal loading, Engrg. Fract. Mech. 22 (1985) 1-7. [4] M. Kikuchi, H. Miyamoto, S. Sugawara and T. Shindo, J-integral evaluation of a crack in the pressure vessel under thermal transient loading, Bull. JSME-28 (1985) 2562-2566. [5] M. Kikuchi, H. Miyamoto and S. Sugawara, Evaluation of the J-integral of a crack in a pressure vessel under thermal transient loading, Trans. ASME. J. Pressure Vessel Technol. 108 (1986) 312-319. [6] M. Kikuchi, H. Miyamoto and S. Sugawara, Evaluation of the J-integral and cladding effect of a crack in the reactor pressure vessel, Bull. JSME 29 (1986) 4026-4030. [7] H. Miyamoto and M. Kikuchi, Three-dimensional J-integral, Proc. 28th Japan National Congress for Applied Mechanics 28 (1978) 195-204. [8] W.-H. Chen and K. Ting, Hybrid finite element analysis of transient thermoelastic fracture problems subjected to general heat transfer conditions, Comput. Mech. 4 (1989) 1-10. [9] R.S. Barsoum, On the use of isoparametric finite elements in linear fracture mechanics, Int. J. Numer Methods Eng. 10 (1976) 25-37. [10] W.-H. Chen and C.-C. Huang, Analysis of three-dimensional thermoelastic fracture problems using path-independent integrals. Engrg. Fract. Mech. 39 (1991) 581-589. [11] W.-H. Chen and K. Ting, Finite element analysis of mixed-mode thermoelastic fracture problems, Nucl. Engrg. Des. 90 (1985) 55-65. [12] B.A. Boley and J.H. Wiener, Theory of Thermal Stresses (John Wiley & Sons, New York, 1960). [13] I.S. Raju and J.C. Newman, Jr., Stress-intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness plates, Engrg. Fract. Mech. It (1979) 817- 829.
143
Three-dimensional thermoelastic analysis of a cylindrical pipe with an internal surface crack under convection cooling Wen-Hwa Chen I and Chin-Cheng Huang 2 Department of Power Mechanical Engineering, National Tsing Hua Unit,ersity, Hsinchu, Taiwan 30043, ROC Received 24 August 1990; revised version 9 April 1991
This paper presents a three-dimensional thermoelastic fracture analysis for a cylindrical pipe containing an internal surface crack subjected to convection cooling on its inner surface. To predict its thermoelastic fracture behaviors, the path-independent integral fl, which is physically the energy release rate per unit area of crack extension along the direction of crack growth, is computed by an accurate three-dimensional finite element model which provides both heat transfer and thermal stress analysis. The influence of realistic convection cooling on the inner surface of the cylindrical pipe on the computation of the temperature and the thermal stress intensity factor is evaluated. The variation of the thermal stress intensity factor along the crack front versus various configuration parameters and Blot numbers is also presented. This work is helpful to the safety evaluation of cylindrical pipes subjected to convection cooling.
I. Introduction
The failure of cylindrical pipes serviced in severe temperature environments, e.g. chemical conduits, pressure vessels and nuclear reactors, etc., has been traced to unavoidable surface cracks or crack like defects arising during the manufacturing process of such structures. The local thermal stresses at the regions near the imperfections are elevated by an intensified temperature gradient and may initiate crack growth or breakdown of the structure. To assure the safety of the structure, the necessity of three-dimensional thermoelastic fracture analysis is obvious. Kobayashi et al. [1] examined a hollow cylinder with a surface crack under internal pressure and thermal shock loadings by using the S c h w a r z - N e u m a n n alternating method and the curvature correction factors obtained from the analogue of edge-cracked strips. The influence of cylinder configuration parameters on the computation of thermal stress intensity factors was studied. However, the realistic three-dimensional effects and the influence of surface heat convection were not considered. Schmitt et al. [2] computed the thermal stress intensity factors using a three-dimensional finite
x Professor. z Graduate student. 0029-5493/91/$03.50
element model via the concept of energy release rates for a hollow cylinder with an axial semi-elliptical surface crack under given temperature or pressure loadings. The temperature field was prescribed in advance and the influence of crack geometry on thermoelastic fracture behaviors was not investigated. In addition, G r e b n e r et al. [3] studied a cylindrical pipe with an internal semi-elliptical surface crack subjected to thermal loadings by weight function method. The stress intensity factors were obtained only at the surface and the deepest point along the crack front. Recently, Kikuchi et al. [4-6] analyzed a pressure vessel with an internal semi-elliptical surface crack under thermal transient loadings by the J-integral introduced by Miyamoto et al. [7]. The temperature distribution in the problem was simulated by a given simple profile. It is mentioned that none of the above studies [1-7] were contributed to the study of convection cooling occurred on the inner surface of the cylindrical pipe. As is known, an infinite heat transfer coefficient is generally not attainable in a practical pipeline cooling system. Hence, the evaluation of convection cooling on thermoelastic fracture mechanics analysis for realistic cylindrical pipes is required. This work is thus to explore the three-dimensional thermoelastic fracture behaviors of a cylindrical pipe with an internal semi-circular or semi-elliptical surface crack subjected to convection cooling on its inner sur-
© 1991 - E l s e v i e r S c i e n c e P u b l i s h e r s B.V. A l l r i g h t s r e s e r v e d
144
IV..-H. Chen, C.-C. Huang / Analysis of a cylindrical pipe with an internal surface crack
face. The influence of various cylinder configuration parameters (e.g. crack depth, crack aspect ratio and thickness of the cylindrical pipe) and the heat transfer coefficient on the computation of the temperature distribution and the thermal stress intensity factor is then discussed. The finite element model for thermal convection and fracture mechanics analysis is extended from the formulation of the principal author's work [8] for two-dimensional problems. To simulate the singularities of the temperature gradient and the thermal stress around the crack front effectively, the collapsed quarter-point singular elements [9] instead of complicated hybrid singular elements [8] are adopted. An accurate path-independent integral fl which is physically the energy release rate per unit area of the crack extension along the direction of crack growth as derived in the authors' earlier work [10] is employed. The intensification of hoop thermal stress near the crack regions and the variation of the thermal stress intensity factor along the crack front are also drawn.
2. Description of the problem An internal surface crack can be induced in a cylindrical pipe during the manufacturing or operating of the structure, such as cooling equipment. The circulation of coolant flowing on the inner surface of the pipe may induce thermal stresses due to the variation of temperature across the thickness of the pipe. Since the heat is assumed to flow in radial direction, without loss of generality, only a mode-I fracture problem is considered here. To study the realistic pipeline system, various convection conditions on the inner surface of the cylindrical pipe are also simulated. Figure 1 displays the problem of a cylindrical pipe with an internal semi-elliptical surface crack subjected to convection cooling. 0 i and 0o denote the temperature of the coolant and the pipe respectively. The temperature Oo is selected as the reference temperature where a stress-free state is assumed (#i < 0o). The ends of the cylindrical pipe are constrained in the z
IY front
t
Z
L
X L
j
'
I~<0.
Fig. l. Finite elemental model for a cylindrical pipe with an internal semi-elliptical surface crack.
145
W.-H. Chen, C.-C. Huang /Analysis of a cylindrical pipe with an internal surface crack
direction and the temperature of the outer surface is prescribed as 0o. For simplicity, 0o can be taken as zero here. Due to convection cooling applied to the inner surface, the shrinkage of the surface may cause the damage of the structure. The effects of different configuration parameters R o / R i , a / b and a / ( R o - R i) are investigated. R o and R i are the outer and inner radius of the pipe (R i is kept constant in all analyses), a and b are the half length of the minor and major axes of the elliptical crack and L is the half length of the pipe respectively. As L > 5Ri, numerical experiments show that the variation of the thermal stress intensity factor due to the change of the length of pipe can be neglected.
derivative with respect to X 1. The relation between E 1 and the path-independent fl-integral, which is the energy release rate per unit area of the crack extension along the X 1 direction in the volume surrounding the crack front increment, can be found as
where ds is the arc length of the crack front increment between s 1 and s 2 (see fig. 2) in the fracture region. After appropriate derivations, the ./l-integral at the midpoint s o of the crack front increment can then be written as [10] 1
Y, ( S o )
[ . OWc
= s2 -
Ou i
[L
dv - fA
3. Calculation of the thermal stress intensity factors One of the three-dimensional path-independent integrals derived by the authors [10] is employed to determine the mode-I thermal stress intensity factor herein. As shown in fig. 2, a slice of cracked body is considered. The rate of energy change E 1 per unit length of crack extension along the X 1 direction in the fracture region can be expressed as Ou i --jAfTi~l .
E l =
dA,
(i = 1, 2, 3),
(1)
where Af is the bounded surface enclosing the fracture region. T/ is the traction acting on the surface and u i is the displacement. O ( . . . ) / O X l denotes the partial
A A.
Fig. 2. Integration domains of the path-independent integral
sl.
c
+ jvO'U-~l d V
],
+Ag
Ti
(i, j = 1, e, 3)
dA
(3)
where We is the elastic strain energy density, ~ij is the stress tensor, * = a t ~ i j A O is the thermal strain tensor, a is the thermal expansion coefficient, 8u is the Kronecker delta and A0 is the temperature variation. V is the volume enclosed by A, A s and Af. A denotes the entire surface enclosing the slice of the cracked body selected except the area of crack surface A s and fracture region Af. The detailed formulation of the Jl-integral has been documented in ref. [10] and, for want of space, is not repeated here. The variation of fl across the crack front increment is assumed to be linear. Using the Gaussian quadrature in the finite element analysis model, the path-independent Jl-integral as stated in eq. (3) can be calculated through simple surface and volume integrations. Based on the relation between the integral and the thermal stress intensity factor [10],
•ij
K 1 = ( E J l / / ( 1 - v 2 )) "'1/2 ,
(4)
the thermal stress intensity factor K 1 along the crack front is determined indirectly. E is Young's modulus and v is Poisson's ratio. For convenience, the same one finite element mesh is employed to compute both the temperature and the thermal stress of the problem. Due to the symmetry of the problem, a quadrant of the problem as seen in fig. 1 is modeled. There are 180 elements (including 24 collapsed quarter-point singular elements and 156 conventional twenty-node isoparametric brick elements) and 1083 nodes adopted in the model. The collapsed quarter-point singular elements [9] are selected to sim-
146
HK-H. Chen, C.-C. Huang
/Analysis o f
a cylindrical pipe with an internal surface crack
0.0
ulate the singularities of temperature gradient and thermal stress around the crack front. Conventional brick elements are taken in the domain far away from the crack region. To calculate the path-independent fl-integral as stated in eq. (3), the integration domains as shown by four shaded regions surrounding the crack front increment perpendicular to the minor axis of the semi-elliptical crack are also displayed in fig. 1.
°~'e'°'~"
°//'/~
- o . 2 - ~"
/ ' /
..# 8*
,.///
A( ' . I
4. Numerical results and discussions
,mj
B~= I00 (" r.~,.,.
B~= ac, )
....
,;/ ,/
B, = =
- ' -
1
"~ J
O0
"
i0
.:: Bt =
--1.0
I 0.2
0.0
.~
A
/
/ // me
-0.0
B, = 1
m
/V Based on the aspect of uncoupled thermoelasticity, the present thermoelastic fracture problem is also solved by two distinct steps [11]: (i) determine the temperature field due to convection cooling and (ii) evaluate the thermal stress based on the computed temperature variations. Since the Biot number denotes the strength of convection cooling, the significance of the magnitude of the Biot number on the computation of the temperature and the thermal stress intensity factor is also discussed.
=1.2
R/R,
,)s/~o/
I
l
[
0.4
0.8
0.8
1.0
r - R, R,-
R,
Fig. 3. Normalized temperature distribution across the wall thickness of a thin pipe (R o / R i = 1.2) for various Biot numbers.
4.1. T h e r m a l a n a l y s i s
The exact solution of the temperature field 0 across the wall thickness of the cylindrical pipe with an internal surface crack subjected to convection cooling on its inner surface can be derived by solving the heat equation and satisfying the corresponding thermal boundary conditions on inner and outer surfaces as follows 0i - 0°
0 = 0o +
In R° , 1
In
because of its bigger thermal resistance. The temperature distribution for the cylindrical pipe with an internal surface crack is thus calculated accurately by the present finite element analysis model. 0.0
.,¢lr'" ~ ,.¢y°°clr"°'O'"I j
(5)
-0.2-
r
/o /
+ B--~ / o°
where B i = h R J k is the Blot number, h and k represent the heat transfer coefficient on the inner surface and the thermal conductivity of the pipe respectively, r is the radial distance from the axis of the pipe. Since the heat flows in radial direction only, the crack surface can be thus considered as insulated. The normalized temperature distributions 0 " ( = ( 0 - 0 o ) / ( 0 o0i)) computed by the present finite element model across the wall thickness of thin ( R o / R i = 1.2) and thick ( R o / R i = 2) pipes are displayed in figs. 3 and 4 respectively. Excellent agreements between the finite element solution and exact solution as shown in eq. (5) are noted. As would be expected, a more profound convection is found for larger Biot numbers. A better convection efficiency between the inner surface of the pipe and the coolant is also observed for a thick pipe
..~ ~ ' * ~ " o °
/
t,•
.m'/
/°
,
-0.4-
/
/"
e"
R./Rj =2
S
. P .~"
-0.6 -
,./ /
/
/'~/ -0.8
•'~"
/
-
o
B, =
[] a
_s,=too
~" ....
Ri=
O0
J
B, =
I
--
J
--
-.--
/
-I.0 0.0
I 0.2
I 0.4 r -
I
.
)
B,= io
~, Zxact
B~= co
i
I 0.6
I 0.8
1.0
R,
R.- R,
Fig. 4. Normalized temperature distribution across the wall thickness of a thick pipe (R o / R i = 2) for various Biot numbers.
t~-H. Chen, C-C, Huang / Analysis of a cylindrical pipe with an internal surface crack
147
3.0 e ..,
2,0
a/(R,-l~)=0.2~ Present a/(R,-RO=O.4S(with crack)
- - - Boley et at. [12] ~ Without
1.0
) crack
--m-- Present
(7"* 0.0
-1.0
-2.0
.....
~........
0.0
~
0.2
014
0.8
0;8 r-
1.0
R~ Rt
R,-
Fig. 5. Variation of the normalized hoop thermal stress across the wall thickness of a thick pipe (R o/R~ = 2).
4.2. Thermoelastic fracture analysis
Based on the temperature distribution computed above, the thermoelastic fracture behaviors can be further investigated. To compute the thermal stress intensity factor K~, the path-independent fFintegral as stated in eq. (3) needs to be first calculated and K~ can
be obtained by the relation as mentioned in eq, (4). Figure 5 shows the variation of the normalized hoop thermal stress ~r*(= ¢ r h / [ E / ( 1 - v)]a(0o - 0i)) across the wall thickness (along x axis) of the thick pipe ( R o / R i = 2) with an internal semi-circular crack of different crack lengths. ~rh is the hoop thermal stress
o.2s T
0.2-
-,,
a/(R.-R,) = o.2/../b,
-
a/(R°-R~)
•
-~ =
0.4
a/(R,-R,) = 0 . 2 ~ / b , a/(R, -R~ )
=
1
J
,n - ~
0,45
t/3
al(Ro-Ri) = 0.2 I a/b = I a/(R,-R~ ) 0.4
0.~,J
A/(R,~-RI )
0.2
a / ( R , , - R ~ ) = 0.4 )
0.15 1
F~ o.1-
0.15 0.05
0.0
t 0.0
0,25
0,5 2 Y'/'~
O,'t5
1.0
Fig. 6. Variation of the normalized thermal stress intensity factor F 1 along the crack front in a thin pipe ( R e / R i = 1.2).
0.0
~
0.0
0,25
l
0.5 2P/'~
~
0,75
........... 1,0
Fig. 7. Variation of the normalized thermal stress intensity factor Ft along the crack front in a thick pipe (R o / R i = 2).
148
W.-H. Chen, C.-C. Huang / Analysis of a cylindrical pipe with an internal surface crack
O'e~l
- ¢ ,8
i
i(l!° 0.4
'L
I
f~
(1)Deformation
(2)Temperature (a)Ro/R,
-
(3)'rhermal s t ~ s s
=t.2
Undeformed
~
- - -
Deformed
,,
J
,
(1)Deformation !
I
e'= I-). !
-o.8
-0.8
-0.4
-0.2
O.
I
(2)Temperature
0 o. !
o
ot
(3)~ermal etreee (b)R./R, =2 Fig. 8. Deformation and distribution of the normalized temperature and the hoop thermal stress near a semi-circular crack for different wall thicknesses.
W.-H. Chen, C.-C. Huang /Analysis of a cylindrical pipe with an internal surface crack
149
o"
L
!)Deformation
(2)Temperature
(3)~nermAI stress
( a ) R . / R , =1.2
Undeformed ---
Deformed
(1)DeformaUon
~.
-0.0
-0.6
-0.4
-0.2
O.
(2)Temperature
0":
Lt
i
O.l 01
-0.1
-0.2
(3)Thermal stress
( b ) R . / ~ =2 Fig. 9. Deformation and distribution of the normalized temperature and the hoop thermal stress near a semi-elliptical crack for different wall thicknesses.
W. -H. Chen, C.-C. Huang / Analysis of a cylindrical pipe with an internal surface crack
150
of the pipe computed for B i = oc. It depicts that ~ * approaches to infinity near the crack tip for both a / ( R o - R i) = 0.2 and 0.4. Further, to display the effect of the existence of a semi-circular crack (a/b = 1), the exact solution of the hoop thermal stress o-h for the pipe without crack under same thermal conditions [12]
0.5
0.45-
i
a/b
=
l
m
a/b
=
I/3
Ea(Oo Oi) -
O"h
=
2(1
Ro p) I n - -
0.3Ro/R~ =2
Ri 0.15
0,0
and the present computed finite element solution are also shown in fig. 5. Tensile and compressive thermal stresses occur at the regions near inner and outer surfaces of the cylindrical pipe with initial traction free boundaries respectively. Little difference of ~ * between the cases with and without crack is found at the region near the outer surface. Figures 6 and 7 display the variation of the normalized thermal stress intensity factor FI(= K I / [ E ( 1 - v)--la(Oo- Oi)(Ri )1/2] along the semi-circular and semi-elliptical surface crack front (a/b = l / 3 ) with a/(Ro - R i) = 0.2 and 0.4 and B i = zc for Ro/R i = 1.2 and 2 respectively. Maximum variation of the J : i n t e g r a l calculated from four selected integration domains for semi-elliptical surface crack front is about 8.3%. In general, good path-independence of the integral is found. These two figures depict that the maximum F L occurs at the intersection of the crack front and the inner surface of the pipe i.e. d~ = 0. F~ then decreases as d~ increases and has a minimum at the deepest point of the crack front i.e. & = ~-/2. It is seen that the mentioned tendency is not true for the shallower crack a / ( R o - R i) = 0 . 2 as a/b = 1 / 3 for which maximum and minimum F~ make less difference. It is worthwhile to mention that, since the boundary layer near the free surface is very small in these problems (d~ = ~-/32) [13], the boundary layer effect is neglected here. Figures 6 and 7 can be used to predict the most probable location along the crack front where crack will propagate. In addition, F I is larger at 4~ = 0 and smaller at d~ = ~ ' / 2 as the aspect ratio a/b increases. That is, the crack region at d~ = 0 for a semicircular crack is more severe and should receive more attention. As the normalized crack depth a / ( R o - R i) increases, FI becomes larger at d, = 0 and smaller at ~h = 7r/2 for both a/b = 1 and 1/3. The unexpected
I
0.0
0.25
I
0,5
I
0.75
1.0
210/~ Fig. 10. Influence of the wall thickness on the normalized thermal stress intensity factor F I along a crack front. results for F~ at d~ = ~ ' / 2 can be explained by the higher tensile thermal stresses found ahead of the crack tip for shallower crack as shown in fig. 5. To further investigate the influence of the wall thickness of the pipe on the evaluation of thermoelastic fracture behaviors (the crack surface of semi-circular or semi-elliptical crack is kept constant here), the deformed shape and the distributions of 0* and o-* on the plane with the semi-circular or semi-elliptical surface crack are displayed in figs. 8 and 9 for different wall thicknesses respectively. The crack regions arc shaded and the undeformed plane is shown by a solid line. Shrinkage of the inner and outer surface of the pipe occurs in the radial direction due to the constraints in the z direction. Larger deformation and a lower 0* distribution accompanied by a larger ~r* near the crack front are also seen for thick pipe (Ro/R i = 2). This explains why the F I value of a thick pipe (Ro/R i = 2) is larger than that of a thin pipe (Ro/R i = 1.2) as shown in fig. 10. That means the safety evaluation of the surface crack existing in the pipe becomes more important as the wall thickness of the pipe increases. Figure 11 shows the variation of F I along the semielliptical crack front versus various Biot numbers. The influence of convection cooling with various surface heat transfer coefficients on the computation of F~ is depicted. Since there is better convection efficiency occurred on the inner surface of the pipe as B i increases, the value of F~ at d~ = 0 thus becomes larger as a lower temperature is induced. The variation of F~
W.-H. Chen, C.-C. Huang / Analysis of a cylindrical pipe with an internal surface crack 0.6
the pipe increases. Since t h e r e is b e t t e r convection efficiency o c c u r r e d as the Biot n u m b e r increases, a larger t h e r m a l stress intensity factor along the crack front is also observed. T h e p r e s e n t analysis should be of help in evaluating the structural integrity of a cylindrical pipe containing internal surface cracks.
Ro/Rt =2
a/b =1/3 0.45 -
1~
a/(R,-R,)
=
0.4
o ®
Bt= Bi =
l 10
= A
Bi = Bl=
t00 OO
151
0.3References
1 Q
0.15
0.0
I
0.0
0.25
I
0.5
I
0.75
t.0
2W~t Fig. If. Influence of the Biot number on the normalized thermal stress intensity factor F t along a crack front.
along the crack front is nearly the same as B i varies. In addition, it is f o u n d that the convection efficiency as B i is over 100 almost achieves the ideal situation with a discrepancy less t h a n 2%. Similar results are also obt a i n e d for a pipe with different configuration p a r a m e ters or crack aspect ratios.
5. Conclusions
A cylindrical pipe with an internal semi-circular or semi-elliptical surface crack subjected to convection cooling on its i n n e r surface has b e e n studied. Based on a t h r e e - d i m e n s i o n a l finite e l e m e n t analysis model and the p a t h - i n d e p e n d e n t f~-integral, the t e m p e r a t u r e distribution a n d t h e t h e r m a l stress intensity factor can be calculated accurately. T h e influence of crack depth, crack aspect ratio, wall thickness of the pipe a n d Blot n u m b e r o n the c o m p u t a t i o n of the t h e r m a l stress intensity factor are thus investigated thoroughly. T h e t h e r m a l stress intensity factor has a m a x i m u m at ~h = 0 a n d a m i n i m u m at ~h = 7 r / 2 except for a shallower crack with aspect ratio a / b = 1 / 3 for which m a x i m u m and m i n i m u m t h e r m a l stress intensity factors m a k e less difference. A s the crack d e p t h or the crack aspect ratio increases, t h e t h e r m a l stress intensity factor b e c o m e s larger at 4, = 0 a n d smaller at ~b = 7r/2. In addition, as the crack surface is kept constant, the t h e r m a l stress intensity factor b e c o m e s larger as the wall thickness of
[1] A.S. Kobayashi, A.F. Emery, N. Polvanich and W.J. Love, Surface flaw in a pressurized and thermally shocked hollow cylinder, Int. J. Press. Vessels and Piping 5 (1977)103-122. [2] W. Schmitt and E. Keim, Linear elastic analysis of semielliptical axial surface cracks in hollow cylinder, Int. J. Press. Vessels and Piping 7 (1979) 105-118. [3] H. Grebner and U. Strathmeier, Stress intensity factors for circumferential semi-elliptical surface cracks in a pipe under thermal loading, Engrg. Fract. Mech. 22 (1985) 1-7. [4] M. Kikuchi, H. Miyamoto, S. Sugawara and T. Shindo, J-integral evaluation of a crack in the pressure vessel under thermal transient loading, Bull. JSME-28 (1985) 2562-2566. [5] M. Kikuchi, H. Miyamoto and S. Sugawara, Evaluation of the J-integral of a crack in a pressure vessel under thermal transient loading, Trans. ASME. J. Pressure Vessel Technol. 108 (1986) 312-319. [6] M. Kikuchi, H. Miyamoto and S. Sugawara, Evaluation of the J-integral and cladding effect of a crack in the reactor pressure vessel, Bull. JSME 29 (1986) 4026-4030. [7] H. Miyamoto and M. Kikuchi, Three-dimensional J-integral, Proc. 28th Japan National Congress for Applied Mechanics 28 (1978) 195-204. [8] W.-H. Chen and K. Ting, Hybrid finite element analysis of transient thermoelastic fracture problems subjected to general heat transfer conditions, Comput. Mech. 4 (1989) 1-10. [9] R.S. Barsoum, On the use of isoparametric finite elements in linear fracture mechanics, Int. J. Numer Methods Eng. 10 (1976) 25-37. [10] W.-H. Chen and C.-C. Huang, Analysis of three-dimensional thermoelastic fracture problems using path-independent integrals. Engrg. Fract. Mech. 39 (1991) 581-589. [11] W.-H. Chen and K. Ting, Finite element analysis of mixed-mode thermoelastic fracture problems, Nucl. Engrg. Des. 90 (1985) 55-65. [12] B.A. Boley and J.H. Wiener, Theory of Thermal Stresses (John Wiley & Sons, New York, 1960). [13] I.S. Raju and J.C. Newman, Jr., Stress-intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness plates, Engrg. Fract. Mech. It (1979) 817- 829.