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Thermal shock in a circumferentially cracked hollow cylinder with cladding
Lnginrering b’raciurz Mcwhmrcs Printed in Great Britmn,
Vol 20, No. 1, pp. 113-137, 1984
0013 7944/84 $3.00+ .XJ Pergmm Press Ltd.
THERMAL SHOCK IN A CIRCU~F~RE~TIALLY CRACKED HOLLOW CYLINDER WITH CLADDING General Electric Company,
Corporate
H. F. NIED Research and Development, U.S.A.
Schenectady,
NY 12301,
Abstract-An analysis is presented which demonstrates the effect that cladding has on the thermal shock resistance of a ~rcumferentially cracked hollow cylinder. The cladding, which in general may have different thermal properties than the base material, is assumed to be bonded to the inner wall of the hollow cylinder. The axisymmetric circumferential crack may either be embedded in the cylinder wall, typically underneath the clad, or may be an edge crack which passes through the clad and opens into the inner wall of the hollow cylinder. The mathematical formulation of the problem results in a singular integral equation of a well known type which is solved numerically. Results, which include transient temperature distributions, thermal stresses in the untracked cylinder, and stress intensity factors as a function of time, are presented for various cladding thickness to cylinder wall thickness ratios. The numerical calculations concentrate on material properties and geometric con~gurations typically seen in nuclear pressure vessel appli~tions. Results of particular interest are the transient stress intensity factors for various crack lengths and the comparison of maximum stress intensity factors in the clad cylinder with those in an unclad cylinder. Assuming the clad has completely yielded. the stress intensity factors for a crack under the clad, oriented in a plane perpendicular to the cylinder axis, are determined using a plastic strip model. It is shown that yielding of the clad under certain conditions can result in a reduction in the magnitude of the stress intensity factor for the crack tip in the elastic base material.
INTRODUCTION IT HAS long been recognized that cladding, when properly bonded to the inner wall of a pressure vessel or pipe, can be very effective in protecting the base metal from severe chemical corrosion. In the nuclear environment, cladding serves the additional role of shielding the base metal from detrimental radiation damage. Typically, this cladding has a significantly lower thermal conductivity than the base metal and a higher coefhcient of thermal expansion. Furthermore, in many applications, the clad has the same elastic modulus and Poisson’s ratio as the base metal For instance, in most nuclear pressure vessel applications the clad is austenitic stainless steel and the base metal is a low alloy steel. When cladding is used to protect the base metal in this manner, an important question arises: What effect does the cladding have on the structural integrity of the system during a severe thermal transient, especially in the presence of preexisting flaws?[l] In a number of cases cracks have been observed underneath the clad, in the base metal, oriented in a plane perpendicular to the cylinder axis and terminating at the bond interface. The purpose of this paper is to investigate the effect of cladding on the thermal shock resistance of a hollow cylinder with an axisymmetric circumferential crack. The severity of the thermal shock is characterized by the stress intensity factor, which varies as a function of time and achieves a maximum value some time after initiation of the transient thermal event. Once the stress intensity factors are known as a function of time, it becomes possible in brittle materials to determine whether catastrophic failure will occur due to unstable crack propagation. Also, the amount of crack growth which occurs after repeated thermal shocks can be estimated using empirical crack growth rate expressions which utilize the stress intensity factor info~ation given as a function of crack length. The problem of interest is depicted in Fig. 1. The initially stress free, infinitely long cylinder is assumed to be composed of two materials, initially at temperature T,,, which are perfectly bonded at the bimaterial interface r = b. The cladding, which we will designate as material I. has a coefficient of thermai conductivity K,, a specific heat q, a mass density pr, a coefficient of thermal expansion q, and a thickness given by b - a. The base material, material II, has properties x2, c2, p2 and t12 associated with the thermal conductivity, specific heat, mass density and coefficient of thermal expansion, respectively, in the region b < r $ c. The inner crack tip radius is given by Y =,f and the outer crack tip radius is designated by r =g. Two special crack configurations are of particular interest: the circumferential crack embedded in the base metal, where the crack tip is 113
114
H. F. NIED
Fig. I. Geometry of a thick-walled cylinder with cladding containing an axisymmetric circumferential crack.
located just underneath the clad (i.e. f = n) and the edge crack which completely passes through the clad. The fracture problem is formulated by first obtaining the solution for the transient thermal stresses in an untracked bimaterial hollow cylinder subjected to a sudden temperature change on the inner wall of the composite cylinder, where the outer radius of the composite cylinder is assumed to be insulated. The thermal stresses calculated from these boundary conditions simulate a severe thermal shock and thus provide a reasonable upper bound solution to the fracture problem of interest. The mixed boundary value problem, which contains all the pertinent fracture information, is then formulated by using the thermal stresses from the untracked composite cylinder, but with opposite sign, as the crack surface tractions along the line of revolution swept out by the crack. Superposition of this result with the untracked cylinder thermoelastic solution results in the correct stress field for the cracked cylinder problem. The mixed boundary value problem that must be solved is simplified by assuming the clad and the base material have the same elastic modulus and Poisson’s ratio. This assumption allows the mathematical formulation to result in a singular integral equation of a well-known type which avoids the difficulties usually associated with bimaterial crack problems, that is, determining the strength of the stress singularity at the crack tip which touches the interface[2]. Another important, but reasonable, assumption in this formulation, which simplifies the analysis greatly, is that the transient thermal stress problem described is quasi-static, that is, the inertia effects are negligible. Since the cylinder and cladding thicknesses are arbitrary, as well as the size and location of the crack, the formulation presented here lends itself ideally to parametric investigations on the general effect of cladding thickness and crack size. It is also recognized that the material properties of most interest are those associated with nuclear pressure vessel applications. Thus, numerical calculations presented in the results rely heavily on those material properties which are representative of pressure vessel steels. When the material properties of the clad and base material are taken to be identical, we of course recover the homogeneous cylinder solution, which is useful for comparison with the solutions where cladding is present. It is important to note that if the cladding is su~ciently ductile, during a severe thermal transient the cladding may yield. Thus, yielding of the clad and the effect this has on the stress intensity factors for an embedded crack underneath the clad is examined by using a plastic strip model which assumes the clad is perfectly plastic. As will be shown in the results, the effect of clad yielding, under certain conditions, can be beneficial and result in a substantial reduction in the stress intensity factor for the crack tip buried in the base material.
Thermal shock in a circumferentially cracked hollow cylinder with cladding
115
FORMULATION Referring again to Fig. 1, the uncoupled transient temperature distribution may be determined from the solution of the diffusion equations
1 &(r,
v*ez(r, t) = o-
t)
iit
2
,
(2)
(h G r d ~1,
’
where H,(r, t) and $,(r, t) are defined by (3) @,(r, t) = Tfr, t) -
T,,,
(b G r G ~1,
(4)
The temperatures, T(r, t) and To, are the temperature in the cylinder at time t and the initial temperature of the composite cylinder. respectively. The thermal diffusivities~ L), and I&, are given by
D2
=
Q/(Pzc~)>
(6)
where K~, K~,pr, pz and cl, cz are the material thermal conductivities, densities and specific heats for materials I (cladding) and II (base), respectively. The initial thermal conditions throughout the cylinder are given by O,(r, 0) = &(r, 0) = 0.
Defining, 8, = T, - r,, where To is the temperature boundary conditions are specified to be
em t>=
(7)
to be maintained
at r = u, the thermal
o
’
ar
where H(t) is the Heaviside step function. These conditions imply a sudden change in temperature on the inner radius a, perfect heat transfer at the bimaterial interface r = b and an insulated outer radius L’. Application of Laplace transforms and condition (7) reduces the partial differential equations (1) and (2) to ordinary di~erentia~ equations of the form
d2t%r,p)+~u’8;(r,p)p -__ - dr -,O,(r,p)=O, r dr2
(i = 1,2)
(12)
where the integral transform is defined as &(r,p)
=
x
Or(r, t) e-P’dt.
(13)
H. F. NIED
116
The general solution to eqn (12) is
with I&y,) and KO(ryi) the usual modified Bessel functions of the first and second kind of order zero. A,(p) and &i(p) are four constants to be determined from conditions (8)-(I I), and yI is given by
After application of the boundary conditions (8~(11) and determination B,(p), the equations in (14) become
of the terms Ai
and
(17) where,
The Laplace inversion to obtain the solutions for 8,(r, t) and O&r, t) may be obtained by evaluating the following integrals with the aid of the contour I- shown in Fig. 2,
(24) It is not difficult to show that the integrands in (23) and (24) have an infinite number of simple poles distributed along the imaginary axis. This can be shown with the aid of the Bessel function identities which relate modified Bessel functions to ordinary Bessel functions[3]; f,Jz
Ky(z
ew21~) = e(vd2)fJ”
e(G)3
=
-! i i.
Substitution
e _.
“n’2)i[- J,,(z)
of (25) and (26) into (18) demonstrates p=l)i&,, c
(25)
(z),
(rz=l,2
-I-
iY"(Z)].
that there are simple poles located at )...,
co),
(27)
Thermal shock in a circumferentially cracked hollow cylinder with cladding
117
Fig. 2. Contour f for evaluation of integrals in eqns (23) and (24).
where the l,s are determined from the roots of the transcendental
equation (28)
Normalizing all dimensions with respect to the outer radius c, we have,
r* =rIc,
a*=a/c,
b*=blc.
(29)
The dimensionless thermal constants K and /I are given by
,=!2
(30)
KI
p=
0 ;
I’*,
(31)
2
and the functions F,, F,, and E,, are defined as
Mx,
Y>=
F,,&Y)
Jo(x)YO(Y >- JO(Y) Y& 1
= J,(x)~,(Y)
J&(X,Y) = J&)Y,(Y)
(32)
- J,(y)Y,(x)
(33)
- J,(Y)Yo(X).
(34)
Recognizing that the integrals in (23) and (24) can be evaluated by using the residue theorem, that is, fA(r, t) ___ = Res,,(O) + f Res,, n=l 0,
,
u-9 t) = Res,,(O) + f Res,,
,
n=l
-e,?
b
(36)
0 x2
where the subscripts 23 and 24 refer to the integrals in eqns (23) and (24) respectively, we obtain after lengthy but straightforward analysis the following closed form solutions for 6, and 0,:
Ur*, t) -=
0,
1 - 2 2 e(-“““‘“n2’[lcBFoo(b*~,, r*/Z,)F,,(b*P&, p&J n=l (37)
The function H(%,) in the denominator
of (37) and (38) is determined
H(j.,) = KP2F,,,(a*&, h*&,)[E,,,(b*/% + E,,(b*&
&) - b*- ‘E&U,,,
- F,(a*L,,
/NJ
from b*/U,)]
b*&) +gF,,(a*&,
b*i.,)
i. - f@‘,,(/%,,
1 - E,,,(u*i,,
b*/%)
b*&) +g
E,,,(b*i,,
a*%,)
i + PEo,(a*&,
b*U
1
- F,,(P&,, b*/N,) + b*-lFoo(~&,,
b*@,)].
(39)
With the aid of the following identities, Foo(x,x>= 0,
(40)
F,,(x, x>= 0,
(41)
Eo,(x,x) = -
&,
(42)
it can easily be shown that the transient solution for a homogeneous cylinder is recovered, by setting fl and K equal to one and letting either b = c in (37) or b = a in (38). In these cases we obtain (43) which can be shown to be identical to the result given by Carslaw and Jaeger[4]. Once the transient temperature is known, it becomes possible to determine the axial thermal stresses in the untracked composite cylinder. The general solution for the axial thermal stresses in the two cylinders can be considered due to two separate effects: first, the axial stresses induced in an unrestrained cylinder with radial temperature variation, and second, the stresses due to the radial interference. The plane strain solution for the axial thermal stresses in the cladding and base material, respectively, can be shown to be[5] olr = E,
-
crT+
(44) + 2v,nb2 P + A2
E2c~2
I-v
(v,T> -
T2h
(45)
2
where subscripts 1 and 2 refer to material I and II. The radial interface pressure P, between the two materials, is given by
with
u;= a,(1 + E;=_
E, 1 -
v,),
a; = cl,(l f v,), I$=---
v,2’
~72 1 -
v22’
(47) (48)
Thermal
shock
in a circumferentially
cracked
hollow
cylinder
with cladding
119
(49) 2 b2-a2
T, = ___
(50)
c T2r dr, b’ s b
(51)
A, = n(b2 - a’), A, = n(c’ - b2).
(52)
T2 = ~
2
b T,r dr, so
C2-
At this point a simplifying assumption is made: it is assumed that the two materials which comprise the composite cylinder have the same values for the elastic constants E and v. Thus the materials have only different coefficients of thermal expansion and, of course, different thermal diffusivities. This assumption is a reasonable one for the cited applications, since the statistical spreads of these properties usually overlap. With this assumption, eqns (44) and (45) can be simplified and the axial thermal stresses become
(a < r < b),
(b 6r
(54)
It turns out that it is possible to evaluate the integrals in (53) and (54) in closed form using the following Bessel identities [6]:
s z
t”J,_ ,(t) dt = zyJy(z),
(55)
0
.?
t’Y,_,(t)dt=z’Y,(z)+--
2”r (v) 7c
s 0
(56)
.
Thus, after substituting (37) and (38) into (53) and (54), evaluating the appropriate Bessel integrals, and normalizing the results with respect to the outer radius c, the axial thermal stresses in the untracked cylinder may be expressed in closed form as (1 - v)ozT(r*, t) EG&
2 C----l-a*=
+
b*= _ .*= 1
2
+ 2 f e’ - D,lc2V.,*1 n=l
a *Eol(b*PA, BW’ll(a *A,,,b *AJ /Lb*~,2H(kJl 1
11 >3
ec-0~ic2)‘~z’F~1(b*p~,, plr,) Pk3b *ff&)
(a Qr cb), EFM
Vol. 20, No.
1 --H
_ e,(r* t)
(57)
120
H. F. NIED
(1 - v)c&*(r*, t) =p EM,
(b
EO,(cc, “- a,) l_v
(ad=b),
(59)
The crack problem may be solved by using the axial stress given in (57) and (58), but opposite in sign, as the crack surface tractions in the hollow cylinder containing a circumferential crack. The details of the formulation of the general nonsymmetric problem is given by Nied and Erdogan[7]. For the axisymmetric problem considered in this paper it is sufficient to use the simpler derivation of the crack problem given by Erdol and Erdogan[8], where it is assumed that the axial stresses do not vary circumferentially %Id are only a function of radial location. In either case, the elasticity problem considered is subject to the following boundary conditions: o,(a,z,t)=O,a,,(a,z,t)=O,
(O
(61)
b,~(C,Z,t)=O,(T,~(C,Z,t)=O,
(O
(62)
@Jr, 0, t ) = 0,
(a~r~c,O~ttos),
(63)
oz.-Jr,0,t) = - oIT(r, t),
u
Cm),
(adrs2f,g
u,(r, 0, t) = 0,
(64)
where cZT(r,t) is given by (57) for (a d r < 6) and (T,~((P, t) is given by (58) for (b < r G c). The mixed boundary condition (64), when applied to the axisymmetric stress and displacement equations of elasticity expressed in terms of the biharmonic Love function[8], results in a singular integral equation of the form
Gr +k(r,s)
I
1- v #(s, t)ds = - jwT(r, P
f 1,
(f
Wf
where 4(r, t) is an unknown function defined by the derivative of the crack surface displacement, namely d u,(r, O+, I) = &(r, t), ar
Cf’
(66)
Thermal shock in a circumferentially cracked hollow cylinder with cladding
121
is the shear modulus and the kernel k(r, s) is given by Erdol and Erdogan[8]. In addition to (6.Q an additional single valuedness condition is necessary for the case of an embedded crack; this is given by In (65) ,u
s /
‘q5(s, r)ds
=O.
(67)
Note that in this problem the time t enters the analysis through o~‘(Y,t) only and (65) must be solved for each value of time separately. The solution of (65) and (67) for the case of an embedded crack is of the form[9]
a-, t>=Q,
f)I(r -f)(g
- J9-‘j2, (,f< s
(68)
for an edge crack the solution can be obtained from[lO]: #(r,t)=;gq;
(u
(69)
where in both cases d is a bounded function. The numerical solution of the integral equations is relatively straightforward and may be obtained by using a Gaussian integration procedure[lO, 111. The quantities of primary interest are the stress intensity factors k, which are the fracture mechanics parameters and are defined by (70)
k(g) = lim,/ma,,(r, r+g
After solving the appropriate
O, 2).
(71)
integral equations, k may be obtained directly from (72)
k(g) = - lim ‘_g &
J20+
(r, 1).
(73)
RESULTS Sample temperature and thermal stress distributions for the untracked cylinder are given in Figs. 3-17. The numerical values used in these plots were calculated directly from eqns (37), (38), (57) and (58). It was found that 50 terms were more than s~cient in the series expressions. The values of 2, needed for these calculations were determined from the roots of eqn (28) using a Newton-Raphson algorithm. For comparison purposes, Figs. 3-5 are the results for a homogeneous cylinder (i.e. K = 1, /?’ = 1, crz/cc,= 1) and give values identical to those obtained by Nied and Erdogan[l2] by a different technique. All other figures are for composite cylinders with two distinct sets of thermal properties. The temperature plots give the ratio @(r*, t*)/8, (0 = 8, for a < r sz b and 0 = 0, for b < r < c) as a function of nondimensional radial location (r - a),/(~ - a), for various values of nondimensional time (Fourier number) given by (74) The nondimensional thermal stresses, (I except that the entire thermal stress history In Figs. 3-14 the ratio of the inner wall Figures 15-17 are from calculations where
v)e=‘(r*,
t*)/~~~~~, are plotted in a similar manner, is shown on two separate graphs for clarity. radius to outer wall radius is chosen to be a/c = 0.9. n/c = 0.7.
122
H. F. NIED
8.8
8.6
0,4
6.2
1.6
8.8
6.6
(r - MC - 8) Fig. 3. Transient temperature distributioa in the homogeneous cylinder. a/e = 0.9, (b - a)/h = 0.5, h = c - il, i* = D,tj$.)
(K2/Kl
=
1, D2/D1
=
1,
=
I,
1‘
0.00001 0.00005
, -
/’ 8: -----bc._
_
0.2
_-___-
‘li
- -.
’
43.0
0.0001
:
I
1
__c__________
--
.___.___._._._._._._.~.~._.~.~
I
’
0.4
if-
8.6
’
I
’ 1.0
0.8
a)/@ - a)
Fig. 4. Transient thermal stress distribution in the homogeneous cylinder. (K2/K1 q/a, = 1, a/c = 0.9, (b - a)/h = 0.5, h = c -a, t* = D,t/c*.)
=
1, D,/D,
Thermal shock in a circumferentially cracked hollow cylinder with cladding
1.90
8.75
8.50
8.S
e.ee
-e.C!s
-9.58
e.e
0.2
8.4
9.6
6.8
1.8
(r - a)/(~ - a) Fig. 5. Transient thermal stress distribution in the homogeneous cylinder. (Q/K, = 1, DJD, = 1, a&, = 1, a/c = 0.9, (6 - a)/h = 0.5, h = c -a, t* = D,t/c2.)
B.8
8.2
Fig. 6. Transient
temperature
distribution
8.8
8.6 8.4 (r-a)/@ -a) in the composite
a/c=0.9,(b-a)/h=O.O4,h=c-a,t*=D,t/c’.)
cylinder.
&/I(,
1.a
=
3, DJD, = 3,
123
H. F. NIED
124
8.0
8.2
8.4 (I -
Fig.
7. Transient
0.6
8.8
1.B
a)l(c - a)
thermal stress distribution in the composite cylinder. (K~/K, a,/a, = 314, a/c = 0.9, (b - a)/h = 0.04, h = c -a, t* = D,t/c2.)
=
3. D,/D, = 3,
8.25
8.88
I -i
d.25
8.8
8.2
8.4
8.6
0.8
1.8
(I - a)@ - a) Fig.
8. Transient
thermal stress distribution in the composite cylinder. (K~/K, a,/~, = 314, a/c = 0.9, (b - a)/h = 0.004, h = c -a, t* = D,t/c’.)
=
3, DJD,
= 3,
Thermal
shock
in a circumferentially
cracked
hollow
cylinder
125
with cladding
1.0
a.8
$ =t o*6 k
. @I
6.2
0.8 0.e
Fig.
9. Transient
0.2
temperature distribution a/c=0.9,(b-a)/h=O.l,
8.8
8.2
0.4
in the composite cylinder, h=c-a, t*=D,t/c2.)
8.4
8.6 v - aW~a_)
Fig.
10. Transient
1.e
e.a
0.6 (r - a)/@- a)
(KJK,
8.8
=
3, D,/D, = 3,
i.e
_
thermal stress distribution in the composite cylinder, (K>/K, = 3, DJD, zJa, = 314, a/c = 0.9, (b - u)/h = 0.1, h = c - a, t* = D,t/c2.)
= 3,
126
H. F. NIED
8.75
8.58
8.25
e.ee
-8.25
8.0
8.2
8.6
8.4
e.8
1,e
(r - a)/(c - a) Fig. 11. Transient thermal stress dist~bution in the composite cylinder, (K& = 3, 1>,/1), = 3, qja, = 3~4, a/c = 0.9, (b - a)/h = 0.1, h = c -a, f* = D,r/c2.)
1.e
0.8
e.2
8.8
0.e
e.2
8.6
8.4 (r -
8.8
1.e
a)& - a)
Fig. 12. Transient temperature distribution in the composite cylinder. (KJK, = 3, Dz/D, = 3, a/c = 0.9, (h - a)/h = 0.25, h = c - a, t* = D,t/c*.)
Thermal shock in a circumferentially cracked hollow cylinder with cladding
l.ee
8.75
s
d 2%
8.58
. -:
+k
P
g.25
t 8.08
L
-9.25
I 6.2
1
I
I
1
8.4 6.6 (r - a)/@- a)
I
I
I
-I
8.8
Fig. 13. Transient thermal stress distribution in the composite cylinder. (K*/K, CQ/U,= 314, a/c = 0.9, (b - a)/h = 0.25, h = c -a, t* = D,t/c2.)
i+e =
3, L&/D, = 3,
8.58
8.25
e.ee
-8.25
8.9
e.2
8.4
8.6
8.8
1.e
(r - a)/@- a) Fig. 14. Transient thermal stress distribution in the composite cylinder. (QK, a,/a, = 314. a/c = 0.9, (b - a)/h = 0.25, h = c -a, I* = D,t/Ez.)
=
3, Dz/DI = 3,
127
H. F. NIED
128
0.9
8.2
6.6
0.4
1.0
6.8
(r - a)/(c - a) Fig.
15. Transient
temperature distribution in the composite cylinder. a/c = 0.7, (b ~ a)ih = 0.04, h = c -a, t* = D,t/c’.)
(KJK,
3, D,/D, = 3,
=
1.00
0.75
-1 I\
0.50
1’
0.25
-
\
\ ‘.\
/-- 0.0001 /’
5
,,-
‘. ‘. ‘.
‘._
0.00
,
’ ----
0.0005
‘i0.002
‘.
i__
________-____----. -._.
-.-._._,
-.-.-.-.-.-.-.
r
-0.25
1 0.0
’
I 0.2
,
I
,
I
0.4
,
0.6
I
I
I 1.0
0.8
(I - a)/(c - a)
Fig. 16. Transient
thermal stress distribution in the composite cylinder. (K~/K, cc2/a, = 314, u/c = 0.7. (b - u)/h = 0.04, h = c -a, t* = D,t/c’.)
=
3, D,/D, = 3,
Thermal shock in a circumferentially cracked hollow cylinder with cladding
129
-8.25
(r- a)@- a) Fig. 17. Transient thermal stress distribution in the compositecylinder. (K~/K, = 3, D,/D, = 3, scz/ct,= 314. a/c = 0.7, (h - a)/h = 0.04, h = c -a, t* = D,tlc’.)
Figures 6-8, where the ratio of cladding thickness to total cylinder wall thickness is given by (h - a)/(~ - a) = 0.04, is representative of geometric configurations encountered in nuclear the nondimensional material constants used in the pressure vessel applications. Furthe~ore, calculations were chosen to be IC= 3, /3’ = l/3 and cc2/cr,= 3/4. These material ratios are fairly representative of the materials encountered in nuclear pressure vessels, that is, austenitic stainless steel clad and low alloy steel base. In all the bimateria1 curves it is clear that discontinuities exist at the material interface. These discontinuities occur in the slope of the thermal field 8, as well as the axial thermal stress az7. The discontinuity in the slope of 0, that is, the heat flux, can be directly attributed to the difference in the thermal conductivity of the two materials [see boundary condition (lo)]. The axial thermal stress discontinuity is directly related to the difference in the coefficient of thermal expansion across the interface, since no discontinuity exists in the temperature field itself. Also, it is immediately clear that for a fixed a/c ratio, as the cladding thickness is increased the thermal gradient in the base material is decreased and, consequently, at any instant in time the thermal stresses in the base material are lower in magnitude. It is interesting to note, when comparing Figs. 6 and 15, that for a fixed cladding thickness to wall thickness ratio (h - a)/(c - a) = 0.04, the same thermal gradient occurs approximately an order of magnitude earlier in nondimensional time t *, for the cylinder with a/c = 0.9 as opposed to the cylinder with u/c = 0.7. The transient stress intensity factors for an edge crack subjected to the thermal history shown in Figs. G-8 (i.e. (h - a)/(c - a) = 0.04) are plotted in Fig. 18. In all fracture calculations, unless otherwise specified, Poisson’s ratio v is prescribed to be 0.3, For any given crack length ratio E/h, the nondimensional stress intensity factor increases, passes through a maximum, and then decreases as a function t*. As would be expected, the maximum stress intensity factor is of greatest magnitude for a very small edge crack and decreases with increasing crack length l/h. The maximum stress intensity factor attained during the thermal transient is the quantity of most interest. Figure 19 is an interesting comparison of the maximum stress intensity factors as a function of crack length for the edge crack problem. The comparison is between the cylinder without cladding and cylinders with two different cladding thicknesses. Noting that the stress intensity factor is normalized with respect to the coefficient of thermal expansion for the base material, that is, Ed, it can be seen that the maximum stress intensity factors for cracks in the cladding are much higher than those in the unclad cylinder. An increase in cladding thickness increases the maximum stress intensity factor attained during thermal shock conditions for edge cracks contained in the cladding. Furthermore, as the edge crack penetrates a short distance into the base material the maximum stress intensity factors are still higher than in the situation when
H. F. NIED
130
C2
Fig. 18. Stress intensity factor k(g) as a function of nondimensional time t * for various length edge cracks. (KJK, = 3, D?/D, = 3, a,/a, = 314, a/c = 0.9, (h ~ a)/h = 0.04, h = c -a, t* = D,t/c2.)
I 0.2
WITHOUT
CLADDING
a = 0.9 C
1 0
I
0
0.1
I
I
I
I
0.2
0.3
0.4
0.5
I
0.6
J
(1.7
P/h Fig. 19. Comparison of maximum without cladding. (With cladding: K>/K,
=
stress intensity factors for edge cracked cylinders KJK, = 3, D,/D, = 3, a,/a, = 3/4, a/c = 0.9. Without 1, D,/D, = t, ~$/a, = 1, U/C =0.9.)
with and cladding:
cladding does not exist. It is only with longer edge cracks that the beneficial effect of having a clad can be clearly seen. The transient behavior for an embedded crack which is directly underneath the cladding (i.e. f = b) is shown in Fig. 20. Again, the stress intensity factors in this plot correspond to the thermal history depicted in Figs. 68. In this situation the two crack tips initially have a negative stress intensity factor, and then independently reach maximum values at different times. Of course, the negative stress intensity factors only have meaning when superposed with a positive stress intensity factor caused by some external loading; for example, uniaxial tensile loading. It can be seen that
Thermal shock in a arcumferentially
cracked
hollow cylinder with cladding
131
a=
0.9
C
0
I 10-e
10-5
0.04
iLp
=
9
= 0.16
I
10-4
10-3
10-2
4t -2
Fig. 20. Stress embedded
factors k(f) and k(g) as a function of nondimensional time $* for an (IQ/K, = 3, D,/D, = 3, cc&, = 314, a/c = 0.9, (b - a)/h = 0.04, f = b, h = c -a. t* = D,r/2.)
intensity
crack.
the crack tip touching the clad always has a higher maximum stress intensity factor. Again, it is also apparent that the most useful quantity from the point of view of fracture is the maximum stress intensity factor for any given crack length. Figure 21 shows these maximum stress intensity factors for embedded cracks, where each point on the curve is obtained by iteration to find when the maximum occurs. There is little variation in the maxims stress intensity factor as a function of
I
I
I
I
0.2 0.3 0.4 0.5 (9 - bYh Fig. 21. Maximum stress intensity factors for embedded cracks of various crack lengths. (&, &fo, = 3, a,/~, = 314, a/c = 0.9, (b -a)/h = 0.04, f = 6, h = c -a.)
= 3,
H. F. NIED
132
crack length for the inner crack tip; however, it can be seen that the maximum stress intensity factor for the outer crack tip drops significantly with increasing crack length. Thus, the propensity for crack propagation is inwards, given that the two materials have the same crack growth rate characteristics. Tables 1-4 give the maximum stress intensity factors and the nondimensional time at which the maximum occurs, for edge cracks in cylinders with various cladding thickness to wall thickness ratios. As would be expected the time at which the maximum stress intensity factor occurs increases with increasing clad thickness and crack length. In these tables the constant gr is given by CT= - (Ec@,)/(l - v). Tables 5-8 contain the maximum stress intensity factors and the nondimensional time for embedded cracks. It can clearly be seen that there is a definite time difference between the maximum values at the two crack tips. Furthermore, for large embedded cracks in cylinders with thick cladding (Table 7) the stress intensity factor may never become positive at either crack tip. If the thermal shock is of sufficient magnitude to cause yielding through the thickness of the clad, the effect this has on the stress intensity factor for an embedded crack can be analyzed using a simple plastic strip model. If the crack is located just underneath the cladding and it is assumed that the entire clad has yielded in the plane which contains the crack, the maximum stress intensity factor at r = g can be determined from the solution of equation (65) withf = a and the stress terms in boundary condition (64) replaced by
a,,(r,O,t)=
-ozT(r.c)+aF,
((I
c:,(r,
-
(h
(75)
Table
Table
1. Maximum
2.
Maximum
0, t) =
azT(r,
t),
stress intensity factors and the nondimensional time at which they occur for internal edge cracks subjected to transient thermal stresses
0 041
0.00014
0.05
0.00014
0 Oh
0 OOOl5
0.07
0 00017
0 08
0 000 IX
0 09
0 rlO91Y
0 IO
0.0001Y
0 20
0.00027
030
0 00033
0 40
oooni:
0.50
0 00039
stress intensity factors and the nondimensional time at which they occur edge cracks subjected to transient thermal stresses b[=
-(En,H,)/(l-u).O/L
=0.9,(b-o)/h=O.l.
h=[~o.I=fi~a.KZ/~,=3.D2/D,=3,U2/a,=3/4)
0 101
0.00027
0 II
0.00027
0.5662
0 1s
0.0003 I
04740
0 20
0 00035
04151
0.25
000037
03732
030
0.00040
03461
0.35
0.00042
03222
0.40
0.00044
0.3034
0.45
0.00045
02896
0.50
0.00046
02122
O.bO90
for internal
Thermal Table 3. Maximum
shock in a circumferentially
cracked
hollow
cylinder
with cladding
stress intensity factors and the nondimensional time at which they occur for internal edge cracks subjected to transient thermal stresses (~r=-(Ea,8,)/(1-v).~l~=09.(h-~)/h=025, h=~-o./=~-a.~~/~,=3.D~/D,=3.a~lu,=3/4)
Table 4. Maximum
0.251
0
00050
0 4760
0.26
0 00050
0 456h
030
o.ono50
0 409'
035
0 00052
0 3720
0.40
0 00053
0 3449
045
0 00054
03224
0.50
0 00056
0 3062
stress intensity factors and the nondimensional time at which they occur for internal edge cracks subjected to transient thermal stresses (or= -(Ea,f?,,)l(I-,,),a/< =o 7,~b-a),!/i=O04. h=~~-o./=y--o.~~l~,=3.D~/Di=3.~~~~~~;=3i41 I h
T-o17-
0041 0.05 006
no0141 0 00152
T
007
Table 5. Maximum
h (,c) <-_ T--j_ ollo134
~
CT,%I --0 73VJ (
06612 0 hlO9
0 OOlhO
0 '751
0 08
0 00173
0 54Xh
0.09
0 OOIS?
05234
0.10
0 no190
0 5018
0.20
0 00265
0 3752
030
00032l
0 3000
040
000361
02410
0.50
00038R
0.1944
stress intensity factors and the nondimensional time at which they occur for embedded cracks subjected to transient thermal stresses
g-b h
k(b)
D,l
-7
CT*
m 0.001
0.00018
0.3899
0.3908
0.01
0.00019
0.3798
0.3883
0.02
0.00020
0.3701
0.3873
0.03
0.0002 1 0.00022
0.3613
0.3874
0.3529
0.3881
0.00023 0.00024
0.3448
03891
0.3369
0.3904
0.00033 0.0004 1
0.2622
0.4006
0.1940
0.4009
0.1309
0.3919
0.46
0.00047 0.00054
0.0728
0.3745
0.56
0.00066
0.0200
0.3494
0.04 0.05 0.06 0.16 0.26 0.36
133
H. F. NIED
134
Table 6. Maximum stress intensity factors and the nondimensional time at which cracks subjected to transient thermal stresses fur = f-b.
they occur
for embedded
-(~a,B,)/(~-~),(1/~-0.9.(b-o)/h=O.l. h-c-a. ~2/~,=3r D>/D,-3.0J/u,-3/4).
e
-Dir
h
c!
m
-
0.001
0.00036
0.2240
0.00036
0.01
0.00031
0.2172
0.00037
0.05
0.00041
0.1897
0.00038
0.10
0.00045
0.1589
0.00039
0.15
0.00049
0.1303
0.00041
0.20
0.00053
0.1030
0.00042
0.25
0.00057
0.0771
0.00044
0.30
0.00061
0.0523
0.00045
0.35
0.00067
0.0288
0.00046
0.40
0.00074
0.0066
0.00048
7
k(b)
mT*
I0.2245 0.2224 0.2150
0.2080 0.2017
0.1951 0.1880 0.1800
0.1713
0.1618
Table 7. Maximum stress intensity factors and the nondimensional time at which they occur for embedded cracks subjected to transient thermal stresses (UT f-b,
-(~o,8,)/(1-~).(1/~-0.9.(b-o)/h-0.25. h=c-a.~~Iu,=3,D~lD,-3,a~/a~-3/4~. k(g)
DI! 7
g-b "
k(b)
_,q
%
_jq
0.001
0.00070
0.0111
0.00070
0.0114
0.01
0.00071
0.0073
0.00070
0.0101
0.05
0.00076
-0.0088’
0.00072
0.0046
0.10
0.00084
-0.0277”
0.00074
-0.0022’
0.15
0.00096
-0.0453*
0.00076
-0.0090’
l
Local maximum
(k - 0
at
f- 0)
Table 8. Maximum stress intensity factors and the nondimensional time at which they occur for embedded cracks subjected to transient thermal stresses -(Ea,B,)l(l-v).olc-0.7,(b-o)lh*0.04, h-c-o. K~/K,-~, D>/D,-3,cq/n,-314).
(CJ f-b.
k(g)
-DlI
g-b h
c&F
2
k(b)
-DlI
c*
cTq-
-
B
-
0.001
0.00173
0.3995
0.00172
0.4003
0.01
0.00182
0.3892
0.00175
0.3980
0.02
0.00193
0.3793
0.00179
0.3971
0.03
0.00203
0.3702
0.00183
0.3973
0.04
0.00214
0.3615
0.00186
0.3980
0.05
0.00223
0.3530
0.00190
0.3990
0.06
0.00233
0.3447
0.00194
0.4002
0.16
0.00316
0.2656
0.0023 1
0.4078
0.26
0.00384
0.1942
0.00264
0.4043
0.36
0.00442
0.1308
0.00293
0.3923
0.46
0.00501
0.0748
0.00317
-
-
0.3737
Thermal shock
in a circumferentiallycracked hollow cylinder with cladding
I35
where crFis the plastic flow stress. The maximum stress intensity factor at the outer crack tip Y= g. normalized with respect to the crack half-length (g - b)/2, is then given by (76)
where the values of k? and kr are given in Tables 9-11, for (b - a)/h ratios of 0.04 and 0.10, when a/c = 0.9 and for (b - a)/h = 0.04, when a/c = 0.7. In effect, kf is the maximum normalized stress intensity factor for an edge crack subjected to - o;‘(r, f) (a < r CCg) and kf is the stress intensity factor for an edge crack subjected to cl” (a < r < b). As can be seen, if or- is not significantly larger than g F, a dramatic reduction in the maximum stress intensity factor is possible. Of course this is not always the case and for a very large value of cr., yielding in the cladding will result in a much higher stress intensity factor at Y= g than in the case where no yielding (Tables 5, 6 and 8) has occurred. It is also seen that the effect of clad yielding on the stress intensity factors for longer cracks is greatly diminished.
Table 9. Plastic strip yield model: parameters for determination of maximum stress intensity factor for an embedded crack
-- ..-
-...
._._-r=
6.68i5 20933 i
1 so31 1 2521 11109 1011: 0 Y3i8 0 6436 (J 5363 0 4665 04135
Table IO. Plastic strip yield model: parameters for determination of maximum stress intensity factor for an embedded crack ~~~=-~Ea,H,i~~1-1~~.Li/~=09.~h-oJ;!7=01. h=c
-Ii,KJK,=3,
n,/D,=3.u>jnr=3/4~ v-.-p-
----‘~~----s-_h I1
h",
X",
~___~ ~__
~-__-
0001
8 hi49
001
2 hS59
-4
0.05
1.1612
-I
5353
0.10
0 xi01
-1
0206
-15
+
0.15
0 6814
0.20
u99;
j
Ii73 1925
~~.~f~~
0.25
EFM
Vol. 20, No. I -I
0.30
0.4955
-0
5872
0.35
0 4643
-0
59.18
0.40
0 4203
-0
55hX
I
136
H. F. NIED Table
11. Plastic
strip yield model:
parameters for determination an embedded crack
of maximum
stress intensity
factor
for
(,r,=-(Lrr,H,)~(l~I~).u:i=O’.(h-~uI/h=O.oJ. /i=~~u.K1~“,~=3,1~~!1~,=3.ir~~~r,~l.ll~I .c_h A_=
~___$
~~ _~
mmm~~7~~~~ :=:~_:__..[ _~~ ;”
-- ~-~ ‘:____ 7902
0 001
hh95?
-8
0 01
2 0909
-2
2168
002
I4963
-I
3669 0161
003
~
12423
-I
0.04
I
IO972
--OX382
0 06
1
09162
0.16
I
005
0 9950
-0 ~
05932
7036
-05x51 -02635
0.26
04558
-0
036
03602
-01497
0 46
I
0 2866
i
-0
IYlh
1280
CONCLUSION By obtaining the transient thermal stresses for the bimaterial cylinder in closed form and using integral equation analytic techniques, a highly accurate parametric study was conducted which examined the diffusive transient characteristics of the thermal shock problem for a circumferentially cracked cylinder with cladding. Examination of Fig. 18 clearly indicates the transient nature of any fracture phenomena which would occur in a thermal shock situation. It is not difficult to envision a thermal shock event during which a small crack will reach a critical stress intensity factor, rapidly advance, and then stop due to the decrease in stress intensity factor with increasing crack length. As the tensile thermal stress field “diffuses” radially outwards, the stress intensity factor would then build up again, attain a critical value, advance, and halt. This start-stop phenomenon could continue during a transient thermal event as long as the maximum stress intensity factor, attainable for any given crack length, is greater than some critical value of the stress intensity factor necessary for the crack to advance at a rate faster than the “diffusing” tensile thermal stress field. Of particular interest are the maximum stress intensity factors and the times at which these maximums occur for various cladding thicknesses and cylinder wall thicknesses. An interesting comparison is made by examining Tables 1 and 5. For example, an edge crack with l/h = 0.2 and an embedded crack with length (g - b)/h = 0.16 both have the outermost crack tip at the same radial location, yet the maximum stress intensity factor is attained sooner for the edge crack than for the embedded crack. In actual pressure vessels these nondimensional times at which the maximum stress intensity factor is reached translate into real times greater than 10 min. In Fig. 19 we clearly see that for an edge crack, when the crack is just through the cladding the maximum stress intensity factor is higher than if there had been no cladding at all. It is when we have deeper crack penetration that cladding causes a reduction in the stress intensity factor. This should not be interpreted as meaning that it is better not to have the cladding. The cladding serves a very important function in protecting the base metal from severe corrosion damage. Thus, even though small edge cracks have a higher maximum stress intensity factor when cladding is present, these elevated values in general would not be high enough to negate the beneficial effects of having a clad. It can be seen that thicker cladding is not necessarily better, at least from a fracture mechanics point of view, which does not consider the possibility of circumferential clad buckling. Thus, if the cladding is not to be breached by an edge crack, care should be taken to ensure that the maximum stress intensity factors remain below acceptable levels for cases where the cladding is thick. The effect of clad yielding on the stress intensity factor for the crack tip embedded in the base material was also investigated, and it was shown, using a plastic strip model, that in certain situations clad yielding will cause a reduction in the stress intensity factor. In long embedded cracks, when the clad yielded, no reduction in the maximum stress intensity factor was seen. The stress intensity factors in these cases are a good deal higher than if no yielding had occurred in the cladding. Of course, if yielding of the clad is the only way to prevent breaching of the clad,
Thermal
shock
in a circumferentially
cracked
hollow
cylinder
with cladding
137
for example due to brittle fracture, the increase in the stress intensity factor at r = g may be quite acceptable. Acknowledgements-A portion of this work was supported by NSF under Grant ENG 78-09737 and by NASA Langley under Grant NGR39-007-011 while the author was at Lehigh University. The author is also greatly indebted to Professor F. Erdogan for his many helpful suggestions.
REFERENCES M. Norris and C. F. Shih, Fracture mechanics related problems in the nuclear power industry. Proc. 9th U.S. National Congress of Applied Mechanics, ASME, p. 143 (1982). PI F. Erdogan, G. D. Gupta and T. S. Cook, Numerical solution of singular integral equations. Mechanics of Fracture 1. Methods of Analysis and Solutions of Crack Problems (Edited by G. C. Sih), pp. 368-425. Noordhoff, Groningen, The Netherlands (1973). and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical [31W. Magnus, F. Oberhettinger Physics, Die Grundlehren der Mathematischen Wissenschaften, Vol. 52. Springer-Verlag, New York (1966). [41H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids. Oxford University Press (1950). [SlD. Burgreen, Elements of Thermnl Stress Analysis. C.P. Press, Jamaica, New York (1971). and I. Stegun, Handbook of Mathematical Functions. Dover, New York (1965). (61M. Abramowitz crack. Znt. [71H. F. Nied and F. Erdogan, The elasticity problem for a thick-walled cylinder containing a circumferential Z. Fracture 22, 277-301 (1983). cylinder with an axisymmetric internal or edge crack. J. Appt. Mech. 45, PI R. Erdol and F. Erdogan, A thick-walled Trans. ASME, 281-286 (1978). Singular Integral Equations. Noordhoff, Groningen, The Netherlands (1953). [91N. I. Muskhelishvili, [lOIG. D. Gupta and F. Erdogan, The problem of edge cracks in an infinite strip. J. Appl. Mech., 41, Trans. ASME, 1001-1006 (1974). value problems in mechanics. Mechanics Today (Edited by S. Nemat-Nasser), Vol. 4, [I11 F. Erdogan, Mixed boundary pp. l-86 (1978). Transient thermal stress problem for a circumferentially cracked hollow cylinder. J. [I21H. F. Nied and F. Erdogan, Thermal Stresses 6, l-14 (1983).
111D.
(Received 29 April 1983)
Vol 20, No. 1, pp. 113-137, 1984
0013 7944/84 $3.00+ .XJ Pergmm Press Ltd.
THERMAL SHOCK IN A CIRCU~F~RE~TIALLY CRACKED HOLLOW CYLINDER WITH CLADDING General Electric Company,
Corporate
H. F. NIED Research and Development, U.S.A.
Schenectady,
NY 12301,
Abstract-An analysis is presented which demonstrates the effect that cladding has on the thermal shock resistance of a ~rcumferentially cracked hollow cylinder. The cladding, which in general may have different thermal properties than the base material, is assumed to be bonded to the inner wall of the hollow cylinder. The axisymmetric circumferential crack may either be embedded in the cylinder wall, typically underneath the clad, or may be an edge crack which passes through the clad and opens into the inner wall of the hollow cylinder. The mathematical formulation of the problem results in a singular integral equation of a well known type which is solved numerically. Results, which include transient temperature distributions, thermal stresses in the untracked cylinder, and stress intensity factors as a function of time, are presented for various cladding thickness to cylinder wall thickness ratios. The numerical calculations concentrate on material properties and geometric con~gurations typically seen in nuclear pressure vessel appli~tions. Results of particular interest are the transient stress intensity factors for various crack lengths and the comparison of maximum stress intensity factors in the clad cylinder with those in an unclad cylinder. Assuming the clad has completely yielded. the stress intensity factors for a crack under the clad, oriented in a plane perpendicular to the cylinder axis, are determined using a plastic strip model. It is shown that yielding of the clad under certain conditions can result in a reduction in the magnitude of the stress intensity factor for the crack tip in the elastic base material.
INTRODUCTION IT HAS long been recognized that cladding, when properly bonded to the inner wall of a pressure vessel or pipe, can be very effective in protecting the base metal from severe chemical corrosion. In the nuclear environment, cladding serves the additional role of shielding the base metal from detrimental radiation damage. Typically, this cladding has a significantly lower thermal conductivity than the base metal and a higher coefhcient of thermal expansion. Furthermore, in many applications, the clad has the same elastic modulus and Poisson’s ratio as the base metal For instance, in most nuclear pressure vessel applications the clad is austenitic stainless steel and the base metal is a low alloy steel. When cladding is used to protect the base metal in this manner, an important question arises: What effect does the cladding have on the structural integrity of the system during a severe thermal transient, especially in the presence of preexisting flaws?[l] In a number of cases cracks have been observed underneath the clad, in the base metal, oriented in a plane perpendicular to the cylinder axis and terminating at the bond interface. The purpose of this paper is to investigate the effect of cladding on the thermal shock resistance of a hollow cylinder with an axisymmetric circumferential crack. The severity of the thermal shock is characterized by the stress intensity factor, which varies as a function of time and achieves a maximum value some time after initiation of the transient thermal event. Once the stress intensity factors are known as a function of time, it becomes possible in brittle materials to determine whether catastrophic failure will occur due to unstable crack propagation. Also, the amount of crack growth which occurs after repeated thermal shocks can be estimated using empirical crack growth rate expressions which utilize the stress intensity factor info~ation given as a function of crack length. The problem of interest is depicted in Fig. 1. The initially stress free, infinitely long cylinder is assumed to be composed of two materials, initially at temperature T,,, which are perfectly bonded at the bimaterial interface r = b. The cladding, which we will designate as material I. has a coefficient of thermai conductivity K,, a specific heat q, a mass density pr, a coefficient of thermal expansion q, and a thickness given by b - a. The base material, material II, has properties x2, c2, p2 and t12 associated with the thermal conductivity, specific heat, mass density and coefficient of thermal expansion, respectively, in the region b < r $ c. The inner crack tip radius is given by Y =,f and the outer crack tip radius is designated by r =g. Two special crack configurations are of particular interest: the circumferential crack embedded in the base metal, where the crack tip is 113
114
H. F. NIED
Fig. I. Geometry of a thick-walled cylinder with cladding containing an axisymmetric circumferential crack.
located just underneath the clad (i.e. f = n) and the edge crack which completely passes through the clad. The fracture problem is formulated by first obtaining the solution for the transient thermal stresses in an untracked bimaterial hollow cylinder subjected to a sudden temperature change on the inner wall of the composite cylinder, where the outer radius of the composite cylinder is assumed to be insulated. The thermal stresses calculated from these boundary conditions simulate a severe thermal shock and thus provide a reasonable upper bound solution to the fracture problem of interest. The mixed boundary value problem, which contains all the pertinent fracture information, is then formulated by using the thermal stresses from the untracked composite cylinder, but with opposite sign, as the crack surface tractions along the line of revolution swept out by the crack. Superposition of this result with the untracked cylinder thermoelastic solution results in the correct stress field for the cracked cylinder problem. The mixed boundary value problem that must be solved is simplified by assuming the clad and the base material have the same elastic modulus and Poisson’s ratio. This assumption allows the mathematical formulation to result in a singular integral equation of a well-known type which avoids the difficulties usually associated with bimaterial crack problems, that is, determining the strength of the stress singularity at the crack tip which touches the interface[2]. Another important, but reasonable, assumption in this formulation, which simplifies the analysis greatly, is that the transient thermal stress problem described is quasi-static, that is, the inertia effects are negligible. Since the cylinder and cladding thicknesses are arbitrary, as well as the size and location of the crack, the formulation presented here lends itself ideally to parametric investigations on the general effect of cladding thickness and crack size. It is also recognized that the material properties of most interest are those associated with nuclear pressure vessel applications. Thus, numerical calculations presented in the results rely heavily on those material properties which are representative of pressure vessel steels. When the material properties of the clad and base material are taken to be identical, we of course recover the homogeneous cylinder solution, which is useful for comparison with the solutions where cladding is present. It is important to note that if the cladding is su~ciently ductile, during a severe thermal transient the cladding may yield. Thus, yielding of the clad and the effect this has on the stress intensity factors for an embedded crack underneath the clad is examined by using a plastic strip model which assumes the clad is perfectly plastic. As will be shown in the results, the effect of clad yielding, under certain conditions, can be beneficial and result in a substantial reduction in the stress intensity factor for the crack tip buried in the base material.
Thermal shock in a circumferentially cracked hollow cylinder with cladding
115
FORMULATION Referring again to Fig. 1, the uncoupled transient temperature distribution may be determined from the solution of the diffusion equations
1 &(r,
v*ez(r, t) = o-
t)
iit
2
,
(2)
(h G r d ~1,
’
where H,(r, t) and $,(r, t) are defined by (3) @,(r, t) = Tfr, t) -
T,,,
(b G r G ~1,
(4)
The temperatures, T(r, t) and To, are the temperature in the cylinder at time t and the initial temperature of the composite cylinder. respectively. The thermal diffusivities~ L), and I&, are given by
D2
=
Q/(Pzc~)>
(6)
where K~, K~,pr, pz and cl, cz are the material thermal conductivities, densities and specific heats for materials I (cladding) and II (base), respectively. The initial thermal conditions throughout the cylinder are given by O,(r, 0) = &(r, 0) = 0.
Defining, 8, = T, - r,, where To is the temperature boundary conditions are specified to be
em t>=
(7)
to be maintained
at r = u, the thermal
o
’
ar
where H(t) is the Heaviside step function. These conditions imply a sudden change in temperature on the inner radius a, perfect heat transfer at the bimaterial interface r = b and an insulated outer radius L’. Application of Laplace transforms and condition (7) reduces the partial differential equations (1) and (2) to ordinary di~erentia~ equations of the form
d2t%r,p)+~u’8;(r,p)p -__ - dr -,O,(r,p)=O, r dr2
(i = 1,2)
(12)
where the integral transform is defined as &(r,p)
=
x
Or(r, t) e-P’dt.
(13)
H. F. NIED
116
The general solution to eqn (12) is
with I&y,) and KO(ryi) the usual modified Bessel functions of the first and second kind of order zero. A,(p) and &i(p) are four constants to be determined from conditions (8)-(I I), and yI is given by
After application of the boundary conditions (8~(11) and determination B,(p), the equations in (14) become
of the terms Ai
and
(17) where,
The Laplace inversion to obtain the solutions for 8,(r, t) and O&r, t) may be obtained by evaluating the following integrals with the aid of the contour I- shown in Fig. 2,
(24) It is not difficult to show that the integrands in (23) and (24) have an infinite number of simple poles distributed along the imaginary axis. This can be shown with the aid of the Bessel function identities which relate modified Bessel functions to ordinary Bessel functions[3]; f,Jz
Ky(z
ew21~) = e(vd2)fJ”
e(G)3
=
-! i i.
Substitution
e _.
“n’2)i[- J,,(z)
of (25) and (26) into (18) demonstrates p=l)i&,, c
(25)
(z),
(rz=l,2
-I-
iY"(Z)].
that there are simple poles located at )...,
co),
(27)
Thermal shock in a circumferentially cracked hollow cylinder with cladding
117
Fig. 2. Contour f for evaluation of integrals in eqns (23) and (24).
where the l,s are determined from the roots of the transcendental
equation (28)
Normalizing all dimensions with respect to the outer radius c, we have,
r* =rIc,
a*=a/c,
b*=blc.
(29)
The dimensionless thermal constants K and /I are given by
,=!2
(30)
KI
p=
0 ;
I’*,
(31)
2
and the functions F,, F,, and E,, are defined as
Mx,
Y>=
F,,&Y)
Jo(x)YO(Y >- JO(Y) Y& 1
= J,(x)~,(Y)
J&(X,Y) = J&)Y,(Y)
(32)
- J,(y)Y,(x)
(33)
- J,(Y)Yo(X).
(34)
Recognizing that the integrals in (23) and (24) can be evaluated by using the residue theorem, that is, fA(r, t) ___ = Res,,(O) + f Res,, n=l 0,
,
u-9 t) = Res,,(O) + f Res,,
,
n=l
-e,?
b
(36)
0 x2
where the subscripts 23 and 24 refer to the integrals in eqns (23) and (24) respectively, we obtain after lengthy but straightforward analysis the following closed form solutions for 6, and 0,:
Ur*, t) -=
0,
1 - 2 2 e(-“““‘“n2’[lcBFoo(b*~,, r*/Z,)F,,(b*P&, p&J n=l (37)
The function H(%,) in the denominator
of (37) and (38) is determined
H(j.,) = KP2F,,,(a*&, h*&,)[E,,,(b*/% + E,,(b*&
&) - b*- ‘E&U,,,
- F,(a*L,,
/NJ
from b*/U,)]
b*&) +gF,,(a*&,
b*i.,)
i. - f@‘,,(/%,,
1 - E,,,(u*i,,
b*/%)
b*&) +g
E,,,(b*i,,
a*%,)
i + PEo,(a*&,
b*U
1
- F,,(P&,, b*/N,) + b*-lFoo(~&,,
b*@,)].
(39)
With the aid of the following identities, Foo(x,x>= 0,
(40)
F,,(x, x>= 0,
(41)
Eo,(x,x) = -
&,
(42)
it can easily be shown that the transient solution for a homogeneous cylinder is recovered, by setting fl and K equal to one and letting either b = c in (37) or b = a in (38). In these cases we obtain (43) which can be shown to be identical to the result given by Carslaw and Jaeger[4]. Once the transient temperature is known, it becomes possible to determine the axial thermal stresses in the untracked composite cylinder. The general solution for the axial thermal stresses in the two cylinders can be considered due to two separate effects: first, the axial stresses induced in an unrestrained cylinder with radial temperature variation, and second, the stresses due to the radial interference. The plane strain solution for the axial thermal stresses in the cladding and base material, respectively, can be shown to be[5] olr = E,
-
crT+
(44) + 2v,nb2 P + A2
E2c~2
I-v
(v,T> -
T2h
(45)
2
where subscripts 1 and 2 refer to material I and II. The radial interface pressure P, between the two materials, is given by
with
u;= a,(1 + E;=_
E, 1 -
v,),
a; = cl,(l f v,), I$=---
v,2’
~72 1 -
v22’
(47) (48)
Thermal
shock
in a circumferentially
cracked
hollow
cylinder
with cladding
119
(49) 2 b2-a2
T, = ___
(50)
c T2r dr, b’ s b
(51)
A, = n(b2 - a’), A, = n(c’ - b2).
(52)
T2 = ~
2
b T,r dr, so
C2-
At this point a simplifying assumption is made: it is assumed that the two materials which comprise the composite cylinder have the same values for the elastic constants E and v. Thus the materials have only different coefficients of thermal expansion and, of course, different thermal diffusivities. This assumption is a reasonable one for the cited applications, since the statistical spreads of these properties usually overlap. With this assumption, eqns (44) and (45) can be simplified and the axial thermal stresses become
(a < r < b),
(b 6r
(54)
It turns out that it is possible to evaluate the integrals in (53) and (54) in closed form using the following Bessel identities [6]:
s z
t”J,_ ,(t) dt = zyJy(z),
(55)
0
.?
t’Y,_,(t)dt=z’Y,(z)+--
2”r (v) 7c
s 0
(56)
.
Thus, after substituting (37) and (38) into (53) and (54), evaluating the appropriate Bessel integrals, and normalizing the results with respect to the outer radius c, the axial thermal stresses in the untracked cylinder may be expressed in closed form as (1 - v)ozT(r*, t) EG&
2 C----l-a*=
+
b*= _ .*= 1
2
+ 2 f e’ - D,lc2V.,*1 n=l
a *Eol(b*PA, BW’ll(a *A,,,b *AJ /Lb*~,2H(kJl 1
11 >3
ec-0~ic2)‘~z’F~1(b*p~,, plr,) Pk3b *ff&)
(a Qr cb), EFM
Vol. 20, No.
1 --H
_ e,(r* t)
(57)
120
H. F. NIED
(1 - v)c&*(r*, t) =p EM,
(b
EO,(cc, “- a,) l_v
(ad=b),
(59)
The crack problem may be solved by using the axial stress given in (57) and (58), but opposite in sign, as the crack surface tractions in the hollow cylinder containing a circumferential crack. The details of the formulation of the general nonsymmetric problem is given by Nied and Erdogan[7]. For the axisymmetric problem considered in this paper it is sufficient to use the simpler derivation of the crack problem given by Erdol and Erdogan[8], where it is assumed that the axial stresses do not vary circumferentially %Id are only a function of radial location. In either case, the elasticity problem considered is subject to the following boundary conditions: o,(a,z,t)=O,a,,(a,z,t)=O,
(O
(61)
b,~(C,Z,t)=O,(T,~(C,Z,t)=O,
(O
(62)
@Jr, 0, t ) = 0,
(a~r~c,O~ttos),
(63)
oz.-Jr,0,t) = - oIT(r, t),
u
Cm),
(adrs2f,g
u,(r, 0, t) = 0,
(64)
where cZT(r,t) is given by (57) for (a d r < 6) and (T,~((P, t) is given by (58) for (b < r G c). The mixed boundary condition (64), when applied to the axisymmetric stress and displacement equations of elasticity expressed in terms of the biharmonic Love function[8], results in a singular integral equation of the form
Gr +k(r,s)
I
1- v #(s, t)ds = - jwT(r, P
f 1,
(f
Wf
where 4(r, t) is an unknown function defined by the derivative of the crack surface displacement, namely d u,(r, O+, I) = &(r, t), ar
Cf’
(66)
Thermal shock in a circumferentially cracked hollow cylinder with cladding
121
is the shear modulus and the kernel k(r, s) is given by Erdol and Erdogan[8]. In addition to (6.Q an additional single valuedness condition is necessary for the case of an embedded crack; this is given by In (65) ,u
s /
‘q5(s, r)ds
=O.
(67)
Note that in this problem the time t enters the analysis through o~‘(Y,t) only and (65) must be solved for each value of time separately. The solution of (65) and (67) for the case of an embedded crack is of the form[9]
a-, t>=Q,
f)I(r -f)(g
- J9-‘j2, (,f< s
(68)
for an edge crack the solution can be obtained from[lO]: #(r,t)=;gq;
(u
(69)
where in both cases d is a bounded function. The numerical solution of the integral equations is relatively straightforward and may be obtained by using a Gaussian integration procedure[lO, 111. The quantities of primary interest are the stress intensity factors k, which are the fracture mechanics parameters and are defined by (70)
k(g) = lim,/ma,,(r, r+g
After solving the appropriate
O, 2).
(71)
integral equations, k may be obtained directly from (72)
k(g) = - lim ‘_g &
J20+
(r, 1).
(73)
RESULTS Sample temperature and thermal stress distributions for the untracked cylinder are given in Figs. 3-17. The numerical values used in these plots were calculated directly from eqns (37), (38), (57) and (58). It was found that 50 terms were more than s~cient in the series expressions. The values of 2, needed for these calculations were determined from the roots of eqn (28) using a Newton-Raphson algorithm. For comparison purposes, Figs. 3-5 are the results for a homogeneous cylinder (i.e. K = 1, /?’ = 1, crz/cc,= 1) and give values identical to those obtained by Nied and Erdogan[l2] by a different technique. All other figures are for composite cylinders with two distinct sets of thermal properties. The temperature plots give the ratio @(r*, t*)/8, (0 = 8, for a < r sz b and 0 = 0, for b < r < c) as a function of nondimensional radial location (r - a),/(~ - a), for various values of nondimensional time (Fourier number) given by (74) The nondimensional thermal stresses, (I except that the entire thermal stress history In Figs. 3-14 the ratio of the inner wall Figures 15-17 are from calculations where
v)e=‘(r*,
t*)/~~~~~, are plotted in a similar manner, is shown on two separate graphs for clarity. radius to outer wall radius is chosen to be a/c = 0.9. n/c = 0.7.
122
H. F. NIED
8.8
8.6
0,4
6.2
1.6
8.8
6.6
(r - MC - 8) Fig. 3. Transient temperature distributioa in the homogeneous cylinder. a/e = 0.9, (b - a)/h = 0.5, h = c - il, i* = D,tj$.)
(K2/Kl
=
1, D2/D1
=
1,
=
I,
1‘
0.00001 0.00005
, -
/’ 8: -----bc._
_
0.2
_-___-
‘li
- -.
’
43.0
0.0001
:
I
1
__c__________
--
.___.___._._._._._._.~.~._.~.~
I
’
0.4
if-
8.6
’
I
’ 1.0
0.8
a)/@ - a)
Fig. 4. Transient thermal stress distribution in the homogeneous cylinder. (K2/K1 q/a, = 1, a/c = 0.9, (b - a)/h = 0.5, h = c -a, t* = D,t/c*.)
=
1, D,/D,
Thermal shock in a circumferentially cracked hollow cylinder with cladding
1.90
8.75
8.50
8.S
e.ee
-e.C!s
-9.58
e.e
0.2
8.4
9.6
6.8
1.8
(r - a)/(~ - a) Fig. 5. Transient thermal stress distribution in the homogeneous cylinder. (Q/K, = 1, DJD, = 1, a&, = 1, a/c = 0.9, (6 - a)/h = 0.5, h = c -a, t* = D,t/c2.)
B.8
8.2
Fig. 6. Transient
temperature
distribution
8.8
8.6 8.4 (r-a)/@ -a) in the composite
a/c=0.9,(b-a)/h=O.O4,h=c-a,t*=D,t/c’.)
cylinder.
&/I(,
1.a
=
3, DJD, = 3,
123
H. F. NIED
124
8.0
8.2
8.4 (I -
Fig.
7. Transient
0.6
8.8
1.B
a)l(c - a)
thermal stress distribution in the composite cylinder. (K~/K, a,/a, = 314, a/c = 0.9, (b - a)/h = 0.04, h = c -a, t* = D,t/c2.)
=
3. D,/D, = 3,
8.25
8.88
I -i
d.25
8.8
8.2
8.4
8.6
0.8
1.8
(I - a)@ - a) Fig.
8. Transient
thermal stress distribution in the composite cylinder. (K~/K, a,/~, = 314, a/c = 0.9, (b - a)/h = 0.004, h = c -a, t* = D,t/c’.)
=
3, DJD,
= 3,
Thermal
shock
in a circumferentially
cracked
hollow
cylinder
125
with cladding
1.0
a.8
$ =t o*6 k
. @I
6.2
0.8 0.e
Fig.
9. Transient
0.2
temperature distribution a/c=0.9,(b-a)/h=O.l,
8.8
8.2
0.4
in the composite cylinder, h=c-a, t*=D,t/c2.)
8.4
8.6 v - aW~a_)
Fig.
10. Transient
1.e
e.a
0.6 (r - a)/@- a)
(KJK,
8.8
=
3, D,/D, = 3,
i.e
_
thermal stress distribution in the composite cylinder, (K>/K, = 3, DJD, zJa, = 314, a/c = 0.9, (b - u)/h = 0.1, h = c - a, t* = D,t/c2.)
= 3,
126
H. F. NIED
8.75
8.58
8.25
e.ee
-8.25
8.0
8.2
8.6
8.4
e.8
1,e
(r - a)/(c - a) Fig. 11. Transient thermal stress dist~bution in the composite cylinder, (K& = 3, 1>,/1), = 3, qja, = 3~4, a/c = 0.9, (b - a)/h = 0.1, h = c -a, f* = D,r/c2.)
1.e
0.8
e.2
8.8
0.e
e.2
8.6
8.4 (r -
8.8
1.e
a)& - a)
Fig. 12. Transient temperature distribution in the composite cylinder. (KJK, = 3, Dz/D, = 3, a/c = 0.9, (h - a)/h = 0.25, h = c - a, t* = D,t/c*.)
Thermal shock in a circumferentially cracked hollow cylinder with cladding
l.ee
8.75
s
d 2%
8.58
. -:
+k
P
g.25
t 8.08
L
-9.25
I 6.2
1
I
I
1
8.4 6.6 (r - a)/@- a)
I
I
I
-I
8.8
Fig. 13. Transient thermal stress distribution in the composite cylinder. (K*/K, CQ/U,= 314, a/c = 0.9, (b - a)/h = 0.25, h = c -a, t* = D,t/c2.)
i+e =
3, L&/D, = 3,
8.58
8.25
e.ee
-8.25
8.9
e.2
8.4
8.6
8.8
1.e
(r - a)/@- a) Fig. 14. Transient thermal stress distribution in the composite cylinder. (QK, a,/a, = 314. a/c = 0.9, (b - a)/h = 0.25, h = c -a, I* = D,t/Ez.)
=
3, Dz/DI = 3,
127
H. F. NIED
128
0.9
8.2
6.6
0.4
1.0
6.8
(r - a)/(c - a) Fig.
15. Transient
temperature distribution in the composite cylinder. a/c = 0.7, (b ~ a)ih = 0.04, h = c -a, t* = D,t/c’.)
(KJK,
3, D,/D, = 3,
=
1.00
0.75
-1 I\
0.50
1’
0.25
-
\
\ ‘.\
/-- 0.0001 /’
5
,,-
‘. ‘. ‘.
‘._
0.00
,
’ ----
0.0005
‘i0.002
‘.
i__
________-____----. -._.
-.-._._,
-.-.-.-.-.-.-.
r
-0.25
1 0.0
’
I 0.2
,
I
,
I
0.4
,
0.6
I
I
I 1.0
0.8
(I - a)/(c - a)
Fig. 16. Transient
thermal stress distribution in the composite cylinder. (K~/K, cc2/a, = 314, u/c = 0.7. (b - u)/h = 0.04, h = c -a, t* = D,t/c’.)
=
3, D,/D, = 3,
Thermal shock in a circumferentially cracked hollow cylinder with cladding
129
-8.25
(r- a)@- a) Fig. 17. Transient thermal stress distribution in the compositecylinder. (K~/K, = 3, D,/D, = 3, scz/ct,= 314. a/c = 0.7, (h - a)/h = 0.04, h = c -a, t* = D,tlc’.)
Figures 6-8, where the ratio of cladding thickness to total cylinder wall thickness is given by (h - a)/(~ - a) = 0.04, is representative of geometric configurations encountered in nuclear the nondimensional material constants used in the pressure vessel applications. Furthe~ore, calculations were chosen to be IC= 3, /3’ = l/3 and cc2/cr,= 3/4. These material ratios are fairly representative of the materials encountered in nuclear pressure vessels, that is, austenitic stainless steel clad and low alloy steel base. In all the bimateria1 curves it is clear that discontinuities exist at the material interface. These discontinuities occur in the slope of the thermal field 8, as well as the axial thermal stress az7. The discontinuity in the slope of 0, that is, the heat flux, can be directly attributed to the difference in the thermal conductivity of the two materials [see boundary condition (lo)]. The axial thermal stress discontinuity is directly related to the difference in the coefficient of thermal expansion across the interface, since no discontinuity exists in the temperature field itself. Also, it is immediately clear that for a fixed a/c ratio, as the cladding thickness is increased the thermal gradient in the base material is decreased and, consequently, at any instant in time the thermal stresses in the base material are lower in magnitude. It is interesting to note, when comparing Figs. 6 and 15, that for a fixed cladding thickness to wall thickness ratio (h - a)/(c - a) = 0.04, the same thermal gradient occurs approximately an order of magnitude earlier in nondimensional time t *, for the cylinder with a/c = 0.9 as opposed to the cylinder with u/c = 0.7. The transient stress intensity factors for an edge crack subjected to the thermal history shown in Figs. G-8 (i.e. (h - a)/(c - a) = 0.04) are plotted in Fig. 18. In all fracture calculations, unless otherwise specified, Poisson’s ratio v is prescribed to be 0.3, For any given crack length ratio E/h, the nondimensional stress intensity factor increases, passes through a maximum, and then decreases as a function t*. As would be expected, the maximum stress intensity factor is of greatest magnitude for a very small edge crack and decreases with increasing crack length l/h. The maximum stress intensity factor attained during the thermal transient is the quantity of most interest. Figure 19 is an interesting comparison of the maximum stress intensity factors as a function of crack length for the edge crack problem. The comparison is between the cylinder without cladding and cylinders with two different cladding thicknesses. Noting that the stress intensity factor is normalized with respect to the coefficient of thermal expansion for the base material, that is, Ed, it can be seen that the maximum stress intensity factors for cracks in the cladding are much higher than those in the unclad cylinder. An increase in cladding thickness increases the maximum stress intensity factor attained during thermal shock conditions for edge cracks contained in the cladding. Furthermore, as the edge crack penetrates a short distance into the base material the maximum stress intensity factors are still higher than in the situation when
H. F. NIED
130
C2
Fig. 18. Stress intensity factor k(g) as a function of nondimensional time t * for various length edge cracks. (KJK, = 3, D?/D, = 3, a,/a, = 314, a/c = 0.9, (h ~ a)/h = 0.04, h = c -a, t* = D,t/c2.)
I 0.2
WITHOUT
CLADDING
a = 0.9 C
1 0
I
0
0.1
I
I
I
I
0.2
0.3
0.4
0.5
I
0.6
J
(1.7
P/h Fig. 19. Comparison of maximum without cladding. (With cladding: K>/K,
=
stress intensity factors for edge cracked cylinders KJK, = 3, D,/D, = 3, a,/a, = 3/4, a/c = 0.9. Without 1, D,/D, = t, ~$/a, = 1, U/C =0.9.)
with and cladding:
cladding does not exist. It is only with longer edge cracks that the beneficial effect of having a clad can be clearly seen. The transient behavior for an embedded crack which is directly underneath the cladding (i.e. f = b) is shown in Fig. 20. Again, the stress intensity factors in this plot correspond to the thermal history depicted in Figs. 68. In this situation the two crack tips initially have a negative stress intensity factor, and then independently reach maximum values at different times. Of course, the negative stress intensity factors only have meaning when superposed with a positive stress intensity factor caused by some external loading; for example, uniaxial tensile loading. It can be seen that
Thermal shock in a arcumferentially
cracked
hollow cylinder with cladding
131
a=
0.9
C
0
I 10-e
10-5
0.04
iLp
=
9
= 0.16
I
10-4
10-3
10-2
4t -2
Fig. 20. Stress embedded
factors k(f) and k(g) as a function of nondimensional time $* for an (IQ/K, = 3, D,/D, = 3, cc&, = 314, a/c = 0.9, (b - a)/h = 0.04, f = b, h = c -a. t* = D,r/2.)
intensity
crack.
the crack tip touching the clad always has a higher maximum stress intensity factor. Again, it is also apparent that the most useful quantity from the point of view of fracture is the maximum stress intensity factor for any given crack length. Figure 21 shows these maximum stress intensity factors for embedded cracks, where each point on the curve is obtained by iteration to find when the maximum occurs. There is little variation in the maxims stress intensity factor as a function of
I
I
I
I
0.2 0.3 0.4 0.5 (9 - bYh Fig. 21. Maximum stress intensity factors for embedded cracks of various crack lengths. (&, &fo, = 3, a,/~, = 314, a/c = 0.9, (b -a)/h = 0.04, f = 6, h = c -a.)
= 3,
H. F. NIED
132
crack length for the inner crack tip; however, it can be seen that the maximum stress intensity factor for the outer crack tip drops significantly with increasing crack length. Thus, the propensity for crack propagation is inwards, given that the two materials have the same crack growth rate characteristics. Tables 1-4 give the maximum stress intensity factors and the nondimensional time at which the maximum occurs, for edge cracks in cylinders with various cladding thickness to wall thickness ratios. As would be expected the time at which the maximum stress intensity factor occurs increases with increasing clad thickness and crack length. In these tables the constant gr is given by CT= - (Ec@,)/(l - v). Tables 5-8 contain the maximum stress intensity factors and the nondimensional time for embedded cracks. It can clearly be seen that there is a definite time difference between the maximum values at the two crack tips. Furthermore, for large embedded cracks in cylinders with thick cladding (Table 7) the stress intensity factor may never become positive at either crack tip. If the thermal shock is of sufficient magnitude to cause yielding through the thickness of the clad, the effect this has on the stress intensity factor for an embedded crack can be analyzed using a simple plastic strip model. If the crack is located just underneath the cladding and it is assumed that the entire clad has yielded in the plane which contains the crack, the maximum stress intensity factor at r = g can be determined from the solution of equation (65) withf = a and the stress terms in boundary condition (64) replaced by
a,,(r,O,t)=
-ozT(r.c)+aF,
((I
c:,(r,
-
(h
(75)
Table
Table
1. Maximum
2.
Maximum
0, t) =
azT(r,
t),
stress intensity factors and the nondimensional time at which they occur for internal edge cracks subjected to transient thermal stresses
0 041
0.00014
0.05
0.00014
0 Oh
0 OOOl5
0.07
0 00017
0 08
0 000 IX
0 09
0 rlO91Y
0 IO
0.0001Y
0 20
0.00027
030
0 00033
0 40
oooni:
0.50
0 00039
stress intensity factors and the nondimensional time at which they occur edge cracks subjected to transient thermal stresses b[=
-(En,H,)/(l-u).O/L
=0.9,(b-o)/h=O.l.
h=[~o.I=fi~a.KZ/~,=3.D2/D,=3,U2/a,=3/4)
0 101
0.00027
0 II
0.00027
0.5662
0 1s
0.0003 I
04740
0 20
0 00035
04151
0.25
000037
03732
030
0.00040
03461
0.35
0.00042
03222
0.40
0.00044
0.3034
0.45
0.00045
02896
0.50
0.00046
02122
O.bO90
for internal
Thermal Table 3. Maximum
shock in a circumferentially
cracked
hollow
cylinder
with cladding
stress intensity factors and the nondimensional time at which they occur for internal edge cracks subjected to transient thermal stresses (~r=-(Ea,8,)/(1-v).~l~=09.(h-~)/h=025, h=~-o./=~-a.~~/~,=3.D~/D,=3.a~lu,=3/4)
Table 4. Maximum
0.251
0
00050
0 4760
0.26
0 00050
0 456h
030
o.ono50
0 409'
035
0 00052
0 3720
0.40
0 00053
0 3449
045
0 00054
03224
0.50
0 00056
0 3062
stress intensity factors and the nondimensional time at which they occur for internal edge cracks subjected to transient thermal stresses (or= -(Ea,f?,,)l(I-,,),a/< =o 7,~b-a),!/i=O04. h=~~-o./=y--o.~~l~,=3.D~/Di=3.~~~~~~;=3i41 I h
T-o17-
0041 0.05 006
no0141 0 00152
T
007
Table 5. Maximum
h (,c) <-_ T--j_ ollo134
~
CT,%I --0 73VJ (
06612 0 hlO9
0 OOlhO
0 '751
0 08
0 00173
0 54Xh
0.09
0 OOIS?
05234
0.10
0 no190
0 5018
0.20
0 00265
0 3752
030
00032l
0 3000
040
000361
02410
0.50
00038R
0.1944
stress intensity factors and the nondimensional time at which they occur for embedded cracks subjected to transient thermal stresses
g-b h
k(b)
D,l
-7
CT*
m 0.001
0.00018
0.3899
0.3908
0.01
0.00019
0.3798
0.3883
0.02
0.00020
0.3701
0.3873
0.03
0.0002 1 0.00022
0.3613
0.3874
0.3529
0.3881
0.00023 0.00024
0.3448
03891
0.3369
0.3904
0.00033 0.0004 1
0.2622
0.4006
0.1940
0.4009
0.1309
0.3919
0.46
0.00047 0.00054
0.0728
0.3745
0.56
0.00066
0.0200
0.3494
0.04 0.05 0.06 0.16 0.26 0.36
133
H. F. NIED
134
Table 6. Maximum stress intensity factors and the nondimensional time at which cracks subjected to transient thermal stresses fur = f-b.
they occur
for embedded
-(~a,B,)/(~-~),(1/~-0.9.(b-o)/h=O.l. h-c-a. ~2/~,=3r D>/D,-3.0J/u,-3/4).
e
-Dir
h
c!
m
-
0.001
0.00036
0.2240
0.00036
0.01
0.00031
0.2172
0.00037
0.05
0.00041
0.1897
0.00038
0.10
0.00045
0.1589
0.00039
0.15
0.00049
0.1303
0.00041
0.20
0.00053
0.1030
0.00042
0.25
0.00057
0.0771
0.00044
0.30
0.00061
0.0523
0.00045
0.35
0.00067
0.0288
0.00046
0.40
0.00074
0.0066
0.00048
7
k(b)
mT*
I0.2245 0.2224 0.2150
0.2080 0.2017
0.1951 0.1880 0.1800
0.1713
0.1618
Table 7. Maximum stress intensity factors and the nondimensional time at which they occur for embedded cracks subjected to transient thermal stresses (UT f-b,
-(~o,8,)/(1-~).(1/~-0.9.(b-o)/h-0.25. h=c-a.~~Iu,=3,D~lD,-3,a~/a~-3/4~. k(g)
DI! 7
g-b "
k(b)
_,q
%
_jq
0.001
0.00070
0.0111
0.00070
0.0114
0.01
0.00071
0.0073
0.00070
0.0101
0.05
0.00076
-0.0088’
0.00072
0.0046
0.10
0.00084
-0.0277”
0.00074
-0.0022’
0.15
0.00096
-0.0453*
0.00076
-0.0090’
l
Local maximum
(k - 0
at
f- 0)
Table 8. Maximum stress intensity factors and the nondimensional time at which they occur for embedded cracks subjected to transient thermal stresses -(Ea,B,)l(l-v).olc-0.7,(b-o)lh*0.04, h-c-o. K~/K,-~, D>/D,-3,cq/n,-314).
(CJ f-b.
k(g)
-DlI
g-b h
c&F
2
k(b)
-DlI
c*
cTq-
-
B
-
0.001
0.00173
0.3995
0.00172
0.4003
0.01
0.00182
0.3892
0.00175
0.3980
0.02
0.00193
0.3793
0.00179
0.3971
0.03
0.00203
0.3702
0.00183
0.3973
0.04
0.00214
0.3615
0.00186
0.3980
0.05
0.00223
0.3530
0.00190
0.3990
0.06
0.00233
0.3447
0.00194
0.4002
0.16
0.00316
0.2656
0.0023 1
0.4078
0.26
0.00384
0.1942
0.00264
0.4043
0.36
0.00442
0.1308
0.00293
0.3923
0.46
0.00501
0.0748
0.00317
-
-
0.3737
Thermal shock
in a circumferentiallycracked hollow cylinder with cladding
I35
where crFis the plastic flow stress. The maximum stress intensity factor at the outer crack tip Y= g. normalized with respect to the crack half-length (g - b)/2, is then given by (76)
where the values of k? and kr are given in Tables 9-11, for (b - a)/h ratios of 0.04 and 0.10, when a/c = 0.9 and for (b - a)/h = 0.04, when a/c = 0.7. In effect, kf is the maximum normalized stress intensity factor for an edge crack subjected to - o;‘(r, f) (a < r CCg) and kf is the stress intensity factor for an edge crack subjected to cl” (a < r < b). As can be seen, if or- is not significantly larger than g F, a dramatic reduction in the maximum stress intensity factor is possible. Of course this is not always the case and for a very large value of cr., yielding in the cladding will result in a much higher stress intensity factor at Y= g than in the case where no yielding (Tables 5, 6 and 8) has occurred. It is also seen that the effect of clad yielding on the stress intensity factors for longer cracks is greatly diminished.
Table 9. Plastic strip yield model: parameters for determination of maximum stress intensity factor for an embedded crack
-- ..-
-...
._._-r=
6.68i5 20933 i
1 so31 1 2521 11109 1011: 0 Y3i8 0 6436 (J 5363 0 4665 04135
Table IO. Plastic strip yield model: parameters for determination of maximum stress intensity factor for an embedded crack ~~~=-~Ea,H,i~~1-1~~.Li/~=09.~h-oJ;!7=01. h=c
-Ii,KJK,=3,
n,/D,=3.u>jnr=3/4~ v-.-p-
----‘~~----s-_h I1
h",
X",
~___~ ~__
~-__-
0001
8 hi49
001
2 hS59
-4
0.05
1.1612
-I
5353
0.10
0 xi01
-1
0206
-15
+
0.15
0 6814
0.20
u99;
j
Ii73 1925
~~.~f~~
0.25
EFM
Vol. 20, No. I -I
0.30
0.4955
-0
5872
0.35
0 4643
-0
59.18
0.40
0 4203
-0
55hX
I
136
H. F. NIED Table
11. Plastic
strip yield model:
parameters for determination an embedded crack
of maximum
stress intensity
factor
for
(,r,=-(Lrr,H,)~(l~I~).u:i=O’.(h-~uI/h=O.oJ. /i=~~u.K1~“,~=3,1~~!1~,=3.ir~~~r,~l.ll~I .c_h A_=
~___$
~~ _~
mmm~~7~~~~ :=:~_:__..[ _~~ ;”
-- ~-~ ‘:____ 7902
0 001
hh95?
-8
0 01
2 0909
-2
2168
002
I4963
-I
3669 0161
003
~
12423
-I
0.04
I
IO972
--OX382
0 06
1
09162
0.16
I
005
0 9950
-0 ~
05932
7036
-05x51 -02635
0.26
04558
-0
036
03602
-01497
0 46
I
0 2866
i
-0
IYlh
1280
CONCLUSION By obtaining the transient thermal stresses for the bimaterial cylinder in closed form and using integral equation analytic techniques, a highly accurate parametric study was conducted which examined the diffusive transient characteristics of the thermal shock problem for a circumferentially cracked cylinder with cladding. Examination of Fig. 18 clearly indicates the transient nature of any fracture phenomena which would occur in a thermal shock situation. It is not difficult to envision a thermal shock event during which a small crack will reach a critical stress intensity factor, rapidly advance, and then stop due to the decrease in stress intensity factor with increasing crack length. As the tensile thermal stress field “diffuses” radially outwards, the stress intensity factor would then build up again, attain a critical value, advance, and halt. This start-stop phenomenon could continue during a transient thermal event as long as the maximum stress intensity factor, attainable for any given crack length, is greater than some critical value of the stress intensity factor necessary for the crack to advance at a rate faster than the “diffusing” tensile thermal stress field. Of particular interest are the maximum stress intensity factors and the times at which these maximums occur for various cladding thicknesses and cylinder wall thicknesses. An interesting comparison is made by examining Tables 1 and 5. For example, an edge crack with l/h = 0.2 and an embedded crack with length (g - b)/h = 0.16 both have the outermost crack tip at the same radial location, yet the maximum stress intensity factor is attained sooner for the edge crack than for the embedded crack. In actual pressure vessels these nondimensional times at which the maximum stress intensity factor is reached translate into real times greater than 10 min. In Fig. 19 we clearly see that for an edge crack, when the crack is just through the cladding the maximum stress intensity factor is higher than if there had been no cladding at all. It is when we have deeper crack penetration that cladding causes a reduction in the stress intensity factor. This should not be interpreted as meaning that it is better not to have the cladding. The cladding serves a very important function in protecting the base metal from severe corrosion damage. Thus, even though small edge cracks have a higher maximum stress intensity factor when cladding is present, these elevated values in general would not be high enough to negate the beneficial effects of having a clad. It can be seen that thicker cladding is not necessarily better, at least from a fracture mechanics point of view, which does not consider the possibility of circumferential clad buckling. Thus, if the cladding is not to be breached by an edge crack, care should be taken to ensure that the maximum stress intensity factors remain below acceptable levels for cases where the cladding is thick. The effect of clad yielding on the stress intensity factor for the crack tip embedded in the base material was also investigated, and it was shown, using a plastic strip model, that in certain situations clad yielding will cause a reduction in the stress intensity factor. In long embedded cracks, when the clad yielded, no reduction in the maximum stress intensity factor was seen. The stress intensity factors in these cases are a good deal higher than if no yielding had occurred in the cladding. Of course, if yielding of the clad is the only way to prevent breaching of the clad,
Thermal
shock
in a circumferentially
cracked
hollow
cylinder
with cladding
137
for example due to brittle fracture, the increase in the stress intensity factor at r = g may be quite acceptable. Acknowledgements-A portion of this work was supported by NSF under Grant ENG 78-09737 and by NASA Langley under Grant NGR39-007-011 while the author was at Lehigh University. The author is also greatly indebted to Professor F. Erdogan for his many helpful suggestions.
REFERENCES M. Norris and C. F. Shih, Fracture mechanics related problems in the nuclear power industry. Proc. 9th U.S. National Congress of Applied Mechanics, ASME, p. 143 (1982). PI F. Erdogan, G. D. Gupta and T. S. Cook, Numerical solution of singular integral equations. Mechanics of Fracture 1. Methods of Analysis and Solutions of Crack Problems (Edited by G. C. Sih), pp. 368-425. Noordhoff, Groningen, The Netherlands (1973). and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical [31W. Magnus, F. Oberhettinger Physics, Die Grundlehren der Mathematischen Wissenschaften, Vol. 52. Springer-Verlag, New York (1966). [41H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids. Oxford University Press (1950). [SlD. Burgreen, Elements of Thermnl Stress Analysis. C.P. Press, Jamaica, New York (1971). and I. Stegun, Handbook of Mathematical Functions. Dover, New York (1965). (61M. Abramowitz crack. Znt. [71H. F. Nied and F. Erdogan, The elasticity problem for a thick-walled cylinder containing a circumferential Z. Fracture 22, 277-301 (1983). cylinder with an axisymmetric internal or edge crack. J. Appt. Mech. 45, PI R. Erdol and F. Erdogan, A thick-walled Trans. ASME, 281-286 (1978). Singular Integral Equations. Noordhoff, Groningen, The Netherlands (1953). [91N. I. Muskhelishvili, [lOIG. D. Gupta and F. Erdogan, The problem of edge cracks in an infinite strip. J. Appl. Mech., 41, Trans. ASME, 1001-1006 (1974). value problems in mechanics. Mechanics Today (Edited by S. Nemat-Nasser), Vol. 4, [I11 F. Erdogan, Mixed boundary pp. l-86 (1978). Transient thermal stress problem for a circumferentially cracked hollow cylinder. J. [I21H. F. Nied and F. Erdogan, Thermal Stresses 6, l-14 (1983).
111D.
(Received 29 April 1983)