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Mode II stress intensity factors for a circular ring or a hollow cylinder with a radial crack
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PII:S~O308-0161(97)00021-5
ELSEVIER
Vex & Piping 72 (1997) 149-156 0 1997 Elsevier Science Limited in Northern Ireland. All rights reserved 030%0161/97/$17.00
Mdde II stress intensity factors for a circular ring or a hollow cylinder with a radial crack A. Y. T. Leung School of Engineering,
The University of Manchester,
Manchester
Ml3
9PL,
UK
J. D. Hu Department
of
Civil and Structural
Engineiering,
The University
of
Hong Kong,
Pokfulam
Road, Hong Kong
(Received 6 March 1997; accepted 3 April 1997) The two-dimensional elasticity problem of a circular ring or a hollow cylinder with a radial crack and subjected to crack surface shear stress is considered. Using dislocation pile-up and singular integral equation techniques, Mode II stress intensity factors are obtained. The crack may be an internal edge crack, an external edge crack, or an embedded crack. Numerical examples are given, for various crack geometries and radius ratios. The numerical results are obtained for a uniform cra.ck surfaceshear stressz,,. 0 1997 Elsevier Science Ltd. using dislocation pile-up and singular integral equation techniques. The crack in the question may be an edge crack or an embedded crack with various loading conditions, i.e. a uniform crack surface pressure, a uniform inside pressure and a rotating disk. In all those studies, however, only Mode I stress intensity factors are considered. In engineering, we often need to consider the problems of Mode II crack and Mixed mode crack. In this paper, we consider the two-dimensional problem of a circular ring or a hollow cylinder with a radial crack and subjected to crack surface shear stress. The problem of interest is the Mode II crack in which the self-equilibrating crack surface shear stresses are the only external loads. The geometry and loading conditions of the problem under consideration are shown in Fig. 1. The approach for Mode II crack employed here is similar to that of Delale and Erdogan’ for Mode I crack, i.e. the corresponding dislocation solution for an infinite plane with a dislocation is sought first, then by superposition and integration, the crack problem is formulated in terms of a singular integral equation. The stress intensity factors of the crack are related to the solution of the singular integral equation and can be evaluated accurately by Gaussian quadrature. Numerical results of the stress intensity factors are
1 INTRODUCTION
The two-dimensional elasticity problems of al circular ring or a hollow cylinder with a radial crack, as typical problems in Linear Elasticity Fracture Mlechanics, have been considered by a number of investigators’” to obtain the stress intensity factors for various crack geometries. The problem of a radial crack located on its inner boundary and subjected to uniform tensile tractions on the outer boundary in a circular ring was considered by Bowie and Freese.’ The problem of multiple radial cracks on the inner or outer boundary of a circular ring was studied by Tracy.’ In R.efs [l, 21 the analytical studies were based on the mappingcollocation technique. The problem of a hollow cylinder containing an external edge crack was considered by Emery and Segedin3 for various loading conditions. A numerical technique similar to finite difference approximation was used for the solution. The problem of a thick cylinder with one or two, internal or external radial edge cracks was considered by Andrasic and Parker.4 The numerical resuhs of the stress intensity factors were obtained by the use of crack weight functions which were generated via the accurate modified mapping collocation metlhod. The problem of a hollow cylinder or a circular ring with a radial crack was considered by Delale and Erdogan’ 149
150
A. Y. T. Leung, .I. D. Hu
tion along 0 = 0 line and of the untracked ring, respectively.
concentric
2.1 Infinite plane with an embedded edge dislocation
Consider an infinite plane with an edge dislocation with Burger’s vector b, = g at Y = t, 8 = 0. We solve the plane elasticity problem by assuming that
(Tee,=o, o~rsw,
i [url(r, 0+) - u,,(r, O-)] = gS(r - t),
Fig. 1. The geometry of the problem.
presented in tabular and graphical crack geometries.
forms for various
Referring to Wang, function of the problem A@7 0) = - .(I”+
2 DISLOCATION
0)
=
aijl(r,
e=n,
(4
0 S r =Sa,
(3)
we express the Airy stress as
K) gr sin 6 ln(r* +
t2-
2rt cos Q),
SOLUTION
(4)
The geometry of the corresponding dislocation problem is shown in Fig. 2. Instead of the crack, an edge dislocation with Burger’s vector (b,, 0,O) is embedded in the circular ring at r = t, 0 = 0. In order to facilitate the application of the boundary conditions, the dislocation problem is expressed in plane polar coordinates. Due to the existence of the dislocation in the ring, the stress function for the ring is constructed as the sum of the general elasticity solution for the circular ring and a cracked or dislocation solution for an infinite plane. The stress state in the cracked ring problem can be obtained by aij(r,
e=o,
0)
+
aijZ.(r2
0)~
(6
i =
r,
01,
where p is the shear modulus, strain and
K
= 3-4~ for plane
3-v
K = -1
+ v for generalized
plane stress, Y
being the Poisson’s ratio. The stresses can be obtained from the Airy stress function (4) as follows
ar,l(r, ,,=1!!h+G% r dr
r2
882 r sin 8
=--
r2+t2-2rtcos0
+ 2t sin 0(r cos 8 - t)(r - t cos f3) ) (5) (r” + t2 - 2rt cos 0)”
1
(1)
where gijl and ajj2 (i, j = r, 6) are stresses of an infinite plane containing an embedded edge disloca-
2%
=
?‘C(l f
-
=
sin 0(2t cos 8 - r) K)
Y2 + t2 - 2rt
COs 6
2rt2 sin3 8 1 (r2 + tZ- 2rt cos 0)’ ’
2Pug 7c(l+~)
r cos 8 - t [ r2+t2-2rtcosf3
- 2t2 sir? f3(r cos 8 - t) (r* + t2 - 2rt cos 0)” ’
1
Fig. 2. The geometry of the dislocationproblem.
In order to combine
(6)
(7)
the infinite plane solution with
Mod,e ZZ stress intensity factors
the circular ring solution, we now express the stresses in eqns (5,7) along the circles r = a and Y = b in the plane in terms of the following Fourier series
D,(t)=
1.51 2b2-t2 -7,
D,(t) = - (n +:l”-’
+ s,
(n 2 2)
(19a,b)
2.2 General ring solution
Due to symmetry, the stresses for general solution should satisfy the following conditions
ring
(ZOa-c) where the Fourier coefficients are determined
Aa
=
X(1
+
K>
=
B,(t) = Co(t) =
41+K)2
-
7$1+K)2
noi
I% +
K)
I-%
1
-
=
Dn(t) =
741+K)2
iug
c,l(a, 0) de, zrel(a, f3)cos no do,
(12a,b) vrrr2(r, 6) = - $! + 2d Ir sin 0 - 2 [a,tz(n - l)r”-’
=
2
urrI (a, @sin n0 de,
(13)
~s,(b,
0)
noI
z,.~~(b, 8)cos ?zOdQ,
(14a,b)
L(t)=
-t”-‘+
na”-’
(22)
+ b,n(n + 1)r” - c,jz(n + l)reflP2 - d,n(n - l)Y]cos
~0,
(23)
where ao, cl, d,, a,, b,,, c, and d, are all arbitrary constants. 2.3 Boundary conditions
= -s+$,
-;,
no,
(15)
>
B,(t)=
(21)
f b,(n + l)(n + 2)r” + c,y1(rz + l)rFfle2
+ d,(n - l)(n - 2)r-“Isin
71
~,.~,(b, B)sin n8 de, xc,I
A,(t) = 0,
A,(t)
n8,
do,
in which r,,, and a,.,, are determined by eqns (5) and (7). With the aid of tables,’ integrals in eqns (12-15) can be evaluated in closed form and the results are obtained as follows
A,(t) =;
+ b,(n + l)(n - 2)r” + c,,n(n + l)rCm2 + d,(n - l)(n + 2)r-“Isin
JE
IT -
t-q
After taking regularity and symmetry conditions into consideration, and by referring to Michell’s general stress functions,’ we can formulate the stresses in the ring as follows
no i
.7C(1+K)2 CJt)
IT
=ai
f-%
7C(l
-
x
no j’
I-%
A,(t)
1
by
(n 2 21,
(16a-c)
(n - 2)an
tn+l
1 (n 3%
(170)
After obtaining the basic form of the solutions for the infinite plane with a dislocation and for the ring, we express the stresses in the circular ring with a dislocation as the sum of the two solutions [eqns (1) (5-7) and (21-23)]. Th e combined stresses must then satisfy the following boundary conditions g,,l(a, 0) + flrr2(al 6) = 0,
Co(t) = 0,
h(a,
2b2 - t2 C,(t) =--p-j
~,,,(b> ‘3 + ~rrz(b> 69 = 0,
4 + h(a,
Q> = 0,
(24a-d)
~rm(b, 0) + c,z(b, 6) = 0, (n z=21,
(Ha-c)
where v~,.~ and rre, are given by eqns (S-15) and u,.,~ and r,.82 are given by eqns (21,23).
152
A. Y. T. Leung, J. D. Hu
Defining the following transformations I-%
b, = -
an = - n(l + K) %I,
” x(1
t-G
d, = -
cn = - n(l + K) Yn,
Pn +
K)
+
K)
” X(1
By integrating the dislocation solution along c 6 r 6 d, 0 = 0 and applying the boundary condition (31), we obtain a singular integral equation as follows (25a-d) [$$dt
S,,
and substituting eqns (8-11) and eqns (21, 23) into eqn (24), we obtain the following equations
+ [k(r,
a,n(n
=A,,
1
a=B
1,
- l)aHp2 + p,n(n
%= b2 2% b3-
Co,
(26a,b)
2S1b = Cl,
- l)a”-’
-$$+,S,b=D,,
(2&b)
+ l)an
(~,n(n - l)bnp2 + p,n(n + l)b-‘-2
a,n(n - l)b”-‘+
+ S,(n + 2)(n - l)a-”
= -B,,
+ y,n(n + l)b-‘-2
For an internal that
crack (i.e. a
= -C,,
+ S&r + 2)(n - l)b-”
= -D,,
where A,, B,, C, and D, are functions of t given by eqns (16-19). From eqns (29), it is found that there are equal numbers of equations and unknown constants. Thus, these unknown constants can be uniquely determined. But in eqns (27, 28), there are four equations and only two unknowns. Equations (27,28) may have a solution only if two of them are arbitrary. If C,+D,=O,
G(t) l?(t) = -\/(t - c)(d - t)’
k2(c) = !ii V@Tj
(c < t < d),
(35)
zre(r, 0)
= -i?k 1 + Klim -g(r) 2(r r-c
Wkb)
(36) k,(d)
= Iii m zz
INTENSITY
(34)
where G(t) is a bounded function at the crack tips.” The Mode II stress intensity factors of the crack can be calculated according to the conventional definitions
then eqns (27) and eqns (28) are identical. Ignoring eqns (28), we uniquely determine the y1 and 6, from eqns (27). After determining all these constants, the stresses for the dislocation problem in Fig. 2 may be obtained by eqns (l), (5-7) and (21-23).
3 STRESS
g(t) dt = 0,
In this case, the index of the singular integral equation (32) is +l. The function g(t) has a square root singularity at the crack tips and we may write
(29a-d)
A,+B,=O,
crack
ic
P,(n - 2)(n + 1)b”
(33)
and CY,, pn, yn and S, (n = 1, 2, . . .) are known functions of t which are determined from eqns (26-29).
f 1)b” - S,n(n - l)b-”
+ l)rn
- ynrz(n + l)rmnp2 - S,n(n - l)r-“I},
3.1 Internal
+ /3,(n - 2)(n + l)a”
+ ynn(n + l)a-“-2 - y,n(n
- 2S,r - jj [(~,n(n - l)rnp2
+ &n(n
Pa,b)
- ynn(n + l)amnp2 - 6,n(n - l)a-n = -A,, a,n(n
(32)
2
Ao,
a2
-%+2S a3
K)g(r),
where k(r, t) = d (5
a0 -=
&-2&a a3
t>g(t) dt = x(::
%(r,
0)
-&lirJ:V@?jg(r) r--f
FACTORS (37)
The dislocation solution obtained above may be used to tackle the crack problem in Fig. 1 using dislocation pile-up and singular integral equation techniques. For the problem under discussion, we write the remaining boundary condition as Ge*(r, 0) + Gn(r, 0) = 4(r), where q(r) is the crack surface shear stress.
(31)
Introducing d-c
r=-s+ 2
the following transformations d+c
-
2 ’
whenc
-l
(380) t=-
d-c
2
Z+d+C
2 ’
whenc
-l
153
Mode II stress intensity factors and substituting obtain
eqns (38) into
eqns (32, 34), we
and substituting
eqns (46) into eqn (32), we obtain
’ g’(r) z--s dz + (d - c>1’0 K(s, %(r> dr
K(s, Qg,(z> dz
= Ir(l
+
K)
2P
q(s), (-1
K)
2P
(0
q(s),
where
g1(r) = c,(z> CT>
g,(z) dz = 0,
(O
a(4 =s(t), q(s)=q(r), K(s,r>=k(r,t),
where a(r) = s(t), q(s) = q(r), K(s, 4 = k(r, t> Accordingly,
n(l+
we may write
Referring to Ref. [lo], and discretizing similar manner, we have
eqn (47) in a
2 ~G,(i)[&+~K rr)] I r i=l Ii &
GI(T~)[&+Cd- C)K(sjj t)]
Using the Gauss-Chebyshev integration formula,’ eqns (39) can be discretized into the arlgebraic equations
l’L(l
=
+
K)
2~
q(Sj)
)
(I=
2,
lr
”
7 n),
(4g)
where
~,n+,(~;) = 0,
X(1
=
+
ds,)
K)
,
2P
(j=
1,2,. . .,I2 -1)
(41)
b&j)
=
T,(q)=O,
(i=1,2
u,~,(sj)=o,
,...,
(42)
(j=l,&...,n-l),
G,(-1) v?
=+L/T7----K
the
a=c
condition required solution. Referring
k,(d)
= -+&V%%
edge
crack
’
G (1) v5’
(43)
to Ref. [lo), we write the function g(t) as G(t) ___ g(r) = qJ-q
(a=c
3.3 External
Y= (d - c)s -t c, t = (d - c)z + c,
whenc
O
(52)
(a
Table 1. Mode II stress intensity factors k,(c) and k,(d) for an embedded crack in a circular ring subjected to uniform crack surface shear stress q(r) = -T,,, (d c)l(b -a) = o-5, (c - a)/@ - d) = 1
a b-a
_ k,(c) d(d - c)/2
k,(d) zov/(d- c)L’
0.05 0.10
1.102403 1.112981 1.143672 l-172981 1.193131 1.200979 1.202374 1.202751 1.202869
1.140259 1.143946 1.158068 1.175292 1.190321 1.198131 1.200159 1.200981 1.201402
(45)
0.25 0.50
(46)
2.0 3.0 4.0 5.0
transformations
edge crack
G(t) s(t) = __ $-q
1.0
the following
l+K
In the case of an external edge crack (i.e. a < c < d = b), like the case of an internal edge crack, the displacement single-valuedness condition (34) does not hold true and also is not required as an additional consition for a unique solution. Referring to Ref. [lo], we write the function g(t) as
(44)
case of an internal edge cra\ck (i.e. the displacement single-valuedness (34) does not hold true and also is not as an additional condition for a unique
Introducing
0)
n),
where Tn(x) and U,-,(x) are Chebyshev polynomials of first and second kinds, respectively. The Mode II stress intensity factors of the crack can be calculated by
In
z&,
-*-G,(l),
with ti and s, as
3.2 Internal
(50)
The Mode II stress intensity factors of the crack can be calculated by k,(d) = lipidm
k*(c)
= 0,
154
A. Y. T. Leung, J. D. Hu
I
1.18
T
a/b 1.1 1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fig. 3. Mode II stressintensity factors for an internal crack in a ring loaded by a uniform crack surfaceshearstress-7,.
Introducing the following transformations r = (d - c)s + d, t = (d - c)z + d,
whenc
where
-l
G,(z) a(~> = CT’
Referring to Ref. [lo], and discretizing similar manner, we have
eqns (53) into eqn (32), we obtain
O---ddz+(d-c) a(4
I-1 K(s, %1(r) =x(1 + K) 2p 4(s),
dr i=, Ii &
t-1,
Wd[
&
(54)
Table 2. The normalized Mode II stress intensity factors c in a circular ring containing an internal wmo-) edge crack (a = c
l/3 1.114205 1.137115 1.184718 1.253483 1.347864 1.480665 1.680663 2.023717 2.813413
a/@ -a) 112 1.113084 1.123781 1.159126 1.219228 1.308061 1.437068 1.633779 1.972012 2.749417
1 1.112207 1.119115 1.141009 1.187572 1.265209 1.385657 1.575502 1.906569 2.669403
+ Cd - C)K(% $1
=
0.1 0.2 0.3 0.4 O-5 0.6 0.7 0.8 0.9
eqn (54) in a
0
I-, z-s
d-a b-a
(55)
g,(z) = g(t), 46) = q(r), K(s, 4 = k(r, t>,
(53)
and substituting
(-l
4Csj)
+
cl]
K)
,
2F
(i = L2, . . . , n>,
(56)
where T2rz+l(-zi)
=
0,
~2n(-sj)
=
0,
(57)
The Mode II stress intensity factors of the crack can be calculated by
2 1~117818 1.136173 1.180698 1.252182 1.366051 1.551078 1.878181 2.635523
=-+GG,(-I), K
4 RESULTS
AND
(58)
DISCUSSIONS
From eqs (41), (49) and (56), we solve numerically the singular integral equation in eqn (32). Thus Mode II
Mode II stress intensity factors
2 t
I 0.5 $
(d-a)/@-a) 0-l
I 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 4. Mode II stress intensity factors for an internal edge crack in a ring loaded by a uniform crack surface shear stress -r,.
stress intensity factors for a circular ring with a radial internal crack, internal edge crack or external edge crack can be obtained by equations (43,44), (51) or (58), respectively. One loading condition, i.e. uniform crack surface shear stress q(r) = -z, (c < r < d), is used in the examples. For various internal crack geometries, normalized Mode II stress intensity factors ;are given in Table 1 and shown in Fig. 3. For various internal edge crack geometries, normalized Mode II stress Table 3. The nonualized Mode II stress intensity factors kz(c)/(70G) in a circular ring containing an external edge crack (a Cc < b = 6) and subjected to uniform crack surface shear stress q(r) = -TV b-c b-a
0.1 0.2 o-3
0”:;
al(b -a) l/3
1.112674 1.135787 1.155528 l-179715 1.214374
0.6 0.7
1.268464
0.8
1.549271 2-022986
0.9
1.362862
112 l-109875 1.132885 1.152163 1.178028 1.217687 1.283583 l-400176 1.628774 2.196203
1
2
1.101673 1.125463 1.145539 1.174962
1.~085728 I.115518
l--136959
3:169969
1.224152
I.226046
1.307923 1.456356 1.739809
1.321528
2.426348
;!-533217
I.486588
Il.795338
intensity factors are given in Table 2 and displayed in Fig. 4. For various external edge crack geometries, normalized Mode II stress intensity factors are given in Table 3 and displayed in Fig. 5. For the internal crack problem, here the crack length is fixed at half the wall thickness (b - a) and the ratio of a/b is varied. From Table 1 and Fig. 3, it can be seen that k,(c) at the inner crack tip Y = c is less than k,(d) at the outer crack tip r = d when the ratio of a/b is small. Stress intensity factors k*(c) and k,(d) raise with the ratio of a/b increasing. The k2(c) increases faster than the k,(d). Finally, k,(c) and k,(d) tend to the same constant. For the internal edge crack and external edge crack problems, we choose that the ratio of the crack length (d - c) to the wall thickness (b - a) and the ratio of the inner radius a to the wall thickness (b - a) are varied. From Tables 2 and 3, Figs 4 and 5, we note that as (d - c)/(b - a) increases, the Mode II stress intensity factors k2(c) and k,(d) increase. Increasing a/(b -a) causes an increase in k*(c), but increasing a/(b - c) causes a decrease in k,(d). The results express that the thinner wall circular ring with the internal edge crack can assume the bigger crack surface shear stress, but the thinner wall circualr ring with the external edge crack can assume the smaller crack surface shear stress.
0.9
156
A. Y. T. Leung, J. D. Hu 3-
k2 (4 r,Jd-c 2.5 +
2i
I I 1.5 t
IT I 0.5
t
(b-NW 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 5. Mode II stressintensity factors for an external edge crack in a ring loadedby a uniform crack surfaceshearstress-r,.
REFERENCES
5. Delale, F. and Erdogan, F., Stressintensity factors in a hollow cylinder containing a radial crack. Int. Journal of 1982,20,2.5-265. 6. Wang, D., Fracture Mechanics, 1. Harbin Institute of Technology Press,1989. 7. Ryshik, I. M. and Gradstein, I. S., Table of Series, Products and Integrals. Academic Press, New York, 1965. 8. Little, R. W., Elasticity. Prentice Hall, 1973. 9. Erdogan, F. and Gupta, G. D., On the numerical solution of singular integral equations.Q. Appl. Math., 1972,29,525-534. 10. Gupta, G. D. and Erdogan, F., The problem of edge cracksin an infinite strip. ASME J. Appt. Mech., 1974, 41,1001-1006. 11. Muskhelishvili, N. I., Singular Integral Equations. P. Noordhoff N.V, Groningen, Holland, 1953. Fracture,
1. Bowie, 0. L. and Freese, C. E., Elastic analysisfor a radial crack in a circular ring. Engng Fracture Mech., 1972,4,315-321. 2. Tracy, P. G., Elastic analysisof radial cracks emanating from the outer and inner surfaces of a circular ring. Engng Fracture Mech., 1979, 11, 291-300.
3. Emery, A. F. and Sedgedin,C. M., The evaluation of the stress intensity factors for cracks subjected to tension, torsion, and flexure by an efficient numerical technique.Journal of Basic Engng, 1972,94,387-393. 4. Andrasic, C. P. and Parker, A. P., Dimensionlessstress intensity factors for cracked thick cylinders under polynomial crack face loadings.Engng Fracture Mech., 1984,19,187-193.
0.9
PII:S~O308-0161(97)00021-5
ELSEVIER
Vex & Piping 72 (1997) 149-156 0 1997 Elsevier Science Limited in Northern Ireland. All rights reserved 030%0161/97/$17.00
Mdde II stress intensity factors for a circular ring or a hollow cylinder with a radial crack A. Y. T. Leung School of Engineering,
The University of Manchester,
Manchester
Ml3
9PL,
UK
J. D. Hu Department
of
Civil and Structural
Engineiering,
The University
of
Hong Kong,
Pokfulam
Road, Hong Kong
(Received 6 March 1997; accepted 3 April 1997) The two-dimensional elasticity problem of a circular ring or a hollow cylinder with a radial crack and subjected to crack surface shear stress is considered. Using dislocation pile-up and singular integral equation techniques, Mode II stress intensity factors are obtained. The crack may be an internal edge crack, an external edge crack, or an embedded crack. Numerical examples are given, for various crack geometries and radius ratios. The numerical results are obtained for a uniform cra.ck surfaceshear stressz,,. 0 1997 Elsevier Science Ltd. using dislocation pile-up and singular integral equation techniques. The crack in the question may be an edge crack or an embedded crack with various loading conditions, i.e. a uniform crack surface pressure, a uniform inside pressure and a rotating disk. In all those studies, however, only Mode I stress intensity factors are considered. In engineering, we often need to consider the problems of Mode II crack and Mixed mode crack. In this paper, we consider the two-dimensional problem of a circular ring or a hollow cylinder with a radial crack and subjected to crack surface shear stress. The problem of interest is the Mode II crack in which the self-equilibrating crack surface shear stresses are the only external loads. The geometry and loading conditions of the problem under consideration are shown in Fig. 1. The approach for Mode II crack employed here is similar to that of Delale and Erdogan’ for Mode I crack, i.e. the corresponding dislocation solution for an infinite plane with a dislocation is sought first, then by superposition and integration, the crack problem is formulated in terms of a singular integral equation. The stress intensity factors of the crack are related to the solution of the singular integral equation and can be evaluated accurately by Gaussian quadrature. Numerical results of the stress intensity factors are
1 INTRODUCTION
The two-dimensional elasticity problems of al circular ring or a hollow cylinder with a radial crack, as typical problems in Linear Elasticity Fracture Mlechanics, have been considered by a number of investigators’” to obtain the stress intensity factors for various crack geometries. The problem of a radial crack located on its inner boundary and subjected to uniform tensile tractions on the outer boundary in a circular ring was considered by Bowie and Freese.’ The problem of multiple radial cracks on the inner or outer boundary of a circular ring was studied by Tracy.’ In R.efs [l, 21 the analytical studies were based on the mappingcollocation technique. The problem of a hollow cylinder containing an external edge crack was considered by Emery and Segedin3 for various loading conditions. A numerical technique similar to finite difference approximation was used for the solution. The problem of a thick cylinder with one or two, internal or external radial edge cracks was considered by Andrasic and Parker.4 The numerical resuhs of the stress intensity factors were obtained by the use of crack weight functions which were generated via the accurate modified mapping collocation metlhod. The problem of a hollow cylinder or a circular ring with a radial crack was considered by Delale and Erdogan’ 149
150
A. Y. T. Leung, .I. D. Hu
tion along 0 = 0 line and of the untracked ring, respectively.
concentric
2.1 Infinite plane with an embedded edge dislocation
Consider an infinite plane with an edge dislocation with Burger’s vector b, = g at Y = t, 8 = 0. We solve the plane elasticity problem by assuming that
(Tee,=o, o~rsw,
i [url(r, 0+) - u,,(r, O-)] = gS(r - t),
Fig. 1. The geometry of the problem.
presented in tabular and graphical crack geometries.
forms for various
Referring to Wang, function of the problem A@7 0) = - .(I”+
2 DISLOCATION
0)
=
aijl(r,
e=n,
(4
0 S r =Sa,
(3)
we express the Airy stress as
K) gr sin 6 ln(r* +
t2-
2rt cos Q),
SOLUTION
(4)
The geometry of the corresponding dislocation problem is shown in Fig. 2. Instead of the crack, an edge dislocation with Burger’s vector (b,, 0,O) is embedded in the circular ring at r = t, 0 = 0. In order to facilitate the application of the boundary conditions, the dislocation problem is expressed in plane polar coordinates. Due to the existence of the dislocation in the ring, the stress function for the ring is constructed as the sum of the general elasticity solution for the circular ring and a cracked or dislocation solution for an infinite plane. The stress state in the cracked ring problem can be obtained by aij(r,
e=o,
0)
+
aijZ.(r2
0)~
(6
i =
r,
01,
where p is the shear modulus, strain and
K
= 3-4~ for plane
3-v
K = -1
+ v for generalized
plane stress, Y
being the Poisson’s ratio. The stresses can be obtained from the Airy stress function (4) as follows
ar,l(r, ,,=1!!h+G% r dr
r2
882 r sin 8
=--
r2+t2-2rtcos0
+ 2t sin 0(r cos 8 - t)(r - t cos f3) ) (5) (r” + t2 - 2rt cos 0)”
1
(1)
where gijl and ajj2 (i, j = r, 6) are stresses of an infinite plane containing an embedded edge disloca-
2%
=
?‘C(l f
-
=
sin 0(2t cos 8 - r) K)
Y2 + t2 - 2rt
COs 6
2rt2 sin3 8 1 (r2 + tZ- 2rt cos 0)’ ’
2Pug 7c(l+~)
r cos 8 - t [ r2+t2-2rtcosf3
- 2t2 sir? f3(r cos 8 - t) (r* + t2 - 2rt cos 0)” ’
1
Fig. 2. The geometry of the dislocationproblem.
In order to combine
(6)
(7)
the infinite plane solution with
Mod,e ZZ stress intensity factors
the circular ring solution, we now express the stresses in eqns (5,7) along the circles r = a and Y = b in the plane in terms of the following Fourier series
D,(t)=
1.51 2b2-t2 -7,
D,(t) = - (n +:l”-’
+ s,
(n 2 2)
(19a,b)
2.2 General ring solution
Due to symmetry, the stresses for general solution should satisfy the following conditions
ring
(ZOa-c) where the Fourier coefficients are determined
Aa
=
X(1
+
K>
=
B,(t) = Co(t) =
41+K)2
-
7$1+K)2
noi
I% +
K)
I-%
1
-
=
Dn(t) =
741+K)2
iug
c,l(a, 0) de, zrel(a, f3)cos no do,
(12a,b) vrrr2(r, 6) = - $! + 2d Ir sin 0 - 2 [a,tz(n - l)r”-’
=
2
urrI (a, @sin n0 de,
(13)
~s,(b,
0)
noI
z,.~~(b, 8)cos ?zOdQ,
(14a,b)
L(t)=
-t”-‘+
na”-’
(22)
+ b,n(n + 1)r” - c,jz(n + l)reflP2 - d,n(n - l)Y]cos
~0,
(23)
where ao, cl, d,, a,, b,,, c, and d, are all arbitrary constants. 2.3 Boundary conditions
= -s+$,
-;,
no,
(15)
>
B,(t)=
(21)
f b,(n + l)(n + 2)r” + c,y1(rz + l)rFfle2
+ d,(n - l)(n - 2)r-“Isin
71
~,.~,(b, B)sin n8 de, xc,I
A,(t) = 0,
A,(t)
n8,
do,
in which r,,, and a,.,, are determined by eqns (5) and (7). With the aid of tables,’ integrals in eqns (12-15) can be evaluated in closed form and the results are obtained as follows
A,(t) =;
+ b,(n + l)(n - 2)r” + c,,n(n + l)rCm2 + d,(n - l)(n + 2)r-“Isin
JE
IT -
t-q
After taking regularity and symmetry conditions into consideration, and by referring to Michell’s general stress functions,’ we can formulate the stresses in the ring as follows
no i
.7C(1+K)2 CJt)
IT
=ai
f-%
7C(l
-
x
no j’
I-%
A,(t)
1
by
(n 2 21,
(16a-c)
(n - 2)an
tn+l
1 (n 3%
(170)
After obtaining the basic form of the solutions for the infinite plane with a dislocation and for the ring, we express the stresses in the circular ring with a dislocation as the sum of the two solutions [eqns (1) (5-7) and (21-23)]. Th e combined stresses must then satisfy the following boundary conditions g,,l(a, 0) + flrr2(al 6) = 0,
Co(t) = 0,
h(a,
2b2 - t2 C,(t) =--p-j
~,,,(b> ‘3 + ~rrz(b> 69 = 0,
4 + h(a,
Q> = 0,
(24a-d)
~rm(b, 0) + c,z(b, 6) = 0, (n z=21,
(Ha-c)
where v~,.~ and rre, are given by eqns (S-15) and u,.,~ and r,.82 are given by eqns (21,23).
152
A. Y. T. Leung, J. D. Hu
Defining the following transformations I-%
b, = -
an = - n(l + K) %I,
” x(1
t-G
d, = -
cn = - n(l + K) Yn,
Pn +
K)
+
K)
” X(1
By integrating the dislocation solution along c 6 r 6 d, 0 = 0 and applying the boundary condition (31), we obtain a singular integral equation as follows (25a-d) [$$dt
S,,
and substituting eqns (8-11) and eqns (21, 23) into eqn (24), we obtain the following equations
+ [k(r,
a,n(n
=A,,
1
a=B
1,
- l)aHp2 + p,n(n
%= b2 2% b3-
Co,
(26a,b)
2S1b = Cl,
- l)a”-’
-$$+,S,b=D,,
(2&b)
+ l)an
(~,n(n - l)bnp2 + p,n(n + l)b-‘-2
a,n(n - l)b”-‘+
+ S,(n + 2)(n - l)a-”
= -B,,
+ y,n(n + l)b-‘-2
For an internal that
crack (i.e. a
= -C,,
+ S&r + 2)(n - l)b-”
= -D,,
where A,, B,, C, and D, are functions of t given by eqns (16-19). From eqns (29), it is found that there are equal numbers of equations and unknown constants. Thus, these unknown constants can be uniquely determined. But in eqns (27, 28), there are four equations and only two unknowns. Equations (27,28) may have a solution only if two of them are arbitrary. If C,+D,=O,
G(t) l?(t) = -\/(t - c)(d - t)’
k2(c) = !ii V@Tj
(c < t < d),
(35)
zre(r, 0)
= -i?k 1 + Klim -g(r) 2(r r-c
Wkb)
(36) k,(d)
= Iii m zz
INTENSITY
(34)
where G(t) is a bounded function at the crack tips.” The Mode II stress intensity factors of the crack can be calculated according to the conventional definitions
then eqns (27) and eqns (28) are identical. Ignoring eqns (28), we uniquely determine the y1 and 6, from eqns (27). After determining all these constants, the stresses for the dislocation problem in Fig. 2 may be obtained by eqns (l), (5-7) and (21-23).
3 STRESS
g(t) dt = 0,
In this case, the index of the singular integral equation (32) is +l. The function g(t) has a square root singularity at the crack tips and we may write
(29a-d)
A,+B,=O,
crack
ic
P,(n - 2)(n + 1)b”
(33)
and CY,, pn, yn and S, (n = 1, 2, . . .) are known functions of t which are determined from eqns (26-29).
f 1)b” - S,n(n - l)b-”
+ l)rn
- ynrz(n + l)rmnp2 - S,n(n - l)r-“I},
3.1 Internal
+ /3,(n - 2)(n + l)a”
+ ynn(n + l)a-“-2 - y,n(n
- 2S,r - jj [(~,n(n - l)rnp2
+ &n(n
Pa,b)
- ynn(n + l)amnp2 - 6,n(n - l)a-n = -A,, a,n(n
(32)
2
Ao,
a2
-%+2S a3
K)g(r),
where k(r, t) = d (5
a0 -=
&-2&a a3
t>g(t) dt = x(::
%(r,
0)
-&lirJ:V@?jg(r) r--f
FACTORS (37)
The dislocation solution obtained above may be used to tackle the crack problem in Fig. 1 using dislocation pile-up and singular integral equation techniques. For the problem under discussion, we write the remaining boundary condition as Ge*(r, 0) + Gn(r, 0) = 4(r), where q(r) is the crack surface shear stress.
(31)
Introducing d-c
r=-s+ 2
the following transformations d+c
-
2 ’
whenc
-l
(380) t=-
d-c
2
Z+d+C
2 ’
whenc
-l
153
Mode II stress intensity factors and substituting obtain
eqns (38) into
eqns (32, 34), we
and substituting
eqns (46) into eqn (32), we obtain
’ g’(r) z--s dz + (d - c>1’0 K(s, %(r> dr
K(s, Qg,(z> dz
= Ir(l
+
K)
2P
q(s), (-1
K)
2P
(0
q(s),
where
g1(r) = c,(z> CT>
g,(z) dz = 0,
(O
a(4 =s(t), q(s)=q(r), K(s,r>=k(r,t),
where a(r) = s(t), q(s) = q(r), K(s, 4 = k(r, t> Accordingly,
n(l+
we may write
Referring to Ref. [lo], and discretizing similar manner, we have
eqn (47) in a
2 ~G,(i)[&+~K rr)] I r i=l Ii &
GI(T~)[&+Cd- C)K(sjj t)]
Using the Gauss-Chebyshev integration formula,’ eqns (39) can be discretized into the arlgebraic equations
l’L(l
=
+
K)
2~
q(Sj)
)
(I=
2,
lr
”
7 n),
(4g)
where
~,n+,(~;) = 0,
X(1
=
+
ds,)
K)
,
2P
(j=
1,2,. . .,I2 -1)
(41)
b&j)
=
T,(q)=O,
(i=1,2
u,~,(sj)=o,
,...,
(42)
(j=l,&...,n-l),
G,(-1) v?
=+L/T7----K
the
a=c
condition required solution. Referring
k,(d)
= -+&V%%
edge
crack
’
G (1) v5’
(43)
to Ref. [lo), we write the function g(t) as G(t) ___ g(r) = qJ-q
(a=c
3.3 External
Y= (d - c)s -t c, t = (d - c)z + c,
whenc
O
(52)
(a
Table 1. Mode II stress intensity factors k,(c) and k,(d) for an embedded crack in a circular ring subjected to uniform crack surface shear stress q(r) = -T,,, (d c)l(b -a) = o-5, (c - a)/@ - d) = 1
a b-a
_ k,(c) d(d - c)/2
k,(d) zov/(d- c)L’
0.05 0.10
1.102403 1.112981 1.143672 l-172981 1.193131 1.200979 1.202374 1.202751 1.202869
1.140259 1.143946 1.158068 1.175292 1.190321 1.198131 1.200159 1.200981 1.201402
(45)
0.25 0.50
(46)
2.0 3.0 4.0 5.0
transformations
edge crack
G(t) s(t) = __ $-q
1.0
the following
l+K
In the case of an external edge crack (i.e. a < c < d = b), like the case of an internal edge crack, the displacement single-valuedness condition (34) does not hold true and also is not required as an additional consition for a unique solution. Referring to Ref. [lo], we write the function g(t) as
(44)
case of an internal edge cra\ck (i.e. the displacement single-valuedness (34) does not hold true and also is not as an additional condition for a unique
Introducing
0)
n),
where Tn(x) and U,-,(x) are Chebyshev polynomials of first and second kinds, respectively. The Mode II stress intensity factors of the crack can be calculated by
In
z&,
-*-G,(l),
with ti and s, as
3.2 Internal
(50)
The Mode II stress intensity factors of the crack can be calculated by k,(d) = lipidm
k*(c)
= 0,
154
A. Y. T. Leung, J. D. Hu
I
1.18
T
a/b 1.1 1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fig. 3. Mode II stressintensity factors for an internal crack in a ring loaded by a uniform crack surfaceshearstress-7,.
Introducing the following transformations r = (d - c)s + d, t = (d - c)z + d,
whenc
where
-l
G,(z) a(~> = CT’
Referring to Ref. [lo], and discretizing similar manner, we have
eqns (53) into eqn (32), we obtain
O---ddz+(d-c) a(4
I-1 K(s, %1(r) =x(1 + K) 2p 4(s),
dr i=, Ii &
t-1
Wd[
&
(54)
Table 2. The normalized Mode II stress intensity factors c in a circular ring containing an internal wmo-) edge crack (a = c
l/3 1.114205 1.137115 1.184718 1.253483 1.347864 1.480665 1.680663 2.023717 2.813413
a/@ -a) 112 1.113084 1.123781 1.159126 1.219228 1.308061 1.437068 1.633779 1.972012 2.749417
1 1.112207 1.119115 1.141009 1.187572 1.265209 1.385657 1.575502 1.906569 2.669403
+ Cd - C)K(% $1
=
0.1 0.2 0.3 0.4 O-5 0.6 0.7 0.8 0.9
eqn (54) in a
0
I-, z-s
d-a b-a
(55)
g,(z) = g(t), 46) = q(r), K(s, 4 = k(r, t>,
(53)
and substituting
(-l
4Csj)
+
cl]
K)
,
2F
(i = L2, . . . , n>,
(56)
where T2rz+l(-zi)
=
0,
~2n(-sj)
=
0,
(57)
The Mode II stress intensity factors of the crack can be calculated by
2 1~117818 1.136173 1.180698 1.252182 1.366051 1.551078 1.878181 2.635523
=-+GG,(-I), K
4 RESULTS
AND
(58)
DISCUSSIONS
From eqs (41), (49) and (56), we solve numerically the singular integral equation in eqn (32). Thus Mode II
Mode II stress intensity factors
2 t
I 0.5 $
(d-a)/@-a) 0-l
I 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 4. Mode II stress intensity factors for an internal edge crack in a ring loaded by a uniform crack surface shear stress -r,.
stress intensity factors for a circular ring with a radial internal crack, internal edge crack or external edge crack can be obtained by equations (43,44), (51) or (58), respectively. One loading condition, i.e. uniform crack surface shear stress q(r) = -z, (c < r < d), is used in the examples. For various internal crack geometries, normalized Mode II stress intensity factors ;are given in Table 1 and shown in Fig. 3. For various internal edge crack geometries, normalized Mode II stress Table 3. The nonualized Mode II stress intensity factors kz(c)/(70G) in a circular ring containing an external edge crack (a Cc < b = 6) and subjected to uniform crack surface shear stress q(r) = -TV b-c b-a
0.1 0.2 o-3
0”:;
al(b -a) l/3
1.112674 1.135787 1.155528 l-179715 1.214374
0.6 0.7
1.268464
0.8
1.549271 2-022986
0.9
1.362862
112 l-109875 1.132885 1.152163 1.178028 1.217687 1.283583 l-400176 1.628774 2.196203
1
2
1.101673 1.125463 1.145539 1.174962
1.~085728 I.115518
l--136959
3:169969
1.224152
I.226046
1.307923 1.456356 1.739809
1.321528
2.426348
;!-533217
I.486588
Il.795338
intensity factors are given in Table 2 and displayed in Fig. 4. For various external edge crack geometries, normalized Mode II stress intensity factors are given in Table 3 and displayed in Fig. 5. For the internal crack problem, here the crack length is fixed at half the wall thickness (b - a) and the ratio of a/b is varied. From Table 1 and Fig. 3, it can be seen that k,(c) at the inner crack tip Y = c is less than k,(d) at the outer crack tip r = d when the ratio of a/b is small. Stress intensity factors k*(c) and k,(d) raise with the ratio of a/b increasing. The k2(c) increases faster than the k,(d). Finally, k,(c) and k,(d) tend to the same constant. For the internal edge crack and external edge crack problems, we choose that the ratio of the crack length (d - c) to the wall thickness (b - a) and the ratio of the inner radius a to the wall thickness (b - a) are varied. From Tables 2 and 3, Figs 4 and 5, we note that as (d - c)/(b - a) increases, the Mode II stress intensity factors k2(c) and k,(d) increase. Increasing a/(b -a) causes an increase in k*(c), but increasing a/(b - c) causes a decrease in k,(d). The results express that the thinner wall circular ring with the internal edge crack can assume the bigger crack surface shear stress, but the thinner wall circualr ring with the external edge crack can assume the smaller crack surface shear stress.
0.9
156
A. Y. T. Leung, J. D. Hu 3-
k2 (4 r,Jd-c 2.5 +
2i
I I 1.5 t
IT I 0.5
t
(b-NW 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 5. Mode II stressintensity factors for an external edge crack in a ring loadedby a uniform crack surfaceshearstress-r,.
REFERENCES
5. Delale, F. and Erdogan, F., Stressintensity factors in a hollow cylinder containing a radial crack. Int. Journal of 1982,20,2.5-265. 6. Wang, D., Fracture Mechanics, 1. Harbin Institute of Technology Press,1989. 7. Ryshik, I. M. and Gradstein, I. S., Table of Series, Products and Integrals. Academic Press, New York, 1965. 8. Little, R. W., Elasticity. Prentice Hall, 1973. 9. Erdogan, F. and Gupta, G. D., On the numerical solution of singular integral equations.Q. Appl. Math., 1972,29,525-534. 10. Gupta, G. D. and Erdogan, F., The problem of edge cracksin an infinite strip. ASME J. Appt. Mech., 1974, 41,1001-1006. 11. Muskhelishvili, N. I., Singular Integral Equations. P. Noordhoff N.V, Groningen, Holland, 1953. Fracture,
1. Bowie, 0. L. and Freese, C. E., Elastic analysisfor a radial crack in a circular ring. Engng Fracture Mech., 1972,4,315-321. 2. Tracy, P. G., Elastic analysisof radial cracks emanating from the outer and inner surfaces of a circular ring. Engng Fracture Mech., 1979, 11, 291-300.
3. Emery, A. F. and Sedgedin,C. M., The evaluation of the stress intensity factors for cracks subjected to tension, torsion, and flexure by an efficient numerical technique.Journal of Basic Engng, 1972,94,387-393. 4. Andrasic, C. P. and Parker, A. P., Dimensionlessstress intensity factors for cracked thick cylinders under polynomial crack face loadings.Engng Fracture Mech., 1984,19,187-193.
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