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Stress intensity factors for radial cracks in circular cylinders and other simply closed cylindrical bodies
Engineering Fracture Mechanics
Printedin Great Britain.
Vol. 32, No. 5, pp.167-114,1989
0013-7944/89 53.00+ 0.00 0 1989Perpmon Presspk.
STRESS INTENSITY FACTORS FOR RADIAL CRACKS IN CIRCULAR CYLINDERS AND OTHER SIMPLY CLOSED CYLINDRICAL BODIES WEILI CHENG
Department of Engineering Mechanics, Jiao-Tong University, Shanghai, People’s Republic of China and IAIN FINNIE Department of M~~nical
En~n~~ng,
University of California, Berkeley, CA 94720, U.S.A.
Abstract-An approach using plane strain solutions is presented to calculate stress intensity factors for one or more internal or external radial cracks in circular cylinders and other simpiy closed cylindrical bodies with symmetric geometry and loading conditions. The results show very good agreement with numerical data in the literature for a number of cracked configurations.
INTRODUCTION STREWSintensity factors for an internal radial crack in a circular ring were originally obtained by Bowie and Freesejl] using a modified mapping collocation method. With the same method, stress intensity factors for more than one internal and external crack were also obtained& 31 for think-walled cylinders for a few ratios of mean radius R to thickness t under several loading conditions. However, the computational procedure for this method is rather complicated to apply and is limited mainly to thick-walled cylinders. Recently an approach[4, 51 has been developed to determine stress intensity factors for either a complete internal circumferential crack or an internal axial crack in thin-walled circular cylinders. The approach is based on the analysis of a small cracked element which is conceptually separated from the main body and satisfies the continuity of displacement and the equilibrium of loading at its ends. This approach leads to a greatly simplified computational procedure and applies to a wide range of geometric configurations. In this paper we extend the approach to circular cylinders with one or more internal and external radial cracks and other simply closed cylindrical bodies with symmetric geometry and loading conditions. The approach was ori~nally developed with the ass~ption that the cylinders were thin-walled with a R/t ratio equal to or larger than ten. However, after using the exact expressions for compliance of the untracked body the present solution procedure obtains satisfactory results for cylinders of R/t ratio as low as 1.5 where R is the mean radius and t the wall thickness. Although the solution presented here focuses on surface tractions normal to the crack faces it may be extended to include shear stress distributions which correspond to mode II loading. A combination of these two loading conditions allows the stress intensity factors to be obtained for nonsymmetric geometry and loading conditions. ANALYSIS
We first consider the case of a single internal or external radial crack as shown in Fig. 1. The stress distribution on the crack faces is arbitrary and from Bueckner’s superposition principle it corresponds to the hoop stress distribution prior to the introduction of the crack. To begin, a small element containing the crack is conceptually separated from the cylinder as shown in Fig. 2. The stress field which exists on the ends of the element consists of two parts. One comprises a moment M and a force F per unit axial distance which are due to the presence of the stress distribution (T&) on the crack faces when subjected to the elastic constraint from the rest of the body. The other one (which is not shown in Fig. 2), corresponds to the stress distribution in a cracked curved 767
WEILI
Fig. 1. A cylinder of mean radius R and thickness single internal or external radial crack.
CHENG
f with a
and I. FINNIE
Fig. 2. An element
containing the crack conceptually rated from the cylinder.
sepa-
beam subjected to only surface tractions on crack faces. This self-equilibrating stress field is assumed to be the same as that in a straight cracked beam and to vanish at a distance from the crack approximately equal to the thickness. This implies that plane strain solutions for a cracked beam may be applied to the cracked element. As shown in Fig. 2, the direction of the forces for the internal crack is opposite to that for the external crack. From superposition the stress intensity factor is given by Kl = KY - Kr” + K;
(1)
where K)‘, KY and Kf are stress intensity factors from the plane strain solutions for surface tractions a,(z) on the crack faces, moments M and forces F on the cracked element, respectively. In eq. (1) and what follows, if two signs are used the upper will apply for an external crack and the lower for an internal crack. If d is a reference stress, for example the value of a&) at the outside or inside wall, eq. (1) may be rewritten as
The functionsfm(s/f) andfo(a/t) are given in the stress intensity factor compilation of Tada, Paris and Irwin[6], while an expression offP(a/r) is given for arbitrary loading conditions by Cheng and Finnie[7]. Hence, to evaluate K,, we need to derive expressions for the moment A4 and force F as a function of crack length. With superscripts m and f denoting the rotation and extension produced by moments and forces on the ends of the cracked element respectively, we can express the rotation and extension of the cracked element as:
c = L’P- vm * VT
(33
where 4” and VPare the rotation and extension due to the surface tractions on the crack faces. They have been derived in[4,5,8] as @(a/t)
6rra = S,(a/t);
E’ a/r S,(Qlt) = (~/~lf”‘(~/~lfP@/~) s0
@lr)
Radialcracksin circularcylinders
769
and
not v (a/t);
vP(alt)= E’
p
where I?’ denotes E/(1 - v’). From eq. (41, making use of cr = ~~~~ and c = F/t the other variables in eq. (3) may be seen to be 367rN q3”(a/t) = MA” = E,r2
S&IO;
The XFare compliances, with extension compliances denoted by a tilde, and, as would be expected, If = /sm and &= V,,,. By continuity the rotation I#Iand extension u at each end of the cracked element must equal those at the split cross section. That is cb =W-4L v=
T&Q ?_I;,
161
where the subscript s denotes the split cylinder. The quantities #F, +I, u,” and ui are found in texts on e~~sti~ity[9]
770
WEILI CHENG and I. FINNIE
where fi = R/r and H,(W), H*(k) are correction
functions for thick-walled cylinders given by
H,(R) = 3 R - (a -0.25)1 n~~~1’) i [ Hz(d)=3R1[In(~)-(B+*.5)~~~W_0.5)2]. For R/t ratios larger than 4.5 the difference between eq. (7) and results found in texts on strength of materials [IO] becomes less than 0.5%. Substituting eqs (4), (5) and (7) into eqs (3) and (6), the moment and force can be found, i.e.
in which G, = Yr+ 18R3(H, + 2~~)/(3~~ Hz);
G2 = 35, + &Hi ;
G, = S, k 2R”2,W,;G4 = 3 V,,, + 6R2/H,. As the R/t ratio approaches infinity, values of M and F in eq. (8) decrease to zero and the stress intensity factor solution coincides with the plane strain solution. Now we consider two cylinders, (as shown in Fig. 3), one with internal radial cracks and the other with external cracks. All cracks are of equal size and evenly spaced along the circumference. Because of the use of plane strain solutions the distance between two radial cracks should be larger than two times of the wall-thickness. In other words, the maximum number of the cracks should be less than x[(R/t) - 0.51. From equilibrium the forces acting on the ends of a cracked element must vanish. This implies that the element is free to extend and we only need to consider the moment. Thus eqs (1) and (2) reduce to
Fig. 3. Cylinders with (a) internal or (b) external radial cracks of equal size evenly spaced along the circumference.
Radial cracks in circular cylinders
111
Fig. 4. A thin-walled simply closed prismatic body with two external radial cracks of equal size.
If there are II radial cracks the compliance of the untracked body becomes I/n of the value given in eq. (7) and the expression for the moment becomes
where A” is given in eq. (5) and A?=
12nR E’nt’H,(R”)’
This equation, when substituted into eq. (lo), indicates that moment M increases as n increases. Thus, eq. (9) predicts smaller values of K, when there are more than two radial cracks. It is noticed that the present solution leads to the same stress intensity factors for internal or external cracks if there is more than one crack evenly spaced along the circumference. As will be seen in the next section, for thin-walled cylinders the difference in K, between internal and external cracks is negligibly small. Also, for thick-walled cylinders of R/t = 1.5, eq. (9) is accurate enough for most engineering applications. We consider another configuration shown in Fig. 4 which is a thin-walled cylindrical simply closed body with two external cracks of equal size. The two cracks lie along the axis of symmetry. The analysis for the simply closed body of uniform wall-thickness is the same as that for a double cracked ring, i.e. only moments M are present at the ends of the element. From Castigliano’s theorem the compliance for the untracked body can be found as
(11) where S is the half length of the simply closed body. A comparison that the stress intensity factor for the thin-walled double cracked double cracked cylinder of the same periphery. This conclusion simply closed body with more than two radial cracks of equal thickness and is symmetric about every crack. RESULTS
of eq, (11) with eq. (7) indicates body is the same as that for a also applies to any thin-walled size if the body is of uniform
AND COMPARISON
Since there are few stress intensity factor solutions available in the literature for thin-walled cylinders and other simply closed bodies with more than one radial crack, we will only compare
WEILI
CHENG
and I. FINNIE 1._
OJ 0 Normalized Fig. 5. Comparison of the by Bowie and Frees: A[l] of R/r = 1.5subjected to a single
thickness
present solution with those given and Grandt O[l l] for a cylinder uniform external pressure P with internal crack.
0.2 0.4 Normalized
' 0.6 ' 0.8 thickness
0
Fig. 6. Comparison with Tracy’s results[2] for a cylinder of R/t = 1.5 with one A, two 0 or three * external radial cracks subjected to internal pressure P.
our results with those obtained for thick-walled cylinders. However, our approach should give better results for thin-walled cylinders. Bowie and Freese[l] first considered a thick-walled cylinder with an internal crack subjected to a uniform external tension. Figure 5 shows a comparison of our result with that given by Bowie and Freese for a cylinder of R/t = 1.5. The result obtained by Grandt[l l] with a weight function technique is also shown in the same figure. It seen that our result is closer to Bowie and Freese’s for small crack sizes while for large crack sizes it is closer to Grandt’s. The modulation in the predicted curve results from the use of different expressions for f”(a/t) and fo(a/t) above and below a/t = 0.6. Tracy[2] calculated stress intensity factors for cylinders with several external cracks subjected to a uniform internal pressure. For a cylinder of R/t = 1.5 the maximum number of radial cracks to which our approach applies is three. Figure 6 shows a comparison of our results with Tracy’s for 1, 2 and 3 cracks. The agreement is good for each case. Andrasic and Parker[3] presented their tabulated results for surface tractions in a form of power functions
in which z is measured from the inner surface for internal cracks and outer surface for external cracks. Figures 7-12 show comparisons of our results with numerical data tabulated in [3] for surface loading given in eq. (12) with i = O-4. It is seen, for R/t = 4.5, that the agreement is almost exact for all radial crack configurations except for a/t = 0.7 in Fig. 9. We suspect that
6 I=0
Oi,O
Normalized
thickness
Fig. 7. Comparison of results of Andrasic and Parker[3] for a cylinder of R/r = 1.5 with two internal radial cracks (A) or two external radial cracks (0) subjected to uniform, linear, quadratic, cubic and quartic stress distributions on crack faces (i = 0,1,2,3,4, respectively) with present predictions shown by solid lines.
04 0
0.2
0.4
Normalized
0.6
08
thickness
Fig. 8. Same as Fig. 7 for R/i = 4.5 and two external internal radial cracks.
or
773
Radial cracks in circular cylinders
I 0
i=O
/
0.2 0.4 Normolized
I
0.6
0.8
04
,
0
0.2
thickness
0.4
Normalized
Fig. 9. Same as Fig. 7 for a single external radial crack for R/t = 4.5.
0.6
I
0.8
thickness
Fig. 10. Same as Fig. 9 for a single internal radial crack for R/t = 4.5. 2.5
p’
i=O
-- 3 ;: z -” 2 a 3.-
E 6
y” 0.5 0
0.2
0.4
Normalized
0.6
0.8
1 4-i
0
0.2
0.4
Normalized
thickness
Fig. 11. Same as Fig. 9 for a cylinder of R/f = 1.833.
i=O
/
1.0
4.
5,
0
I
-z 2.0 \ c) _c 1.5
0.6
0.8
a0
thickness
Fig. 12. Same as Fig. 7 for single external radial crack for Rlt = 1.5.
Andrasic and Parker’s value at that point may be in error. Also, our results agree well with theirs for R/t ratios as low as 1.5 for all kinds of loading. For a double cracked cylinder of R/t = 4.5 Andrasic and Parker found almost no difference between an internal crack and an external crack and for a cylinder of R/t = 1.5 the difference becomes about seven per cent. Our results lie between their predictions. It is of interest to note that with the present approach it is possible to explain why an external crack leads to a larger value of the stress intensity factor than an internal one, why the value of Ki increases when a single cracked cylinder becomes a double cracked one and why it decreases when the number of cracks becomes more than two. CONCLUSION An approach is presented to calculate the stress intensity factors for radial cracks in circular cylinders and simply closed cylindrical bodies with geometry and loading conditions symmetric about each crack. It applies to a wide range of cylinders from thin-walled ones to those with R/t ratio as low as 1.5. Very good agreement with other numerical solutions is obtained for a number of configurations and loading conditions. A major advantage of the present method over commonly used methods is its simple computational procedure. Also, the present approach may be extended to include mode II loading. This should allow stress intensity factors to be obtained for simply closed cylindrical bodies with nonsymmetric geometry and loading conditions. REFERENCES [l] 0. L. Bowie and C. E. Freese, Analysis for a radial crack in a circular ring. Engng Fracture Me& 4, 315-321 (1972). [2] P. G. Tracy, Elastic analysis of radial cracks emanating from the outer and inner surfaces of a circular ring. Engng Fracfure Mech. 11, 291-300 (1979).
774
WEILI CHENG and I. FINNIE
[3] C. P. Andrasic and A. P. Parker, Dimensionless stress intensity factors for cracked thick cylinders under polynomial crack face loadings. Engng Fracfure Mech. 19, 187-193 (1984). [4] W. Cheng and I. Finnie, On the prediction of the stress intensity factors for axisymmetric cracks in thin-walled cylinders from plane-strain solutions. J. Engng Mater. Technol. 107, 227-234 (1985). [5] W. Cheng and I. Finnie, Determination of stress intensity factors for partial penetration axial cracks in thin-walled cylinders. J. Engng Mufer. Technol. 108, 83-86 (1986). [6] H. Tada, P. Paris and G. Irwin, The Sfress Anai_vsis of Cracks Handbook. Del Research Corporation, PA (1973). [7] W. Cheng and f. Finnie, Ki solutions for an edge-cracked strip. Engng Fracture Neck. 31, 201-207 (1988). f8] H. F. Bueckner, The end rotation of a notched bar in mode I deformation. J. &gng Marer. Technol. 107,24&247 (1985). 191S. P. Timoshenko and J. N. Goodier, Theory o/’ ~/as~~cjf~. McGraw-Hill, New York (1970). [IO] J. P. Den Hartog, Advanced Sfrengfh qf Mafer&s. McGraw-Hill, New York (1952). [l I] A. F. Grand& Jr, Stress intensity factors for cracked holes and rings loaded with polynomial crack face pressure distributions. Inc. J. Fracfure 14, R221 (1978). (Received 22 March 1988)
Printedin Great Britain.
Vol. 32, No. 5, pp.167-114,1989
0013-7944/89 53.00+ 0.00 0 1989Perpmon Presspk.
STRESS INTENSITY FACTORS FOR RADIAL CRACKS IN CIRCULAR CYLINDERS AND OTHER SIMPLY CLOSED CYLINDRICAL BODIES WEILI CHENG
Department of Engineering Mechanics, Jiao-Tong University, Shanghai, People’s Republic of China and IAIN FINNIE Department of M~~nical
En~n~~ng,
University of California, Berkeley, CA 94720, U.S.A.
Abstract-An approach using plane strain solutions is presented to calculate stress intensity factors for one or more internal or external radial cracks in circular cylinders and other simpiy closed cylindrical bodies with symmetric geometry and loading conditions. The results show very good agreement with numerical data in the literature for a number of cracked configurations.
INTRODUCTION STREWSintensity factors for an internal radial crack in a circular ring were originally obtained by Bowie and Freesejl] using a modified mapping collocation method. With the same method, stress intensity factors for more than one internal and external crack were also obtained& 31 for think-walled cylinders for a few ratios of mean radius R to thickness t under several loading conditions. However, the computational procedure for this method is rather complicated to apply and is limited mainly to thick-walled cylinders. Recently an approach[4, 51 has been developed to determine stress intensity factors for either a complete internal circumferential crack or an internal axial crack in thin-walled circular cylinders. The approach is based on the analysis of a small cracked element which is conceptually separated from the main body and satisfies the continuity of displacement and the equilibrium of loading at its ends. This approach leads to a greatly simplified computational procedure and applies to a wide range of geometric configurations. In this paper we extend the approach to circular cylinders with one or more internal and external radial cracks and other simply closed cylindrical bodies with symmetric geometry and loading conditions. The approach was ori~nally developed with the ass~ption that the cylinders were thin-walled with a R/t ratio equal to or larger than ten. However, after using the exact expressions for compliance of the untracked body the present solution procedure obtains satisfactory results for cylinders of R/t ratio as low as 1.5 where R is the mean radius and t the wall thickness. Although the solution presented here focuses on surface tractions normal to the crack faces it may be extended to include shear stress distributions which correspond to mode II loading. A combination of these two loading conditions allows the stress intensity factors to be obtained for nonsymmetric geometry and loading conditions. ANALYSIS
We first consider the case of a single internal or external radial crack as shown in Fig. 1. The stress distribution on the crack faces is arbitrary and from Bueckner’s superposition principle it corresponds to the hoop stress distribution prior to the introduction of the crack. To begin, a small element containing the crack is conceptually separated from the cylinder as shown in Fig. 2. The stress field which exists on the ends of the element consists of two parts. One comprises a moment M and a force F per unit axial distance which are due to the presence of the stress distribution (T&) on the crack faces when subjected to the elastic constraint from the rest of the body. The other one (which is not shown in Fig. 2), corresponds to the stress distribution in a cracked curved 767
WEILI
Fig. 1. A cylinder of mean radius R and thickness single internal or external radial crack.
CHENG
f with a
and I. FINNIE
Fig. 2. An element
containing the crack conceptually rated from the cylinder.
sepa-
beam subjected to only surface tractions on crack faces. This self-equilibrating stress field is assumed to be the same as that in a straight cracked beam and to vanish at a distance from the crack approximately equal to the thickness. This implies that plane strain solutions for a cracked beam may be applied to the cracked element. As shown in Fig. 2, the direction of the forces for the internal crack is opposite to that for the external crack. From superposition the stress intensity factor is given by Kl = KY - Kr” + K;
(1)
where K)‘, KY and Kf are stress intensity factors from the plane strain solutions for surface tractions a,(z) on the crack faces, moments M and forces F on the cracked element, respectively. In eq. (1) and what follows, if two signs are used the upper will apply for an external crack and the lower for an internal crack. If d is a reference stress, for example the value of a&) at the outside or inside wall, eq. (1) may be rewritten as
The functionsfm(s/f) andfo(a/t) are given in the stress intensity factor compilation of Tada, Paris and Irwin[6], while an expression offP(a/r) is given for arbitrary loading conditions by Cheng and Finnie[7]. Hence, to evaluate K,, we need to derive expressions for the moment A4 and force F as a function of crack length. With superscripts m and f denoting the rotation and extension produced by moments and forces on the ends of the cracked element respectively, we can express the rotation and extension of the cracked element as:
c = L’P- vm * VT
(33
where 4” and VPare the rotation and extension due to the surface tractions on the crack faces. They have been derived in[4,5,8] as @(a/t)
6rra = S,(a/t);
E’ a/r S,(Qlt) = (~/~lf”‘(~/~lfP@/~) s0
@lr)
Radialcracksin circularcylinders
769
and
not v (a/t);
vP(alt)= E’
p
where I?’ denotes E/(1 - v’). From eq. (41, making use of cr = ~~~~ and c = F/t the other variables in eq. (3) may be seen to be 367rN q3”(a/t) = MA” = E,r2
S&IO;
The XFare compliances, with extension compliances denoted by a tilde, and, as would be expected, If = /sm and &= V,,,. By continuity the rotation I#Iand extension u at each end of the cracked element must equal those at the split cross section. That is cb =W-4L v=
T&Q ?_I;,
161
where the subscript s denotes the split cylinder. The quantities #F, +I, u,” and ui are found in texts on e~~sti~ity[9]
770
WEILI CHENG and I. FINNIE
where fi = R/r and H,(W), H*(k) are correction
functions for thick-walled cylinders given by
H,(R) = 3 R - (a -0.25)1 n~~~1’) i [ Hz(d)=3R1[In(~)-(B+*.5)~~~W_0.5)2]. For R/t ratios larger than 4.5 the difference between eq. (7) and results found in texts on strength of materials [IO] becomes less than 0.5%. Substituting eqs (4), (5) and (7) into eqs (3) and (6), the moment and force can be found, i.e.
in which G, = Yr+ 18R3(H, + 2~~)/(3~~ Hz);
G2 = 35, + &Hi ;
G, = S, k 2R”2,W,;G4 = 3 V,,, + 6R2/H,. As the R/t ratio approaches infinity, values of M and F in eq. (8) decrease to zero and the stress intensity factor solution coincides with the plane strain solution. Now we consider two cylinders, (as shown in Fig. 3), one with internal radial cracks and the other with external cracks. All cracks are of equal size and evenly spaced along the circumference. Because of the use of plane strain solutions the distance between two radial cracks should be larger than two times of the wall-thickness. In other words, the maximum number of the cracks should be less than x[(R/t) - 0.51. From equilibrium the forces acting on the ends of a cracked element must vanish. This implies that the element is free to extend and we only need to consider the moment. Thus eqs (1) and (2) reduce to
Fig. 3. Cylinders with (a) internal or (b) external radial cracks of equal size evenly spaced along the circumference.
Radial cracks in circular cylinders
111
Fig. 4. A thin-walled simply closed prismatic body with two external radial cracks of equal size.
If there are II radial cracks the compliance of the untracked body becomes I/n of the value given in eq. (7) and the expression for the moment becomes
where A” is given in eq. (5) and A?=
12nR E’nt’H,(R”)’
This equation, when substituted into eq. (lo), indicates that moment M increases as n increases. Thus, eq. (9) predicts smaller values of K, when there are more than two radial cracks. It is noticed that the present solution leads to the same stress intensity factors for internal or external cracks if there is more than one crack evenly spaced along the circumference. As will be seen in the next section, for thin-walled cylinders the difference in K, between internal and external cracks is negligibly small. Also, for thick-walled cylinders of R/t = 1.5, eq. (9) is accurate enough for most engineering applications. We consider another configuration shown in Fig. 4 which is a thin-walled cylindrical simply closed body with two external cracks of equal size. The two cracks lie along the axis of symmetry. The analysis for the simply closed body of uniform wall-thickness is the same as that for a double cracked ring, i.e. only moments M are present at the ends of the element. From Castigliano’s theorem the compliance for the untracked body can be found as
(11) where S is the half length of the simply closed body. A comparison that the stress intensity factor for the thin-walled double cracked double cracked cylinder of the same periphery. This conclusion simply closed body with more than two radial cracks of equal thickness and is symmetric about every crack. RESULTS
of eq, (11) with eq. (7) indicates body is the same as that for a also applies to any thin-walled size if the body is of uniform
AND COMPARISON
Since there are few stress intensity factor solutions available in the literature for thin-walled cylinders and other simply closed bodies with more than one radial crack, we will only compare
WEILI
CHENG
and I. FINNIE 1._
OJ 0 Normalized Fig. 5. Comparison of the by Bowie and Frees: A[l] of R/r = 1.5subjected to a single
thickness
present solution with those given and Grandt O[l l] for a cylinder uniform external pressure P with internal crack.
0.2 0.4 Normalized
' 0.6 ' 0.8 thickness
0
Fig. 6. Comparison with Tracy’s results[2] for a cylinder of R/t = 1.5 with one A, two 0 or three * external radial cracks subjected to internal pressure P.
our results with those obtained for thick-walled cylinders. However, our approach should give better results for thin-walled cylinders. Bowie and Freese[l] first considered a thick-walled cylinder with an internal crack subjected to a uniform external tension. Figure 5 shows a comparison of our result with that given by Bowie and Freese for a cylinder of R/t = 1.5. The result obtained by Grandt[l l] with a weight function technique is also shown in the same figure. It seen that our result is closer to Bowie and Freese’s for small crack sizes while for large crack sizes it is closer to Grandt’s. The modulation in the predicted curve results from the use of different expressions for f”(a/t) and fo(a/t) above and below a/t = 0.6. Tracy[2] calculated stress intensity factors for cylinders with several external cracks subjected to a uniform internal pressure. For a cylinder of R/t = 1.5 the maximum number of radial cracks to which our approach applies is three. Figure 6 shows a comparison of our results with Tracy’s for 1, 2 and 3 cracks. The agreement is good for each case. Andrasic and Parker[3] presented their tabulated results for surface tractions in a form of power functions
in which z is measured from the inner surface for internal cracks and outer surface for external cracks. Figures 7-12 show comparisons of our results with numerical data tabulated in [3] for surface loading given in eq. (12) with i = O-4. It is seen, for R/t = 4.5, that the agreement is almost exact for all radial crack configurations except for a/t = 0.7 in Fig. 9. We suspect that
6 I=0
Oi,O
Normalized
thickness
Fig. 7. Comparison of results of Andrasic and Parker[3] for a cylinder of R/r = 1.5 with two internal radial cracks (A) or two external radial cracks (0) subjected to uniform, linear, quadratic, cubic and quartic stress distributions on crack faces (i = 0,1,2,3,4, respectively) with present predictions shown by solid lines.
04 0
0.2
0.4
Normalized
0.6
08
thickness
Fig. 8. Same as Fig. 7 for R/i = 4.5 and two external internal radial cracks.
or
773
Radial cracks in circular cylinders
I 0
i=O
/
0.2 0.4 Normolized
I
0.6
0.8
04
,
0
0.2
thickness
0.4
Normalized
Fig. 9. Same as Fig. 7 for a single external radial crack for R/t = 4.5.
0.6
I
0.8
thickness
Fig. 10. Same as Fig. 9 for a single internal radial crack for R/t = 4.5. 2.5
p’
i=O
-- 3 ;: z -” 2 a 3.-
E 6
y” 0.5 0
0.2
0.4
Normalized
0.6
0.8
1 4-i
0
0.2
0.4
Normalized
thickness
Fig. 11. Same as Fig. 9 for a cylinder of R/f = 1.833.
i=O
/
1.0
4.
5,
0
I
-z 2.0 \ c) _c 1.5
0.6
0.8
a0
thickness
Fig. 12. Same as Fig. 7 for single external radial crack for Rlt = 1.5.
Andrasic and Parker’s value at that point may be in error. Also, our results agree well with theirs for R/t ratios as low as 1.5 for all kinds of loading. For a double cracked cylinder of R/t = 4.5 Andrasic and Parker found almost no difference between an internal crack and an external crack and for a cylinder of R/t = 1.5 the difference becomes about seven per cent. Our results lie between their predictions. It is of interest to note that with the present approach it is possible to explain why an external crack leads to a larger value of the stress intensity factor than an internal one, why the value of Ki increases when a single cracked cylinder becomes a double cracked one and why it decreases when the number of cracks becomes more than two. CONCLUSION An approach is presented to calculate the stress intensity factors for radial cracks in circular cylinders and simply closed cylindrical bodies with geometry and loading conditions symmetric about each crack. It applies to a wide range of cylinders from thin-walled ones to those with R/t ratio as low as 1.5. Very good agreement with other numerical solutions is obtained for a number of configurations and loading conditions. A major advantage of the present method over commonly used methods is its simple computational procedure. Also, the present approach may be extended to include mode II loading. This should allow stress intensity factors to be obtained for simply closed cylindrical bodies with nonsymmetric geometry and loading conditions. REFERENCES [l] 0. L. Bowie and C. E. Freese, Analysis for a radial crack in a circular ring. Engng Fracture Me& 4, 315-321 (1972). [2] P. G. Tracy, Elastic analysis of radial cracks emanating from the outer and inner surfaces of a circular ring. Engng Fracfure Mech. 11, 291-300 (1979).
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WEILI CHENG and I. FINNIE
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