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Elastic analysis of radial cracks emanating from the outer and inner surfaces of a circular ring
liR#iw+
Fmdm
MJchmdcJ Vol. Il. pp. 29l-MO l%edh!QreatBritain
@P~F%~~Ltd..1979.
ELASTIC ANALYSIS OF RADIAL CRACKS EMANATING FROM THE OUTER AND INNER SURFACES OF A CIRCULAR RING PETER G. TRACY Army Materialsand MechanicsResearchCenter, Watertown,MA 02172,U.S.A. Abstract-The elastic solutionfor one to four radialcracks emanatingfrom eitherthe inneror outer surface of a circular ring is considered. For cracks emanati~ from the outer surface, internal pressure is considered,while for cracks emanati~ from the innersurface, the loadingis uniformexternaltension.The technique used for the solution is the modifiedmapping-collocationapproach.Data is presented for two wall ratios for each geometry.
INTRODUCTION
A CON~G~~ON commonly employed to contain materials at high pressures is the hollow cylinder. The classical solution developed by Lame (seeill) shows that the largest tangential stresses occur at the inner boundary. The problem of cracks emanating from the inner boundary with internal pressure has been previously studied for one crack[2] and for two equally spaced cracks [3]. Techniques which induce a residual compressive stress at the internal boundary have been developed to increase the amount of pressure which can be withstood by a cylinder. However, these techniques can also induce a residual tensile ~ngenti~ stress at the outer surface which combined with the stress due to the internal pressure may cause the formation of cracks at the outer surface. Several investigators have studied the problem of one crack emanating from the outer surface of a cylinder under internal pressure[44]. The purpose of this paper is the calculation of stress intensity factors for one to four radial cracks em~ating from the outer surface of a ring subjected to internal pressure. The stress intensity factors are also calculated for three and four cra$ks emanating from the inner surface of a ring subjected to uniform tension at the outer boundary which gives the same stress intensity factors as the case of pressure acting on the cracks and the inner surface of the ring. The technique used is the modified mapping-collocation method[9] combined with partitioning[lO, 111. This method combines facets of complex variable theory and the collocation method to provide the analyst with a ~ompu~tional tool of great versatility. ANALYSIS The complex variable formulation of Muskhelishvili[ 121is utilized and is briefly summarized. The functions (j?l(z) and t&(z) in the Muskhelishvih formulation are analytic functions of the complex variable z = x + iy, with x and y being the physical coordinates. In terms of cbl(z) and r/+(z) the stresses and displacements in Cartesian coordinates are expressable as a, + 0; = 2[4Xz) + &WI 0;
-
cr,
+
2irxy= 2[%$%2)+ #Xz)]
-2G(u + iv) = 7jqMz) - zt)Xz)- &(zf
11)
where -- E G - 2(1+ V) and r) = 3 - 4v
(plane strain),
3-v t+v
(plane stress),
E and v being Young’s Modulus and Poisson’s ratio, respectively. Here and throughout primes 291
P.GTRACY
292
denote differentiation and bars complex conjugation. The resultant force along an arc s can be expressed as -f,+if2=~~(z)+z~~z)+~~(z)=i (X,+iY,)ds (2) I s where X,ds and Y.ds are the x- and y-components, respectively, of the force acting normal to the element ds. If the f-plane along with the conformal mapping, z = o(p), is introduced, the equations for displacements and resultant force become f,+if,=
f$(l)+fl,-&
(0 + -W)
(3)
2G(u + iv) = q+(J) -fly 0’0 4 (5) - -4(I) where +(5) = W4))
and $(5) = +A45)).
The analytic continuation arguments developed by Muskhelishvili can be utilized to guarantee traction-free conditions on a surface in the physical plane. Let S’ denote the region in the t-plane above the real axis (i.e. where the imaginary part of l is greater than zero) and S- the region below the real axis, and define
4(l) = -
vm- g
where
q(p),
5E
s-
f(l) = f(4).
Then $(l) can be written as
and the resultant forces and displacement equations become
If c$({) is analytic in the part of the l-plane corresponding to the physical region (S’) plus the extended region (S), then the surface in the physical region corresponding to the real axis in the l-plane will be traction-free as can be seen by evaluating eqn (6) for real values of & The preceding argument provides the added advantage of eliminating 4(l) from the analysis, thus reducing the problem to finding only one unknown function, 4(t). It will be convenient to consider o(t) as the composition of two mapping functions w2and 0,. Thus 4)
=
02(43)).
(8)
and eqns (6) become (9) (10) In the Muskhelishvili formulation c$~(z) is considered to be analytic in z. Thus, b(l) is analytic in [ since o(c) is a conformal mapping and 1#45)can be represented as W) = %l”.
(11)
The method used to apply the complex variable formulation to the cracked ring problem is
293
Elastic analysis of radial cracks emanating from the outer and inner surfaces of a circular ring
the modified mapping-collocation method [9] combined with partitioning [ 10, 111. To carry out this plan, the geometry is subdivided into one or more zones, each having its own mapping function and expansion for d(l). The local expansions for #J(L) of adjacent zones are “stitched” along surfaces separating the zones by requiring continuity of integrated forces and displacements, giving conditions on the unknown coefficients of the expansions for 4(l) in adjacent zones. The boundary conditions on the boundaries and cracks are used to obtain further conditions on the coefficients in the expansions. A larger number of conditions than unknown coefficients, a,, is generated and a least squares minimization of error is employed to obtain a linear system of sumultaneous equations, which is then solved to find the coefficients, a., of eqn (11) for each zone. Once the unknown coefficients of t$(l) in the zones containing cracks are known it is a simple matter to calculate the stress intensity factors. Sih et al. [ 141have shown that
(12)
K = K, - iK2 = 2(2~)“* lim z:‘*&(z) Z-U,
where 4 is the coordinate of the crack tip in the z-plane referred to a coordinate system whose origin is .zcand whose real axis is parallel to the crack. Equation (12) can be written as K = KI - iK2 = 2(27r)“*e-” ,“T (w(l) - r,)“*$$ -c
(13)
where z, = ~(5) and a is the angle between the real axis and the line from the center of the crack tip of interest. MAPPING FUNCTIONS Several mapping functions will be used. to describe the geometries for the ring/crack configurations. In considering problems with cracks emanating from the inner surface of the ring we use the following two mappings: o,(l) = iue-ceie
wr(l) =
(14)
+tr -tr’).
(15)
As illustrated in Fig. 1, the first mapping function (14) maps the real axis in the t-plane onto an arc of radius a, with points above the real axis in the f-plane becoming points in the z-plane with radius greater than. a. The second mapping (15) takes the unit circle in the t-plane onto the crack of length 21, centered about the origin in the z-plane and lying on the imaginary axis, while points with radius larger than unity in the I-plane become points exterior to the crack in
6
F
A
Fig. 1. The mapping function Z = @r(g)= iae-6”.
E
B
D
C
294
P. G. TRACY
the z-plane as illustrated in Fig. 2. The mapping employed in zones with cracks emanating from the inner surface of the ring; is a composition of or and wrr with 8 = 0 and I, = In (a + I/a). That is, o(l)
= or(wrr(l))
=
~ae-(iW(c-c-l)
(16)
for these zones. This combination of mappings has been previously used in[l3]. In considering problems with cracks emanating from the outer surface we use on(l) along with orrr({) = We”.
(17)
As shown in Fig. 3, this mapping takes the real axis in the l-plane into points with radius b in the z-plane. Points above the real axis in the l-plane correspond to points in the z-plane with radius less than b. For zones with cracks emanating from the outer surface, a composition of wrlr and oIr is the mapping function. For these zones we use 8 = 0 and I, = In(b/b - I) to get ~(5) = OJ~~~(W~([))= ibei”.JZKC-C-‘).
(18)
Cracked ring problem definition The problems under consideration are shown in Fig. 4. The mappings in eqns (16) and (18) lead to the z- and C-planes as shown in Fig. 5. When w > 4L or L > w, convergence diflIculties can arise, particularly for longer cracks, as has been noted for similar situations in [ 101.For this reason the partitioning plans for one and two cracks emanating from the outer surface shown in Figs. 6-8 are introduced.
I
0 Fig. 2. The mapping function Z = or,([) = (UZ)(l- 5-l).
z
D
F F
A
Fig. 3. The mapping function Z = orI&) = ibe’%“.
E
8
D
c
Elastic analysis of radialcracks emanatingfrom the outer and innersurfaces of a circularring
Fig. 4. Circularring witb Muftipk equal lengthedge cracks emanatingfrom the innerand outer boundaries. b/a = 2,3; 1 s n ~4; II= no. of cracks.
Fii. 5. Physical and parameterplanes.
The p~itio~ plan in Fi. 6 is used for b/a = 3, The mapping function used for zone one is that of eqn (18).A simple pole in 4(g) at S = - i in this zone is anticipated and requires altering the form for d(l) in eqn (11) as follows: (B(I;)= ~$$bC
+ k+P’) &+i
(19)
with uk and ub+l both real. Only the real part of the coefficientsfor even A and tbe imaginary parts for odd n are retained since the others parts of the coefficients must be zero due to stress symmetry about the imaginary axis. The mapping function used for zone 2 of the partitioning plan of Fii. 6 is mm(g) with B= - W.That is, w(g) = - ibe?
(zo)
2%
Fig. 6. Partitioning planfor single edge crack at exterior surface of ring with b/a = 3.
Fii. 7. Partitioning plan for single edge crack at exterior surface of ring with b/a = 3.
For +5(l) the form in eqn (11) is used retaining only non-negative powers since the expansion point is in the region of interest. Thus, we get
(21) with ati and u&+~ both real and only the terms consistent with stress symmetry about the imaginary axis retained. When 6/a = 2, for the case of one crack coming from the outer surface, w and L of Fig. 5 are such that w > 4L. Therefore, the partitioning plan in Fig. 7 is introduced. Zones 1 and 2 of the partitioning plan of Fig. 7 have the same form for 4(g) and w(5) as do zones 1 and 2, respectively, of the partitioning plan of Fig 6. Zone 3 of this con&ration uses a mapping
Elastic analysis of radial cracks emanating from the outer aad inner surfaces of a circdar
0 * x/2
- w/n,
i?= number
ring
297
of crocka
Fig. 8. Partitioning plan for rings with multiple cracks. (I = ?r/2- n/n, II = no. of cracks.
function of the form of (17) with 6 = - ~12,giving o(C) = beY
(221
Because the region of interest contains the expansion point, the following form of 4(g) is appropriate for zone 3 of Fig. 7: (231 with a,, being complex, For problems with more than one crack we turn to the plan of Fig. 8, If the angle ~1as shown in Fig. 8 is taken such that the Q = nf:! - win, then along the surface A33 there is no shear ,and no tangential displacement due to the symmetry of the problem and the problems for ,rn~ltip~ecracks can be solved using only one zone. The mappi~ function and stress functions for this con~g~ation are defined in eqns (18) and (19) for problems with cracks coming from the outer surface. For cracks emanating from the inner surface of the hollow cylinder, the mapping of eqn (16) is used with (6(i) as defined in (19). For the case of b/a = 2, and two cracks coming from the outer surface, we again find that w and L of Fig* 5 are such that w > 4L which leads to convergence difficulties.Therefore, the p~tioning ptan of Fig. 9 will be required. Zone 1 of this plane as before uses the mapping function and stress function forms of (18)and (19).Zone 2 has the mapping function of (22),but due to stress symmetries in this problem, the stress function has the following form:
with 4% and as+% both real and only the non-negative‘powers of 5 retained because the expansion point is in the region of interest. Once the coefficients a, for each zone are determined, we can calculate K. In the above co&gurations for zones containing cracks, it can be seen that the angle (I in the expression for K in (13) must be
298
P. G.
TRACY
Fig. 9. Partitioningplan for ring with two edge cracks emanatingfrom exteriorboundarywith b/a = 2.
for an edge crack emanating from the outer and inner surfaces of the cylinder, respectively, This leads to the foUowing expressions for the stress intensity factors:
(W where lzClis the magnitude of z,.
For the case of cracks emanating from the external boundary, a pressure is applied on the inner surface, while for cracks coming from the internal boundary a positive stress normal to the outer surface is applied. It can be shown by evaluating (2) that for either of these boundary conditions the integrated forces, f f if*, at points on the boundary on which the load is applied can be expressed as fl+ifi=x+iy+c=z+c
127)
where c is complex and is the value of the integral in (2) on a path from the unloaded boundary to the loaded one. The cracks and unloaded boundary give f, + ifi = 0.
(281
Note that force conditions need only be written for the loaded surface and crack as the mapping functions and analytic continuation arguments used assure traction-free conditions on the unloaded surface. In addition, for the configuration of Fii. 8, the conditions of zero normal displacement and zero shear must be applied to surface AB. This implies that on that surface ff sina-ucosa=O and f, sina - ft cosa = 0. The stress intensity factors for one to four cracks emanating from either the outer or inner surface were calculated for wall ratios (Ma) of two and three. It is convenient to introduce the
Elastic analysis of radial cracks emanating from the outer and inner surfaces of a circular ring
299
function
where a, is the tangential stress, in the absence of a crack, at the surface from which the crack emanates. Thus, for cracks emanating from the outer surface.
and for cracks emanating from the inner surface
Results for small cracks at a free edge were checked using the well-known result that HI + 1.12 as I + 0. Results for cracks emanating from the inner surface when i + a 4 b were checked Table 1. Values of H,t for cracks at outer surface of a ring subjected to internal pressure (b/a = 3) 1 b-a
0.1 0.2 0.3 0.4 0.5 0.6 0.7
1 crack
2 cracks
3 cracks
4 cracks
1.23 1.37 1.54 1.82 2.12 2.43 3.11
I.23 1.37 1.58 1.80 2.06 2.38 2.75
1.23 1.34 I.48 1.61 1.78 1.99 2.36
1.23 I.31 1.38 1.46 1.58
Table 2. Values of H,f for cracks at outer surface of a ring subjected to internal pressure (b/a = 2) I b-a 0.1 0.2 0.3 0.4 0.5 0.6 0.7 f&
=
1 crack
2 cracks
3 cracks
4 cracks
1.20 1.37 1.56 MO 2.10 2.49
1.22 1.37 1.59 1.84 2.15 2.48 2.91
1.21 1.32 1.50 1.67 1.87 2.12 2.43
1.21 1.30 1.42 1.54 1.68 1.89 2.16
K1(b2 - “),
&
= 2
pfl2as Table 3. Values of HI? for cracks at inner surface of a ring subjected to uniform tension at outer surface (b/a = 3) 1 b-a 0.1 0.2 0.2 0.4 0.5
t 1 crack
0 2 cracks
3 cracks
4 cracks
0.941 0.871 0.841 0.836 0.847
0.974 0.974 1.01 1.08 1.18
0.954 0.904 0.907 0.937 1.07
0.932 0.854 0.827 0.837 0.908
tH,=Kl(b2--,b,a=3 p’lf;;72b2
SFrom Ref. [2] OFrom Ref. [3]
300
P. G. TRACY
using the calculations of Tweed and RookeM. The values of HI for the geometries for which K, was calculated are shown in Tables l-4. Results for one and two cracks coming from the inner surface of the ring are taken from[2, 31 and are included for comparison purposes. Table 4. Values of Hit for cracks at inner surface of a ring subjected to uniform tension at outer surface (b/o = 2) I
i
b-a
1 crack
0 2 cracks
4 cracks
I .05 1.04 I .08 1.12 1.18
1.06 1.11 1.21 1.33 1.50
1.03 1.03 1.04 1.12 1.31
0.1
0.2 0.3 0.4 0.5 tH,
=
Kl(b2-a2), bla
=
2
t&i2b2 *From Ref. [2] %FromRef. [3]
REF’ERENCES [1] C. T. Wang, Applied Elaslicity. McGraw-Hill, New York (1953). [2] 0. L. Bowie and C. E. Freese, Elastic analysis for a radial crack in a circular ring. U.S. AMMRC monograph M-70-3, Watertown, MA (1970). [3] 0. L. Bowie and C. E. Freese, private communication. [4] J. A. Kapp, The effect of autofrettate on fatigue crack propagation in externally flawed thick-walled disks, U.S. ARAJXOM Tech. Rep. ARCLB-TR-77025,Watervliet, N.Y. (1977). [S] A. F. Emery and C. M. Segedin, The evaluation of the stress intensity factors for cracks subjected to tension, torsion, and tlexure by an efficient numerical technique. 1. Basic Engng 94,387-393 (1972). 161A. S. Kobayashi, A simple procedure for estimating stress intensity factor in region of hiih stress gradient, Interim Tech. Rep. No. 1, U.S. Army Res. Grant No. DA-ARG-D31-124-73-638(1973). I71 P. S. Choora, Finite element fracture mechanics analysis of creep_ rupture of fuel element cladding. Nuclear Engng _ - _ and Des&t 29 (1974). PI A. S. Kobayashi,D. E. Maiden and B. J. Simon, Application of finite element analysis method to two-dimensional nrobkms in fracture mechanics. ASME Faner No. 69- WAIPVP-12 (1%9). [9] b. L. Bowie and D. M. Neal, A modified mapping collocation technique for the accurate calculation of stress intensity factors. In?. J. Fracture Mech. 6, 199-206(1970). [lo] 0. L. Bowie, and C. E. Freese and D. M. Neal, Solutions of plane problems of elasticity utilizing partitioning concepts. L 4~1. hfech. %,767-772 (1973). [ll] 0. L. Bowie, Application of partitioning to problem solving in elasticity, Proc. Hong Kong Conf. (24-25 March 1977). [12] N. I. MuskheIishviIi, Some Basic Problems of the Mathematical Theory of Elasticity. Noordboff, Groaingen, Holland (1953). [I31 P. G. Tracy, Analysis of a radial crack in a circular ring segment. Engng Fracture Mech. 7,253-260 (1975). [14] G. C. Sih, P. C. Paris and F. Erdogan, Crack tip stress-intensity factors for plane extension and plate bending problems. 1. Appl. Mech. 29,306 (1%2). [15] J. Tweed and D. P. Rooke. The stress intensity factor for a crack in a symmetric array originating at a circular hole in an infinite elastic solid. Int. J. Engng Sci. 13,653-661 (1975). (Received 13 January 1978;received for publication 6 March 1978)
Fmdm
MJchmdcJ Vol. Il. pp. 29l-MO l%edh!QreatBritain
@P~F%~~Ltd..1979.
ELASTIC ANALYSIS OF RADIAL CRACKS EMANATING FROM THE OUTER AND INNER SURFACES OF A CIRCULAR RING PETER G. TRACY Army Materialsand MechanicsResearchCenter, Watertown,MA 02172,U.S.A. Abstract-The elastic solutionfor one to four radialcracks emanatingfrom eitherthe inneror outer surface of a circular ring is considered. For cracks emanati~ from the outer surface, internal pressure is considered,while for cracks emanati~ from the innersurface, the loadingis uniformexternaltension.The technique used for the solution is the modifiedmapping-collocationapproach.Data is presented for two wall ratios for each geometry.
INTRODUCTION
A CON~G~~ON commonly employed to contain materials at high pressures is the hollow cylinder. The classical solution developed by Lame (seeill) shows that the largest tangential stresses occur at the inner boundary. The problem of cracks emanating from the inner boundary with internal pressure has been previously studied for one crack[2] and for two equally spaced cracks [3]. Techniques which induce a residual compressive stress at the internal boundary have been developed to increase the amount of pressure which can be withstood by a cylinder. However, these techniques can also induce a residual tensile ~ngenti~ stress at the outer surface which combined with the stress due to the internal pressure may cause the formation of cracks at the outer surface. Several investigators have studied the problem of one crack emanating from the outer surface of a cylinder under internal pressure[44]. The purpose of this paper is the calculation of stress intensity factors for one to four radial cracks em~ating from the outer surface of a ring subjected to internal pressure. The stress intensity factors are also calculated for three and four cra$ks emanating from the inner surface of a ring subjected to uniform tension at the outer boundary which gives the same stress intensity factors as the case of pressure acting on the cracks and the inner surface of the ring. The technique used is the modified mapping-collocation method[9] combined with partitioning[lO, 111. This method combines facets of complex variable theory and the collocation method to provide the analyst with a ~ompu~tional tool of great versatility. ANALYSIS The complex variable formulation of Muskhelishvili[ 121is utilized and is briefly summarized. The functions (j?l(z) and t&(z) in the Muskhelishvih formulation are analytic functions of the complex variable z = x + iy, with x and y being the physical coordinates. In terms of cbl(z) and r/+(z) the stresses and displacements in Cartesian coordinates are expressable as a, + 0; = 2[4Xz) + &WI 0;
-
cr,
+
2irxy= 2[%$%2)+ #Xz)]
-2G(u + iv) = 7jqMz) - zt)Xz)- &(zf
11)
where -- E G - 2(1+ V) and r) = 3 - 4v
(plane strain),
3-v t+v
(plane stress),
E and v being Young’s Modulus and Poisson’s ratio, respectively. Here and throughout primes 291
P.GTRACY
292
denote differentiation and bars complex conjugation. The resultant force along an arc s can be expressed as -f,+if2=~~(z)+z~~z)+~~(z)=i (X,+iY,)ds (2) I s where X,ds and Y.ds are the x- and y-components, respectively, of the force acting normal to the element ds. If the f-plane along with the conformal mapping, z = o(p), is introduced, the equations for displacements and resultant force become f,+if,=
f$(l)+fl,-&
(0 + -W)
(3)
2G(u + iv) = q+(J) -fly 0’0 4 (5) - -4(I) where +(5) = W4))
and $(5) = +A45)).
The analytic continuation arguments developed by Muskhelishvili can be utilized to guarantee traction-free conditions on a surface in the physical plane. Let S’ denote the region in the t-plane above the real axis (i.e. where the imaginary part of l is greater than zero) and S- the region below the real axis, and define
4(l) = -
vm- g
where
q(p),
5E
s-
f(l) = f(4).
Then $(l) can be written as
and the resultant forces and displacement equations become
If c$({) is analytic in the part of the l-plane corresponding to the physical region (S’) plus the extended region (S), then the surface in the physical region corresponding to the real axis in the l-plane will be traction-free as can be seen by evaluating eqn (6) for real values of & The preceding argument provides the added advantage of eliminating 4(l) from the analysis, thus reducing the problem to finding only one unknown function, 4(t). It will be convenient to consider o(t) as the composition of two mapping functions w2and 0,. Thus 4)
=
02(43)).
(8)
and eqns (6) become (9) (10) In the Muskhelishvili formulation c$~(z) is considered to be analytic in z. Thus, b(l) is analytic in [ since o(c) is a conformal mapping and 1#45)can be represented as W) = %l”.
(11)
The method used to apply the complex variable formulation to the cracked ring problem is
293
Elastic analysis of radial cracks emanating from the outer and inner surfaces of a circular ring
the modified mapping-collocation method [9] combined with partitioning [ 10, 111. To carry out this plan, the geometry is subdivided into one or more zones, each having its own mapping function and expansion for d(l). The local expansions for #J(L) of adjacent zones are “stitched” along surfaces separating the zones by requiring continuity of integrated forces and displacements, giving conditions on the unknown coefficients of the expansions for 4(l) in adjacent zones. The boundary conditions on the boundaries and cracks are used to obtain further conditions on the coefficients in the expansions. A larger number of conditions than unknown coefficients, a,, is generated and a least squares minimization of error is employed to obtain a linear system of sumultaneous equations, which is then solved to find the coefficients, a., of eqn (11) for each zone. Once the unknown coefficients of t$(l) in the zones containing cracks are known it is a simple matter to calculate the stress intensity factors. Sih et al. [ 141have shown that
(12)
K = K, - iK2 = 2(2~)“* lim z:‘*&(z) Z-U,
where 4 is the coordinate of the crack tip in the z-plane referred to a coordinate system whose origin is .zcand whose real axis is parallel to the crack. Equation (12) can be written as K = KI - iK2 = 2(27r)“*e-” ,“T (w(l) - r,)“*$$ -c
(13)
where z, = ~(5) and a is the angle between the real axis and the line from the center of the crack tip of interest. MAPPING FUNCTIONS Several mapping functions will be used. to describe the geometries for the ring/crack configurations. In considering problems with cracks emanating from the inner surface of the ring we use the following two mappings: o,(l) = iue-ceie
wr(l) =
(14)
+tr -tr’).
(15)
As illustrated in Fig. 1, the first mapping function (14) maps the real axis in the t-plane onto an arc of radius a, with points above the real axis in the f-plane becoming points in the z-plane with radius greater than. a. The second mapping (15) takes the unit circle in the t-plane onto the crack of length 21, centered about the origin in the z-plane and lying on the imaginary axis, while points with radius larger than unity in the I-plane become points exterior to the crack in
6
F
A
Fig. 1. The mapping function Z = @r(g)= iae-6”.
E
B
D
C
294
P. G. TRACY
the z-plane as illustrated in Fig. 2. The mapping employed in zones with cracks emanating from the inner surface of the ring; is a composition of or and wrr with 8 = 0 and I, = In (a + I/a). That is, o(l)
= or(wrr(l))
=
~ae-(iW(c-c-l)
(16)
for these zones. This combination of mappings has been previously used in[l3]. In considering problems with cracks emanating from the outer surface we use on(l) along with orrr({) = We”.
(17)
As shown in Fig. 3, this mapping takes the real axis in the l-plane into points with radius b in the z-plane. Points above the real axis in the l-plane correspond to points in the z-plane with radius less than b. For zones with cracks emanating from the outer surface, a composition of wrlr and oIr is the mapping function. For these zones we use 8 = 0 and I, = In(b/b - I) to get ~(5) = OJ~~~(W~([))= ibei”.JZKC-C-‘).
(18)
Cracked ring problem definition The problems under consideration are shown in Fig. 4. The mappings in eqns (16) and (18) lead to the z- and C-planes as shown in Fig. 5. When w > 4L or L > w, convergence diflIculties can arise, particularly for longer cracks, as has been noted for similar situations in [ 101.For this reason the partitioning plans for one and two cracks emanating from the outer surface shown in Figs. 6-8 are introduced.
I
0 Fig. 2. The mapping function Z = or,([) = (UZ)(l- 5-l).
z
D
F F
A
Fig. 3. The mapping function Z = orI&) = ibe’%“.
E
8
D
c
Elastic analysis of radialcracks emanatingfrom the outer and innersurfaces of a circularring
Fig. 4. Circularring witb Muftipk equal lengthedge cracks emanatingfrom the innerand outer boundaries. b/a = 2,3; 1 s n ~4; II= no. of cracks.
Fii. 5. Physical and parameterplanes.
The p~itio~ plan in Fi. 6 is used for b/a = 3, The mapping function used for zone one is that of eqn (18).A simple pole in 4(g) at S = - i in this zone is anticipated and requires altering the form for d(l) in eqn (11) as follows: (B(I;)= ~$$bC
+ k+P’) &+i
(19)
with uk and ub+l both real. Only the real part of the coefficientsfor even A and tbe imaginary parts for odd n are retained since the others parts of the coefficients must be zero due to stress symmetry about the imaginary axis. The mapping function used for zone 2 of the partitioning plan of Fii. 6 is mm(g) with B= - W.That is, w(g) = - ibe?
(zo)
2%
Fig. 6. Partitioning planfor single edge crack at exterior surface of ring with b/a = 3.
Fii. 7. Partitioning plan for single edge crack at exterior surface of ring with b/a = 3.
For +5(l) the form in eqn (11) is used retaining only non-negative powers since the expansion point is in the region of interest. Thus, we get
(21) with ati and u&+~ both real and only the terms consistent with stress symmetry about the imaginary axis retained. When 6/a = 2, for the case of one crack coming from the outer surface, w and L of Fig. 5 are such that w > 4L. Therefore, the partitioning plan in Fig. 7 is introduced. Zones 1 and 2 of the partitioning plan of Fig. 7 have the same form for 4(g) and w(5) as do zones 1 and 2, respectively, of the partitioning plan of Fig 6. Zone 3 of this con&ration uses a mapping
Elastic analysis of radial cracks emanating from the outer aad inner surfaces of a circdar
0 * x/2
- w/n,
i?= number
ring
297
of crocka
Fig. 8. Partitioning plan for rings with multiple cracks. (I = ?r/2- n/n, II = no. of cracks.
function of the form of (17) with 6 = - ~12,giving o(C) = beY
(221
Because the region of interest contains the expansion point, the following form of 4(g) is appropriate for zone 3 of Fig. 7: (231 with a,, being complex, For problems with more than one crack we turn to the plan of Fig. 8, If the angle ~1as shown in Fig. 8 is taken such that the Q = nf:! - win, then along the surface A33 there is no shear ,and no tangential displacement due to the symmetry of the problem and the problems for ,rn~ltip~ecracks can be solved using only one zone. The mappi~ function and stress functions for this con~g~ation are defined in eqns (18) and (19) for problems with cracks coming from the outer surface. For cracks emanating from the inner surface of the hollow cylinder, the mapping of eqn (16) is used with (6(i) as defined in (19). For the case of b/a = 2, and two cracks coming from the outer surface, we again find that w and L of Fig* 5 are such that w > 4L which leads to convergence difficulties.Therefore, the p~tioning ptan of Fig. 9 will be required. Zone 1 of this plane as before uses the mapping function and stress function forms of (18)and (19).Zone 2 has the mapping function of (22),but due to stress symmetries in this problem, the stress function has the following form:
with 4% and as+% both real and only the non-negative‘powers of 5 retained because the expansion point is in the region of interest. Once the coefficients a, for each zone are determined, we can calculate K. In the above co&gurations for zones containing cracks, it can be seen that the angle (I in the expression for K in (13) must be
298
P. G.
TRACY
Fig. 9. Partitioningplan for ring with two edge cracks emanatingfrom exteriorboundarywith b/a = 2.
for an edge crack emanating from the outer and inner surfaces of the cylinder, respectively, This leads to the foUowing expressions for the stress intensity factors:
(W where lzClis the magnitude of z,.
For the case of cracks emanating from the external boundary, a pressure is applied on the inner surface, while for cracks coming from the internal boundary a positive stress normal to the outer surface is applied. It can be shown by evaluating (2) that for either of these boundary conditions the integrated forces, f f if*, at points on the boundary on which the load is applied can be expressed as fl+ifi=x+iy+c=z+c
127)
where c is complex and is the value of the integral in (2) on a path from the unloaded boundary to the loaded one. The cracks and unloaded boundary give f, + ifi = 0.
(281
Note that force conditions need only be written for the loaded surface and crack as the mapping functions and analytic continuation arguments used assure traction-free conditions on the unloaded surface. In addition, for the configuration of Fii. 8, the conditions of zero normal displacement and zero shear must be applied to surface AB. This implies that on that surface ff sina-ucosa=O and f, sina - ft cosa = 0. The stress intensity factors for one to four cracks emanating from either the outer or inner surface were calculated for wall ratios (Ma) of two and three. It is convenient to introduce the
Elastic analysis of radial cracks emanating from the outer and inner surfaces of a circular ring
299
function
where a, is the tangential stress, in the absence of a crack, at the surface from which the crack emanates. Thus, for cracks emanating from the outer surface.
and for cracks emanating from the inner surface
Results for small cracks at a free edge were checked using the well-known result that HI + 1.12 as I + 0. Results for cracks emanating from the inner surface when i + a 4 b were checked Table 1. Values of H,t for cracks at outer surface of a ring subjected to internal pressure (b/a = 3) 1 b-a
0.1 0.2 0.3 0.4 0.5 0.6 0.7
1 crack
2 cracks
3 cracks
4 cracks
1.23 1.37 1.54 1.82 2.12 2.43 3.11
I.23 1.37 1.58 1.80 2.06 2.38 2.75
1.23 1.34 I.48 1.61 1.78 1.99 2.36
1.23 I.31 1.38 1.46 1.58
Table 2. Values of H,f for cracks at outer surface of a ring subjected to internal pressure (b/a = 2) I b-a 0.1 0.2 0.3 0.4 0.5 0.6 0.7 f&
=
1 crack
2 cracks
3 cracks
4 cracks
1.20 1.37 1.56 MO 2.10 2.49
1.22 1.37 1.59 1.84 2.15 2.48 2.91
1.21 1.32 1.50 1.67 1.87 2.12 2.43
1.21 1.30 1.42 1.54 1.68 1.89 2.16
K1(b2 - “),
&
= 2
pfl2as Table 3. Values of HI? for cracks at inner surface of a ring subjected to uniform tension at outer surface (b/a = 3) 1 b-a 0.1 0.2 0.2 0.4 0.5
t 1 crack
0 2 cracks
3 cracks
4 cracks
0.941 0.871 0.841 0.836 0.847
0.974 0.974 1.01 1.08 1.18
0.954 0.904 0.907 0.937 1.07
0.932 0.854 0.827 0.837 0.908
tH,=Kl(b2--,b,a=3 p’lf;;72b2
SFrom Ref. [2] OFrom Ref. [3]
300
P. G. TRACY
using the calculations of Tweed and RookeM. The values of HI for the geometries for which K, was calculated are shown in Tables l-4. Results for one and two cracks coming from the inner surface of the ring are taken from[2, 31 and are included for comparison purposes. Table 4. Values of Hit for cracks at inner surface of a ring subjected to uniform tension at outer surface (b/o = 2) I
i
b-a
1 crack
0 2 cracks
4 cracks
I .05 1.04 I .08 1.12 1.18
1.06 1.11 1.21 1.33 1.50
1.03 1.03 1.04 1.12 1.31
0.1
0.2 0.3 0.4 0.5 tH,
=
Kl(b2-a2), bla
=
2
t&i2b2 *From Ref. [2] %FromRef. [3]
REF’ERENCES [1] C. T. Wang, Applied Elaslicity. McGraw-Hill, New York (1953). [2] 0. L. Bowie and C. E. Freese, Elastic analysis for a radial crack in a circular ring. U.S. AMMRC monograph M-70-3, Watertown, MA (1970). [3] 0. L. Bowie and C. E. Freese, private communication. [4] J. A. Kapp, The effect of autofrettate on fatigue crack propagation in externally flawed thick-walled disks, U.S. ARAJXOM Tech. Rep. ARCLB-TR-77025,Watervliet, N.Y. (1977). [S] A. F. Emery and C. M. Segedin, The evaluation of the stress intensity factors for cracks subjected to tension, torsion, and tlexure by an efficient numerical technique. 1. Basic Engng 94,387-393 (1972). 161A. S. Kobayashi, A simple procedure for estimating stress intensity factor in region of hiih stress gradient, Interim Tech. Rep. No. 1, U.S. Army Res. Grant No. DA-ARG-D31-124-73-638(1973). I71 P. S. Choora, Finite element fracture mechanics analysis of creep_ rupture of fuel element cladding. Nuclear Engng _ - _ and Des&t 29 (1974). PI A. S. Kobayashi,D. E. Maiden and B. J. Simon, Application of finite element analysis method to two-dimensional nrobkms in fracture mechanics. ASME Faner No. 69- WAIPVP-12 (1%9). [9] b. L. Bowie and D. M. Neal, A modified mapping collocation technique for the accurate calculation of stress intensity factors. In?. J. Fracture Mech. 6, 199-206(1970). [lo] 0. L. Bowie, and C. E. Freese and D. M. Neal, Solutions of plane problems of elasticity utilizing partitioning concepts. L 4~1. hfech. %,767-772 (1973). [ll] 0. L. Bowie, Application of partitioning to problem solving in elasticity, Proc. Hong Kong Conf. (24-25 March 1977). [12] N. I. MuskheIishviIi, Some Basic Problems of the Mathematical Theory of Elasticity. Noordboff, Groaingen, Holland (1953). [I31 P. G. Tracy, Analysis of a radial crack in a circular ring segment. Engng Fracture Mech. 7,253-260 (1975). [14] G. C. Sih, P. C. Paris and F. Erdogan, Crack tip stress-intensity factors for plane extension and plate bending problems. 1. Appl. Mech. 29,306 (1%2). [15] J. Tweed and D. P. Rooke. The stress intensity factor for a crack in a symmetric array originating at a circular hole in an infinite elastic solid. Int. J. Engng Sci. 13,653-661 (1975). (Received 13 January 1978;received for publication 6 March 1978)