External cracks under longitudinal shear

External Cracks Under Longitudinal Shear by

GEOI~GE C. SII-I a

Cali[ornia Institute o[ Technology Pasadena, California ABSTRACT: The theory of cracks under longitudinal shear, as discussed in previous publications (1, 2, 3), is extended to bodies containing external cracks. It is possible, by

an adaptation of the Schwartz-Christoffel transformation, to derive a general formula for finding the stress-intensity factor, a parameter governing the onset of rapid fracture in the theory of Gri:~th-Irwin (4). Specific examples of fundamental interest are solved in closed-form. The results are useful in predicting the remaining strength of structural members with crack-like imperfections.

Introduction

Considered in this paper are some two-dimensional problems of cracks originating from the external surface of a cylindrical body owing to out-ofplane shear directed along the generators of the cylinder. This type of loading will henceforth be referred to as "longitudinal shear." General solutions of longitudinal shear cracks were given in (1) and (3) for the two-dimensional region outside a simply connected crack boundary. Using the technique of conformal mapping, the comparison region was outside a circle (1). The half-plane was used in (3) because of its convenience in developing certain basic properties of the mapping function. It was pointed o u t that the Schwartz-Christoffel transformation can be slightly modified to facilitate the analytical developments. For cracks extending from an external surface, the Schwartz-Christoffel transformation can be applied in a direct manner without loss in mathematical expedience. If the crack boundaries are multiply connected, it is far simpler to use the complex function theory of Muskhelishvili (5) compared with the method of conformal transformation. In an earlier paper (2), it was shown that, in the case of a finite or infinite number of collinear cracks under longitudinal shear, the problem may be formulated as one in linear relationship by means of the Plemelj formulae and Cauchy integrals. Related to the problem of longitudinal shear cracks is the more complicated one of cracks in cylindrical bars subjected to flexural and torsional loads. On the basis of a complex variable method, stress solutions for the flexure and torsion of circular and elliptical bars imbedded with cracks were obtained (6). * On leave of absence from Lehigh University, Bethlehem, Pa.

139

George C. Sih

The case of a rectangular beam with longitudinal surface cracks under flexure is dicussed in (7). In (6) and (7), the results are complicated functions of the geometry of the problem. Hence, it is difficult to examine certain effects, such as the influence of section dimensions on the critical crack length. For cracks under longitudinal shear, it is possible to obtain relatively simple solutions and to provide insight for analogous but more difficult problems of cracks in flexure and torsion, as well as in plane extension. In this paper, attention is focused on the out-of-plane shear stresses in the vicinity of the crack tip since they are of importance in the formulation of fracture theories. More specifically, they are of the forms (4), Fig. 1, ka Txt

0 sin ~ + 0 (r'~),

(1)

k3 0 ~'~ = ~ r cos ~ + 0 (r~l~),

Y

/~?Ext ~yt

CRACK

X

0

Fio. 1. Rectangular components of longitudinal shear stresses.

where higher order terms in r, the radial distance from the crack front, have been neglected. The parameter k3 is generally referred to as the stress-intensity factor. Since k3 is not dependent on the polar coordinates r and 0, it effectively controls the intensity of the crack-tip stress field in a manner found later. It is well-known that unstable crack extension is governed by the critical values of k3. As originally proposed by Irwin (4), k3 may be computed through the work done by the crack surface tractions times the corresponding displacement required to close the crack by an increment ~, i.e., ka2 = 2G Lim 1 7¢

e---*o

f:

ryt(e -- x, O)w(x, 7r)dx.

(2)

Here, G is the shear modulus of elasticity and w is the displacement along the t-axis which coincides with the center line of the cylinder. For the crack problem, w is given by 0 w = ~k~ (2r)1/2 sin 2"

140

(3)

Journal of The Franklin Institute

External Cracks Under Longitudinal Shear

Alternatively, a more convenient way of finding ka has been derived in (6). By transforming the singular crack points zj into regular points f~. in the mapped plane, k~ can be computed from k3 = i G

) E~"F ' ((ff j~)-]'~"

(4)

Equation 4 indicates that the evaluation of ka requires a knowledge of the stress function F' (f) and the mapping function o~(f) evaluated at the crack tip. We shall further show that the Schwartz-Christoffel transformation may be directly incorporated into Eq. 4 without loss in generality. The function F' (i') is then derived for the case of concentrated shear loads. Therefore, once the mapping function co(f) is known, k3 is completely determined. The method of evaluation is illustrated in a number of basic example problems. The results are also discussed in connection with the Griffith-Irwin theory of fracture.

Stress-Intensity Factor Formula Consider the problem of mapping a polygonal boundary with a crack in the z-plane onto the real axis in the f-plane. To do this, each vertex of the polygon is transformed to a point on the G-axis, and at the same time convert the corresponding angle of the polygon into a straight angle. In general, it is permissible to have one or more of the vertices lie at infinity. The resulting transformation is known as the Schwartz-Christoffel transformation which may be written in the form (5)

where p ~ = - - - - 1. 7F

The image points G, ~'1, • • ", G, on the real axis in the mapped plane correspond to the vertices Zo, z l , • • . , z~, with interior angles no, a i , • • ", an, in the physical plane. The complex constants A and B, associated with the size and position of the polygon, may always be chosen so that any prescribed polygon in the z-plane is made to correspond point by point to the upper half of the f-plane. In fact, the correspondence can be set up in infinitely many ways, in that three of the numbers fs can be determined arbitrarily. Upon substitution of Eq. 5 into Eq. 4, there results a general formula for finding k3. Since the derivation involves a considerable amount of algebra, only the final result will be given, k~ = 2 - m F (fo)

{-E'~,H=o

(to - f ; P ; J

(fo - f;)

]I

(6)

where ~'o is the image of the crack tip zo with ao = 2~r. Note that the leading term, j = o, in the summation series has a simple pole at fj = G. To remove

Vol. 280, No. 2, August 1965

14I

George C. Sih this singularity, the summation series may be multiplied by the first term in the product series for j = o, where the exponent po = 1. Hence, with the exception of the term containing po, the remaining terms in the summation series have no contribution. As a result, Eq. 6 simplifies to ka = ~

F'(fo)[~[ (~'o -- ~i)~] -1/2.

(7)

j=l

Thus, the problem is now reduced to determining fj and the stress function F(~) corresponds to a given load. This will be discussed in the following sections. C o n c e n t r a t e d Shear Loads The determination of F(~) involving concentrated loads in elasticity is analogous1 to the problem of sources and sinks in hydrodynamics. As long as the concentrated loads, P, are self-equilibrating and applied to the boundary of the polygon in the z-plane, it suffices to consider, say, m sources and m sinks located at ~, on the negative ~-axis and ~8 on the positive ~-axis, respectively. Both the source and sink are assumed to have the same strength IP/lrGI. In view of the simple analogy between plane hydrodynamics and the longitudinal shear problem, F' (i') can be constructed without difficulty,

P[~

F'(0=V0

~;+~

1

~=~ 1 ] oEr-~.

•

(8)

If the sources and sinks are placed symmetrically with respect to the ~-axis in the f-plane, then Eq. 8 becomes F ! (~') "~

2P ~m ~-2 f~ "/I'(~ r=l

--

~.j.

(9)

In Eq. 9, m stands for the total number of pairs of source and sink. The case of uniform shear loads will not be treated separately since it may be regarded as a special case of the more general solution of concentrated loads. It will be subsequently shown that the concentrated load solution can be taken as a Green's function to obtain the results of crack problems involving uniform loads. Cracked Wedge Figure 2 shows two intersecting radial edges with included angle 2~ (1 -- a). A straight crack of length a enters at the point of intersection and is placed symmetrically with respect to the radial edges. Two equal and opposite concentrated forces P are applied at the points z = =t= 0. This configuration is to be mapped onto the upper half of the F-plane in accordance with the SchwartzChristoffel transformation. The required information is given as follows: 1 T h e d i s p l a c e m e n t w c o r r e s p o n d s to t h e velocity p o t e n t i a l a n d t h e stress v e c t o r t to t h e velocity vector.

142

Journal of The Franklin Institute

External Cracks Under Longitudinal Shear TABLE I j

zi

'~i

~'J

0

2v ~ (1 - - a ) Ir (1 - - a )

-- 1

--a

2

a 0 0

1

--a

3

a¢

--

1

PJ

0

1

59

--

Inserting Eq. 9 for m = 1 into Eq. 7 gives

2iP

ks =

~

A~/2

FII (~o - ~j)pf]-l/2.

o2 _ i-2

(10)

j=t

F o r this problem, ~o = 0, ~', = 1 and A = -- 2a(1 -- a)ei~% Using T a b l e I, it is found t h a t ks

-

~/2P 7r~fa(1 - - a ) '

0 _< a <

(11)

1.

T w o configurations of interest are readily attained b y setting a = 0 and a = ½. T h e first corresponds to the special case of a semi-infinite crack imbedded in an infinite m e d i u m and the second to the case of a half-plane with a crack extending perpendicularly f r o m the straight edge. F o r other values of a, the stress-intensity factor ks m a y be obtained from the graph shown in Fig. 2. Figure 2 shows t h a t as the half wedge angle, 7r(1 - a), decreases, the value of ks increases appreciably. T h e physical meaning of this result is that, according to the Griffith-Irwin fracture theory, unstable crack extension 5.0

Y

4,C n 5.C

.2 -,-- 2.0 N 1.0

0.0 0°

2__ I

[

20 °

40 =

"~,L-.>,P \

..--.'co "'T = ,

"

o

/ L

I 60 =

.

.

.

I

F--k-j"G~ .

.

.

.

.

.

.

.

.

.

x

l .

.

.

.

.

'[

1

I

I

80 °

io0 °

120 °

140 °

-J

I 160"

180 °

FIG. 2. Wedge with a crack at the~apex.

Vol. 280, No. 2, August 1965

143

George C. Sih

Y

4..C

I

5.C Za

2£

X

ZlOZ2 1.0

O.O 0°

1

I

1

I

I

I

I

I

I0°

20 °

50 °

40 °

50 °

60 °

70 °

80 °

90 °

lTcc

FIG. 3. Crack entering a half-plane. is more likely to occur at the larger values of ka for a given material or at the smaller values of ~(1 -- a). Half-Plane with an Inclined Crack

A slight variation of the preceding example is t h a t of a crack which leaves the free surface of a half-plane at an arbitrary angle ~r% Fig. 3. The crack is stressed by oppositely directed concentrated forces P. Without going into details, the properties of the mapping function are given b y Table I I and the constant A in the Schwartz-Christoffel transformation is

Recall that Eq. 7 is valid only when the crack tip lies on the positive x-axis, Fig. 1. Hence, it must be modified to accommodate the present case, where the tip of thd crack is situated at an angle ~a with reference to the x-axis. A rotation of coordinate axes shows t h a t a factor of e ~"/2 should be inserted in Eq. 7. TABLE I I

j

zj

aj

i'i

p~

0

ae i~a

2~r

0

1

o

144

2

0

3

co

--

~ra - -

--Ol

1

a--1

¢o

- -

Journal of The Franklin. Institute

External Cracks Under Longitudinal Shear I t follows t h a t letting j = 1, 2 in Eq. 7 yields P

1

4~(i

~~ = ~

~)'

-

i.

0 ~ ~ <

(12)

N o t e t h a t in the special case of a = ½, Eq. 12 agrees with: Eq. 11 as the crack configurations in b o t h problems are the same. T h e variation of k~ with the angle 7ra is plotted in Fig. 3. Semi-Infinite

C r a c k in a B e a m

The problem of a rectangular b e a m of finite height 2~ in the y-direction and of infinite width in the x-direction is considered. The b e a m is cracked along the line y = ~, 0 _> x > - ~ , on which two forces P are applied at x = - a. T h e locations of P in the m a p p e d plane are ~- (1 -- e-~) 1/2. The other points of interest are shown in T a b l e I I I . . TABLE

j

z]

aj

0

~ri

1

-- ~o

2

--oo

2~ 0 0

3

oo

III ~'j

p~"

0

1

--I

--I

--i

1

--

oo

--

Using Eqs. 7 and 9 b y substituting A = 2 and j = 1,2, k3 is evaluated as V2pe~/2 k~ = ~ r 4 J - - 1

(13)

I t is interesting to note how the height of the b e a m affects the magnitude of the stress-intensity factor k3. For this purpose, refer to the curves in Fig. 4. 3.C

y

2o

z,[ "~\ -,~ \

I~. -~w2.C

Fini ,.c 0,0

Z

,,~

---

Infinite

I 0.1

I 0.2

'

, ~ !

Height

®,

iz°

,..vv 7~ I---Cl---t --

o

I I

I

'

"'"'"'"-

~.......

Height ,,I 0.3

I 0.4

I 0.5

X

I 0.6

I 0.7

I 0,8

I 0.9

1,0

El

FIG. 4. Longitudinal crack in a beam.

Vol. 280, No. 2, AuguSt 196S

145

George C. Sih

Zs

_

Ua

_

Z4

I

:s ~

Z20 Z3

b

1

X

FIG. 5. Concentrated forces on an edge crack• The solid curve was drawn for a b e a m of finite height from Eq. 13 and the dotted curve was intended as a b e a m of infinite height according to Eq. 11 for a = 0. As expected, k3 for a b e a m of finite height is always higher t h a n t h a t for a b e a m of infinite height. I n fact, a significant difference m a y be observed when a becomes sufficiently large. I t is seen f r o m Eq. 13 t h a t as a --+ oo, k3 for the finite-height b e a m tends to a limiting value of V 2 P / z whereas in the case of an infinite-height b e a m k8 approaches zero. Rectangular

Beam

with

a Saw-Cut

T h e fracture strength of a rectangular b e a m containing a saw-cut or crack at right angles to one of its edges can be estimated b y a stress analysis based on the concept of stress-intensity factor. T h e b e a m occupies the two-dimensional region 0 < y < 1 and - ~¢ < x < ~ with a crack running from z = 0 to z = ia, Fig. 5. T h e position of the concentrated forces P at z = ib on the crack TABLE IV

j

zy

c~j

~'g

pj

0 1

ia --~

27r 0

0 --1

1 --1

2

0

~/2

- sin

3

0

~/2

4

~

0

•

~'a

•

7ra

_

sm 2

1

½

- ½

-1

2 ira

surface corresponds to a source and sink at ~'~= 4- [ 1-- (cos - 2 - ) / ( c o s 2 - ~ ) ] 1/2. The correspondence between the various points of the cracked b e a m in the z-plane to those in the ~'-plane can be best shown in t a b u l a t e d form• Here, A is found to be 2

~ra

A = --lrcos~.

146

Journal of The Franklin Institute

External Cracks Under Longitudinal Shear

Following through the same procedures of evaluation as before by having j = 1, 2, 3, 4 in Eq. 7, k3 is obtained. k3

7rb COS ~- ( t a n 2)1]2

( 2 ) 1'2

(14)

P/cos~ ~~'b- - cos 2 ~7ra \~/2" - ) L

=

The generality of this result is best illustrated by taking some limiting cases of it. Referring to Fig. 5, if the dimensions a and b are small compared to unity, i.e., the height of the beam, then the denominator of Eq. 14 may be approximated by

(cos 2 ?

- c o s

2_

=~(

a2

_,...

In such a case, Eq. 14 reduces to the solution of an edge crack in a semiinfinite body subjected to shear, i.e.,

~

-

2P~/a 7r~/a2 -- b2

(15)

Moreover, when b = 0, Eq. (15) agrees with the known result obtained earlier by putting a = ½ in either Eq. 11 or Eq. 12. In addition, the solution of a beam of finite height with a free crack surface and subjected to uniform stresses r%t, Fig. 6, may also be deduced from Eq. 14. This is done by solving the original problem for the shear stresses at the crack site with no crack present and then superimposing tractions equal and opposite to the crack site stresses. Hence, setting P = -- r % t d y and using Eq. 14 as a

o[ ~?,

®

Y

I ,°II

×/

~,o

........

~

o.s

inite (b= I)

.~"

~'-~---'~-'o.o 0,0

I 0.1

I 0,2

/

~?,

I 0.3

Semi-Infinite (b=+Qo) t 0.4

I 0.5 O

I 0,6

! 0,7

I 0.8

I 0.9

1,0

FIG. 6. Beamunderuniformloads. Vol. 280, No. 2, August 1965

147

George C. S i h

Green's function, the result is r® =

--

~r

~ra

)1/2

~o~

cos ~ dy

)i/2"

(16)

After integration, k8 = ( 2 ) 1 ' : ~%~ (tan 2 ) 1'2.

(17)

It can be easily shown that Eq. 17 using the concept of stress-intensity factor is equivalent to Eq. 9 in Field's paper (8) based on energy considerations. When the crack length a is sufficiently small compared to the height of the beam, which is taken to be unity in Fig. 6, Eq. 17 may be further simplified as ka = T~xt a l/2.

(18)

Equation 18 is now valid for all lengths of the crack in a semi-infinite body. In order to study the effect of a straight edge approaching the tip of a crack, both Eqs. 17 and 18 are plotted in Fig. 6. For crack length up to one half of the beam height, Eq. 18 may serve as a good approximation to the finiteheight beam problem with errors of less than 11.5 per cent. Appreciable deviations between the two curves are apparent for values of a > 0.5. Concluding R e m a r k s

With the aid of the Schwartz-Christoffel transformation, a general formula for finding the stress-intensity factor for external cracks under longitudinal shear is derived. Examples have been given for various basic configurations, which illustrate the power of the technique presented. Cracked bodies having finite dimensions can also be treated by the method given in this paper. Some additional results are listed in the Appendix.

Appendix The following stress-intensity factor solutions are given to supply the reader with additional examples. ¥

:ai

Fro. 7. C r a c k e d b e a m of p a r a bolic cross section.

148

Fro. 8. R a d i a l crack in a r o u n d bar.

FIG. 9. Circular b a r c o n t a i n i n g t w o collinear cracks.

Journal Of The Franklin Institute

External Cracl~s Under Longitudinal Shear 1) S u r f a c e c r a c k in a p a r a b o l i c b e a m , Fig. 7 P ka -- ~2a"

2)

R o u n d b a r w i t h a r a d i a l c r a c k , Fig. 8

÷a) 3) E x t e r n a l c r a c k s in a c i r c u l a r shaft, Fig. 9

l~3(ac)

~ P - /~- + ~

1+ d

= 7ra/-d~ / 1 -- c ~/(c -t- d)(1 + cd) k3(-

a~) = ~P

./!

$

-

$

d

1 -

c

$

The results presented in this paper were obtained in the course of research sponsored bY the National Science Foundation under Grant No. G-24145.

References (1) G. C. Sih, "Stress-Intensity Factors for Longitudinal Shear Cracks," Jour. Amer. Inst. Aero. and Astro., Vol. 1, No. 10, pp. 2387-2388, 1963. (2) G. C. Sih, "Boundary Problems for Longitudinal Shear Cracks," Proceedings of the Second Southeastern Conference on Theoretical and Applied Mechanics, Vol. 2, New Yorl~, Pergamon Press, pp. 117-130, 1965. (3) G. C. Sih, "Stress Distribution Near Internal Crack Tips for Longitudinal She~r Problems," Jour. Appl. Mech., Series E, Vol. 32, pp. 51-58, 1965. (4) G~ R. Irwin, "Fracture," in Encyclopedia of Physics, Vol. 6, "Elasticity and Plasticity," Berlin, Springer Verlag, pp. 551-590, 1958. (5) N. I. Muskhelishvili, "Some Basic Problems of the Mathematical Theory of Elasticity," Groningen, P. Noordhoff, 1953. (6) G. C. Sih, "Strength of Stress Singularities at Crack Tips for Flexural and[ Torsional Problems," Jour. Appl. Mech., Series E, Vol. 30, pp. 419-425, 1963. (7) G. C. Sih, "Fracture Strength of a Rectangular Beam With Surface Cracks," Jour. Soc. for Indus. and Appl. Math., Vol. 12, No. 2, pp. 403-412, 1964. (8) F. A. Field, "Yielding in a Cracked Plate under Longitudinal Shear," Jour. Appl. Mech., Series E, Vol. 30, pp. 622-623, 1963.

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149