Dynamic stress intensity factor for an unbounded plate having collinear cracks

DYNAMIC UNROUNDED

STRESS PLATE

INTENSITY FACTOR HAVING COLLINEAR

M. A. HUSSAIN Benet R & E Laboratories,

U.S. Army,

and

Watervliet

FOR AN CRACKS?

S. L. PU

Arsenal,

Watervliet,

New York

12189, U.S.A.

Abstract-The steady-state vibration of an infinite plate with collinear cracks is considered for low frequency cyclic loading. The formulation of the mixed boundary value problem leads to a dual trigonometric series. The Schwinger’s method gives an automatic perturbation scheme. The dynamic stress intensity factor is found to be higher than the corresponding static one. The inertial effect on the stress intensity factor becomes significant only when the frequency of the external load is close to that of the shear wave.

INTRODUCTION present paper we consider a row of identical collinear cracks in an unbounded plate under low frequency cyclic loading. The static problems of collinear cracks have been studied extensively by many investigators e.g. Westergaard [ 1I, Koiter 121, England and Green[3] and Sneddon and Srivastav[41. A comprehensive survey of such problems can be found in Sneddon’s book[51. The corresponding dynamic problem has recently been considered by Kudriavtsev and Parton[61. The problem is formulated in terms of dual series involving trigonometric functions. Due to the inertia effect the dual series cannot be solved in a closed form as its static counterpart. Hence approximate methods have to be used in solving the dual series. Kudriavtsev et ul.[6] have expanded these series in terms of frequency parameters and then reduced the single set of dual series into an infinite set of dual trigonometric series. Certain of these infinite sets of dual series are then solved by Tranter’s methodr71, which itself is quite complicated. In the present paper we have used a simple perturbation scheme which primarily depends upon our ability to solve the corresponding zero order system (static case) in a simple manner. The method for solving the zero order system is based upon an ingenious transformation devised by Schwinger to solve wave guide problems [81. This method is extremely simple and does not seem to have received the attention it deserves. Unlike the usual perturbations about the exact solution (e.g. see Ames [9] p. 209) the present perturbation scheme does not require the solution of a set of systems. Once the order of perturbation is fixed, only one set of dual series must be solved. The method is quite general and can be used in many problems of mathematical physics which lend to trigonometric dual series. Based on this method the dynamic stress intensity factor for the present problem can be simply expressed by the static stress intensity factor and a multiplicative factor which is a function of material constants, crack length and the frequency of the cyclic loading. The multiplicative factor is evaluated from a matrix of order N. By increasing the value of N the multiplication factor can be as close to the exact value (N --+ x) as we desire. Numerically it is found that very good accuracy can be.obtained without using large values of N because of the fast convergence for the case of low frequency loadings. IN

THE

ir’l~eaenled at the Symposium tieorge

Washington

University,

on Fracture and Fatigue at the School of Washington, D.C., May 3-5, 1972. 865

EFM.Vol.4.No

4-R

Engineering

and

Applied

Science,

866

M. A. HUSSAIN

and S. L. PU

DUAL SERIES RELATION

Consider an infinite row of identical cracks of length 217equally spaced at a distance 27~.apart along the real axis. An oscillatory load (r, = - qe+ is applied at each crack. Assume that w is small in comparison with c, and czr the longitudinal and transverse wave speeds, and that the two edges of each crack are separated by a small distance due to a uniform tensile field so that they do not come into contact when the plate is vibrating under the cyclic loading. The latter assumption was discussed by Ma1 El01. Since the problem is symmetric with respect to the real axis, we consider only the semi-infinite plate y G 0. Let & = iu +JU be the displacement vector. The governing equations for plane strain are:

(h$_/“&)VV+bpV2ii=p~

! 1)

where A, p are Lame constants, p is the density of the plate and the dot designates the partial derivative with respect to time. The boundary conditions on the plane 4’= 0 are

T&K, 0, t) = T,,(x+~~T,

0, t) = 0

0 G 1x1s

7r

u,(x, 0, t) = cr,(x+ 2kT, 0, t) = - qeeiof

0

u(x,O,t)= u(x+2k~,O,t) = 0

q s 1x1s G?

For a steady-state

/XI s q

k=t-1,42,. t>O

*. 121

solution, we let a(x, y, t) = .fi*

then the amplitudes of the displacements, 2(1--v)aZu*+a”u*+ l- 2V r3x* ay2 2{ 1 -v) 1 -2v Introducing

G

fx, y)e+’

(3)

U* and IJ*, satisfy 0J* -~ 1 Pv” =_-Iu* l-22v&xdy cz2

a%* +d%* + -- 1 d2U” 1-22va.dy=-c,Z” ay2 ax*

6J2 *

(4)

the finite Fourier transform U = U(n, y) = JOXU* (x, y) sin nx dx (5) V= V(n,y)

- J”VV*(x, y) cos nx dx

equations (4) reduce to the ordinary differential equations

(6) where primed quantities are derivatives with respect to y and Oi

=

OJ(nCi)-'9

i= 1,2

(7)

Dynamic stress intensity factor for unbounded plate

867

in which

2-h-P -

Cl

c22 =

-)

2 -cz Z--C

0

P

P

l-2v

x+2/x

Cl

II-

P

(8)

2(1-v)’

Equation (6) can be uncoupled in the form,

[$- (+$)][$- (+$)] {&y;;)J} =0. The appropriate

solution of which is given by

U (n, y ) = a,,e*lv

+ anZennzv

V(n, y) = a,, cos z

+ ao2sin z

where a,, and armare superposition

-anlfi,ennlu

- un2f122-1e”2y

(9)

constants and

ni=

(1-C$)1’2,

i= 1,2.

(10)

The inversion of (5) gives U*(x,y)

=$

I/* (x, y) = b

C {a,,e”“‘“+a,,e”“‘y}sinnx Yl=l

[

a,, cos y+

(11)

+z ao2sin z ( 11

The zero order terms in the above correspond of the stresses are given by:

(1 - 2~)

2P

(1-2v) 2P

v

mr

Cl

* _ (1 -VI

_‘T?J

+i and

(

uaOl

*=- 77 --

7r

1

. 0~

sine+1

WaOl . ----_~n-++ L‘l

to plane waves at infinity. The amplitude

WY

waoz (‘1

2 {-anl~n,e~nlY-an2rR2-1eRnZY}. ??=I

cosc

0~

wag2

Cl

(‘1

11

WY cosc_

1 I

i {[- (I-v)~,z+v]a,,e~R~u+[-l+2v]a,2e”n~y}ncosnx ?I=,

M. A. HUSSAIN

868

and S. L. PU

From the first boundary condition in (2) we have, from (12), (13) Using (13) in (12) and letting y + 0, we have the boundary values for normal stress and displacement: 2(1-V)PCfJ 2/.L m uo2+ - x d,F,, cos nx c+,*(X,o) = (1-2V)CTC, 7r ,,=1 TV* (x, 0) =

cl01 +

2 d, cos

(14) nx

where d, = - an2m22fl-1,

(15) (16)

Applying the mixed boundary conditions of equation (2) we have from (14) the following dual series relation for the unknown constants d, (n 2 1) , uo2and uol, 2( 1 -v)po (I-2u)m,

~,,+q+-

2P Ca dnFncosnx= 7T n=,

a,, + i d,,cos nx = 0, 7l=,

0,

for0 s 1-d s r)

forr, < 1x1C 7~

(17)

where F, is given by (16). Because of the complex nature of F, the above dual series cannot be solved in a closed form for the general case. When w * 0 the above dual series reduce to that of the static problem and can be solved exactly. For boundedness of displacements it is required that F, should not vanish for general loading condition. Simple manipulation shows that this condition is satisfied if o # (Y, where (Yis the Rayleigh wave speed. Expanding F, in the powers of w2/n2c12 and 02/n22c22 we have: 1 Fn = 2(1_“)

n+O

1 ; . !I

(18)

Hence, we write 1 Fn = 2(1 _v) n (1 +Pn)

(19)

for the static case pn + 0. In the next section we solve this static case by Schwinger’s method for illustration and in the subsequent section we use this method to set up the perturbation scheme. It should be noted that in (18) we used the expansion of F, to obtain the zero order term (static case). The perturbation scheme does not require any further expansion of F, or p n, as was necessary in reference [6].

Dynamic stress intensity factor for unbounded plate

869

SOLUTION OF THE STATIC CASE (o + 0) : For pn = 0, (17) reduces to (1 -v)7r P

for 0 G 1x1G 7j

4+c. nd,cosrzx=O, ll=,

a,,+

5 d,cosnx=O, ?l=,

for 7) G 1x14 7r

(20)

(21)

Let (1 -v)rr

q+ 5 nd,cosnx= 7l=l

CL

h,,(x),

fern

s 1x1c 7~

(22)

where P (1

_

V)~ h,(x) -q is the unknown stress UC (x, 0) , along r) s 1x1c 7~.

Inversion of (22) gives: (23) and h,(t) cos ntdr.

(24)

Substitution of (24) into (22) and interchange of the order of summation and integration give the following integral equation for ho(t) : a01+-

2

7r h,(r) 5 ‘OS nyS I ?r 9 ?l=1

nt dt = 0,

n 4 lx1 s 7r.

(25)

The Kernel in the above integral equation can be summed Oc cos nxcos nt =-~log2/cosx-costl. n

z

(26)

?l=l

Equation (25) is a singular integral equation with logarithmic Kernel (26) and can be solved by the usual techniques, see for example reference [l 1, p. 1431. For our general dynamic case, however, the corresponding integral equation cannot be reduced to such a simple form. Hence we use Schwinger’s method. The advantage of this method will become clear in the subsequent sections. Let cosx=

b+b’cos[

cost=b+b’cos[

(27)

where b and 6’ are constants such that new variables 6 and 5 cover the entire range from 0 to rr while x, t vary from n to r. Hence b = (COST-

1)/2,

b’ = (cos “II+ 1)/2.

(28)

870

M. A. HWSSAIN

and S. I.. PU

Substituting (27) in (26), we have

cccos

22 n=1

nxcos nf rt

= -$og

2lcos x-cos

m cos ne cos n< b’-t 2 n a=1

t] = -+og

(2%

From (27), we have dt

1 (cos q - cos t) 1’2

z=.q$

sin (t/2)

(30)

.

The unknown stress has at most a singularity of order one half as t * q. Hence, from (30), ho(t (5) ) (dtldl;) satisfies Dirichelet’s condition in the range 0 6 5 G rr, and due to the symmetry of the problem we can write, ([ 121, p. 72) 1y,cos (m--l){,

for0 6 5 G rr,

(31)

Substituting (29) and (3 1) into the integral equation (25) we have a01+;

2

1

?r

(c

0

O”

amcos (in-I){){-;logb’+i

0 s 5 == T.

+osn&osr$}d<=O,

t7L=,

Tl=J

(32) Using the orthogonality cl01 -

conditions, we reduce (32) to, ix1

log b’ + f:

a,+,

=-Lo,

()<(sr.

(33)

n

?l=l

Consequently, 1.

(34)

r)
(35)

ffl=a,,jflogb’),tr,,,=O,n~ Substitution

of (30), (34) into (3 1) yields tisin (t/2) /Z”(l) =-A?!log b’ (cos q- cos f)““’

The unknown coefficient sol is determined a 01=

by putting (35) into (23)

cl-v)T*ogb’ P *

(36)

4

The normal stress is then obtained

a;(x,o)

=

4

v?sin (xm x) 1/Z

[ (cos

7) -

co.5

1

I

)

77

<

Ix/

6

T

(37)

and the stress intensity factor is given by K, = h-5 (27r) If2(x - 77)““ug (x, 0) = VG This is the well known result.

$ tanIf (7712).

(38)

871

Dynamic stress intensity factor for unbounded plate

In the above method we were able to solve the dual series relation by reducing it to an algebraic equation. The simplicity of this method becomes clear when it is compared to Tranter’s method [6,7]. SOLUTION OF THE DYNAMIC CASE From (18) and (19) it is seen that pn = O(n-‘) for large n, so the series 8 n dnpn cos nx may be considered as a perturbation of the exact solution given in the previous section. Taking the first N terms of the perturbation in the first of equation (17), we have

P (l-2V) 1

Tr(l-V)

2(1-v)

/.MJ 7X1

-a,,+-

N

+gnd,cosnx+

2

ndGPncosnx=O,

n=1

a=1

osxsr). (39)

The solution of the dual series given by (39) and the second equation of (17) is the N-th order approximation to the solution of (17). By increasing N the approximation may be made as close to the excact solution as desired. As before, let the left hand side of (39) be equal to the unknown function hN(x) for 7 G Ix] G 7~. The equilibrium condition gives

71(1-y)q=i IT

h.v(x) dx. T1

P

(40)

The coefficient ao2 must vanish as a result of comparing (40) with the inversion of (39) and the latter inversion of (39) also gives d,, =-&

= hN(f) cos ntdr, I t)

n > N (41)

/VThN(f) cosnrdr,

dam= (I-$-)$

n=

1,2,.

. . .,N.

Substitution of (40), (4 1) into the second equation of (17) gives the integral equation for the unknown function hN(x) : a,,,+$_/”

by(r) i

9

‘OS n~snf 1

dt-

i

-l-!$-y%

I1=1

c

h,(f) cos nrdt = 0,

n

We use the same change of variables defined by (27) and replace cos nx in the finite series of (42) by IL+1

cosn-45) =kg, Pk(n)cos(k-l)t,cosnr([)

n+1

=kzl pk(n)cOs (k-l){.

(43)

Based on the same reasoning as discussed in obtaining (3 l), we let

Substituting (44) and (43) into the integral equation (42) and using the same procedure

872

M. A. HUSSAIN

and S. L. PU

outlined in the previous section (see 32 and 33), we have:

OS[ST

where B, = log h’. BI,=--k_,,

6, =2

1

&=

1.

k=2,3

,....

Since left hand side of (45) vanishes for all values oft in the range (0,7~),each coefficient of cos k( fork = 0, 1,2, . . . in (45) must vanish. Hence a,(S) = 0 for m > N + 1 and CX,,(.~)for m = 1,2, . . ., N + 1 can be determined simultaneous equations C, (BiSij+Aij)aj(iV)= i=1 where 6ij is the Kronecker

Ri,

(47) from the following

i= 112,. . ., N+ 1

(N + I !

(48)

delta and R, = a,,,; Ri = 0, i > 1 (49)

From (43), it is obvious

Insertion of (Y,(,~)obtained from (47) and (48) into (44) yields the unknown function in terms oft : .v+1

where de dx=

[4-((1+cos77)(1+cos5)]1’2~ (1 +cos 7)1’2( 1 -cos 5)“”

(52)

The normal stress on the boundary y = 0 may be written as

77

=s 1x1=s7-r.

(53)

813

Dynamic stress intensity factor for unbounded plate

In (5 3) the following relation has been used: hN(t) $ df = alcn’).

(54)

The dynamic stress intensity factor is given by D1 = lim (27r)1’2(x-~)1’2’+~(x, *+‘I

0).

It can be shown easily that D,/ ( V’% q) =

w)

(cY,(~))-I

.hTCX,(~“’

(56)

VI=1

or N+l

DI/KI = (a,‘“‘) -I C amcN). ??I=1 NUMERICAL RESULTS As a first approximation, only one term is taken in the perturbation ing (43) and (27), the coefficients in (43) are @,(l)= 6,

series. Compar-

p,“’ = 6’.

(57) The coefficient czl(l) is known from (54), the remaining two unknown coefficients CQ(~) and a,, are determined from two simultaneous equations written in matrix form as

(log6’) + Wwdl(l +P,)

bb’pJ(l +A

2bb’d( 1+A

-1

+

(b’)$,/(

1 +pl)

= I[ HI (1) a1

a01

czp(l)

0

(58)

where p, is given by (19) and (16) with n = 1. For the given material constant V, the frequency of vibration of the external loading o and the crack length n, we can obtain numerical values of the dynamic stress intensity factor from (58) and (56). In order to compare results with those shown in Fig. 1 of [61, we used I/ = l/3 and (w/c,) = 0.214, 0.224 and 0.316. The numerical results thus obtained are quite close to the corresponding curves obtained by Kudriavtsev and Pat-ton in [6]. Increasing the number of terms in the perturbation series we obtain the second, third and fourth approximations. We stop at N = 4 because very good convergence has been obtained for low frequencies. For N = 4 the five unknown coefficients aO,, and cyi(*), i = 2,3,4,5, are to be solved from the following system of equations E mijat4’ = Ri, j=1

i= 1,2,. . .,5

where the elements of the (5 X 5) matrix mij are given by (48) and (49) e.g.

(59)

874

M. A. HUSSAIN

++&w

p4 (

and S. L. PU

etc.

l+P,

>

in which the PJN), defined in (43), are given explicitly by (57) and (60). PI(“) = 2b2 + b’” -

1,

/I,‘“‘= 4b”-3b+6bbt2, /3,(4)= 8b4-8b2+ p,‘4’=4bf2(8b2+b’2-1),

p,c”

=

4bb’,

p,“’

=

/,,I’

p,‘“’ = 3b’(4bZ + b’” _ 1).

1 +4b”(8b2-

1) +3b’“, /jq(4)=

&;:‘I = 6bb’2, fi,C”,= b’:’

pzc4)= 8bb’(4b’-2+3b”)

$jb/,‘:‘.

@5(4)=

b’4. (60)

The numerical results obtained from (59) and (56) for the case of plane strain are shown in Fig. 1 for v = l/3 and w = 0*316c,. Also shown in Fig. 1 are results for N = 0 (static case) and N = 1 and the curve 3 in figure 1 of reference [6]. The dynamic stress intensity factor increases with increasing frequency of the external load as shown in Fig. 2 for v = l/3 and for various crack lengths. The ratio of the dynamic stress intensity factor to the corresponding static one is plotted as a function of W/C, in Fig. 3 for several crack lengths. The ratio increases monotonically

Y =I/3 w :o x

316c,

Kudrmvtsev

results

i

875

Dynamic stress intensity factor for unbounded plate

Q 3

1

0 N

in -

L

” 0

876

M. A. HUSSAIN

0.81

0

3

0 I

02

and S. L. PU

/

0.3

o-4

(

Fig, 4. Effect of Poisson’s ratio on (D&s)

with uIcz. When the crack length (+T) is close to O-6, the inertia effect reaches its max~um. The effect of Poisson’s ratio on the dynamic stress intensity factor is shown in Fig. 4 for two crack lengths, n/rr = O-4 and O-6 and for the frequency w/c, = 0.2. The stress intensity factor increases with Poisson’s ratio. The increase is slight for Poisson’s ratio less than 0.3 and the increase is rapid as v approaches O-5. REFERENCES [I ] H. M. Westergaard, Bearing Pressures and Cracks. J. a&. Me&. 6, (1939). [Z] W. T. Koiter, An Infinite Row of C&near Cracks in an Infinite Elastic Sheet. (ngen. Arch. 28, (1959). [3] A. I-I. England and A. E. Green, Some Two-Dimensional Punch and Crack Problems in Classical Elasticity. Proc. Camb. phi/. Sot. 59, (1963). [4] 1. N. Sneddon and R. P. Srivastav, The Stress in the Vicinity of an Infinite Row of Collinear Cracks in an Elastic Body. Pruc. R. Sot. Edin. A67, (I 965). [5] I. N. Sneddon and M. Lowengrub, Crack Prohfoms in the Classical Theory qf Etasticity. John Wiley, New York (1969). [6] B. A. Kudriavtsev and V. Z. Parton, Dual Trigonometric Series in Crack and Punch Problems. PMM 33 ( 1969). 171 C. J. Tranter, Dual Trigonometric Series. Proc. Glasg. mafiz. Assn. 4 i 1959). Theory ofWavegui&s. Iliffe, London, (195 1). WI L. Lewin,Aduanced Academic Press. New York r91 W. F. Ames, Nonlinear Partial Differential Equations in Engineering. (1965). A. K. Mal, Interaction of Elastic Waves with a Penny-Shaped Crack. Int. 1. Engng. Sci. 8. 381-388 ( 1970). W. Magnus and F. Oberhettinger, Formulas and Theorems the Funcfions ofMathematical Physics. Chelsea (1949). I. N. Sneddon, Fourier Transforms McGraw-Hill. (195 1).

for

(Rareived

24April1972)