The elastostatic problem of an infinite strip containing a periodic row of line cracks

Ldf. Appl Enm Sci Vol. 20.No. 9, pp. 10574070,1982 P?intcdin Great Britain.

0020-722.5/&2/081057-lucunWo.WO @ 1982Pel.gc4mon Press Ltd.

~jj~LAST0S-fATIC

PROM&b4 OF AN lNFINlTE

STRIP CONTAINING A PERIODIC ROW OF LINE CRAChS

I

R. Kant

Bell Telephone Laboratories Whippany, N. J. 07981

The two dimensional each

subjected

solutions: infinite

with

to arbitrary

the problem strip loaded

equations. the

The help

problem

of an infinite

but identical

strip containing

pressure

of a row of line cracks

at the edges.

solution

in an infinite

This procedure

9-F

leads

by reduction

of these

expansion

of involved

functions.

of Fourier

stress-intensity

factor

is also derived.

of the problem

of an infinite

Boundary

sheet containing

conditions doubly

is solved medium

to a system

is obtained

1057

CES Vol. 2% No.

distribution,

a periodic

equations

and

the solution

into algebraic

An analytic

array

by superposing

of simultaneous

are later modified

periodic

row of line cracks,

expression

two of an

integral equations for the

to get the solution

of line cracks.

1058

R. KANT OF AN INFINIE srRlPCONTAININGOlJ]C

ROW OF LINE

CRAQj.S

INTRODUCTION

Many

problems

in a strip.

have been solved involving

one or more cracks, with various orientations,

The most recent solution of such a problem

which concerns with finite number

of

line cracks, located randomly

on the center line, is given by G. G. Adams [l].

The technique

given in [l] is not appropriate

for handling the case of infinite number of cracks.

The reference

[l] also contains an extensive bibliography

on this subject.

The present paper deals with the solutions of the problems of an infinite strip containing row of periodic cracks and an infinite The solution

is obtained

sheet containing

by superposition

exists in the literature

in two different

for use in the present investigation. different

solution.

a doubly periodic array of line crocks.

of two solutions:

strip and others is the row of periodic collinear

line cracks.

forms [1,2], Utilizing

a

one is the solution of an infinite The solution of the later problem

but none of these forms was found suitable

the periodicity

of structure,

Sections I and II contain the solution of the infinite

we have obtained a

strip and the periodic

row of collinear cracks in an infinite medium.

1. THE

PROBLEM

OF INFINITE

STRIP

1. Infinite Strip with Periodic Tractions lt tbc Edges

Because of assumed periodicity of both the problems for which the solutions are sought, the constituent an infinite

stress and strain fields of the superposed solutions must also be periodic. strip occupying the region R(]y]

loaded at y -

the superscript

functions

f’(x)

b, 1x1 < co). where 2b is the width of the strip,

-t b as

aA(x,b)

where

5

Consider

‘I”

and g’(x)

= f’(x),

indicates quantities

ujr(x,b)

= g’(x)

belonging

are even and odd functions,

the inhnite

respectively;

(1) strip solution

[S’].

The

they are periodic of period

The elastostatic problem of an infinite strip containing a periodic row of line cracks 2~ and they satisfy all the necessary conditions periodic

stress and displacement

so a Fourier

field

expansion

of each is possible.

which satisfies the equations

1059 A

of elasticity is

i3.41:

ej,(x,y)

=

m--l

u&(x*y)

-

2 m sin mx m-1

a,!+.~)

-

A: -

2W,(X.Y)

5

cos mx

[

m*AA cash my + CA(2m

mAA sinh my + CA(sinh

my + ym cash my)

2 [ n-1 m* cos mx

AA cash my + Cl y sinh my

m-l

- $(r+l)A;y-

msinh myA:+

z

’ sinh my)

cash my + ym

1

p

generalized Ck

is the

shear

modulus;

I = 3 - 4~

mycosh my-+vsinh

for

plane stress and Y is the Poisson ratio.

are determined

by satisfying

the

boundary

plane

strain

In the equation conditions

(2)

1 my

I

where

1

(1).

and (2). This

,!, cos mx (3)

kl

I = (3-v)/(l+v)

for

the constants AA and procedure

yields the

following set of equations for AA, Cj,

(4)

where

SEsg =

sinh mb +*bm cash mb

, slm2 =

m -mThbm

A, - -m2(s~nh

mb siAnhbm m

,%,=

-myhr&

(5)

mb + bm cash m:) c,lsh mb + m’b sinh* bm

and f&g,!, are given by

(6)

R. KANT

1060 2. Tractions

on the Imagined Crack Faces From the Strip Solution

To accomplish Figure

superposition,

1). In this problem,

we need to know tractions

however,

the line y = 0 is the line of symmetry,

e,,(x,O)

and a,,(x,O)

is obtained

of this traction function

which

into a Fourier

from (2). acting

at the imagined

- 0 and u,(x,O)

For superposition Ix-2nrl

along

crack

faces (See

therefore,

= 0,

(7)

purposes,

we need to know only that part

5 a.

Therefore

we expand

fhe following

Series. I

m

Ad ayJx.O)

2 m*A:cos m-1

mx

I

a

, n=O,fl,&2 , otherwise

= 0

Thus on the imagined

, Ix-2nal

w

,...

crack face, we have

where

1

sin ‘(k+m)a (k+m)

(k = O.I....,

II. INFINITE

ROW OF COLLINEAR

(101

m = I,2 ,.,.)

CRACKS

1. Basic Equations The equations

Kolosov

- Muskhelishvili

for two dimensional

to this theory

theory

elasticity

we seek an analytic

[51 has

been

used

here.

in the form used by Green

function

fl (z) of a complex

We quote

and Zerna

variable,

[6].

here

basic

According

x = x + iy, 5 = x - iy,

(11)

The elastostatic problem of an infinite strip con~g

=w -

ie,r-

cracks

,

2 R’(z)+O’(Z)+(z-Qn”(Z)

2io,,

1061

(12)

I

@xx- *yy +

Furthermore,

a periodic row of lie

= 4(Ez)Q”(Z).

this function should satisfy the following conditions:

n (2) - 0 5 II

(1%

for large 1~1, and

R(z) =:

T(z), O’(z)

Q(z) -8

ir(z’),t2’(z)= F(q

=

F(z)

for y c 0

(14)

so that

for y > 0

At y = 0, from (8) we find,

aJx.0)

= 4 Re Q’(r)

ru,(X,O) =

provided y?(Z)

2.

and ylii’(Z)

(n+l)Im

tend to 0 as y -

(1% Q(z),

Ixl5a

0.

statementof the Problem WC consider an infinite

figure

1).

row of colinear

For the sake of convenience

equal line cracks, located along the real axis (see

only, we shall assume that unit of length is such that

the distance between the centers of any two neighboring cracks is 2n. and according to this unit of length,

the length of each crack is 2a.

equal and opposite

I

p1a.u = -1

quantities

are then

We further

consider that each crack is opened by

pressure

on each side of the crack.

The

stress and displacement

periodic

functions

2r.

The

of

x with

period

superscript

fields

“2” indicates

belonging to the solution infinite row of collinear crack solution [S’f.

u~(x.0) =

-f2(x)

(

Ix-2rnl

e:(x,O)

- 0

,

-oo
u,z(x,O)

= 0

7

a 5

1 a

Ix-2rrn/

n = 0. Fl,...

5

tt

-a

(16)

1062

R. KANT

We assume Fourier

that

r(x)

satisfies

all the necessary

conditions

so that it can be expanded

into

Series.

We now assume

that the analytic

function

fi (z) has the form:

Q(z) = Q,(z) +

2

%2,(z),

m-1.3....

where

and

(18)

cos

We will discuss is referred variable. (Ii)

briefly the validity of the form of S(z)

to [7,81. Hence

We note, however,

the stress

and strain

and (12) are also periodic.

except

that

are absolutely

is periodic

field [~~,u_~]

Furthermore,

on the real line and their values

stress and strain

n(z)

convergent

function

obtained

these quantities

at the end points

fields, on any line other than y=O,

and uniformly

for OUTproblem.

from

(-*

2rr in the real

(17) in conjunction

of the period

can be expanded

z = .+r’r

of period

are continuous

[S]. Also on examining

M = v’cos u-cos

For details the reader

with

in entire Z-plane

2rr are equal.

into Fourier

Thus

series which

the square root

5 T 5 *)

(19)

where

t2 cost - cos u - cos x cash y I’sin

e-3

r = sinh y sin x

We find on the real axis:

M -

&OS u-cos

M - -&OS

u-cos

M-i\/cosx-cosu M--i

Finally

cosx-~osu

Q(z) = O(1) for large /yl.

(16) is satisfied

for all line segments

(r

x

x

> x > u, y -

1

*to)

(n > 1x1 > II, x < 0, y --+ *o) (1x1 (1x1

(21)


In view of (21). outside cracks.

the displacement

condition

specified‘ in

The elastostatic problem of an in6nite strip cot&ining a periodic row of line cracks

1063

Substituting (17) in (15) we find that on the interval 0 5 x -5 a, the rr$x.O) is given as 1 4d

I

du

14,

dx I

m-1.3....

0

Integration of the left hand side of (22) is elementary.

From (16) we note that f*(x) is a

periodic function of period 2%. Thus both side of (22) can be expanded into similarly convergent

Fourier series to find coefficients ae,a,(m-?1,3,..,).

This procedure leads

to a

following system of algebraic equations

T+

5

a,*,“---!-Q

m-1*3....

(23)

2v5

where

a? = 8, - 1

i {m sinme’ f x-(mfl)sinm+t

-$X/XX kxdx,

(24)

r/2

&np

sin= #de and

fi e 1 * l-t&a I

i 13

f*(x)

cos

kxdx.

Equations (23) can be truncated at the desired accuracy: thus th:y represent n equations in n unknowns.

3.

Tbr

We also note that we obtain results of Green and Ergland [21 when f*(x) = pa, a

Stress-In&a&yFactor.

Displacement of tbs Crack Faces and the Energy Associated nitb a

Singlt Crack

In

this section we calculate the stress-intensity

cracks along the real axis.

The stress-intensity

factor for the problem of a row of collinear

factor is defined as

Kt - Lim & - 0 o$x,L)m.

(25)

R. KANT

1064 In (25).

16-1 is the distance from the crack tip.

From (15) and (17) we find

asinm-’ $3

From

(22),

and ae -

-

1

it follows that when the cracks are opened by constant pressure -

(26)

pea,

- O(m#O)

Thus, for this special case

pd2fi.

K,-$&p,,,

a result obtained

by Sneddon

(7).

known result for a single crackin

Since the cracks faces is calculated

when a
*Jan

on the real axis y-0,

a. Kt -

6

pe, a well-

the vertical

displacement

of the crack

(15) as: a0

, WY(X)

a =Z f

f

an in finite medium.

are located

by relation

Also,

(27)

aotanf

u +

2 &osu

I

The elastic energy associated

+

1 ucos- u du

m-l.odd

= cos + L J

(I+r)

a,sinm

with opening

2

- cosx

of a crack is obtained

, IxlSa.

from the relation

AE = ; n,+;dx -#

where

ug is the vertical

co, are determined 4. Tractions

displacement

and a,

These quantities

for the case b =

= 0. for all m.

and Dirpleccmcnts WI the Imegiaed Lines in the Z-Plane

For the strip problem, displacements

(29)

of the crack face (28).

by setting a0 = -2 d

(28;)

for superpositioning

along a line y e;

purposes,

& b. From (11) and

- 4[Ret2’(x*)

we need to evaluate

tractions

and

(12) it follows that

+ bImO”(x*)]

(30)

a&(x,b)

= - 4bReR”(z*) 5 m-1.3,...

a,fi”mL

in kx

(31)

The elastostatic problem of an infinite strip containing a periodic row of line cracks ru,‘(x,b) - (x+l)lmtJ(z*) * g

1065

- ZbRctI’(z*)

[(x+l)tI,

- ZbQ’& + ~,[(r+l)I).ylbn’.r]].ol _

kx

(32)

where

Z* - x + ib. and

(33)

nmdbh fi ,db)

I

= &

Cl;,,

R ik etc.

S ,(x*)(cos

(34)

kx,sin kx)dx

, 1.2 *...

m,k-0 and

j

of Fourier expansion

are the coefficients

of the first and second

derivatives

respectively.

III.

SUPERPOSITIONING

OF THE SOLUTIONS

IS’] AND IS’]

1. The Infinite Strip Problem

We consider

an infinite

periodic

row of line cracks,

length.

According

following

strip occupying with centers

2~ distance

to this unit of length,

(lyl 5 b; - co < x < co), containing apart,

which is assumed

each crack is of length

a

to be the unit of

2a. The strip is subject to the

edge conditions.

~,(x,‘-Q = fVx)

. . , ,

~,,(x.W = g,(x) u,(x,O)

e,y(x) The function function

a domain

f’(x)

0 = h’(x)

= 0

and h’(x)

-co
--cotx
Ix-2nrl

I

a < tr,n=0,+1,+2

,... (35)

-03
are even periodic

functions

of period

2r

while g’(x)

is an odd

with the same period.

The solution as depicted

to this problem

can be obtained

in Fig. 1 and demanding c$f2;b)

+ f’(x) = f(x),

The condition 3 unknown

ury(x,O) functions

•t P(x)

= h’(x),

-co

< x < co

-a~

< x < co

lx-2nrl5

= 0 is automatically f’(x),g’(x)

the solutions

[S’] and [S2]

that

afY[f2;bl + g’(x) = g’(x). c,!r[f’.g’;Ol

by superpositioning

satisfied.

and f2(x).

(36)

a < *. n -0,~ Equations

l,f2,...

(36) represent

With help of (9), (23).

5

a,

m--1.1.. I

R,t+bR,I:

+_-_f: I

4

for

(30) - (34). the equations

(36) can be written as

’

3 equations

f: 4

R. KANT

1066

1

(37)

k - 0,1,2,... These

equations

determine

the

coefficients

f&g:,aoaltl

for

prescribed

tractions

(1 II

gi and hi .

2. Tbc Pkrc Confaiaiag A Doubly Peridic Amy of Line Crrcks

As a corollary, we can use the present superpositioning scheme to solve the problem of a plane containing a doubly periodic array of line cracks, each subjected to the same but arbitrary pressure.

This problem is equivalent to solving a problem a strip containing a periodic row of

line cracks, with its edges subjected to the conditions of symmetry: u,r(x,bI = O,u,(x,b) - 0

(38)

and the conditions at the crack faces are (39)

~&.O) = -p(x),

The supe~sition

lx-2*nl

ts a, n = O,* I,*2 ,...

scheme outlined before can be advantageously

used for solving this

problem. In this case the equations for the unknown functions f’, g’ end f’ become

u~,[f*;bl + g’(x) = 0,

-co
u2(f2*b] I ’ + uI‘if’ .tg’*bl = 0 ,

--c0
P(X)+ ujJP,g';Ol = -p(x),

(40)

Ix-2rnl

da

< v, n =O,+l,f2,,.,

For the sake of brevity only, we shall omit the longhand version of (40), which is similar to (37). The calculations implied in (37) and (40) have been carried out for several particular cases. The results are discussed in the next section.

The elastostatic

1067

problem of an infinite strip containing a periodic row of line cracks

IV. RESULTS AND DISCUSSION Numerical various

results,

crack lengths

pressure

f’(x)-1

Figure

after solving

and strip-widths;

and g’(x)=0

from

for smaller

crack lengths

higher.

of the stress-intensity

other

a>0.75*, cracks

the effect of diminishing

known

is approximately

For the strip-width

the case of a line of periodic

for various crack lengths.

with previously

(at0.125x)

For larger crack lengths

lengths,

have been obtained

for

in all cases the strip is loaded at the edges with normal

(26) and coincides

medium.

of superposition,

for all x.

2 shows the variation

is obtained

infinite

for the coefficients

than

however,

the stress-intensity

factor

the stress-intensity

factor approaches

medium.

between

The stress-intensity

equal to the case of a single crack in an

b=w,

in an infinite

distance

results.

The case b=cc

This is because

factor

is

the value for at these

the crack tips is more pronounced

crack

than that

of the strip-wid:h. Figure

3 shows

calculations

were

the vertical carried

out

normal

displacement

increase

in the crack length.

The change the

energy

containing

unity.

of the crack

in the

strip

a line of periodic For larger

crack

a doubly

periodic

The results

of these calculations

factor

with respect

cases.

Table I. In this table, the numbers

with

1.5.03.

the energy

These

As expected,

the

of the strip-width

and

is shown in the Figure 4, where change

in the infinite

This ratio, for all strip-widths,

peaks at of about

to the fact that the effect of crack interaction

is more

Finally,

and approaches

plane

the value

the calculations implied

for the problem in (40),

has also been considered

have been compared of change values

b=1.25,

of cracks,

array of line cracks,

K,. and the amount

faces for the same loading.

with the narrowing

the ratio decreases

This problem

to the corresponding

0.75n;

of the same lengths.

be attributed

particular

intensity

is compared

lengths,

for several

0.625~.

due to presence

than that of the strip-width.

sheet containing

of the crack

faces increases

cracks

This can, once again,

pronounced

for a=OSr.

in the elastic energy,

change

az0.625r.

displacement

by Delameter

are taken from (91.

in [9].

per crack

for a single crack in an infinite

in parentheses

have been carried

with those obtained

in the elastic energy

of an infinite

sheet,

out

et al. 191. The stress-

El, normalized are shown in

1068

R.

a

KANT

El

I

a

a

0.5

1.25

0.3914 (0.3921)

0.3002 (0.2890)

1.0

1.25

0.536 I (0.5572)

0.5125 (0.5087)

I.0

1.61

0.5561 (0.5650)

0.5196 (0.5181)

I.5

1.25

0.6875 (0.6683)

+ rpjaa’(l-v)H-’

0.6228 (0.6386)

(

Table 1. The values of K, and El

As can be seen from

in [9].

Such

assurance

the table.

comparison

the results

is needed

obtained

in order

that series have been properly

here are in agreement

to test the suitability

with those reported

of this

method

and for

truncated.

CONCLUSIONS

Using superposition periodic

technique,

the solutions

row of line cracks and an infinite sheet containing

- have been obtained.

The later problem

results

for the second

problem

found

to be in agreement.

The results

width,

the distance

the two neighboring

effect becomes the distance

between

between

the centers

with the results

for the first problem

when a/x

periodic

z 0.6.

given SUggCSt

et al. 191. The results

that in addition role.

were

to ship-

In fact, this

the crack length

the critical

approaches

to the value for the infinite

with those

for a periodic

a

array of line cracks

in [91: these

At this ratio between cracks,

strip containing

by Delameter

crack tips plays an important

of two neighboring

is incipient,

- an infinite

a doubly

h:rs also been considered

are compared

more pronounced

which crack propagation periodic

of two problems

internal

and

pressure

at

sheet containing

a

row of line cracks.

The results

obtained

sheet if b/a becomes

here agree

large.

If. in addition,

a/r

becomes

row of line crack

small the results approach

in an infinite those for a

single crack in a plane.

ACKNOWLEDGEMENTS

The author

is thankful

work on this problem. Center,

Bell Laboratories,

to Messrs.

In addition, Whippany

T. G. Grau

and J. M. Segelken

I would like to thank for program

counselling.

Ms. Arline

for the opportunity Cowell of the Computer

to

The elastostatic problem of an infinite strip containing a periodic row of line cracks REFERENCES

111

G. G. Adams, Inf. J. Engg. Science, 18.455 (1980).

(21 A. E. Green and A. H. England, Proc. Comb. Phil. Sot. 59,489 (1963). 131 S. P. Timoshenko. et al., Theory ojElasriciry. MC Graw Hill, N.Y. (1951). [41 Y. C. Fung. Foundorionr oJSOM Mechanics, Prentice Hall, Englewood Cliffs, N.J. (1961). 151 N. 1. Muskbelishvilli. Some Basic Problems 01 the Mathemarical Theory of Elanidy, Noordhoof, Leyden (1953).

161 A. E. Green, et al.. Theoretical &iastialy. Oxford, London (1954). I71 1. N. Sneddon. et al.. Crack Problems in the Classical Theory of ElaJridy. Wiley, N.Y. (1969). I81 W. T. Koiter. Ingenier-Achiv,

28.168 (1959).

191 W. R. Delameter. et al.. J of Appl. Mech.. 42,74 (1975).

LEGEND: -------IMAGINED

SURFACES

Figure 1 The Superposition of solutions S’ and S?

4.0

3.0

-

Y 2.0 -

I.0

0

0

0.2sa

057

D7577

(!

Figure 2 Variation with the crack length of the stressintensity factor for various strip-widths.

1069

1070

R. KANT

0

0.25n

0.5w X

0 757

Figure 3 Vertical displacement of the crack faces for various crack lengths and striwvidths.

5

I

I

IL

m I

1

b.1.5

b.2

0

Figure 4 Change in the elastic strain energy (after normalization) with crack length for various strip-widths.

b= 1.25

-..-

b =I 25

____

b=l5 b ;@,

-