Periodic elliptic holes in anisotropic elasticity

Mechanics Research Commumcations, Vol. 25, No. 2, pp. 171-178, 1998 Copyl~lht © 1998 Elsevier Science Lid Printed in the USA. AU tights t~served 0093-6413/98 $19.00 + .00

Pergamon

PII S 0 0 9 3 - 6 4 1 3 ( 9 8 ) 0 0 0 2 1 - 4

Periodic elliptic holes in anisotropic elasticity By Yiantai Hu I , Y u y m g H u a n g 1 , Lingzlu C h e n g z I

,

Siping Li I .Weifiang ZJaong l

Dcparunent ofMechalfiez, l t u a z h o n g Umversity o f Science

mtd

"leclmology.

W u h a n 430074, P. R. China

Henan Designing Institute o f Ci .ly and countryside Construction, Zhengzhou 450000, P. g. C'lmm

(Received 17 March 1997; accepted for print 17 December 1997) Abstract

The problem of cotlinear periodic elliptic holes ill an anisntropic medium is examined in Ihis paper. By means of Stroh lozlnalism and the conlurmal tnapping Incthod, explicit full domain solutions for the periodic hole problems are presented. The solutions are valid not only for plane problems but also for anliplane problems and the problems whose inplane and antiplane deformations are coupled. The stre~s concentration around the holes is analysed. Key'a, nrds:

© 1998 Elsevier Sc"ienceLtd

Stroh formalism, conformal maplfing, stress concentration, hole

1. I n t r o d u c t i o n

In the past m a n y i n v e s t i g a t o r s have c o n s i d c r e d the elastic fields o f periodic m i c r o d c f c c t s [I to 7]. Ill particular, p r o b l e m s o f .this type for collincar periodic dcl'cc,ts, such as i n c l u s i o n s or cracks, in an isotropic m e d i u m wcrc s t u d i e d by Lu a n d Chai[4], a n d collincar c r a c k s in a n i s o t r o p i c m a t e r i a l s by Hwu[5]. Periodic i n c l u s i o n s or c r a c k s in an a n i s o t r o p i c m e d i u m were also researched by H u et a1.[6,7]. So far thcrc are no simplc cxplicit s o l u t i o n s lot pcriodic holes in an a n i s o t r o p i c m c d i u m d u c to the c o m p l e x i t y o f c o n s t i t u t i v e e q u a t i o n s a n d the dillqculties in m a t h e m a t i c a l o p c r a t i o n s . H o w e v e r the stress c o n c e n t r a t i o n a l o n g the holes is very c o m p l e x a n d critical, the m a t e r i a l s here m a y d a m a g e attd cvcn crack. T h e r c l o r c this re, a r c h is necessary I'or e n g i n e e r i n g a p p l i c a t i o n s . In c o n t r a s t to the chtssical [ o r m a l i s m , S t r o h f o r m a l i s m h a s bccn s h o w n to bc e l e g a n t a n d p o w c r l ' u l i n solving p r o b l c m s o f t w o - d i m c n s i o a a l a n i s o , t r o p i c c l a s t i c i t y [ 8 - 1 3 ]

In t h i s p a p c r

',vc deal with gcnczal collincar periodic elliptic holes in all a n i s o t r o p i c tncdiulll by Stroh l o m a l i s m a n d ,the c o n l o r m a l m a p p i n g mc,thod, q'hc b o u n d a r y values o1" elastic I'iclds arc I0ttnd out First by S t r o h f o r m a l i s m . T h e n t h r o u g h the c o a l ' o r m a l m a p p i n g m e t h o d a n d Cauchy-(ioursat

t h e o r y wc o b t a i , the explicit solutioll o f the a b o v e p r o b l e m . Finally, the

s'trcss c o n c e n t r a t i o n a r o u n d ,the holes is a n a l y s e d . 2.

Collinear periodic elliptic holes

C o n s i d e r an infinite row o f collincar periodic elliptic holes lying a l o n g the x j - a x i s w h o s e b o u n d a r i c s 1. l~(k = 0,1,2,..-) are given by

x~ = I__kr+acosO,

x2=bsinO,

(I)

171

172

Y. HU, Y HUANG, L. CHENG, S. LI and W. ZHONG where 2a,2b

arc the two m a i n axes o1 ellipses a n d 0 is a real p a r a m e t e r , r

is the

periodicity. W i t h o u t loss o f generality, let the origin be the middle p o i n t o f L o , Fig.I. T h e d e f o r m a t i o n is a s s u m e d to be t w o - d i m e n s i o n a l in the sense that e33 = 0. I-

--'-~,t,,) "1 ( x / ~

l''

FIG, 1 Collincar I:~r*odic elliptic holes. Based upon Stroh Iormalism for t w o - d i m e n s i o n a l anisotropic elasticity [5 to 13], the general analytical solution for collinear periodic holes in an anisotropic body can be written as

[91 u = AF

I. AF',

O'h

(Di,2,

--

@ = BF o'2J

-

t- BF, i

~J,I,

(2)

'

where

A =(al, az, a~), I"(z)=(q~fa(zl),

B=(bl, b2, b~),

q2/~(z2), q~/i(z~)) 7",

In t h e a b o v e , u a n d @ a r e 3 x function) a n d o,,

/ z~ = x l

~ p, xz,

~t= 1,2,3 j'

(3)

I matrices w h o s e elements arc u, (displacement) a n d @l(stress

is the stress tensor a n d q = ( q l , q 2,q 3)r is an u n k n o w n c o m p l e x vector.

The o v c r b a r d e n o t e s c o m p l e x c o n j u g a t e a n d the superscript T

s t a n d s for transpose; a

c o m m a d e n o t e s differentiation; /:, ( • ), 0t = 1,2,3, arc arbitrary f u n c t i o n s which c a n be determined from the k n o w n b o u n d a r y c o n d i t i o n ; p°

arc t h c c o m p l e x e i g e n v a l u e s w i t h

posi-

tive i m a g i n a r y p a r t s delined by the elasticity c o n s t a n t s C,,~z[9], a n d a,, a n d b,, are the eigcnvcctors associated with p , . It s h o u l d be n o t e d that the d i s c u s s i o n in this p a p e r is based upon the a s s u m p t i o n that p, (a = 1,2,3) arc distinct. T h e elliptic surfaces arc subjected to loadings which can be expressed as

'

@u _ 2 ~ i l n a f ~

/!

[ v . o ' " ~" ~ , a

"l,

(4)

m-0

whereof = e '°, f i s t h c resultant I o r c e a l o n g L o ; v ~ a r c t h e

known constant vcctorsrelated

with the d i s t r i b u t i o n s o l ' l o a d i n g s . T h c stresses a p p r o a c h zero as z - *

t-leo. F r o m (2) a n d

(4) wc o b t a i n the b o u n d a r y value ol ~ F on 1. ~ BF+ BF= 2nilno'f4

~ [v,~o'" -t-~,,o -~1,

(5)

when f ~ 0, (5) b e c o m e s as ,o

BF4 BF=

~ {v~a" t-~a

~],

(6)

m-0

With rclercnce to T i n g ' s work[10], @ can also be written as @ = 2Rc/BhU(z)A r } g + 2 R e { B W ( z ) B r }h,

(7)

where hO(z) = diagl fl (z ~),/~(z 2 ), f~ (z ~ )}, g a n d h are two u n k n o w n real c o n s t a n t vectors,

PERIODIC ELLIPTIC HOLES

173

q = A r g + B r h . C o m b i n i n g (5) a n d (7) a n d e x p a n d i n g their values a l o n g L ik in F o u r i e r series, wc have 2 ~ [(Re{BM.Arlg

+ Rel'BM.Br}h)cosn0 + (Rc{BN.Arlg

+ Re{BN.Br~h)sinn0]

= y" [(v. + % ) c o s n O 4- i(v. - V.)sinn0l,

(8)

n-0

where |

2It

M,

2n f o W(z)cosnOdO,

N,, =

~;

n = 0,1,2,.--.

(9)

E q u a t i n g the coefficients o f similar t e r m s in the two sides oi"(8) leads to 2Re{BM.Ar}g

+ 2 R e { B M . B r }h = v. + ~ . ,

(10)

2 R c { B N . A r }g + 2 R e { B N . B r } h = i(v. - ~.). /

It follows from the a b o v e e q u a t i o n that the n t h loading, v . o " + ~ . o - ' , only affects M . a n d N~ ; therefore in the following, a t t e n t i o n is focused o n d i s c u s s i n g the elastic fields d u e tov.a ~ +~.cr-'.

In this case, M . = N , . = 0

(m ~ n). In a d d i t i o n , b e c a u s e g

andh

can

be adjusted a p p r o p r i a t e l y , c h o o s i n g M. = l=diag{I,l,I}, N . = - il = - i d i a g { I , l , I } ,

(11)

in (8) a n d (10) a n d s u b s t i t u t i n g (11) into (10), one Iinds that g=v. where S =

+~.,

h~L-I[i(v.-~,,)+Sr(v.+~,,)],

(12)

i(2AB r - I),L = - 2iBB r .

O b v i o u s l y the values o f f . ( z . ) ( a = l , 2 , 3 ) o n L i K

c a n be written by

h ( z . ) = e -,.0

(13)

D u e to the periodic p r o p e r t y , we can a n a l y s e only the region: [Rezl < I / 2r (see Fig. 2) in which Re

d e n o t e s the real part o f z = x j + i x 2 . D e s i g n a t e the interior d o m a i n over a peri-

od as S ~ , then the w h o l e region S is the s u m m a t i o n o f Sk

(S = ~ ' _

_ ~ S~ ).

X,

- ~ 'r .

m ). (/,"~- ~- - - i ~ .. _

So

Lo

: lr -~

FIG. 2. Elliptic hole in a periodic array If W(z) is the solution in the w h o l e region S, the projection udo(z) o f W(z) on So is piecewise h o l o m o r p h i c a n d ~d0(z) will be c o n t i n u o u s to x 1 = _+ I / 2r. It s h o u l d satisfy thc following c o n d i t i o n s :

Y. H U , Y H U A N G ,

174

lo,(z°) = e .... o,

L. C H E N G .

a = 1,2,3,

q%(z)l,-,.,2 .....

S. LI arid W. Z H O N G

on I-o,

= q%(z)l,-

/

(14)

,x21
~2 . . . . . .

On the o t h e r h a n d , if tt°0(z) is the solution on S o , we easily o b t a i n the whole region solu. tion q'(z) t h r o u g h an e x t e n s i o n o f W0 with a periodicity r.

3. D e t e r m i n a t i o n

o f the u n k n o w n f u n c t i o n s

T h e con formal isotropic function

~1 = tan( "~ z),

(15)

r

can m a p the region IRez] < l / 2 r

o n t o the region ~ 0

in ~ - p l a n e f o r m e d by the whole

q - p l a n e inside o f the region [--i, i] (see Fig.3). M o v e o v e r the p o i n t s z = 0, 4- I / 2r,ioo a n d tcx~" arc m a p p e d o n t o q = 0 , c o , i a n d L I /2r

-- i, respectively. T h e two straight lines x

~

=

arc m a p p e d o n t o the left a n d right faces. Lo is m a p p e d o n t o a c l o s e d curve F0

whichcncloscsthcncworigin0,

buti

and

i arc o u t s i d e o f f 0 .

.fi

'j/,o

FIG. 3. q-plane

For a m s o t r o p i c elasticity, therc arc three S t r o h ' s c o m p l e x variables, z , = x ~

¢ p,x2,

besides the physical variable z This requires a n o t h e r thrce a d d i t i o n a l m a p p i n g f u n c t i o n s which can bc c o n s t r u c t e d in the light o f ( 1 5 ) it is easily f o u n d that [6, 71 /t

~l, - tan( r z , ) ,

a - 1,2,3,

(16)

possess similar properties to that o f (15)(see A p p e n d i x A). A s s u m i n g that W'(z) is m a p p e d to W"

by the a b o v e m a p p i n g s , W"

will have the s a m e limitations a l o n g the c u t - l i n e ,

thercforc W " m u s t be piecewisc h o l o m o r p h t c m ~ 0

(they are also b o u n d e d at intinity).

Using C a u c h y - G o u r s a t t h e o r e m m )/--plane (Fig. 4), wc o b t a i n

tlJg (q')

,c !!'_; !~°--~d,.,

= 2hi ". z,

(~7)

:l,

where .Q is the integration c o n t o u r m q--plane(see A p p e n d i x A). R c t u r n i n g to z , - p l a n e , we obtairl I

]&(z.)= ~rri ~

n(t,

/~.(/.)cot.--r

z~)

..... dt. t

B ~s the integration c o n t o u r in z - p l a n e

I

nto

.

2ri -~n/o,,(t.)tan.sOt.,

(18)

PERIODIC ELLIPTIC HOLES

175

x~ hi

So

#-

ml

m2

ht

FIG. 4. Integration contours It is seen from (14) that equating the integrand on m = to that on m 2, the s u m m a t i o n o f the integrals along these two lines vanishes. Therelore integral along B reduces to f

= B

f I +

hi

+ ml

f f +

AI

ml

.¢" = LI

; i +

~l

k~

f.

(19)

LQ

The stress fields at infinity are zero and the b o u n d a r y value o f f'o.(z, ) can be obtained from (8) as irte

f0, (z.)

- ,t0

(20)

p . b c o s O - asin0 '

so that (18) becomes n f / ~ . ( z , ) = 2r

e-'"°asinoCOtn(t,---Z,)dt" p,bcosO r

+ n f 2r

e-i"° tan n t " d , . . ( 2 1 ) p . b c & O - asin0 r

l.Jp to the present we have obtained the elastic solution to collinear periodic elliptic hole in an anisotropic medium subjected to loadings, v , o " + ~ , a - " , along the holes. Based on the solution, when the b o d y is subjected to arbitrary loadings along L ± , , we can also find its solution by s u m m a t i o n on n, namely

*" q.ft~(z.)=

. n

fp.bcosO-asinO

e -

x [A • ( v . i -

v . t ) + B n L i , l(i(v.* - v . * ) + S I k ( v . i - v.t))]}.

(22)

Since q . f o . ( z . ) , a = I, 2, 3, are essentially periodic functions o f z . , the whole region solution q . , ~ ( z . )

arc oo

. n

e -`"o

x [A . ( v . i -

[corn(t,

z°)+tan_~_]d/,

v . t ) + B ~ L i , t ( i ( v . , - V . , ) + S~k(v.t -- v.I))]},

in S.

(23)

When (23) is substituted into (2) and (3) the stress and displancement fields in the medium are very easily obtained and the stress concentration along L i* can also be found. If

v. = 0 , ( n > l ) , v~ =

-

~(at~

+

ibiS),

(24)

and r ---, co, the elastic fields in (23) will become as F = diag{~t,~2,~3}{Argt + B r h l ) ,

(25)

176

Y. HU, Y HUANG, L. CHENG, S. LI and W. ZHONG wtth z , - ~'-: ~---? a : ¢" =

......

gl -

-.ate.

"

p;,b ~

i-iP.b

i

(26)

hi = [ , l ( b t ? - a S r t ~ ) , i

t'l' and t~/ are two real constant vectors. This is the result obtained by H w u and Ting [10].

4. The hoop stress Let n(¢.,), m(~,) bc, rcspcctivcly, the umt vectors tangent and normal to the hole b o u n d a r y l,u. Fig. 2. Wc havc nr(@) = (cos@,sin~O,0) r, m r ( @ ) - ( then thc traction vector t . t.

:

---~.~ -

-. sin¢, cos@,0) r,

on the surface normal t o L o

2RelBW.,,,(z)Ar}g

(27)

will be

2Re{BW~(z)Br}h,

(28)

where m ~s the arc length measured normal to L0 in the direction o f m . The h o o p stress t..

is determined from t..

=

n7(¢)

•

t..

(29)

If /.0 ts only subjected to the action o f v l ( s e e (24)), the solution functions q . ~ ( z , )

be-

comes

q, L ( Z . )

- -

2r ~ p-.bcosi)-- asinli L x [

cot t[(l~

nt r

~ tanr~-

dr,,

aA~,t~,4 Bt, l.j,l(btY~ • aSikt~t)].

(30)

In addition,

"~ [ . ( z , ) ~ [p.cosqJ

sm¢]/,~(z,L

(31)

Thus by combining (28-31), we easily obtain the h o o p stress distribution. To illustratc the procedure, wc cite a special cxample in point that the medium is alunumum (cubic crystal) and the holcs is applied by uniform load as fllows (I) e l a s t i c c o n a t a n t s : C . x

~ 10.8 × 101°Nm ~, ("12 = 6 . 1 3 x 101°Nm -2, C 44 = 2 . 8 5

10~° Nm -2 (2) Uniform load: ty ~ ( I , I , 0 )

~,t~ =,.~(I,I,0) r , a is a constant. The results is shown

in ]:iS.5

1:1(;.5. The stress concentration along L0

5. Conclusion The elastic lields o f collinear periodic elliptic holes in an anisotropic medium are given by

PERIODIC ELLIPTIC HOLES

explicitly form. The solution presented here is simple and tractable for obtaining the stress concentration distribution. References I. R.C. ]. Howland, Proc. R. Soc. Lond. A148(|935)471-491. 2. B. L. Kariha]oo, EngnR. Fract. Mech. 12(1979)49-77.

3. N. I. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity (Noordhoff, Groningen, 1953). 4. Jianke Lu and Haitao Chai, The Periodic Problems of Plane Elasticity (Hunan Press, 5. 6. 7. 8.

Changha 1986). Chyanbin Hwu, Int. J. Fract. 52(1991)239-256. Hu Yiantai and Zhao Xinghua, Int. J. Fract. 76(1996)207-219. Yian-Tai Hu and Yu-Ying Huang, Int. J. Enl~ng. Sci.34(1996)1623-1630. A. N. Stroh, Phil. Mag. 30958)625-646.

9. T. C. T. Ting, [Q. J.] Mech. Appl. Math. 49(1996)1-8. 10. Q. Li and T. C. Ting, [Q. J.] Mech. Appl. Math. 42(1989)553-572. I I. Q. Li and T. C. T. Ting, J. Appl. Mech. 560989)556-563. 12. P. Chadwick and G. C. Smith, Adv. Appl. Mech. 17(1977)303-376. 13. I-Iuajian Gao, [Q. J.] Mech. Appt. Math. 45(1992)149-168.

177

178

Y. HU, Y HUANG, L. CHENG, S. LI and W. ZI-IONG Appendix A F o r a m s o t r o p i c elasticity there are there S t r o h ' s c o m p l e x variables, z ° = x~ ÷ p , x 2 ,(~ = I, 2, 3), which m a p 1.0

and So

in z - p l a n e o n t o the ellipses L °, a n d D U - in za-plane,

(scc Fig. A I ).

'

-

tr

l.~,

r .

~

D:

l:'

_r

X.

_

.

'.

i

Fig A[ The

two

straight

lines,

x~ -- +

I /2r,

in

XI-B,/A,X~ThB,/2A~,

y. ~ tan - ~ ( B , / A , ). T h u s , l ; ( z , ) m a p p i n g o f D ,° t o .~,

So inz-plane a n d D ,° m~l-plane in

z--plane

z.-plane,

where

are

changed

p,=A,+iB,,

to

l~t a n d

/~,

B,>

0;

is readily o b t a i n e d , p r o v i d e d the a p p r o p r i t e c o n f o r m a l

is k n o w n . In the following attention is focused on the m a p p i n g func-

tion q, - t a n ( n z , / ti). which m a p s l ) ° i n t o ~ ,

a -- 1,2.3.

(B.I)

rormcd by the whole tl, - p l a n e outside o f the r e g m n l - i . i];

M o r e o v e r the point ( i / p,}z° = 0. ( i / p , ) r / 2 .

ico a n d + i o o , which c o r r e s p o n d to the

points z = 0 . + I / 2, i,:x:, and - i o c . arc m a p p e d o n t o t/, = 0 , c o

i and

• i, respectively;

The two straight lines, l~ and l ] . are m a p p e d o n t o the left a n d r i g h t l h c e s ; I.~ is m a p p e d onto a closed curve F]

which encloses the origin but i and

i are outside or F~. Com-

paring the n a t u r e t)['r/, with q in cqn.(15)ywc find that ~l, h a p p e n s to be the m a p p i n g functlonrequired. Becauscq,

is cntireiy similar to ~l, thc solving process in the paper is suitablc.

0

/

:

i

)

-I

Fig. A2.

~l,-plane