Crack Paths 2012

the plate. Thus, these faces will be denoted with signs <<+>> and <<->>, respectively. Then

for the cracked domain (0,l) of the plate displacements (8) for the plate part under the

crack can be written as,

4

II

_ _ v p w1 = A 1 r 2 + K 1w+T=fiA,T—pr2/(4Kl), u 1 = F 1 r + fi n(9) _ _ pr _ _ , _

For the uncracked domain (l;R) displacements are equal to:

4 W =Ar2+Br2lnL+K + qr ;

10

2

2

2

R 2

( )

II wiz) IA?) +Bi2)ln%—qr2/4K', u2 =F2r+V—qr.

Integration constants AHBPKI, Biz), AS), F. are determined within the boundary

conditions at the edge r = R of the plate. In the case of a hinge supported plate these

boundary conditions write as,

w2(R)=0, w,(R)=0,M,2(R)=0,Nr2(R)=0.

(11)

Satisfying conditions (11) one can obtain that

_ qR2(3+V) _ cohzq

A2 = — 4 A?) : q2R2;/4K'; F2 : 0; 32D(l+v) 2D(l+v)

(l2)

qR4 5 + V 3280 h2 = — 1+ — ,

qR2

r2

= 3+v 1-_;

2 64D1+vl 5+vR2 ’2

6( )( R2)

,dw‘” qR R = K — ’I — D —Aw ———. Q”( ) dr rzR dr( 2L, 2 d

Here it is assumed that the constants 3,, Biz) can be determined from the equilibrium of

the shear forces Q, and under the given load they are zero (B2 = Biz) = 0).

Except the conditions (11) at the edge of the plate, it is necessary to satisfy the

contact conditions between the cracked and uncracked domains at r =l :

w;(l)=W(l,h(1-p));

0',‘(l,,5h)=0',(l,h);

N;(l)= o',(l,z)dz,

(13)

where

0'r_(r,Zl)=N (V)+MV('")Z r

G !

ho

r

2,2 — 0.15h02)[q,2 — 0.25Aq,,h§

;

l

z, = z — h + hO / 2 is a local coordinate of the plate part under the crack, which is directed

downwards.

Hence, the normal and shear forces and the bending momentacting at the bottom

part of the plate under the crack are defined as [6],

N; = hOEKuf)’ + Vu1_/r]+ h0A'q,, ,

_ _ d2w_ v d w Qrl : K I MrWl : ’_ D 1{ — + — — drz r dr ]—0.25£0q,2h02. ,dw; _

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