Crack Paths 2012
where
N,_1(l)=3,5(1—fl)M,,(l)/h—0.1qhG*f/4;
M,,(l)=qll;
(3+v)(1—62);
2
M;,(o) =3l+—6v pr + ,B3M,,(l)+0.4fl2(q—
p)h2G*/3+0.1,Bqh2fG*/12;
f(fl)=1—(1—2,5)2(1+20,5(1—,5));G*=4(liv)(g—v”(3+v)).
Substituting the values of N,_1(l) and M,_1(0) into Eq. (15) one can obtain the
closed-form formulae for the maximal stress of, at the external surface of the plate
under the crack:
3 3 . 0;‘,(0, h 0 / 2 ) = % [ 1 + [ § — 1 ) 6 2 ] + 0 2 G * q ; (17) R2
3 3+v R2 0',_1(0,— hO/Z) = % [ ( 1 _ 6 2 ) ( 1 — 2 fl ) — § 6 2 ] — 0 . 2 G * q —0.025G*qf(,B)/,B.
For determination of stresses in the plate part above the crack, one can utilize Eq. (6)
in the local coordinates (21,, r):
+ 2 G’ 0' r,z = ’ + ’ z + “ — z — 01—. 6 h —0.6A h 1— — , 18 r ( u ) h o 1 + u 3 1 +( u ( q u 2 ( E ; ( ) N + M + Z G * 2 2 2 2
where z“ = 2 +,Bh is a transverse coordinate of the upper part of the plate above the
crack, directed downwardtso its mediansurface.
The values of Nf, M : are obtained from the contact conditions on the interface of
cracked and uncrackeddomainsat r :1 :
+
_ (1-2/1)
_
+ _ _ _ _
N, (1)-[h o',(l,z)dz,
o',(l, (1 ,B)h)-0',(l,
h).
(19)
Satisfying conditions (19), one obtains
N,+1(l)=—3,B(l—,B)M,,(l)/h+0.lqhG*f/4;
(20)
Wm)=h<1—r>N;<1)+3<1—/r>
M..<1>+0.4<1—rrpha<
2
A n dhence,
M; (l) = (1 -/3)3 M,, (l) + 0.4(1-fl)2 ph2G* /3+ 0.1(1—,B)qh2fG*/12.
(21)
Consequently, maximal stress of, (0,i h, / 2) can be obtained from Eq. (18) with the
account of Eqs. (20), (21),
Nari) . 3M.l(0)
0',+1(0,i h,/2)=
i0.2G*(q—p),
(22)
2(1 _ ,B)h _ 2(1 — ,B)2h2
where h, =2h+ =2h(1—,5); M,+,(o) =M,+,(l)+3l+—6V(q-p)r.
The problem is solved under the assumption that the applied load causes crack faces
to be in a smooth contact, thus, the opening mode stress intensity factor (SIF) KI is
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