Crack Paths 2012
D re 0;]
2
1
where D,: 1
( ),i=1for re[0;l),andi=2 for re(l;R];AId—2— i ;
D2 re (l;R)
dr rdr
I _ K_=—Ghi; g,=__,; 1),: 2=IE;I=2h /3; K,=K =4Gh/3; q,,=—0,5q; 4 I 2 E "‘ 3 I I I
‘
3
5 G
W10) : _ § 8 1 h ‘ 2 A _W 'i h i 2 q i_ £2 _ 2 '4Aq'2; qrz : q_; ui’W"Wir’hzi u ’ w ’ w r ’ fhor the
4 l
l
l
l
l
domain r>l; hi=h+=h(l—fl); ,BIhO/2h; qil=qul=0,5(0'Z(h—h0)—q_),
q,2 = qu2 = q_ +0'Z(h—hO), u,,w,,w,, =u+,w+,w:; D1. = D1+ = I+E ; 1+ = 2h3(1—,B)3 /3 ,
K,’=K; :4G’h(1—,B)/3
or
D, =1); =r-E; r =h03/l2;
K,'=2G’h0/3;
q,1 = q,1 = 0,50'Z(h—h0); q,2 = q,2 = —0'Z(h—h0) and ul.,wl.,wl.,,hl.
= u_,w_,w;,h0 /2 for
the top and bottom parts of the plate in the domain r S l ,
respectively;
a, =3(1—0,75v*)5,;
e, =l(1—v*)5,; E=E/(1—v2); A’=V—; =0,5v"G'/G;
5
G 20
1 — v
E, E’, G , G’, v, v” are the elastic and shear moduli and Poisson ratios of the plate in
the longitudinal and transverse (with primes) directions; q- = q = const is the
distributed load applied to the top surface of the plate ( z = — h); ul. are horizontal
displacements of the median surfaces of the upper and lower parts of the plate; w and
w, are the entire and shear vertical displacements of the median surface of an
uncracked part of the plate; the Romannumerals at superscripts of w,wT, u and ql, q2
denote the order of a derivative on the variable r; subscripts <> and <
respectively upper and lower parts of the plate at the cracked domain; 2h is a thickness
of the plate; h0 is a thickness of the plate part which is under the crack. In the
formulated problem one assumes that the bottom face (2 = h) of the plate is traction
free, hence, q+ I 0 , and the stress 0'Z(h—h0) equals to the contact pressure between
crack faces.
One can obtain the value of normal contact pressure p (or the stress 0'Z(h—h0))
within the framework of the Kirchhoff — Love hypotheses for thin plates, or based on
the Timoshenkoplates theory. Both states that vertical displacement w (together with
their derivatives) does not depend on the transverse coordinate z , i.e. w, =wu= w .
Therefore, the first equation of the system (2) for the upper and lower parts of the plate
can be written as:
Dl+A2W:qu2:q_pD;fA2W:ql2:p~
(3)
Thus, the approximate value of the contact pressure between crack faces, according
to Eq. (3), is equal to
qDf
D + + D _ A 2 w = ;
i3/5,
= 4
(l l)
q P D1.+D1_qfl
()
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