Crack Paths 2012

in x.,_k

x,,+i

x.,_r

x,,+i

T ]

102,, +

02, +

2 92, +

12,

sin Mr (8)

2

#1

#1

M2

#2

n I O

The formulation of the problem shown in Figure 1 can be assessed by establishing

continuity, in terms of stress and displacement, when0 I 0 (the two materials are bonded

along the line of the crack extension), and w h e n0 I :lI7l' (cohesive crack surfaces). The

stresses at the cohesive crack tip are non-singular (because the stress intensity factors are

K 1 I K2 I 0). The above mentionedconditions can be summarisedas follows.

Cohesive frictional crack with normal cohesive separation

The following conditions need to be satisfied (6 I i 0 , two materials are bonded):

uli9:0+ = ulozoI , Ulo:0+ = vlozoI , Uyli9:0+ = UyloIOI , TzyldIo‘” = TxylozoI

(9,10,11,12)

Eqs (9) , (l0) , (11) and (12) give:

1

1

lli

#2

— > \ n ) a l n— +

I

—

— +

l

l

m [ ( —+k> \1n ) a 2 n+ +

: , u — 2 [ ( — k+2

+ +

a l n ‘ l ' b l n z g l n ‘ l ' h l n ( A n — 1)a2n+ + I — + +

Thecontinuity of u guarantees that of 5,6. For each value of A”, the asymptotic fields

in material 1 are characterized by a vector of 4 unknowns[a1,.,,a2n,b1n,b2n];

similarly, in

material 2 they are characterized by a second vector [g1,,,g2n,h1,,,h2n].

The following conditions need to be satisfied along the cohesive zone (9 I in):

O-yli9I21' : O-yl19II7r

5 TzyldIir : TrylblIIir : —p/fUyldIir

where ,uf equals the positive or negative value of the of kinetic friction coefficient, which

is assumed to be constant, and to depend on the relative sliding direction of the two crack

edges. In other words, ,uf > 0 when6 < 0 and ,uf < 0 when6 > 0.

Eqs (17) and (18) give:

(62., + 02, + 92,, + 02,) sin()\,.,

_ m )= 0

(19)

[(411 — 1)(aln + gln) + ( M+ 1X51” + hid] Sin((>\n — Dir) I 0

(20)

{19211 + h2n + ,u/f(a1n + b1n)]>\n + [—g2n + h2n + ,uf(a1n + 5119]} 005G411 I 1)7T)+

ilgln + hi1. + 11,402.. + b21014” + l—.g1n + hi1. + M01211 + b21011 Sin((/\n — 1)ir)= 0 (21)

Eqs(l9), (20) and (21) showthat the asymptotic solution is composedof two parts:

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