Crack Paths 2009

components on the materials plane element. The unit vector i specifies the orientation

of plane. A general critical plane condition can be formulated as follows

F I I n a X f ( 0 ' , I ’ T n , 5 , I ’ 7 / n ) — f g : O

where f; represents the critical value reached by the failure condition generally

depending on both stress and strain components associated with the respective planes.

The comprehensive review was recently presented by Karolczuk and Macha[42].

In this section, the local critical plane models will be extended by introducing

non-local failure criteria applicable to both regular and singular stress regimes and also

for monotonic and cyclic loading cases.

Consider an arbitrary physical plane A and the local coordinate system

(§I,§2,§3), Fig. 8. In the global coordinate system (101,102,103)

the origin of the local

system is specified by the position vector x0(x01,x02,x03) and the unit normal vector

n(nI,n2,n3) specifies the plane orientation, where n, I cos(§3,x,.).

Fig. 8. The 3D system in R-criterion.

The resulting shear stress and strain in the plane A are expressed as follows:

in I (131 + 132T”, y" I (yj1 + 7,2,)”. Assume that crack initiation

and propagation

process depends on the contact stress and strain components and also on the damage

accumulation on the physical plane. Consider an elliptical condition for 0", > 0 and the

Coulombcondition for on s 0 , thus

l/2

[ 2

2

6.]

+ — —1,

o,,>0,

F : R U [ fi , T _ nm]a_x1 :01

T0

(17)

o' r

11

L‘

C

l —(

In +0‘, tano)—1,

0'” <0,

T C

where 06,16 denote the failure stress of material in tension and shear. For large stress

gradients or singular stress regimes such as those occurring at vertices of wedgeshaped

notches, the non-local stress failure conditions is applied by averaging the failure stress

function over an area dO >< d0 , thus

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