Crack Paths 2009

can be represented in a way similar to that employed in standard plasticity-like FE

numerical approaches. The present formulation does not introduce discontinuous or

modified shape functions to reproduce strain localisation, but it simply relaxes the stress

field in an appropriate fashion by considering crack bridging and shearing laws to

evaluate the normal and tangential stresses transmitted across the crack faces.

Furthermore, the uncracked material is allowed to behave as a linear elastic or an

elastic-plastic one.

The proposed approach is presented in the context of a variational FE formulation.

Then, the behaviour of brittle structures as well as the crack paths inside the material in

2D problems are predicted. Finally, some comparisons with literature and experimental

results are discussed to assess the capability of this approach.

D I S C O N T I N U OFUOSR M U L A T I O N

The discontinuous displacement field in a solid Ω (Fig. 1a) where a displacement

discontinuity takes place along the line S can be written as follows [12]:

(1)

x δ x δ

x δ x δ x

4 x24w13x H ⋅ ) ( ) ( ) ( ) ] ] ( ) [ [ ( ) ( ) H = + = +

) ( x δ d

where the total displacement field )(xδ is written as the sumof the continuous )(xδ and

) ( ) ( ) ] ] ( ) [ [ ( ) x( w x x δ x x δ ⋅ = = H H d

the discontinuous part

(where )(xH is the

)( )]](x[[ w x δ = is the discontinuity

Heaviside jump function across the crack line, and

displacement jump vector across the line S ). The corresponding strain field is given by:

∇⋅

δ

s = ∇ +

H

+

⊗

(2)

s

s

s

( ) 4 4 3 4 4 2 1 4 4 4 4 3 14 4 42 ) ( ε x i x w x w x x δ u b

x ε

)( x ε

)(

)( ) ( ) (

) (

is the Dirac delta function in S , ()s•

()•

denotes the symmetric part of

,

where

s δ

)(xεb and )(xεu are the bounded and unbounded part of the strains, respectively.

By considering a finite element (Fig. 1b), in which a discontinuity occurs along a

straight line S crossing the element in a direction identified by the unit vector j and

centred in its geometrical centre C, the displacement field )(xu can be written [12, 16]:

[ ] w x N x )( + H

δ x N x u )( ) ( ) ( ⋅ + − =

(3)

where the discontinuous part is given by []wxNx)()(+−H.

The displacement jump

vector is v= +u, wherwe u and v are the displacements jumps normal and parallel to

the crack line, respectively (Fig. 1b).

The corresponding small strain field can be obtained from:

[ ] 3 2 1 4 4 4 4 3 14 4 42 4 ) ( )( )( ) ( ) ( ) ( ) ( ) ( x ε x ε x w x w x B x δ x B x ε u b s s δ+ − ⊗ = + with

∑ + Ω ∈

)( x B

(4)

=

x

i B ) (

+

e i

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