Issue 29

D. Addessi et al., Frattura ed Integrità Strutturale, 29 (2014) 178-195; DOI: 10.3221/IGF-ESIS.29.16

where E and  are the material Young's modulus and Poisson ratio, and I is the 6 6  identity matrix. The damage variable is bounded in the range   0 ,1 , where D = 0 correponds to the initial undamaged state of the material and D = 1 to the complete degraded state.

x  ( x , y , z ) over the cross - section in the channel-shaped cantilever beam with warping restraints at the fixed end.

Figure 8 : Axial stress

For a given strain state of the material, the value assumed by D is computed by defining a damage associated variable, which governs its evolution. In particular, an equivalent strain eq,t  is defined on the basis of the positive part of the principal strains k  as:

2



 





3

  

k

k



(39)

 

eq,t

2



k

=1

To model the unilateral effect, two different damage variables are considered: D t

describing the damage related due to

tensile deformation states, and D c evolution laws are stated for both:   1 A  

reproducing the damage in presence of compressive deformation states. The following













B

  ,    D 0          =  

i

0

 



A  e

i t , c 

(40)

 D 1

i

t

eq,

0

i

i

i



eq,t

 is the initial damage threshold, and A i

and B i

where 0

are material parameters. The resulting damage variable D is a

combination of the two variables D t

and D c

defined as:

 D D D t t c c      

(41)

 are defined as follows:

The weighting coefficients i

















3

3

(

)

(

)





          and           t,k t,k c,k

c,k t,k c,k





t 

c 

H

H

(42)

k

k

2

2





eq,t

eq,t

k

k

=1

=1

with:

     



1     if      0      if     

0

  

t,k c,k

H

k



0



t,k c,k

 and c,k

 are the principal strains evaluated on the basis of the two tensors t E and c

where t,k

E , defined as:

-1   E Λ Σ

-1  E Λ Σ

(43)

           and          

t

t

c

c

190