Issue 24

G. Cricrì, Frattura ed Integrità Strutturale, 24 (2013) 161-174; DOI: 10.3221/IGF-ESIS.24.17

Figure 1 : Physical vs. equivalent voids distribution (plane representation).

The homogenized constitutive law is defined by (tensors are indicated with bold characters):     2 2 2 1 3 3 , , , 2 1 0 2 eq m eq m q f q fcosh q f                         

(1)

  



p d d  ε

(2)

σ





: p

1 f d   ε

df

(3)

I

p

(1 ) f 

d  







d

:

(4)

  p   σ C ε ε

(5)

0        N N 

1









if

0   and d

0

0          0 0



(6)

d

otherwise

Where:

Eq. (1) defines the plastic surface; Eq. (2) is the plastic flow rule; Eq. (3) is the void growth rate definition; Eq. (4) imposes the equivalence between micro and macro-mechanical plastic work; Eq. (5) is the global stress-strain relationship; Eq. (6) is the plastic hardening power law for the matrix material.

Further, the symbols indicate: 

: stress tensor;

: total and plastic strain tensors;

   p

: constitutive elastic law;

C

: Von Mises equivalent global stress;

 eq  m

: global mean stress;



: current matrix flow stress (internal variable of the model); : current matrix equivalent strain (internal variable);



: voids volume fraction (internal variable);

f

N

: hardening coefficient;

: yield equivalent stress and strain; : Tvergaard correction coefficients.

 0

  0

, q 2

, q 3

q 1

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