Issue 29

N.A. Nodargi et alii, Frattura ed Integrità Strutturale, 29 (2014) 111-127; DOI: 10.3221/IGF-ESIS.29.11

whence it follows that p k    α ε

(11)

and









ˆ σ ε

p    ε i  ,

p     q

i 

D

    i ˆ , ˆ , sup q q σ σ

(12)

i



f

0

i

With the assumption that the yield function is isotropic, it can be represented, with a slight abuse of notation, as   ˆ ˆ ˆ i , , , f q    σ σ σ , where ˆ  σ , ˆ 0   σ ,   ˆ 0, 3    σ are the Haigh-Westergaard coordinates of ˆ σ (see Appendix A). Since the maximum of the scalar product of two symmetric tensors is attained when they are coaxial (see, e.g., [34]), in (12) it is possible to assume that ˆ σ is coaxial with p  ε and to express their scalar product making use of the Haigh-Westergaard representation. Accordingly, the dissipation function is given by         p p p ˆ ˆ ˆ i ˆ ˆ ˆ i p ˆ ˆ ˆ i i i , , , , , , 0 , sup cos( ) q f q D q                          σ σ σ σ σ σ σ σ σ ε ε ε ε (13)





,

,

p  ε .









 



0,

3

where

are the Haigh-Westergaard coordinates of

0

p

p

p

 ε



ε

ε

Deviatoric yield function, perfect plasticity or kinematic hardening The yield function is assumed to be of the form     0 ˆ ˆ ˆ ˆ ˆ y , , f g c         σ σ σ σ σ

(14)

0 g g    . The latter condition y  and c are positive constants;

2 C , positive, even, periodic function with period 2 3  , such that

where g is a

  ˆ σ

ˆ g  σ

 

are convex. Moreover,

guarantees that curves of polar equation

const

0

y  can be interpreted as the initial yield limit in tension if   0 1 g  , or in compression if

c 

: in that case,

usually,

2 3

0

  3 1 g 

 . According to (14), the dissipation function is given by           p p p ˆ ˆ ˆ ˆ ˆ y 0 p ˆ ˆ ˆ , , sup cos g c D                     σ σ σ σ σ σ σ σ ε ε ε ε

(15)

This implies:

  p   ε

  p 0, D c   ε

D 

 

 



(16)

y

p

p



ε

ε

0

where





  p   ε





D 



  ˆ ˆ ˆ { , } 1 sup cos g       σ σ σ σ σ ˆ ˆ

ˆ  σ





(17)

p



ε

or, equivalently,   p D   ε 





 

 





cos

   

  p ˆ ˆ   σ ε σ

  ˆ sup      σ

(18)

g





It is worth noting that D  is the support function of the convex set of polar coordinates   ,   the supremum is attained on the boundary of that set, a simple computation yields         p p ˆ ˆ ˆ ˆ sin cos 0 g g              σ σ σ σ ε ε ˆ σ σ ˆ

  ˆ σ

ˆ  σ

 

g

:

1

. Because

(19)

which can be recast as:

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