Issue 29

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

    ˆ ˆ   σ



g

ˆ  σ





 σ

(20)

arctan

p

g



ε

, which implicitly defines a function   p ˆ    σ ε





ˆ  σ

0, 3 



This is a scalar equation in the unknown theorem, its first derivative turns out to be:

. By applying Dini’s

2 

 g g g g g 2

ˆ  σ

d



(21)





 

d

p

ε

where g and its derivatives are computed at   p ˆ    σ ε

p ε can be assumed to belong to   









ˆ  σ



3,

3

. Noting that

,

by (20) it follows that   p ˆ cos      σ ε



g

g





p   

ˆ  σ

, sin

(22)



ε





2 g g 

2

2 g g 

2

( D   

p  and its first and second derivatives with respect to p    )

whence, recalling (18) and (21),

easily follow:







 

4

2 2

3 g g







g

g

g g

1 ,

2

  p   ε

  p   ε

  p   ε

D 

D 

D 











,

(23)











2 g g 

2

2 g g g 

2

3 g g g 

2 g g 

2

and computed at   p ˆ    σ ε

where the derivatives on the right-hand sides of (23) are performed with respect to ˆ  σ the gradient and the Hessian of   p D  ε are straightforwardly obtained from (16):

. Finally

D  D 

D 



   

 

 

y   y

p     ε

 





D c

p

p



ε

ε

0

(24)





D 

D 

D 

 







 

       

p      p



 

 





 

 

 

 

D c

p

p

p

p

p

p

p

p











ε

ε

ε

ε

ε

ε

ε

ε

ε

0

Simple expressions for the gradient and the Hessian of p   ε and p   

are given in Appendix A. The same argument holds

if kinematic hardening is present. Deviatoric yield function, isotropic hardening The yield function is assumed to be      ˆ ˆ ˆ ˆ ˆ i , , , f q g c       σ σ σ σ σ whence the dissipation function follows:          ˆ ˆ ˆ i ˆ ˆ y i p ˆ i , , , , sup q g c q D σ σ σ σ σ σ ε            



y   

q

(25)

i

0







 



ˆ σ   

ˆ σ 

p    q 

i 

cos

(26)

i

p

p





ε

ε

ε

0

This implies:









  p D q   ε 





p    ε i  ,

 



y 



 

 

i 

D

c

q

0,

sup

(27)

i

i

p

p

ε

ε

0









q

i

y 0

Accordingly, the dissipation function turns out to be:

    p p p p D D     ε ε  

y   

i if otherwise c c       i if

i 

   

 p     ε i  , c

ε

0

  p   ε



D 

 

D

(28)

y   

p



ε

ε

0

115