Issue 57

R. Fincato et alii, Frattura ed Integrità Strutturale, 57 (2021) 114-126; DOI: 10.3221/IGF-ESIS.57.10

  that can be obtained substituting the expressions in Eqn.

The only unknown in Eqn. (15) is represented by the term

  can be

(15) into the consistency condition of Eqn. (4) 1 . After mathematical manipulations (details can be found in [9])

computed as follows:

3 2

 

  





2

   1 k



k n

1

2

   4 r

3 : h

3 : h

:

χ β

χ β

χ χ





n

n

n

n

n n

1

1

  



1

(16)

2

r

2

n

Lastly, the back stress  1 n q and the radius r n+1 are updated by substituting the computed   into Eqn. (15) 3 and then  1 n χ in Eqn. (15) 1 . Novel algorithm for the definition of the workhardening stagnation The novel approach consists in the observation that, if the condition in Eqn. (14) is fulfilled, the back stress of the bounding surface  1 n β , computed at the k+1 sub-iteration of the CPPM, should lies on the non-IH surface. Therefore, it is possible to estimate the updated  g by means of a series of additional sub-iteration j within the same k+1 sub-step. In particular,

 j n g can be obtained from the previous   , 1 j  1 , 1 

n g by a Taylor expansion that neglects the quadratic terms:

j

j





g

g



   1 j q



j

j

j

1

1





  , 1 n   g



g

r

 , 1 n







n

n

1

1





r

q



n

1



n

1

         , 1 1 1 j n j n j n g r   

(17)

g



n

,



j

with

for

0

q q

n

r

n

Figure 3: schematic representation of the evolution of the non-IH surface.

The only terms that need to be estimated are the increments of the back stress    1 1 j n q

 j n r . To compute   1 1

and of the radius

  is introduced:

those terms the following variable











k n

j

k n

1

1

1





3

:

β

n r q

β







n

1

1

1

   1 j 

(18)



n

1

 2, 1 1 j 

2

From Eqs. (5) and (6) it is possible to write:











  1 k

j

j

j

1

1

    1 h q

 n



β

q









n

n

n

1

1

1

1

(19)





j

j

j

1

1

    r h 1 n

 n

r

  n

1 1

120