Issue 62

A. Baryakh et alii, Frattura ed Integrità Strutturale, 62 (2022) 585-601; DOI: 10.3221/IGF-ESIS.62.40

Combining the above, the total yield surface "Parabolic envelope of Mohr circles/Rankine" (PMC/R) can be written as a piecewise function for the sextant  max >  mid >  min :

1 2 1 2

 

PMC



   

   

P

( ,{ , }),

( , )



c

t

c

t

t

min

PMC R /



 

   ( ,{ , })

(37)

c

t

 

R



   

  ( , ),



P

( , )

t

c

t

t

min



where the parabolic criterion (30) describes the "compression-compression" regime of the stress-strain state (SSS) and partly "tension-compression", and the Rankine criterion (35) limits the tensile stresses. The total yield surface (37) is illustrated in Fig. 8. The set A for (37) contains two parameters   c   t  . The numerical simulation of uniaxial loading of a cubic salt specimen was carried out implying the associated law of plastic strain. The corresponding parameters of the TCP algorithm in principal stresses for the parabolic envelope of Mohr circles are:                                           1 1 1,3 1,3 2 2 2,3 2,3 3 3 1,2 1,2 max,min max min N N 0 N N 0 N N 0 2( ) ( , ) 2( ) ( , ) T T PMC PMC T T PMC PMC T T PMC PMC c t a b a b a b a P b P (38)

c

t

max,min

max

min

Also for the Rankine criterion they can be written as:                   1 1 2 2 3 3 N N 1 0 0 N N 0 1 0 N N 0 0 1 . T T R R T T R R T T R R         

(39)

Similar to the linear Mohr-Coulomb criterion (22), the material is also dilatant. The linear isotropic hardening was incorporated. Hardening was implemented by evolution of the uniaxial compressive strength similar to the Tresca criterion (20). Clearly, this effect works only in the region where the parabolic criterion (30) is satisfied. The TCP algorithm parameters associated with the hardening are:

    

  P Q

 Κ

  max

   (

)

min

c

c



h

(40)

Η Κ

PMC



N



The derivatives in expression (40) 1 can be written using the chain rule:

              c c c c p P P r p r p Q Q r p r        

(41)

596