PSI - Issue 23

Nikitin I.S. et al. / Procedia Structural Integrity 23 (2019) 119–124 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

120

2

a version of the theory of plastic flow by Nikitin (2009). For the proposed local slip conditions, the description of active and partial plasticity is obtained for the first time in an explicit form. Also, the zones of active and partial loading are clearly defined in the range of loading parameters. The yield function and plastic potential are found. For the plastic potential function, obtained values express the associated law of plastic flow. Ranges of orientation changes for slip planes are defined for some of certain loading paths in cases of two- and three-axial stress states. Comparison between calculated plastic deformations and experimental ones is performed. 2. Local slip conditions and constitutive integrals of the theory. Let’s suppose that at each point of the medium the slip (plastic shear) can occur along any plane with a normal n passing through this point. In the Cartesian coordinate system 1 2 3 , , x x x the stress state is given by the stress tensor σ . A slip (plastic shear) vector γ is equal to a relative displacement [u] along the selected plane. A vector of tangential stress τ on the plane is      τ = σ n- n σ n n . The loading process occurs as a certain parameter  (for example, time) changes. The derivative with respect to this parameter is denoted by dot on top of the function, so d d    γ γ . The main assumptions about the nature of the slip on a unit plane are similar to those from Mokhel et al. (1983). Slip (plastic shear) occurs when the critical condition 0   τ and the active loading condition   2 0    are satisfied. One can normalize the first condition by 0  to obtain the following form of condition: 1  τ . In any other case a plastic shear does not occur. The slip condition on a unit plane satisfying accepted hypotheses is

γ

2 2 2 ( ) ( -1) (( ) ) H H      τ

0 = с



where 0 c is a model’s parameter , H is the Heaviside function. Contribution of the shear’s increment d d    γ γ to the plastic deformation tensor’s increment d   ε is

(       ε n γ γ n 

) / 2

or

ε

( n τ τ n   

0 = с

2 )( ) ( -1) (( ) ) / 2 H H      2 2



.

Let’s introduce the coordinate system associated with the principal axes of the stress tensor, such that principal values of the stress tensor satisfy the inequalities 1 2 3      , and spherical coordinate system R ,  ,  , related to it. Integral components of the tensor  ε are obtained by integration over various slip planes. It is to notice that their normal unit-vectors are within the solid angle sin d d d     . The integral components are



c H H

2           2 )sin cos sin 2 1 0 d d

2 ( -1) (( ) )( ) (    2

 



11 0

,  



c H H

2           2 )sin sin sin 2 2 0 d d

2 ( -1) (( ) )( ) (    2

 



(1)

22 0

,  



c H H

2          2 )cos sin 3 0 d d

2 ( -1) (( ) )( ) (    2

 



33 0

,  





2 sin cos sin sin cos             , 2 2 2 2

3 12 cos2 A S S    ,

where

,

A

B

2 2 cos

sin











2

2

2

0

1

2

3

















12 sin2 B S   ,

S

S

S

1 2 / 2 0      ,

1 3 / 2 0      ,

1 3         2 3 / 2

/ 2

. The limits of

12

13

3