PSI - Issue 40

Artyom A. Tokarev et al. / Procedia Structural Integrity 40 (2022) 426–432 Artyom A. Tokarev, Anton Yu. Yants / Structural Integrity Procedia 00 (2022) 000 – 000

428

3

ˆ

ˆ

T z v v r               f f r r r r r T in p T , ω z l T

– , ω l ω

(2)

r

p p 1 T 

,



e z l

f f

e   

e e 1 

,

where ˆ  is a nabla operator defined in current configuration, r v is displacement rate relative to the mobile CS rotating at a rate determined by the spin tensor , ω v is absolute displacement rate, ω is a spin of the coordinate system related to the lattice, T ˆ   l v is the displacement rate gradient (transposed). The additive decomposition is valid for the introduced strain rate measure:

e in .   z z z

(3)

In this model, it is assumed that at the initial moment of time, the volume of the material contains a sufficient number of edge dislocations necessary to implement the prescribed process of inelastic deformation. The dislocations movement along crystallographic slip systems (SS) are described through shear deformations (Hirth and Lothe (1982), Honeycombe (1984)). Each SS is characterized by a unit normal vector of the slip plane n and the direction of edge dislocations slip, i.e. the normalized Burgers vector b , which have fixed directions relative to the crystallographic coordinate system (CCS) throughout the entire process. At each moment of time, the inelastic component of the relative displacement rate gradient is determined as follows:

K

in k    z b n 1 γ

      , k k k

(4)

where   γ k is the shear rate along the SS with a number k . Shear rate in the slip system is determined by viscoplastic ratio (Asaro and Needleman (1985)):

1 m

    j j c       τ τ





  γ γ j 

  j

  j

(5)

H τ τ , 1,K, j  

0

c

where   0 , γ j is shear rate in SS when the acting shear stress reaches the critical stress value, m is material rate sensitivity parameter, K is the total amount of SS in the crystal. Critical shear stress on a slip system with number k are determined by taking into account intragrain hardening, based on the interaction of dislocations of different slip systems with each other. Hardening due to the interaction of dislocations is described by introducing a matrix of dimensionless coefficients (Franciosi (1985)). Each coefficient is responsible for a specific type of interaction, and its value is responsible for a relative contribution to hardening. The possibility of transition of dislocations to a neighboring crystallite with the formation of orientational mismatch dislocations is also taken into account (Kondratev and Trusov (2012)). In this work, for the correct recording of the constitutive relation independent of any superposed rigid body motions, the approach described in Trusov, Shveykin, Nechaeva et al. (2012), Trusov, Shveykin (2017) and Yants (2016) is used. This approach is based on "binding" the introduced rigid mobile coordinate system, which is responsible for quasi-rigid motion, to one crystallographic direction and the crystallographic plane of a monocrystal grain containing it throughout the entire process (Yants (2016)). Hooke's anisotropic law in a mobile coordinate system (MCS) in a velocity form will take the following form:   cr in : ,        σ σ σ ω ω σ п z z (6)   c τ , τ j j are acting and critical shear stresses in SS with number