Issue 75

N. S. Kondratev et alii, Fracture and Structural Integrity, 75 (2026) 373-389; DOI: 10.3221/IGF-ESIS.75.27

where   γ k s  is the shear rate acting on the k -th intragranular slip system,     , k k s s  b n are the unit vectors of the slip direction and the normal to the k- th slip system, 0 γ  is the shear rate acting on the slip system when the shear stress reaches the critical shear stress, m s is the strain rate sensitivity exponent of the material at dislocation slip and twinning,     τ , τ k k s cs  are the shear and critical shear stress of the k- th slip system, and H is the Heaviside function. The critical shear stress evolution is determined by the known quasi-linear relationship [10]:

N

s

  k     τ 1 cs j

  j

( ) kl

h

(8)

γ 

s

a

  τ τ l

 

 

( ) kl

( ) kl

( ) l

( ) l h h

s



(1  





  

h

q

q

h

) δ

,

1 /

(9)

_ lat s

_ lat s

_ sat s

0

cs

where ( ) kl h is the matrix describing crystal hardening that arises because of the interactions between dislocations and forest dislocations, q lat_s is the latent hardening parameter characterizing the location of slip systems relative to the initial twins formed in SLM, ( ) δ kl is the Kronecker delta, τ _ sat s are the slip saturation stresses, h 0 , a s are the slip system hardening parameters, and К s , is the number of slip and twinning systems in the crystallite under consideration. The dot above the corresponding variables shows the time derivative t , the superscript T represents the transpose operation of the value of the second rank tensor, the superscript (–1) represents the operation of determination of an inverse, and is the meso-level volume averaging operator. The present work focuses on the early stage of plastic deformation, and therefore the hardening law is applied with sufficient accuracy to slip (8). Here, the interaction of dislocations with twins generated during deformation is ignored. In relation to initial critical shear stresses   0 τ k cs , the developed model mostly relies on the assumptions presented in [12]. The initial critical shear stresses   0 τ k cs depend on the status of grain and cell boundaries, initial twin boundaries and lattice resistance. The value of   0 τ k cs is determined by the relations from [12], which describe the initial critical shear stresses according to the Hall-Petch relationship: where 0 τ is the lattice resistance of the k -th slip system, d is the mean free path length of dislocations, and k y is the Hall Petch coefficient; options for determining d and k y depending on the morphology of the considered grain are described below. The presence of twins' boundaries is one of the main factors promoting an increase in critical stresses for the initiation of slipping and twinning mechanisms and consequently the yield stress. According to the obtained data, the twins of the SLM produced samples have a lamellar structure [18] and are characterized by a sharp change in the crystal lattice orientation [14]. The orientation of the crystallographic coordinate system of the twinning part of the crystal of the system k with respect to the crystallographic coordinate system of the original crystal is determined by the orthogonal vector   k tw r [14]:   k   0 τ cs 0.5 0 τ   y k d  (10)

  k

    2 k k tw tw



 n n I

(11)

r

tw

    , k k tw tw  b n are the unit vectors of the twin shear direction and the normal to the habitus plane

where I is the unit tensor, and

k . Just similar to [12], we assume that the slip and twinning systems in which the Burgers vector (perfect dislocation for slip systems, and twinning dislocation for twin systems) lies in the habitus plane of the initial annealing twins (Fig. 4) are soft or the slip plane is parallel to the habitus plane of these twins. The rest of the slip systems are considered being rigid. For instance, for the twins with a habitus plane (111), both slip   011 111   ,   011 111   ,   110 111   ,   110 111   ,   101 111   ,   101 111   and twinning systems   121 111   ,   112 111   ,   211 111   are soft.

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