PSI - Issue 43

Sergiy Kotrechko et al. / Procedia Structural Integrity 43 (2023) 228–233 Author name / Structural Integrity Procedia 00 (2022) 000 – 000

230

3

( )   =   r C t t

cb g t t d 2 ,

(1)

where cb  is carbide particles density and function g ( t ) is given by equation:

  

   .

2 t

1

(2)

( )

exp

g t

=

2

2



In this equation, t is the magnitude of normalised shear microstresses ns  , acting in slip systems:



ns

t

=

.

(3)

ns D 

Here ns D  is the variance of shear microscopic stresses ns  . Then C t and r t are the critical values of normalized microstresses at which the CN forms and which there is relaxation of incompatibilities in intergranular/interphase boundaries. Dependences of these parameters on the plastic strain magnitude and temperature pre-determine the influence of these factors on the density of crack nuclei generating within the “process zone”. In the first approximation, expression for the value of critical microstress of the CN formation, C t , may be represented as follows: (4) where  and ̅ are the equivalent macroscopic stresses and strains; d and max d are the average and maximum (with a given probability) ferritic grain sizes; C  is the critical stress of СN formation as a result of the carbide particle cleavage; М is the orientation factor (for  - Fe M = 0.36);  k , e k , C, β are the coefficients (for ferritic steels  k = 0.225; e k = 1.52 MPa; С = 0.0336 m/N ; β ≈ 2.57 MPa m 0.5 (Kotrechko (2013));  is the parameter that takes the value 0 or 1. If the value of the equivalent plastic strain ̅ is less than or equal to the critical value C e , then  = 0 and e * = ̅ , otherwise  = 1 and e * =e C . The physical meaning of the last two terms in expression (4) lies in the fact that they describe the effect of microstresses arising at grain boundaries on the process of crack nuclei formation. They characterize the value of shear microstresses caused by the incompatibility of microplastic deformations at grain boundaries. The magnitude of this incompatibility changes non-monotonically with an increase in macroplastic strain e . In the early stages of plastic deformation, it grows and then begins to decrease as a result of rearrangements of the crystal structure in the vicinity of facets of grain boundaries (in the near-boundary regions). Transition from growth (  = 0) to a decrease (  = 1) in these incompatibilities occurs at critical strain C e . At the macroscale, the value of C e can be estimated based on the dependence of the cleavage fracture stress of smooth (unnotched) specimens,  f , on the value of strain preceding fracture. When the strain C e is achieved, this stress  f reaches its minimum value. For common structural steels with basic ferritic microstructure, the value if critical strain C e ≈0.02. With an increase in ferrite grain size by annealing, the critical strain C e can increase to ≈0.05 (Kotrechko et al. (2007)). Generalized expression for the critical value of microstress relaxation due to the incompatibility of microplastic deformations, r t is:                                e k e − −   +      + =  1 1 1 C e C C d e Cd M k t _ * max _ ,

    

    

    

    

    

    

_

*



1

1

r

e

Y

,

(5)

1

t

M

 e k e e

=



+



−  

+  

−

r

_

k

m

d

d

max

b

C





where Y  is the critical stress of the relaxation beginning; b m is orientation factor for relaxation slip systems in intergranular boundaries; r is the distance from the grain boundary to the origin of microstress relaxation.