PSI - Issue 28

Fedor S. Belyaev et al. / Procedia Structural Integrity 28 (2020) 2110–2117 Author name / Structural Integrity Procedia 00 (2019) 000–000

2114

5

oriented defects vector b p on the p- th shear plane was introduced. It can be associated with densities of piled-up dislocations and dislocation loops. The components b pi of the vector b p were included into the expression for G mix :

  

  



4

3

4

(5)

2



 p p Φ b  

) 2    b (

mix

,

G



pi

pi

2

1  

1

1

p i

p



where μ = q 0 (( M f– M s)/ T 0 ) is a coefficient of proportionality; α is the parameter of the model The thermodynamic force causing the growth of the i -th variant of martensite in the p -th zone (i.e. the increase of the variable  pi ) is the partial derivative of the the Gibbs’ potential with respect to  pi :

G

 

.

pi F

 

pi

The necessary condition of the martensitic transformation was written in the terms of the correspondent dissipative force F fr as F pi =  F fr , (6) where the plus sign is for the direct and minus – for the reverse transformation. The value of F fr is derived from the transformation characteristics: F fr = q 0 ( M s– T 0 )/ T 0 . To take into account the process of the mechanical twinning (reorientation) of martensite the following assumptions were made. (1) The reorientation of martensite in a grain can occur only if this grain is purely martensitic (  gr = 1) and the condition of the reverse martensitic transformation is not satisfied for every variant of martensite. (2) The reorientation occurs inside a zone in the way providing the maximum decrease of the Gibbs’ potential. (3) The thermodynamic driving force of the reorientation in the zone � �� � ���� ��� � must achieve the critical value F fr tw that is different from the dissipative force F fr for the progress of the transformation. So the reorientation in any zone can be considered independently and the total amount of martensite in the zone during the reorientation does not change. The formulation of the evolution law for variables b pi is described in Belyaev et al. (2018). For this purpose the micro-plastic flow conditions are formulated in the form similar to that for the classic plastic flow:

MP – F

  = F

y ,

p  > 0,

(7)

 F

 F pi

pi

n

n

MP is the generalized force conjugated with the parameter b pi : . pi MP pi b F G    

where F pi

y and F

pi  are the forces describing the isotropic and kinematic hardening.

Here F pi

The conception of two types of the deformation defects, oriented defects b pi mentioned above and scattered defects f pi was used. According to Evard et al. (2006) and Volkov et al. (2015) the evolution equations for the defect densities were formulated as ), H( 1 * MP pi pi MP pi pi MP pi pi b b b            (8)

  

 

(9)

( r f f          ) H( MP f

).

0

pi

pi

pi

pi

pi

where β * and r are the material constants, H( x ) is the Heaviside’s step function. The last term in (9) describes softening caused by the reverse transformation. The closing equations connecting the defect densities with the parameters of hardening were chosen in the simplest linear form