PSI - Issue 28

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

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4

The approximation of the small-strain theory and the Reuss’ hypothesis was used for calculation of the strain tensor  of the representative volume:   k k gr k f ( )  ε ε , (1) gr (  k ) are the volume fraction and the strain of a grain with the orientation  k of the crystallographic axes, the sum is taken over all grains. The grain strain  gr is the sum of elastic  gr e , thermal  gr T , phase  gr Ph and micro plastic  gr MP strains: where f k and 

gr Ph ε ε ε ε ε     gr T gr e gr

gr MP

.

(2)

The elastic strain  gr e was calculated by the Hook’s law. The thermal strain  gr T was calculated by the isotropic heat expansion law. The phase strain for each martensite variant is the Bain’s deformation D (p,i) realizing the transformation of the lattice taken with the correspondent weight Φ pi . A total phase strain in a grain is calculated by neutralisation over all the martensitic variants belonging to the grain:

.

(3)

  

  

   4 1 3 1 p i

ε

gr Ph

( , )

p i D





1

12

pi

The main assumption for the calculation of the micro plastic strains caused by the incompatibility of the phase strains is that the phase strain of a Bain’s variant activates a combination of slips producing a strain proportional to the deviator of the phase strain. Thus, for the total micro plastic strain of a grain one can assumes:

   (4) where the internal variables ε �� � � are measures of the micro-plastic strains, dev D ( p , i ) is the deviator part of the tensor D (p,i) ,  is a material constant. To formulate the evolution equations for the variables Φ �� and ε �� � � generalized forces conjugated with these parameters were used. For a two-phase medium, the Gibbs’ thermodynamic potential for one grain can be written as: mix eig G G G   , where G eig is the eigen potential of the phases as if they were not interacting, and G mix is the energy of their mixing. The eigen potentials can be found as        4 1 3 1 p i 12 1 dev ( , ) p i D MP pi gr MP  ε , where superscript ɑ =A stands for austenite and ɑ = M pi – for the variant of martensite appeared in the p -th zone by the i -th shear; T 0 is the phase equilibrium temperature (i.e. such temperature, at which G A = G M n ); G 0 a and S 0 a are the Gibbs’ potentials and the entropies at stress  =0 and temperature T = T 0 ;  ij 0T a are strains of the phases at  =0; c  a are the specific heat capacities at constant stress and D a ijkl are the elastic compliances. For T 0 we use an estimate T 0 = ( M s+ A f)/2 (hereinafter M s, M f, A s, A f are the characteristic temperatures and q 0 is the latent heat of the transformation). According to Evard et al. (2018) the mixing potential G mix was written as a quadratic form taking into account the interaction of the martensite variants related to each of the closed-packed planes. A decrease of the elastic inter-phase stress energy caused by the accommodative micro-plastic deformation was taken into account. For this purpose an 2 0 ij 0 0 0 0 0 ( c T T   a ) 1 2 ( G G S T T a a a ) ( ) T       , a = A, M pi , a a ijkk ij kl D 2 ij T    