PSI - Issue 13

Fedor S. Belyaev et al. / Procedia Structural Integrity 13 (2018) 988–993 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

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of the phase strain. For FeMn-based alloys this assumption is corroborated by the fact that the phase transformation and dislocation plastic deformation are both realized by means of shears on the close-packed {111} fcc and {0001} hcp planes in the same directions. Thus, for the total micro plastic strain of a grain one can write: (4) where internal variables ε are measures of the micro-plastic strains, dev D ( p , i ) is the deviator part of tensor D (p,i) ,  is a material constant . To formulate the evolution equations for the variables Φ and ε we consider generalized forces conjugated with these parameters. For a two- phase medium consisting of martensite and austenite, 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). The choice of the mixing potential G mix was discussed in detail by Evard et al. (2018). It was proposed to calculate the energy of the variants interaction by a quadratic form taking into account the interaction of the martensite variants related to each of the closed-packed planes. In this work we additionally account for a decrease of the elastic inter phase stress energy caused by the accommodative micro-plastic deformation. For this purpose an oriented defects vector b p on the p- th shear plane is introduced, which can be associated with densities of piled-up dislocations and dislocation loops. The components b pi of the vector b p are included into the expression for G mix : where μ = q 0 (( M f – M s)/ T 0 ) is a coefficient of proportionality; α is the parameter of the model. The first term in (5) describes the energy of the martensite-austenite interaction and does not depend on the interaction between the variants of martensite. The second term takes into consideration contributions of uncompensated shears: the less is the difference between  p 1 ,  p 2 ,  p 3 for each p -th plane, the less is the correspondent term. Carrying out calculations of the vector moduli in (5) one should take into account that the components  pi of the vector  p and b pi of the vector b p are written in linearly dependent (not Cartesian) basis reflecting the threefold symmetry of the close-packed planes {111} fcc (see Evard et al. (2018)). 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 : , )  − + b ( 2 4 1 = = 3 1 4 1 2 2       − p p b Φ  =   = p i p pi pi mix G  (5) 2 0 ij 0 0 0 0 0 ( c T T  − a ) 1 2 ( G G S T T a a ) ( ) T −   −   , a = A, M pi , a a ijkk ij kl D 2 a = − − − ij T

G

 

.

pi F

= −

pi

To formulate the transformation condition considering the existence of the thermal and mechanical hysteresis, the correspondent dissipative force F fr is introduced. So, the necessary condition of the martensitic transformation can be written as F pi =  F fr , (6)