PSI - Issue 2_B

Margarita E. Evard et al. / Procedia Structural Integrity 2 (2016) 1546–1552

1548

Author name / Structural Integrity Procedia 00 (2016) 000–000

3

characterizing the amount of martensite belonging to a zone, (1/4) Φ z being the volume fraction of martensite of this zone. Quantities Φ n themselves do not have physical meaning of any volume fractions of martensite. Still, Φ n serve as the measures of the volumes transformed by the phase deformation D n . The approximation of the small-strain theory and the Reuss’ hypothesis were used for calculation of the strain tensor ε of the representative volume: ε = Σ ε gr ( ω i ) , (2 a ) ε gr = ε gr e + ε gr T + ε gr Ph + ε gr MP (2 b ) Here f i and ε gr ( ω i ) are the volume fraction and the strain of a grain with the orientation of the crystallographic axes ω i , the sum is taken over all grains and a grain strain ε gr is considered as the sum of elastic ε gr e , thermal ε gr T , phase ε gr Ph and micro plastic ε gr MP deformations. The elastic strain ε gr e was calculated by the Hook’s law and the use of the “mixture rule” corresponding to the Reuss’ approach. : The thermal strain ε gr T was calculated in a similar way by the isotropic expansion law. The phase strain for each martensite variant is the Bain’s deformation D n realizing the transformation of the lattice and (1/ N ) Φ n is the weight of the n -th variant in the total phase strain: ε gr Ph = 1/ ( ∑ Φ Nn=1 n D n ) . (3) Micro plastic strains due to the accommodation of martensite are caused by the incompatibility of the phase strains. According to (Volkov et al. (2015)) we assumed 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 write: MP gr p 1 N

(4)

N = ε = κε ∑

n n D

n

1

p are measures of the micro-plastic strains, dev D n is the deviator part of tensor D n , κ is a

where internal variables ε n

material constant setting the scale of the microplastic strain measures ε n p . To formulate the evolution equations for the variables Φ n and ε n

p we consider the Gibbs’ potential of a grain

consisting of the two-phases: = (1 −

Φ gr ) A + 1 ∑ Φ M + mix =1 ,

(5)

where G A and G M n are the eigen potentials of austenite and n -th variant of martensite (potentials of the phases if they were not interacting), G mix is the potential of mixing, which is the elastic inter-phase stress energy. The eigen potentials can be written as

( c T T σ − a

2

)

1 2

0 ij ( ) T − ε σ − σ σ , a = A, M n, a a ijkk ij kl D ij

(6)

( G G S T T a a

a = − − −

)

0

0

0

0

T

2

0

where superscript ɑ=A stands for austenite and ɑ= M n – for the n -th variant of martensite; 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 ( ɑ= A , M n ) are the Gibbs’ potentials and the entropies at stress σ =0 and temperature T = T 0 ; ε ij 0T a ( ɑ= A , M n ) are strains of the phases at σ =0; c σ a and D a ijkl ( ɑ= A , M n ) are the specific heat capacities at constant stress and the elastic compliances. For T 0 we use an estimate proposed by Salzbrenner and Cohen (1979): 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). To estimate the inter-phase stress energy G mix we take into account that it grows with the phase deformations characterized by variables Φ n and it is decreases by oriented defects b n . Thus, mix = 1 2 µ ( − ) 2 , (7) where µ = q 0 (( M f– M s)/ T 0 ).