PSI - Issue 13

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

989

2

Development of devices using FeMn-based SMA requires reliable tools for calculating their deformation behavior and prediction of the fatigue life. In this work the microstructural model and criterion of fracture developed by Belyaev et al. (2018) for TiNi is adjusted for FeMn-based SMAs. Following to Evard et al. (2016, 2018) specific features of fcc  hcp martensitic transformation are taken into account. 2. Model 2.1. The internal variables, Gibbs’ potential and conditions of martensitic transformation Following to the work Evard et al. (2016), the constitutive model assumes the following hierarchy of the structural levels: (1) a representative volume, (2) a grain belonging to this volume and (3) a sub-volume (domain) inside a grain occupied either by austenite or by any of the N crystallographically equivalent orientation variants of martensite. As the fcc ↔ hcp martensitic transformation is realized by one of the three possible simple shears by 1/6   112 fcc vector on each second of any of the four sets of {111} fcc crystallographic planes, for FeMn-based SMA there exist N = 12 possible variants of the martensitic transformation. All martensite can be divided into four triplets (zones). According to Evard et al. (2018) the total amount of martensite belonging to zone p ( p = 1,2,3,4 is a number of the sets of {111} fcc close-packed planes) is characterized by vector  p with components  pi ( i = 1,2,3 is the number of the shear direction in the plane). The sum of the components characterizes the total amount of martensite in the zone:  =  =  3 1 i pi zon p . The total amount of martensite in a grain is calculated as the sum of martensite over all zones:  =  =  4 1 p zon p M . For FeMn-based alloys two obvious necessary conditions of the transformation were formulated: (a) direct fcc → hcp transformation cannot make the volume fraction of martensite in a grain  M more than 1; (b) martensite fraction in a zone  p zon cannot become less than zero. Note, that though the quantities  pi are the measures of the volume transformed into martensite with the transformation strain tensor D ( p , i ), individually they do not have physical meaning of any volume fractions of martensite. Thus, the c omponents Φ pi of the vector  p can be negative. It means that there is a possibility of the reverse transformation of a martensite crystal by a deformation not equal to the inverse of that, by which this martensite crystal has appeared. This possibility reflects the multi-variance of the reverse hcp → fcc martensitic transformation, when each of three shears   1/ 3 1120 hcp restores the initial orientation of austenite. 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) where f k and  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 considered as the sum of elastic  gr e , thermal  gr T , phase  gr Ph and micro plastic  gr MP strains: gr MP gr Ph gr T gr e gr ε ε ε ε ε = + + + . (2) The elastic strain  gr e and thermal  gr T strains were calculated according to the linear isotropic Duhamel-Neumann's law for austenite and martensite belonging to this grain and then using the “mixture rule” in accordance to the Reuss’ approach. 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 calculation of the micro plastic strains provoked 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