PSI - Issue 6

Sviatoslav M. Lobanov et al. / Procedia Structural Integrity 6 (2017) 90–94 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

92

3

   

   

r

s μ

P ε  



  

  

r I

P ε

  

  

N

M

      c r I I 1 1 



,

 

f





r

P

 

I (3) where µ α and s α are Schmid tensor and vector for α switching from I -system to J -system; γ α and P α are material constants that characterizing the magnitude of the difference in the remanent strain and the remanent polarization under switching domains;  f  is the function that define the switching rate. The later depends on driving force G c and volume fraction c I . Similarly to visco-plasticity case presented by Huber et al. (2001) the function  f  can be defined as following: m I n c c c c G G G f f G         0 1 0      ,

(4)

where the driving force for each switching angle is defined by

.

(5)

σ     3

σ μ   

E s

d E

G

P



 



 

 

Thus, G c , P α , n, m are model parameters, which can be identified by the experimental hysteresis curves. The use of the evolution equation in the form (4) ensures the fulfillment of the dissipative inequality. The proposed model has been implemented in the program CES 4.1 (Semenov, 2008) and used in further computations.

2.2. Domain structure and model parameters

There are 6 directions of polarization for tetragonal phase, 8 directions for rhombohedral and 12 directions for orthorhombic. One of them for each phase is shown on fig. 2. So the general number of polarization directions, which is equal to the number of domains in а crystal, is N = 26. Each domains coordinate system can be obtained by rotating of the basic cubic crystal coordinate system. The overall number of switching directions is M=N(N-1)=650 .

Fig. 2. Polarization direction for (a) tetragonal. (b) orthorhombic, (c) rhombohedral crystal structure (after Vainstein, 1981).

In tetragonal phase the switching can occur via two angles: 90° and 180 ° , for rhombohedral possible switching angles are 71 °, 109 ° and 180 °, for orthorhombic – 60 °, 90° , 120 ° and 180 °. The switching between phases (phase transition) is not taken in account in the present work. Even though, all together for 3 phases there are 6 switching directions which are defined in the model by 9 G c parameters. Under the assumption of the simultaneous start of switching on every angle presented by Huber et al. (1999) in the case of pure electrical loading ( σ = 0 ), the main part of G c can be simplified from (5) to: G P      E s . (5.1) From (5.1) for the case of uniaxial electric loading the further relations for six G c can be obtained due to geometrical reasons: