PSI - Issue 6

Pudeleva Olga et al. / Procedia Structural Integrity 6 (2017) 309–315 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

3

311

Fig.1. Domain orientations in a) tetragonal variant, b) rhombohedral variant, c) mixed tetragonal/rhombohedral variant.

3. Micromechanical and phenomenological models

The motion of the domain walls with the velocity of f  generates in the single crystal an increment of remanent strain and remanent polarization. The total remanent strain and polarization are summation over all switching systems as proposed in Huber (1999) and Huber (2001):

˙          ε P ˙ r

P         s    μ

˙ I   I c     ε  M N

˙

 

f

,

2



I   P

I

1

1







r

where  μ is identical to the Schmid orientation tensor, while  s is a unit vector in the direction of the change in remanent polarization. The scalars   and P  define the maximum shear strain and polarisation change due to the transformation of system  . Let introduce the Helmholtz free energy per unit volume of material:       Ψ Ψ , , , Ψ , , 3 s r r r r r   ε ε D P ε P where Ψ s is the stored elastic energy, Ψ r – part of the free energy associated only with the internal state of material. Landis (2003) proposed the formulation of mechanical part of remanent free energy in the following way:

2

     

     

  

  

e

m

J

m

1 2

r

m H 

Ψ

exp

,



2



c

0

(4)

c 

1     

c 





3 r 

r

r   

m H

, m c m  are parameters of the material,

e

,

ij 

0 ,

kk

 is the dependence between remanent

where

ij

ij