PSI - Issue 6

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

312

4

strain and the type of strain rate. It is assumed that  is the function of the form:

3 e e J f J J        2 2 e

.



(5)

4. Formulation of the problem and boundary conditions In this paper a micromechanical finite element model for the representative volume element of polycrystalline piezoceramics is considered in the same way with Liskowsky (2005), Pathak (2008), Semenov (2011) and Neumeister (2011). As the representative volume element in form of a cube with a regular partition on finite elements was taken. Each Gaussian point was considered as a single crystal. The orientation of single crystals was generated randomly. In this paper ferroelectric models with only rhombohedral variants, only tetragonal variants and mixed tetragonal/rhombohedral variants were considered. The piezoceramic material PZT-5H was considered in the computations. The formulation of coupled electro-mechanical boundary value problem in the volume of the representative volume element is defined by equation: 0, 0,     σ D (6) , . s      E ε u

u D S S S S S       are:

The boundary conditions for these equations on the outer surface

 n σ n σ  

, n D n D   

,

(7)

S

S

D



u

ε r

  φ E r

,

,

 

S

S

u

φ

where σ is the external stress tensor, D is the electric displacement vector, ε is the external strain tensor, E is the external electric field intensity vector. The vector potential finite-element formulation (see for details Landis (2002), Semenov (2006), Semenov (2010-1)) is used in simulations. The effective methods of integration of constitutive equations are used in a similar way with Semenov (2010-1), Kitaeva (2007) and Semenov (2010-2). The numerical results are obtained with help of finite-element program PANTOCRATOR described in Semenov (2003).

5. Results

Computations have been made for different values of the ratio 3 2 / e e J J to obtain the function (5). For the rhombohedral and mixed tetragonal-rhombohedral variants within the single crystal calculations were carried out for three orientations. After these calculations an approximating function   3 2 / e e f J J .was yielded with the least squares method. For the rhombohedral variant function   3 2 / : e e f J J

3

6

3 e     e J J 2  

3 e     e J J 2  

3 e     e J J 2  

e

J J

rh

f

0.0972

0.0071

0.8444,

0

 







3

(8)

e

2