PSI - Issue 59

M. Levchenko et al. / Procedia Structural Integrity 59 (2024) 724–730 M. Levchenko et al. / Structural Integrity Procedia 00 (2019) 000 – 000

725

2

1. Introduction Piezoelectric materials are widely used in practice, but piezoelectric ceramics are very brittle and prone to destruction. In addition, delamination of the piezoelectric materials interfaces is possible. It can lead to the appearance of cracks at the interface, which can be the reason for the destruction of devices. Therefore, the investigation of a crack between two piezoelectric materials is important. When studying such cracks, their electrically permeable and insulated models (Suo et al., 1992) are most often used, which are extreme cases of real cracks. For real cracks in a homogeneous piezoelectric material and between two piezoelectric materials, the model of a crack with finite electrical permeability, proposed by Hao and Shen (1994), is the most adequate reality. Subsequently, this model was developed mainly concerning the cracks in a homogeneous piezoelectric material. Its application to an interface crack in a piezoelectric bimaterial has been carried out by Govorukha et al. (2006), Li and Chen (2007, 2008), Lapusta et al. (2011) and Loboda et al. (2023). However, the study of this model concerning the spatial case of an interface crack in a 3-D body is unknown to the authors of this paper. Such problem investigation is the main purpose of the present paper. 2. Formulation of the problem region 1 b x b    of the material interface 3 0 x  there is a tunnel crack that has finite electrical permeability and is free of stresses and electrical charges on its faces. Tensile stresses 33    and shear stress 23    act on the upper and lower faces of the parallelepiped and electric flux 3 D d  passes through the body. It is assumed that   m ijkl c ,   m lij e ,   m ij  ( 1,2 m  ) are the matrices of the modulus of elasticity, piezoelectric constants and dielectric constants for the top and bottom materials, respectively, and both materials have a symmetry class 6mm with the direction of polarization 3 x . We assume that the crack filler has dielectric permeability A bimaterial piezoelectric parallelepiped 1 1 1 l x l    , 2 2 2 l x l    , 3 3 3 l x l    (Fig. 1) is considered. In the

0 a r     ,

(1)

and the electric field inside the cracks can be found as

u u        

for 1 ( , ) x b b   ,

a E



3

3

where r  is the relative dielectric constant,

is the dielectric constant of vacuum.

12 0 8.85 10 / C Vm    

Taking into account that

3 a a D E   , we arrive to the following electrical condition

u u        

for 1 ( , ) x b b  

(2)

D



a 

3

3

3

along the crack region which was previously used by Hao and Chen (1994)]. Thus, the boundary conditions at the material interface can be written as













1 , x b b   :

(3)

for

,

V

t

1 2 , ,0 0, x x 

1 2 ,,0 0 x x 





   13 1 m



   33 1 m



 

1 , x b b   :

, x x

2 ,0 0, 

, ,0 0

x x





 ,

0  ,

for

3 1 D x

2