PSI - Issue 5

Tomasz Trzepieciński et al. / Procedia Structural Integrity 5 (2017) 562 – 568 Mojtaba Biglar et al./ Structural Integrity Procedia 00 (2017) 000 – 000

565

4

2 t

1 u

2 u

1 t

Fig. 1. Simple example of using the boundary element equation in order to obtain displacement and traction.

The following linear algebraic equation can be achieved:        T u U t   

(4)

In Eq. (4), [Δ T ] and [Δ U ] can be defined as follows:

       

       

       

       

2 1

2 1

1

2

1

2

   T

T

U U

  





1

1

1

1

    

     u u

   

2 1      t t

2 1

2 1

2 1 1 2

2

1 2

2

(5)

T

T

U

U



 



 





2

2

  

  

By solving these algebraic equations, the unknown parameters can be found. One of the most striking applications of the BEM is in solving the problem of a plate that contains different kinds of crack and holes. So far, some researchers have investigated this kind of problem via different numerical and analytical methods. Modelling the singular integral near the crack tip is key, and it is important to pay attention to it. This issue will be discussed in future chapters of the current work. The displacement equation of BEM for polycrystalline aggregate cracked plate can be written according to Benedetti and Alibaldi (2013a) as:

2 1 

( ) s u Z U X Z t X j ij i   0 0

0 s ds X T i X Z u X j j  

( , ) ( ) ( ) ds X

( , ) ( ) ( )

(6)

U X Z

( , ) t X ds X T i X Z u X ds X   ( ) ( ) ( ) ( )

( , )



 

 

ij

j

j

0

0

j

4. RVE of grains

In order to analyse the microstructure of piezoelectric ceramics, it is essential to have exact information on the grains such as their possible shapes, coordinates, and so on. Although some researchers have used artificial grains made according to different theories such as the Poisson – Voronoi tessellation method, in this study, a photograph of barium titanate obtained by SEM (Fig. 2a) has been used to analyse the microstructure of the model (Fig. 2b). In this part, the procedures that were applied to obtain the final model from the SEM image will be described shortly.