PSI - Issue 80

Miroslav Hrstka et al. / Procedia Structural Integrity 80 (2026) 471–492 M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000 – 000

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3

The piezoelectric layer of PZT-5H exhibits the tetragonal symmetry. However, because the crystallographic reference frame can be arbitrary rotated about x 3 axis with respect to global coordinate system in Fig.1, the elasticity and piezoelectricity matrices possess the structure of monoclinic materials in the global coordinate system, hence the problem is formulated for the monoclinic materials with the symmetry plane at x 3 = 0.

Fig. 1. Geometry of a bi-material notch characterized by two regions I and II. Notch faces are defined by angles ω 1 and ω 2 . Material interface is always considered at θ = 0. Angle α 1 denotes poling direction of the materials I The constitutive laws for a linear elastic piezoelectric material which include homogeneous initial strains due to uniform temperature change ΔT can be written in the matrix form as

   

   

   

   

T

T

      σ D

− ε α

− ε α

      σ D

T

T





  

,      

  

C e

D S g

E

,

=

=

(1)

E

E

−

−

e

ω

− g β

−





where σ is the stress tensor, ε is the strain tensor (both written in the Voigt notation in the vector form), α is the thermal expansion tensor, E and D are the vectors of electric intensity and electric flux density, C E is the elastic stiffness tensor at constant electric field, e is the stress/charge piezoelectric tensor, and ω ε is the dielectric permittivity tensor at constant strain, S D is the elastic compliance tensor at constant electric induction, g is the strain/voltage piezoelectric tensor, and β σ is the dielectric non-permittivity tensor at constant stress (Hwu and Ikeda (2008), Hwu and Kuo (2009), 2010)). The material matrices are related through an inversion as

     

           

     

T

T

C e

D S g

E

.

I

=

(2)

e

− ω g β −





The elasticity and piezoelectricity matrices for a monoclinic material poled in the x 1 x 2 -plane have the following structure:

                 

                 

11 12 13 E E E E E E 12 22 23 E E E 13 23 33 0 0 0 0 0 0 E E E C C C C C C 16 26 36 C C C C C C

E

44 45 E E E E 45 55 0 0 0 0 0 0 0 0

C C C

2                     6        

1              2  

11           22  

11           22     

1             3 4 5 1

16

E

    

26

E

33           = =                                                 ε 23 3 33 4 23 5 3 13 6 12 12 , 2 2 2   

36

,

C

σ

=

=   =

(3)

E

0 0

C C C C

E

C

66