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

476 6

( ͳʹ ͶͶ ͶͶ ͳʹ ͶͶ ͳʹ ͶͶ ǡ ͵ ʹ Ǥ ʹ E E E E E E E E C C C E C C C C C  + = = + + ) ͳʹ

The in-plane problem of the stress singularity at the sharp notch composed of the monoclinic piezoelectric material and the isotropic non-piezoelectric material requires some care. The monoclinic piezoelectric material is described in terms of the LES formalism while the isotropic non-piezoelectric material needs to modify the LES formalism by employing the Muskhelishvili complex potentials. We start with the description of the asymptotic field in the monoclinic piezoelectric material referred to as the Material I. The following vectors are introduced both in the case of anisotropic piezoelectric material and isotropic non-piezoelectric one

1 u     2       u 

1 T     2       D T T

,

,

u

T

=

=

(14)

where u 1 , u 2 are the displacement components,  is the electric potential, T 1 , T 2 and T D are the components of the generalized stress function vector of the mechanical and electrical quantities. The generalized stress function components T 1 , T 2 and T D are related to the stresses and electric displacements by

(15)

,

,

1,2,

, D T D T = =−

.

T

T

i



=−



=

=

i1

,2 i2 ,1 i i

1

,2

2

,1

D

D

The complex potential for expression of the stress singularity has the form

 

 



1 2 3    z z z

diag , ,

,

, 1,2,3, =

Z

1 i  z x x i = + 2 i

=

(16)

where i  are the material eigenvalues and they are obtained by solving the characteristic equation Eq. (A5) (see Appendix A). The coupled electromechanical field near the notch vertex is sought in the form of the ansatz for unknown singularity exponents and corresponding eigenvectors as ( ) ( ) ( ) ( ) ( ) ( ) δ δ δ δ , , , , r r       = + = + u AZ v AZ w T LZ v LZ w (17)  Z and matrices A and L are defined in Appendix A, the bar above the symbols denotes complex conjugate quantities. v and w are eigenvectors pertinent to the singularity exponent eigenvalue problem of the considered sharp notch in Fig. 1. The unknown singularity exponents and eigenvectors are determined through the satisfaction of the boundary conditions at the notch tip. The material eigenvectors matrices A , L in Eq. (17), see also Eq. (A7), are non-degenerate when the poling direction is perpendicular to the x 3 axis and semi-degenerate or degenerate otherwise. Only non-degenerate cases are considered thereinafter. In case of the asymptotic field in the isotropic non-piezoelectric material, the structural and electrical constitutive equations are decoupled due to the assumption of zero piezoelectric coefficient ' ˆ g , see Eq. (5). Thus, the modification of the formalism unifies the relations for pure isotropic elasticity with equations describing the electrostatic field. By substituting the material parameters into the in-plane characteristic equation (A5), triple complex conjugate roots 1,2,3 = are obtained. The complex potentials of an isotropic media (marked with a star) have the form ( ) ( ) ( ) ( ) * d , d z z z z z z = + − f f f Q (18) where the complex function