PSI - Issue 23

Miroslav Hrstka et al. / Procedia Structural Integrity 23 (2019) 419–424 Author name / Structural Integrity Procedi 00 (2019) 000 – 00

421 3

[ 1 1 2 2 0 − 0 − 0 0 − − ] {

} = ,

(5)

where the matrix elements are defined as 1 = 1 ( 1 )( ) −1 , 1 = 1 ( 1 ) ( ) −1 , 2 = 2 ( 2 )( ) −1 , 2 = 2 ( 2 ) ( ) −1 . This system can be reduced to the algebraic system of three equations ( ) = , where is a 3 × 1 zero vector and is expressed by = 0 + 0 1 − ( 0 + 0 2 ) ( − 2 ) −1 ( − 1 ),

(6a)

(6b)

(7)

(8) where 1 I = ( 1 I ) −1 1 I , 2 II = ( 2 II ) −1 2 II . The unknown exponents δ in (2) are determined from the nonlinear characteristic equation det[ ( )] = 0. (9) The back-substitution of the received exponents δ to the eigenvalue problem (5) leads to the evaluation of the corresponding eigenvectors I , II , I and II . The exponents δ ̂ = − δ are also the solution of (9) and with eigenvectors ̂ I , ̂ II , ̂ I and ̂ II obtained from (5) form the so-called auxiliary solutions ̂ and ̂ . The generalized stress intensity factors in (2) can be evaluated as the ratio of the so-called Ψ -integrals = ( FEM , ̂ ̂ ( )) ( ( ), ̂ ̂ ( )) , (10) where is a radii of the Ψ -integral integration path enclosing the notch tip. The Ψ -integral is discussed generally in Hwu (2010) and applied to the piezoelectric materials in Hrstka (2019). The FEM in (10) is the solution of the FEM model, see Fig. 2, of the piezoelectric bi-material notch constrained according to the boundary conditions (1), i.e. displacements and electric potential are set to zero along the notch face of the domain II .