PSI - Issue 23

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

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another phenomenon – the two state Ψ -integral. In the following, the generalization of the anisotropy to the piezoelectricity from the point of view of the asymptotic analysis of the bi-material notch is discussed.

Fig. 1. Clamped piezoelectric bi-material notch.

2. Problem formulation

Let us consider a piezoelectric bi-material notch with geometry and boundary conditions depicted in Fig. 1. The notch face corresponding to the notch angle ω 2 is clamped. The boundary conditions have the form ( 1 ) = 0, ( 2 ) = 0, (1) where = { 1 , 2 , ϕ } and = { 1 , 2 , D } are vectors of displacements, the electric potential, the stress functions and the electric charge along the semi-infinite line passing through the origin of the coordinate system. The superscripts I and II mean the domains above and under the interface of the bi-material respectively. The vector and matrix formulation is followed from the expanded Lekhnitskii-Eshelby-Stroh formalism for piezoelectric media summarized in Hwu (2010). The in-plane complex representation of displacements and stresses for a piezoelectric bi material wedge with the generally complex singularity exponent δ ‹• …‘•‹†‡”‡† ( , ) = ∑ ( ) , ( , ) = ∑ ( ) , (2) where = I, II and ( ) = ( ) + ( ) , ( ) = ( ) + ( ) , (3) = [ 11 12 14 21 22 24 41 42 44 ] , = [ − 1 − 2 − 4 4 1 1 4 − 1 − 2 −1 ] , = { 1 2 3 } , = { 1 2 3 }. (4) A matrix , components of the matrix , material eigenvalues μ and numbers ξ depends on electromechanical material properties of the domain I and II of the notch, see Hwu (2010). From the structure of the generalized displacement vector (2) 1 can be seen that the bottom face is mechanically clamped and electrically closed. Another combination of the boundary conditions, such as clamped and electrically open notch face, cannot be covered by the following eigenvalue problem. The asymptotic solution of that kind of problems was discussed for example in Xu and Rajapakse (2000), Chue and Chen (2003), Huang and Hu (2013) and Weng and, Chue (2004). Substituting (2) into (1) and interface continuity conditions along the bi-material interface we obtain a system of twelve algebraic equations for the exponent δ written in the matrix form as