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Fracture analysis of mode III crack problems for the piezoelectric bimorph

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Abstract

In this paper, a symplectic method based on the Hamiltonian system is proposed to analyze the interfacial fracture in the piezoelectric bimorph under anti-plane deformation. A set of Hamiltonian governing equations is derived from the Hamiltonian function by introducing dual variables of generalized displacements and stresses which can be expanded in series in terms of the symplectic eigensolutions. With the aid of the adjoint symplectic orthogonality, coefficients of the series are determined by the boundary conditions along the crack faces and along the external geometry. The stress\electric displacement intensity factors and energy release rates (G) directly relate to the first few terms of the nonzero eigenvalue solutions. The two ideal crack boundary conditions, namely the electrically impermeable and permeable crack assumptions, are considered. Numerical examples including the complex mixed boundary conditions are considered to show fracture behaviors of the interface crack and discuss the influencing factors.

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Abbreviations

\({C_{44}^i,\,e_{15}^i,\,\kappa _{11}^i}\) :

Elastic stiffness, piezoelectric constant and dielectric constant

\({D_r^i,\,D_\theta ^i}\) :

Electric displacements

\({E_r^i,\,E_\theta ^i}\) :

Electric fields

F i, Q i :

Body force and electric charge density

G I, G P :

Total energy release rate

H :

Hamiltonian function

\({K_{\rm S}^I,\,K_{\rm S}^P}\) :

Strain intensity factor for two types of electric boundary conditions

\({K_{\rm E}^I,\,K_{\rm E}^P}\) :

Electric field intensity factor

\({K_3^I,\,K_3^P}\) :

Stress intensity factor

\({K_{\rm D}^I,\,K_{\rm D}^P}\) :

Electric displacement intensity factor

L i :

Lagrangian function

U i, Π i :

Potential energy density function and the total potential energy

W i :

Anti-plane mechanical displacement

(x, y):

Cartesian coordinates

(r, θ):

Polar coordinates

\({\varepsilon _{rz}^i,\,\varepsilon _{\theta z}^i}\) :

Shear strains

\({\sigma _{rz}^i,\,\sigma _{\theta z}^i}\) :

Shear stresses

μ j :

Eigenvalue of Hamiltonian matrix

θ 0 :

Angle between electrode plate and the x-axis

ω :

Loading angle

Φ i :

In-plane electric potential

f i :

The vector of generalized external forces

H i :

Hamiltonian operator matrix

J :

symplectic identity matrix

q i, p i :

Mutually dual vectors

\({{\bf \Psi}^{i}}\) :

Complete solution in the symplectic space

\({{\bf \Psi}_{\rm P}^i}\) :

Particular solution of the non-homogenous part

\({{\bf {\varphi}}_n^{i(\alpha )},\,{\bf {\varphi}}_n^{i(\beta )}}\) :

Two groups of zero-eigenvalue solutions

\({{\bf {\psi}}_j^{i(\alpha )},\,{\bf {\psi}}_j^{i(\beta )}}\) :

Symmetric and anti-symmetric eigensolution

\({{\bf {\Lambda}}^{{\rm I}},\,{\bf {\Lambda}}^{{\rm P}}}\) :

Matrix of coefficients for the impermeable and permeable cracks

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Authors and Affiliations

  1. State Key Laboratory of Structure Analysis of Industrial Equipment and Department of Engineering Mechanics, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    C. H. Xu, Z. H. Zhou & X. S. Xu

  2. Department of Civil and Architectural Engineering, City University of Hong Kong, Kowloon, Hong Kong

    A. Y. T. Leung

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  1. C. H. Xu
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  2. Z. H. Zhou
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  3. X. S. Xu
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Correspondence to Z. H. Zhou.

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Xu, C.H., Zhou, Z.H., Xu, X.S. et al. Fracture analysis of mode III crack problems for the piezoelectric bimorph. Arch Appl Mech 84, 1057–1079 (2014). https://doi.org/10.1007/s00419-014-0848-8

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  • DOI: https://doi.org/10.1007/s00419-014-0848-8

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