Dynamic fracture of a partially permeable crack in a functionally graded one-dimensional hexagonal piezoelectric quasicrystal under a time-harmonic elastic SH-wave

Abstract

The dynamic fracture of a crack in a functionally graded one-dimensional (1D) hexagonal piezoelectric quasicrystals (PEQCs) subjected to a time-harmonic elastic SH-wave is studied by using integral transform technique and Copson method. It is assumed that the crack surface is a partially permeable boundary condition and the material properties vary continuously as an exponential function. With the help of the Fourier transform, the boundary value problem of partial differential equation describing fracture problem is formulated to three pairs of dual integral equations, which are numerically solved by Copson method. Explicit expressions for the electroelastic field including phonon and phason stresses and electric field on the crack face are determined. The dynamic intensity factors of the electroelastic field are obtained in closed form, and some special cases of obtained results are discussed. On the basis of theoretical analysis and numerical simulation of the established models, numerical analysis was then conducted to discuss crack length, gradient parameter, electric boundary condition, electric loading, incident angle, amplitude, and wave number on the fracture characteristics of material. The research of this paper will provide a theoretical basis for nondestructive testing, optimal design, and reliability analysis of materials and will enrich the research content of fracture mechanics of multi-field coupling materials.

Interaction of anti‐plane shear waves with two collinear cracks in 1D hexagonal piezoelectric quasicrystals

“…This implies that an electrically impermeable crack tends to expand more readily than one that is filled with air. This impact of the ECCP on the normalized DSIFs of the phason field was similarly identified by Yang et al [26] Figures 6 and 7 are the plots of normalized DSIFs of phonon field for QCs at inner and outer crack tip respectively. It is obvious from both figures that normalized DSIFs decrease very slowly with the increase of the values of 𝑘 𝑠 .…”

“…This impact of the ECCP on the normalized DSIFs of the phason field was similarly identified by Yang et al. [26]…”

“…The constitutive equations () are simplified as follows Yang et al. [25, 26]: …”

“…Using Fourier transform, the general solution of Equation () is given by (Yang et al. [25, 26]) $$$$\begin{equation} \begin{aligned} &\varphi _1(x,y)=\frac{2}{\pi }\int _{0}^{\infty }\mathcal {A}_1(s)e^{-\gamma _1y}\cos (sx)ds\\ &\varphi _2(x,y)=\frac{2}{\pi }\int _{0}^{\infty }\mathcal {A}_2(s)e^{-\gamma _2y}\cos (sx)ds\\ &\varphi _3(x,y)=\frac{2}{\pi }\int _{0}^{\infty }\mathcal {A}_3(s)e^{-sy}\cos (sx)ds \end{aligned} \end{equation}$$$$where, $$ \gamma _1=\sqrt {s^2-\frac{k_s^2}{\mathcal {K}+a}},\;\gamma _2=\sqrt {s^2-\frac{k_s^2}{\mathcal {C}-a}}$$ and $$ \mathcal {A}_1(s),\;\mathcal {A}_2(s)$$ and $$ \mathcal {A}_3(s)$$ are unknown functions...…”

“…A Yoffe type running crack in 1D hexagonal PQCs under anti-plane shear load is examined by Zhou and Li [24]. Yang et al [25,26] studied the dynamic crack problem in a non-homogeneous 1D hexagonal PQCs under SH-wave. Shear wave interaction with two collinear dynamic cracks in 1D hexagonal PQCs is not studied yet according to the authors knowledge.…”