Fundamental solution for extended dislocation in one‐dimensional piezoelectric quasicrystal and application to fracture analysis

2019 ◽  
Vol 99 (6) ◽  
pp. e201800232 ◽  
Author(s):  
CuiYing Fan ◽  
Shuai Chen ◽  
GuangTao Xu ◽  
QiaoYun Zhang
Keyword(s):  
Fundamental Solution ◽  
Fracture Analysis ◽  
One Dimensional ◽  
Extended Dislocation

Accurate fracture analysis of electrically permeable/impermeable cracks in one-dimensional hexagonal piezoelectric quasicrystal junction

2019 ◽  
Vol 24 (12) ◽  
pp. 4032-4050 ◽  
Author(s):  
Zhenting Yang ◽  
Xiong Yu ◽  
Wang Xu ◽  
Chenghui Xu ◽  
Zhenhuan Zhou ◽  
...  
Keyword(s):  
Electric Displacement ◽  
Fracture Analysis ◽  
The Finite Element Method ◽  
Crack Face ◽  
Impermeable Crack ◽  
One Dimensional ◽  
Fracture Parameters ◽  
Small Set ◽  
Singular Domain ◽  
Good Agreement

An accurate fracture analysis of a multi-material junction of one-dimensional hexagonal quasicrystals with piezoelectric effect is performed by using Hamiltonian mechanics incorporated in the finite element method. Two idealized electrical assumptions, including electrically permeable and impermeable crack-face conditions, are considered. In the Hamiltonian system, the analytical solutions to the multi-material piezoelectric quasicrystal around the crack tip (singular domain) are obtained and expressed in terms of symplectic eigensolutions. Therefore, the large number of nodal unknowns in the singular domain is reduced into a small set of undetermined coefficients of the symplectic series. The unknowns in the non-singular domain remain unchanged. Explicit expressions of phonon stresses, phason stresses, and electric displacement in the singular domain and newly defined fracture parameters are achieved simultaneously. Comparisons are presented to verify the proposed approach and very good agreement is reported. The key influencing parameters of the crack are discussed in detail. The effects of electrical assumptions and positions of the crack on the fracture parameters are discussed in detail.

scholarly journals Analysis of Gegenbauer kernel filtration on the hypersphere

2021 ◽  
Vol 8 (11) ◽  
pp. 1-9
Author(s):  
Omenyi et al. ◽  
Keyword(s):  
Fundamental Solution ◽  
Beltrami Operator ◽  
Gegenbauer Polynomials ◽  
Wave Problem ◽  
Linear Combinations ◽  
One Dimensional ◽  
Unit Hypersphere ◽  
Finite Linear

In this study, we aim to construct explicit forms of convolution formulae for Gegenbauer kernel filtration on the surface of the unit hypersphere. Using the properties of Gegenbauer polynomials, we reformulated Gegenbauer filtration as the limit of a sequence of finite linear combinations of hyperspherical Legendre harmonics and gave proof for the completeness of the associated series. We also proved the existence of a fundamental solution of the spherical Laplace-Beltrami operator on the hypersphere using the filtration kernel. An application of the filtration on a one-dimensional Cauchy wave problem was also demonstrated.

Sign in / Sign up

Export Citation Format

Share Document