Green’s functions of infinite and semi-infinite plane problems of one-dimensional quasicrystals with piezoelectric effect are obtained in a closed form by Stroh formalism. Some numerical examples under different loading conditions, such as line forces, line dislocations, and a line charge, are given to explain the mechanical and electric behaviors of the quasicrystals. Various elastic–electric constants of quasicrystals are analyzed and the coupling effects between the phonon and phason fields are studied. The presented solutions will be useful for many boundary value problems of one-dimensional quasicrystals with piezoelectric effect. Moreover, the numerical results can be used to verify the accuracy of the solutions by some numerical methods, such as the finite element and boundary element methods. Furthermore, the Stroh formalism can be generalized to the researches on more complex problems of quasicrystals.
AltayGDökmeciMC (2012) On the fundamental equations of piezoelasticity of quasicrystal media. International Journal of Solids and Structures49(23–24): 3255–3262.
2.
BachtelerJTrebinHR (1998) Elastic Green’s function of icosahedral quasicrystals. The European Physical Journal B: Condensed Matter and Complex Systems4(3): 299–306.
3.
ChenWQMaYLDingHJ (2004) On three-dimensional elastic problems of one-dimensional hexagonal quasicrystal bodies. Mechanics Research Communications31(6): 633–641.
4.
DingDYangWHuC. (1993) Generalized elasticity theory of quasicrystals. Physical Review B: Condensed Matter48(10): 7003–7010.
5.
FanTY (2011) Mathematical Theory of Elasticity of Quasicrystals and Its Applications. Berlin; Heidelberg: Springer.
6.
FujiwaraTLaissardièreGTYamamotoS (1994) Electronic structure and electron transport in quasicrystals. Materials Science Forum150–151: 387–394.
7.
GaoCFFanWX (1998) Green’s functions for the plane problem in a half-infinite piezoelectric medium. Mechanics Research Communications25(1): 69–74.
8.
GaoYRicoeurAZhangLL. (2014) Crack solutions and weight functions for plane problems in three-dimensional quasicrystals. Archive of Applied Mechanics84(8): 1103–1115.
9.
GuoJHLuZX (2011) Exact solution of four cracks originating from an elliptical hole in one-dimensional hexagonal quasicrystals. Applied Mathematics and Computation217(22): 9397–9403.
10.
GuoJHYuJSiR (2013) A semi-inverse method of a Griffith crack in one-dimensional hexagonal quasicrystals. Applied Mathematics and Computation219(14): 7445–7449.
11.
HouPFHeSChenCP (2011) 2D general solution and fundamental solution for orthotropic thermoelastic materials. Engineering Analysis with Boundary Elements35(1): 56–60.
12.
HwuC (2010) Anisotropic Elastic Plates. New York: Springer.
13.
LeeJSJiangLZ (1996) Exact electroelastic analysis of piezoelectric laminae via state space approach. International Journal of Solids and Structures33(7): 977–990.
14.
LejcekL (1969) The Green function of the theory of elasticity in an anisotropic hexagonal medium. Czechoslovak Journal of Physics B19(6): 799–803.
15.
LiCLLiuYY (2004) The physical property tensors of one-dimensional quasicrystals. Chinese Physics13(6): 924–931.
16.
LiLH (2013) Complex potential theory for the plane elasticity problem of decagonal quasicrystals and its application. Applied Mathematics and Computation219(19): 10105–10111.
17.
LiLHLiuGT (2012) Stroh formalism for icosahedral quasicrystal and its application. Physics Letters A376(8–9): 987–990.
18.
LiXFFanTY (1999) A straight dislocation in one-dimensional hexagonal quasicrystals. Physica Status Solidi B212(1): 19–26.
19.
LiXY (2014) Elastic field in an infinite medium of one-dimensional hexagonal quasicrystal with a planar crack. International Journal of Solids and Structures51: 1442–1455.
20.
LiXYDengH (2013) On 2D Green’s functions for 1D hexagonal quasi-crystals. Physica B: Condensed Matter430: 45–51.
21.
LiXYLiPDWuTH. (2014) Three-dimensional fundamental solutions for one-dimensional hexagonal quasicrystal with piezoelectric effect. Physics Letters A378(10): 826–834.
22.
LiuGTGuoRPFanTY (2005) Plane elasticity and dislocation of one-dimensional hexagonal quasicrystals with point group 6. Journal of Beijing Institute of Technology (English Edition)14(1): 87–91.
23.
PanYCChouTW (1976) Point force solution for an infinite transversely isotropic solid. Journal of Applied Mechanics43(4): 608–612.
24.
RickerMBachtelerJTrebinHR (2001) Elastic theory of icosahedral quasicrystals—application to straight dislocations. The European Physical Journal B: Condensed Matter and Complex Systems23: 351–363.
25.
SladekJSladekVPanE (2013) Bending analyses of 1D orthorhombic quasicrystal plates. International Journal of Solids and Structures50(24): 3975–3983.
26.
TingTCHorganCO (1996) Anisotropic Elasticity: Theory and Applications. New York: Oxford University Press.
27.
WangTLiXYZhangX. (2015) Fundamental solutions in a half space of two-dimensional hexagonal quasicrystal and their applications. Journal of Applied Physics117(15): 154904.
28.
YangLZGaoYPanE. (2014) Electric-elastic field induced by a straight dislocation in one-dimensional quasicrystals. Acta Physica Polonica A126(2): 467–470.
29.
YuJGuoJHPanE. (2015) General solutions of plane problem in one-dimensional quasicrystal piezoelectric materials and its application on fracture mechanics. Applied Mathematics and Mechanics36(6): 793–814.
30.
ZhangLLZhangYMGaoY (2014) General solutions of plane elasticity of one-dimensional orthorhombic quasicrystals with piezoelectric effect. Physics Letters A378(37): 2768–2776.
