The Decomposed Theorem of Torsional Circular Shaft with Two-Dimensional Dodecagonal Quasicrystal

Gregory’s decomposed theorem of isotropic plate is extended to investigate torsional circular shaft for two-dimensional dodecagonal quasicrystal (2D dodecagonal QCs)with homogeneous boundary conditions, and the theory of equivalence that Cheng’s refined theory and Gregory’s decomposed theorem is extended to the cylindrical coordinate. The decomposed theorem of torsional circular shaft of 2D dodecagonal QCs with homogeneous boundary conditions is proposed on the basis of the classical elasticity theory and stress-displacement relations of 2D dodecagonal QCs without ad hoc assumptions. At first expressions are obtained for all the displacements and stress components in term of some 1D functions. Using Lur’e method, the exact equations were given. And the exact equations for the torsional circular shaft on 2D dodecagonal QCs without surface loadings consist of four governing differential equations: two harmonic equations and two transcendental equations.