In this paper, the axisymmetric deformation of one cylinder with transversely isotropic is researched. According to the general solution of deformation body with transversely isotropic and the Lur’e method, the exact deformation field and exact stress field are represented by unknown functions with single independent variables. Based on boundary conditions of radial direction surface loading, the unknown functions can be ascertained. By dropping terms of high order, the approximate solution is derived, and the department field and the stress field for a circular cylinder under radial direction surface loading can be obtained. After simplifying, the states with isotropic can be obtained.

In this paper, the axisymmetric general steady-state solution for porous media is presented. And the completeness of the P-Ns representation for the axisymmetric displacement field equation is proved directly from the equations governing the displacement field, which can be applied to homogeneous and isotropic poroelastic materials. At last, the Boussinesq general solution and Timpe general solution are obtained from P-N General Solution.

Gregory’s decomposed theorem of isotropic plate is extended to investigate torsional circular shaft of cubic quasicrystal 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 general solution of torsional circular shaft on cubic quasicrystal with homogeneous boundary conditions is proposed on the basis of the classical elasticity theory and stress-displacement relations of cubic quasicrystal 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 cubic quasicrystal without surface loadings consist of four governing differential equations: two harmonic equations and two transcendental equations. Using basic mathematic method and the general solutions, an example is examined.

Through generalizing the method of a decay analysis technique determining the interior solution developed by Gregory and Wan, a set of necessary conditions on the end-data of axisymmetric circular cylinder in cubic quasicrystal for the existence of a rapidly decaying solution is established. By accurate solutions for auxiliary regular state, using the reciprocal theorem, these necessary conditions for the end-data to induce only a decaying elastostatic state (boundary layer solution) will be translated into appropriate boundary conditions for the circular cylinder with axisymmetric deformations in cubic quasicrystal. The results of the present paper enable us to establish a set of correct boundary conditions, and mix boundary conditions of which are obtained for the first time.

Cheng’s refined theory is extended to investigate torsional circular shaft of cubic quasicrystal, and Lur’e method about harmonic function is extended to harmonic function in the respective cylindrical coordinate. The refined theory of torsional circular shaft of cubic quasicrystal under reverse direction surface loading is proposed on the basis of the classical elasticity theory and stress-displacement relations of cubic quasicrystal, and the refined theory provides the solutions about torsional deformation of a circular shaft without ad hoc assumptions. Exact solutions are obtained for circular shaft from boundary conditions. Using Taylor series of the Bessel functions and then dropping all the terms associated with the higher-order terms, we obtain the approximate expressions for circular shaft of cubic quasicrystal under reverse direction surface. To illustrate the application of the theory developed, one example is examined.