Elastodynamic Solutions Of Hollow Cylinders And Hollow Spheres Under Radial Deformation

The elastodynamic solutions of othotropic hollow cylinder for axisymmetric plane strain problems are obtained in this paper. For homogeneous, othotropic elastic hollow cylinder, firstly, a special function is introduced to transform the inhomogeneous boundary conditions into homogeneous ones. Secondly, by using the method of separation of variables, the quantity that the displacement subtracts the special function is expanded as the multiplication series of Bessel function and the unknown functions of time. Thirdly, by virtue of the orthogonal properties of Bessel functions, the equations about these unknown functions are derived and the solutions are obtained. Finally, the elastodynamic solution of the hollow cylinder is obtained. For homogeneous, piezoelectric and pyroelectric hollow cylinders, by introducing a special function and by using orthogonal expansion technique, the equation about a function with respect to time is derived. Then by means of the initial conditions and electric boundary conditions, the dynamic problems of piezoelectric and pyroelectric hollow cylinders are transferred to a second kind Volterra integral equation about a time function which is related to electric displacement. And by using interpolation method, two recursive formula are constructed, which can be employed to solve the Volterra integral equation of the second kind efficiently and quickly. The solutions of displacement, stresses, electric placement as well as electric potential are finally obtained. For multilayered hollow cylinders, by using state space method and by following the solving procedure for homogeneous hollow cylinder, the elastodynamic solutions of multilayered orthotropic elastic, piezoelectric as well as pyroelectric hollow cylinder are finally obtained. For functionally graded material (FGM) orthotropic hollow cylinder, a special case that the material constants have a power-law dependence on the radial coordinate is considered. By introducing a new dependent variable, the elastodynamic problems of a FGM orthotropic hollow cylinder are then transferred into those of homogeneous ones which can be solved as above mentioned. The solutions of a special functionally graded, elastic, piezoelectric and pyroelectric, orthotropic hollow cylinders are obtained at the end. The elastodynamic problems of the orthotropic hollow cylinders for the case that the axial strain is considered are also investigated. For elastic hollow cylinder, by introducing a special function and by using the orthogonal expansion technique, the equation about a function of time is derived. And by using the initial conditions as well as the end conditions, the dynamic problem is then transferred to a second kind Volterra integral equation about the function of the axial strain with respect to time which can also be solved successfully by the interpolation method. For piezoelectric and pyroelectric hollow cylinders, by following the solving procedure for elastic hollow cylinder and by using the electric boundary conditions, the dynamic problems are transferred to two Volterra integral equations about two functions of time, one is axial strain and the other is related to electric displacement, which can also be solved efficiently and quickly by employing interpolation method.The elastodynamic solutions of hollow spheres, which are made of elastic, piezoelectric and pyroelectric materials, respectively, for spherically symmetricproblems are also obtained. And for each of the materials, the elsatodynamic solutions of the homogeneous, multilayered as well as a special FGM (the material constants have a power-law dependent on radial coordinate) hollow sphere are presented at the end. The solving method for hollow spheres is just similar with that for hollow cylinders.The elastodynamic solutions of homogeneous, isotropic solid cylinder and solid sphere are also presented. And the stress-focusing effect phenomena in the solid cylinder and solid sphere are discussed. Additionally, the general solution of piezoelectric hollow sphere for equilibr...