Research On State Space Solution For Spatial Axisymmetry Of Transversely Isotropic Materials

The mechanical analysis of the axisymmetric problem of transversely isotropic materials is one of the most important and widely used branches in the field of elasticity.Theoretical calculation and experiment are two important methods to solve this kind of engineering application problems,but most of the solutions to this problem involve potential function and the expression of the solution is basically integral form,which makes the calculation complex and not intuitive.Moreover,with the increasing of composite materials and the increasing of engineering specifications,the difficulty and cost of experiment are also increasing.Studying a simple and high-precision solution method will be popular in the engineering field.In this paper,a new state space method is proposed to solve the axisymmetric problem of transversely isotropic materials,which combines matrix theory and Hankel transformation system.The specific research contents are as follows:(1)In cylindrical coordinates,the equation of state is established by using Hankel transformation and Bessel function theory from the axisymmetric stress displacement relationship and equilibrium equation of transversely isotropic materials.Then,the eigenvalues of the coefficient matrix are solved,and the two cases of equal and unequal eigenvalues are discussed respectively.The state equation is then decoupled to obtain a general expression of the state vector,and the integral constant in the equation is determined by the boundary condition.Finally,the general expressions of displacement and stress components are obtained by Hankel inverse transformation.(2)For three cases,the concentrated load is applied to the surface of the transversely isotropic material half-space body,to the interior of the transversely isotropic material half-space body,and to the interior of the transversely isotropic material.The Boussinesq solution of the transversely isotropic material half-space body,the Mindlin solution of the half-space body and the Kelvin solution of the full-space body are derived by the above solution steps and the corresponding boundary conditions under three load cases,the expression of displacement and stress components in three cases.(3)The verification of the correctness of the formula is based on the ANSYS simulation software,and the ANSYS command stream file is written.The spatial axisymmetric model of the transversely isotropic material under three load conditions is established,and the stress and displacement data are derived.Then using MATLAB software to draw the stress and displacement data obtained by ANSYS finite element analysis and the stress and displacement data obtained by the derivation formula,the coincidence curve is obtained,and the accuracy of the derived formula is verified.