On The Stress Concentrations In Centrally Symmetric Inhomogeneous Hollow Spheres

Stress concentration is an important factor affecting the bearing capacity of the structure.Unlike homogeneous materials,the material properties of functionally graded materials vary in gradient with spatial coordinates.The existing research results show that if properly designed,the functionally graded components can alleviate the stress concentration intensity to a certain extent.However,due to the non-uniform characteristics of functional gradient members,the mechanical response of such structures is more complicated.Therefore,sufficient calculation analysis must be carried out and reflected back to the design process.In this thesis,the stress analysis of hollow spheres with functionally graded spherical shells is carried out.It is assumed that the shear modulus of functionally gradient coatings is power function distribution and pois-son's coefficient is constant.Based on the displacement system of three-dimensional elasticity theory,the stress field of hollow spheres with functionally graded spherical shells under three conditions of hydrostatic stress,uniaxial tension and biaxial tension is solved.For the solution under hydrostatic stress conditions,firstly,using the spherical symmetry of the model and the load,the displacement equilibrium equation of the variable coefficient is obtained from the radial displacement by the governing equation.Secondly,the displace-ment equilibrium equation is directly solved to obtain the radial displacement general solution.Finally,the boundary conditions are matched,and the stress field of the spherical shell with functionally graded spherical shells under hydrostatic stress is obtained by the geometric and physical equations.For the solution under uniaxial tension conditions,a functional spherical shell is simulated by homogeneous layers with gradient of shear modulus.Secondly,based on the displacement theory of elastic theory,the Boussinesq displacement potential function is used to solve the stress field of a single finite homogeneous spherical shell under uniaxial tension.Thirdly,according to the stress field of a single homogeneous spherical shell,the matrix field is used to obtain the stress field of multiple perfectly connected homogeneous spherical shells.Finally,the shear modulus of a number of perfectly connected homogeneous spherical shells was controlled to simulate a functionally-graded hollow sphere model,and the stress field of a functionally graded spherical shell homogeneous hollow sphere under uniaxial tension was obtained.For the solution under the equibiaxial tension condition,the same displacement po-tential function as in the uniaxial tension condition is firstly used to re-match the stress boundary conditions under the biaxial stretching condition,and then the analysis on each homogeneous sublayer is obtained,solution.Secondly,the shear modulus of each homogeneous sublayer is adjusted to fit the shear modulus distribution of the functionally graded spherical shell.Fi-nally,the stress field of a functionally graded spherical shell homogeneous hollow sphere under biaxial stretching is obtained.On the basis of the analytical solution,this thesis quantitatively analyzes the effect of the material distribution characteristics of the functionally graded spherical shell on the stress con-centration factor around the hole,and compares the calculated results with the stress concen-tration factor around the hole containing the homogeneous spherical shell.The analysis of the parameters of the functionally graded material system for the purpose of reducing the stress concentration factor is realized.The results show that the stress concentration factor of the thick-walled spheres can be significantly reduced by the reasonable design of the functionally graded spherical shell material.Among them,the hydrostatic internal pressure decreases by 91.0%,and the hydrostatic outer pull decreases by 29.0%.Under uniaxial tension condition,the pole stress concentration factor decreased by 44.5%,the circumferential stress concentration factor decreased by 75.2%.Under the biaxial tension condition,the hoop stress concentration factor is reduced by 48.0In order to verify the correctness of the theoretical solution,this thesis uses ABAQUS software to carry out finite element simulation of the model.First,due to the spherical symmetry of the model and the axial symmetry of the load,1/8 thick-walled spherical shell is created.Secondly,functional gradient coatings are simulated using 10 perfectly connected homogeneous sub-layers;again,meshing is performed using C3D8R elements Again,the load and boundary conditions under various conditions are applied to the section;finally,the results of the finite element numerical simulation are compared with the results of the theoretical solution.The results show that the results of finite element are in good agreement with the theoretical solution.In this thesis,the stress field of a hollow sphere with a functionally graded reinforcement shell under uniaxial and biaxial tension is solved for the first time.In this process,the mathemat-ical mechanical model is established and solved.The homogeneous spherical shell simulation using infinite discretization is used.The continuous functionally graded spherical shell realizes the theoretical solution of the spherical shell with functional gradient under uniaxial tension.Stress concentration around the geometric defects such as microscopic holes and inclusions is one of the important factors leading to the failure of metal foam materials.Therefore,the re-search results of this thesis have guiding significance for the application of functionally graded materials to improve the strength of metallic foam.