State Space Solutions Of A Functionally Graded Sphere Subjected To Electro-magneto-thermo-mechanical Loads

In this paper,the elastic mechanics of hollow spheres with arbitrary radial graded of material parameters under different combined loads of electro-magneto-thermo-mechanical is studied.By stratifying the sphere into a large number of sub-spherical shells and making the material parameters with arbitrary graded change discrete into piecewise constant functions according to the stratification situation,the state space theory is established based on the continuity of the state vector at the sub-shell interface,thus the state space solution of the arbitrary graded sphere under different loads is obtained.Based on the equilibrium equation,thermoelastic constitutive relation and heat conduction equation of the spherical symmetric problem,and introducing the heat flux,the state space theory is established by taking the radial displacement,radial stress,temperature and heat flux as the state vectors,and the accurate solution of the steady state thermoelastic problem of the hollow sphere with arbitrary graded under the action of thermo-mechanical load is obtained.On the basis of the spherical symmetric heat conduction problem and considering the equilibrium equation in the case of Lorentz force,the corresponding state space theory is established to obtain the exact solution of the thermoelastic problem of a spherical sphere with arbitrary graded under magneto-thermo-mechanical load.Based on the equilibrium equation,constitutive relation and electrostatic control equation of the spherical symmetry problem of the hollow piezoelectric sphere,the state space theory is established by taking the radial displacement,radial stress,electric potential and electric displacement as the state vectors,and the exact solution of the piezoelectric sphere with arbitrary graded under the action of electro-mechanical load is obtained.The results show that the state space solution is correct,accurate and efficient.The effects of the material parameters of functionally graded spheres and laminated spheres on the stresses and displacements in the spheres are discussed.