The natural frequency values of free vibrations are generally dependent on properties of materials, boundary conditions, displacement types, vibration matter with respect to axes direction, environment temperature and shear correction in thick composited shells. Advanced natural frequency investigation of thick functionally graded material (FGM) spherical shells is presented by considering the advanced shear correction coefficient. When the more thicker shells are used, it is more necessary to consider the transverse shear effect on the vibrations. It is novel to consider the effects of nonlinear coefficient in third-order shear deformation theory (TSDT) on the advanced shear correction coefficient. The nonlinear coefficient term of TSDT is included the fully homogeneous equation according to the homogeneous matrix under free vibration. The numerical solution can be solved for the five degree polynomial equation derived from zero determinant of the fully coefficient matrix by using the numerical method, then the natural frequency can be obtained. Three effects of nonlinear coefficient term, environment temperature and power law index on the frequency of thick FGM spherical shells with advanced shear correction coefficient are studied. The numerical values of natural frequencies are calculated and investigated. These numerical values of natural frequencies are very important in the designs of structures used to prevent resonance in the vibration of mechanisms.
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