Elasticity Analysis Of Functionally Graded Hollow Disks Or Cylinders And Its Applications

In this thesis, based on the piecewise and continuously gradient models, the functionally graded hollow disks or cylinders are studied for the axisymmetric and non-axisymmetric problems. The analytical solutions of the stresses and displacements are derived explicitly with the aids of the piecewise model and the complex function theory when the elastic constants vary arbitrarily along the radial direction, and the corresponding closed-form solutions are obtained by using the special functions or the Frobenius method when Young’s modulus is expressed as one of some continuous functions of the radial coordinate. According to these solutions, the elastic problems of the isotropic plates and beams, in which a functionally graded ring is embedded, are analyzed under different loads. Meanwhile, an equivalent calculation model for the hole-drilling strain gauge method, which is used to measure residual stresses, is presented in the light of a work hardened layer with multilayered or graded composite structures formed on the wall of the hole due to drilling. Based on this model, the formulae of residual stress field are explicitly derived, and a special test method is presented to determine the equivalent Young’s modulus of the work hardened layer. The calibration coefficients are redefined by adding corrective terms. A case of numerical simulation is provided to verify the correctness of the obtained formulae. Assisted by some discussions and the corresponding tests, the fruits of the study in this thesis are obtained as follows:(1) The functionally graded hollow disk or cylinder, whose Young’s modulus varies along the radial direction as a linear, rational, Dryden’s, or arbitrary order polynomial function, is analyzed theoretically under radially symmetric loads, and the stresses and displacements are derived explicitly. For Young’s modulus varying linearly or as Dryden’s model, the analytical solutions obtained by using the Gauss function and the Whittaker function in this thesis will include the missing solutions in the literature.(2) The functionally graded hollow disk or cylinder, whose Young’s modulus is a continuous function of the radial coordinate, is analyzed theoretically under nonaxisymmetric loads. When the continuous function can be expanded into the Taylor series or is a polynomial function, the unified series solution is obtained by using the Frobenius method, and the solutions for the linear function, the exponential function, and the Dryden model are given in detail.(3) The stress and displacement fields for the axisymmetric and nonaxisymmetric problems of the functionally graded hollow disk or cylinder are also obtained by the complex function theory and the piecewise mode. The solutions are universal which can be used for the elastic constants varying arbitrarily along the radial direction. Otherwise, they can degenerate into the solutions for the cases of homogeneous materials.(4) When the functionally graded hollow disk or cylinder is subjected to the uniform or nonuniform load, the stresses and displacements are related not only to the ratio of the inner elastic modulus to the outer one but also to the variations of Young’s modulus along the radial direction. Relatively, the different radial gradient of the elastic constants will have a great influence on the tangential stress and the radial displacement. The maximum tangential stress may appear inside the disk or cylinder instead of at the boundary. If the disk or cylinder is loaded nonuniformly, the maximum radial stress may also appear inside it. In addition, when the disk or cylinder is subjected to uniform internal pressure, the magnitude of the tangential stress at the inner boundary may less than that of the radial stress in some cases, which is different than the magnitude of the tangential stress is always larger than that of the radial stress for the isotropic material.(5) For the homogeneous plate and pure bending homogeneous beam embedded a functionally graded circular ring under different loads, the distribution of the tangential stress inside the ring is quite different when the gradient of Young’s modulus is different along the radial direction, especially for the case of the stiffness at the outer edge of the ring being larger than that at the inner edge. If the functionally graded ring is used to reinforce the edge of hole in the plate or beam, it is better to adopt the functionally graded ring with the outer stiffness being larger than the inner one.(6) The work hardened layer formed in measuring the residual stress by the hole-drilling method has no influence on the distribution of relieved stresses, but has some effects on the magnitude of the relieved stresses near the hole. The equivalent calculation model, in which the work hardened layer with multilayered or graded composite structures is simplified and regarded as an isotropic circular ring of another material, is sound and valid, and can improve the measurement accuracy to some extent. When the equivalent elastic modulus is determined by the presented test method, the thickness of the work hardened layer can be assumed arbitrary in the proper range, which hardly affects the calculated results.