On the Optimal Choice of Young's Modulus for a Functionally Graded Material

The problem of maximizing the value of the first natural frequency for a functionally graded material depending on the variation law of Young's modulus is considered. It is assumed that there is a limitation on the average integral value of Young's modulus. The effect of variable material properties on the value of the first natural frequency is shown using the finite element method for the numerical solution of a two-dimensional axisymmetric problem of free oscillations of a cylinder. The optimality condition is obtained using the methods of variational calculus based on the general formulation of the problem for an inhomogeneous elastic isotropic body. It is noted that the left-hand side of this condition has a quadratic form. The problem of finding the optimal variation law of Young's modulus is essentially nonlinear in the general case and special numerical methods must be used to solve it. Three special cases are considered using the obtained optimality condition: bending vibrations of a circular solid plate, longitudinal vibrations of a rod and radial vibrations of a solid thin disk, taking into account the corresponding hypotheses. The optimal variation laws of Young's modulus and the displacement function are obtained in analytical form for all problems. Particularly, in the problem for the disk, a representation is proposed for the radial component of the displacement field, which is described by a linear law. It is shown that in this case the corresponding radial and tangential components of the stress tensor are equal. The sought-for function of the change in Young's modulus along the radial coordinate is found in analytical form from the equation of motion and the boundary condition on the outer boundary. An analytical expression is obtained for determining the value of the natural frequency, corresponding to the found solution. The accuracy of this formula is estimated by comparing it with the numerical solution obtained using the finite element method in the FlexPDE package. A comparison of the values of the natural frequency for homogeneous and inhomogeneous disks is made.