Matrix Green's function solution of closed-form singularity for functionally graded and transversely isotropic materials under circular ring force vector

Abstract

The paper develops the Green's function solution in matrix form for functionally graded and transversely isotropic material (FGTIM) due to the loading of a circular ring force vector. The force vector is concentrated along a horizontal circular ring and acts at any location in the interior of the FGTIM material occupying either a fullspace or a halfspace. Without loss of generality, the variations of the FGTIM's five elastic parameters are expressed as five arbitrary step functions with the depth as the independent variable. The two-dimensional Fourier integral transforms in the cylindrical coordinate system are used for the mathematical formulation and derivation in matrix form. The Green's function solution of the displacement and stress fields is explicitly expressed in the matrix form in terms of classical improper Hankel transform integrals of the orders of 0 to 3. The kernel functions of the Hankel transform integrals are explicitly expressed in the forms of backward transfer matrix. Their mathematical properties are analyzed analytically and numerically. The singular terms associated with the improper Hankel transform integrals are analytically isolated and expressed in the exact closed-form in terms of the complete elliptic integrals of the first, second and third kind. Such closed-form singular terms can be expressed as the matrix Green's function solution for the corresponding bi-material due to the same circular ring force vector. Numerical results show that the computation of the matrix Green's function solution can be achieved with high accuracy and efficiency and the heterogeneity and anisotropy can have significant effects on the elastic fields in the FGTIM induced by the circular ring force vector.