The method of ray expansions for solving boundary-value dynamic problems for spatially curved rods of arbitrary cross-section

Abstract

The method of ray expansions is developed for solving boundary-value problems connected with the propagation of planes of strong and weak discontinuity in spatially curved linearly elastic rods of arbitrary cross section. The equations of the three-dimensional theory of elasticity are utilized, which are written on the wave surface using the theory of discontinuities and then are integrated over the cross-sectional area. It is assumed that the plane of discontinuity remains perpendicular to the centroidal axis of the rod all the time during its propagation; the discontinuities in the normal stresses in the sections with the normals perpendicular to the centroidal axis can be ignored as compared to the discontinuities in stresses in the sections with the normals parallel to the centroidal axis. The cross-sections of the rod remain plane during the process of the rod deformation. These assumptions lead to the generation of two wave surfaces propagating in the spatially curved rod with the velocities of longitudinal-flexural and transverse-torsional waves of elastic rods. As this takes place, on the longitudinal-flexural wave, the bulk deformations experience a discontinuity not only at the sacrifice of shortening-elongation of the medium’s element locating along the centroidal axis, but also at the expense of thickening–thinning of this element in the directions of the principle axes of the rod cross-section. For the transverse-torsional wave, there exist discontinuities in the components of the velocities directed along the principle axes of the cross-section, in the angular velocity of the cross-section rotation as a rigid whole with respect to the centroidal axis, as well as in transverse deformations occurring due to the inhomogeneity of the transverse displacements. During the solution of boundary-value problems, the values to be found are represented in terms of the power series, the coefficients of which are the discontinuities in arbitrary order partial time-derivatives of the desired functions, while the time of arrival of the wave front is the independent variable; in so doing the order of the partial time-derivative coincides with the power exponent of the independent variable. The ray series coefficients are determined from the recurrent equations of the ray method within the accuracy of arbitrary constants, while the arbitrary functions themselves are found from the boundary conditions. Examples illustrating the efficiency of the ray method for solving the problems of dynamic contact interaction, resulting in the propagation of transient waves of strong discontinuity in spatially curved rods, are presented.