Longitudinal oscillations of layered and structurally heterogenic composite rods

The equations of longitudinal oscillation of a rod for all possible combinations of linearly elastic and rheological properties of homogeneous, longitudinally layered, cross-layered and structurally inhomogeneous composite rods were derived. An equation of longitudinal oscillation of nonlinear deformable rod in the sense of an arbitrary form of nonlinearity was obtained. In determining the effective properties of the composite rod of different structure the volume fractions of materials were used. It corresponds to the application of a discrete random variable with an appropriate distribution. In this sense the average values of the stresses and strains that occur in the composite rod were considered. Method of obtaining a multi-homogenized non-linear function of the laminate or structurally inhomogeneous composite elastic rod in the case of the use of bilinear Prandtl diagram or power function to describe the deformation of each component is described by the authors in earlier publication which applied to deformation without unloading of elastic-plastic and hyper elastic materials. Because in the article reader deals only with the elastic deformation of the rod, i.e. especially without unloading, then the previous results of studies can be transferred to this case is up to a changing constants names. This will implicitly use hypothesis that the tension / compression of nonlinear elastic material of rod are centrally symmetric around the origin. It was found that the Voigt approximation of effective properties of the rod corresponds to longitudinal layered structure of the rod, and the Reiss approximation corresponds to its cross-layered structure. The effective properties of structurally inhomogeneous composite rod are obtained as Hill approximations. The linear or nonlinear equations of hereditary theory (in the linear case with aging), the technical theory of aging, non-linear and linear Voigt equation of relaxation were used in the paper. According to these theories the equation of oscillations of the homogeneous rod were constructed. It is further generalized to the case of the composite rheological active rod. The density of the composite rod is calculated as the sum of the densities of the components multiplied by the corresponding volume fractions, regardless of the structure of the composite rod. In some cases, the analytical expressions for Eigen frequencies of oscillation of composite rods were obtained.