Elastodynamics of Linearized Isotropic State-Based Peridynamic Media

The peridynamic theory has been used to model and simulate numerically various kinds of mechanical behavior of solids. This work is devoted to analytical solutions of the elastodynamic behavior of linearized isotropic state-based peridynamic materials. First, we present the solutions of the dispersion relations, group velocities, and phase velocities of longitudinal and transverse waves, and examine in detail the effects of the Poisson's ratio on these properties. It is shown that the elastodynamic behavior of the state-based peridynamic material with a negative Poisson's ratio is remarkably different from that of the material with a positive Poisson's ratio. We then derive the general solutions of initial-value problems, and obtain the Green's function in a closed form. Finally, we study the evolution of a displacement discontinuity in the state-based peridynamic medium, and find that each component of the discontinuity in the three-dimensional theory varies independently according to the same vibrational mode. The results may have implications in investigations of wave propagations, including discontinuities such as phase transitions and kink propagations.