Displacement potential functions for elastodynamic problems in transversely isotropic media based on nonlocal strain gradient theory

Today nanotechnology has become important in many fields, including industry, medicine, engineering, aerospace, national security and electronics. As the dimensions of the structures decrease, the effects of size play a crucial role in properties of the media. Mechanical properties, electrical conductivity, thermal properties and other known chemical and physical properties are some examples that differ on nanoscales. Classical continuum mechanics are impotent to cover the effects of dimensions of the constituents of the media on nanoscales. Hence, several non-classical continuum theories, including non-local elasticity theory, strain gradient theory, and non-local strain gradient theory, have been developed by researchers to explain size-dependent mechanical behavior on a nanoscale. In this research, governing equations in terms of displacement potential functions based on nonlocal strain gradient theory are introduced for elastodynamic problems in homogeneous Transversely isotropic media. To this end, the three-dimensional equations of motion of the homogeneous Transversely isotropic media are first calculated using the nonlocal strain gradient theory. Then, using a systematic method, a set of complete displacement potential functions will be presented to solve elastodynamic problems in these media. By use of potential functions, the governing equations of motion will be decoupled. The proposed potentials include two scalar functions. One of them satisfies an 8th-order partial differential equation and 4th-order PDE is governed on the other. These potential functions are obtained in the form of a combination of wave operators, non-local parameter, and characteristic length, which are functionally and physically meaningful. These potential functions are obtained in the form of a combination of wave operators, non-local parameter, and characteristic length, which are functionally and physically meaningful. In addition, potential functions for limiting cases namely strain gradient theory and Eringen nonlocal elasticity theory are presented, separately. Also, by neglecting non-local parameters and characteristic length, the solution is degenerated to the Eskandari-Ghadi solution for classical theory of elasticity. Moreover, a new set of potential functions is presented to solve the elastodynamics of nonlocal strain gradient theory for the simpler case of isotropic materials.