Periodic solutions for nonlinear deformation waves based on new Boussinesq-type equation within strain gradient elasticity and reductive perturbation technique

Abstract

In this article, we present a detailed variational formulation to study and analyze the geometrically nonlinear behavior of elastic materials at small scales, where the classical theory of elasticity becomes inadequate. The presented technique is based on the variation of the internal energy of the elastic material and the virtual work of the external forces for the dynamical system in the framework of the first strain gradient theory of elasticity. This approach simultaneously determines the equilibrium equations in nonlinear form and the complete set of nonlinear boundary conditions. A new hyperbolic Boussinesq-type equation is derived based on the presented model in one-dimensional space and time to examine the impact of geometrical nonlinearity on wave propagation in an isotropic elastic material. The exponential reductive perturbation technique (ERPT) is employed to obtain an approximate and hierarchical solution for the new nonlinear partial differential equation (NPDE). The obtained results are plotted and analyzed, moreover, the results reveal that geometric nonlinearity significantly influences wave propagation in polycrystalline silicon.