Solitary waves in elastic materials based on modified strain gradient elasticity

In the present study, we derive a Boussinesq–type nonlinear partial differential equation to describe solitary wave propagation in isotropic elastic materials. The mathematical formulation is based on the modified strain gradient elasticity (MSGE) framework, which accounts for micro–deformations arising from micro–structural effects as well as macro–scale deformation due to surface effects. The derivation is based on Hamilton's principle, which equates the variation of the strain energy functional to the virtual work done by external forces. The resulting mathematical model is formulated in tensor form to maintain generality and is subsequently specialized to the one–dimensional case to elucidate the nonlinear nature of solitary wave propagation and the influence of micro–structural effects on the material's dynamic response. A key result of this study is the demonstration that the type of wave propagation in the medium can be controlled by appropriately selecting the length–scale parameter associated with micro–inertia, as well as the material length–scale parameters. Three types of initial and boundary conditions are considered: (ⅰ) Constant initial and boundary conditions, (ⅱ) dynamic boundary conditions, and (ⅲ) static initial conditions, moreover; all physical quantities are plotted and discussed in detail.