Nonlinear Deformation Waves Based on New Boussinesq‐Type Equation Within Simplified Strain Gradient Elasticity

ABSTRACT In this paper, we derive a nonlinear partial differential equation to describe wave propagation in an isotropic elastic material, based on the simplified strain gradient theory of elasticity. The variational principle of the internal energy functional and the virtual work for the external forces are used to derive the field equations and boundary conditions. Additionally, we show that the derived nonlinear partial differential equation can be considered a Boussinesq‐type equation, which includes only one nonlinear term, whereas the new equation contains three nonlinear terms. Therefore, this equation can be used to study the dispersive behavior of wave propagation in various fields where both higher order spatial derivatives and nonlinear terms influence partial differential equations. Analyzing such an equation often requires advanced mathematical techniques or numerical methods, depending on the specific initial and boundary conditions. Furthermore, kink and soliton solutions for the new nonlinear partial differential equation were introduced using d'Alembert's solution, and the results are plotted and analyzed.