Nonlinear wave propagation in graphene incorporating second strain gradient theory

The exploration of graphene has attracted extensive interest owing to its significant structural and mechanical properties. In this research, we numerically investigate wave propagation in defect -free single -layer graphene, considering its geometrically nonlinear behavior through second -strain gradient elasticity. To capture the geometric nonlinearity, firstly, the nonlinear strain-displacement relations are introduced. The governing equation and associated boundary conditions are derived using Hamilton's principle. Then, the weak form, including the element matrices, is established. The eigenvalue problem is solved for 2D wave propagation through periodic structures theory. Finally, the dynamical properties such as band structures, mode shapes, energy flow, and wave beaming effects are analyzed. The numerical results reveal that the geometric nonlinearity through the second strain gradient influences the wave propagation characteristics in graphene. The findings are significant and contribute to the understanding of graphene's dynamic response, with implications for the engineering applications of graphene-based nanostructures.