Beyond Classical Continua: Third-Gradient Effects on Wave Transmission in 1D Metamaterials

The study of wave propagation in higher-order continua is crucial for understanding the mechanical behavior of metamaterials and architected micro structures. In this work, we analyze plane wave solutions in a one-dimensional third-gradient continuum within the small deformation regime. Starting from a rigorous variational formulation, we derive the governing equations through the principle of stationary action, incorporating higher-order strain gradients that naturally emerge from homogenization of complex microstructures. The dispersion relations obtained reveal the presence of multiple wave modes, including propagative and damped waves, whose behavior is controlled by the characteristic length scales associated with third-gradient effects. We rigorously derive boundary conditions from the variational principles, ensuring a physically consistent formulation. Special attention is given to the reflection and transmission of waves at discontinuity interfaces, particularly when a third-gradient continuum is coupled to a second-gradient continuum, highlighting the role of higher-order boundary interactions in wave energy transfer. By computing the mean energy flux we get solutions for the reflection and transmission coef`cients. Our results demonstrate that third-gradient effects signi`cantly alter wave propagation characteristics, enabling tunable energy transmission across interfaces, a feature with potential applications in wave control, impact mitigation, and metamaterial design. The theoretical framework presented is validated through numerical simulations, providing insights into how wave propagation is affected by microstructural parameters. The `ndings offer a solid foundation for engineering materials with enhanced dynamic properties, paving the way for new applications in advanced mechanical systems and next-generation structural materials.