Influence of first to second gradient coupling energy terms on the wave propagation of three-dimensional non-centrosymmetric architectured materials

In the current work, we analyze the role of the coupling energy between the first and second order gradient kinematic terms on the wave propagation characteristics of three-dimensional (3D) architectured materials with a non-centrosymmetric inner structure. To that scope, we elaborate an energy based method that computes the network material's unit-cell total deformation energy, based on its full linear 3D beam kinematics. We use a continualization method to provide the effective second gradient constitutive law, where a coupling energy tensor of order five appears, due to the lack of a center of inversion. Thereupon, we consider the case study of a pyramid shaped unit-cell periodic microstructure. We formulate the dynamic equilibrium equations and compute the architectured materials’ wave propagation attributes. We analyze the effect of the coupling energy terms on the propagating longitudinal and shear modes, presenting the corresponding phase velocities for different directions of propagation. We compute the influence of the coupling energy terms on the wave propagation characteristics as a function of the propagation direction. We observe considerable differences between second gradient media description with and without the consideration of the coupling energy term contributions for propagating modes along the non-centrosymmetric inner material direction. We assess the effect of coupling energy over a wide range of propagating directions, deriving useful overall conclusions its role in the wave propagation characterization of 3D architectured media.