| Popis: |
A metamaterial is defined as a class of substances with a complex internal structure and unique physical and mechanical properties. As a rule, such a material is a complex periodic system, in the nodes of which there are not material points, but bodies of small but finite sizes, which have internal degrees of freedom. To describe metamaterials, gradient continuums are often used, which are obtained by continuumizing the equations of motion of discrete lattices consisting of identical masses and springs of different stiffness. However, it should be noted that the gradient continuum model must be dynamically consistent, i.e. stable and providing a finite rate of energy transfer, while in most gradient models the group velocity of waves increases indefinitely with frequency. To achieve dynamic consistency of the gradient continuum model, the continuum method proposed by A.V. Metrikine and H. Askes, the essence of which is the assumption of a nonlocal connection between the displacements of the lattice nodes and the resulting continuum (the method of alternative continualization). In this paper, this method is generalized to the case of finite deformations and applied to obtain a nonlinear dynamically consistent model of a metamaterial (gradient elastic medium). Within the framework of the obtained model, the formation of spatially localized nonlinear waves, which are strain solitons and their periodic analogs, in gradient-elastic media is studied. The sign of the dimensionless parameter, which is the ratio of the nonlinear addition to the spring stiffness to its linear stiffness, affects the polarity of the soliton. For positive values of the parameter (hard nonlinearity), the soliton has a negative polarity. For negative values of the parameter (soft nonlinearity), the soliton has a positive polarity. The magnitude of the nonlinearity does not affect the speed of wave propagation and their width, but affects their amplitude. |
