A minimal low-dimension model simulating the nonlinear dynamics of pretensioned textile metamaterials

Within the theoretical and applied research on nonlinear dynamics of periodic microstructured systems, the amplitude-dependent dispersion properties of mechanical metamaterials are attracting increasing interest. The paper investigates the nonlinear free and forced oscillations of a minimal two degrees of freedom model simulating the local interplay between plain woven yarns in pretensioned textile metamaterials. Numerical solutions are obtained by time-integrating the equations of motion, which are characterized by nonlinear forces. Nonlinearities arise from the piece-wise constitutive law determined by the superposition of geometric stiffness caused by pretension and unilateral elastic stiffness due to inter-yarn contact. Frequency-response curves of the harmonically forced system are numerically obtained and the nonlinear softening behavior arising from the piece-wise stiffness characteristics is discussed. Particular attention is given to the effective possibility of consistently describing the unilateral elastic stiffness governed by smooth Hertzian laws with linear and cubic non-smooth approximations. It is shown that numerical solutions exhibit time periodicity disturbed by significant distortions of the linear solutions as soon as the cubic nonlinearities are activated or the oscillation amplitude exceeds a certain detachment threshold. Finally, parametric analyses disclose a marked softening trend in the frequency response functions, as well as the birth of superharmonic components in the time-histories.