Nonlinear non-smooth dynamics of a minimal two degrees of freedom model for pretensioned textile metamaterials

Within the theoretical and applied research on periodic microstructures, increasing attention is being directed towards the nonlinear dynamics of mechanical metamaterials. The present study investigates the nonlinear free and forced dynamics of a minimal two degree-of-freedom model that captures the local mechanical interactions between warp and weft yarns in pretensioned plain-weave textile metamaterials. The analysis incorporates the high-amplitude range of oscillations, including the yarn-to-yarn detachment events. The governing equations of motion are characterized by geometric stiffness induced by self-equilibrated yarn pretension and piecewise nonlinear elastic stiffness arising from unilateral inter-yarn contact. Frequency-response curves under harmonic excitation exhibit a systematic softening behavior, initially driven by the nonlinear constitutive law in the attached configuration and further amplified by the stiffness reduction associated with detachment. Direct numerical integration combined with computational continuation reveals a rich nonlinear response scenario, characterized by coexistence of multistable and unstable solutions, onset of bifurcations and generation of superharmonics. The role of the fundamental mechanical parameters-namely yarn pretension and slenderness-is investigated to clarify their influence on the interplay between weak constitutive nonlinearities and strong non-smooth detachment effects. Pretension is identified as a passive tuning parameter for the local dynamic response, providing promising perspectives towards the spectral design of complex textile metamaterials. Finally, exact solutions are qualitatively and quantitatively compared with solutions resulting from third-order asymptotic approximations of the contact law, highlighting the accuracy and limitations of linearized and cubic approximate models.