This thesis derives a mechanical model of wrinkling of a growing elastic beam with geometric imperfections resting on a nonlinear viscoelastic substrate, combining within a unified framework three features of real systems: film imperfections and the time-dependent and nonlinear response of the substrate to mechanical loads. The transverse displacement of the beam is represented in a spectral basis, while the substrate is described by coupling a relaxation model and a nonlinear model incorporating quadratic and cubic terms. Particular attention has been paid to the conservativeness of the elastic part of the formulation. In limiting cases, the model reduces to known results from the literature. Stability analysis yields the critical values, while numerical simulations provide the evolution of deformation patterns, the energy landscape and the transitions between stable states. The results show that the substrate modulus ratio does not alter the set of equilibrium states, but only the selection and the time scale of the path between them, that the nonlinearity governs deformation localization through the effective substrate stiffness and that the sensitivity to imperfections is resonant, peaking in the critical band of deformation eigenmodes. The energy landscape proves an effective tool for understanding the formation and evolution of wrinkling patterns.
|
