Period doubling and quadrupling of wrinkles in film/substrate bilayers: a nonlinear symplectic analysis

Abstract

Under compression, a stiff film bonded to a soft substrate can progress from sinusoidal wrinkling to period doubling and quadrupling before localization. We develop an analytical-depth Rayleigh–Ritz method for this finite-amplitude cascade. Hamiltonian–Stroh modes represent the homogeneous depth response, nonlinear product rates enrich the finite state, and signed Floquet spaces test secondary stability through the constitutive second variation. For µf/µs = 100, the method gives a period-doubling threshold T24 = 0.179 and a retained-space period-four neutral point T45 = 0.244. A fixed-grid lattice calculation gives corresponding neutral crossings at 0.176 and 0.243 and produces similar period-two and period-four patterns. The two transitions arise through repeated subharmonic coupling: the primary wrinkle couples the signed half-wavenumber pair, and the doubled state couples the quarter-wavenumber pair. Across modulus ratios from 14 to 375, the primary onset strain varies nearly ninefold, whereas the doubling boundary remains near 0.18. Published incompressible finite-element calculations show the same weak dependence of the secondary boundary on stiffness contrast. The method therefore follows two successive wavelength multiplications with out substrate-depth nodal unknowns.