Wavelength selection in the twist buckling of pre-strained elastic ribbons

A competition between short- and long-wavelength twist buckling instabilities has been reported in experiments on thin elastic ribbons having pre-strain concentrated in a rectangular region surrounding the axis. The wavelength of the twisting mode has been reported to either scale (i) as the width of the ribbon when the pre-strain is large (short-wavelength case) or (ii) as the length of the ribbon when the pre-strain is small (large-wavelength case). Existing one-dimensional rod or ribbon models can only account for large-wavelength buckling. We derive a novel one-dimensional model that accounts for short-wavelength buckling as well. It is derived from non-linear shell theory by dimension reduction and captures in an asymptotically correct way both the non-convex dependence of the strain energy on the twisting strain τ (which causes buckling) and its dependence on the strain gradient τ^′. The competition between short- and long-wavelength buckling is shown to be governed by the sign of the incremental elastic modulus B₀ associated with the twist gradient τ^′. The one-dimensional model reproduces the main features of equilibrium configurations generated in earlier work using 3D finite-element simulations. In passing, we introduce a novel truncation strategy applicable to higher-order dimension reduction that preserves the positiveness of the strain energy even when the gradient modulus is negative, B₀<0.