Elastic rods with incompatible strain: Macroscopic versus microscopic buckling

We consider the buckling of a long prismatic elastic solid under the combined effect of a pre-stress that is inhomogeneous in the cross-section, and of a prescribed displacement of its endpoints. A linear bifurcation analysis is carried out using different structural models (namely a double beam, a rectangular thin plate, and a hyper-elastic prismatic solid in 3-d): it yields the buckling mode and the wavenumber q_c that are first encountered when the end-to-end displacement is progressively decreased with fixed pre-stress. For all three structural models, we find a transition from a long-wavelength (q_c=0) to a short-wavelength first buckling mode (q_c ≠ 0) when the inhomogeneous pre-stress is increased past a critical value. A method for calculating the critical inhomogeneous pre-stress is proposed based on a small-wavenumber expansion of the buckling mode. Overall, our findings explain the formation of multiple perversions in elastomer strips, as well as the large variations in the number of perversions as a function of pre-stress and cross-sectional geometry, as reported by Liu et al. (2014).