With the aim to characterize the formation and propagation of bulges in cylindrical rubber balloons, we carry out an expansion of the nonlinear axisymmetric membrane model assuming slow axial variations. We obtain a diffuse interface model similar to that introduced by van der Waals in the context of liquid–vapour phase transitions. This provides a quantitative basis to the well-known analogy between propagating bulges and phase transitions. The diffuse interface model is amenable to numerical as well as analytical solutions, including linear and nonlinear bifurcation analyses. Comparisons to the original membrane model reveal that the diffuse interface model captures the bulging phenomenon very accurately, even for well-localized phase boundaries.

We propose a general method for deriving one-dimensional models for nonlinear structures. It captures the contribution to the strain energy arising not only from the macroscopic elastic strain as in classical structural models, but also from the strain gradient. As an illustration, we derive one-dimensional straingradient models for a hyper-elastic cylinder that necks, an axisymmetric membrane that produces bulges, and a two-dimensional block of elastic material subject to bending and stretching. The method offers three key advantages. First, it is nonlinear and accounts for large deformations of the cross-section, which makes it well suited for the analysis of localization in slender structures. Second, it does not require any a priori assumption on the form of the elastic solution in the cross-section, i.e., it is Ansatz-free. Thirdly, it produces one-dimensional models that are asymptotically exact when the macroscopic strain varies on a much larger length scale than the cross-section diameter.

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International audienceWe 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