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Growth-induced blisters in a circular tube. (English) Zbl 1349.74131
Summary: The growth of an elastic film adhered to a confining substrate might lead to the formation of delamination blisters. Many results have been derived when the substrate is flat. The equilibrium shapes, beyond small deformations, are determined by the interplay between the sheet elastic energy and the adhesion potential due to capillarity. Here, we study a non-trivial generalization to this problem and consider the adhesion of a growing elastic loop to a confining circular substrate. The fundamental equations, i.e., the Euler Elastica equation, the boundary conditions and the transversality condition, are derived from a variational procedure. In contrast to the planar case, the curvature of the delimiting wall appears in the transversality condition, thus acting as a further source of adhesion. We provide the analytic solution to the problem under study in terms of elliptic integrals and perform the numerical and the asymptotic analysis of the characteristic lengths of the blister. Finally, and in contrast to previous studies, we also discuss the mechanics and the internal stresses in the case of vanishing adhesion. Specifically, we give a theoretical explanation to the observed divergence of the mean pressure exerted by the strip on the container in the limit of small excess-length.

##### MSC:
 74G05 Explicit solutions of equilibrium problems in solid mechanics 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
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