A diffuse interface model for the analysis of propagating bulges in cylindrical balloons

Abstract

<jats:p>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.</jats:p>

Asymptotically exact strain-gradient models for nonlinear slender elastic structures: A systematic derivation method

“…We have recovered the model established by Lestringant and Audoly (2018). Typical solutions predicted by the 1d model are compared to those of the full axisymmetric model in figure 4: the 1d models appears to be highly accurate, even in the regime where the bulges are fully localized.…”

“…Standard dimension reduction without gradient terms yields a non-convex elastic potential W hom and thus fails at describing the details of localization. Localized solutions can be analyzed using the full membrane model (Fu et al, 2008;Pearce and Fu, 2010), but are more easily and very accurately described based on a 1d strain-gradient model, as recently shown by the authors, starting from the theory of axisymmetric elastic membranes and using a typical constitutive law for rubber (Lestringant and Audoly, 2018). This 1d model is rederived here as a first illustration of the general reduction method presented in section 2.…”

“…The axisymmetric membrane model, which we used as a starting point was already a 1d model: it does not make use of any transverse variable, and has discrete degrees of freedom (Z, R) in each cross-section. The Lestringant and Audoly (2018). The material model for rubber proposed by Ogden (1972) is used, with the same set of material parameters as used in the previous experimental work of Kyriakides and Chang (1991), see also §2 in Lestringant and Audoly (2018).…”

“…reduction method led us to another 1d model and it therefore is improper to speak of dimension reduction in this case. The reduction method is still useful, as reduced model is simpler and, more importantly, much more standard: it is the well-known diffuse-interface model introduced by van der Walls in the context of liquid-vapor phase transition, as discussed by by Lestringant and Audoly (2018). Even when bulges are fully formed, the typical length of the interface between the bulged and unbulged regions never gets much less than ∼ ρ, i.e., remains always much larger than than the membrane's thickness t (assuming the membrane is thin in a first place ρ t).…”

“…In prior work, asymptotic 1d strain-gradient models have been obtained as refinements over the standard theory for linearly elastic beams (Trabucho and Viaño, 1996;Buannic and Cartaud, 2000), inextensible ribbons (Sadowsky, 1930;Wunderlich, 1962) and thin-walled beams (Freddi et al, 2004). The possibility of using 1d models to analyze localization in slender structures easily and accurately has emerged recently in the context of necking in bars and bulging in balloons (Audoly and Hutchinson, 2016;Lestringant and Audoly, 2018). Several other localization phenomena could be better understood if 1d models were available.…”