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Register a new userAn analytic proof of the bifurcation condition for localized bulging in an inflated hyperelastic tube
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We provide a simple proof of the bifurcation condition for localized bulging in a hyperelastic tube of arbitrary thickness that is subjected to combined axial loading and internal pressure. Using analytic tools, we prove that the bifurcation condition is equivalent to the vanishing of the Jacobian of the internal pressure $P$ and the resultant axial force $N$, with each of them viewed as a function of the azimuthal stretch on the inner surface and the axial stretch. Previously this was only established by numerical calculations. The method should be applicable to any bifurcations that depend on a slowly varying variable, concluding that they share the same bifurcation conditions with bifurcations into uniform/homogeneous states as long as the equations determining the bifurcation condition are not trivially satisfied by uniform/homogeneous solutions.
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Previous linear bifurcation analyses have evidenced that an axially stretched soft cylindrical tube may develop an infinite-wavelength (localised) instability when one or both of its lateral surfaces are under sufficient surface tension. Phase transition interpretations have also highlighted that the tube admits a final evolved two-phase state. How the localised instability initiates and evolves into the final two-phase state is still a matter of contention, and this is the focus of the current study. Through a weakly non-linear analysis conducted for a general material model, the initial sub-critical bifurcation solution is found to be localised bulging or necking depending on whether the axial stretch is greater or less than a certain threshold value. At this threshold value, an exceptionally super-critical kink-wave solution arises in place of localisation. A thorough interpretation of the anticipated post-bifurcation behaviour based on our theoretical results is also given, and this is supported by Finite Element Method (FEM) simulations.
We propose an analytic proof of the Malgrange-Sibuya theorem concerning a sufficient condition of the convergence of a formal power series satisfying an ordinary differential equation. The proof is based on the majorant method and allows to estimate the radius of convergence of such a series.
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