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This and the next 3 images are from:
Amit Patil, Arne Nordmark and Anders Eriksson (KTH Mechanics, Royal Institute of Technology, Stockholm, Sweden),
“Instability investigation on fluid-filled pre-stretched cylindrical membranes”, Proceedings of the Royal Society A, Vol. 471, 20150016, 2015, http://dx.doi.org/10.1098/rspa.2015.0016
ABSTRACT: This paper discusses the evaluation of instabilities on the quasi-static equilibrium path of fluid-loaded pre-stretched cylindrical membranes and the switching to a secondary branch at a bifurcation point. The membrane is represented by only the in-plane stress components, for which an incompressible, isotropic hyperelastic material model is assumed. The free inflation problem yields governing equations and boundary conditions, which are discretized by finite differences and solved by a Newton–Raphson method. An incremental arclength-cubic extrapolation method is used to find generalized equilibrium paths, with different parametrizations. Limit points and bifurcation points are observed on the equilibrium path when fluid level is seen as the controlling parameter. An eigen-mode injection method is employed to switch to a secondary equilibrium branch at the bifurcation point. A limit point with respect to fluid level is observed for a partially filled membrane when the aspect ratio (length/radius) is high, whereas for smaller aspect
ratios, the limit point with respect to fluid level is observed at over-filling. Pre-stretch is observed to have a stiffening effect in the pre-limit zone and a softening effect in the post-limit zone.
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