The effects of parametric uncertainties on the nonlinear vibrations of a pressure-loaded spherical hyperelastic membrane

Resumo

Hyperelastic membranes are found in many engineering fields. A key step in their mathematical
modelling is the choice of an appropriate constitutive law and, subsequently, the determination of the associated
material parameters, usually obtained from experimental results, with the occurrence of multiple sets of optimal
material parameters for the same data sets, depending on the fitting process. The influence of the constitutive law
of a hyperelastic material on its nonlinear behavior is well known and both static and dynamic responses can vary
greatly, depending on the parameter values. Thus, the application of the uncertainty propagation analysis, capturing
the expected outcome in a probabilistic sense, constitutes an interesting tool in the analysis of hyperelastic
structures, particularly in the dynamic case. Here it is applied to the analysis of a pressure-loaded spherical
hyperelastic membrane with constitutive uncertainty. Different hyperelastic constitutive models are addressed,
comparing the static and dynamic nonlinear response under parametric uncertainty. Finally, the global stability is
developed by expanding a previous concept of (Q, Λ)-attractors to the dual space where basins are defined.
Specifically, the inclusion of parametric uncertainty is responsible for the diffusion of attractors and basins
boundaries, drastically changing the phase-space topology. A novel methodology of phase-space discretization is
also considered to represent the phase-space topology under uncertainty. A global dynamical analysis framework
in the context of parametric uncertainty is yet to be addressed, specifically for hyperelastic constitutive models,
constituting the novelty of the present work.