Keywords: space membranes, inflation, compressible medium, quasi-static equilibrium, parameter dependence.

A large variety of thin three-dimensional pressurised structures, balloons, are used in engineering and medical contexts. These structures show large displacements and deformations when pressurised. Several analytical and numerical treatments for more or less general situations are available in literature, but treatment of these structures also leads to accompanying aspects, such as the load description, contact formulations, dynamics, wrinkling under compression, and instability aspects. A particular aspect is related to the medium used to pressurise the membrane, where fluids and gases or combinations give different situations of density and compressibility, significantly affecting the response.

A common assumption in a numerical treatment is that the response can be based purely on the membrane behaviour, avoiding the problems of very thin shell formulations. The problems are, however, both geometrically non-linear due to finite deformations and materially non-linear through, e.g., hyper-elastic material assumptions. It is common also to only consider quasi-static equilibrium situations, where any combination of pressure and volume can be immediately introduced, without dynamic or thermal effects. The dynamical aspects of the behaviour are in these cases primarily related to the filling processes of the membranes.

Instability investigations of any class of optimised structures demand elaborate solution methods. Previous work by the author has described generalised path-following methods for non-linear quasi-static discretised structural problems, as developments of the equilibrium path methods for load-displacement relations. The methods thereby allow the calculation of the parameter dependence in different aspects of structural response, e.g., deflections, stresses, critical loads. The parameter dependence evaluations also go beyond sensitivity investigations. It will be shown how also non-obvious parameters such as the ambient temperature can be used in mechanical response evaluations.

The primary objective in using these general paths is the detailed phenomenological description of structural instabilities. Such evaluations are useful tools for understanding the behaviour of many structures, particularly as the quasi-static critical states are often the attractors of dynamic response.

From the application viewpoint, the parameterised formulation means that a studied load-carrying structure is just one instance of a class of similar structures, where different responses are relevant for different instances. The borders between these behaviour regions also provide important results from simulations. An interesting possibility is to use one- and two-dimensional manifolds for the visualization of response aspects.

The presentation will describe a simple finite element model, but primarily the solution methods for parameterised non-linear equilibrium problems. The pressurised balloon problem is used to demonstrate the generalised setting and applications of these models.

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