Nonlinear Viscoelastic and Viscoplastic Behavior of Multi-Layer Polymeric Films Used in Super-Pressure Balloon Envelopes
Hirsekorn, M.¹; Petitjean, F.²; Deramecourt, A.^1
1Centre National d'Etudes Spatiales (CNES); ²Institute Catholique d'Arts et Métiers (ICAM)

Super-pressure balloons provide a cheap, non-polluting, and quickly operable platform for in-situ measurements at altitudes up to about 25 km, and for remote-sensing of earth, atmosphere, and space from regions inaccessible to other means of transportation over similar observation periods. They are lifted by a helium filled, approximately spherical, closed envelope made of a PET based multi-layer polymeric film. The filling quantity of lifting gas is adjusted such that the envelope remains pressurized throughout the flight, once float altitude is reached. Compared to open balloons, the volume of the envelope is relatively insensitive to variations of internal and external pressure, and gas leakage is limited to a minimum. Thus, the altitude variations between day and night lie within a tolerable range. Adjustments by dropping ballast or deflating the envelope rarely needed, permitting long term flights of up to several months.

Nevertheless, the volume changes considerably throughout the flight, mainly because of variations of the difference between internal and external pressure during day and night cycles, which cause a varying deformation of the envelope material. Moreover, even at constant pressure difference (i.e., at constant stress in the polymeric film), the volume evolves due to creep and recovery effects, arising from the viscous properties of the polymer. In order to determine the optimum quantity of lifting gas needed to reach a specific altitude with a balloon carrying a given payload and the amount of ballast necessary for projected altitude adjustments during flight, it is essential to know precisely the evolution of the material deformation in time under constant and varying stress. Accurate simulation of the evolution of the envelope volume is also crucial to the calculation of the balloon trajectory.

We have analyzed the mechanical behavior of the polymer film by means of a series of creep and recovery tests at different stress levels using rectangular specimens. The experiments show that the response of the material is strongly time dependent, including viscoelastic and viscoplastic deformation, which both depend nonlinearly on stress. Moreover, the observed time dependence of strain during creep at constant stress depends strongly on the applied stress level. Simple rescaling along the time and strain axes is not sufficient to superpose the strain curves at different stress levels. Standard models of nonlinear viscoelasticy like the model of Schapery [Polym. Eng. Sci. 9,4 (1969), pp. 295-310] do therefore not entirely describe the mechanical behavior of the envelope material.

We present a model for the viscoelastic and viscoplastic behavior of the film that takes into account the dependence of the strain response on creep stress. For easy numerical implementation, the observed strain response is represented by a Prony series, whose coefficients form a continuous spectrum on the logarithmic retardation time scale. The spectrum is well approximated by an exponential power law distribution with exponent 3. The distribution is fully characterized by three stress dependent parameters: center, width, and amplitude of the distribution. The first two parameters fully describe the strain evolution in time at constant stress, whereas the third parameter quantifies the total amount of creep strain accumulated after a certain time.

The representation by a Prony series is equivalent to a generalized Kelvin-Voigt model, where each term of the series corresponds to one Kelvin-Voigt element. Since the strain responses both in purely viscoelastic and in highly viscoplastic stress ranges are well approximated by the same type of distribution, we reproduce the viscoplastic strain by a generalized Kelvin-Voigt model with elements that do not recover. The coefficients of the corresponding Prony series are represented by a separate spectrum of the same type of distribution.

The stress dependence of the model parameters can be expressed in terms of simple analytic functions. The experiments show that both viscoelastic and viscoplastic strain are highly stress dependent over a limited stress range only, and are approximately linear at low stresses and around the maximum stress reached during flight. A continuous threshold function is proposed that approximates well the observed stress dependence of the distribution amplitudes. It is assumed that the other viscoelastic (viscoplastic) parameters change around the same threshold as the amplitude of the viscoelastic (viscoplastic) spectrum and are approximately constant elsewhere. The model reproduces very well the experimentally observed strain response and provides a good prediction of the response at other stress levels.