Inflation And Bifurcation Of A Spherical Membrane Subjected To Uniform Internal Pressure

The inflation and bifurcation of spherical membrane has received the attention of researchers at home and abroad because of its important theoretical significance and engineering background.It is well known that when a spherical membrane is inflated under the uniform internal pressure, initially the pressure increases as the principal stretch increases, then it becomes aspherical after the pressure maximum is reached. Upon further inflation, the spherical configuration is regained.Based on a kind of strain energy function proposed by Y. C. Gao, an incompressible spherical membrane subjected to uniform internal pressure is analyzed with the membrane assumption. Its inflation and bifurcation is given, and the influence of compressibility on inflation is also included. For an incompressible membrane, when the constitutive parameter n is less than 1.5, internal pressure has only a single minimum; when n is equal to 1 and a is equal toμ/2 (where μ is the shear modulus), the material becomes to the Neo-Hookeanmodel; when n is greater than 1.5, internal pressure increases monotonically with the principal stretch. For a compressible membrane, the explicit relationship between the internal pressure and principal stretch is derived by means of formulating the Poisson function for a biaxial tension. The influence of Poisson's ratio on the inflation of membrane is discussed. It is demonstrated that when the Poisson's ratio approaches to 0.5, the behavior of compressible membrane is the same as that of incompressible membrane.As for the bifurcation analysis, the general theory of small elastic deformations superposed on finite elastic deformations is introduced with the assumption that perturbation deformation is small compared with the finite deformation. A non-axisymmetric bifurcation mode is given. A mode which is consistent with the experimental results corresponds to local thickening in theupper hemisphere and thinning in the lower hemisphere. The corresponding bifurcation criteria are also deduced. Some bifurcation modes are excluded by reasonable derivation. The results in present paper may be used to predict the performance of a meteorological balloon during flight and they are also helpful in the production of inflatable gels as a theoretical reference.