Multi-lobed inflated membranes: Their stability under finite deformation

Abstract The stability of inflated membraneous spheres of rubber-like materials is examined theoretically and stability criteria are confirmed by experiment. A negative pressure-volume gradient is found to be a critical condition for instability. Another condition derives from geometric considerations and usually involves a critical volume enclosed within the entire system. These two conditions together determine the stability of any multi-lobed inflatable membraneous structure, regardless of material. Equilibrium states of two and three inter-connected spherical balloons, whether identical or not, are presented with corresponding stability diagrams and the inflation history of such configurations is adduced. The scope of poly-stable states is described and confirmed experimentally. Specific results are presented for rubber-like materials that are described by the exponential-hyperbolic elasticity parameters developed elsewhere [1]. Inadequacies of other theories [3–11] are traced either to improper elastic representations or to the neglect of one of the dual conditions for stability.

[1]  P. D. Soden,et al.  The discharge characteristics of confined rubber cylinders , 1966 .

[2]  A. C. Burton.  On the physical equilibrium of small blood vessels. , 1951, The American journal of physiology.

[3]  L. J. Pollock,et al.  CEREBROSPINAL FLUID PRESSURE , 1936 .

[4]  R. Kubo.  Large Elastic Deformation of Rubber , 1948 .

[5]  R. T. Shield,et al.  Finite deformation of elastic membranes with application to the stability of an inflated and extended tube , 1961 .

[6]  F. H. Newman,et al.  The general properties of matter , 1929 .

[7]  D'arcy W. Thompson.  On growth and form i , 1943 .

[8]  J. Plateau,et al.  Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires , 1873 .

[9]  A. C. Burton.  Relation of structure to function of the tissues of the wall of blood vessels. , 1954, Physiological reviews.

[10]  L. J. Hart-Smith,et al.  Elasticity parameters for finite deformations of rubber-like materials , 1966 .

[11]  A. J. Carmichael,et al.  Phenomenological Elastomechanical Behavior of Rubbers over Wide Ranges of Strain , 1961 .

[12]  William Sutherland,et al.  The Elasticity of Rubber Balloons and Hollow Viscera , 1909 .

[13]  The Application of the Principle of Stationary Potential Energy to Some Problems in Finite Elasticity , 1965 .

[14]  Ch. Delaunay,et al.  Sur la surface de révolution dont la courbure moyenne est constante. , 1841 .

[15]  L. Hart-Smith,et al.  Large elastic deformations of thin rubber membranes , 1967 .