Abstract
This paper studies the inflation and interaction mechanics of a flat circular membrane inside an elastic cone under the action of uniform gas pressure. The membrane is assumed to be a homogeneous and isotropic Mooney–Rivlin hyperelastic material, while the conical surface is taken to be a distributed linear stiffness in the direction normal to the undeformed surface. The set of coupled second order nonlinear ordinary differential equations that governs the constrained inflation mechanics is reduced to a set of four first order ordinary differential equations by change of variables. A two dimensional grid search technique using the bisection method is employed to determine the equilibrium configuration of the inflated membrane. The principal stretches and curvatures have been obtained which exhibit some interesting trends. It is observed that the limit point instability can completely disappear (even in the case of neo-Hookean membrane material model) when an inflating membrane interacts with a constraining surface. Most remarkably, pre-stretching the membrane can revive the occurrence of the limit point instability in certain cases leading to a softening behavior. This counterintuitive effect appears to be a shadow of the stretch induced softening behavior observed recently in literature.
Similar content being viewed by others
References
Adkins JE, Rivlin RS (1952) Large elastic deformation of isotropic materials. ix. Philos Trans R Soc Ser A 244:505–531
Antonio JG, Bonet J (2006) Finite element analysis of prestressed structural membranes. Finite Elem Anal Des 42:683–697
Arruda E, Boyce MC (1993) A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J Mech Phys Solids 41(2):389–412
Campbell JD (1956) On the theory of initially tensioned circular membranes subjected to uniform pressure. Q J Mech Appl Math 9(1):84–93
Charrier JM, Shrivastava S, Wu R (1987) Free and constrained inflation of elastic membranes in relation to thermoforming-axisymmetric problems. J Strain Anal Eng 22(2):115–125
Charrier JM, Shrivastava S, Wu R (1989) Free and constrained inflation of elastic membranes in relation to thermoforming-non-axisymmetric problems. J Strain Anal Eng 24(2):55–74
Christensen RM, Feng WW (1986) Nonlinear analysis of the inflation of an initially flat, circular, elastic disk. J Rheol 30:157–165
Eriksson A, Nordmark A (2012) Instability of hyper-elastic balloon-shaped space membranes under pressure loads. J Comput Methods Appl Mech Eng 237:118–129
Evans E, Needham D (1987) Physical properties of surfactant bilayer membranes: thermal transitions, elasticity, rigidity, cohesion, and colloidal interactions. J Phys Chem 91:4219–4228
Feng WW, Huang P (1974) On the general contact problem of an inflated nonlinear plane membrane. Int J Solids Struct 11:437–448
Feng WW, Huang P (1974) On the inflation problem of a plane nonlinear membrane. J Appl Mech 40:9–12
Feng WW, Yang WH (1973) On the contact problem of an inflated spherical nonlinear membrane. J Appl Mech 40:209–214
Foster HO (1967) Inflation of a plane circular membrane. J Eng Ind 89:403–407
Foster HO (1967) Very large deformations of axially symmetrical membranes made of neo-Hookean materials. Int J Eng Sci 5:95–117
Fung YC (ed) (1990) Biomechanics: motion, flow, stress, and growth. Springer, New York
Goncalves PB, Soares RM, Pamplona D (2009) Nonlinear vibrations of a radially circular hyperelastic membrane. J Sound Vib 327:231–248
Green AE, Adkins JE (1970) Large elastic deformation. Oxford University Press, London
Hart-Smith LJ, Crisp JDC (1967) Large elastic deformations of thin rubber membranes. Int J Eng Sci 5(1):1–24
Holzapfel GA, Eberlein R, Wriggers P, Weizsacker HW (1996) Large strain analysis of soft biological membranes:formulation and finite element analysis. J Comput Methods Appl Mech Eng 132:45–61
Humphrey JD (ed) (2002) Cardiovascular solid mechanics: cells, tissues and organs. Springer, New York
Hung ND, Saxce G (1980) Frictionless contact of elastic bodies by finite element method and mathematical programming. Comput Struct 11:5567
Jenkins CHM (ed) (2001) Gossamer spacecraft: membrane and inflatable structures technology for space applications, vol 191. American Institute of Aeronautics and Astronautics Inc., Reston
Klingbeil WW, Shield RT (1964) Some numerical investigation on empirical strain energy functions in the large axisymmetric extension of rubber membranes. J Appl Math Phys 15:608–629
Kumar N, DasGupta A (2013) On the contact problem of an inflated spherical hyperelastic membrane. Int J Non-Linear Mech 57:130–139
Kumar N, DasGupta A (2014) Contact mechanics and induced hysteresis at oscillatory contacts with adhesion. Langmuir 30:9107–9114
Khayat RE, Derdouri A (1994) Inflation of hyperelastic cylindrical membranes as application to blow molding: Part I—Axisymmetric case. J Numer Methods Eng 37(22):3773–3791
Lardner T, Pujara P (1980) Compression of spherical cells. Mech. Today 5:161–176
McGarry GJ, Prendergast PJ (2004) A three dimensional finite element model of an adherent eukaryotic cell. J Eur Cell Mater 7:27–34
Mooney M (1940) A theory of large elastic deformation. J Appl Phys 11(9):582–592
Nadler B (2010) On the contact of spherical membrane enclosing a fluid with rigid parallel planes. J Non-linear Mech 45(3):294–300
Needleman A (1977) Inflation of spherical rubber balloons. Int J Solids Struct 13:409–421
Ogden RW (1972) Large deformation isotropic elasticity: on the correlation of theory and experimental for compressible rubber like solids. Philos Trans R Soc Ser A 326(1567):567–583
Ogden RW (1997) Non-linear elastic deformations. Dover, New York
Patil A, DasGupta A (2013) Finite inflation of an initially stretched hyperelastic circular membrane. Eur J Mech A Solids 41:28–36
Patil A, Nordmark A, Eriksson A (2014) Free and constrained inflation of a pre-stretched cylindrical membrane. Proc R Soc A 470, No. 20140282
Pujara P, Lardner TJ (1978) Deformation of elastic membranes: effect of different constitutive relations. J Appl Math Phys 29:315–327
Rao PVM, Dhande SG (1999) Deformation analysis of thin elastic membranes in multiple contact. Adv Eng Softw 30:177–183
Rivlin RS (1948) Large elastic deformations of isotropic materials. i. Fundamental concepts. Philos Trans R Soc Ser A 240(822):459–490
Schweizerhof K, Rumpel T, Habler M (2005) Efficient finite element modelling and simulation of gas and fluid supported membrane and shell structures. Comput Methods Appl Sci 3:153–172
Shrivastava S, Tang J (1993) Large deformation finite element analysis of non-linear viscoelastic membranes with reference to thermoforming. J Strain Anal Eng 28(1):115–125
Tamadapu G, DasGupta A (2012) In-plane surface modes of an elastic toroidal membrane. Int J Eng Sci 60:25–36
Tamadapu G, DasGupta A (2013) Finite inflation analysis of a hyperelastic toroidal membrane of initially circular cross-section. Int J Non-Linear Mech 49:31–39
Tamadapu G, DasGupta A (2013) In-plane dynamics of membranes having constant curvature. Eur J Mech A Solids 39:280–290
Tielking JT, Feng WW (1974) The application of the minimum potential energy principle to nonlinear axisymmetric membrane problems. J Appl Mech 41:491–496
Wong FS, Shield RT (1969) Large plane deformations of thin elastic sheets of neo-Hookean material. J Appl Math Phys 20(2):176–199
Yang WH, Feng WW (1970) On axisymmetrical deformations of nonlinear membranes. J Appl Mech 37(4):1002–1011
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Patil, A., DasGupta, A. Constrained inflation of a stretched hyperelastic membrane inside an elastic cone. Meccanica 50, 1495–1508 (2015). https://doi.org/10.1007/s11012-015-0102-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-015-0102-7