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Inhomogeneous spherical configurations of inflated membranes

Abstract

Based on physically meaningful choice of the strain measures, we study the equilibrium and stability of an inflated spherical membrane. First, we obtain general results deduced by global geometric properties and then we analyze the possibility of inhomogeneous configurations. The stability analysis shows that under special constitutive assumptions the global energy minimum can be attained by inhomogeneous spherical configurations that we analytically describe. We argue that these deformations can reproduce well-known experimental results.

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References

  1. Alexander H.: Tensile instability of initially spherical balloons. Int. J. Eng. Sci. 9, 151–162 (1971)

    Article  Google Scholar 

  2. Batra R.C.: Instabilities in biaxially loaded rectangular membranes and spherical balloons made of compressible isotropic hyperelastic materials. Math. Mech. Sol. 10, 471–485 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  3. Beatty F.M., Johnson M.A.: The Mullins effect in equibiaxial extension and its influence on the inflation of a balloon. Int. J. Eng. Sci. 33(2), 223–245 (1995)

    MATH  Article  Google Scholar 

  4. Chen Y.C., Healey T.: Bifurcation to pear-shaped equilibria of pressurized spherical membranes. Int. J. Nonlinear Mech. 26, 279–291 (1991)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  5. D’Ambrosio P., De Tommasi D., Ferri D., Puglisi G.: A phenomenological model for healing and hysteresis in rubber-like materials. J. Eng. Sci. 46(4), 293–305 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  6. Deseri L., Piccioni M.D., Zurlo G.: Derivation of a new free energy for biological membranes. Continum Mech. Thermodyn. 20(5), 255–273 (2008)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  7. De Tommasi D., Puglisi G., Saccomandi G.: A micromechanics based model for the Mullins effect. J. Rheol. 50, 495–512 (2006)

    ADS  Article  Google Scholar 

  8. De Tommasi D., Marzano S., Puglisi G., Zurlo G.: Damage and healing effects in rubber-like balloons. Int. J. Solids Struct. 46(22–23), 3999–4005 (2009)

    MATH  Article  Google Scholar 

  9. De Tommasi, D., Puglisi, G., Zurlo, G.: A note on strong ellipticity in two-dimensional isotropic elasticity. J. Elast. (2012). doi:10.1007/s10659-011-9370-1. Published online

  10. De Tommasi D., Puglisi G., Saccomandi G.: Localized versus diffuse damage in amorphous materials. Phys. Rev. Lett. 100, 085502 (2008)

    ADS  Article  Google Scholar 

  11. Do Carmo M.: Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs, NJ (1976)

    MATH  Google Scholar 

  12. Ericksen J.L.: Introduction to the Thermodynamics of Solids. Chapman & Hall, London (1991)

    MATH  Google Scholar 

  13. Guillemin V., Pollack A.: Differential Topology. Prentice Hall, Englewood Cliffs, NJ (1974)

    MATH  Google Scholar 

  14. Gurtin M.E., Murdoch I.A.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)

    MathSciNet  MATH  Article  Google Scholar 

  15. Haughton D.M.: Post-bifurcation of perfect and imperfect spherical elastic membranes. Int. J. Solids Struct. 16, 1123–1133 (1980)

    MATH  Article  Google Scholar 

  16. Knowles J.K., Sternberg E.: On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch. Ration. Mech. Anal. 63(4), 321–336 (1976)

    MathSciNet  Article  Google Scholar 

  17. Molzon R., Man C.S.: Residual stress in membranes. J. Elast. 20, 181–202 (1988)

    MathSciNet  MATH  Article  Google Scholar 

  18. Müller, I., Strehlow, P.: Rubber and rubber balloons: paradigms of thermodynamics. In: Lecture Notes in Physics. Springer, Berlin, Heidelberg, GmbH and Co. K (2004)

  19. Pagitz M.: The future of scientific ballooning. Philos. Trans. R. Soc. A 365(1861), 3003–3017 (2007)

    ADS  Article  Google Scholar 

  20. Rudykha, S., Bhattacharyac, K., de Botton, G.: Snap-through actuation of thick-wall electroactive balloons. Int. J. Nonlinear Mech. (2011, in press). doi:10.1016/j.ijnonlinmec.2011.05.006

  21. Sewell M.J: Mathematics Masterclasses: Stretching the Imagination. Oxford University Press, Oxford (1997)

    MATH  Google Scholar 

  22. Truesdell C., Noll W.: The Non-linear Field Theories of Mechanics, Handbuch der Physik, Band III/3. Springer, Berlin (1965)

    Google Scholar 

  23. Tsunoda, H., Senbokuya, Y.: Rigidizable membranes for spaceinflatable structures, vol. 1367. American Institute of Aeronautics and Astronautics, Reston, VA (2002)

  24. Verron E., Marckmann G.: Numerical analysis of rubber balloons. Thin Walled Struct. 41, 731–746 (2003)

    Article  Google Scholar 

  25. Yoda M., Konishi S.: Acoustic impedance control through structural tuning by pneumatic balloon actuators. Sens. Act. A 95, 222–226 (2002)

    Article  Google Scholar 

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Correspondence to D. De Tommasi.

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Communicated by Francesco Dell'Isola and Samuel Forest.

This work is dedicated to Professor Gianpietro Del Piero in occasion of his retirement, with deep gratitude and friendship.

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Tommasi, D.D., Puglisi, G. & Zurlo, G. Inhomogeneous spherical configurations of inflated membranes. Continuum Mech. Thermodyn. 25, 197–206 (2013). https://doi.org/10.1007/s00161-012-0240-2

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  • DOI: https://doi.org/10.1007/s00161-012-0240-2

Keywords

  • Membranes inflation
  • Nonlinear elasticity
  • Ballooning instability
  • Energy minimization