Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Large finite strain membrane problems

Author: Chien H. Wu
Journal: Quart. Appl. Math. 36 (1979), 347-359
MSC: Primary 73G05
DOI: https://doi.org/10.1090/qam/520120
MathSciNet review: 520120
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Abstract: Nonlinear membrane problems involving large finite strains are considered. It is found that explicit asymptotic solutions are possible for a rather large class of problems. Two distinct types of asymptotic solutions, roughly depending on whether the strain energy density function is dominated by $ {I_1}$ or $ {I_2}$, are found to exist.

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  • [1] R. S. Rivlin and A. G. Thomas, Large elastic deformations of isotropic materials. VIII. Strain distribution around a hole in a sheet, Philos. Trans. Roy. Soc. London. Ser. A. 243 (1951), 289–298. MR 0040942, https://doi.org/10.1098/rsta.1951.0005
  • [2] J. E. Adkins and R. S. Rivlin, Large elastic deformations of isotropic materials. IX. The deformation of thin shells, Philos. Trans. Roy. Soc. London. Ser. A. 244 (1952), 505–531. MR 0049761, https://doi.org/10.1098/rsta.1952.0013
  • [3] A. E. Green and J. E. Adkins, Large elastic deformations, Second edition, revised by A. E. Green, Clarendon Press, Oxford, 1970. MR 0269158
  • [4] A. H. Corneliussen and R. T. Shield, Finite deformation of elastic membranes with application to the stability of an inflated and extended tube, Arch. Rational Mech. Anal. 7 (1961), 273–304. MR 0126993, https://doi.org/10.1007/BF00250766
  • [5] J. J. Stoker, Topics in non-linear elasticity, Courant Institute of Mathematical Sciences (1964)
  • [6] A. E. Green and W. Zerna, Theoretical elasticity, Oxford, at the Clarendon Press, 1954. MR 0064598
  • [7] W. H. Yang, Stress concentration in a rubber sheet under axially symmetric stretching, J. Appl. Mech 34, 943-947 (1967)
  • [8] E. Varley and E. Cumberbatch, The finite deformation of an elastic material surrounding an elliptical hole, in Symposium on finite elasticity theory, AMD-Vol. 27 (1977)
  • [9] C. H. Wu, Tube to annulus--an exact nonlinear membrane solution, Quart. Appl. Math. 27, 489-496 (1970)
  • [10] A. D. Kydoniefs and A. J. M. Spencer, Finite axisymmetric deformations of an initially cylindrical elastic membrane, Quart. J. Mech. Appl. Math. 22, 87-95 (1969)
  • [11] A. D. Kydoniefs, Finite axisymmetric deformations of an initially cylindrical elastic membrane enclosing a rigid body, 22, 319-331 (1969)
  • [12] A. C. Pipkin, Integration of an equation in membrane theory, ZAMP 19, 818-819 (1968)
  • [13] H. O. Foster, Very large deformations of axially symmetric membranes made of neo-Hookean material Int. J. Eng. Sci. 5, 95-117 (1967)
  • [14] L. K. Yu and K. C. Valanis, The inflation of axially symmetric membranes by linearly varying hydrostatic pressure, Trans. Soc. Rheology 14, 159-183 (1970)
  • [15] F. S. Wong, and R. T. Shield, Large plane deformations of thin elastic sheets of neo-Hookean material, SAMP 20, 176-199 (1969)
  • [16] Eugene Isaacson, The shape of a balloon, Comm. Pure Appl. Math. 18 (1965), 163–166. MR 0175383, https://doi.org/10.1002/cpa.3160180115
  • [17] C. H. Wu, Spherelike deformations of a balloon, Quart. Appl. Math. 30, 183-194 (1972)
  • [18] C. H. Wu and D. Y. P. Perng, On the asymptotically spherical deformations of arbitrary membranes of revolution fixed along an edge and inflated by large pressure--a nonlinear boundary layer phenomenon, SIAM J. Appl. Math. 23, 133-152 (1972)
  • [19] D. Y. P. Perng and C. H. Wu, Flattening of membranes of revolution by large stretching--asymptotic solution with boundary layer, Quart. Appl. Math. 32, 407-420 (1973)
  • [20] C. H. Wu, Infinitely stretched Mooney surfaces of revolution are uniformly stressed catenoids, Quart. Appl. Math. 33, 273-284 (1974)
  • [21] Fritz John, Plane strain problems for a perfectly elastic material of harmonic type, Comm. Pure Appl. Math. 13 (1960), 239–296. MR 0118022, https://doi.org/10.1002/cpa.3160130206
  • [22] P. J. Blatz and W. L. Ko, Application of finite elastic theory to the deformation of rubbery materials, Trans. Soc. Rheology 6, 223-251 (1962)
  • [23] L. J. Hart-Smith, Elasticity parameters for finite deformations of rubber-like materials, ZAMP 17, 608-626 (1966)
  • [24] R. W. Dickey, Dynamic behavior of soap films, Quart. Appl. Math. 24, 97-106 (1966)
  • [25] H. Hopf, Selected topics in differential geometry in the large, Courant Institute of Mathematical Sciences (1955)
  • [26] J. E. Adkins, A reciprocal property of the finite plane strain equations, J. Mech. Phys. Solids 6 (1958), 267–275. MR 0093972, https://doi.org/10.1016/0022-5096(58)90002-4

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DOI: https://doi.org/10.1090/qam/520120
Article copyright: © Copyright 1979 American Mathematical Society

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