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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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)
C. H. Wu, Tube to annulus—an exact nonlinear membrane solution, Quart. Appl. Math. 27, 489–496 (1970)
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)
A. D. Kydoniefs, Finite axisymmetric deformations of an initially cylindrical elastic membrane enclosing a rigid body, 22, 319–331 (1969)
A. C. Pipkin, Integration of an equation in membrane theory, ZAMP 19, 818–819 (1968)
H. O. Foster, Very large deformations of axially symmetric membranes made of neo-Hookean material Int. J. Eng. Sci. 5, 95–117 (1967)
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)
F. S. Wong, and R. T. Shield, Large plane deformations of thin elastic sheets of neo-Hookean material, SAMP 20, 176–199 (1969)
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C. H. Wu, Spherelike deformations of a balloon, Quart. Appl. Math. 30, 183–194 (1972)
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)
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)
C. H. Wu, Infinitely stretched Mooney surfaces of revolution are uniformly stressed catenoids, Quart. Appl. Math. 33, 273–284 (1974)
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L. J. Hart-Smith, Elasticity parameters for finite deformations of rubber-like materials, ZAMP 17, 608–626 (1966)
R. W. Dickey, Dynamic behavior of soap films, Quart. Appl. Math. 24, 97–106 (1966)
H. Hopf, Selected topics in differential geometry in the large, Courant Institute of Mathematical Sciences (1955)
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R. S. Rivlin and A. G. Thomas, Large elastic deformations of isotropic materials VIII. Strain distribution around a hole in a sheet, Phil. Trans. Roy. Soc. (London) A-243, 289–298 (1951)
J. E. Adkins and R. S. Rivlin, Large elastic deformations of isotropic materials IX. The deformation of thin shells, Phil. Trans. Roy. Soc. (London) A-244, 505–531 (1952)
A. E. Green and J. E. Adkins, Large elastic deformations and nonlinear continuum mechanics, Oxford University Press, London (1960)
A. H. Corneliussen and R. T. Shield, Finite deformation of elastic membranes with application to the stability of an inflated and extended tube, Arch. Rat. Mech. Anal. 7, 273–304 (1961)
J. J. Stoker, Topics in non-linear elasticity, Courant Institute of Mathematical Sciences (1964)
A, E. Green and W. Zerna, Theoretical elasticity, Oxford University Press, London (1954)
W. H. Yang, Stress concentration in a rubber sheet under axially symmetric stretching, J. Appl. Mech 34, 943–947 (1967)
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)
C. H. Wu, Tube to annulus—an exact nonlinear membrane solution, Quart. Appl. Math. 27, 489–496 (1970)
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)
A. D. Kydoniefs, Finite axisymmetric deformations of an initially cylindrical elastic membrane enclosing a rigid body, 22, 319–331 (1969)
A. C. Pipkin, Integration of an equation in membrane theory, ZAMP 19, 818–819 (1968)
H. O. Foster, Very large deformations of axially symmetric membranes made of neo-Hookean material Int. J. Eng. Sci. 5, 95–117 (1967)
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)
F. S. Wong, and R. T. Shield, Large plane deformations of thin elastic sheets of neo-Hookean material, SAMP 20, 176–199 (1969)
E. Isaacson, The shape of a balloon, Comm. Pure Appl. Math. 18, 163–166 (1965)
C. H. Wu, Spherelike deformations of a balloon, Quart. Appl. Math. 30, 183–194 (1972)
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)
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)
C. H. Wu, Infinitely stretched Mooney surfaces of revolution are uniformly stressed catenoids, Quart. Appl. Math. 33, 273–284 (1974)
F. John, Plane strain problems for a perfectly elastic material of harmonic type, Comm. Pure Appl. Math. 13, 239–296 (1960)
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)
L. J. Hart-Smith, Elasticity parameters for finite deformations of rubber-like materials, ZAMP 17, 608–626 (1966)
R. W. Dickey, Dynamic behavior of soap films, Quart. Appl. Math. 24, 97–106 (1966)
H. Hopf, Selected topics in differential geometry in the large, Courant Institute of Mathematical Sciences (1955)
J. E. Adkins, A reciprocal property of the finite plane strain equations, J. Mech. Phys. Solids 6, 267–275 (1958)
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Article copyright:
© Copyright 1979
American Mathematical Society
