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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 or , are found to exist.

**[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

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
73G05

Retrieve articles in all journals with MSC: 73G05

Additional Information

DOI:
https://doi.org/10.1090/qam/520120

Article copyright:
© Copyright 1979
American Mathematical Society