Some estimates for the maximum shear stress in plane, isotropic elasticity
Author:
M. Aron
Journal:
Quart. Appl. Math. 52 (1994), 545-551
MSC:
Primary 73C50; Secondary 73G05
DOI:
https://doi.org/10.1090/qam/1292204
MathSciNet review:
MR1292204
Full-text PDF Free Access
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Abstract: Upper and lower bounds for the maximum shear stress in a configuration corresponding to a purely distortional deformation originating from a given undistorted (ground) state are obtained in the framework of plane, isotropic, nonlinear elasticity. The bounds are shown to be expressible in terms of the deformation and the boundary traction that is required for maintaining the purely dilatational deformation in the ground state.
P. J. Flory, Thermodynamic relations for high elastic materials, Trans. Faraday Soc. 57, 829–838 (1961)
R. W. Ogden, Nearly isochoric elastic deformations: volume changes in plane strain, Quart. Appl. Math. 36, 337–345 (1979)
M. Aron, On a minimum property in nonlinear elasticity, Internat J. Engng. Sci. 29, 1471–1478 (1991)
M. Baker and J. L. Ericksen, Inequalities restricting the form of the stress-deformation relations for isotropic elastic solids and Reiner-Rivlin fluids, J. Wash. Acad. Sci. 44, 33–35 (1954)
R. W. Ogden, On non-uniqueness in the traction boundary-value problem for a compressible elastic solid, Quart. Appl. Math. 42, 337–344 (1984)
F. John, Plane strain problems for a perfectly elastic material of harmonic type, Comm. Pure Appl. Math. 13, 239–296 (1960)
S. C. Hunter, Mechanics of continuous media, Ellis Horwood, Chichester, 1976
R. W. Ogden, Non-linear elastic deformations, Ellis Horwood, Chichester, 1984
M. Aron, A note on undistorted states of isotropic elastic solids, J. Elasticity 19, 179–187 (1988)
C. Truesdell and W. Noll, The Non-linear field theories of mechanics, Encyclopedia of Physics III/3, Springer, Berlin, Heidelberg, and New York, 1965
E. Varley and A. Day, Equilibrium phases of elastic materials at uniform temperature and pressure, Arch. Rational Mech. Anal. 22, 253–269 (1966)
P. Podio-Guidugli, G. Vergara Caffarelli, and E. G. Virga, Cavitation and phase transitions of hyperelastic fluids, Arch. Rational. Mech. Anal. 92, 121–136 (1986)
J. L. Ericksen, Deformations possible in every compressible isotropic, perfectly elastic material, J. Math. Phys. 34, 126–128 (1955)
M. F. Beatty, Stability of hyperelastic bodies subject to hydrostatic loading, Internat. J. Non-Linear Mech. 5, 367–383 (1970)
J. K. Knowles and S. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rational Mech. Anal. 63, 321–335 (1977)
J. K. Knowles and S. Sternberg, On the singularity induced by certain mixed boundary conditions in linearized and nonlinear elastostatics, Internat. J. Solids Structures 11, 1173–1201 (1975)
E. Varley and E. Cumberbatch, Finite deformations of elastic materials surrounding cylindrical holes, J. Elasticity 10, 341–405 (1980)
A. H. Jafari, R. Abeyaratne, and C. O. Horgan, The finite deformation of a pressurized circular tube for a class of compressible materials, J. Appl. Math. Phys. (ZAMP) 35, 227–246 (1984)
D. J. Steigmann and A. C. Pipkin, Stability of harmonic materials in plane strain, Quart. Appl. Math. 46, 559–568 (1988)
P. J. Flory, Thermodynamic relations for high elastic materials, Trans. Faraday Soc. 57, 829–838 (1961)
R. W. Ogden, Nearly isochoric elastic deformations: volume changes in plane strain, Quart. Appl. Math. 36, 337–345 (1979)
M. Aron, On a minimum property in nonlinear elasticity, Internat J. Engng. Sci. 29, 1471–1478 (1991)
M. Baker and J. L. Ericksen, Inequalities restricting the form of the stress-deformation relations for isotropic elastic solids and Reiner-Rivlin fluids, J. Wash. Acad. Sci. 44, 33–35 (1954)
R. W. Ogden, On non-uniqueness in the traction boundary-value problem for a compressible elastic solid, Quart. Appl. Math. 42, 337–344 (1984)
F. John, Plane strain problems for a perfectly elastic material of harmonic type, Comm. Pure Appl. Math. 13, 239–296 (1960)
S. C. Hunter, Mechanics of continuous media, Ellis Horwood, Chichester, 1976
R. W. Ogden, Non-linear elastic deformations, Ellis Horwood, Chichester, 1984
M. Aron, A note on undistorted states of isotropic elastic solids, J. Elasticity 19, 179–187 (1988)
C. Truesdell and W. Noll, The Non-linear field theories of mechanics, Encyclopedia of Physics III/3, Springer, Berlin, Heidelberg, and New York, 1965
E. Varley and A. Day, Equilibrium phases of elastic materials at uniform temperature and pressure, Arch. Rational Mech. Anal. 22, 253–269 (1966)
P. Podio-Guidugli, G. Vergara Caffarelli, and E. G. Virga, Cavitation and phase transitions of hyperelastic fluids, Arch. Rational. Mech. Anal. 92, 121–136 (1986)
J. L. Ericksen, Deformations possible in every compressible isotropic, perfectly elastic material, J. Math. Phys. 34, 126–128 (1955)
M. F. Beatty, Stability of hyperelastic bodies subject to hydrostatic loading, Internat. J. Non-Linear Mech. 5, 367–383 (1970)
J. K. Knowles and S. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rational Mech. Anal. 63, 321–335 (1977)
J. K. Knowles and S. Sternberg, On the singularity induced by certain mixed boundary conditions in linearized and nonlinear elastostatics, Internat. J. Solids Structures 11, 1173–1201 (1975)
E. Varley and E. Cumberbatch, Finite deformations of elastic materials surrounding cylindrical holes, J. Elasticity 10, 341–405 (1980)
A. H. Jafari, R. Abeyaratne, and C. O. Horgan, The finite deformation of a pressurized circular tube for a class of compressible materials, J. Appl. Math. Phys. (ZAMP) 35, 227–246 (1984)
D. J. Steigmann and A. C. Pipkin, Stability of harmonic materials in plane strain, Quart. Appl. Math. 46, 559–568 (1988)
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Article copyright:
© Copyright 1994
American Mathematical Society
