Solutions for an infinite compressible nonlinearly elastic body under a line load
Authors:
Paul G. Warne and Debra A. Polignone
Journal:
Quart. Appl. Math. 54 (1996), 317-326
MSC:
Primary 73G05; Secondary 73C50
DOI:
https://doi.org/10.1090/qam/1388019
MathSciNet review:
MR1388019
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Abstract: The axisymmetric deformation of a nonlinearly elastic isotropic compressible infinite elastic body subjected to a concentrated vertical line load is considered. We first derive the solution to this problem within the context of the linear theory of elasticity. We then obtain the governing equations for the nonlinear problem via the Principle of Stationary Potential Energy, and use these equations to obtain classes of compressible finite elasticity solutions for the line load problem. Finally, a comparison with finite anti-plane shear of compressible isotropic materials is made.
S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, third edition, McGraw-Hill, Inc., New York, 1970
R. W. Little, Elasticity, Prentice-Hall, Inc., 1973
I. S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw-Hill, Inc., New York, 1956
J. L. Sanders, Note on the Mindlin problem, Mechanics of Material Behavior, Elsevier, New York, 1984, pp. 345–349
P. G. Warne, Ph. D. Dissertation, The Nonlinearly Elastic Boussinesq Problem and Lie Groups, University of Virginia, Charlottesville, VA, May 1993
J. G. Simmonds and P. G. Warne, Notes on the nonlinearly elastic Boussinesq problem, Journal of Elasticity 34, 69–82 (1994)
J. G. Simmonds, A necessary condition on the string-energy density for a circular, rubber-like plate to have a finite deflection under a concentrated load, J. Appl. Mech. 56, 484–486 (1989)
J. G. Simmonds and M. A. Horn, Asymptotic analysis of the nonlinear equations for an infinite, rubber-like slab under an equilibrated vertical line load, J. Elasticity 24, 105–127 (1990)
D. A. Polignone and C. O. Horgan, Axisymmetric finite anti-plane shear of compressible nonlinearly elastic circular tubes, Quart. Appl. Math. 50, 323–341 (1992)
A. E. Green and W. Zerna, Theoretical Elasticity, second edition, Oxford, 1968
J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63, 337–403 (1977)
R. W. Ogden, Non-Linear Elastic Deformations, Ellis Horwood Limited, England, 1984
S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, third edition, McGraw-Hill, Inc., New York, 1970
R. W. Little, Elasticity, Prentice-Hall, Inc., 1973
I. S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw-Hill, Inc., New York, 1956
J. L. Sanders, Note on the Mindlin problem, Mechanics of Material Behavior, Elsevier, New York, 1984, pp. 345–349
P. G. Warne, Ph. D. Dissertation, The Nonlinearly Elastic Boussinesq Problem and Lie Groups, University of Virginia, Charlottesville, VA, May 1993
J. G. Simmonds and P. G. Warne, Notes on the nonlinearly elastic Boussinesq problem, Journal of Elasticity 34, 69–82 (1994)
J. G. Simmonds, A necessary condition on the string-energy density for a circular, rubber-like plate to have a finite deflection under a concentrated load, J. Appl. Mech. 56, 484–486 (1989)
J. G. Simmonds and M. A. Horn, Asymptotic analysis of the nonlinear equations for an infinite, rubber-like slab under an equilibrated vertical line load, J. Elasticity 24, 105–127 (1990)
D. A. Polignone and C. O. Horgan, Axisymmetric finite anti-plane shear of compressible nonlinearly elastic circular tubes, Quart. Appl. Math. 50, 323–341 (1992)
A. E. Green and W. Zerna, Theoretical Elasticity, second edition, Oxford, 1968
J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63, 337–403 (1977)
R. W. Ogden, Non-Linear Elastic Deformations, Ellis Horwood Limited, England, 1984
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© Copyright 1996
American Mathematical Society
