Buckling and barrelling instabilities on non-linearly elastic columns
Author:
Penny J. Davies
Journal:
Quart. Appl. Math. 49 (1991), 407-426
MSC:
Primary 73H05; Secondary 73G05, 73V25
DOI:
https://doi.org/10.1090/qam/1121674
MathSciNet review:
MR1121674
Full-text PDF Free Access
References |
Similar Articles |
Additional Information
- Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI https://doi.org/10.1002/cpa.3160120405
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. MR 162050, DOI https://doi.org/10.1002/cpa.3160170104
- Gilles Aubert and Rabah Tahraoui, Sur la faible fermeture de certains ensembles de contraintes en élasticité non-linéaire plane, Arch. Rational Mech. Anal. 97 (1987), no. 1, 33–58 (French). MR 856308, DOI https://doi.org/10.1007/BF00279845
- J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. London Ser. A 306 (1982), no. 1496, 557–611. MR 703623, DOI https://doi.org/10.1098/rsta.1982.0095
- J. M. Ball, Differentiability properties of symmetric and isotropic functions, Duke Math. J. 51 (1984), no. 3, 699–728. MR 757959, DOI https://doi.org/10.1215/S0012-7094-84-05134-2
M. F. Beatty and P. Dadras, Some experiments on the elastic stability of some highly elastic bodies, Internat. J. Engrg. Sci. 14, 233–238 (1976)
M. F. Beatty and D. E. Hook, Some experiments on the stability of circular rubber bars under end thrust, Internat. J. Solids and Stuctures 4 623–635 (1968)
- P. J. Davies, Buckling and barrelling instabilities in finite elasticity, J. Elasticity 21 (1989), no. 2, 147–192. MR 1002449, DOI https://doi.org/10.1007/BF00040894
- Penny J. Davies, A simple derivation of necessary and sufficient conditions for the strong ellipticity of isotropic hyperelastic materials in plane strain, J. Elasticity 26 (1991), no. 3, 291–296. MR 1135480, DOI https://doi.org/10.1007/BF00041893
L. HÖrmander, On the regularity of the solutions of boundary problems, Acta. Math. 99, 225–264 (1958)
- Léon Van Hove, Sur le signe de la variation seconde des intégrales multiples à plusieurs fonctions inconnues, Acad. Roy. Belg. Cl. Sci. Mém. Collect. 8$^{\rm o}$ (2) 24 (1949), no. 5, 68 (French). MR 35406
- J. K. Knowles and Eli Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rational Mech. Anal. 63 (1976), no. 4, 321–336 (1977). MR 431861, DOI https://doi.org/10.1007/BF00279991
- J. K. Knowles and Eli Sternberg, On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics, J. Elasticity 8 (1978), no. 4, 329–379. MR 516599, DOI https://doi.org/10.1007/BF00049187
M. Levinson, Stability of a compressed neo-Hookean rectangular parallelepiped, J. Mech. Phys. Solids 16, 403–415 (1968)
- R. W. Ogden, Nonlinear elastic deformations, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1984. MR 770388
- Henry C. Simpson and Scott J. Spector, On barrelling instabilities in finite elasticity, J. Elasticity 14 (1984), no. 2, 103–125. MR 747056, DOI https://doi.org/10.1007/BF00041660
- Henry C. Simpson and Scott J. Spector, Some monotonicity results for ratios of modified Bessel functions, Quart. Appl. Math. 42 (1984), no. 1, 95–98. MR 736509, DOI https://doi.org/10.1090/S0033-569X-1984-0736509-2
- Henry C. Simpson and Scott J. Spector, On barrelling for a special material in finite elasticity, Quart. Appl. Math. 42 (1984), no. 1, 99–111. MR 736510, DOI https://doi.org/10.1090/S0033-569X-1984-0736510-2
- Henry C. Simpson and Scott J. Spector, On the positivity of the second variation in finite elasticity, Arch. Rational Mech. Anal. 98 (1987), no. 1, 1–30. MR 866722, DOI https://doi.org/10.1007/BF00279960
H. C. Simpson and S. J. Spector, On bifurcation in finite elasticity: buckling of a rectangular rod, written communication
- G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975
S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12, 623–727 (1959)
S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17, 35–92 (1964)
G. Aubert and R. Tahraoui, Sur la faible fermeture de certains ensembles de contraintes an elasticite non-lineaire plane, Arch. Rational Mech. Anal. 97, 33–58 (1987)
J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. Lond. Ser. A 306, 557–611 (1982)
J. M. Ball, Differentiability properties of symmetric and isotropic functions, Duke Math. J. 51, 699–728 (1984)
M. F. Beatty and P. Dadras, Some experiments on the elastic stability of some highly elastic bodies, Internat. J. Engrg. Sci. 14, 233–238 (1976)
M. F. Beatty and D. E. Hook, Some experiments on the stability of circular rubber bars under end thrust, Internat. J. Solids and Stuctures 4 623–635 (1968)
P. J. Davies, Buckling and barrelling instabilities in finite elasticity, J. Elasticity 21, 147–192 (1989)
P. J. Davies, A simple derivation of necessary and sufficient conditions for the strong ellipticity of isotropic hyperelastic materials in plane strain, to appear in J. Elasticity
L. HÖrmander, On the regularity of the solutions of boundary problems, Acta. Math. 99, 225–264 (1958)
L. van Hove, Sur la signe de la variation seconde des integrales multiple a plusieurs fonctions inconnues, Mem. Acad. Roy. Belg. (5) 24 (1949)
J. K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rational Mech. Anal. 63, 321–336 (1977)
J. K. Knowles and E. Sternberg, On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics, J. Elasticity 8, 329–379 (1978)
M. Levinson, Stability of a compressed neo-Hookean rectangular parallelepiped, J. Mech. Phys. Solids 16, 403–415 (1968)
R. W. Ogden, Non-linear Elastic Deformations, Ellis-Horwood Series: Mathematics and its Applications, Ellis-Horwood Ltd., Chichester, 1984
H. C. Simpson and S. J. Spector, On barrelling instabilities in finite elasticity, J. Elasticity 14, 103–125 (1984)
H. C. Simpson and S. J. Spector, Some monotonicity results for ratios of modified Bessel functions, Quart. Appl. Math. 42, 95–98 (1984)
H. C. Simpson and S. J. Spector, On barrelling for a special material in finite elasticity, Quart. Appl. Math. 42, 99–111 (1984)
H. C. Simpson and S. J. Spector, On the positivity of the second variation in finite elasticity, Arch. Rational Mech. Anal. 98, 1–94 (1987)
H. C. Simpson and S. J. Spector, On bifurcation in finite elasticity: buckling of a rectangular rod, written communication
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, London, 1922
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
73H05,
73G05,
73V25
Retrieve articles in all journals
with MSC:
73H05,
73G05,
73V25
Additional Information
Article copyright:
© Copyright 1991
American Mathematical Society