Abstract
The assumption of incompressibility brings about a considerable simplification in finite elastostatics for homogeneous isotropic solids.1 Most notably, it affords a set of closed form solutions, called controllable or universal deformations, which are independent of material properties and thus are possible in all incompressible materials. Solutions of this type were first discovered by Rivlin [1–4] and later by others [5–9]. They include torsion of cylinders, bending or straightening of blocks and cylindrical sectors, and inflation or eversion of hollow cylinders and spheres. Some related controllable or universal motions have been discussed [10–13] and all of these results have been extended to include inelastic response [14, 15].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. S. Rivlin, Large elastic deformations of isotropic materials. IV. Further developments of the general theory, Phil. Trans. Roy. Soc. A241, 379 (1948).
R. S. Rivlin, A note on the torsion of an incompressible highly-elastic cylinder, Proc. Cambridge Phil. Soc. 45, 485 (1948).
R. S. Rivlin, Large elastic deformation of isotropic materials. V. The problem of flexure, Proc. Roy. Soc. London A195, 463 (1949).
R. S. Rivlin, Large elastic deformation of isotropic materials. VI. Further results in the theory of torsion, shear and flexure, Phil. Trans. Roy. Soc. London A242, 173 (1949).
A. E. Green and R. T. Shield, Finite elastic deformations in incompressible isotropic bodies, Proc. Roy. Soc. London A202, 407 (1950).
J. L. Ericksen, Deformations possible in every isotropic, incompressible, perfectly elastic body, Z. angew. Math. Phys. 5, 466 (1954).
J. L. Ericksen, Inversion of a perfectly elastic spherical shell, Z. angew Math. Mech. 35, 382 (1955).
W. W. Klingbeil and R. T. Shield, On a class of solutions in plane finite elasticity, Z. angew Math. Phys. 17, 489 (1966).
M. Singh and A. C. Pipkin, Note on Ericksen’s problem, Z. angew Math. Phys. 16, 706 (1965).
J. A. Knowles, Large amplitude oscillations of a tube of incompressible elastic material, Q. Appl. Math. 18, 71 (1960).
J. A. Knowles, On a class of oscillations in the finite-deformation theory of elasticity, J. Appl. Math. 29, 283 (1962).
Z.-H. Guo and R. Solecki, Free and forced finite-amplitude oscillations of an elastic thick-walled hollow sphere made of incompressible material, Arch. Mech. Stosow. 15, 427 (1963).
C. A. Truesdell, Solutio generalis et accurata problematum quamplurimorum de motu corporum elasticorum incomprimibilium in deformationibus valde magnis, Arch. Rat. Mech. Anal. 11, 106 (1962).
M. M. Carroll, Controllable deformations of incompressible simple materials, Int. J. Engng. Sci. 5, 515–525 (1967).
R. L. Fosdick, Dynamically possible motions of incompressible, isotropic, simple materials, Arch. Rational Mech. Anal. 29, 272 (1968).
J. L. Ericksen, Deformations possible in every compressible, isotropic perfectly elastic material, J. Math, and Phys. 34, 126 (1955).
M. M. Carroll, Finite strain solutions in compressible isotropic elasticity, J. Elasticity 20, 65 (1988).
M. M. Carroll, Controllable deformations for special classes of compressible elastic solids, SAACM 1, 309 (1991).
M. M. Carroll, Controllable deformations in compressible finite elasticity, SAACM 1, 373 (1991).
F. John, Plane strain problems for a perfectly elastic material of harmonic type, Comm. Pure Appl. Math. 13, 239 (1960).
R. W. Ogden and D. A. Isherwood, Solution of some finte plane-strain problems for compressible elastic solids, Q. J. Mech. Appl. Math. 31, 219 (1978).
R. Abeyaratne and C. O. Horgan, The pressurized hollow sphere problem in finite elastostatics for a class of compressible materials, Int. J. Solids Struct. 20, 715 (1984).
P. K. Currie and M. Hayes, On non-universal finite elastic deformations, in Finite Elasticity (D. E. Carlson and R. T. Shield, eds.), p. 143, Martinus Nijhoff 1982.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to my colleague, mentor, and friend, Paul M. Naghdi, on his 70th birthday with much gratitude, deep affection, and highest regard
Rights and permissions
Copyright information
© 1995 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Carroll, M.M., Brown, G.R. (1995). On obtaining closed form solutions for compressible nonlinearly elastic materials. In: Casey, J., Crochet, M.J. (eds) Theoretical, Experimental, and Numerical Contributions to the Mechanics of Fluids and Solids. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9229-2_7
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9229-2_7
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9954-3
Online ISBN: 978-3-0348-9229-2
eBook Packages: Springer Book Archive