Abstract
The simplest theory of spatial rods is presented in a variational setting and certain necessary conditions for minimizers of the potential energy are derived. These include the Weierstrass and Legendre inequalities, which require that the vector describing curvature and twist belong to a domain of convexity of the strain energy function.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G.A. Bliss,Lectures on the Calculus of Variations. University of Chicago Press, Chicago (1946).
R.L. Fosdick and R.D. James, The elastica and the problem of the pure bending for a non-convex stored energy function.J. Elast. 11 (1981) 165–86.
R.D. James, The equilibrium and post-buckling behaviour of an elastic curve governed by a non-convex energy.J. Elast. 11 (1981) 239–69.
A.E.H. Love,A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press (1927).
L.D. Landau and E.M. Lifshitz,Theory of Elasticity, 3rd edn. Pergamon Press, Oxford (1986).
H. Cohen, A non-linear theory of elastic directed curves.Int. J. Engng. Sci. 4 (1966) 511–24.
A.E. Green and N. Laws, A general theory of rods.Proc. R. Soc. Lond. A293 (1966) 145–55.
A.E. Green, P.M. Naghdi and M.L. Wenner, On the theory of rods I. Derivations from three-dimensional equations.Proc. R. Soc. Lond. A337 (1974) 451–83.
A.E. Green, P.M. Naghdi and M.L. Wenner, On the theory of rods II. Developments by direct approach.Proc. R. Soc. Lond. A337 (1974) 485–507.
P.M. Naghdi, Finite deformations of elastic rods and shells. In D.E. Carlson and R.T. Shield (eds),Proceedings of the IUTAM Symposium on Finite Elasticity. Martinus Nijhoff, The Hague. (1980) pp. 47–103.
S.S. Antman, The theory of rods. In C. Truesdell (ed.),S. Flügge's Handbuch der Physik, VI a/2. Springer-Verlag, Berlin, Heidelberg and New York (1972) pp. 641–703.
S.S. Antman, Kirchhoff's problem for non-linearly elastic rods.Quart. Appl. Math. 33 (1974) 221–40.
E. Reissner, On one-dimensional large-displacement finite strain beam theory.Studies in Applied Maths. 52 (1973) 87–95.
J.J. Stoker,Differential Geometry. Wiley-Interscience, New York (1969).
P. Cicala, Helicoidal buckling of an elastic rod.Meccanica 20 (1985) 124–6.
Y. Lin and A.P. Pisano, The differential geometry of the general helix as applied to mechanical springs.J. Appl. Mech. 55 (1988) 831–6.
C.B. Kafadar, On the nonlinear theory of rods.Int. J. Engng. Sci. 10 (1972) 369–91.
A.E. Green and N. Laws, Remarks on the theory of rods.J. Elast. 3 (1973) 179–84.
P.M. Naghdi and M.B. Rubin, Constrained theories of rods.J. Elast. 14 (1984) 343–61.
S.S. Antman, Monotonicity and invertibility conditions in one-dimensional nonlinear elasticity. In R.W. Dickey (ed.),Proceedings of the Symposium on Nonlinear Elasticity. Academic Press, New York (1973) pp. 57–92.
S.S. Antman, Ordinary differential equations of non-linear elasticity I: Foundations of the theories of non-linearly elastic rods and shells.Arch. Ration. Mech. Anal. 61 (1976) 307–51.
S.S. Antman, Ordinary differential equations of non-linear elasticity II: Existence and regularity theory for conservative boundary value problems.Arch. Ration. Mech. Anal. 61 (1976) 353–93.
J.H. Maddocks, Stability of nonlinearly elastic rods.Arch. Ration. Mech. Anal. 84 (1984) 312–54.
L.M. Graves, The Weierstrass condition for multiple integral variation problems.Duke Math. J. 5 (1939) 656–60.
M.G. Hilgers and A.C. Pipkin, Energy-minimizing deformations of elastic sheets with bending stiffnessJ. Elast. 31 (1993) 125–39.
A.B. Whitman and C.N. DeSilva, An exact solution in a nonlinear theory of rods.J. Elast. 4 (1974) 265–80.
M. Farshad, On general conservative end loading of pretwisted rods.Int. J. Solids Structures 9 (1973) 1361–71.
M. Farshad, On general conservative loading of naturally curved and twisted funicular rods.Int. J. Solids Structures 10 (1974) 985–91.
L. Elsgolts,Differential Equations and the Calculus of Variations. MIR, Moscow (1977).
H. Ziegler, On the concept of elastic stability, inAdvances in Applied Mechanics 4 (1956) 351–403.
G.M. Ewing,Calculus of Variations with Applications. W.W. Norton, New York (1969).
L.M. Graves, On the Weierstrass condition for the problem of Bolza in the calculus of variations.Annals of Mathematics 33 (1932) 747–52.
E.J. McShane, On multipliers for Lagrange problems.Am. J. Maths. 61 (1939) 809–19.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Steigmann, D.J., Faulkner, M.G. Variational theory for spatial rods. J Elasticity 33, 1–26 (1993). https://doi.org/10.1007/BF00042633
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00042633