General solutions for plane extensible elasticae having nonlinear stress-strain laws

Antman, Stuart

doi:10.1090/qam/99868

Author: Stuart Antman
Journal: Quart. Appl. Math. 26 (1968), 35-47
DOI: https://doi.org/10.1090/qam/99868
MathSciNet review: QAM99868
Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: A general finite deformation solution is obtained for the equilibrium of hydrostatically loaded elasticae whose underformed shapes are circular arcs. The nonlinear stress-strain relations employed give the bending moment and the axial force as derivatives of a strain energy function with respect to suitable strain measures. The representation of the solution involves arbitrary constants of integration which can accommodate any end conditions consistent with equilibrium. Examples are given. The constraint of inextensibility is examined and a perturbation procedure for small extension is developed. In an appendix, the stress-strain laws are derived by an appropriate reduction of the equations for three-dimensional hyperelasticity.

I. Tadjbakhsh, The variational theory of the plane motion of the extensible elastica, Int. J. Eng. Sci. 4, 433–450 (1960)

C. Truesdell and R. Toupin, Static grounds for inequalities in finite strain of elastic materials, Arch. Rational Mech. Anal. 12 (1963), 1–33. MR 144511, DOI https://doi.org/10.1007/BF00281217
A. E. Green, The equilibrium of rods, Arch. Rational Mech. anal. 3 (1959), 417–421 (1959). MR 0108055, DOI https://doi.org/10.1007/BF00284190
John L. Synge and Byron A. Griffith, Principles of mechanics. 3rd ed, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1959. MR 0106562
A. E. H. Love, A treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944. Fourth Ed. MR 0010851
Alf Pflüger, Stabilitätsprobleme der Elastostatik, Springer-Verlag, Berlin-New York, 1964 (German). Zweite neubearbeitete Auflage. MR 0178635

R. Frisch-Fay, Flexible bars, Butterworths, Washington, 1962

A. J. M. Spencer, Finite deformations of an almost incompressible elastic solid, Second-order Effects in Elasticity, Plasticity and Fluid Dynamics (Internat. Sympos., Haifa, 1962) Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1964, pp. 200–216. MR 0166989
Stuart S. Antman and William H. Warner, Dynamical theory of hyperelastic rods, Arch. Rational Mech. Anal. 23 (1966), 135–162. MR 199998, DOI https://doi.org/10.1007/BF00251729
C. Truesdell and R. Toupin, The classical field theories, Handbuch der Physik, Bd. III/1, Springer, Berlin, 1960, pp. 226–793; appendix, pp. 794–858. With an appendix on tensor fields by J. L. Ericksen. MR 0118005
J. L. Ericksen and R. S. Rivlin, Large elastic deformations of homogeneous anisotropic materials, J. Rational Mech. Anal. 3 (1954), 281–301. MR 63232, DOI https://doi.org/10.1512/iumj.1954.3.53014
P. M. Naghdi and R. P. Nordgren, On the nonlinear theory of elastic shells under the Kirchhoff hypothesis, Quart. Appl. Math. 21 (1963), 49–59. MR 145743, DOI https://doi.org/10.1090/S0033-569X-1963-0145743-4