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

Author:
Stuart Antman

Journal:
Quart. Appl. Math. **26** (1968), 35-47

DOI:
https://doi.org/10.1090/qam/99868

MathSciNet review:
QAM99868

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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.

**[1]**I. Tadjbakhsh,*The variational theory of the plane motion of the extensible elastica*, Int. J. Eng. Sci.**4**, 433-450 (1960)**[2]**C. Truesdell and R. Toupin,*Static grounds for inequalities in finite strain of elastic materials*, Arch. Rational Mech. Anal.**12**, 1-33 (1963) MR**0144511****[3]**A. E. Green,*The equilibrium of rods*, Arch. Rational Mech. Anal.**3**, 417-421 (1959) MR**0108055****[4]**J. L. Synge and B. A. Griffith,*Principles of mechanics*, 3rd ed., McGraw-Hill, New York, 1959 MR**0106562****[5]**A. E. H. Love,*A treatise on the mathematical theory of elasticity*, 4th ed., Dover, New York, 1944 MR**0010851****[6]**A. Pflüger,*Stabilitätsprobleme der Elastostatik*, Springer-Verlag, Berlin, 1964 MR**0178635****[7]**R. Frisch-Fay,*Flexible bars*, Butterworths, Washington, 1962**[8]**A. J. M. Spencer,*Finite deformation of an almost incompressible elastic solid*, I.U.T.A.M. International Symposium, Haifa, Israel, 1962;*Second order effects in elasticity, plasticity, and fluid dynamics*, edited by M. Reiner and D. Abir, Macmillan, New York, 1964 MR**0166989****[9]**S. Antman and W. Warner,*Dynamical theory of hyperelastic rods*, Arch. Rational Mech. Anal.**23**, 135-162 (1966) MR**0199998****[10]**C. Truesdell and R. Toupin,*The classical field theories*, Handbuch der Physik,**III/1**, Springer-Verlag, Berlin, 1960 MR**0118005****[11]**J. L. Ericksen and R. S. Rivlin,*Large elastic deformations of homogeneous anisotropic materials*, J. Rational Mech. Anal.**3**, 281-301 (1954) MR**0063232****[12]**P. M. Naghdi and R. P. Nordgren,*On the nonlinear theory of elastic shells under the Kirchhoff hypothesis*, Quart. Appl. Math.**21**, 49-59 (1963) MR**0145743**

Additional Information

DOI:
https://doi.org/10.1090/qam/99868

Article copyright:
© Copyright 1968
American Mathematical Society