Principal stress and strain trajectories in nonlinear elastostatics

Author:
R. W. Ogden

Journal:
Quart. Appl. Math. **44** (1986), 255-264

MSC:
Primary 73C50

DOI:
https://doi.org/10.1090/qam/856179

MathSciNet review:
856179

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Abstract: The Maxwell-Lamé equations governing the principal components of Cauchy stress for plane deformations are well known in the context of photo-elasticity, and they form a pair of coupled first-order hyperbolic partial differential equations when the deformation geometry is known. In the present paper this theme is developed for nonlinear isotropic elastic materials by supplementing the (Lagrangean form of the) equilibrium equations by a pair of compatibility equations governing the deformation. The resulting equations form a system of four first-order partial differential equations governing the principal stretches of the plane deformation and the two angles which define the orientation of the Lagrangean and Eulerian principal axes of the deformation. Coordinate curves are chosen to coincide locally with the Lagrangean (Eulerian) principal strain trajectories in the undeformed (deformed) material.

**[1]**H. J. Jessop,*Photoelasticity*, in Handbuch der Physik, Vol. VI (Edited by S. Flügge), Springer, 1958 MR**0093121****[2]**A. Franek, J. Kratochvil and L. Travnicek, ZAMM**63**, 156-158 (1983)**[3]**R. W. Ogden,*Inequalities associated with the inversion of elastic stress-deformation relations and their implications*, Math. Proc. Cambridge Philos. Soc.**81**, 313-324 (1977) MR**0431884****[4]**R. W. Ogden,*Non-linear elastic deformations*, Ellis Horwood, 1984**[5]**A. E. Green and W. Zerna,*Theoretical elasticity*, Oxford University Press, 1968 MR**0245245****[6]**A. E. Green and J. E. Adkins,*Large elastic deformations*, Oxford University Press, 1970 MR**0269158****[7]**R. T. Shield,*The rotation associated with large strains*, SIAM J. Appl. Math.**25**, 483-491 (1973)**[8]**A. Franek, J. Kratochvil, and L. Travnicek,*Inhomogeneous inverse problem in finite elasticity*, J. Elasticity**14**, 363-372 (1984) MR**770297**

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DOI:
https://doi.org/10.1090/qam/856179

Article copyright:
© Copyright 1986
American Mathematical Society