There are two proportional invariants for a linear isotropic material, the hydrostatic invariant, and the deviatoric invariant. The former is proportional to the trace of the tensor and the latter is proportional to the trace of the square of the associated deviatoric tensor. The hydrostatic stress and strain and the von Mises stress and strain are directly related to the hydrostatic and deviatoric proportional invariants, respectively, for an isotropic, linear elastic material. For each anisotropic linear elastic material symmetry there are up to six proportional invariants. In this paper we illustrate the six proportional invariants of an orthotropic elastic material using the elastic constants for spruce as the numerical example. The proportional elastic invariants play a role in anisotropic linear elasticity similar to the roles played by the hydrostatic stress and strain and the von Mises stress and strain in isotropic elasticity. They are the unique parameters whose contours represent both the stress and the strain distributions. They also have potential for representing failure or fracture criteria.
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March 1991
Research Papers
The Proportional Anisotropic Elastic Invariants
A. M. Sadegh,
A. M. Sadegh
Department of Mechanical Engineering, City College of the City University of New York, New York, NY 10031
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S. C. Cowin
S. C. Cowin
Department of Mechanical Engineering, City College of the City University of New York, New York, NY 10031
Search for other works by this author on:
A. M. Sadegh
Department of Mechanical Engineering, City College of the City University of New York, New York, NY 10031
S. C. Cowin
Department of Mechanical Engineering, City College of the City University of New York, New York, NY 10031
J. Appl. Mech. Mar 1991, 58(1): 50-57 (8 pages)
Published Online: March 1, 1991
Article history
Received:
November 16, 1989
Revised:
May 7, 1990
Online:
March 31, 2008
Citation
Sadegh, A. M., and Cowin, S. C. (March 1, 1991). "The Proportional Anisotropic Elastic Invariants." ASME. J. Appl. Mech. March 1991; 58(1): 50–57. https://doi.org/10.1115/1.2897178
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