Abstract
The problem considered here is that of extremizing the strain energy density of a linear anisotropic material by varying the relative orientation between a fixed stress state and a fixed material symmetry. It is shown that the principal axes of stress must coincide with the principal axes of strain in order to minimize or maximize the strain energy density in this situation. Specific conditions for maxima and minima are obtained. These conditions involve the stress state and the elastic constants. It is shown that the symmetry coordinate system of cubic symmetry is the only situation in linear anisotropic elasticity for which a strain energy density extremum can exist for all stress states. The conditions for the extrema of the strain energy density for transversely isotropic and orthotropic materials with respect to uniaxial normal stress states are obtained and illustrated with data on the elastic constants of some composite materials. Not surprisingly, the results show that a uniaxial normal stress in the grain direction in wood minimizes the strain energy in the set of all uniaxial stress states. These extrema are of interest in structural and material optimization.
Similar content being viewed by others
References
P. Petersen, On optimal orientation of orthotropic materials. Structural Optimization, 1 (1989) 101.
M. P. Bendsøe and N. Kikuchi, Generating optimal topologies in structural design using a homogenization method. Comp. Meths. Appl. Mech. & Engr. 71 (1988) 197–224.
K. Suzuki, Shape and Topology Optimization Using the Homogenization Method, Ph.D. Dissertation, 1991, University of Michigan.
D. P. Fyhrie and D. R. Carter, A unifying principle relating stress to trabecular bone morphology. J. Ortho. Res. 4 (1986) 304.
R. F. S. Hearmon, An Introduction to Applied Anisotropic Elasticity. Oxford University Press, Oxford (1961).
S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body. Holden Day, San Francisco (1963).
F. I. Fedorov, Theory of Elastic Waves in Crystals. Plenum Press, New York (1968).
S. C. Cowin and M. M. Mehrabadi, On the identification of material symmetry for anisotropic elastic materials. Quart. J. Mech. Appl. Math. 40 (1987) 451.
Y. Z. Huo and G. del Piero, On the completeness of the crystallographic symmetries in the description of the symmetries of the elastic tensor, preprint (1987).
H. Hancock, Theory of Maxima and Minima. Ginn & Co., Boston (1917).
F. F. P. Kollmann and W. A. Côté, Jr., Principles of Wood Science and Technology, I Solid Wood. Springer Verlag (1968).
S. C. Cowin, Bone Mechanics. CRC Press, Boca Raton, FL (1989).
M. F. Beatty, Vector analysis of rigid rotations. J. Appl. Mech. 44 (1977) 501.
M. F. Beatty, Kinematics of finite rigid body displacements. Am. J. Phys. 34 (1966) 949.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cowin, S.C. Optimization of the strain energy density in linear anisotropic elasticity. J Elasticity 34, 45–68 (1994). https://doi.org/10.1007/BF00042425
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00042425