Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-19T00:24:15.418Z Has data issue: false hasContentIssue false
  • English
  • Français

Explicit Expression of the Stationary Values of Young's Modulus and the Shear Modulus for Anisotropic Elastic Materials

Published online by Cambridge University Press:  05 May 2011

T. C. T. Ting
T. C. T. Ting*
Affiliation:
Division of Mechanics and Computation, Stanford University, Durand 262, Stanford, CA 94305–4040, U.S.A.
*
* Consulting Professor, also Professor Emeritus of University of Illinois at Chicago
Get access

Abstract

Explicit expressions of the direction n and the stationary values (maximum, minimum and saddle point) of Young's modulus E(n) for orthotropic, tetragonal, trigonal, hexagonal and cubic materials are presented. For the shear modulus G(n, m), explicit expressions of the extrema (maximum and minimum) and the two mutually orthogonal unit vectors n, m are given for cubic and hexagonal materials. We also present a general procedure for computing the extrema of G(n, m) for more general anisotropic elastic materials. It is shown that Young's modulus E(n) can be made as large as we wish for certain n without assuming that the elastic compliance s11, s22 or s33 is very small. As to the shear modulus G(n,m), it can be made as large as we wish for certain n and m without assuming that any one of the elastic compliance sαβ is very small. We also show that Young's modulus E(n) can be independent of n for orthotropic and hexagonal materials while the shear modulus G(n, m) can be independent of n and m for hexagonal materials.

Keywords

Anisotropic elasticityYoung's modulusShear modulusOrthotropic materialsTetragonal materialsTrigonal materialsHexagonal materialsCubic materials
Type
Articles
Information
Journal of Mechanics , Volume 21 , Issue 4 , December 2005 , pp. 255 - 266
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Turley, J. and Sines, G., “The Anisotropy of Young's Modulus, Shear Modulus and Poisson's Ratio in Cubic Materials,” J. Phys., D4, pp. 264–271 (1971).Google Scholar
2.Baughman, R. H., Shacklette, J. M., Zakhidov, A. A. and Stafstrom, S., “Negative Poisson's Ratios as a Common Feature of Cubic Materials,” Nature, 392, pp 362–365 (1998).CrossRefGoogle Scholar
3.Hayes, M. and Shuvalov, A., “On the Extreme Values of Young's Modulus, the Shear Modulus, and Poisson's Ratio for Cubic Materials,” J. Appl. Mech., 65, pp. 786–787 (1998).CrossRefGoogle Scholar
4.Li, Y., “The Anisotropic Behavior of Poisson's Ratio, Young's Modulus, and Shear Modulus in Hexagonal Materials,” Phys. Stat. Sol, (a)38, pp. 171–175 (1976).Google Scholar
5.Theocaris, P. S. and Philippidis, T. P., “True Bounds on Poisson's Ratios for Transversely Isotropic Solids,” J. of Strain Analysis, 27(1), pp. 43–44 (1992).CrossRefGoogle Scholar
6.Lempriere, B. M., “Poisson's Ratio in Orthotropic Materials,” AIAA Journal, 6(11), pp. 2226–2227 (1968).CrossRefGoogle Scholar
7.Boulanger, Ph. and Hayes, M., “Poisson's Ratio for Orthotropic Materials,” J. Elasticity, 50, pp. 87–89 (1998).CrossRefGoogle Scholar
8.Zheng, Q. S. and Chen, T., “New Perspective on Poisson's Ratios of Elastic Solids,” Acta Mech., 150, pp. 191–195 (2001).CrossRefGoogle Scholar
9.Ting, T. C. T. and Chen, T., “Poisson's Ratio for Anisotropic Elastic Materials Can Have no Bounds,” Q. J. Mech. Appl. Math., 58(1), pp. 73–82 (2005).CrossRefGoogle Scholar
10.Lakes, R. S., “Advances in Negative Poisson's Ratio Materials,” Adv. Mater., 5, pp. 293–296 (1993).Google Scholar
11.Ting, T. C. T. and Barnett, D. M., “Negative Poisson's Ratios in Anisotropic Linear Elastic Media,” J. Appl. Mech., 72, pp. 929–931 (2005).Google Scholar
12.Ting, T. C. T., “Very Large Poison's Ratio with a Bounded Transverse Strain in Anisotropic Elastic Materials,” J. Elasticity, 77(2), pp. 163–176 (2004).CrossRefGoogle Scholar
13.Saint-Venant, B., “Mémoire sur la distribution des elasticies,” J. Math. Pures er Appl., (Liouville) II, 10, pp. 297–349 (1863).Google Scholar
14.Ravinovich, A. L., “On the Elastic Constants and Strength of Aircraft Materials,” Trudy Trentr. Aero-Gidvodin., 582, pp. 1–56 (1946).Google Scholar
15.Lekhnitskii, S. G., Theory of Elasticity of an Anisotropic Body, Holden-Day, San Francisco (1963).Google Scholar
16.Goens, E., “Uber Eine Verbsserte Apparatur Zur Statischen Bestimmung Des Drillungsmodulus Von Kristallstaben Und Ihre Anwendung Auf Zink-Einkristalle,” Annalen der Physik, 5(16), pp. 793–809 (1933).Google Scholar
17.Schmid, E. and Boas, W., Kristallplastizitat mit Besonderer Berücksichtigung der Metalle., Springer, Berlin (1935).Google Scholar
18.Cazzani, Antonio and Rovati, Marco, “Extrema of Young's Modulus for Cubic and Transversely Isotropic Solids,” Int. J. Solids Structures, 40, pp. 1713–1744 (2003).Google Scholar
19.Cazzani, Antonio and Rovati, Marco, “Extrema of Young's Modulus for Elastic Solids with Tetragonal Symmetry,” Int. J. Solids Structures, 42, pp. 5057–5096 (2005).Google Scholar
20.Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, University Press, Cambridge (1927).Google Scholar
21.Nye, J. F., Physical Properties of Crystals, Clarendon Press, Oxford (1985).Google Scholar
22.Chadwick, P., Vranello, M. and Cowin, S. C., “A New Proof that the Number of Linear Elastic Symmetries is Eight,” J. Mech. Phys. Solids, 49, pp. 2471–2492 (2001).Google Scholar
23.Ting, T. C. T., “Generalized Cowin-Mehrabadi Theorems and a Direct Proof that the Number of Linear Elastic Symmetries is Eight,” Int. J. Solids Structure, 40, pp. 7129–7142 (2003).Google Scholar
24.Ting, T. C. T., “The Stationary Values of Young's Modulus for Monoclinic and Triclinic Materials,” J. Mechanics (formerly the Chinese Journal of Mechanics-Series A), 21(4), pp. 249–253 (2005).CrossRefGoogle Scholar
25.Ting, T. C. T., Anisotropic Elasticity: Theory and Applications, Oxford University Press, New York (1996).Google Scholar
26.Sirotin, Yu. I. and Shakol'skaya, M. P., Fundamentals of Cry stal Physics, MIR Pub., Moscow (1982).Google Scholar
27.Voigt, W., Lehrbuch der Kristallphysik, Leipzig (1910).Google Scholar
28.McConnell, A. J., Applications of Tensor Analysis, Dover, New York (1957).Google Scholar
29.Hohn, F. E., Elementary Matrix Algebra, London, Macmillan (1965).Google Scholar
30.Ting, T. C. T., “On Anisotropic Elastic Materials for Which Young's Modulus E(a) is Independent of n or the Shear Modulus G(n,m) is Independent of n and m,” J Elasticity, 81(1), in press (2006).CrossRefGoogle Scholar
31.Ting, T. C. T., “Anisotropic Elastic Constants that are Structurally Invariant,” Q. J. Mech. Appl. Math., 53(4), pp. 511–523 (2000).CrossRefGoogle Scholar