Poisson’s ratio of anisotropic systems
Institute of Molecular Physics, Polish Academy of Sciences
M. Smoluchowskiego 17, 60-179 Poznań, Poland
Received:
Rec. 30 May 2005
DOI: 10.12921/cmst.2005.11.01.73-79
OAI: oai:lib.psnc.pl:584
Abstract:
The Poisson’s ratio of anisotropic materials depends, in general, both on a “longitudinal” direction along which the stress is changed and on a “transverse” direction in which the transverse deformation is measured. For cubic media there exist “longitudinal” directions, parallel to the 4-fold and 3-fold axes, for which the Poisson’s ratio does not depend on the “transverse” direction. Depending on the tensor of elastic compliances (or elastic constants), crystals of cubic symmetry can exhibit negative Poisson’s ratio in both these directions (they are called strongly auxetics), in one of them (i.e. either along the 4-fold axis or along the 3-fold one; they are called partially auxetic) or in none of them. For crystals exhibiting 3-fold symmetry axis the Poisson’s ratio along this axis does not depend on the “transverse” direction. For other “longitudinal” directions the Poisson’s ratio depends, in general, on the “transverse” direction. The Poisson’s ratio averaged with respect to the “transverse” direction depends only on the “longitudinal” direction and can be conveniently presented graphically. As an example the f.c.c. hard sphere crystal is considered. It is shown that the average (with respect to “transverse” direction) Poisson’s ratio of the hard sphere crystal is positive for all “longitudinal” directions. One should add, however, that there exist directions for which the (not averaged) Poisson’s ratio of hard spheres is negative.
Key words:
auxetics, elastic properties of crystals of cubic symmetry, negative Poisson’s ratio
References:
[1] L. D. Landau and E. M. Lifshits, Theory of elasticity, Pergamon Press, London, 1986.
[2] R. S. Lakes, Science 235, 1038 (1987).
[3] J. Glieck, The New York Times, 14 April 1987.
[4] K. E. Evans, M. A. Nkansah, I. J. Hutchinson, Nature 353, 124 (1991).
[5] G. Milton, J. Mech. Phys. Solids 40, 1105 (1992).
[6] D. A. Konyok, K. W. Wojciechowski, Yu. M. Pleskachevskii, S. V. Shilko, Mech. Compos. Mater. Construct. 10, 35 (2004), in Russian.
[7] W. G. Cady, Piezoelectricity, Dover, New York, 1964.
[8] Y. Li, Phys. Status Solidi (a) 38, 171 (1976).
[9] J. H. Weiner, Statistical Mechanics of Elasticity, Wiley, New York, 1983.
[10] K. W. Wojciechowski and K. V. Tretiakov, Comp. Phys. Commun. 121-122, 528 (1999).
[11] M. V. Jaric and U. Mohanty, Phys. Rev. Lett. 58, 230 (1987).
[12] D. Frenkel and A. J. C. Ladd, Phys. Rev. Lett. 59, 1169 (1987).
[13] K. V. Tretiakov and K. W. Wojciechowski, J. Chem. Phys. (2005).
[14] F. Milstein and K. Huang, Phys. Rev. B19, 2030 (1979).
[15] R. H. Baughman, J. M. Shacklette, A. A. Zakhidov and S. Stafstrom, Nature 392, 362 (1998).
[16] K. J. Runge and G. V. Chester, Phys. Rev. A36, 4852 (1987).
The Poisson’s ratio of anisotropic materials depends, in general, both on a “longitudinal” direction along which the stress is changed and on a “transverse” direction in which the transverse deformation is measured. For cubic media there exist “longitudinal” directions, parallel to the 4-fold and 3-fold axes, for which the Poisson’s ratio does not depend on the “transverse” direction. Depending on the tensor of elastic compliances (or elastic constants), crystals of cubic symmetry can exhibit negative Poisson’s ratio in both these directions (they are called strongly auxetics), in one of them (i.e. either along the 4-fold axis or along the 3-fold one; they are called partially auxetic) or in none of them. For crystals exhibiting 3-fold symmetry axis the Poisson’s ratio along this axis does not depend on the “transverse” direction. For other “longitudinal” directions the Poisson’s ratio depends, in general, on the “transverse” direction. The Poisson’s ratio averaged with respect to the “transverse” direction depends only on the “longitudinal” direction and can be conveniently presented graphically. As an example the f.c.c. hard sphere crystal is considered. It is shown that the average (with respect to “transverse” direction) Poisson’s ratio of the hard sphere crystal is positive for all “longitudinal” directions. One should add, however, that there exist directions for which the (not averaged) Poisson’s ratio of hard spheres is negative.
Key words:
auxetics, elastic properties of crystals of cubic symmetry, negative Poisson’s ratio
References:
[1] L. D. Landau and E. M. Lifshits, Theory of elasticity, Pergamon Press, London, 1986.
[2] R. S. Lakes, Science 235, 1038 (1987).
[3] J. Glieck, The New York Times, 14 April 1987.
[4] K. E. Evans, M. A. Nkansah, I. J. Hutchinson, Nature 353, 124 (1991).
[5] G. Milton, J. Mech. Phys. Solids 40, 1105 (1992).
[6] D. A. Konyok, K. W. Wojciechowski, Yu. M. Pleskachevskii, S. V. Shilko, Mech. Compos. Mater. Construct. 10, 35 (2004), in Russian.
[7] W. G. Cady, Piezoelectricity, Dover, New York, 1964.
[8] Y. Li, Phys. Status Solidi (a) 38, 171 (1976).
[9] J. H. Weiner, Statistical Mechanics of Elasticity, Wiley, New York, 1983.
[10] K. W. Wojciechowski and K. V. Tretiakov, Comp. Phys. Commun. 121-122, 528 (1999).
[11] M. V. Jaric and U. Mohanty, Phys. Rev. Lett. 58, 230 (1987).
[12] D. Frenkel and A. J. C. Ladd, Phys. Rev. Lett. 59, 1169 (1987).
[13] K. V. Tretiakov and K. W. Wojciechowski, J. Chem. Phys. (2005).
[14] F. Milstein and K. Huang, Phys. Rev. B19, 2030 (1979).
[15] R. H. Baughman, J. M. Shacklette, A. A. Zakhidov and S. Stafstrom, Nature 392, 362 (1998).
[16] K. J. Runge and G. V. Chester, Phys. Rev. A36, 4852 (1987).
