Cowin-Mehrabadi Theorem in six dimensions

• Autorzy: Ahmad F.
• Języki publikacji: EN

Abstrakty

EN

The Cowin-Mehrabadi Theorem concerning normals to the planes of symmetry of an anisotropic material is generalized to six dimensions. Commutation of the reflection matrix with the 6×6 matrix representing the elasticity tensor in the six-dimensional formulation of the elasticity tensor, provides the condition for the existence of a plane of symmetry. This condition implies the existence of at least two isochoric states for every class except the triclinic one. A simple proof is presented of the fact that an axis of symmetry An, with n > 4 must be an axis of isotropy.

Słowa kluczowe

EN

planes of symmetry; anisotropic material; necessary and sufficient condition

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Archives of Mechanics
• Rocznik: 2010
• Tom: Vol. 62, nr 3
• Strony: 215--222
• Opis fizyczny: Bibliogr. 19 poz.

Twórcy

autor

Ahmad F.

• Centre for Adyanced Mathematics and Physics National Unwersity of Sciences and Technology EME College Campus Peshawar Road, Rawalpindi. Pakistan,

Bibliografia

• 1. W.K. THOMSON (Lord Kelvin), On six principal strains of an elastic solid, Phil. Trans.R. Soc., 166, 495-498, 1856.
• 2. J. RYCHLEWSKI, On Hooke's law, Prik. Matem. Mekhan., 48, 303-314, 1984.
• 3. M.M. MEHRABADI, S.C. COWIN, Eigentensors of linear amsotropic elastic materials, Q. Jl. Mech. Appl. Math., 43, 15-41, 1990.
• 4. J. RYCHLEWSKI, Unconventional approach to linear elasticity, Arch. Mech., 47, 149-171, 1995.
• 5. S.C. COWIN, M.M. MEHRABADI, Anisotropic symmetries of linear elasticity, Appl- Mech. Rev., 48, 247-285, 1995.
• 6. F AHMAD R.A. KHAN, Eigenvectors of a rotation matrix, Q. Jl. Mech. Appl. Math. 62, 297-310, 2009.
• 7. A. BLINOWSKI, J. OSTROWSKA-MACIEJEWSKA, On the elastic orthotropy, Arch. Mech., 48, 129 141, 1996.
• 8. S.C. COWIN, G. YANG, M.M. MEHRABADI, Bounds on the effective anisotropic elastic constants, J. Elasticity, 57, 1-24, 1999.
• 9. S.C. COWIN, G YANG, Material symmetry optimization by Kelvin modes. Journal of Engineering Mathematics, 37, 27-43, 2000.
• 10. M.M. MEHRABADI, S.C. COWIN, J. JARIC. Six-dimensional orthogonal tensor representation of the rotation about an axis in three dimensions, Int. J. Solids Structures, 32, 439-449, 1995.
• 11. A.N. NORRIS, Optimal orientation of anisotropic solids, Q. Jl. Mech. Appl. Math. 59, 29-53, 2006.
• 12. A. BLINOWSKI, J. RYCHLEWSKI, Pure shears in the mechanics of materials, Mathematics and Mechanics of Solids, 4, 471-503, 1998.
• 13. M. HAMERMESH, Group Theory and its Applications to Physical Problems, Addison-Wesley, London 1964.
• 14. C. HERMANN, Tensoren und Kristallsymmetrie, Z. Kristallogr. A., 89, 32-48, 1934.
• 15. S.C. COWIN, M.M. MEHRABADI On the identification of material symmetry for anisotropic materials, Q. Jl. Mech. Appl. Math., 40, 451-476, 1987.
• 16 T.C.T. TING, Anisotropic Elasticity. Theory and Applications, Oxford University Press, Oxford 1996.
• 17. T.C.T. TING. Generalized Cowin-Mehrabadi theorems and a direct proof that the number of linear elastic symmetries is eight, Int. J. Solids Structures, 40, 7129-7142, 2003.
• 18. B. FRIEDMAN, Principles and Techniques of Applied Mathematics, John Wiley and Sons, New York 1966.
• 19. E. DIEULESAINT, D. ROYER, Elastic Waves in Solids, John Wiley and Sons, New York 1980.