Invariants of 3D Anisotropic Elasticity Tensor: Measure of Symmetry Defects

Nametchougle Dampotime¹, Kossi Atchonouglo^2, and Edo-Owodou Ayeleh³

¹Université de Lomé, Laboratoire d’Analyse de Modélisations Mathématiques et Applications (LAMMA), Lomé, Togo

²Université de Lomé, Laboratoire sur l’Energie Solaire, Lomé, Togo

³Université de Lomé, Laboratoire de Structures et de Mécanique des Matériaux (LaS2M), Lomé, Togo

Cite this paper:
Nametchougle Dampotime, Kossi Atchonouglo and Edo-Owodou Ayeleh. Invariants of 3D Anisotropic Elasticity Tensor: Measure of Symmetry Defects. International Journal of Physics. 2024; 12(5):206-211. doi: 10.12691/ijp-12-5-3

Abstract

Hooke’s Law Formula is given as , where F denotes the force applied, is the displacement extension and k is the spring constant or force constant. In classical linear elasticity, with the stress tensor σ and the strain tensor ε, the generalized Hooke's law is written: , where the tensor of fourth order is called tensor of elasticity. It is a tensor generalization of the stiffness constant k of a spring. The invariants of the elasticity tensors represent mechanical characteristics of the anisotropic materials (such as elasticity, ductility, resistance to deformations), help to classify materials following their symmetries, which generalize the concept of “stiffness of a spring”. In this paper, we perform the calculation of invariants for the anisotropic elasticity tensor under the rotation action of SO(3) groups. The tools developed by G. de Saxé et al. [1] to determine the independent invariants in 2D, essentially consisting of changing reference base and Kelvin’s decomposition of the elasticity tensor, are borrowed for the 3D case. In total, eighteen independent invariants also called global invariant emerge including 5 for the first order and thirteen for higher order. At the end, we give the physical signification of these invariants for isotropic materials.

Keywords:
Representation theory Anisotropy Linear elasticity

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References:

[1]	De Saxé G. and Vallée C., 2012. Invariant Measures of the Lack of Symmetry with Respect to the Symmetry Groups of 2D Elasticity Tensors, J Elast, pp. 21–39.
[2]	Cowin S., 1989. Properties of the anisotropic elasticity tensor, The Quarterly Journal of Mechanics and Applied Mathematics, pp. 249–266.
[3]	Hehl F. W. and Itin Y., 2002. The Cauchy Relations in Linear Elasticity Theory, Journal of elasticity and the physical science of solids, pp. 185–192.
[4]	Olive M. and Kolev B. and Auffray N., 2015. Les invariants du tenseur d’élasticité, 22ème congrès français de Mécanique [CFM2015], hal–01576369.
[5]	Ahmad M., 2002. Invariants and structural invariants of the anisotropic elasticity tensor, Q. Jl Mech. Appl. Math., pp. 597–606.
[6]	Norris A., 2007. Quadratic invariants of elastic moduli, Q. Jl Mech. Appl. Math., pp. 367–389.
[7]	Thomson, W. (Lord Kelvin), 1890, Mathematical and Physical Papers. Elasticity, Heat, Electromagnetism, vol.3. Cambridge University Press, Cambridge.
[8]	Ting T., 1987. Invariants of anisotropic elastic constants, The Quarterly Journal of Mechanics and Applied Mathematics, pp. 431–448.
[9]	Thomson W., 1856. Elements of a Mathematical Theory of Elasticity,” Philosophical Transactions”, pp. 481–498 146 (Part II).
[10]	Boehler J.-P., Kirillov A. A., Onat E. T., 1994. On the polynomial invariants of the elastic tensor. J. Elast. 34, 97–110.
[11]	Forte S. and Vianello M. «Symmetry classes for elasticity tensors». In: Journal of Elasticity 43.2 (mai 1996), p. 81-108. (cf. p. 10).
[12]	Auffray N., Ropars P., 2016. Invariant-based reconstruction of bidimensionnal elasticity tensors. International Journal of Solids and Structures, Elsevier, 87, pp.183-193.
[13]	Atchonouglo K. and de Saxcé G. and Ban M., 2021. 2d elasticity tensor invariants, invariants definite positive criteria, Advances in Mathematics: Scientific Journal, pp.2999–3012.
[14]	Olive M., 2017. About Gordan’s Algorithm for Binary Forms, Found Comput Math, pp. 1407–1466.
[15]	Olive M. and Desmorat R. and Auffray N. and Desmorat B. and Kolev B., 2022. Minimal functional bases for elasticity tensor symmetry classes, Journal of Elasticity.
[16]	Monteghetti F., 2012. Quaternions, orientation et mouvement, [Rapport de recherche] ISAE-SUPAERO, hal–01618257.
[17]	Euler L., 1771. Problema algebraicum ob affectiones prorsus singulars memorabile, Novi Commentarii academiae scientiarum Petropolitanae, pp. 75–106 15.
[18]	Lekhnttskii S. G., 1981. Theory of elasticity of an anisotropic body, Editions Mir, pp. 39–94.
[19]	Hearmon R. F. S., 1961. An Introduction to Applied Anisotropic Elasticity, Oxford University Press, vol.13, pp. 136.
[20]	Tsai S. W.,1966. Introduction to mechanics of composite materials, part ii—theoretical aspects, Air Force Materials Laboratory, pp. 66–149.
[21]	Zuber J., «Introduction à la théorie des groupes et de leurs représentations». Lecture. Sept. 2006. url: https:// cel.hal.science/cel-00092968 (cf. p. 22).50.
[22]	Dieulesaint, E & Royer, D. (Daniel). (1974). Ondes elastiques dans les solides : application au traitement du signal / par E. Dieulesaint et D. Royer ; pref. du Pr P. Grivet. Paris: Masson.