Abstract
An invariant (with respect to rotations) formalization of equations of linear and nonlinear elasticity theory is proposed. An equation of state (in the form of a convex generating potential) for various crystallographic systems is written. An algebraic approach is used, which does not require any geometric constructions related to the analysis of symmetry in crystals.
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References
S. K. Godunov and E. I. Romensky, Elements of Continuum Mechanics and Conservation Laws, Kluwer Acad. Publ., Dordrecht (2003).
S. K. Godunov, “The equations of elasticity with dissipation as a nontrivial example of thermodynamically consistent hyperbolic equations,” J. Hyperbol. Differ. Eq., 1, No. 2, 235–249 (2004).
S. P. Kiselev, Mechanics of Continuous Media [in Russian], Novosibirsk State Technical University, Novosibirsk (1997).
G. Leibfried and N. Brauer, Point Defects in Metals. Introduction to the Theory, Springer, Heidelberg (1978).
S. K. Godunov and T. Yu. Mikhailova, Presentations of Rotation Groups and Spherical Functions [in Russian], Nauchnaya Kniga, Novosibirsk (1998).
S. K. Godunov and V. M. Gordienko, “Clebsch-Gordan coefficients for various choices of the bases of unitary and orthogonal presentations of the SU(2) and SO(3) groups,” Sib. Mat. Zh., 45, No. 3, 540–557 (2004).
S. V. Selivanova, “Elements of the theory of presentations of the group of rotations and their application in the elasticity theory,” MA Dissertation, Novosibirsk (2007).
S. K. Godunov and V. M. Gordienko, “Complicated structures of Galilean-invariant conservation laws,” J. Appl. Mech. Tech. Phys., 43, No. 2, 175–189 (2002).
N. I. Ostrosablin, “On the structure of the elastic tensor and the classification of anisotropic materials,” J. Appl. Mech. Tech. Phys., 27, No. 4, 600–607 (1986).
R. M. Garipov, “Hooke’s law for single crystals,” Preprint, Inst. Hydrodynamics, Sib. Div., Russ. Acad. of Sci., Novosibirsk (2005).
B. I. Delone, N. Padurov, and A. D. Aleksandrov, Mathematical Fundamentals of the Structural Analysis of Crystals [in Russian], Gostekhteorizdat, Moscow (1934).
N. I. Ostrosablin, “On invariants of the fourth-rank tensor of elasticity moduli,” Sib. Zh. Indust. Mat., 1, No. 1, 155–163 (1998).
L. D. Landau and E. M. Lifshits, Theory of Elasticity, Pergamon Press, Oxford-New York (1970).
F. I. Fedorov, Theory of Elastic Waves in Crystals [in Russian], Nauka, Moscow (1965).
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Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 127–142, September–October, 2008.
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Selivanova, S.V. Invariant recording of elasticity theory equations. J Appl Mech Tech Phy 49, 809–822 (2008). https://doi.org/10.1007/s10808-008-0101-8
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DOI: https://doi.org/10.1007/s10808-008-0101-8