Acoustic waves in hypoelastic solids. II. Anisotropic materials
https://doi.org/10.22405/2226-8383-2024-25-2-334-349
Abstract
For hypoelastic anisotropic materials, the constitutive relations of nonlinear elasticity are written out. The relations establish a connection between the generalized Yaumann derivatives of the stress tensor and the nonholonomic strain measure introduced in the works of A. A. Markin. The relation is concretized for anisotropic materials with cubic symmetry
of properties. The relations are written in projections into elastic eigen subspaces of a cubic material. The analysis of the mutual influence of finite deformation processes belonging to different eigen subspaces is carried out.
Deformation processes in which the main axes of strains coincide with the same material fibers, whose position relative to the main axes of anisotropy does not change, are considered. Elastic potentials for such processes are written out for two material models: a general one containing nine elastic constants, and a model satisfying the generalization of A. A. Ilyushin
particular postulate to anisotropic materials and containing six constants.
The results of solution of the problem of acoustic wave propagation in hypoelastic anisotropic materials with cubic symmetry of properties are presented. As initial strains of a material, deformations located entirely in its elastic eigen subspaces are considered: purely volumetric
strain, tension-compression in the main axes of anisotropy, pure shear in the plane of the main
axes of anisotropy. Preliminary finite purely volumetric deformations do not affect the shape
of the angular dependences of the phase velocities of wave propagation in the vertical plane, but only affect the values of the phase velocities. At preliminary shaping, the material acquires additional anisotropy, and the shape of the angular dependencies of the phase velocities changes. The results show that in some cases, the six-constant model of the material does not predict a change in the propagation velocities of transverse waves at initial finite strains.
About the Authors
Marina Sokolova Yur'evna
Tula State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
Dmitriy Viktorovich Khristich
Tula State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
References
1. Thurston, R. N. & Brugger, K. 1964. “Third-order elastic constants and the velocity of small
2. amplitude waves in homogeneously stressed media”, Phys. Rev., vol 133, pp. A1604–A1610.
3. Brugger, K. 1965, “Pure modes for elastic waves in crystals”, J. Appl. Phys., vol. 36, no. 3, pp.
4. -768.
5. Chernykh, K. F. 1988. “Introduction to anisotropic elasticity”, Nauka, Moscow. [in Russian].
6. Knowles, K. M. 2016. “The biaxial moduli of cubic materials subjected to an equi-biaxial elastic strain”, J. Elast., vol. 124, p. 1-25.
7. Duffy, T. S. 2018. “Single-crystal elastic properties of minerals and related materials with cubic symmetry”, American Mineralogist, vol. 103, no. 6, p. 977-988.
8. Kambouchev, N., Fernandez, J. & Radovitzky, R. 2007. “A polyconvex model for materials with cubic symmetry”, Modelling and Simulation in Materials Science and Engineering, vol. 15, n. 5, p. 451-468.
9. Kube, C. M. & Turner, J. A. 2016. “Estimates of nonlinear elastic constants and acoustic
10. nonlinearity parameters for textured polycrystals”, J. Elast., vol. 122, no. 2, pp. 157-177.
11. Kube, C. M. 2017. “Scattering of harmonic waves from a nonlinear elastic inclusion”, J. Acoust. Soc. Am., vol. 141, no. 6.
12. Markin, A.A. & Sokolova, M.Yu. 2015. “Thermomechanics of elastoplastic deformation”,
13. Cambridge International Science Publishing, Cambridge.
14. Markin, A. A., Sokolova, M.Yu. & Khristich, D. V. 2011. “A.A. Il’yushin’s postulate for
15. anisotropic materials and a version of constitutive relations”, Mech. Solids, vol. 46, no. 1, p.
16. -35.
17. Markin, A. A. & Tolokonnikov, L. A. 1987. “Measures of the finite deformation processes”, News of the North Caucasus Scientific Center of Higher Education. Natural Sciences, n. 2, p. 49-53 [in Russian].
18. Markin, A. A., Sokolova, M.Yu. & Khristich, D. V. 2021. “Nonlinear elasticity of cubic crystals”, Elasticity and anelasticity. Proceedings of the International scientific symposium on problems of the mechanics of deformable bodies dedicated to the 110th anniversary from the birthday of A. A. Ilyushin (Moscow, January 20-21, 2021). Edited by Prof. G. L. Brovko, Prof.
19. D. V. Georgievskii, Prof. I. N. Molodtsov. Moscow, Moscow University Press, p. 100-110 [in
20. Russian].
21. Ostrosablin, N. I. 1984. “On the structure of the elastic moduli tensor. Eigen elastic states”,
22. Dynamics of a solid deformable body, vol. 66, p. 113-125 [in Russian].
23. Rykhlevsky, J. 1984. “About Hooke’s Law”, J. Appl. Math. Mech., vol. 48, n. 3, p. 420-435 [in Russian].
24. Sokolova, M.Yu. & Khristich, D. V., 2021. “Finite strains of nonlinear elastic anisotropic
25. materials”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika [Tomsk
26. State University Journal of Mathematics and Mechanics], no. 70, p. 103–116 [in Russian].
27. Guz’, AN. 1986. “Elastic waves in bodies with initial stresses”, in two volumes, Naukova Dumka, Kiev, [in Russian].
28. Lurie, A. I. 2012. “Non-linear theory of elasticity”, North Holland.
Review
For citations:
Yur'evna M.S., Khristich D.V. Acoustic waves in hypoelastic solids. II. Anisotropic materials. Chebyshevskii Sbornik. 2024;25(2):334-349. (In Russ.) https://doi.org/10.22405/2226-8383-2024-25-2-334-349
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