Identification of a model of a nonlinear elastic anisotropic material with cubic symmetry of properties
https://doi.org/10.22405/2226-8383-2023-24-3-320-332
Abstract
The propagation of acoustic waves in nonlinear elastic anisotropic media with finite preliminary strains is considered. The media in the initial state are homogeneous with an elastic potential in which the first two nonzero terms of expansion in a series by degrees of the strain tensor are preserved. The dynamic equations are written as the equations of propagation of small displacement perturbations imposed on finite strains. The equations are concretized for
the case of propagation of plane monochromatic waves.
An anisotropic material with the symmetry of properties inherent in crystals of cubic symmetry is considered. Constitutive relations of the nonlinear model are written in terms of the basis tensors of the eigen elastic subspaces of the fourth and sixth ranks. The relations include three second-order constants and six third-order constants. A program of experiments for determining the constants of elasticity of a cubic material is proposed.
To determine the elasticity constants of the second order, it is proposed to fulfill an experiment to measure the phase velocities of longitudinal and two transverse waves propagating along the edge of a prismatic sample. To determine the elasticity constants of the third order, the phase velocities of acoustic wave propagation are measured in two samples differing in the orientation of the main axes of anisotropy. In the samples preliminary tension-compression
strains are created along the two edges.
The results of numerical simulation of the proposed experiments for niobium crystals, whose elastic properties are known from sources, are presented. Sections of the surfaces of the phase velocities of longitudinal (quasi-longitudinal) and transverse (quasi-transverse) waves found at different levels of preliminary deformations proposed in the experimental program are constructed. It is shown that not only the values of the wave propagation velocities depend on the level of strains, but also the shape of the cross sections of the phase velocity surfaces with different planes.
About the Authors
Marina Yurievna Sokolova
Tula State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
Dmitry 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 amplitude waves in homogeneously stressed media“, Phys. Rev., vol 133, pp. A1604–A1610.
2. Brugger, K. 1965, “Pure modes for elastic waves in crystals“, J. Appl. Phys., vol. 36, no. 3, pp. 759-768.
3. Eastman, D. E. 1966, “Measurement of third order elastic moduli of ittrium iron garnet“, J. Appl. Phys., vol. 37, no. 6, pp. 2312-2316.
4. Pham, H. H. & Cagin, T. 2010, “Lattice dynamics and second and third order elastic constants of iron at elevated pressures“, Computers, materials and continua CMC, vol. 16, no. 2, pp. 175-194.
5. Kube, C. M. & Turner, J. A. 2016, “Estimates of nonlinear elastic constants and acoustic nonlinearity parameters for textured polycrystals“, Journal of Elasticity, vol. 122, no. 2, pp. 157-177.
6. Kube, C. M. 2017, “Scattering of harmonic waves from a nonlinear elastic inclusion“, J. Acoust. Soc. Am., vol. 141, no. 6.
7. Li, X. 2017, “First-principles study of the third-order elastic constants and related anharmonic properties in refractory high-entropy alloys“, Acta Materialia, vol. 142.
8. Telichko, A. V., Erohin, S. V., Kvashnin, G. M., Sorokin, P. B., Sorokin, B.P. & Blank, V.D˙ . 2017, “Diamond’s third-order elastic constants: ab initio calculations and experimental investigation“, J. Mater. Sci., vol. 52, no. 6, pp. 3447–3456.
9. Lubarda, V. A. 1997, “New estimates of the third-order elastic constants for isotropic aggregates of cubic crystals“, J. Mech. Phys. Solids, vol. 45, no. 4, pp. 471-490.
10. Zhang, H., Lu, D., Sun, Y., Fu, Y. & Tong, L. 2022, “The third-order elastic constants and mechanical properties of 30∘ partial dislocation in germanium: A study from the first-principles calculations and the improved Peierls–Nabarro model“, Crystals, vol. 12, no. 1, 4.
11. Guz’, AN. 1986, “Elastic waves in bodies with initial stresses“, in two volumes, Naukova Dumka, Kiev, [in Russian].
12. Lurie, A. I. 2012, “Non-linear theory of elasticity“, North Holland.
13. Sirotin, Yu. & Shaskolskaya, M. 1982, “Fundamentals of crystal physics“, Mir Publishers, Moscow, 656 p.
14. Fedorov, F. I. 1965, “Theory of elastic waves in crystals“, Nauka, Moscow, 388 p., [in Russian]
15. Sokolova, M.Yu. & Khristich, D. V. 2021, “Finite strains of nonlinear elastic anisotropic materials“, Tomsk State University Journal of Mathematics and Mechanics, vol. 70, pp. 103-116.
16. Markin, A. A. & Sokolova, M.Yu. 2015, “Thermomechanics of elastoplastic deformation“, Cambridge International Science Publishing, Cambridge.
17. Pau, A. & Vestroni, F. 2019, “The role of material and geometric nonlinearities in acoustoelasticity“, Wave Motion, vol. 86, pp. 79-90.
18. Sokolova, M., Astapov, Y. & Khristich, D. 2021, “Identification of the model of nonlinear elasticity in dynamic experiments“, Int. J. Appl. Mech., vol. 13, no. 2, 2150025.
Review
For citations:
Sokolova M.Yu., Khristich D.V. Identification of a model of a nonlinear elastic anisotropic material with cubic symmetry of properties. Chebyshevskii Sbornik. 2023;24(3):320-332. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-3-320-332
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