Acoustic waves in hypoelastic solids. I. Isotropic materials

M. Yu. Sokolova ; D. V. Khristich

Čebyševskij sbornik, Tome 25 (2024) no. 2, pp. 318-333.

Résumé

Two models of hypoelastic isotropic materials based on the use of a nonholonomic strain measure, the generalized Yaumann derivative of which coincides with the strain rate tensor, are considered. The conditions under which elastic strain potential exists in such models are formulated. The elastic potential and the constitutive relations are written in terms of elastic eigen subspaces of an isotropic material. The models differ in the number of elastic constants. It is shown that the four-constant model satisfies the requirements of A.A.Ilyushin particular postulate of isotropy, and the five-constant model does not satisfy them. The equation of acoustic wave propagation in such materials is obtained. The influence of the use of the particular isotropy postulate as a hypothesis on the results of dynamic problems solution is investigated. The phase velocities of acoustic wave propagation under various types of initial strains are determined for two models. At preliminary purely volumetric strains, calculations using five-constant and four-constant models give the same result. For strains located in the deviatoric subspace, the stress tensor has a component located in the first elastic eigen subspace, and its projection into the second subspace when using the five-constant model is misaligned to the strain deviator. At the same time, the initially isotropic material acquires anisotropy with respect to acoustic properties. The material model satisfying the particular postulate of isotropy in the case under consideration also describes the anisotropy of longitudinal waves propagation velocities.

Détail

Keywords: acoustic waves, finite strains, phase velocities of wave propagation, isotropic materials, hypoelastic materials, particular postulate of isotropy.

Bibliographie

[1] Pasmanik, L. A., Kamyshev, A. V., Radostin, A. V., Zaitsev, V. Yu., “Acoustic inhomogeneity parameters for non-destructive evaluation of the influence of manufacturing technology and operational damage on the metal structure”, Defectoscopy, 2020, no. 12, 24–36 | DOI

[2] Glushkov, E. V., Glushkova, N. V., Golub, M,V., Eremin, A. A., “Resonant Method for Detection and Identification of Delamination in Composite Plates by Elastic Guided Waves”, Mech. Solids, 55:6 (2020), 865–871 | DOI | DOI

[3] Jiang Y., Li G., Qian L.-X., Liang S., Destrade M., Cao Y., “Measuring the linear and nonlinear elastic properties of brain tissue with shear waves and inverse analysis”, Biomech. Model. Mechanobiol., 14:5 (2015), 1119–1128 | DOI

[4] Truesdell C., “General and exact theory of waves in finite elastic strain”, Arch. Ratio. Mech. Anal., 8:1 (1961), 263–296 | DOI | MR | Zbl

[5] Guz', A. N., Elastic waves in bodies with initial stresses, in two volumes, Naukova Dumka, Kiev, 1986 (in Russian)

[6] Lurie, A. I., Non-linear theory of elasticity, North Holland, 2012

[7] Haupt P., Pao Y.-H., Hutter K., “Theory of incremental motion in a body with initial elasto-plastic deformation”, J. Elasticity, 28 (1992), 193–221 | DOI | MR | Zbl

[8] Destrade M., Ogden R. W., “On stress-dependent elastic moduli and wave speeds”, J. Appl. Math., 78:5 (2013), 965–997 | MR | Zbl

[9] Romenskii, E. I, Lys', E. V., Cheverda, V. A., Epov, M. I., “Dynamics of deformation of an elastic medium with initial stresses”, J. Appl. Mech. Tech. Phys., 58:5 (2017), 914–923 | DOI | MR | Zbl

[10] Belyankova, T. I., Kalinchuk, V. V., Sheidakov, D., “Higher-order moduli in the equations of dynamics of a prestressed elastic body”, Mech. Soldis, 2019, no. 3, 3–15 (in Russian) | DOI

[11] Yang H., Fu Li-Yun, Fu Bo-Ye, Müller T. M., “Acoustoelastic FD simulation of elastic wave propagation in prestressed media”, Front. Earth Sci., 10 (2022)

[12] Zhu Q., Burtin C., Binetruy C., “Acoustoelastic effect in polyamide 6: Linear and nonlinear behaviour”, Polym. Test, 40 (2014), 178–186 | DOI | MR

[13] Brovko, G. L., Constitutive relations of continuum mechanics: The development of the mathematical apparatus and the foundations of general theory, Nauka, M., 2017 (in Russian)

[14] Markin, A. A., Tolokonnikov, L. A., “Measures of the finite deformation processes”, News of the North Caucasus Scientific Center of Higher Education. Natural Sciences, 1987, no. 2, 49–53 (in Russian) | Zbl

[15] Markin, A. A., Sokolova, M. Yu., Thermomechanics of elastoplastic deformation, Cambridge International Science Publishing, Cambridge, 2015

[16] Ostrosablin, N. I., “On the structure of the elastic moduli tensor. Eigen elastic states”, Dynamics of a solid deformable body, 66, 1984, 113–125 (in Russian) | MR

[17] Rykhlevsky, J., “About Hooke's Law”, J. Appl. Math. Mech., 48:3 (1984), 420–435 (in Russian) | MR

[18] Ilyushin, A. A., “Questions of the general theory of plasticity”, J. Appl. Math. Mech., 24:3 (1960), 399–411 | MR | Zbl

[19] Markin, A. A., Sokolova M. Yu., “A variant of relations of the nonlinear elasticity”, Mech. Solids, 2019, no. 6, 68–75 (in Russian) | DOI

[20] Astapov Y., Khristich D., Markin A., Sokolova M., “The construction of nonlinear elasticity tensors for crystals and quasicrystals”, Int. J. Appl. Mech., 9:6 (2017), 1750080-1-1750080-15 | DOI

[21] Markin, A. A., Sokolova, M. Yu., Khristich, D. V., “A.A. Il'yushin's postulate for anisotropic materials and a version of constitutive relations”, Mech. Solids, 46:1 (2011), 30–35 | DOI | MR