Modelling one-dimensional elastic diffusion processes in~an~orthotropic solid cylinder under unsteady volumetric perturbations

N. A. Zverev ; A. V. Zemskov ; D. V. Tarlakovskii

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 26 (2022) no. 1, pp. 62-78.

Résumé

A polar-symmetric elastic diffusion problem is considered for an orthotropic multicomponent homogeneous cylinder under uniformly distributed radial unsteady volumetric perturbations. Coupled elastic diffusion equations in a cylindrical coordinate system is used as a mathematical model. The model takes into account a relaxation of diffusion effects implying finite propagation speed of diffusion perturbations. The solution of the problem is obtained in the integral convolution form of Green's functions with functions specifying volumetric perturbations. The integral Laplace transform in time and the expansion into the Fourier series by the special Bessel functions are used to find the Green's functions. The theory of residues and tables of operational calculus are used for inverse Laplace transform. A calculus example based on a three-component material, in which two components are independent, is considered. The study of the mechanical and diffusion fields interaction in a solid orthotropic cylinder is carried out.

Détail

Mots-clés : elastic diffusion, Laplace transform
Keywords: Fourier series, Green's functions, polar symmetric problems, unsteady problems, Bessel functions, cylinder.

Bibliographie

[1] Aouadi M., “Variable electrical and thermal conductivity in the theory of generalized thermoelastic diffusion”, ZAMP, 57:2 (2005), 350–366 | DOI

[2] Bachher M., Sarkar N., “Fractional order magneto-thermoelasticity in a rotating media with one relaxation time”, Math. Models Eng., 2:1 (2016), 57–68 https://www.extrica.com/article/17103

[3] Deswal S., Kalkal K., “A two-dimensional generalized electro-magneto-thermoviscoelastic problem for a half-space with diffusion”, Int. J. Thermal Sci., 50:5 (2011), 749–759 | DOI

[4] Kumar R., Chawla V., “Fundamental solution for two-imensional problem in orthotropic piezothermoelastic diffusion media”, Mater. Phys. Mech., 16:2 (2013), 159–174 https://mpm.spbstu.ru/en/article/2013.27.7/

[5] Zhang J., Li Y., “A two-dimensional generalized electromagnetothermoelastic diffusion problem for a rotating half-space”, Math. Probl. Eng., 2014 (2014), 1–12 | DOI

[6] Abbas A. I., “The effect of thermal source with mass diffusion in a transversely isotropic thermoelastic infinite medium”, J. Meas. Eng., 2:4 (2014), 175–184 https://www.extrica.com/article/15667

[7] Abbas A. I., “Eigenvalue approach on fractional order theory of thermoelastic diffusion problem for an infinite elastic medium with a spherical cavity”, Appl. Math. Model., 39:20 (2015), 6196–6206 | DOI | Zbl

[8] Aouadi M., “A generalized thermoelastic diffusion problem for an infinitely long solid cylinder”, Int. J. Math. Math. Sci., 2006 (2006), 1–15 | DOI

[9] Aouadi M., “A problem for an infinite elastic body with a spherical cavity in the theory of generalized thermoelastic diffusion”, Int. J. Solids Struct., 44:17 (2007), 5711–5722 | DOI | Zbl

[10] Bhattacharya D., Kanoria M., “The influence of two-temperature fractional order generalized thermoelastic diffusion inside a spherical shell”, IJAIEM, 3:8 (2014), 096–108

[11] Xia R. H., Tian X. G., Shen Y. P., “The influence of diffusion on generalized thermoelastic problems of infinite body with a cylindrical cavity”, Int. J. Eng. Sci., 47:5–6 (2009), 669–679 | DOI | Zbl

[12] Bhattacharya D., Pal P., Kanoria M., “Finite element method to study elasto-thermodiffusive response inside a hollow cylinder with three-phase-lag effect”, Int. J. Comp. Sci. Eng., 7:1 (2019), 148–156 | DOI

[13] Elhagary M. A., “Generalized thermoelastic diffusion problem for an infinitely long hollow cylinder for short times”, Acta Mech., 218:3–4 (2011), 205–215 | DOI | Zbl

[14] Elhagary M. A., “Generalized thermoelastic diffusion problem for an infinite medium with a spherical cavity”, Int. J. Thermophys., 33:1 (2012), 172–183 | DOI

[15] Shvets R. M., “On the deformability of anisotropic viscoelastic bodies in the presence of thermodiffusion”, J. Math. Sci., 97:1 (1999), 3830–3839 | DOI

[16] Minov A. V., “Study of the stress-strain state of a hollow cylinder subjected to the thermal diffusion effect of carbon in an axisymmetric thermal field variable in length”, Izv. Vuzov. Mashinostroenie, 2008, no. 10, 21–26 (In Russian)

[17] Deswal S., Kalkal K. K., Sheoran S. S., “Axi-symmetric generalized thermoelastic diffusion problem with two-temperature and initial stress under fractional order heat conduction”, Phys. B – Condensed Matter., 496 (2016), 57–68 | DOI

[18] Kumar R., Devi S., “Deformation of modified couple stress thermoelastic diffusion in a thick circular plate due to heat sources”, CMST, 25:4 (2019), 167–176 | DOI

[19] Olesiak Z. S., Pyryev Yu. A., “A coupled quasi-stationary problem of thermodiffusion for an elastic cylinder”, Int. J. Eng. Sci., 33:6 (1995), 773–780 | DOI | Zbl

[20] Tripathi J. J., Kedar G. D., Deshmukh K. C., “Generalized thermoelastic diffusion in a thick circular plate including heat source”, Alexandria Eng. J., 55:3 (2016), 2241–2249 | DOI

[21] Poroshina N. I., Ryabov V. M., “Methods for Laplace transform inversion”, Vestnik St. Petersb. Univ. Math., 44:3 (2011), 214–222 | DOI | Zbl

[22] Zemskov A. V., Tarlakovskii D. V., “Polar-symmetric problem of elastic diffusion for isotropic multi-component plane”, IOP Conf. Ser.: Mater. Sci. Eng., 158 (2016), 012101 | DOI

[23] Zemskov A. V., Tarlakovskii D. V., “Polar-symmetric problem of elastic diffusion for a multicomponent medium”, Problems of Strength and Plasticity, 80:1 (2018), 5–14 (In Russian) | DOI

[24] Zverev N. A., Zemskov A. V., Tarlakovskii D. V., “Modeling of unsteady coupled mechanodiffusion processes in a continuum isotropic cylinder”, Problems of Strength and Plasticity, 82:2 (2020), 156–167 (In Russian) | DOI

[25] Koshlyakov N. S., Gliner E. B., Smirnov M. M., Osnovnye differentsial'nye uravneniia matematicheskoi fiziki [Basic Differential Equations of Mathematical Physics], Nauka, Moscow, 1962, 768 pp.

[26] Ditkin V. A., Prudnikov A. P., Spravochnik po operatsionnomu ischisleniiu [Guide to operational Calculus], Vyssh. Shk., Moscow, 1965, 568 pp. (In Russian)

[27] Babichev A. P., Babushkina N. A., Bratkovskii A. M., et al., Fizicheskie velichiny: Spravochnik [Physical Quantities: Handbook], Energoatomizdat, Moscow, 1991, 1232 pp. (In Russian)

[28] Nachtrieb N. H., Handler G. S., “A relaxed vacancy model for diffusion incrystalline metals”, Acta Metal., 2:6 (1954), 797–802 | DOI

[29] Petit J., Nachtrieb N. H., “Self-diffusion in liquid gallium”, J. Chem. Phys., 24:5 (1956), 1027–1028 | DOI