Unsteady electromagnetic elasticity of piezoelectrics considering diffusion

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

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 2, pp. 193-204.

Résumé

The paper considers a model of the linear theory of deformation of elastic continuum with diffusion and piezoelectric effect taken into account, which describes the relationship between mechanical deformations, mass transfer, and the internal electric field. A one-dimensional model of electromagnetic diffusion in a rectangular Cartesian coordinate system is used. At the present level, the methods of solving the corresponding initial-boundary value problems based on the application of the integral Laplace transform and decomposition into trigonometric Fourier series are described. Based on the solution of model problems, the effect of the fields coupling on the processes of dynamic deformation are shown. The results of the calculations are presented in analytical form and in the form of graphs.

Bibliographie

[1] Afram A. Y., Khader S. E., “2D Problem for a Half-Space under the Theory of Fractional Thermoelastic Diffusion”, American Journal of Scientific and Industrial Research, 6:3 (2014), 47–57 | DOI

[2] Atwa S. Y., Egypt Z., “Generalized Thermoelastic Diffusion with Effect of Fractional Parameter on Plane Waves Temperature-Dependent Elastic Medium”, Journal of Materials and Chemical Engineering, 1:2 (2013), 55–74 | MR

[3] Belova I. V., Murch G. E., “Thermal and diffusion-induced stresses in crystalline solids”, Journal of Applied Physics, 77:1 (1995), 127–134 | DOI

[4] Choudhary S., Deswal S., “Mechanical loads on a generalized thermoelastic medium with diffusion”, Meccanica, 45 (2010), 401–413 | DOI | MR | Zbl

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

[6] El-Sayed A. M., “A two-dimensional generalized thermoelastic diffusion problem for a half-space”, Mathematics and Mechanics of Solids, 21:9 (2016), 1045–1060 | DOI | MR | Zbl

[7] Knyazeva A. G., “Model of medium with diffusion and internal surfaces and some applied problems”, Mater. Phys. Mech., 7:1 (2004), 29–36

[8] Kumar R., Chawla V., “Green's Functions in Orthotropic Thermoelastic Diffusion Media”, Engineering Analysis with Boundary Elements, 36:8 (2012), 1272–1277 | DOI | MR | Zbl

[9] Olesiak Z. S., Pyryev Yu. A., “A coupled quasi-stationary problem of thermodiffusion for an elastic cylinder”, International Journal of Engineering Science, 33:6 (1995), 773–780 | DOI | MR | Zbl

[10] Pidstryhach Ya. S., “Differential equations of the problem of thermodiffusion in a solid deformable isotropic body”, Dopov. Akad. Nauk Ukr. RSR, 1961, no. 2, 169–172 (in Ukrainian)

[11] Sherief H. H., El-Maghraby N. M., “A Thick Plate Problem in the Theory of Generalized Thermoelastic Diffusion”, Int. J. Thermophys., 30 (2009), 2044–2057 | DOI

[12] Aouadi M., “Variable electrical and thermal conductivity in the theory of generalized thermoelastic diffusion”, Zeitschrift fur Angewandte Mathematik und Physik, 57:2 (2006), 350–366 | DOI | MR | Zbl

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

[14] Tarlakovskii D. V., Vestyak V. A., Zemskov A. V., “Dynamic Processes in Thermo-Electro-Magneto-Elastic and Thermo-Elasto-Diffusive Media”, Encyclopedia of Thermal Stresses, 6, ed. Hetnarski R. B., Springer, Dordrecht–Heidelberg–N. Y.–London, 2014, 1064–1071 | DOI

[15] Zhang J., Li Y., “A Two-Dimensional Generalized Electromagnetothermoelastic Diffusion Problem for a Rotating Half-Space”, Mathematical Problems in Engineering, 2014 (2014), 964218, 12 pp. | DOI | MR

[16] Chu J. L., Lee S., “Diffusion-induced stresses in a long bar of square cross section”, J. Appl. Phys., 73:7 (1993), 3211–3219 | DOI

[17] Freidin A. B., Korolev I. K., Aleshchenko S. P., Vilchevskaya E. N., “Chemical affinity tensor and chemical reaction front propagation: theory and FE-simulations”, Int. J. Fract., 202:2 (2016), 245–259 | DOI | MR

[18] Hwang C. C., Chen K. M., Hsieh J. Y., “Diffusion-induced stresses in a long bar under an electric field”, J. Phys. D : Appl. Phys., 27:10 (1994), 2155–2162 | DOI

[19] Indeitsev D. A., Semenov B. N., Sterlin M. D., “The Phenomenon of Localization of Diffusion Process in a Dynamically Deformed Solid”, Doklady Physics, 57:4 (2012), 171–173 | DOI

[20] Zemskov A. V., Tarlakovskii D. V., “Statement of the one-dimensional problem of thermoelectromagnetoelastic diffusion”, Dynamic and Technological Problems of a Mechanics of Constructions and Continuous Mediums, Proc. XXIV Int. Symposium Dedicated to A. G. Gorshkov, v. 2, TRP, M., 2018, 157–163 (in Russian)

[21] Davydov S. A., Zemskov A. V., Tarlakovskiy D. V., “Surface Green's function in non-stationary problems of thermomechanical diffusion”, Journal Problems of Strength and Plasticity, 79:1 (2017), 38–47 (in Russian) | DOI

[22] Ditkin V. A., Prudnikov A. P., Handbook of operational calculus, Vysshaya shkola, M., 1965, 466 pp. (in Russian) | MR

[23] Zverev N. A, Zemskov A. V., Tarlakovskii D. V., “One-dimensional problem of piezoelectric electromagnetic diffusion for a layer”, Journal of Physics : Conference Series, 1129 (2018), 012040 | DOI

[24] Bardzokas D. I., Zobnin A. I., Senik N. A., Fil'shtinskii M. L., Mathematical modeling in problems of mechanics of related fields, in 2 vols, v. 1, Introduction to the theory of piezoelectricity, KomKniga, M., 2005, 312 pp. (in Russian)