Model of bending of an orthotropic cantilever beam of Bernoulli--Euler under the action of~unsteady thermomechanodiffusion loads

A. V. Zemskov ; V. H. Le ; D. O. Serdyuk

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 28 (2024) no. 4, pp. 682-700.

Résumé

The study investigates the interaction of mechanical, thermal, and diffusion fields during nonstationary bending of a cantilevered beam. The mathematical formulation of the problem is based on a system of equations describing nonstationary flexural vibrations of a Bernoulli–Euler beam, accounting for heat and mass transfer. This system is derived from the general thermomechanodiffusion model for continuum media using the generalized principle of virtual displacements. The research assumes a finite velocity of thermal and diffusive perturbation propagation. The interaction of mechanical, thermal, and diffusion fields is analyzed using a cantilevered three-component beam composed of a zinc–copper–aluminum alloy under the action of a nonstationary load applied to its free end.

Détail

Keywords: thermomechanical diffusion, Green's function, equivalent boundary conditions method, unsteady problems
Mots-clés : Bernoulli–Euler beam, console

Bibliographie

[1] Flyachok V. M., Shvets R. N., “Theorems of the mechano-thermodiffusion theory of anisotropic shells”, Mat. Metody Fiz.-Mekh. Polya, 1985, no. 21, 32–37 (In Russian) | Zbl

[2] Shvets R. N., Flyachok V. M., “Equations of the mechano-thermodiffusion of anisotropic shells taking into account transverse deformations”, Mat. Metody Fiz.-Mekh. Polya, 1984, no. 20, 54–61 (In Russian) | Zbl

[3] Shvets R. N., Flyachok V. M., “Variational approach to the solution of dynamical problems of thermoelastic diffusion for anisotropic shells”, Mat. Fiz. Nelin. Mekh., 1991, no. 16, 39–43 (In Russian)

[4] Ravrik M. S., “A variational mixed formula for contact problems of thermal diffusion theory of deformation of laminated shells”, Mat. Metody Fiz.-Mekh. Polya, 1985, no. 22, 40–44 (In Russian) | Zbl

[5] Bhattacharya D., Kanoria M., “The influence of two temperature generalized thermoelastic diffusion inside a spherical shell”, Int. J. Eng. Technol. Res., 2:5 (2014), 151–159

[6] Ravrik M. S., Bichuya A. L., “Axisymmetric stress state of a heated transversal-isotropic spherical shell with circular opening at diffusion saturation”, Mat. Metody Fiz.-Mekh. Polya, 1983, no. 17, 51–54 (In Russian) | Zbl

[7] Aouadi M., Copetti M. I. M., “Analytical and numerical results for a dynamic contact problem with two stops in thermoelastic diffusion theory”, ZAMM, 96:3 (2016), 361–384 | DOI | Zbl

[8] Aouadi M., Copetti M. I. M., “A dynamic contact problem for a thermoelastic diffusion beam with the rotational inertia”, Appl. Numer. Math., 126 (2018), 113–137 | DOI | Zbl

[9] Aouadi M., Copetti M. I. M., “Exponential stability and numerical analysis of a thermoelastic diffusion beam with rotational inertia and second sound”, Math. Comput. Simul., 187 (2021), 586–613 | DOI | Zbl

[10] Aouadi M., Miranville A., “Smooth attractor for a nonlinear thermoelastic diffusion thin plate based on Gurtin–Pipkin's model”, Asymptotic Anal., 95:1–2 (2015), 129–160 | DOI | Zbl

[11] Copetti M. I. M., Aouadi M. A, “A quasi-static contact problem in thermoviscoelastic diffusion theory”, Appl. Numer. Math., 109 (2016), 157–183 | DOI | Zbl

[12] Shevchuk P. R., Shevchuk V. A., “Mechanodiffusion effect in bending a two-layer bar”, Mater. Sci., 23:6 (1987), 604–608 | DOI

[13] Huang M., Wei P., Zhao L., Li Y., “Multiple fields coupled elastic flexural waves in the thermoelastic semiconductor microbeam with consideration of small scale effects”, Compos. Struct., 270 (2021), 114104 | DOI

[14] Kumar R., Devi S., Sharma V., “Resonance of nanoscale beam due to various sources in modified couple stress thermoelastic diffusion with phase lags”, Mech. Mech. Eng., 23:1 (2019), 36–49 | DOI

[15] Aouadi M., “On thermoelastic diffusion thin plate theory”, Appl. Math. Mech., 36:5 (2015), 619–632 | DOI | Zbl

[16] Miranville A., Aouadi M., “Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory”, Evol. Equ. Control Theory, 4:3 (2015), 241–263 | DOI | Zbl

[17] Mikolaychuk M. A., Knyazeva A. G., Grabovetskaya G. P., Mishin I. P., “Research of the stress influence on the diffusion in the coating plate”, PNRPU Mechanics Bulletin, 2012, no. 3, 120–134 (In Russian)

[18] Zemskov A. V., Le V. H., Serdyuk D. O., “A model of bending of an orthotropic cantilevered Bernoulli–Euler beam under the action of nonstationary thermomechanodiffusion loads”, Innovatsionnoe razvitie transportnogo i stroitel'nogo kompleksov [Innovative Development of Transport and Construction Complexes], v. 2, Belarusian State University of Transport, Gomel', 2023, 90–92 (In Russian)

[19] Vol'fson E. F., “To the 80th anniversary of the famous opening of V.S. Gorskii”, Izv. Vyssh. Uchebn. Zaved. Chernaya Metallurgiya, 59:5 (2016), 357–359 (In Russian) | DOI

[20] Kukushkin S. A., Osipov A. V., “The Gorsky effect in the synthesis of silicon-carbide films from silicon by topochemical substitution of atoms”, Tech. Phys. Lett., 43:7 (2017), 631–634 | DOI | DOI

[21] Knyazeva A. G., Vvedenie v termodinamiku neobratimykh protsessov. Lektsii o modeliakh [Introduction to Thermodynamics of Irreversible Processes. Lectures on Models], Ivan Fedorov, Tomsk, 2014, 172 pp. (In Russian)

[22] Keller I. E., Dudin I. S., Mekhanika sploshnoi sredy. Zakony sokhraneniia [Continuum Mechanics. Conservation Laws], Perm', PNIPU, 2022, 142 pp. (In Russian)

[23] Zemskov A. V., Tarlakovskii D. V., Modelirovanie mekhanodiffuzionnykh protsessov v mnogokomponentnykh telakh s ploskimi granitsami [Modeling of Mechanodiffusion Processes in Multicomponent Bodies with Flat Boundaries], Fizmatlit, Moscow, 2021, 288 pp. (In Russian)

[24] Zemskov A. V., Le V. H., Tarlakovskii D. V., “Bernoulli–Euler beam unsteady bending model with consideration of heat and mass transfer”, J. Appl. Comp. Mech., 9:1 (2023), 168–180 | DOI

[25] Zemskov A. V., Tarlakovskii D. V., Faykin G. M., “Unsteady bending of the orthotropic cantilever Bernoulli–Euler beam with the relaxation of diffusion fluxes”, ZAMM, 102:10 (2022), e202100107 | DOI | Zbl

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

[27] Hirano K., Cohen M., Averbach B. L., Ujiiye N., “Self-diffusion in alpha iron during compressive plastic flow”, Trans. Metall. Soc. AIME, 227 (1963), 950–957