Bending of an orthotropic cantilevered Bernoulli-Euler beam under an unsteady distributed transverse load considering heat and mass transfer


Аuthors

Zemskov A. V.*, Hao L. V.**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: azemskov1975@mail.ru
**e-mail: vanhaovtl@gmail.com

Abstract

Beams, plates, and shells constitute the foundational elements of most structures designed for various purposes. A crucial stage in advancing the physical understanding of material strength is the consideration of the influence of thermal and diffusive motion of particles within solid bodies on the deformation process, particularly when designing structures operating under multifactor external conditions. The paper investigates the coupled thermomechanodiffusion processes arising during the non-stationary bending of a cantilever beam subjected to a distributed load. It is assumed that the free end of the beam is thermally and mass-isolated, while the fixed end maintains constant temperature and concentrations of the substances constituting the beam. The deformations of the beam induce heating and generate upward diffusive fluxes, directed from the compression zones to the tension zones (Gorsky effect). The resulting heat and mass transfer directly affects the mechanical field, thereby altering the stress-strain state of the beam. The mathematical formulation of the problem includes the system of equations for the non-stationary bending vibrations of the Bernoulli-Euler beam, taking into account heat and mass transfer, which is derived from the general model of thermomechanodiffusion for continuous media using the generalized principle of virtual displacements. It is assumed that the speed of propagation of thermal and diffusive disturbances is finite. The solution algorithm is based on the use of Laplace transforms, Fourier series expansions, and the method of equivalent boundary conditions. This approach significantly simplifies the task of inverting the Laplace transform, reducing it to the inversion of rational functions through residue calculus and operational calculus tables. Using the example of a cantilever three-component beam made of a zinc, copper, and aluminum alloy under a distributed non-stationary transverse load, the interaction of mechanical, thermal, and diffusive fields was studied. The study demonstrates that non-stationary bending initiates the process of heat and mass transfer. At the same time, static bending affects only the mass transfer process and does not cause temperature changes within the beam. The calculation results are consistent with those obtained by the authors in their previous works. It should also be noted that, on a qualitative level, the obtained results align with the conclusions of experimental studies, which indicate that the interaction of mechanical and diffusive fields becomes significant only under plastic deformation, while in the elastic deformation regime, this interaction remains negligibly small.

Keywords:

thermoelastic diffusion, Bernoulli-Euler beam, cantilever beam, Green's function, equivalent boundary conditions method, unsteady problems

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