An Elastodiffusive Orthotropic Euler–Bernoulli Beam Considering Diffusion Flux Relaxation

Тарлаковский, Д. В.; Земсков, А. В.

doi:10.3390/mca24010023

Cited by 6 publications

(12 citation statements)

References 26 publications

(64 reference statements)

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“…We consider the unsteady plane bending problem of elastic diffusion homogeneous Bernoulli-Euler orthotropic cantilever beam (Figure 1). The mathematical formulation is a closed system of equations for beam transverse vibrations taking into account diffusion, which is obtained from an elastic diffusion model using the d'Alembert variational principle [19,25]:…”

Section: Problem Formulationmentioning

confidence: 99%

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Unsteady bending of the orthotropic cantilever Bernoulli–Euler beam with the relaxation of diffusion fluxes

2022

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The unsteady bending problem of a cantilever elastic-diffusion homogeneous orthotropic Bernoulli-Euler beam is considered, taking into account the relaxation of diffusion fluxes. The solution to the problem is sought using the method of equivalent boundary conditions. For this, an auxiliary problem is additionally considered. The solution to the auxiliary problem is obtained using the integral Laplace transform in time and the trigonometric Fourier series expansions in spatial coordinates. Next, relations are constructed connecting the right-hand sides of the boundary conditions of the original and the auxiliary problems. These relations represent a system of Voltaire integral equations of the 1st kind. This system is solved numerically using quadrature formulas. The limit transitions to the static problem of elastic diffusion, as well as to the unsteady elastic problem for a cantilever beam, are investigated. Using a three-component material as an example, a numerical study of unsteady mechanical and diffusion fields interaction in an orthotropic beam is done. The calculation results are presented in the form of graphs. They show the time and coordinate dependencies of the sought displacement fields and components concentration increments of the continuum. At the end of the publication, the main conclusions about the coupling effect of the stress-strain state and mass transfer in the beam are represented.

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Section: Problem Formulationmentioning

confidence: 99%

“…The components of the stress tensor and the diffusion flux vector are determined using the equalities [19]:…”

Section: Problem Formulationmentioning

confidence: 99%

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Unsteady bending of the orthotropic cantilever Bernoulli–Euler beam with the relaxation of diffusion fluxes

2022

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“…Transverse deflections are considered small. Then the linearization of the unknown quantities with respect to the variable x 3 will have the form [7,8,9,10]…”

Section: General Especificationsmentioning

confidence: 99%

“…This article considers the effects of the interaction of mechanical and diffusion fields in a Kirchhoff-Love plate on an elastic foundation. A mathematical model of plate elastic diffusion vibrations is obtained based on variational principles as well as the well-known plate theory relations presented in the works [7,8,9,10].…”

Section: Introductionmentioning

confidence: 99%

Rectangular Isotropic Kirchhoff-Love Plate on an Elastic Foundation under the Action of Unsteady Elastic Diffusion Perturbations

Земсков¹,

Тарлаковский²

2021

14th WCCM-ECCOMAS Congress

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We study unsteady elastic diffusion vibrations of a freely supported rectangular isotropic Kirchhoff-Love plate on an elastic foundation, which is under the action of a distributed transverse load. A model that describes coupled elastic diffusion processes in multicomponent continuum is used for the mathematical problem formulation. The longitudinal and transverse vibrations equations of a rectangular isotropic Kirchhoff-Love plate with diffusion were obtained from the model using the d'Alembert variational principle.The problem solution of unsteady elastic diffusion plate vibrations is sought in integral form. The bulk Green's functions are the kernels of the integral representations. To find the Green's functions, we used the Laplace transform in time and the expansion into double trigonometric Fourier series in spatial coordinates. Green's functions in the image domain are represented in the form of rational functions depends on the Laplace transform parameter. The transition to the original domain is done analytically through residues and tables of operational calculus. The bulk Green's functions analytical expressions are obtained.Using a two-component continuum, a numerical study of unsteady mechanical and diffusion fields interaction is done for an isotropic plate. The solution is presented in analytical form, as well as in the form of three-dimensional graphs of the displacement fields and concentration increments on time and coordinates.

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Unsteady Elastic–Diffusion Vibrations of a Simply Supported Euler–Bernoulli Beam Under the Distributed Transverse Load

Земсков

Okonechnikov

Тарлаковский

2020

Multiscale Solid Mechanics

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An Elastodiffusive Orthotropic Euler–Bernoulli Beam Considering Diffusion Flux Relaxation

Cited by 6 publications

References 26 publications

Unsteady bending of the orthotropic cantilever Bernoulli–Euler beam with the relaxation of diffusion fluxes

Unsteady bending of the orthotropic cantilever Bernoulli–Euler beam with the relaxation of diffusion fluxes

Rectangular Isotropic Kirchhoff-Love Plate on an Elastic Foundation under the Action of Unsteady Elastic Diffusion Perturbations

Unsteady Elastic–Diffusion Vibrations of a Simply Supported Euler–Bernoulli Beam Under the Distributed Transverse Load

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