Nonlinear resultant theory of shells accounting for thermodiffusion
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O R I G I NA L A RT I C L E
Victor A. Eremeyev
· Wojciech Pietraszkiewicz
Nonlinear resultant theory of shells accounting for thermodiffusion In occasion of the 65th birthday of Prof. Holm Altenbach
Received: 22 August 2020 / Accepted: 11 September 2020 © The Author(s) 2020
Abstract The complete nonlinear resultant 2D model of shell thermodiffusion is developed. All 2D balance laws and the entropy imbalance are formulated by direct through-the-thickness integration of respective 3D laws of continuum thermodiffusion. This leads to a more rich thermodynamic structure of our 2D model with several additional 2D fields not present in the 3D parent model. Constitutive equations of elastic thermodiffusive shells are discussed in more detail. They are formulated from restrictions imposed by the resultant 2D entropy imbalance according to Coleman–Noll procedure extended by a set of 2D constitutive equations based on heuristic assumptions. Keywords Thermodiffusion · Nonlinear shell · Thermodynamics · Resultant shell theory
1 Introduction Different approaches to description of thermodiffusion phenomena in 3D bodies were proposed in the literature, see the historic review by [38]. A deeper physical understanding of diffusion indicates that it is caused by Brownian motion of atoms and molecules, which can jump in large numbers between various interstitial sites in the body structure. Thus, most theories of diffusion in 3D bodies are based on the random walk theory and laws of statistical physics. They are used to model and analyse many processes in modern micro- and nanotechnology leading to development of such devices as computer chips, accumulator batteries, solid-state lasers, elements of mobile phones and so on. This approach is partly summarized, for example, by [18,37,41,53], where many references are available. In the present paper, we are interested in developing the alternative nonlinear phenomenologic continuum model of thermodiffusion as applied to thin-walled solid shell structures. The first empirical law for the 3D diffusion flux of the species in the body was proposed already by [25] as the linear function of species concentration gradient field. Later experiments indicated that in various circumstances, diffusion coefficients in the Fick law may depend also on temperature, direction, internal stress, deformation and structure of the Communicated by Andreas Öchsner. V.A.E. acknowledges the support of the Government of the Russian Federation (contract No. 14.Z50.31.0046) V. A. Eremeyev (B) · W. Pietraszkiewicz Faculty of Civil and Environmental Engineering, Gda´nsk University of Technology, ul. Gabriela Narutowicza 11/12, 80-233 Gda´nsk, Poland E-mail: [email protected] W. Pietraszkiewicz Don State Technical University, Gagarina sq., 1, Rostov-on-Don, Russia 344000 E-mail: [email protected]
V. A. Eremeyev, W. Pietraszkiewicz
body, speed and/or relaxation of thermodynamic process, electrical field, possible chemical reactions and other factors. The contemporary linear phenomenologic model of stre
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