Functional kinetic equations in mathematical modeling of coupled processes in solids
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O R I G I NA L A RT I C L E
Taras Nahirnyj · Kostiantyn Tchervinka
Functional kinetic equations in mathematical modeling of coupled processes in solids
Received: 13 November 2019 / Accepted: 10 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we consider the role of functional kinetic equations in models of solid mechanics. It is shown that the choice of time-non-locality kernels allows both the finite speed of signal propagation and the inhomogeneity of material to be taken into account. Using analysis of the rheological kinetic equation for the entropy flux, we propose the motion energy, defined in the space of the momenta of considered motions. We also propose a method for constructing mathematical models that take into account inertia of various physical processes. By interposing a component, proportional to the Dirac’s delta function, into the kernel of the rheological kinetic equation for mass flux, we can introduce the energy of an inhomogeneous medium and describe the near-surface inhomogeneity and its associated size effects. The latter is illustrated by the example of an elastic layer and its strength. Keywords Non-equilibrium thermodynamics · Kinetic equations · Finite speed of signal propagation · Material inhomogeneity · Size effects
1 Introduction The mathematical models of solid mechanics are the basis for investigating the deformation, strength and other operational parameters of structural elements and parts of machines and devices. Such models should be adequate to the real objects whose behavior they describe, that is, they must satisfactorily include the structure and properties of material, as well as the nature of the external action, which is often intense. Each real physical process has a finite speed of propagation in space. However, many models take no account of it. These models include, in particular, the classical models of thermoelasticity and mechanics of solid solutions [1–4]. The standard Fourier’s proposal to consider the heat flux proportional to the gradient of the phenomenological temperature leads to a well-known paradox: the instantaneous propagation of temperature disturbances in a rigid conductor. Several analyses have been aimed at overcoming such a physical inconsistency, among them, the rational extended thermodynamics [5], Green and Naghdi theory [6–9] and Communicated by Andreas Öchsner. T. Nahirnyj University of Zielona Góra, ul. prof. Z. Szafrana 4, 65-516 Zielona Góra, Poland T. Nahirnyj Centre of Mathematical Modeling, IAPMM of NASU, Dudaeva Street 15, Lviv 79005, Ukraine E-mail: [email protected] K. Tchervinka (B) Ivan Franko National University of Lviv, 1, Universytetska St., Lviv 79000, Ukraine E-mail: [email protected]
T. Nahirnyj, K. Tchervinka
other papers concerning the generalized thermoelasticity [10–14]. The relation that ties up the heat flux and the temperature gradient has been generalized to reflect time-non-locality (see [15,16]), eventually leading to the heat conduction equation of
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