Models of Continuum Mechanics and Their Deficiencies

Abstract

AbstractIn order to proceed to the consideration of the peculiarities of complex non-equilibrium processes induced by shock loading in condensed media, one must first have a good idea of what is meant by the macroscopic response of a system to an external action from the generally accepted viewpoint within the framework of continuum mechanics. At the beginning of the first chapter, we briefly look at the fundamental aspects of continuum mechanics, with particular attention to the assumptions underlying the continuum modeling. Section 1.4 describes the problem of closing the system of macroscopic equations for the transport of mass, momentum, and energy. A lot of profound and thorough papers are devoted to these issues ( J.C. Slattery, Momentum, Energy, and Mass Transfer in Continua, McGraw-Hill Co (1971).; L.I. Sedov, Mechanics of Continuous Medium, World Scientific (1997).; Chadwick in Continuous Mechanics, Allen & Unwic, London, 1976;Gurtin in An Introduction in Continuous Mechanics, Academic press, New York, 1981; Reddy in An Introduction to Continuous Mechanics, Cambridge University Press, Cambridge, UK, 2006; A.Ian Murdoch, Physical Foundations of Continuum mechanics, Cambridge University press (2012).; Gurtin et al. in Mechanics and Thermodynamics of continua, Cambridge University Press, New York, 2010; R.I. Nigmatulin, Mechanics of Continuous Medium, Moskow: GEOTAR-MEDIA, (2014) (in Russian)). The concept of a medium model used in continuum mechanics and its shortcomings in modeling transient processes are discussed in Sect. 1.5. Hypotheses and relationships connecting macroscopic fields of continuous densities with the microscopic behavior of real molecules and other elements of physical systems are considered. The statistical description of macroscopic systems considers the behavior of microscopic elements of the medium as a random process ( Henk Tijms, Understanding Probability, Cambridge University Press (2004).; Murray Rosenblatt, Random Processes, Oxford University Press (1962).; Joseph L. Doob, Stochastic processes, Wiley. pp. 46, 47 (1990).; N.G. Van Kampen, Stochastic Processes in Physics and Chemistry, Elsevier (2011).; Samuel Karlin; Howard E. Taylor, A First Course in Stochastic Processes, Academic Press (2012).; Ionut Florescu, Probability and Stochastic Processes. John Wiley & Sons. pp. 294, 295 (2014)). A number of hypotheses about the nature of such processes can significantly simplify the approaches to substantiating the continuum mechanics and the interpretation of experimental results (Peter Walters, An Introduction to Ergodic Theory, Springer (1982)). The connecting basis between the micro and macro levels of description is the averaging procedure. In mechanics, various averaging methods have been developed: in space, in time, statistical methods, etc. (A.I. Murdoch & D. Bedeaux, A microscopic perspective on the foundations of continuum mechanics. 1. Macroscopic states, reproducibility, and macroscopic statistics, at prescribed scales of length and time, Int. J. Eng. Sci., 34, pp.1111–1129 (1996).; Kiarash Gordiz, David J. Singh, and Asegun Henry, Ensemble averaging vs. time averaging in molecular dynamics simulations of thermal conductivity. Journal of Applied Physics, 117, 045,104 (2015)). Among them, the weight averaging methodology plays an important role ( A.I. Murdoch & D. Bedeaux, Continuum Equations of Balance via weighted averages of microscopic quantities, Proc. R. Soc., London, 445, pp.157–179 (1994)). This procedure is discussed in Sect. 1.6. The mathematical apparatus of continuum mechanics is a system of partial differential equations that relate the gradients of macroscopic fields and their rates of change at the same spatial point and at the same time moment under the assumption that the system has forgotten its history and is not related to the conditions of its loading. In the last sections of the chapter, issues related to the insufficiency of this mathematical apparatus and the need to develop new, more universal approaches to describing macroscopic systems in real conditions of interaction with their surroundings are considered.KeywordsContinuum mechanicsTransport processesMedium modelTransientsAveraging