Mathematical Modeling of Shock-Wave Processes in Condensed Matter: From Statistical Thermodynamics to Control Theory (Shock Wave and High Pressure Phenomena) 981192404X, 9789811924040

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Shock Wave and High Pressure Phenomena

Tatiana Aleksandrovna Khantuleva

Mathematical Modeling of Shock-Wave Processes in Condensed Matter From Statistical Thermodynamics to Control Theory

Shock Wave and High Pressure Phenomena Founding Editor Robert A. Graham

Honorary Editors Lee Davison, Tijeras, NM, USA Yasuyuki Horie, Santafe, NM, USA Series Editors Frank K. Lu, University of Texas at Arlington, Arlington, TX, USA Naresh Thadhani, Georgia Institute of Technology, Atlanta, GA, USA Akihiro Sasoh, Department of Aerospace Engineering, Nagoya University, Nagoya, Aichi, Japan

Shock Wave and High Pressure Phenomena The Springer book series on Shock Wave and High Pressure Phenomena comprises monographs and multi-author volumes containing either original material or reviews of subjects within the field. All states of matter are covered. Methods and results of theoretical and experimental research and numerical simulations are included, as are applications of these results. The books are intended for graduate-level students, research scientists, mathematicians, and engineers. Subjects of interest include properties of materials at both the continuum and microscopic levels, physics of high rate deformation and flow, chemically reacting flows and detonations, wave propagation and impact phenomena. The following list of subject areas further delineates the purview of the series. In all cases entries in the list are to be interpreted as applying to nonlinear wave propagation and high pressure phenomena. Development of experimental methods is not identified specifically, being regarded as a normal part of research in all areas of interest. Material Properties Equation of state including chemical and phase composition, ionization, etc. Constitutive equations for inelastic deformation Fracture and fragmentation Dielectric and magnetic properties Optical properties and radiation transport Metallurgical effects Spectroscopy Physics of Deformation and Flow Dislocation physics, twinning, and other microscopic deformation mechanisms Shear banding Mesoscale effects in solids Turbulence in fluids Microfracture and cavitation Explosives Detonation of condensed explosives and gases Initiation and growth of reaction Detonation wave structures Explosive materials Wave Propagation and Impact Phenomena in Solids Shock and decompression wave propagation Shock wave structure Penetration mechanics Gasdynamics Chemically Reacting Flows Blast waves Multiphase flow Numerical Simulation and Mathematical Theory Mathematical methods Wave propagation codes Molecular dynamics Applications Material modification and synthesis Military ordnance Geophysics and planetary science Medicine Aerospace and Industrial applications Protective materials and structures Mining

Tatiana Aleksandrovna Khantuleva

Mathematical Modeling of Shock-Wave Processes in Condensed Matter From Statistical Thermodynamics to Control Theory

Tatiana Aleksandrovna Khantuleva Department of Physical Mechanics Saint Petersburg State University Saint-Petersburg, Russia

ISSN 2197-9529 ISSN 2197-9537 (electronic) Shock Wave and High Pressure Phenomena ISBN 978-981-19-2403-3 ISBN 978-981-19-2404-0 (eBook) https://doi.org/10.1007/978-981-19-2404-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

For a long time, mechanics believed that shock loading of a solid induces a steady wave in which an elastic–plastic transition occurs. In continuum mechanics, which describes the behavior of macroscopic systems, various combinations of an elastic medium model and a hydrodynamic model of a viscous medium were used to describe the transient process. But how the transition itself from the elastic response of the medium to the hydrodynamic one occurs such a model could not explain. It has now become generally accepted that the nature of shock-induced processes in solids makes it unsuitable for their description of both models of continuum mechanics based on the concept of local thermodynamic equilibrium and dislocation models describing microstructural processes without their direct connection with dynamic properties of the material. An analysis of the suitability of continuum mechanics for describing non-equilibrium processes is given in Chaps. 1 and 2. Modern understanding of non-equilibrium processes based on experimental data obtained with the use of high-precision instruments is fundamentally different from the previously common opinion that non-equilibrium processes are dissipative nonstationary processes described by nonlinear partial differential equations. Differential equations relate the response of the system to a disturbance at the same spatial point and time moment, which is possible only in a state of local equilibrium. In real systems, disturbances propagate at a finite speed, and the response of the system out of equilibrium always lags behind the disturbance. The collective effects typical to the system response to external loading, which deflects its state far from equilibrium, make it impossible to localize the relationship between them. Only at low speeds, when the system reaches a state near local equilibrium, the effects of retardation and nonlocality can be neglected. Numerous attempts to construct a general theory of the non-equilibrium transport encountered an obstacle: the inadaptability of the close-to-equilibrium thermodynamics and continuum mechanics to the processes far from equilibrium on the one hand and the lack of an approach to describing the processes of structure formation on the other hand. After Prigogine, it is believed that structures can be formed as a result of self-organization only by dissipative processes. However, it is clear that

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the slow diffusive transport mechanism cannot play an important role in high-rate short-duration processes such as shock-induced deformation. Without a basic theoretical approach, until recently, many researchers stubbornly continued to use variants of the elastic–plastic model with the assumptions that the material under planar shock loading is homogeneous and that the stress and strain tensors can be separated into elastic and plastic components. The emergence of a new trend, mesomechanics, which deals with the formation of multi-scale defect structures in a material at intermediate levels between macro and micro, did not answer the main questions for shock-wave processes. Although the concept of mesomechanics has been introduced more than 30 years ago, it is still not included in theoretical studies of shock-wave processes. The diversity of mechanisms of multi-scale and multi-stage impulse and energy exchange makes it impossible to derive general relationships linking the dynamic macroscopic properties of the material with its mesoscopic structure. As early as 20 years ago, researchers asked a number of questions regarding the processes responsible for the shock-induced waveform, risetime, and amplitude decay. Even then, there was enough experimental evidence to indicate that the transition zone was highly heterogeneous, unstable, and turbulent. Despite the many experimental results on shock loading of solids, a complete understanding of the processes inside the waveform has not been achieved. For the interpretation and analysis of experimental data, it is necessary to have a mathematical model based on rigorous theoretical foundations that could provide answers to the questions posed. However, the development of theoretical approaches to the description of such highly non-equilibrium processes as shock waves in solids came up against three main unsolved problems of modern physics: (i) description of the processes of mass, momentum, and energy transport in macroscopic systems far from local thermodynamic equilibrium from the first principles, (ii) creation of thermodynamics of an arbitrary non-equilibrium state, (iii) creation of a general theory of turbulence. By now, successful steps have been taken in each separate direction, leading our understanding further and further away from thermodynamic equilibrium. However, the existing far-reaching differentiation of science leads to the fact that the achievements of one science often become inaccessible to the understanding of representatives of other sciences. Shock loading mechanics are not only unaware of new advances in these areas of physics, but also often simply do not understand why they need them. When high-rate and short-duration processes began to be used in modern technology and engineering, the lack of modeling based on the formalism of nonlinear differential equations became more and more obvious. The attempts to include integral effects in a macroscopic description of the system behavior out of equilibrium have been undertaken for a long time. Back in the second half of the last century, in the framework of non-equilibrium statistical mechanics, D. N. Zubarev (Steklov Institute of Mathematics, Moscow) proved from the first principles that far from equilibrium the macroscopic transport equations cannot be localized either in space or in time. By method of a nonequilibrium statistical operator, he derived generalized integral-differential macroscopic transport equations that are valid far from local equilibrium. According to his

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results, the medium response to high-rate deformation should be nonlinear, nonlocal, and history-dependent. A brief description of Zubarev’s results, which served as the basis for the approach developed by the author of the book, is presented in Chap. 3. However, the generalized description was also incomplete because the spatiotemporal correlation functions in the nonlocal transport equations with memory were unknown nonlinear functionals of macroscopic gradients. This circumstance became an obstacle to using the nonlocal models in practical problems for several decades. Although the nonlocal transport equations obtained by Zubarev are the only fundamentally new universal mathematical models obtained in science over the past two hundred years, they have not received further development until now. However, the fact that the dynamics of space-time correlations determines the macroscopic behavior of the system far from equilibrium indicates the possibility of their use to describe the self-organization of turbulent structures. These structures should introduce into the system additional information about the system’s evolution far from equilibrium. This idea was used by the author of the book in a new approach, presented in Chap. 5, to describe highly non-equilibrium processes in real open systems. Based on the results of statistical mechanics obtained by Zubarev and cybernetical physics, a new self-consistent nonlocal theory of non-equilibrium transport was proposed to describe the system evolution far from equilibrium. Within the framework of this theory, an integral mathematical model has been built that links the macroscopic response of the system to an external action with the dynamics of spacetime correlations on the mesoscale. Revealing the relationships between space-time correlations and the dynamic structure of the system at an intermediate level between macro and micro was a big step into the field of highly non-equilibrium processes. It has been shown that the non-equilibrium correlation function generates vectors associated with rotational modes and polarization of the medium. Boundary conditions and external constraints imposed on the system lead to a discrete-size spectrum of dynamic structures formed due to self-organization, which accompanies transport processes far from equilibrium. Cybernetic methods developed within the control theory of adaptive systems are used in Chap. 6 to govern the system’s temporal evolution far from local equilibrium. In the process of the system evolution, between the response of the macroscopic system to an external load and dynamic structural transformations on the mesoscale, the information-control feedbacks are formed, which makes the system behavior far from local equilibrium more stable. The speed-gradient principle defines the fastest way to a more stable state, admissible by the constraints imposed, and plays a role in the engine of system evolution in the internal control mechanism. Without internal control, new degrees of freedom associated with the formed structure can cause fluctuations and oscillations, which, in turn, make the behavior of the system unstable and poorly predictable. Unlike traditional synergetic models, where the type of the mathematical model is set in advance, nonlocal models are “soft”, capable of changing their type depending on external conditions and adapting to changing mechanisms of interaction at the mesoscale. So, self-organization and internal control with feedback are necessary for adequate modeling of non-equilibrium processes.

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Only such a multi-disciplinary approach at the crossroads of mechanics, physics, and cybernetics allows us to complete the self-consistent formulation of the problems for highly non-equilibrium transport processes in open systems. The description of the temporal evolution on the mesoscale incorporated in the mathematical model of the highly non-equilibrium process makes it possible to predict the dynamic medium properties by tracing the evolutionary paths of dynamic structures under high-rate loading. The possibilities of applying the proposed approach in various fields of science and practice go far beyond the scope of this book. Although this approach to modeling highly non-equilibrium processes was proposed more than 15 years ago and was successfully tested on several problems of hydromechanics, it remained far from the mainstream. Most people do not like the heavy formalism of integral equations; they do not want to leave the comfort zone of their narrow specialization. Numerous actively developing trends of nonequilibrium thermodynamics and statistical theory are trying to move into the area of non-equilibrium processes, away from equilibrium. But without revising the generally accepted concepts, it is not possible to break free from the “gravitation” of equilibrium. All the concepts of thermodynamics, continuum mechanics, and classical physics were formed under equilibrium conditions, which were then considered normal and are still implicitly preserved in many areas of modern science. Attempts to extend local equilibrium concepts to non-equilibrium situations using such artificial methods as splitting the system into small parts, introducing hidden variables, and other cumbersome constructions that do not have physical meaning, as shown in Chap. 4, lead to an impasse. Besides, there is confusion in terminology. It is believed that the concept “far from equilibrium” means “far from complete equilibrium”. This implies that states close-to-local equilibrium are considered to be “far from equilibrium”. Therefore, in this book we are forced to explain everywhere that we are dealing with processes “far from local equilibrium”. It became clear that only the need to describe the processes that are really far from equilibrium, such as shock waves, can force one to go beyond the generally accepted concepts. Shock-induced wave processes in condensed matter are really the most nonequilibrium of those that belong to the field of mechanics. The specific feature of the response of a solid to shock loading is a very strong interaction of atoms with each other that corresponds to a very high degree of spatial correlation and memory about the loading history. Results of experimental research on the shock loading of solid materials had demonstrated that the revealed dependences of the shock-induced waveforms on strain rate, target thickness, and state of the material structure could not be described in the framework of the conventional concepts. It became obvious that in order to develop predictive mathematical models, it is necessary to critically examine the fundamental postulates used to interpret shock compression phenomena and to develop a first-principle approach capable to describe a whole complex of non-equilibrium processes in a deformable medium. Since the experimental results required an explanation, and the developed approach to such processes was awaiting

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application, the problem of the propagation of a shock-induced elastic–plastic wave in condensed matter was formulated. Chapter 7 proposes a new concept of shock-wave processes in condensed matter based on the nonlocal theory of non-equilibrium transport, which describes the transition from an elastic medium response to a hydrodynamic one depending on the strain rate and loading duration. It was shown that the hydrodynamic response cannot be achieved as a result of the shock of moderate intensity. Shock on a solid breaks correlations into mesoscopic parts making the material less solid. However, its solid state is partially restored on the so-called plastic front. Moving after shock at different speeds, like wave packets, mesoparticles form the forefront and plateau of the compression pulse. When the unloading wave arrives, synergistic formation of vortex-wave structures on the mesoscale can occur. If the solid state is restored only partially on the plateau, an irreversible part of the mesoscopic structures remain frozen into material behind the wave. An integral mathematical model of an elastic–plastic wave is constructed to describe both the elastic precursor relaxation and the plastic front evolution during the wave propagation. Analysis of the experimental waveforms showed that for the shock-induced processes, it is a priori incorrect to divide components of stress and strain into elastic and plastic parts. The explicit approximate solution to the problem obtained in Sect. 7.8 allowed us to decipher the information about the relaxation of stresses involved in the evolution of the waveform during its propagation through the material. The processing of experimental oscillograms obtained at different distances from the impact surface for different materials in a wide range of impact velocities revealed the general regularities of the relaxation processes. In Chap. 8, the described evolution of the mesostructure within a waveform of finite duration is an example of a process in which, as a result of the selforganization of new mesoscopic structures, the generalized integral entropy production can become negative. As shown in Chap. 3, this does not contradict the second law of thermodynamics. In Chap. 9, we show that the experimentally recorded waveform is the result of a superposition of wave packets moving in a dispersive medium. In Sect. 9.8, we show that the effects arising on the mesoscale under the shock loading of solid materials have much in common with quantum effects. In particular, the self-organization of turbulent structures on the mesoscale can be considered as the capture of a quasiparticle by an entropy well (an analog of a potential well). In Chap. 10, we obtained the stress–strain relationship for the case of uniaxial high-rate deformation without dividing it into elastic and plastic parts. The fundamental difference between shock and quasi-static loading is shown. To summarize, we can say that the results obtained in this book do not represent a completely finished study. There are not enough specific examples, compared with experimental and numerical calculations. This is so, but at the present time there are simply no systematic experimental results that allow one with sufficient accuracy to compare the recorded waveforms with the behavior of the mass velocity dispersion and data on the mesostructures remaining in the targets. Ready-made

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modern software packages are not suitable for numerical modeling of processes far from local equilibrium, since they incorporate traditional models. Moreover, in contrast to quasi-static processes, for which the apparatus of nonlinear functionals of a special type can be used, an adequate mathematical apparatus for solving dynamic problems has not been developed at all. In the book, we could only look beyond traditional concepts and approaches and see a huge new area awaiting its researchers. This is just the beginning. However, we can argue that as long as theorists use “rigid” models of complex, rapidly changing processes without internal control and the evolution of information structures synergistically formed in open systems, they will never bridge the gap between theory and modern practical needs. I am grateful to my colleagues in the Department of Physical Mechanics and Theoretical Cybernetics (St. Petersburg State University) for many years of discussions on related topics. I would especially like to thank Prof. Yu. I. Meshcheryakov (Institute of Mechanical Engineering Problems, St. Petersburg, Russia) for access to his experimental data and information on the possibility of publishing the book. Saint-Petersburg, Russia

Tatiana Aleksandrovna Khantuleva

Contents

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2

Models of Continuum Mechanics and Their Deficiencies . . . . . . . . . . 1.1 Description of Macroscopic Systems; Macroscopic Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Macroscopic Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Equation of Mass Transport . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Equation of Momentum Transport . . . . . . . . . . . . . . . . . . . . 1.2.3 Equation of Energy Transport . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Validity of Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Problem of Closure of the Transport Equations . . . . . . . . . . . 1.5 Medium Models and Transient Processes . . . . . . . . . . . . . . . . . . . . 1.6 Averaging in Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Deficiencies of the Continuum Mechanics Concept . . . . . . . . . . . . 1.8 The Problem of a Uniform Description of the Media Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Short Review of Approaches to Extension of Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specific Features of Processes Far from Equilibrium . . . . . . . . . . . . . . 2.1 Experimental Difficulties in Studying Non-Equilibrium Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Anomalous Medium Response to Strong Impact . . . . . . . . . . . . . . 2.3 The Internal Structure Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Fluctuations, Pulsations, and Instabilities . . . . . . . . . . . . . . . . . . . . 2.5 Multi-Scale Energy Exchange Between Various Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Multi-Stage Relaxation Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Finite Speed of Disturbance Propagation and the Delay Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Influence of the Loading Duration and Inertial Effects . . . . . . . . .

1 2 4 5 6 8 11 13 14 17 21 22 24 28 31 31 34 37 39 44 46 49 53

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2.9

Dynamic Self-Organization of New Internal Structure in Open Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Predictive Ability of Modeling Non-Equilibrium Processes . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

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Macroscopic Description in Terms of Non-Equilibrium Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Fundamentals of Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . 3.2 Bogolyubov’s Hypothesis of Attenuation of Spatiotemporal Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Rigorous Statistical Approaches to Non-Equilibrium Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Description of Macroscopic Systems from the First Principles. Local-Equilibrium Distribution Function . . . . . . . . . . . 3.5 Description of Macroscopic Systems from the First Principles. Non-equilibrium Distribution Function . . . . . . . . . . . . 3.6 Non-Equilibrium Statistical Operator by Zubarev . . . . . . . . . . . . . 3.7 The Nonlocal Thermodynamic Relationships with Memory Between the Conjugate Macroscopic Fluxes and Gradients . . . . . 3.8 Two Types of the Spatiotemporal Nonlocal Effects . . . . . . . . . . . . 3.9 Main Problem of Non-Equilibrium Statistical Mechanics . . . . . . . 3.10 The Disadvantages and New Opportunities to Close Transport Equations for High-Rate Processes . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Concepts Out of Equilibrium . . . . . . . . . . . . . . . . . . . 4.1 Basic Concepts and Principles of Thermodynamics . . . . . . . . . . . . 4.2 Entropy Production in Transport Processes . . . . . . . . . . . . . . . . . . . 4.3 Linear Thermodynamics of Irreversible Processes . . . . . . . . . . . . . 4.4 Revision of the Generally Accepted Thermodynamic Concepts Out of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Thermodynamic Entropy, Information Entropy, and Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Local Entropy Production Near and Far from Equilibrium . . . . . . 4.7 Total Entropy Generation and the Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Maximum Entropy Principle by Jaynes . . . . . . . . . . . . . . . . . . . . . . 4.9 Influence of the Constraints Imposed on the System . . . . . . . . . . . 4.10 Thermodynamic Temporal Evolution Out of Equilibrium . . . . . . . 4.11 Self-organization of New Structures in Thermodynamics . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56 59 61 65 66 71 72 74 79 83 86 89 90 92 93 95 96 102 105 111 115 119 121 124 126 128 131 134

New Approach to Modeling Non-equilibrium Processes . . . . . . . . . . . 137 5.1 Generalized Constitutive Relationships Based on Non-Equilibrium Statistical Mechanics . . . . . . . . . . . . . . . . . . . 138 5.2 Modeling Spatiotemporal Correlation Functions . . . . . . . . . . . . . . 141

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5.3 5.4

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Temporal Stages of the Correlation Attenuation . . . . . . . . . . . . . . . Deficiencies of the Generally Accepted Models for the Media with Complicated Properties . . . . . . . . . . . . . . . . . . . 5.5 Requirements to New Approach to Modeling Shock-Induced Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Foundations of New Approach to Modeling Transport Processes Far from Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 New Interdisciplinary Approach to Modeling Highly Non-Equilibrium Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Interrelationship Between Spatiotemporal Correlations and Dynamic Structure of the System . . . . . . . . . . . . . . . . . . . . . . . 5.9 Modeling Correlation Functions in Boundary-Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Boundary Conditions for Nonlocal Equations . . . . . . . . . . . . . . . . 5.11 Discrete-Size Spectrum of the Dynamic Structure of a Bounded System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 The Mathematical Basis for the Self-Consistent Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Distinctive Features of New Approach from Semi-Empirical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Description of the Structure Evolution Using Methods of Control Theory of Adaptive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Methods of Control Theory in Physics. Cybernetical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Speed Gradient Principle by Fradkov for Non-Stationary Complex Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Description of the System Temporal Evolution at Macroscopic Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Temporal Evolution of Statistical Distribution Functions . . . . . . . 6.5 The System Temporal Evolution Out of Local Equilibrium on the Mesoscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Internal Control on the Mesoscale Based on Speed Gradient Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Entropy Production in a Stationary State Out of Equilibrium . . . . 6.8 Paths of Structural Evolution and Reduction of Irreversible Losses Due to Self-Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 A New Look at the Problem of Stability of Non-Equilibrium Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Influence of Feedbacks on the Paths of the System Evolution and Prediction of the Final States . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147 150 151 152 153 157 159 162 165 166 171 175 176 178 180 186 188 189 194 197 203 204 207

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The Shock-Induced Planar Wave Propagation in Condensed Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Thermodynamic Properties of Solids . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Wave Processes in Crystal Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Elastic Properties of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Plastic Deformation. Deficiencies of Continuum Mechanics . . . . 7.5 Shock Wave as a Transient Highly Non-equilibrium Process . . . . 7.6 The Integral Model for the Stress Tensor Without Separation into Elastic and Plastic Parts . . . . . . . . . . . . . . . . . . . . . 7.7 Formulation of the Problem of the Shock-Induced Wave Propagation in Condensed Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Self-similar Quasi-Stationary Solution to the Problem . . . . . . . . . 7.9 The Relaxation Model of the Shock-Induced Waveforms During Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Decryption of the Information Recorded in the Experimentally Observed Waveforms . . . . . . . . . . . . . . . . . . 7.11 Comparison of the Model Waveforms with Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209 210 212 215 218 221 224 231 236 239 243 244 247

Evolution of Waveforms During Propagation in Solids . . . . . . . . . . . . 8.1 Waveforms Evolution Within the Integral Model . . . . . . . . . . . . . . 8.2 Speed-Gradient Principle for the Waveforms Evolution . . . . . . . . 8.3 The Waveform Evolution During Quasi-Stationary Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Paths of the Waveform Evolution and Experimental Results . . . . 8.5 Modeling Finite-Duration Waveforms . . . . . . . . . . . . . . . . . . . . . . . 8.6 Shock-Induced Waveform as a Result of Non-monotone Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Entropy Production in Finite-Duration Waveforms . . . . . . . . . . . . 8.8 Evolution of the Waveforms During Their Propagation . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251 252 255

Abnormal Loss or Growth of the Wave Amplitude . . . . . . . . . . . . . . . 9.1 Mass Velocity Dispersion and Interference of Shock Waves on the Mesoscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Shock-Induced Waveform as a Wave Packet . . . . . . . . . . . . . . 9.3 Behavior of the Mass Velocity Dispersion and the Waveform Amplitude Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Dependence of the Waveform Amplitude Loss on the Impact Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Multi-scale Momentum and Energy Exchange in Wave Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 The Shock-Induced Structure Instability . . . . . . . . . . . . . . . . . . . . 9.7 Self-organization of Turbulent Structures in the Entropy Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

283

258 263 265 269 272 276 280

284 289 293 295 297 301 304

Contents

xv

9.8 Quantum Effects on the Mesoscale . . . . . . . . . . . . . . . . . . . . . . . . . . 306 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 10 The Stress–Strain Relationships for the Continuous Stationary Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Stress–Strain Relationships for Continuous Quasi-Static Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Stress–Strain Relationship for Continuous Planar Loading with Accounting Only Shear Relaxation . . . . . . . . . . . . . 10.3 The Stress–strain Relationship for Continuous Plane High-Rate Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Reversible and Irreversible Loading–Unloading Processes . . . . . 10.5 Entropy Production Surfaces for Various Duration Loading and Possible Evolutionary Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Probable Evolutionary Paths and Final States . . . . . . . . . . . . . . . . . 10.7 Fundamental Difference Between Shock and Quasi-Static Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311 312 315 320 323 327 330 332 335

Chapter 1

Models of Continuum Mechanics and Their Deficiencies

Abstract In 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 Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 T. A. Khantuleva, Mathematical Modeling of Shock-Wave Processes in Condensed Matter, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-981-19-2404-0_1

1

2

1 Models of Continuum Mechanics and Their Deficiencies

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. Keywords Continuum mechanics · Transport processes · Medium model · Transients · Averaging

1.1 Description of Macroscopic Systems; Macroscopic Variables Physical systems of macroscopic size can be described at different scale levels. Because of the huge number of atoms or molecules in a macroscopic system, its microscopic description is accessible only in the framework of statistical mechanics. The macroscopic description is used in continuum mechanics. Instead of a real system consisting of microscopic particles, continuum mechanics considers structureless systems with continuously distributed mechanical values. Macroscopic variables in continuum mechanics are averaged densities of macroscopic fields of distributed mechanical values over the system volume. The introduction of the averaged densities imposes certain restrictions on the scale of averaging. Below, we consider the choice of the averaging scale. For macroscopic quantities proportional to the volume, the concept of macroscopic density at a given space point in the system volume and at a given instant of time is introduced. Any point of the system with a radius-vector r can be surrounded shape with characteristic dimensions of the same by some volume δV of arbitrary  1   r order so as to obtain r = δV δV dr . The volume δV contains a mass δ M equal to the sum of the masses of all molecules inside this volume. Then the averaged mass

1.1 Description of Macroscopic Systems; Macroscopic Variables

3

density in the volume δV is ρ = δδVM . The choice of the volume δV of the medium assumes that it contains so many structural elements of the medium (for example, molecules) that all the macroscopic properties of the medium are preserved in this volume. This means that although δV  V , the number of particles in the volume itself δ N must be large in order to neglect fluctuations and to consider the volume δV macroscopic 1  δ N  N . For such a choice to be possible, the volume of the whole system V must be sufficiently large. If there is a sufficiently large range of sizes of the volume δV , in which an averaged value does not depend on the size and shape of the volume, then the averaged value is considered stable. The distribution of the macroscopic density ρ(r, t) over the volume V of the system characterizes a macroscopic field which is considered representative if the mass M in any volume V can be calculated through the integral  ρ(r, t)d V.

M(t) =

(1.1.1)

V

Taking into account d M = ρd V , we can define the density as follows ρ = . Just the same definitions can be obtained for any other volumetric quantities. lim δM δV

δv→0

Introduction of bulk and specific densities For macroscopic quantities proportional to volume, the concept of bulk or volumetric density is introduced, that is, the density of a quantity per unit volume. For example, the bulk density introduced by formula (1.1.1) is the bulk density of the mass. The volumetric momentum and energy densities are introduced in the same way. For all densities except the mass one, the concept of specific density can be introduced, i.e. density value per unit mass. If A is the amount of a certain quantity distributed in a volume V with a specific density a(r, t), then the relation between A and its densities takes a form  A(t) = ρa(r, t)d V (1.1.2) V

In this case, ρa(r, t) represents the bulk density of the quantity A. As the quantity of A, we can choose any quantity proportional to the volume. In problems of continuum mechanics, mass, momentum, and energy transport play the main role. (1) (2)

If the quantity A is the mass distributed in the volume V , then a(r, t) = 1, and the relation (1.1.2) turns into (1.1.1). The relation (1.1.1) can be generalized to the case of multi-component media, if to take the specific concentration of the ith component as a(r, t) = ci (r, t),  i ci = 1. The total mass of the ith component of the medium  contained in the volume V is determined by the formula (1.1.2) Mi (t) = V ρci (r, t)d V .

4

(3)

(4)

1 Models of Continuum Mechanics and Their Deficiencies

Considering the momentum transport, one must remember that, unlike mass, momentum is a vector. The bulk momentum density is pulse p(r, t) = ρv, in which the quantity a(r, t) = v corresponds to the vector value of the specific pulse density v, i.e. the mass velocity. In particular, the x-projection of momentum per unit mass is determined by the x-projection of velocity in the volume V is determined vi . Then the x-projection of the momentum  (t)= ρv (r, t)d V , and the total momentum, P by the formula (1.1.2): x x V  respectively, P(t) = V ρv(r, t)d V . Passing to the energy transport, one should introduce the total volumetric 2 energy density E = ρ v2 + ρ E, which consists of the volumetric densities of the kinetic and internal energies, respectively. The quantity E is the specific thetotal energy in the volume is given density of the internal energy. Similarly,    v2 = V ρ 2 + E (r, t)d V . by the formula (1.1.2):

1.2 Macroscopic Transport Equations Consider an immobile spatial volume V in a continuous medium. The change in the quantity A with time, distributed in the volume V with a specific density a(r, t), is due to the penetration of this quantity through the boundaries of the region with its surroundings and to an origin or loss of a certain amount of quantity A inside the volume V . Then in the laboratory coordinate system, i.e. in Euler coordinates, one can write    ∂ρa dA (1.2.1) = d V = − Jan d S + σa d V . dt ∂t V

S

V

In formula (1.2.1), Jan is the normal projection of the vector J a representing the volumetric flux density of the quantity A through the surface S of the volume V . The second integral on the right σa is the volume density of the source of the quantity A inside the volume V . The minus sign in front of the first integral on the right is due to the choice of a positive flux direction (inflow through the surface) opposite to the direction of the outer normal to the surface S. Note that the definition of flux densities and sources in specific problems is largely arbitrary. By the Gauss– Ostrogradsky theorem, the surface integral can be expressed in terms of the volume integral of the vector divergence 

 Jan d S = S

∇ · JadV . V

Now the change of the quantity A in the volume V can be written as follows:

1.2 Macroscopic Transport Equations

 V

∂ρa dV + ∂t

5



 ∇ · JadV = V

σa d V .

(1.2.2)

V

Equation (1.2.2) is the integral form of the transport equation of quantity A in the volume V . Since the selected volume V is arbitrary, you can go from the integral form to the differential one ∂ρa + ∇ · J a = σa . ∂t

(1.2.3)

However, the transition of the transport Eqs. (1.2.2) to the differential form (1.2.3) is not always correct in real conditions. For the transition, the condition (1.1.2) must be met that limits the selected volume from below and requires that it contains such a large number of structural elements of the medium that the influence of the internal structure of the medium can be neglected.

1.2.1 Equation of Mass Transport To obtain the mass transport equation from the general transport Eq. (1.2.3), we take the specific density a(r, t) = 1 that corresponds to the bulk density of the mass ρ(r, t). In this case, the Eq. (1.2.3) takes the form ∂ρ + ∇ · J ρ = σρ . ∂t

(1.2.4)

In continuum mechanics, a flux of any magnitude through a surface can be carried out using two different transport mechanisms: convection and diffusion. Convective transport of mechanical quantities (mass, momentum, and energy) occurs at the macroscopic level when the medium flows through the volume boundary in a reversible manner without a loss for dissipation of mechanical energy into a thermal form, while diffusion transport at the microscopic (molecular) level is always accompanied by an irreversible loss of momentum and energy. In a one-component medium, mass can only be transported by convection. Therefore, the bulk density of the mass flux is determined as follows: J ρ = ρv.

(1.2.5)

If the mass is conserved in the system, the mass source is absent σρ = 0. In this case, the mass transport equation takes the form of the well-known continuity equation in the Euler coordinates ∂ρ + ∇ · ρv = 0. ∂t

(1.2.6)

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1 Models of Continuum Mechanics and Their Deficiencies

Introducing the total derivative of the mass density along the trajectory of motion = ∂ρ + v · ∇ρ and taking into account the equality ∇ · ρv = of the liquid particle dρ dt ∂t v · ∇ρ + ρ∇ · v, the Eq. (1.2.6) can be written in the Lagrangian coordinates attached not to the laboratory frame of reference but to the moving together with liquid particle dρ + ρ∇ · v = 0. dt

(1.2.7)

It should be noted that, unlike the Euler approach, the Lagrangian approach is not always applicable. If there are mixing in the flow of a continuous medium, jumps of field values, and disintegration of the region occupied by the liquid particle into parts during its movement, then it is impossible to trace the trajectory of a liquid particle. Therefore, the Lagrangian approach is incorrect for describing turbulent flows, as well as flows with shock waves and many other wave processes.

1.2.2 Equation of Momentum Transport Bulk impulse density is defined as p = ρv, and specific pulse density is the mass velocity v. Take as a x-projection of the velocity v a(r, t) = vx (r, t), and x-projection of impulse px = ρvx as the bulk density of the momentum. The volumetric flux density of the x- component of the impulse is a vector consisting of two parts: convection (due to transport with mass) and diffusion J (d) vx (due to molecular transport) Jvx = ρvvx + J (d) vx . According to Newton’s law, only the bulk density of the x-component of the force σx = ρ Fx can be a volume source of a change in momentum. Then the Eq. (1.2.3) for momentum transport can be written in the form ∂ρvx + ∇ · (ρvvx + J (d) vx ) = ρ Fx . ∂t Disclosing the derivatives of the products in the Eq. (1.2.8) ∂ρvx ∂ρ ∂vx =ρ + vx , ∂t ∂t ∂t ∇ · (ρvvx ) = ρv · ∇vx + vx ∇ · (ρv), we get ρ

∂ρ ∂vx + vx + ρv · ∇vx + vx ∇ · (ρv) + ∇ · J (d) vx = ρ Fx . ∂t ∂t

(1.2.8)

1.2 Macroscopic Transport Equations

7

If the mass is conserved in the system, then, taking into account the continuity Eq. (1.2.6), we obtain the transport equation for the x-component of the momentum in the Eulerian coordinates ρ

∂vx + ρv · ∇vx + ∇ · J (d) vx = ρ Fx . ∂t

(1.2.9)

Introducing the total derivative along the trajectory of the liquid particle, similarly x x = ∂v + v · ∇vx , we obtain the equation to the total derivative of the mass density dv dt ∂t for the transport of the x-component of the momentum in the Lagrangian coordinates ρ

dvx + ∇ · J (d) vx = ρ Fx . dt

(1.2.10)

The transport equations for y- and z-momentum projections can be obtained similarly. Since for scalar quantities, such as mass or the x-projection of the momentum, the flux is a vector quantity, the flux of a vector quantity is a tensor of the second rank. In the general case, the equation of momentum transport in the Lagrangian coordinates can be written in the vector form ρ

dv + ∇ · J (d) v = ρ F, dt

(1.2.11)

where J (d) v is the diffusion part of the momentum flux density tensor, and F is the specific density of the body force vector. Introducing the stress tensor Π = − J (d) v instead of the diffusive part J (d) v of the momentum flux, we get (1.2.11) in the form ρ

dv = ∇ · Π + ρ F. dt

(1.2.12)

Like any tensor, the stress tensor can be decomposed into spherical and deviatoric parts: Π = − p I + P,

(1.2.13)

3 ii , I is the unit tensor, and P is the viscous stress tensor. For where − p = 31 i=1 simple droplet media, the value p has the meaning of hydrostatic pressure and characterizes the reversible processes of all-round compression and expansion that occur without dissipation. In this case, the viscous stress tensor has zero diagonal elements, and the deviator elements characterize the shear stresses caused by viscous friction arising from the diffusion transport of momentum through the surface tangent to the streamlines. Shear processes accompanied by viscous dissipation are irreversible. It is obvious that the tensor of viscous stresses occurs P = 0 only in the presence of motion in the system with an inhomogeneous field of mass velocity. For a steady state or the medium moving as a whole, there is no viscous friction P = 0, and the stress

8

1 Models of Continuum Mechanics and Their Deficiencies

tensor has only a spherical reversible component Π = − p I which is completely determined by hydrostatic pressure p. Equation (1.2.12) for droplet media (1.2.13), taking into account the relation ∇ · p I = ∇ p, can be rewritten as follows: ρ

dv = −∇ p + ∇ · P + ρ F. dt

(1.2.14)

It should be noted that, in the general case, under all-round compression, dissipation also can occur and the viscous stress tensor will acquire a spherical part associated with the effects of volumetric friction. All processes that are accompanied by friction lead to the dissipation of mechanical energy into thermal form and are irreversible in time from the point of view of thermodynamics. This means that by turning back time and letting liquid particles in the opposite direction, we will never return the system to its original state. In the general case, to achieve the initial state, additional energy consumption from the outside will be required. It is known from thermodynamics that reversible processes occur only in the equilibrium state of the system. Full thermodynamic equilibrium for macroscopic systems is characterized by a small number of thermodynamic quantities, such as system volume, temperature, and pressure, which do not depend on time and are related to each other by the equations of state of the medium. For example, Clapeyron’s ideal gas equation of state is pV = RT (Here R is the universal gas constant, and V = 1/ρ is the specific volume). In the equilibrium state of the system, the transport of mass, momentum, and energy is completely absent, and, therefore, there is no motion, and hence, no mass velocity. Individual molecules of the medium move chaotically, and their average mass velocity is zero while their average kinetic energy determines the temperature of the medium. It is generally accepted that in a reversible process, the listed quantities can vary infinitely slowly over time without violating the equations of state and without creating flows. In all other cases, the state of the system is considered non-equilibrium, and the processes in such a system are considered irreversible. As a rule, these are dynamic processes in which the macroscopic fields of mass, momentum, and energy change rapidly both in space and in time due to the fluxes of these quantities and their sources. Therefore, in the general case, it is more correct to speak not about the nonequilibrium state of the system, but about the non-equilibrium process occurring in the system. In reality, all processes are non-equilibrium and irreversible. Reversible can be considered processes that proceed so slowly and smoothly that the dissipation that accompanies all real processes can be neglected under the given conditions.

1.2.3 Equation of Energy Transport The total energy of the system consists of mechanical and internal parts which can be transformed into each other in the process of performing work. According to the second law of thermodynamics, these transformations are irreversible due to

1.2 Macroscopic Transport Equations

9

dissipation. This means that a part of the mechanical energy is irreversibly lost transforming into internal energy. It is usually considered that mechanical energy consists of kinetic and potential parts. In reversible processes without dissipation, the total mechanical energy is conserved. In the presence of dissipative processes, a part of the mechanical energy is converted into thermal energy. However, the physical nature of the potential energy of the medium in the absence of external potential fields is due to the interaction of the molecules of the medium. The thermal energy of a medium is due to the kinetic energy of molecules moving in the coordinate system in which the medium is at rest. Both potential energy and thermal energy are not associated with the macroscopic motion of the medium and, therefore, can be attributed to the internal part of the energy. In this case, internal energy consists of cold internal energy (potential energy of interaction on the microscale) and thermal internal energy. In non-equilibrium processes, these two energy components are difficult to strictly separate and correctly determine the potential energy of the medium. Therefore, when describing non-equilibrium processes, as is done, for example, in calculations by molecular dynamics methods, it is impossible to determine the interaction potential for the medium under external loading. For an ideal gas, due to low densities, the potential energy of interaction of molecules is neglected and only the thermal energy of molecules is considered as the internal energy. For condensed media, it is the interaction energy that determines all their properties. Slow flows of simple liquids that can be considered as weakly non-equilibrium processes are described similarly to gas flows. The cold internal energy manifests itself only through elastic properties while the compressibility of a fluid is often neglected. The discipline based on this approach to describing the flows of gaseous and liquid media is called hydromechanics. Adhering to the same approach, we will assume that the specific density of 2 the total energy a = E = v2 + E contains only the density of the kinetic part of mechanical energy and only the thermal part of internal energy. We substitute the specific density in the general transport Eq. (1.2.3) and obtain the equation for the total energy transport in the Euler coordinates   2  ∂ ρ v2 + E ∂t

2  v +∇ · ρ + E v + ∇ · J (d) E = ρ Fv + ∇ · (Π · v). 2 (1.2.15)

In the Eq. (1.2.15), the bulk density  2 of the  total energy flux consists of two parts: v convection and diffusion: J E = ρ 2 + E v + J (d) E , and the diffusion component refers only to the internal energy. The volumetric density of the total energy source is determined by the power densities of the volumetric and surface forces, σ E = ρ Fv + forces appeared ∇ · (Π · v), where the divergence sign for the surface  when passing  from the surface integral to the volume integral S (Π · v)n d S = V ∇ · (Π · v)d V . Expanding the derivatives on the left and taking into account the relation

10

1 Models of Continuum Mechanics and Their Deficiencies



v2 ∇ · ρv +E 2



2  2  v v =v·∇ ρ +E + + E ∇ · (ρv), 2 2

we obtain an equation in the form

ρ

 ⎧  2 ⎨ ∂ v2 + E ⎩

∂t



⎫ ⎬

v +v·∇ +E + ⎭ 2 2



v2 +E 2



 ∂ρ + ∇ · (ρv) + ∇ · J (d) E = ∂t

= ρ Fv + ∇ · (Π · v), where the first term on the left is the total time derivative along the trajectory of the total energy density, and the second vanishes by virtue of the continuity equation. As a result, we obtain the equation for the total energy transport in the Lagrangian coordinates  2  d v2 + E ρ (1.2.16) + ∇ · J (d) E = ρ F · v + ∇ · (Π · v). dt Now we can get the transport equation for the internal energy. For this, we subtract term-by-term the momentum transport Eq. (1.2.12) scalar multiplied by the velocity vector from the Eq. (1.2.16) d v2 = ρ F · v + (∇ · Π) · v. dt 2

ρ

Taking into account the relation ∇ ·(Π · v) = (∇ · Π)·v+Π ◦∇v (in the last term, ∂vi ), the circle denotes the double convolution of rank 2 tensors Π ◦ ∇v = i, j i j ∂r j we obtain the equation for the internal energy transfer in the Lagrangian coordinates ρ

dE + ∇ · J (d) E = Π ◦ ∇v. dt

(1.2.17)

For droplet liquids with the stress tensor in the form (1.2.13), the Eq. (1.2.17) can be rewritten as dE (1.2.18) + ∇ · J (d) E = − p∇ · v + P ◦ ∇v. dt  ∂vi It was taken into account that p i, j δi j ∂r = p∇ · v where the Kronecker symbol j denotes the components of the unit tensor I Please provide appropriate citation instead of 2.24. We introduce a new notation that defines the vector of the heat flux q = − J (d) E in terms of the density of the diffusion flux of internal energy and rewrite the Eq. (1.2.18) in the new notation ρ

1.3 Validity of Continuum Mechanics

ρ

11

dE = − p∇ · v + P ◦ ∇v + ∇ · q. dt

(1.2.19)

According to (1.2.19), the internal energy at a given point of the system can change both due to reversible processes of volumetric compression and expansion (first term on the right) and due to irreversible dissipative processes accompanied by viscous friction (second term) and diffusion heat transfer (third term) that is called thermal conductivity. However, we must take into account that the strong volumetric compression and expansion can also be irreversible due to dissipative processes accompanying the rearrangement of the medium internal structure.

1.3 Validity of Continuum Mechanics In the framework of continuum mechanics, the introduction of macroscopic densities and the transition from integral transport equations to the differential ones are correct only on the condition that the volume V of the macroscopic system under consideration is so large that there exists such a differentially small but still macroscopic volume that meets the condition d V  V . It means that the number of particles in the differential volume d N must be rather large but simultaneously small compared to the total number of particles in the system 1  d N  N.

(1.3.1) 1

Let the typical linear size of the system is L ∼ (V ) 3 , the differential size is 1 dl ∼ (d xd ydz) 3 , and the typical microscopic length is λ (for example, the average distance between the molecules of the medium or the average free path of a gas molecule). It must be noticed that complicated media can be characterized by a whole spectrum of typical lengths connected to the medium internal structure. In this case, we must choose the largest of them. The relations between these three scales must satisfy the inequality λ  dl  L .

(1.3.2)

This means that all the effects associated with the internal structure of the medium can be neglected. The concept of a continuous or structureless medium is valid only when the inequality (1.3.2) is satisfied. However, all of the above is true provided that the change in density on the length scale dl is negligible |a|  1. |a|

(1.3.3)

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1 Models of Continuum Mechanics and Their Deficiencies

It means that there exists such an averaging scale that is rather large to provide insensitivity to microscopic fluctuations and rather small to be independent of its own size. This is assumed that the density field gradients are sufficiently small. In the general case, the typical macroscopic scale L is not always comparable to the system size. The area of validity of continuum mechanics depends on the gradients of macroscopic densities fields. Therefore, the typical macroscopic scale L should be defined by macroscopic gradients as follows: L=

|a| . |grada|

(1.3.4)

 1, the typical macroscopic When the density gradients are small |grada| ∼ |a| L length L is rather large compared to the characteristic size of the structural elements of the medium. In this case, we can use the macroscopic size of the system as the typical macroscopic scale L. When the gradients of the density field a(r, t) are large in a certain zone of the system, the typical macroscopic scale L is determined by its gradients according to (1.3.4). In the general case, the condition for the applicability of continuum mechanics which is based on differential transport Eqs. (1.2.3) can be formulated as follows: λ  dl 

|a| . |grada|

(1.3.5)

If the condition (1.3.5) is not satisfied for at least one of the densities, then the classical continuum mechanics is inapplicable. In zones with large gradients of macroscopic fields, the transition to a differential form of the transport equations may be incorrect. A similar condition can be written for the characteristic times τ

a . |∂a/∂t|

(1.3.6)

According to condition (1.3.6), the characteristic time of the processes at the level of the internal structure of the medium τ should be much less than the characteristic time of the macroscopic process. The condition (1.3.6) is satisfied for slow processes when a typical time for a change of the density field a(r, t) is large. For fast processes characterized by small typical times, the continuum mechanics is unsuitable. So, the differential transport equations of continuum mechanics are correct to describe slow processes with small spatial gradients in macroscopic systems. Thermodynamic states of the system during the process remain close to local equilibrium. High-rate and large gradient processes are non-equilibrium; the effects of the medium internal structure cannot be neglected because their scale is compatible with the size of the zone (1.3.4). Just the same takes place for the processes in media with complicated internal structures. Thin layers near interphase boundaries and initial stages of processes characterized by small space-time typical scales are also out of the validity

1.4 The Problem of Closure of the Transport Equations

13

conditions of continuum mechanics. Therefore, non-equilibrium processes cannot be described by the equations of continuum mechanics.

1.4 The Problem of Closure of the Transport Equations The set of transport Eqs. (1.2.7), (1.2.14), and (1.2.19) consists of 5 equations for 18 unknowns: ρ, vi , p, Pi j , E, qi , i, j = 1, 2, 3. Since the set is incomplete, there is a problem with closing it. To solve this problem, it is necessary to obtain the missing number of relationships between the quantities in the set, i.e. closing relationships. Thermodynamic equations of state for equilibrium systems connecting quantities ρ, p, E, and constitutive relations for dissipative flows arising in various media during their motion can be used as the closing relationships. There are various approaches to obtain the closing relationships for the macroscopic transport equations. Some of them listed below are theoretically grounded. Others are empirical or semi-empirical. They will be discussed in the next section. All of them have different ranges of applicability limited by various restrictions and conditions. For example, in the absence of motion in the system when the mass velocity v = 0, there is no viscous friction P = 0, but at the same time, the vector of the heat flux can be nonzero q = 0. In some cases, in flows of liquid media, it is possible to neglect thermal conductivity and to take q = 0. The first way to close the transport equations suitable only for locally equilibrium systems under the condition P = 0, q = 0 is closure using thermodynamic equations of state. For example, thermal equations of state p = (ρ, T ), E = (ρ, T ), where T is the thermodynamic temperature. Equations of state in equilibrium thermodynamics will be described in Chap. 4. Although it is generally incorrect to use the equilibrium equations of state to describe non-equilibrium processes, under close-toequilibrium conditions, as the practice has shown, the equilibrium equations of state work. The method is applicable only for media described by the ideal fluid model. The second method is based on linear thermodynamics of irreversible transport processes [19], which allows one to obtain constitutive relationships for Pi j , qi directly at the macroscopic level for the processes in any media near local thermodynamic equilibrium. This method will be described in Chap. 3. The problem of constitutive relationships is solved on the base of linear thermodynamics only in two limiting cases: for low velocity gradients or low strain-rates (classical hydrodynamics) and for small deformations (elastic theory). Under these constraints, continuum mechanics adequately describes real processes. Besides the linear thermodynamic, there are various extended thermodynamic approaches [20] that describe the nonlinear properties of certain media with complicated internal structures. There are empiric and semi-empiric models, such as rheological models of non-Newtonian [21], multi-component and multi-phase media, models of turbulent flows [8, 22–24], and others. However, such kind models are not universal; each of them has its own rather narrow range of applicability. The models are not predictive because they cannot describe the temporal evolution of

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1 Models of Continuum Mechanics and Their Deficiencies

the medium properties. The cause of these deficiencies lies in the fact that specific properties of medium with complicated structure appear only under non-equilibrium conditions while the macroscopic description of non-equilibrium processes is always incomplete. The third method is based on the kinetic description of the motion of gaseous media using the velocity distribution functions of gas molecules [25–27]. Such a distribution function makes it possible to calculate all macroscopic densities ρ, vi , p, Pi j , E, qi when a solution to the kinetic equation describing the evolution of the distribution function under the given conditions is found. The kinetic method of description is the most general and rigorous since it provides closure at the microscopic level, but its applicability, in contrast to the above two methods, is limited only to rarefied gaseous media. Various statistical and kinetic approaches to describe condensed media are incomplete and have to use different hypotheses for their closure.

1.5 Medium Models and Transient Processes So, the equations of state and the constitutive relations for the dissipative fluxes Pi j , qi allow us to close the system of transport equations. The resulting closed transport equations, depending on the form of the constitutive relations, can differ significantly from each other. Differential relationships result in differential equations in partial derivatives of various orders, and integral relations in integro-differential transport equations. In the framework of continuum mechanics, a choice of the constitutive relationships determines a concept of the medium model. For the transport processes in liquid media in the local equilibrium state, differential equations of the first order for an ideal fluid P = 0 q = 0 (Euler’s equations) are used. Describing transport processes in a viscous fluid, the Navier–Stokes [28] equations are partial differential equations of the second order of parabolic or elliptic type depending on the parameters. The constitutive relationships resulting in the Navier–Stokes equations are determined by a linear dependence between the corresponding components of the viscous stress tensor and the strain-rate tensor. This is the Newtonian model of a fluid. Linear relationships between the components of the stress and strain tensors like, for example, Hooke’s law, result in the wave equations describing elastic vibrations in solids. Nonlinear relations define non-Newtonian models of liquids. Integral constitutive relationships define, for example, the model of the medium with memory. Such models to describe real flows of liquid and gas are successfully applied both in the interests of science development and for solving practical problems, the development of new technologies. When describing steady state of media with a complicated internal structure, the concept is used correctly provided the steady state is stable. However, the stability of such media states depends on the constraints imposed to maintain the steady state of the system out of equilibrium. Close to equilibrium, any medium behaves like a continuum. When describing processes in media with complicated internal

1.5 Medium Models and Transient Processes

15

structures or in simple media under intensive external influence, the medium state and its properties can essentially change during the process and the problem of transient models appears. For the processes far from equilibrium, the concept of the medium model becomes incorrect. For example, with impact on a medium, a liquid can behave like a solid and exhibit elastic properties, while a solid can exhibit plastic properties like a liquid. When the constitutive relations describe the medium behavior in the intermediate zone between the elastic and hydrodynamic responses, then one can talk about models of transient processes. The main difficulty in describing transients is that the initial and final properties of the system are so different that their descriptions can relate to different disciplines. Before the transition, the system exhibits properties inherent to one type of model, and after the transition, its properties can correspond to a quite different model. During transient mode, the initial properties are replaced by others. To describe the transient process, models of new media uniting both of these properties simultaneously are often introduced in the form of a linear combination of initial and final models, such as the Voigt model of a viscoelastic medium; Maxwell’s model, which also takes into account the relaxation properties of the medium, semiempirical models of non-Newtonian media; turbulence models of different types, etc. However, it turned out that all such models have a very narrow range of applicability and are completely unsuitable for predicting the medium response when external conditions change. All attempts to generalize them to wider classes of processes, as a rule, lead to very cumbersome constructions that lose the physical visibility inherent in basic models. Consider, for example, the well-known Voigt model of viscous–elastic medium [29] that consists of a Newtonian damper and Hookean elastic spring connected in parallel. In the case of one-dimensional deformation of the medium along the x-axis e, the corresponding stress component J exceeding the elastic limit can be written as follows: 



2 ∂e 4 (1.5.1) , J = K + G e+ λ+ μ 3 3 ∂t where K , G are the bulk and shear elastic modules and λ, μ are the bulk and shear viscosities correspondingly. The model (1.5.1) assumes that both elastic and plastic parts contribute simultaneously while we know that the plastic flow develops gradually over time. Besides, the elastic and plastic parts of stress and strain are separated in advance. It is not quite correct for short-duration processes when the stress depends on strain-rate. The Maxwell relaxation model is partially devoid of these shortcomings. In the same case of one-dimensional deformation, the Maxwell relaxation model [30] can be written in the form of a differential constitutive relationship 4 ∂e J − J∗ ∂J = (K + G) − , ∂t 3 ∂t tr (J )

(1.5.2)

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1 Models of Continuum Mechanics and Their Deficiencies

where tr is a typical relaxation time. It is assumed that the medium has a well-defined plasticity threshold, that is, the critical value of the compression stress J ∗. When J > J ∗, the medium begins to exhibit the plastic properties. In the case when the relaxation time is independent of the stress, tr = const, the Eq. (1.5.2) can be easily integrated over time as a linear differential equation of the first order with respect to J on the initial condition J (t = 0) = 0

 t 

 t 1 ∂e(x, t  )  4 t − t J = J ∗ 1 − exp − dt . + K + G tr exp − tr 3 tr tr ∂t 0

(1.5.3) From relation (1.5.3), it follows that the elastic, viscous, or plastic properties of the medium, as mentioned above, can be determined only depending on the ratio between the loading time and the relaxation time of the system. At long times t >> tr in the limit of complete relaxation, the relationship (1.5.3) leads to the viscous–plastic model of the medium





 2 ∂e 2 4 J → J ∗+ λ+ μ ; λ + μ = K + G tr t/tr → ∞. 3 ∂t 3 3 In the absence of dissipation, when the viscosity can be neglected after the threshold J * is exceeded, we get the model of ideal plasticity. In the case of frozen relaxation at small times t  tr , the model of the elastic medium follows from (1.5.3)



  4 2 1 J → λ+ μ ex x = K + G ex x ; ε → Ctr → ∞, t/tr → 0. 3 ε 3 If the loading time and the relaxation time are not separated by the scale t ∼ tr , the model of the viscous–elastic–plastic medium is used. We can see that the transient model (1.5.3) even in the simplified form is much more meaningful than the Voigt model. However, as experiments show, the model also has restrictions on the applicability to shock-induced processes. In the relationship (1.5.3), the integral kernel has a simple exponential form that results from the linear Eq. (1.5.2). The dependence tr (J ) makes the Eq. (1.5.2) nonlinear but leaves it differential in the framework of continuum mechanics. Now we know that the high-rate plastic deformation is a multi-scale and multi-stage process that cannot be described by differential models of continuum mechanics. It must be noticed that in the general case, the problem of transients is not resolved because thermodynamics of the processes far from equilibrium is also not yet developed. Mathematical models of transient processes should lead to the transport equations, which can change their type over time depending on external conditions. Such

1.6 Averaging in Continuum Mechanics

17

models can be classified as flexible mathematical models, while traditional models of continuum mechanics and extended thermodynamics are rigid models.

1.6 Averaging in Continuum Mechanics The problem of averaging is one of the central problems in the mechanics of continuous media. The choice of the method and scale of averaging determines the adequacy of one or another mathematical model. Equations describing motions of atoms and molecules or movements of small volumes of real media at different scale levels can be considered within the framework of the averaging procedure as microscopic ones. Various procedures for averaging such microscopic equations reduce the initial description of the system, making it simpler and more approximate. Equations obtained by averaging can have various forms: from integral equations and equations in finite differences to differential equations of various types. The averaged description can be considered macroscopic. At the same time, in complex (multiphase, turbulized, and composite) media, movements of an intermediate, mesoscopic scale appear. The procedure of spatial averaging leads to the natural introduction of additional degrees of freedom. The most important point of spatial averaging is the possibility of the appearance of new variables associated with statistical moments of higher orders. Movements at the intermediate between macro and micro levels associated with higher moments lead to a significant change in the macroscopic behavior of the medium. For example, taking rotational motions on the mesoscale into account makes the medium anisotropic, stress tensors in it lose their symmetry properties, etc. It is the principle of spatial averaging that makes it possible to uniformly describe a very wide class of media with a complex internal structure. However, the growing number of new degrees of freedom can lead to going beyond the limits of the applicability of the entire concept of a continuous medium. Macroscopic densities of mechanical quantities included in the equations of continuum mechanics are averaged values of microscopic quantities. At the microscopic level, the carriers of these mechanical quantities are such structural elements of the medium as molecules, atoms, and small dispersed particles of different phases. The averaging principle offers a recipe for replacing a complicated disturbed system with a simpler one taking into account the contribution of these disturbances. In continuum mechanics, the averaging process makes it possible to replace a discrete system of a huge number of particles by a continuous medium in which mechanical quantities are distributed continuously with certain densities [6, 8]. As a result of averaging, the number of variables is essentially reduced; the description of the system becomes coarser and simpler. In contrast to the microscopic movements of molecules, which are described statistically as a random process, the macroscopic behavior of the system becomes deterministic, more stable, and predictable. However, the field of applicability of the models of continuum mechanics is significantly narrowed in comparison with the full microscopic description. This averaging

18

1 Models of Continuum Mechanics and Their Deficiencies

procedure is possible if the scales of the spatial and temporal averaging satisfy conditions (1.3.5)–(1.3.6). As discussed in the previous sections, continuum mechanics models are not suitable for describing high-speed and fast processes, as well as the behavior of complex systems. In mechanics, there are several methods for introducing average characteristics of motion: spatial averaging, temporal, space-time, probabilistic, etc. [31]. All these methods lead to practically the same systems of averaged equations. The difference is manifested only in the choice of the main hypotheses in their substantiation and experimental measurement of the averaged parameters. Averaging over time and volume in continuum mechanics is considered equivalent. This property of systems is called ergodicity [15]. Ergodicity means that the average time behavior on a trajectory of movement is independent of the particular trajectory chosen. This means that the mathematical expectation of temporal data is equal to the mathematical expectation of spatial data; therefore, when modeling or calculating, temporal data can be replaced with spatial data. Many systems are ergodic, but some are not. The modern concept of ergodicity has found many applications in mathematics. The averaging method makes it possible to overcome the difficulties of solving singularly perturbed problems and to obtain uniformly suitable solutions. Various methods of averaging over time have been used in the theory of nonlinear oscillations when there is a separation of variables into fast and slow ones [32]. For example, in turbulence, motion is divided into pulsating and main, averaged. The idea of separating the fast and slow components of the solution turned out to be very fruitful. The method of multi-scale expansions makes it possible to use descriptions of different levels in areas of different scales and stitch them together at the boundaries of these areas. Many statistical theories accept the ergodic hypothesis when the system “forgets” its initial state and behaves chaotically. In some cases, a part of the parameters of a dynamical system at sufficiently long time intervals can take constant values. Statistical dynamical systems can have varying degrees of stochastic stationarity [9–14]. A stationary process is one whose probability distribution is the same at all times. Modeling real processes, it is often assumed for simplification that the random processes describing them have stochastic stationarity to the required extent. Macroscopic quantities are mathematical expectations or the zeroth-order moment functions of a random process 1 m x = x(t) = lim T →∞ T

T /2 x(t)dt. −T /2

The first-order moment function of a random process is the variance that determines the power of fluctuations in the system

1.6 Averaging in Continuum Mechanics



19



1 Dx = (x(t) − m x ) = lim T →∞ T 2

T /2 x 2 (t)dt − m x −T /2

The most important characteristic of a random process is the correlation function which characterizes the statistical relationship between its values, separated by a time interval Rx = x(t)x(t − τ ) − m 2x . The slower this function decreases with increasing time intervals, the longer its values are related. In a stationary process, the probability density does not change when the time arguments are shifted by the same amount. Its first moments do not depend on time, and the correlation function is an even function and depends only on the duration of the time interval Rx (τ ). It can be noticed that Rx (0) = Dx and the normalized correlation function meets the inequality Rx (τ > 0)/Dx < Rx (0)/Dx . To calculate the estimates of these functions for different values of the argument, you need to have a set of process implementations. In some cases, in practice, it is not possible to obtain a sufficient number of such implementations. Often, the researcher has only one implementation of the process. It turns out that when certain conditions are met, one such single implementation can be used to calculate moment functions. In particular, one of these conditions is the ergodicity of a random process in terms of its main characteristics. Stationary stochastic processes may or may not have an ergodic property. An ergodic random process is always stationary, but not every stationary process is ergodic. A sufficient condition for the ergodicity of a stationary process is its correlation function tending to 0: Rx (τ → ∞) → 0. In practice, ergodicity is used without proof. Since averaging over a set of realizations is equivalent to averaging over time of one infinitely long realization, all characteristics are counted from one realization, averaging over a finite time interval, taking into account that with an increase in its duration, the accuracy increases. However, long-term prediction of ergodic systems is impossible: a small measurement error will lead to a serious discrepancy between the real trajectory and the predicted one. In the general case, the averaging operation can be considered as a certain  εT smoothing operator u(τ ) = εT1 0 x(τ + ξ )dξ . The concept of time evolution is closely related to a smoothing operator. The thermodynamic evolution of physical systems also can be described by a certain smoothing operator since all gradients and rates in the system decrease approaching thermodynamic equilibrium. According to Bogolyubov’s hypothesis [33], in the process of the system evolution toward equilibrium, correlations decay, gradients are smoothed out, and, with them, the power of fluctuations described by the variance, the first-order moment function of the random process, falls. Since moments of a random process change with time, the random process cannot be considered stationary. It means that a non-equilibrium process is not ergodic. The process far from equilibrium is not chaotic; it remembers

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1 Models of Continuum Mechanics and Their Deficiencies

the information about the initial state of the system and new information about regular modes of motion appearing during the process. Substitution of temporal probability for ensemble one in such processes will lead to a discrepancy between theory and practice. Therefore, it is impossible to apply modeling methods based on averaging over spatial data to non-ergodic dynamical systems. The systems in the framework of continuum mechanics are all considered ergodic. Beyond the validity of continuum mechanics, we deal with non-ergodic processes. For example, consider the thermal diffusivity equation obtained within continuum mechanics in the case when the temperature fluctuations are not small. ∂T ∂2T − κ 2 = 0. ∂t ∂y We average over space the equation in the one-dimensional case using the smoothing operator with a weight function in the form 1 ε

∞ −∞

 

y − y dy  exp − ε2

2  .

We take the integral of the right-hand side of the equation by parts, and the derivative of the integral kernel can be taken outside the integral sign, taking into account the equality     2  2  y − y y − y ∂ ∂ exp − = − exp − . ∂ y ε2 ∂y ε2 By repeating this procedure twice, we obtain the equation for averaged T over the spatial scale ε ∂T ε ∂ 2 T ε = 0. −κ ∂t ∂ y 2 In an unbounded domain, the averaged equation has the same form as the original one but in a real system, initial and boundary conditions should be different. In the case ε → 0, the averaged equations come back to the original one, and in the case ε → ∞, we get a stationary homogeneous distribution of the temperature over the y-axis. Then the growing scale of averaging leads the system to thermodynamic equilibrium. In a semi-bounded domain, the averaged equation changes

  2y ∂T ε ∂ 2 T ε 1 (y − 0)2 (y − 0)2 ∂ T exp − = 0. exp − (0) + −κ T (0) − 3 2 2 ε ε ε ε ∂y ∂t ∂ y 2

1.7 Deficiencies of the Continuum Mechanics Concept

21

This means that a layer of thickness of the order O(ε) has appeared near the boundary of the domain, where the averaged description is inapplicable. Averaging over time results in a quite different equation related to the initial temperature distribution. So, we can see that formulation of initial and boundary problems for averaged equations essentially depends on the higher order moments of the random fluctuations which are not included in the description of continuum mechanics. Even the stationary random process near the system boundaries cannot be considered ergodic. Just the same one can assert about non-stationary processes in their initial stage. Both the cases are out of the validity domain of the continuum mechanics model.

1.7 Deficiencies of the Continuum Mechanics Concept The transport equations of continuum mechanics describe macroscopic properties of the system near thermodynamic equilibrium. The slow motion of liquids and gases is described by the equations of classical hydrodynamics; the behavior of solids under load is described by the mechanics of a deformable solid. The apparatus of continuum mechanics is based on differential equations for the transport of mass, momentum, and energy, which are obtained under the condition that the size of the structural elements of the medium is small in comparison with the inhomogeneity scale of the macroscopic fields. These equations are closed using various relations between the flux densities and the gradients of macroscopic fields, which determine the model of the medium (Newtonian fluid and elastic solid). On close-to-equilibrium conditions, the constitutive relationships are linear. The linear mathematical models of continuum mechanics describe the behavior of media deterministically. When the medium has a coarse-grained structure comparable with the scale of the macroscopic inhomogeneity of the system, the assumption that the structural elements of the medium are small does not work, and the concept of a continuous medium loses its meaning. Therefore, the well-known nonlinear models describing the flows of non-Newtonian fluids, multi-phase media, and plastic flows of solids have limited validity areas. Just the same takes place on small typical scales of length and time when the effects of the internal structure of the medium begin to appear in high-speed processes in real media. Atomic–molecular effects in macroscopic systems require a transition to the microscopic level of description, where statistical approaches are used. For high-speed and fast-flowing processes far from local equilibrium, the models of continuum mechanics are no longer suitable. These processes are accompanied by a variety of turbulent, inertial, and relaxation effects, the space-time scales of which are intermediate between the macro and microscale levels. These processes are already characterized by the highest moments of statistical distributions, which are not included in the set of macroscopic densities. Besides, the response of the system to an external action depends significantly on a set of parameters characterizing the properties of the loading itself and especially on its duration, pulse input rate,

22

1 Models of Continuum Mechanics and Their Deficiencies

boundary conditions, etc. These parameters should also be included in the description of non-equilibrium processes. This means that the set of macroscopic densities in the equations of continuum mechanics is incomplete. The inclusion of additional variables in it does not expand the possibilities of such a description if the equations themselves remain within the concept of a continuous medium. Differential equations, in principle, do not correspond to the physics of such phenomena as collective effects, inertia, turbulence, and structure formation.

1.8 The Problem of a Uniform Description of the Media Motions The problem of a uniform description of the medium motions in a wide range of conditions cannot be considered completely solved. On the conditions that lie beyond the limits of applicability of commonly used models, the choice of one or another closing equation is a very nontrivial problem. Active development of numerical methods and the creation on their basis of universal software packages that use the latest achievements in the field of computer technology is undoubtedly a significant advance in the development of the instrumentation of mechanic researchers to solve practical problems. However, the problem of closing the transport equations in such software packages is essentially reduced to the voluntary choice of one or another model built into a given computing complex without indications of under what conditions a particular model should be applied. The models of media with complex properties, such as rheological models of non-Newtonian fluids [21], most of which are tied to the properties of a particular substance, and various turbulence models [22–24] are valid only in a fairly narrow range of parameters, and attempts to generalize them staying within the concepts of continuum mechanics to wider classes of problems, as a rule, lead to very cumbersome constructions that lose their physical meaning. Transient processes, such as elastic–viscous–plastic flows of deformable solids and flows with a laminar–turbulent transition, are characterized by a very complex interaction mechanism in the medium, which can significantly change over time depending on the stage of the process, its mode, and external loading conditions. It is obvious that such transient phenomena cannot be described within the framework of models developed for their limiting situations. Therefore, the task of determining the limits of validity of certain approaches and the development of a more general and universal apparatus seems to be very urgent at present. It is necessary to develop fundamental theories based on rigorous results of nonequilibrium statistical mechanics [34, 35], as a flexible apparatus for solving problems, which, due to its self-consistency, is also applicable in those cases when a rigorous model of the phenomenon is either not known a priori, but should be determined directly in the course of solving the problem. To describe transient transfer processes, it is necessary to develop approaches that could take into account the possibility of changing the regularities of the process at its various stages. It is essential

1.8 The Problem of a Uniform Description of the Media Motions

23

that such a change in the characteristic internal properties of the process should occur self-consistently, that is, it should be linked to the history of the system. In the framework of continuum mechanics, it is not possible to implement such an approach. The point is that the local theory does not have an internal control mechanism responsible for the structural features of the ongoing process. As a rule, the change in the transport mechanism is set manually by dividing the space-time continuum into regions where the transport mechanism is fixed, and then “splicing” the solution at the fictitious boundaries of these regions. This approach leads to an unjustified complication of theoretical constructions and at the same time imposes very strict restrictions on the degree of generality of the results obtained. Models based on the results of statistical and phenomenological theories of turbulence are being replaced by models in which great importance is attached to dividing the scales of description, identifying intermediate mesoscopic levels, and taking into account the evolution of large structures. Thus, a gradual abandonment of “rigid” models tied to fixed modes in favor of models with more flexible properties is already beginning. A partial solution to this problem was obtained within the framework of the theory of dissipative processes with self-organization, in which the methods of external control are partially applied. Within the framework of this theory, the integral conditions imposed on the system, which determine some of the so-called order parameters, played an important role. The introduction of these parameters leads to the appearance of a new (intermediate between microscopic and macroscopic) level of description and makes it possible to adjust the structure of the system to external conditions. Within the framework of this approach, the macroscopic system becomes capable of self-organization. In works in this direction, very interesting and meaningful results were obtained [36–38]. The main problem that hinders the further development of synergetic approaches in mechanics is the lack of a clear understanding of the physical essence of the order parameters, which are introduced, as a rule, formally. The change in these parameters, that is, the evolution process itself, has to be modeled in most cases based on intuitive ideas about the nature of non-equilibrium processes. There is no complete self-consistency in this approach. Therefore, it is necessary to develop more general approaches that would take into account the possibility of changing the patterns of the medium response to external loading, for example, by generating the corresponding internal structure. Despite quite different initial states, any medium under loading tends to move closer to the equilibrium state as far as the imposed constraints allow. It means that there are regularities common to all media that are inherent in the transition processes not only between qualitatively different states of the system, but also phase transitions, changes in modes, and mechanisms of mass, momentum, and energy transport. In order to predict when and under what conditions such a transition will occur, it is necessary to understand what processes occur in the zone of the transition itself. And here another general problem arises, associated with the description of processes occurring far from thermodynamic equilibrium. High-rate and rapidly occurring processes are characterized by small spatiotemporal scales on which the internal structure effects appear while the continuum mechanics models adequately describe processes in distributed systems at a rather

24

1 Models of Continuum Mechanics and Their Deficiencies

large, macroscopic scale when the system state is approaching local equilibrium. So, a description of the narrow transient zone and highly non-equilibrium processes are closely related. Therefore, in order to describe transient processes, we need to understand the physical nature of the processes far from equilibrium, generalize and revise all the concepts of thermodynamics, and develop a unified mathematical description valid for various media in a wide range of loading conditions with accounting self-organization effects.

1.9 Short Review of Approaches to Extension of Continuum Mechanics Formally, the generalization of the transport equations within the framework of continuum mechanics to high-rate processes in real media can be carried out at various stages of the reduction of the description depending on the deviation of the system state from thermodynamic equilibrium. Attempts to build such extended approaches have been undertaken for a long time [39–70]. They can be divided into three main types. (1). The approaches of the first type contain hydrodynamic models with higher derivatives, obtained using expansions in a small parameter (inhomogeneity or in the Knudsen number) with the subsequent termination of the series at a finite number of terms. Since asymptotic expansions do not have the property of uniform convergence to the limit, attempts to move toward large values of the parameter using a larger number of terms of the asymptotic series, as a rule, do not lead to success. So, for example, for rarefied gases, there is such a Knudsen number Kn *: that the equations obtained in the framework of the Chapman–Enskog method and containing a finite number of terms of the series for some problems simply do not have a solution [59]. Gradient models and moment theories can be attributed to the same type since the presence of a small parameter that allows one to neglect higher order terms and close the model is still implicitly implied in these theories]26[. In addition, it is worth noting that higher order equations are too complicated, cumbersome, and require the setting of non-physical conditions for higher derivatives. The earliest attempts to generalize the classical theory were made for the equations of diffusion and heat conduction [42, 47, 55] and allowed to take into account the finite rate of the transfer process. The so-called telegraph equation was obtained in this way. It differs from the usual parabolic diffusion equation by an additional term with the second time derivative of the concentration, which makes the equation hyperbolic. This equation can describe the process of fast diffusion, for which the parabolic equation is unsuitable. It was proved that for t → ∞, the solution of the telegraph equation asymptotically tends to the solution of the usual diffusion equation. The hyperbolic equation of heat conduction for a medium in which there is no transfer of mass and momentum was derived by Khonkin [60] from the Boltzmann

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25

equation and by Robertson from the Liouville equation [51]. Hyperbolic equations were derived and used to describe the propagation of disturbances in a medium with a finite velocity by many authors [42, 48, 55, 61]. In work [62], hydrodynamic equations describing high-speed and highly gradient transport processes were obtained on the basis of the Boltzmann kinetic equation. These hyperbolic equations are uniformly valid up to the boundaries and the initial moment of time, including the Knudsen layers. However, these equations are correct only as long as these layers are thin. As the thickness of the near-wall layers grows with increasing Knudsen numbers, the obtained equations also become unusable. Previously, only phenomenological approaches were used to close the transport equations. For example, the various constitutive relations in classical hydrodynamics, linear theory of elasticity, theory of multi-phase flows, and rheology were obtained in this way. Determining the response of a specific medium to an external disturbance, all these relationships introduce the medium model with its specific empiric characteristics (in the form of transport coefficients or elastic modules) into the description of the system. A complete classification of the models of media with complicated properties proposed by W. Prager [63]: σ + c1 σ˙ + c2 e + c3 e˙ + c4 = 0. Here σ is stress, e is deformation, μ is viscosity, E is the elastic module, dots indicate the first time derivatives of stress and strain, and ci are various constants of the medium. For simplicity, we consider the one-dimensional case. Giving a physical meaning to these constants, we can get 8 constitutive equations that determine the models of the media: σ + μe˙ = 0—Newton’s viscous fluid, σ − σ0 + μe˙ = 0—viscoplastic medium, (F.N. Shvedov, B.G. Vainberg, A.A. Ilyushin), σ − Ee + μe˙ = 0—viscoelastic Voigt medium, σ − σ0 − Ee + μe˙ = 0—elastic–viscoplastic medium, σ + τ σ˙ + μe˙ = 0—viscoelastic relaxing medium, (Maxwell’s model), σ − σ0 + μe˙ + τ σ˙ = 0—elastic–viscoplastic relaxing environment, σ − Ee + μe˙ + τ σ˙ = 0— (A.Yu. Ishlinsky), σ − σ0 − Ee + μe˙ + τ σ˙ = 0—(V.V. Sokolovsky, L. Malvern). Differential constitutive relationships that increase the differential order of the transport equations can be referred to as the first type of model. It is generally accepted that for each medium there should be its own governing equation. However, the question of the applicability of this or that model is associated not only with the properties of the medium, but also with the mode of its loading. During high-rate loading, the coefficients of the models cease to be the medium constants and turn into nonlinear functionals of macroscopic gradients. This means that modeling based on differential models with empirical coefficients does not adequately describe non-equilibrium transport processes in real media with an internal structure.

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(2). Understanding the shortcomings of the first type of approach led to the development of integral models in which the integrals are the convolution of an infinite series and do not contain a small parameter. For gases, such models are valid for arbitrary values of the Knudsen numbers. Since the constitutive relations for dissipative flows are integral, the corresponding transport equations will be integrodifferential. Such equations, including the effects of memory and spatial nonlocality, are very complex, and only some special cases of their solution are known. For example, in [47], a theorem was proved for the existence and uniqueness of the solution of the one-dimensional integro-differential heat equation. From a mathematical point of view, nonlocal equations lead to so-called pseudo-differential operators, for which a rigorous theory has not yet been sufficiently developed. In [53], some nonclassical effects associated with nonlocality are considered, when anisotropy and boundary layers arise. In the presence of boundary effects, the operators become anisotropic near the boundaries or even over the entire region. In [52], the mathematical meaning of nonlocality in physical theories is determined, and relations are established between the classical, non-classical, and pseudo continuum; integral operators; and differential operators of infinite order. It is shown that most of the non-physical results of the classical theory are due to the freedom with which the classical continuum forms singularities, high-frequency oscillations, etc. In [54], it was proposed to proceed not from the dynamics of a material point, but from the dynamics of a structural element of a medium of finite size. At the same time, it was emphasized that nonlocality is a price to pay for those effects that affect macroscopic fields, but are not included in the system of parameters that describe the process. And first of all, the above applies to the near-boundary effects. Strictly speaking, nonlocal macroscopic models should be built on the basis of nonlocal kinetic equations. However, the reduction of the description even from the level of the local kinetic equation, for example, the Boltzmann equation, to the hydrodynamic level should still lead to nonlocal hydrodynamic equations that are valid for arbitrary Knudsen numbers. In the general case, the transition from the kinetic level to the hydrodynamic description, being a part of the fundamental problem of non-equilibrium statistical mechanics, is not completed without using some additional simplifying assumptions. As models of the second type, we can name nonlocal models using such additional assumptions at the kinetic level of description, which make it easier to reach a hydrodynamic description, but do not have a rigorous theoretical basis [54, 62, 64]. As a rule, the accepted simplifications are quite serious and severely limit their area of applicability. Therefore, such approaches cannot serve as a basis for constructing sufficiently universal macroscopic models. In [64], in order to obtain expressions for the stress tensor and the heat flux vector, suitable for arbitrary Knudsen numbers, a kinetic approach was developed based on the concept of the mean free path, which belongs to Maxwell. The result is a nonlocal theory in which the mean free paths for mass, momentum, and energy are the only phenomenological elements. A significant simplifying assumption is made about the nature of the averaging performed in accordance with the Maxwellian function over velocities, which imposes a restriction on the degree of non-equilibrium of the processes. In addition, in order to take

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into account the interaction with a solid boundary, it is necessary to introduce additional mean free paths near the boundaries, which are different from those already introduced, although they do not completely describe the effects of anisotropy. In a number of works [65, 66], the injection of a beam of test particles into the background gas is considered on the basis of the model Boltzmann equation. Under certain assumptions concerning the nature of the interaction of particles, integration along the trajectories of particles allows expressing the velocity distribution function of particles in terms of the density of test particles. Further integration leads to a nonlinear integro-differential equation for the density of test particles in a closed form. The use of the V-shaped function as the collision probability density makes it possible to obtain explicitly nontrivial space-time distributions for the density of test particles. In paper [67], a semi-phenomenological theory of high-speed and high-gradient transport processes in media with a microstructure is developed on the basis of the kinetic equation for the N-particle distribution function of structural elements of the medium by the projection operator method using general principles, model considerations, and experimental data. However, this theory is suitable only for unlimited problems, since the model does not take into account the influence of boundary effects. Generally speaking, the formulation of boundary value problems in a nonlocal theory requires a separate consideration. (3). Approaches of the third type also lead to nonlocal macroscopic equations, but are based on the rigorous foundations of non-equilibrium statistical theory, that is, they are derived from the first principles [34, 35, 50, 51, 68–70]. For gases, there are approaches based on the kinetic theory [25–27]. However, in the general case, the macroscopic expressions for integral kernels are unknown. Moreover, to date, even nontrivial approximations have not been found for them; therefore, in almost all cases, in order to get to the number, it is necessary to involve some model or empirical reasoning. In contrast to the models of the second type, the modeling of the third type is carried out directly at the macroscopic level of description and is reduced to the construction of explicit space-time dependences for integral kernels of nonlocal equations, which are space-time functionals of macroscopic fluxes [40, 56–58]. However, the use of simple space-time dependences as transport kernels in the general case makes the model too rough and does not allow satisfying the real boundary conditions imposed on the system. The point is that this modeling does not take into account the dependence of the integral kernels on the macroscopic gradients. In particular cases, integration over time of the constitutive relationships given in the form of differential equations leads to the integral kernels of the simplest form [70]. The relaxation-type constitutive relations lead to exponential kernels, while the Gaussian kernels are given by the telegraph-type constitutive equations. Then it is clear that exponential kernels describe only the processes of monotonic relaxation without taking into account the rate of propagation of perturbations in the medium. The telegraph equation of the hyperbolic type already contains a finite velocity of propagation of disturbances and combines both wave processes at small

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characteristic times and diffusion processes at large ones. However, although all linear differential equations can be formally written in integral form using Green’s function, not all integral equations can be reduced to differential equations [70, 71]. In the general case, nonlocal hydrodynamic correlations always existing under non-equilibrium conditions [35] lead to a non-monotonic relaxation process and a non-analytical dependence of integral kernels on macroscopic gradients. Therefore, for an arbitrary deviation from equilibrium, it is incorrect to construct differential models of constitutive relations. Within the framework of generalized hydrodynamics, modeling of relaxation transfer kernels for non-stationary processes should include both nonlocal and memory effects. If the spatial and temporal scales of relaxation differ in the order of magnitude or both are small, then it is possible to split the spatial and temporal effects and build different models in time and coordinates. If the effects of nonlocality and memory cannot be separated, the problem of modeling relaxation kernels becomes much more complicated because of the need to take into account the correlations between these effects. The approach proposed to modeling shock-induced processes can also be referred to as the third type. It will be considered in detail in Chap. 5 of the book.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

17. 18. 19.

Slattery JC (1971) Momentum, energy, and mass transfer in continua. McGraw-Hill Co Sedov LI (1997) Mechanics of continuous medium. World Scientific Chadwick P (1976) Continuous mechanics. Allen & Unwic, London Gurtin ME (1981) An introduction in continuous mechanics. Academic press, New York Reddy JN (2006) An introduction to continuous mechanics. Cambridge University Press, Cambridge, UK Ian Murdoch A (2012) Physical foundations of continuum mechanics. Cambridge University press Gurtin ME, Fried E, Anand L (2010) Mechanics and thermodynamics of continua. Cambridge University Press, New York Nigmatulin RI (2014) Mechanics of continuous medium. GEOTAR-Media, Moskow. Tijms H (2004) Understanding probability. Cambridge University Press Rosenblatt M (1962) Random processes. Oxford University Press Doob JL (1990). Stochastic processes. Wiley, pp 46–47 Van Kampen NG (2011) Stochastic processes in physics and chemistry. Elsevier Karlin S, Taylor HE (2012) A first course in stochastic processes. Academic Press Florescu I (2014) Probability and stochastic processes. John Wiley & Sons, pp 294–295 Walters P (1982) An introduction to ergodic theory. Springer Murdoch AI, Bedeaux D (1996) 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, 1111–1129 Gordiz K, Singh DJ, Henry A (2015) Ensemble averaging vs. time averaging in molecular dynamics simulations of thermal conductivity. J Appl Phys 117:045104 Murdoch AI, Bedeaux D (1994) Continuum equations of balance via weighted averages of microscopic quantities. Proc R Soc Lond 445:157–179 De Groot S, Mazur P (1963) Nonequilibrium thermodynamics. Nort-Holland publ. Co., Amsterdam

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20. Lebon G, Jou D, Casas-Vazquez J (2008) Understanding non-equilibrium thermodynamics. Springer-Verlag 21. Astarita G, Marucci G (1974) Principles of non-Newtonian fluid mechanics. McGraw-Hill, New York 22. Hintze T (1962) Turbulence. Mc.Grow, New York 23. Pope S (2000) Turbulent flows. Cambridge University Press, 24. Piquet J (2001) Turbulent flows. (revised 2nd printing) Springer-Verlag, Berlin 25. Chapman S, Cowling TG (1970) The mathematical theory of non-uniform gases, 3rd edn, Cambridge University Press 26. Kogan MN (1969) Rarefied gas dynamics. Plenum, New York 27. Cercignani C (1990) Mathematical methods in kinetic theory, 2nd edn. Plenum Press, New York 28. Landau LD, Lifshitz EM (1987) Fluid mechanics: course of theoretical physics, vol 6 29. Lakes R (1998) Viscoelastic solids. CRC Press 30. Dimitrienko Yu (2011) Nonlinear continuum mechanics and large inelastic deformations. Springer 31. Monin AS, Yaglom AM (1971) Statistical fluid mechanics: mechanics of turbulence, vol 1. In: Lumley JL (ed). MIT, Cambridge, Mass 32. Mitropolsky YA (1971) Averaging method in nonlinear mechanics. Moskow 33. Bogoliubov NN (1960) Problems of dynamic theory in statistical physics. Technical Information Service, Oak Ridge TN 34. Richardson JM (1960) The hydrodynamical equations of a one-component system derived from nonequilibrium statistical mechanics. J Math Anal Appl 1:12–60 35. Zubarev DN (1974) Non-equilibrium statistical thermodynamics. Springer 36. Glansdorff P, Prigogine I (1972) Thermodynamic theory of structure, stability and fluctuations. Wiley Interscience 37. Nicolis G, Prigogine I (1977) Self-organization in nonequilibrium systems. From dissipative structure to order through fluctuations, NY 38. Klimontovich YL (1993) From the Hamiltonian mechanics to a continuous media. Dissipative structures. Criteria of self-organization. Theoret Math Phys 96(3):1035–1056 39. Ailavadi N, Rahman A, Zwanzig R (1971) Generalized hydrodynamics and analysis of current correlation functions. Phys Rev 4a(4):1616–1625 40. Bixon M, Dorfman JR, Mot KC (1971) General hydrodynamic equations from the linear Boltzmann equation. Phys Fluids 14(6):1049–1057 41. Chung CH, Yip S (1965) Generalized hydrodynamics and time correlation functions. Phys Rev 182(1):323–338 42. De Facio B (1987) Heat conduction model with finite signal speed. J Math Phys 16(4):971–974 43. Doering CR, Burshka MA, Horsthenike W (1991) Fluctuations and correlations in a diffusionreaction system: exact hydrodynamics. J Stat Phys 65(5/6):953–970 44. Edelen DG (1976) Nonlocal field theories in continuum physics, 4. Press Inc., Acad 45. Kadanoff LP, Martin PC (1963) Hydrodynamic equations and correlation functions. Ann Phys 24:419–460 46. Kawasaki K, Ganton JD (1973) Theory of nonlinear transport processes: nonlinear shear viscosity and normal stress effects. Phys Rev A 8(4):2048–2064 47. MacCamy RC (1977) An integro-differential equation with application in heat flow. Quart Appl Math 35(1):1–19 48. Mogen GA (1979) Nonlocal theories or gradient type theories: a matter of convenience. Arch Mech 31(1):15–26 49. Mori H (1965) Transport, collective motion and Brownian motion. Progr Theor Phys 33(3):423– 454 50. Piccirelli R (1968) Theory of the dynamics of simple fluid for large spatial gradients and long memory. Phys Rev 175(1):77–98 51. Robertson B (1967) Equations of motion in nonequilibrium statistical mechanics. Phys Rev 35(1):160–183

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52. Rogula D (1979) Geometrical and dynamical nonlocality. Arch Mech 15(1):66–75 53. Rymarz Z (1974) Boundary problems of the nonlocal theory. Proc Vibrat Probl 15(4):355–372 54. Wilmanski N (1979) Localization problem of nonlocal continuum theories. Arch Mech 31(1):77–89 55. Weynann HO (1979) Finite speed of propagation in heat conduction, diffusion and viscous shear motion. Am J Phys 35(2):488–496 56. Filippov BV, Khantuleva TA (1982) Boundary problems of nonlocal hydrodynamics. Leningrad University 57. Khantuleva TA (1984) Nonlocal hydrodynamical models of gas flows in the transition regime. In: Papers 13th international symposium on raref gas dynamics, vol 1. Plenum Press, pp 229– 236 58. Khantuleva TA, Mescheryakov YuI (1999) Nonlocal theory of the high-strain-rate processes in a structured media. Int J Solids Struct 36:3105–3129 59. Grad H (1949) On the kinetic theory of rarefied gases. Comm Pure Appl Math 2(4):331–340 60. Khonkin AD (1976) Paradox of infinite perturbation propagation velocity in the hydrodynamics of a viscous heat-conducting medium and equations of hydrodynamics of fast processes. In: Aeromechanics, Nauka, Moscow, pp 289–299 61. Davis PL (1980) On the hyperbolicity of second order hydrodynamic equations. J Non-Equilibr Thermodyn 5(6):377–377 62. Khonkin AD (1973) Hydrodynamic equations of fast processes. Rep USSR Acad Sci 210(5):1033–1035 (in Russian) 63. Prager VW (1955) Probleme der plastizitatstheorie. Birkhauser, Basel-Stuttgart 64. Woods LC (1979) Transport processes in dilute gases over the whole range of Knudsen numbers. Part 1. General theory. J Fluid Mech 93(3):585–607 65. Oggioni S, Spiga G (1991) On exact solution to a discrete-velocity model of the extended kinetic equations. Nouvo Cimento 106B(1):9–20 66. Boffi VC, Spiga G (1988) Spatial effects in the study of nonlinear evolution problems of particle transport theory. Transp Theory Stat Phys 17(2&3):241–255 67. Skvortsov GE (1975) To the theory of high-speed and strongly gradient processes of small amplitude. Rep USSR Acad Sci 68(3):956–973 (in Russian) 68. Zubarev DN (1961) Statistical operator for nonequilibrium systems. Rep USSR Acad Sci 140(1):92–95 69. Zubarev DN, Tishchenko SV (1972) Nonlocal hydrodynamics with memory. Physica 59(2):285–304 70. Rudyak VY (1987) Statistical theory of dissipative processes in gases and liquids. Nauka, Novosibirsk (in Russian) 71. Volterra V (1930) Theory of functionals and of integral and integro-differential equations. Dover Publications Inc, New York

Chapter 2

Specific Features of Processes Far from Equilibrium

Abstract Modern understanding of non-equilibrium processes in distributed systems based on experimental data, obtained with the use of high-precision instruments, is fundamentally different from the previously common opinion that nonequilibrium processes are irreversible non-stationary processes that can be described by partial differential equations. Attempts to apply conventional mathematical models of continuum mechanics far from thermodynamic equilibrium led to serious errors. The main problem is that all physical concepts are related to the system states near local equilibrium and a generalization of one concept implies the revision of all fundamentals of thermodynamics. In order to avoid contradictions, an understanding of special features characterizing the system response to an external perturbation out of equilibrium is needed for mathematical modeling. In this chapter, we list and briefly describe those features that distinguish dynamic processes far from local equilibrium from quasi-static near-equilibrium processes. The influence of each of these non-equilibrium effects will be considered separately and the information currently available to describe it is given. Keyword Non-equilibrium · Internal structure · Multi-scale processes · Multi-stage relaxation · Inertia · Instability · Self-organization

2.1 Experimental Difficulties in Studying Non-Equilibrium Processes The processes occurring far from thermodynamic equilibrium are of high speed and are fast; their typical spatiotemporal scales are small. Since neither models of continuum mechanics nor classical thermodynamics are suitable for their description, a direct experimental study of these processes comes to the fore. This is an extremely difficult task because the nature of real non-equilibrium processes and their characteristics is infinitely diverse. It is clear that for their experimental study, it is necessary to have the appropriate tools. First of all, measuring instruments are required with high accuracy, which will allow tracking very rapid changes in system parameters both in time and in space. All measuring devices have a different degree © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 T. A. Khantuleva, Mathematical Modeling of Shock-Wave Processes in Condensed Matter, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-981-19-2404-0_2

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of averaging of measurement results. If the averaging scale during measurement is greater than or of the order of the typical scale of the process itself, then the result will correspond to a more averaged, near-equilibrium state of the system, from which it is impossible to obtain any information about a real non-equilibrium process. Therefore, in the old days, when neither high-speed technology nor high-precision instruments existed yet, the models of continuum mechanics and equilibrium thermodynamics were quite enough for science and practice. Non-equilibrium processes were considered as some of the few exceptions to the rule. Now the situation has changed significantly. With the advent of laser technology, high-precision measurements have become available. An important step in the diagnosis of fast processes in solids was the registration of the mass velocity waveforms in real time [1–7]. The shape and behavior of the waveforms proved to be very sensitive to structural inhomogeneities of the dynamically deformable material. A particularly great contribution to the study of wave propagation processes and the shock-wave behavior of materials was made by the method of recording wave processes with the help of laser interferometers [2, 4, 7–12]. The high temporal and spatial resolution of interferometric methods made it possible to reveal many features of the shock-wave behavior of materials, including phase and structural transformations, spall processes, etc. The next important step in improving the methods of recording shock-wave processes was the development of a technique for determining the variation of the mass velocity D (the square root of the dispersion of the mass velocity), as a quantitative characteristic of the velocity inhomogeneity of the wave process [5–7]. It turned out that individual parts of cross-sections of the deformable medium have different values of the mass velocity. Intensive shear processes between them lead to the appearance of relative displacements and/or rotations of the structural elements relative to each other and further to the structural transition initiated by shock loading. The observed occurrence of velocity pulsations and the rotational modes are a display of turbulence [13]. It becomes clear that turbulent effects accompany non-equilibrium processes in various media. However, the turbulent effects make non-equilibrium processes even more difficult to study. The implementation of very laborious and detailed experimental and theoretical research has become possible due to advances in the field of electronic and optical instrumentation and computer technology. Methods of direct visualization of flows have acquired an important role in the physical understanding of turbulent processes [14]. It is high-speed visualization methods that can help to see the processes taking place at the intermediate scale levels between macro- and microscales. Back in the second half of the last century, it was shown that transport processes in shear turbulent flows are determined by organized movements, the shape, intensity, and scale which vary greatly depending on conditions [15–17]. However, when studying them, very serious difficulties arose associated with measuring organized motion, separating it from the surroundings and averaging. The study of these movements required, first of all, the direct measurement of spatiotemporal correlations in the fields of various macroscopic quantities. But most of the experimental studies in this direction have a large scatter of data around the poorly defined mean. Only a few

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properties of organized structures can be considered established. Further research in this direction requires improvement and standardization of procedures for obtaining and processing experimental data. The greatest difficulty is the transient regimes from laminar to turbulent flow [18, 19]. It was found that all turbulent shear flows are very sensitive to small perturbations in the transition zone. Previously, it was believed that far from the transition zone, a certain asymptotic state would be achieved, in which the system forgets its initial state. But further studies have shown that it is precisely organized structures that carry the memory of the initial disturbances in the transition zone. The very existence of such information structures suggests that some forms of turbulent flow control are possible. With the advent of improved computational resources over the past few decades, researchers have made significant progress in the study of turbulent flows. Using the methods of computer simulation, it was possible to identify new mechanisms of the origin and development of turbulence near solid boundaries. However, and in this area of research, problems arose with the scale of the calculation cells and the accuracy of the results. For example, accounting for the dynamics of the small-scale motions and their effect on the resolved flow fields requires the use of subgrid-scale models near rigid boundaries [20]. The problem of the accuracy of measurements and models of non-equilibrium processes is common in the study of processes in various media in a wide range of conditions. It should also be noted that its solution requires a huge investment of funds, labor, and time. In the experimental study of non-equilibrium processes in media, in addition to purely technical difficulties associated with the need, as a rule, to carry out measurements on small space-time scales, there are also fundamental difficulties in processing and interpreting the results obtained. Any experiment always gives the value of the quantity, averaged both over a certain spatial region and over a certain time interval. The more the accurate value we want to get at a given point at a given time, the smaller should be the area over which the device performs averaging. However, when studying the dynamics of fields of macroscopic quantities, one cannot constrict the averaging region to the point. If the size of the vicinity of the point covered by the device becomes commensurate with the size of the structural element of the medium, then the result of averaging already refers not to the macroscopic scale level, but to the level of the internal structure of the medium, which is most often characterized by intermediate scales between macro and micro levels [2, 13, 21, 22]. Here, another problem arises. On the one hand, in order to control fast processes with large spatial inhomogeneities in real time, it is necessary to decrease the averaging scales, and on the other hand, the measured values of mechanical quantities cease to be macroscopic; their fields do not satisfy the continuum mechanics models. To resolve this issue, it becomes necessary to redefine the concept of macroscopic density. This can be done correctly only from the standpoint of non-equilibrium statistical mechanics, rethinking the distributed values from the probabilistic standpoint. In non-equilibrium statistical mechanics, macroscopic quantities are determined independently of the space-time vicinity of a point by using a non-equilibrium statistical

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distribution (in kinetic theory, for example, this is the velocity distribution function). However, out of local equilibrium this distribution is generally unknown [23]. The point is that the distribution function describing near-equilibrium processes in media that behave as structureless or continuous is close to a Gaussian distribution, and the macroscopic value is defined by its maximum with a probability close to 1. Therefore, continuum mechanics describes the medium behavior in a deterministic way. Far from equilibrium, statistical distributions can be asymmetric or even have several maxima. Therefore, far from equilibrium, the macroscopic quantities are determined only in the probabilistic sense and do not coincide with their classical definition. Naturally, the reliability of experimental measurements decreases; and it is fundamentally impossible to increase it for the processes out of equilibrium, because the law of large numbers does not take place. Since the processes far from equilibrium will always show their probabilistic nature, good repeatability of the results obtained cannot be required. Hence, it becomes impossible to obtain a reliable statistical sample and predict the results with sufficient accuracy. Thus, it becomes clear why continuum mechanics stops working in nonequilibrium conditions. But it should be noted that any set of averaged values will not describe processes far from equilibrium in a deterministic manner.

2.2 Anomalous Medium Response to Strong Impact For several centuries, a linear approach dominated science. According to the approach, the system response is proportional to the external action and the response to several actions is the sum of all individual responses. The linear mathematical models due to the uniqueness of solutions to the linear sets describe physical processes deterministically, i.e. in a predictable manner. The processes adequately described by the linear models are stable and well reproduced in experiments. The linear approach is the foundation of classical physics. At a rather large, macroscopic scale, the linear laws adequately describe processes in distributed systems near local equilibrium. The development of science and technology brings to the fore the research of high-rate and short-duration processes which do not correspond to the linear approach. Experimental results, obtained in the study of non-equilibrium processes in different branches of mechanics (hydrodynamics of turbulent flows, multi-phase flows, shock-induced wave processes in solids, biomechanical processes, etc.), show many similar characteristics of the non-classical system response to external perturbations [13, 24–26]. First of all, the nonlinear properties of the system violate the regularities of the behavior of the system from the standpoint of traditional models of continuum mechanics and make the response of the system poorly predictable. Moreover, taking into account the nonlinear properties of the system allows for the multiplicity of the system’s response to external influences. Since traditional models cannot determine

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what choice the system will make, the response becomes probabilistic [27]. The probabilistic nature is due to the fact that out of equilibrium, the macroscopic description becomes incomplete because it does not take into account the higher moments, and the mean is poorly defined. In continuum mechanics, when linear laws are violated, anomalous behavior of parameters is observed that specify the properties of a particular medium, such as transport coefficients (viscosity, thermal conductivity) in fluids or elastic modules in solids. In continuous medium models, these parameters are constants characterizing the properties of a specific medium. However, as experiments show, in nonequilibrium conditions they cease to be constants and begin to depend on the integral properties of the system and on the conditions imposed on the system from outside. For example, the viscosity of even a simple fluid with an increase in flow velocities begins to depend first on the temperature and pressure fields, then on the gradients of macroscopic fields, the distance to solid boundaries, and the geometry of the system [28]. The viscosity of a liquid flowing through a nanosized channel depends not only on the channel size but also on the nature of the interaction of liquid molecules with the channel wall material [29, 30] and differs significantly from its value in a largescale channel. On small scales, the propagation velocities of disturbances differ from their equilibrium values [31]. Elastic modules in the region of nonlinear elasticity begin to depend on the loading duration of the system, and in the more general case, on the loading history of the system [32]. The emergence of size effects in the process under study is a sign that the system under the given conditions goes beyond the validity of continuum mechanics. The dependence of the system properties on its size means that the traditional similarity criteria used in continuum mechanics are becoming insufficient to predict the behavior of systems of different sizes. Experimentally studying the processes of mass, momentum, and energy transport in various media has shown that high-speed and fast processes are accompanied by the emergence of various kinds of collective effects that lead to the formation of some organized movements at intermediate scales between macro and micro levels. Such movements can be extremely diverse depending on specific conditions, but all of them can be attributed to such phenomena as turbulence. In experiments on shock loading of solid materials, it was found that after a high-speed impact, traces of rotational structures frozen into the material remain in the target material and the propagation of shock-induced waves is accompanied by a number of anomalous effects that cannot be explained from the standpoint of continuum mechanics [4–7, 13]. Thus, turbulent effects are an inherent property of transport processes far from equilibrium. However, there is still no theory that would satisfactorily predict the behavior of turbulent processes. The inability to make reliable predictions about the possible development of a turbulent process is a significant obstacle to technological progress. The former hope for the creation of a universal model of turbulence was gradually replaced by the understanding that the turbulence phenomena should be studied from a more general standpoint.

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Back in the last century, an approach to the problem of turbulence based on the theory of self-organization has already been proposed. The theory of selforganization of dissipative structures was developed by Prigogine and others [33, 34]. However, this theory is not suitable for the description of dynamic vortexwave structures. The point is that in fast processes, slow diffusive mechanisms of momentum transport do not ensure dissipation of mechanical energy into a thermal form during the typical time of the process and therefore cannot play the main role, as in near-equilibrium processes [35]. Instead of dissipation, the energy received during the impact first goes to the formation of dynamic structures in the medium and only then begins to dissipate [36]. In experiments on shock loading of solid materials, an anomalous loss of the wave amplitude was detected that cannot be explained by dissipation. The loss arose in a threshold manner with an increase in the impact velocity and its appearance turned out to be associated with the discovery of new mesoscopic structures frozen into the target material [37, 38]. The observed behavior of the amplitude loss, in turn, closely correlated with a change in macroscopic properties of the medium, in particular, with the spall strength of the material. From the point of view of continuum mechanics, the response of the material to shock loading is anomalous and cannot be described within the concept. The observable self-organization effects are characterized not only by the medium properties (composition, phase state) but also by the loading regime, boundary conditions, and the system size and geometry. The collective effects, turbulent structures, and other processes on the mesoscale that accompany non-equilibrium transport spread the system response over its volume. The collective effects lead to inadequacy of localized differential mathematical models and require integral approaches to describe the non-equilibrium processes. The interaction between the formed mesoscopic structure elements causes the system to evolve. Therefore, a steady state of the system under the constraints imposed far from equilibrium cannot be stable. The stability of the system state out of equilibrium should be closely connected to the temporal evolution of the system taking into account its history. Unlike the linear approach, the system response to the high-rate impact should retard the impact itself due to the finite speed of propagating disturbances. The delaying is closely associated with inertia and memory that determine the response of the system. The retarding responses to multiple impacts can result in oscillations, system instability, transient modes, and structural transformations. An influence of the individual factor among all the rest effects becomes indistinguishable because of the close loops formed in the system. The system behavior becomes ambiguous, poorly predictable, and poorly reproducible in experiments. In various fields of mechanics, the experimental researches of high-rate processes show a lot of common features which characterize an anomalous response of the medium to an intense external impact. The main ones are the self-organization of mesoscopic dynamic structures and the temporal evolution with feedbacks. It is these features that make the response of the system nonlocal and history-dependent and therefore do not fit the concept of continuum mechanics.

2.3 The Internal Structure Effects

37

2.3 The Internal Structure Effects At high speeds and large deformations, the medium begins to manifest the effects of its internal structure, which is confirmed by many experiments. However, even under normal conditions the internal structure of the system can influence the macroscopic properties of the system. It is possible on small spatiotemporal scales commensurate with the size of the internal structure element. In Chap. 1, we have already shown that the medium cannot be considered continuous close to the boundaries of the system and near the initial time moment. The medium behavior in these zones depends on microscopic characteristics connected to the molecular interaction and their initial microstates which require the higher moments introduced into the medium description. Viscosity and thermal conductivity of simple liquids in channels of nanosizes differ significantly from their values in large channels [29, 30]. The size and near-boundary effects make the Navier–Stokes’s hydrodynamic equations invalid to describe flows in thin capillary systems. The behavior of the media with complicated internal structure is entirely determined by their internal structure effects. For example, macroscopic characteristics of multi-phase dispersed media depend on the sizes of dispersed particles, and volumetric and mass concentrations of different phases. Slow macroscopic size flows of multi-phase media with rather small effective viscosity can be described within the scope of continuum mechanics [39]. Highly viscous media go beyond continuum mechanics due to the collective effects occurring at high concentrations of the dispersed phase. Therefore, rheological models of highly viscous media have narrow areas of applicability. Flows of liquids with nanoparticles also exhibit collective effects even at small volumetric concentrations due to molecular interaction between phases through large total interphase surfaces [29, 30]. With an increase in the velocities of the medium, the flows become nonequilibrium, the influence of initial inhomogeneities and perturbations, which could be neglected at low velocities, begins to grow and significantly change the behavior of the entire system. In this case, first random fluctuations grow, and over time they organize into some quasi-periodic pulsations of mass velocity and pressure. Then, between the elements of the medium moving at different speeds, intense shear processes arise, which can lead to the appearance of rotational motions of the medium. The movement of the medium becomes turbulent. Under the constraints imposed on the system, the collective correlated movements of the medium can form largescale ordered structures, the shape and size of which are determined by the entire complex of imposed conditions, including external influences, boundary conditions, the shape and size of the system. Besides, they significantly depend on the initially perturbed state of the system and its further evolution. Thus, the sizes of structures resulting from self-organization can be different. In contrast to the structural elements of multi-phase media, dynamic turbulent structures cannot be described within the framework of continuum mechanics, since the long-range correlations and memory effects require spatiotemporal nonlocal description. It is because of the infinite variety

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of conditions determining turbulent processes that it is impossible to build a universal turbulence model. High-rate deformation of solids is also accompanied by a restructuring of the internal structure of the material, which is characterized by a high concentration and complicated organization of lattice defects, the formation of internal boundaries, and structures at the mesoscopic scale level [22]. Observable evidence of this process is localized shear bands, cracks, and traces of rotational movements. The evolution of mesostructure during plastic deformation of solids and its effect on their mechanical properties is an urgent problem for fracture mechanics, materials science, and the development of new fine technologies. At low strain rates and stresses not exceeding the yield point, the process of deformation of crystalline solids can be described with sufficient accuracy by classical mechanics of a continuous medium. The evolution of the microstructure within a wide range of conditions is explained in terms of dislocation–disclination dynamics [22]. At very high strain rates and stresses (strong shock loading, explosions), the deformation process is usually described in the framework of the hydrodynamics of an incompressible viscous fluid (Navier–Stokes equations). It should be noticed that it is believed that in both limiting cases, the concept of continuum mechanics describes plastic deformation quite satisfactorily. In the transition region between these limits, when non-equilibrium processes of mass, momentum, and energy transfer are accompanied by self-organization on the mesoscopic scales, the continuum mechanics becomes inadequate. It has long been known that an increase in the velocity of a medium first causes an increase in velocity fluctuations, and the magnitude of the velocity variation is proportional to the gradient of the average velocity [15]. This pattern was observed for some regimes of turbulent flows of liquid media. However, it turned out that in a solid body after shock loading, velocity dispersion is observed at the propagating wavefront. Experimental studies have confirmed that in the case of quasi-stationary wave propagation, the velocity variation has a maximum at the center of the front where the maximum deformation rate of the medium takes place [4–7]. Later, these results were confirmed by the results of numerical simulation of the process of shock loading of a polycrystalline material [40]. Previously, it was thought that the mesostructure comprises grains or intragranular defect structure of the material, which are studied in isolation from the momentum transport in the medium [22]. Experiments on shock loading of solid materials [4–7, 13] found out a new type of mesostructure formed by shock-induced plastic deformation in the metal targets. The observed traces of rotational structures of the mesoscopic scale show that the turbulent effects are inherent to transients in various media. Shock loading of crystalline materials has shown that under certain conditions, an impulsive action on the material forces to rotate individual grains whole whereas in another case, several traces of the mesorotations in the target metal pass through the grain boundaries without noticing them [41]. The above experimental research works differ significantly in the methods of pulse injection into the material, the pulse duration, and its amplitude. It can be assumed that in the first case, for a weak pulse, the

2.4 Fluctuations, Pulsations, and Instabilities

39

initial grain structure in the material plays the main role. In this case, rotation turned out to be possible only due to resonance, when the size of the material promoted by the pulse coincided with the grain size. In the second case, for a stronger impact, the initial structure of the medium ceased to affect the momentum transport induced by impact. The individual elements of the medium, which received the highest velocity after the shock, move forward by inertia through the original structure, forming new shear and rotational structures. Experimental studies have shown that the evolution of the mesostructure of the medium under plastic deformation changes its mechanical properties, in particular, strength and spall properties. Predicting such changes is required for solving many scientific and practical problems. Since the theoretical description of the plasticity phenomenon is fraught with the same principal difficulties as the description of turbulence, up to the present time no such fundamental theory has been developed that would make it possible, on a unified theoretical basis, to explain both the formation of mesoscopic structures and the medium response to dynamic loading.

2.4 Fluctuations, Pulsations, and Instabilities With increasing velocities and gradients, the processes of mass, momentum, and energy transport are accompanied by the appearance of a whole series of effects, which were negligible under close-to-equilibrium conditions. Random fluctuations of macroscopic quantities that always exist in physical systems begin to play an essential role under non-equilibrium conditions. In turbulent modes of motion of media, pulsations of velocity and pressure arise among which periodic oscillations appear that radically change the nature of the movement of the medium at the macroscopic level. The further development of such processes of different scales leads to various kinds of instabilities. New degrees of freedom appear in the system; the scales of dynamic structures change; transitions occur between different modes of motion of the medium. Let’s consider these effects in more detail. Fluctuations are random deviations of physical quantities from their mean values. In the general case, they always exist in real physical systems including the state of thermodynamic equilibrium [42, 43]. The classical thermodynamic description, i.e. the description of the state of the system  solely with the help of average values is 1 valid when the relative fluctuations X  (δ X )2 of thermodynamic quantity X are from it is δ X = X − X , δ X  = 0; negligible. Here, X  is its mean; thedeviation  its standard deviation is (δ X )2 = X 2 − X 2 . For example, in Fig. 2.1 we can see a schematic picture of fluctuations of the x-component mass velocity υx where υx = X, υx  = υcp , υx = δ X . The quantities can evolve over time. From a mathematical point of view, neglecting fluctuations of macroscopic parameters is equivalent to the passage to the so-called thermodynamic limit which is understood as the replacement of a finite-size system with an infinitely large one maintaining the particle density V → ∞, N → ∞, N /V = const. Only in this

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2 Specific Features of Processes Far from Equilibrium

Fig. 2.1 Fluctuations of the x-component mass velocity υx for the interval of time [t1 , t2 ]

limit, we can expect an exact correspondence with the laws of phenomenological thermodynamics. So, the thermodynamic transition makes the system under consideration isolated, whereas all real systems are open in which the transport processes are determined by boundary conditions. The transition to thermodynamic limit creates an obstacle for the use of the results of such fundamental theories as statistical physics in practical problems of mechanics, where boundary conditions play a major role. This is especially true of processes far from equilibrium, in which processes of interaction with the system boundaries of various natures determine the response of the system to external influences. Therefore, differential equations of continuum mechanics in real problems should be used with great care. The first theory of fluctuations was developed by Einstein in 1910. He expressed the relative probability of fluctuations W in terms of entropy S from the Boltzmann formula S = k ln W, W = C exp(S/k), S = S − S0 < 0. The entropy in thermodynamic equilibrium S0 is maximum; the value S determines the deviation of the system state from thermodynamic equilibrium. The expansion near equilibrium gives S = S − S0  21 δ 2 S < 0. When the entropy S(X ) depends on the thermodynamic quantity with X   ≡ X = 0 in equilibrium state, its maximum value is reached on the conditions ∂∂ XS X =0 =  2  ∂ S = −β < 0. Close to equilibrium the value X is very small. Series 0; ∂ X2 X =0

expansion in powers of X approximately takes a form S(X ) ≈ S(0) − β2 X 2 where the second term of the series determines a change in entropy due to fluctuations. The probability of the entropy value in the state2 with X in the interval [X, X + d X ] is P(X ) = K exp{S(X )}; d P ≈ C exp − β 2X d X, C = K exp{S(0)}. The constant  ∞ β C = 2π can be found in the normalization condition −∞ P(X )d X = 1. Then we of the quantity  get Gaussian X : 2P(X

probability distribution of fluctuations

)=  2  ∞ 2 β β β X2 βX exp − 2 . Having calculated the value X , −∞ X 2π exp − 2 d X = 2π   1 , the distribution of fluctuations can be expressed in terms of X 2 β

2.4 Fluctuations, Pulsations, and Instabilities

P(X ) =

41

 X2 1   exp −   . 2π X 2 2 X2

This theory is valid provided (1) (2)

fluctuations are small and higher moments can be neglected; fluctuations correspond to such deviations from equilibrium, which are described by one of the classical thermodynamic potentials.

These conditions are met only by the state of local thermodynamic equilibrium. Prigogine et al. [33, 34] believed that non-equilibrium thermodynamics should be based on the theory of fluctuations. They develop a phenomenological theory of the stability of non-equilibrium states, in which δ 2 S plays the role of the Lyapunov function. Such an approach works near local equilibrium under stationary boundary conditions. In practice, the region of validity of classical thermodynamics can be determined from the condition that fluctuations are negligible in comparison with the measurement accuracy. Pulsations of velocities and pressures always accompany turbulent motions of media of various natures [16, 17]. Pulsation is a continuous change in the magnitude and direction of the velocities, the pressure of the medium flow near solid surfaces. At different points of the flow, the nature of the change in these values over time is different. The measure of the intensity of the pulsation is the degree of flow turbulence. The degree of turbulence, which is a kind of criterion for the kinematic similarity of turbulent flows, is customarily understood as the ratio of the mean square pulsation velocity over time T to the average velocity at the same point over the same time  t∗+T interval v = T1 t∗ vdt:  ⎛ t∗+T ⎞−1  t∗+T    1 1 (v  )2 dt ⎝ vdt ⎠ . ε= T T t∗

t∗

The pulsating field and vortex motion promote the mixing of the medium and the occurrence of additional transport processes. In this case, shear stresses and energy fluxes increase; their influence significantly changes the flow fields. The problem of determining the characteristics of the turbulent transfer becomes urgent. The pulsations are most intense in the border zones where the velocity gradients are at their maximum. Far away from the boundaries of the system, the velocity pulsations gradually decay. Pulsations can be random with a significant change in amplitudes and frequencies and periodic with clearly defined prevailing frequencies and amplitudes. Random pulsation occurs during uniform or almost uniform turbulent motion due to saturation of the flow by particles interacting with the surface of the wall with a roughness of different sizes and shapes (clouds of smoke, dust). Periodic pulsation occurs when the flow breaks off the surface of a bluff body.

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The pulsation of velocities entails a pulsation of pressure, and the latter causes the pulsation of forces acting on the streamlined surface [44]. The amplitude growth changes the frequency of the velocity and pressure pulsations. The larger is the amplitude, the lower is the frequency, i.e. the longer the period of oscillation. The pulsation of the pressure force can cause intense vibration of constructions and technical objects when flowing around them in a turbulent flow. The less is the mass of the object, the greater is the range of its vibrations. Therefore, for lightweight objects, it is necessary to take into account the dynamic loads during vibration when calculating the strength. Since the amplitude of the pulsating pressure and force pulsations is proportional to the square of the flow rate, the dynamic load grows very rapidly by increasing the flow rate. Vibrations of streamlined structures caused by pulsations in the boundary layer, in turn, emit secondary noise into the surrounding space or into objects [45]. Methods for predicting such noise and developing ways to reduce it require taking into account the elastic and other mechanical properties of objects in a streamlined flow. In view of the extreme complexity of this problem, the existing methods for calculating acoustic radiation induced by turbulence are far from perfect. For predicting the acoustic radiation of objects, the initial information about the primary field of turbulent pressure pulsations in the boundary layer is needed [46–48]. Turbulent pulsations of velocity and other hydrodynamic quantities in a turbulent flow lead to the fact that it is impossible to accurately describe the instantaneous values of velocities, pressures, and forces. Therefore, the use of the exact equations of hydrodynamics in the case of turbulent motion becomes impossible. In solving practical problems, the analysis of turbulent flow is carried out at the level of the hydrodynamic parameters averaged over a sufficient time interval. The pulsation of velocities, pressures, and forces in turbulent flows are studied experimentally. Pulsation modeling helps to establish the characteristics of dynamic pulsations, and to select rational forms and masses of objects that provide their stability and strength. In contrast to the laminar flows of the medium in which random fluctuations are negligible and the arising small disturbances quickly decay, retaining the flow stable, in a turbulent flow, small disturbances will grow with time due to the huge number of degrees of freedom (Fig. 2.2). The transition from laminar to turbulent regime as a result of energy inflow is characterized by the Reynolds number Re = ρU L/μ, which determines the comparative contributions of the effects of energy dissipation and inertia. For small numbers Re > 1, on the contrary, the fluid motion turns out to be practically dissipationless, since in this case there is no viscous friction mechanism. Since the growing

Fig. 2.2 The growth of the pulsation during the transition to turbulent jet flow

2.4 Fluctuations, Pulsations, and Instabilities

43

Fig. 2.3 Scheme of the transition to turbulent jet flow

disturbances ultimately destroy the flow, it becomes unstable. The size of the system does not allow the growth of perturbations of a scale larger than the system size L. Therefore, smaller modes will be excited which, in turn, will also be unstable. Thus, a whole hierarchy of unstable disturbances is formed which completely changes the nature of the fluid motion. The process of the birth of motions of smaller and smaller scales will stop only after reaching the minimum scale at which the diffusion mechanism of the momentum and energy transport plays a decisive role. The motions of the minimum scale are stable and practically do not disintegrate further, since their energy is spent mainly on overcoming viscous forces and dissipate into heat. While the instability of the main flow leads to the appearance of more and more perturbations, the process of successive fragmentation of all non-dissipative perturbations does not stop and creates a continuous flow of energy down the scales. The scenario of the development of disturbances described above gives an idea of the development mechanism and structure of the turbulent flow (Fig. 2.3). The energy spectrum of turbulent motions can be conventionally divided into three regions. The short-wavelength region of the spectrum corresponds to the dissipation of the kinetic energy of turbulence into heat under the influence of viscosity. This region contains a relatively small part of the turbulence energy. The corresponding small-scale pulsations have a complex statistical structure; their distribution is a substantially non-Gaussian one. The region of developed turbulence belongs to the inertial range of the spectrum and is characterized by a locally isotropic structure of turbulent pulsations. The statistical properties of the pulsations of the inertial region are described with good accuracy by the equilibrium (Gaussian) probability density function, and the motion of liquid particles in the field of isotropic turbulence is similar to Brownian motion. The energy region corresponding to long-wave (large-scale) pulsations, on the contrary, contains the main part of the turbulence energy and, basically, determines the nature of the turbulent transport. Long-wave

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pulsations correspond to large-scale vortex structures, which are characterized by a relatively long relaxation time and contain information about the prehistory and structure of the averaged flow (memory effect). Therefore, as a rule, this region of the spectrum is characterized by an anisotropic, essentially non-Gaussian probability distribution function. Thus, the long-range pulsations and the inertial region of the energy spectrum are the main objects of research in the problems of modeling turbulent flows.

2.5 Multi-Scale Energy Exchange Between Various Degrees of Freedom Far from thermodynamic equilibrium, transport processes are often accompanied by the formation of new multi-scale dynamic structures at the mesoscopic scale level. The mesoscopic scale is an intermediate level between micro- and macroscales. Microscopic processes defined by interaction between atoms and molecules are outside our area of interest. Macroscopic sizes of the system under consideration can vary from mm sample materials to the ocean scale. Therefore, the range of real scales of mesoscopic structures can be very wide. However, depending on the influx of energy from the outside, the transport processes in a finite-size system can be accompanied by mesoscopic motions in a rather narrow range of sizes. In classical continuum mechanics, macroscopic transport processes are accompanied by the conversion of kinetic energy into heat. So, only the exchange between the two limiting levels is taken into account. However, the validity conditions of the continuum mechanics limit the spatiotemporal scales of the processes under consideration. On large spatiotemporal scales, when the system state is close to equilibrium and stable, any small disturbances rapidly fade and cannot induce new degrees of freedom except microscopic ones. A quite different situation occurs out of local equilibrium. Due to large gradients in non-equilibrium processes, their typical scales are small. Over the small time interval, the slow diffusive transport mechanisms of dissipative processes cannot provide the full transition of kinetic energy to a microscopic scale. As a result, a part of the energy trying to dissipate reaches an intermediate level and remains on it in the form of new mesoscopic degrees of freedom. The size spectrum of turbulent motions of fluids considers being continuous. A small part of turbulent energy determined by the small-scale pulsations can reach the microscale and dissipate into heat. However, the large-scale pulsations carry the main part of the turbulence energy in the form of the large-scale vortex structures. These structures must be supported by a constant influx of energy from the outside. Without this support, vortex structures are fragmented, their sizes decrease and in the end, their energy dissipates due to viscosity. High-strain-rate deformation of solids is a non-equilibrium process that is accompanied by a whole complex of multi-stage and multi-scale relaxation and energy

2.5 Multi-Scale Energy Exchange Between Various Degrees of Freedom

45

exchange processes that make the medium response to an external loading abnormal from the point of view of continuous mechanics [4, 7, 22]. The mechanisms of the impulse and energy exchange between different degrees of freedom at mesoscopic scales are various and not entirely understood till the present. There is sufficient experimental evidence to indicate that the shock structure is highly unsteady and turbulent. The transition zone has a cellular structure with internal boundaries, collections of interacting transverse waves, shear layers, and density interfaces. Apart from the mass velocity pulsations, strong pressure pulsations induce intensive pressure waves that make the turbulent structure in a compressible medium far more complex compared to the turbulent structures in liquids. It must be noticed that among the whole multi-scale spectrum of dynamic structures existing during and just after the shock, in the target material one can find only traces of some structures frozen into the material after passing the wavefront. In experimental research [4–7], the mass velocity pulsations in a form of velocity dispersion are detected in real time during the shock-induced wavefront reaches the backside of the target. It was found that the velocity variance is proportional to the velocity gradient and takes its maximum in the center of the front. Just the same regularity is observed in turbulent flows of liquids. The velocity pulsations that behave in this way are reversible. However, the irreversible shear and rotational structures that remained in the target metal after the waveform passed are significantly larger though both types of structures can be related to the mesoscopic scale. Their sizes correlate with the size of the laser spot on the target surface whose displacement is registered by an interferometer. The measurements showed that different spots of the target moved at different velocities. The dispersion of the velocity distribution over the target surface induces shear processes which are able to result in rotational modes. The irreversible part of the structures can be observed in the shocked material after the stress relaxation. All the structures discovered in experiments can be attributed to the mesoscopic scale. Previously, it was generally accepted that dislocations realize only translational deformation modes. However, when the density of dislocations and other defects increases, shear instability can arise. As a result, new types of defects (microbands, disclinations, rotations, etc.) can arise on the mesoscale (Figs. 2.4 and 2.5). Mesodefects emerge and propagate over considerable distances regardless of the initial structure of the medium. In contrast to the micro level, shear and rotational mesoscopic structures can move bulk elements of different sizes (grains, their individual parts, and whole blocks of material) inside a deformable solid. The dimensions and nature of mesoscopic structures substantially depend not only on the initial defect structure of the material but also on the strain rate during loading, its duration, and also on the shape and dimensions of the loaded sample. The movement of such mesoscopic elements determines the mechanisms of plastic deformation of the medium. In the zone of the stress and deformation concentration, the loss of shear stability leads to material fragmentation and finally to its destruction. Thus, plastic deformation and fracture of a material are determined by the self-consistent behavior of a whole set of structural levels of deformation.

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2 Specific Features of Processes Far from Equilibrium

Fig. 2.4 New defect structure of shocked metal on the mesoscale

Fig. 2.5 Nncleation of rotations on the mesoscale

2.6 Multi-Stage Relaxation Processes A non-equilibrium state of a physical system caused by mechanical action from another system arises due to the inflow of momentum and energy. The interaction of systems of a finite size does not occur instantaneously because the process of impulse and energy transmission takes a finite period of time during which a force is applied to the body imparting the corresponding acceleration to it. Depending on where the force is directed, to which part of the body it is applied, how it is distributed, and what are the initial properties of the system and the striker itself, the response of the system to external influence can be extremely diverse. As soon as the process of impulse transfer begins, deformation processes occur as a result of which stresses are redistributed in the system. At the same time, the system begins to relax, i.e. the stresses arising in it gradually attenuate, if any restrictions or boundary conditions are not imposed on the

2.6 Multi-Stage Relaxation Processes

47

system. So, two oppositely directed processes simultaneously proceed in the system: the loading process, accompanied by the appearance of stresses, and the second one, parallel to the first, which redistributes them, reducing their magnitude. The system tends to approach an equilibrium state, but the continued inflow of impulse from the outside counteracts this. The result is determined by the interrelation between the intensity and speed of the incoming impulse and the ability of the system to redistribute them, i.e. the relaxation rate. Depending on the initial state of the system and the imposed conditions, the relaxation time for different degrees of freedom and scale levels is different. If the impulse injection process has accomplished, you can trace the relaxation process, which proceeds in stages. Back in the 1940s, N. N. Bogolyubov [23] laid the foundations for the dynamic theory of kinetic phenomena which made it possible to consistently obtain the equations of hydrodynamics “from first principles”. In the framework of non-equilibrium statistical mechanics, he mathematically formulated the hypothesis of decay of spacetime correlations in a system tending to thermodynamic equilibrium which is associated with the concept of successive stages of the relaxation process each of which is characterized by its own time scales and abbreviated sets of variables describing non-equilibrium systems. According to this hypothesis, at the initial stage of relaxation, the system retains its initial state for some time which further the system forgets during the relaxation process. At the final stage, the system completely forgets its history and approaches thermodynamic equilibrium. In the process of relaxation, the set of variables describing the state of the system is gradually reduced from a complete description at the microscopic level to a small set of macroscopic variables which is sufficient to describe the behavior of the system near the state of local thermodynamic equilibrium. Let us consider in detail what properties the system can exhibit at different stages of the relaxation process. It is known that weak perturbations cause the propagation of sound or elastic waves in a macroscopic system. If the wavelength corresponding to the perturbation duration is much larger than the structural scale of the medium, the medium of any nature can be considered as structureless. In theory, long waves do not decay during propagation and their waveform created by the disturbance does not change during propagation. It means that the system remembers its origin infinitely long though in real media any waves ultimately decay. Such a process can be considered completely reversible, since the wave propagates without dissipative losses. This stage of relaxation is described by the wave equation, which belongs to the models of classical mechanics of continuous media. However, in reality, the initial stage and the waves induced may be too short to be able to neglect the initial structure of the medium. If the stress caused by an external action exceeds the elastic limit of the medium, the rate of impulse input into the system plays an important role. Unlike quasi-static processes for which, as is commonly believed, the amplitude of the transmitted pulse is reached instantly, the nature of the wave processes induced by a high-rate impact is determined by the finite rate of the impulse entering the system. With a high-rate impact at the initial stage of loading, while the elastic limit has not yet been reached, the system will respond to

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the action by inducing a collection of various short and, in the general case, nonlinear waves interacting with each other and with inhomogeneities of the medium. Thus, the wave transport mechanisms determine the initial stage of relaxation of any physical system, although the nature of the wave processes and the duration of this stage for different media and conditions can differ significantly. Dissipative effects at the initial stage are very small, since slow diffusion transport mechanisms do not have time to develop in too short a time interval. Further, due to dispersion on small-scale inhomogeneities of the medium, waves propagating with close frequencies form wave packets that transport mass and travel at a group velocity which can be much less than the phase one. Wave packets spread out when propagating in a medium. Interacting with each other and with interphase boundaries, they acquire a phase lag and begin to slow down more and more. At this stage, the formed structures become partially dissipative, and the irreversible energy losses grow. As soon as the arising stresses exceed the elastic limit, stress relaxation begins immediately. As a result, the waveforms begin to evolve and spread, momentum and energy tending to dissipate on the microscopic level go deeper and first reach an intermediate scale between macro- and micro levels that can be called mesoscopic one. New mesoscopic degrees of freedom generate new types of motion that are typical to the next stage of relaxation. First, random fluctuations of macroscopic quantities grow, and then partially ordered pulsations of velocity and pressure can form. As a result, shear processes are intensified which, in turn, can lead to the appearance of vortex structures. We can say that the transition to turbulent motion occurs in the system. The scale and character of the turbulent motions are already determined both by the initial structural inhomogeneities of the medium itself and by the structure of the wave processes induced during the initial stage of relaxation. In the future, these movements can self-organize and form ordered structures. Thus, the emerging dynamic structures become carriers of information about the loading history of the system. This transient stage between the initial and final ones is characterized by the greatest deviation of the system states from equilibrium. Its typical feature is the self-organization of new dynamic structures which makes it impossible to use the models of continuum mechanics to describe it. However, turbulent structures require a lot of energy to develop and maintain such processes. If the received energy is not enough, then the process will not go further and turbulent structures will degrade, break up, and ultimately dissipate into heat at the micro level. The final stage of relaxation begins. When the state of the system approaches local equilibrium, the velocities and gradients of macroscopic fields become small. Further redistribution of momentum and energy is described by a set of several macroscopic variables within the framework of continuum mechanics. If the initial state of the medium after loading was close to local equilibrium and the relaxation of the initial microstate has already ended, then the hydrodynamic stage of relaxation will be the only one. However, the hydrodynamic stage of relaxation does not always take place. If a system has an ordered microstructure, such as a crystal lattice in solids, its state

2.7 Finite Speed of Disturbance Propagation and the Delay Effects

49

is considered far from equilibrium. Intense external influence destroying this order shifts the state of the system toward equilibrium making the medium response liquidlike. In this case, turbulent structures on the mesoscopic scale may begin to develop in the system. But if the system has not received enough energy to reach the hydrodynamic stage, the relaxation process can go in the opposite direction. Since the vortex-wave structures are partially reversible, the system will strive to restore, at least partially, its original ordered state. When the mesoscale motions cease, the final state of the system will again be ordered, but with traces of turbulent structures in the form of new defects. And with a sufficient supply of energy, any physical system can reach the hydrodynamic stage and behave like a liquid or gas. For example, prolonged intense pressure on a solid can induce plastic flow which is described by hydrodynamic models. Different limiting states can be studied only if there is a model of the thermodynamic evolution of the system [49, 50]. In subsequent chapters, we will consider these issues from a thermodynamic point of view and propose a new approach to modeling the thermodynamic evolution of a macroscopic system using the principles of the control theory of adaptive systems.

2.7 Finite Speed of Disturbance Propagation and the Delay Effects In real media, all interactions propagate at finite speeds. If the rate of momentum transferis also finite u < ∞, then there is a certain trajectory of momentum transfer t r(t) = 0 udt, which connects spatial coordinates and time into a single complex. It is well known that the transition to space-time variables coupled together in the form ζ = t ± r/u where r = |r| is the distance traveled by the wave in the direction of wave propagation provides a solution to the wave equation. When the rate of momentum transfer is finite, then at small times, not the entire volume occupied by the medium is enveloped by the perturbation but only a part of it that the wave had time to travel at the speed u. Weak perturbations propagate in a medium with a constant speed close to the equilibrium speed of sound, while the mass speed is small u = C + v, v/C tr , the system forgets a part of the initial conditions, and then its evolution is completely determined by the quasi-equilibrium function f q . In this sense, tr determines the relaxation time during which the system reaches a quasi-equilibrium state. For gases, this time corresponds to the mean free travel time. This shortens the description and the evolution of the system is determined by several macroscopic parameters. This description corresponds to the hydrodynamic stage of evolution. In more complex systems, there can be several stages corresponding to a certain hierarchy of relaxation processes. Moreover, each subsequent stage is characterized by a greater degree of chaotization of the system and a less detailed description of its evolution. The non-equilibrium statistical operator method is based on the concept of a quasi-equilibrium distribution function which differs from the local-equilibrium one. Following [37], we briefly describe the plan for deriving nonlocal equations of hydrodynamics from the generalized Liouville equation (3.6.1) without touching on all the details of this method. The non-equilibrium statistical operator method is based on the analogy with nonlinear mechanics. Since a nonlinear system gradually forgets its initial state, the variables describing it should be constructed by taking a part of the statistical operator that is invariant with respect to motion with the given H . This operation leads to a smoothing of the oscillating terms. Of great importance is Bogolyubov’s concept

84

3 Macroscopic Description in Terms of Non-Equilibrium …

of quasi-averages [30], according to which infinitesimal perturbations can have a significant effect on the system if they break any symmetry. If the perturbations tend to 0 after the thermodynamic transition, then their final influence remains. Zubarev has shown that this concept is also very important in the theory of irreversible processes. In the method of constructing the non-equilibrium statistical operator, a small source was introduced that breaks the symmetry of time in the Liouville equation. He showed that the effects of irreversibility are closely related to the breaking of the symmetry of time in the Liouville equation. In order to determine dissipative processes, we will seek a solution to the Eq. (3.6.1) in the form f = fq + f1 ,

(3.6.2)

where f q ( p, r, r, t) = f q ( p, r, H ( p, r, r)t , n(r, r)t ,  p( p, r, r)t ) is the quasiequilibrium distribution function that determines the chosen level of the system description. The explicit form of this function needs to be defined. We consider a relaxation process in a molecular system that is initially out of equilibrium. Over time, in the vicinity of each point of the system r, a certain quasi-equilibrium state is established that is completely determined by the selected set of mean hydrodynamic densities. In this state, the values of these quantities can vary significantly from point to point. In this case, the quasi-equilibrium distribution function corresponds to the extremum of the information entropy. It should be emphasized that the entire thermodynamics of a non-equilibrium state is completely determined by the quasiequilibrium distribution function. In this case, the macroscopic values (averaged densities of particle number, momentum, and energy) must coincide with their quasiequilibrium values f q ( p, r, r, t) = f q ( p, r, H ( p, r, r)t , n(r, r)t ,  p( p, r, r)t )

(3.6.3)

In this case, the quasi-equilibrium distribution function has the same form as the local-equilibrium one (3.4.10), but the averaged densities fields over the distribution can correspond to not small deviations from the local equilibrium state. At the hydrodynamic level of description, the state of the system is completely determined by the average values of the mass or particle number density n(r, t), momentum p(r, t), and energy E(r, t) which depend on time and spatial variables. In contrast to the thermodynamic properties of the system, its transport properties are not described by the quasi-equilibrium distribution function. The viscous stress tensor P(r, t) and the heat flux vector q(r, t) are determined only by the true non-equilibrium distribution function, while their quasi-equilibrium values are zero. This fact indicates that the quasi-equilibrium distribution function does not describe dissipative hydrodynamic processes. The non-equilibrium distribution function f 1 for dissipative processes satisfies the equation

3.6 Non-Equilibrium Statistical Operator by Zubarev

∂ fq ∂ f1 + i L f1 = + i L fq . ∂t ∂t

85

(3.6.4)

The formal solution of Eq. (3.6.4) and its transformation by using the technique of projection and evolution operators ultimately results in a non-equilibrium distribution function in the same form (3.5.11). However, it was shown that the obtained distribution function can describe not only the states close to local-equilibrium but in a more general case out of equilibrium. The method introduces different scales of time from the very beginning. We represent the Hamiltonian H = H0 + V as the sum of the unperturbed Hamiltonian H0 and a small perturbed part V . The choice of H0 determines the fast time t0 during which the system forgets its initial state and depends only on the mean values of a certain finite set of variables. This state is defined by the quasi-equilibrium distribution function. Macroscopic fields determined by the average densities over the quasi-equilibrium distribution function first rapidly oscillate at a short time t ∼ t0 , then t > t0 evolve slowly over time. For example, relaxation processes in systems consisting of subsystems between which exchange processes are prevented proceed in two stages: first, partial equilibrium is established inside the subsystems, and then a slow evolution between the subsystems continues toward complete equilibrium. Adequate description of multicomponent systems such as mixtures with particles of different masses, with internal degrees of freedom, plasma requires multi-temperature and multi-velocity models. The completeness of the set of macroscopic parameters characterizes the representativeness of the statistical ensemble. For example, an ensemble characterized only by mean values of velocities is not representative of describing turbulent flows. If the fluctuations are large, then it is necessary to take into account not only the average but also their variance. So, we can conclude that the further the state of the system deviates from equilibrium, the more the information about it is required to describe the non-equilibrium state. The equilibrium distribution functions and statistical operators for all Gibbs ensembles correspond to the maximum information entropy under given external conditions. The local equilibrium distribution corresponds to the maximum information entropy for given macroscopic fields of mass, momentum, and energy densities at a given point r and at a given time t. Attempts to construct a non-equilibrium operator from the extremum of the information entropy [35] have been made repeatedly. However, only quasi-equilibrium distributions were obtained, which do not describe irreversible processes [2, 3]. From the extremum of information entropy, it is possible to construct a statistical operator describing irreversible processes, if only we impose the condition that macroscopic fields are given not only at the time moment t, but also at all previous moments of time, i.e. all the information about the prehistory of the system [1]. In the framework of the non-equilibrium statistical operator method, the causality principle is taken into account for the choice of the Lagrangian factors in the functional like (3.4.7) that make the statistical operator a delayed solution to the Eq. (3.6.4).

86

3 Macroscopic Description in Terms of Non-Equilibrium …

3.7 The Nonlocal Thermodynamic Relationships with Memory Between the Conjugate Macroscopic Fluxes and Gradients In Sect. 3.5, the non-equilibrium distribution function (3.5.11) was derived under the condition that all macroscopic prehistory of the system is given. In the previous section, we have understood that this function is valid in a rather wide range of deviations of the system states from local equilibrium. However, in order to use the distribution, we have to describe the evolution of macroscopic fields. Averaging the microscopic conservation laws (3.4.12)–(3.4.14) with the weight function (3.5.11), we obtain the macroscopic equations of hydrodynamics of viscous and thermally conductive fluid d n(r, r, t)loc = −n(r, r, t)loc divv(r, t); dt

(3.7.1)

d 1 (−∇ p(r, t) − ∇P(r, t) ); v(r, t) = dt mn(r, r, t)loc

(3.7.2)

d (3.7.3) E(r, t) = − p(r, t)div v(r, t) − P(r, t) · ∇v(r, t) − divq (r, t). dt   Here, the averaged macroscopic stress tensor T( p, r, r)t = P(r, t)   values for and heat flux vector Q( p, r, r)t = q(r, t) are expressed as follows: 

 T( p, r, r)t = −

t

dt0 eε(t0 −t)

−∞ t

dt0 eε(t0 −t)

+





" # dr1 T( p, r, r)t , ei L(t0 −t) T( p, r, r1 )t β∇v(r1 , t0 )+ " # dr1 T( p, r, r)t , ei L(t0 −t) Q( p, r, r1 )t ∇β(r1 , t0 );

−∞

(3.7.4) 

t



Q( p, r, r) = t

dt0 e

ε(t0 −t)



  dr1 Q( p, r, r)t , ei L(t0 −t) Q( p, r, r1 )t ∇β(r1 , t0 )+

−∞

t +

dt0 e

ε(t0 −t)



  dr1 Q( p, r, r)t , ei L(t0 −t) T( p, r, r1 )t ∇β(r1 , t0 ).

−∞

(3.7.5) Unlike the Eqs. (3.4.12)–(3.4.14), the hydrodynamic equations (3.7.1)–(3.7.3) include the dissipative parts of the momentum and energy fluxes that have the meaning of the viscous stress tensor P and the heat flux vector q .

3.7 The Nonlocal Thermodynamic Relationships with Memory …

87

Neglecting small cross-terms and deviations from equilibrium in the local equilibrium function (3.5.11), the dissipative fluxes are expressed through the integrals t P(r, t) = −

dt0 eε(t0 −t)

 drRP,P (r − r1 , t − t0 )β∇v(r1 , t0 );

−∞

t q(r, t) =

dt0 eε(t0 −t)

(3.7.6)

 drRq,q (r − r1 , t − t0 )∇β(r1 , t0 ).

−∞

In the relations (3.7.4) and (3.7.5), correlations are taken into account between all components of tensors. Each component of tensors is a correlation function. The relaxation kernels of momentum transport RP,P (tensor of the 4th rank) and energy Rq,q (tensor of the 2nd rank) are the space–time correlation functions of fluxes (3.4.17)   RP,P (r − r1 , t) = P( p, r, r)t , e−i Lt P( p, r, r1 )t ;   Rq,q (r − r1 , t) = q( p, r, r)t , e−i Lt q( p, r, r1 )t .

(3.7.7)

Relationships (3.5.6) between thermodynamic fluxes and forces are nonlocal in space and delayed in time. It is the nonlocality and memory that allow us to include the additional information about the collective and inertial effects observed in the processes far from local equilibrium. This is the fundamental difference between the description of non-equilibrium and local-equilibrium processes described by the models of continuum mechanics. It means that it is impossible to use differential equations to describe high-rate and fast processes where the spatiotemporal nonlocal effects play the main role. If the velocity and temperature change little over the correlation length and over the relaxation time, then the nonlocality and delay in (3.5.6) can be neglected and their gradients can be taken out of the integral sign at the point (r, t). The result will be the usual linear relationships between thermodynamic fluxes and gradients P(r, t) = −λ · β∇v(r, t); q(r, t) = κ∇β(r, t).

(3.7.8)

Here, the components of the tensor of the 4th rank λ are momentum transport coefficients convoluted with the velocity gradient. The components of the tensor of the 2nd rank κ are thermal conductivity coefficients convoluted with the inverse temperature gradient. The expressions for these coefficients in terms of correlation functions are the so-called Green–Kubo formulas

88

3 Macroscopic Description in Terms of Non-Equilibrium …

t λ=



  dr1 P( p, r, r)t , P( p, r, r1 )t ;

dt1 −∞

t κ=

(3.7.9)





 t

dr1 q( p, r, r)t , e−i Lt q( p, r, r1 ) .

dt1 −∞

The form of tensors λ, κ is greatly simplified due to the spatial isotropy of the liquid. In this case, the usual coefficients of volumetric λ, shear viscosity μ, and thermal conductivity κ are obtained. Sometimes, it is possible to neglect nonlocality and retain only memory effects, then we get t P(r, t) = −

dt0 MP,P (t − t0 )β∇v(r, t0 );

(3.7.10)

dt0 Mq,q (t − t0 )∇β(r, t0 ).

(3.7.11)

−∞

t q(r, t) = −∞

If we integrate (3.7.10) by parts and add the local equilibrium part of the stress, we obtain a relaxation relationship between stress and strain. However, in contrast to the phenomenological theory, the statistical approach yields explicit expressions for the relaxation functions. In this case, we can always assess the admissibility of various assumptions and simplifications. For a simple fluid, the coefficients λ, μ, κ are sufficient but in complex media with an internal structure, the off-diagonal components of tensors are nonzero. Some general properties of the relaxation functions are consistent with the properties of the transport coefficients. In particular, it is proved that these functions ensure the fulfillment of the Onsager reciprocity relations and the positiveness of the transport coefficients although the relaxation functions themselves, generally speaking, do not have to be monotonic and positive for all times. Close to local equilibrium, the correlations of dissipative fluxes rapidly decay and the correlation functions tend to 0. In this case, the Maxwellian relaxation function R(t) ∼ exp(−t/τ ) is often used for linear systems. If the mass velocity and temperature vary little on the correlation scales, then nonlocality and delay can be neglected and hydrodynamic equations for a viscous heat-conducting medium can be obtained. In contrast to phenomenological equations, the transport coefficients can be calculated if the potential of the interaction of molecules is known. But in practice, various assumptions and approximations are used because the rigorous explicit expressions are too cumbersome. However,

3.8 Two Types of the Spatiotemporal Nonlocal Effects

89

by using them it becomes possible to derive all results of linear thermodynamics of irreversible transport processes.

3.8 Two Types of the Spatiotemporal Nonlocal Effects Generalized constitutive relations between conjugate thermodynamic flows and forces (cross-effects neglected) obtained by averaging over the distribution functhe non-equilibrium tion (3.5.11) (or using   statistical operator) of the microscopic  flux expressions T( p, r, r)t = P(r, t), Q( p, r, r)t = q(r, t) originally had the form [1] t P(r, t) = −

t dt1

−∞

t q(r, t) =

dt2

 dr1

dr2 RP,P (r, r1 , r2 , t, t1 , t2 )β∇v(r1 , t1 );

−∞

t dt1

−∞



 dt2

 dr1

dr2 Rq,q (r, r1 , r2 , t, t1 , t2 ) ∇β(r1 , t1 ).

−∞

(3.8.1) The resulting constitutive relationships are nonlinear, nonlocal in space, and historydependent. There are two pairs of the space–time integrals determining spatiotemporal nonlocal effects of different nature. The scales of the effects described by each integral in (3.8.1) can be different. The first type of physical nonlocality is connected to the nonlocal molecular interaction which is characterized by finite spatiotemporal scales. This type of nonlocality influences only the integral kernels form. Therefore, spatiotemporal nonlocal effects take place even in the local distribution function. In the relationships (3.7.6), the nonlocal effects connected with the intermolecular interaction were neglected. The second pair of integrals in the relationships (3.8.1) is characterized by larger spatiotemporal scales. Spatial nonlocality is due to the statistical effects of the interaction of medium volumes of an intermediate scale between macroscopic and molecular levels. This is the scale of the spatial correlations. The time lag is caused by the finite propagation velocity of small perturbations in the medium. This type of nonlocality determining the spatiotemporal correlations between dissipative fluxes and forces is associated with the incompleteness of the macroscopic densities set of variables to describe a non-equilibrium state of the system. For gases, the first type of nonlocality has a typical scale of the order of the molecular interaction radius. The scale of nonlocality of the second type is much larger. It has the order of the mean length of the molecule-free path. Therefore, for gases and even liquids under normal conditions, the nonlocality of the first type can be neglected at the hydrodynamic level of description. Just the same we can say about the time lag or memory effects that are caused by the finite time of interaction of the

90

3 Macroscopic Description in Terms of Non-Equilibrium …

medium structural elements and the finite propagation speed of small perturbations in the medium. In this case, the first type of delay can also be neglected. So, under normal conditions, the scales of these two types of nonlocal effects differ greatly. In deformable solids, it is rather difficult to estimate the two scales in the general case. Under extreme conditions, when the intermolecular potential is not defined, we cannot distinguish the two types of nonlocalities for any medium. It means that far from equilibrium, we can unify both types of nonlocalities and consider the relationships (3.8.1) in the form t P(r, t) = −

 dt1

dr1 RP,P (r, r1 , t, t1 )β∇v(r1 , t1 );

−∞

t q(r, t) =

(3.8.2)

 dt1

dr1 Rq,q (r, r1 , t, t1 ) ∇β(r1 , t1 ).

−∞

In Chap. 5 we discuss this question in detail.

3.9 Main Problem of Non-Equilibrium Statistical Mechanics The goal of statistical mechanics is to create a consistent and efficient formalism for describing the macroscopic behavior of many-particle systems based on microscopic theory. In statistical mechanics, a very effective and reliable approach has been formulated that allows one to describe a wide variety of phenomena in such systems as liquids, gases, and solids. Statistical mechanics also provides the foundation for thermodynamic concepts such as heat, temperature, and entropy based on the laws of microscopic particle behavior. Non-equilibrium statistical mechanics provides approaches and tools for describing irreversible processes in real dissipative systems within the framework of a unified theoretical method allowing one to calculate, albeit approximately, the transport coefficients that characterize the system relaxation to equilibrium. However, as we have already seen, all the developed approaches are valid only not far from equilibrium. Now, the main problem of non-equilibrium statistical mechanics is the consistent derivation of the evolution equations of a real finite-size open system based on the reversible equations of particle motion. An essential condition for describing macroscopic systems in the framework of statistical mechanics is the transition to thermodynamic limit that eliminates the direct influence of the boundary conditions of the interaction of an open system with its surroundings. However, the setting of the boundary conditions determines the formulation of real problems in mechanics. Real systems are open ones that can be considered closed only approximately if the

3.9 Main Problem of Non-Equilibrium Statistical Mechanics

91

system state does not deviate much from the local equilibrium. It must be noticed that in all practical problems in mechanics, boundary conditions play a principal role because they determine the interaction of the system with its surroundings and the system’s response to it. Another problem in statistical theory arises in the description of impulse processes when the forces of inertia can exceed the forces of potential interaction. The concept of the interaction potential underlies the formalism used in statistical mechanics to describe macroscopic systems. It is for these reasons that a gap has arisen between the results of statistical physics and their applications in mechanics. In the general case, the deviation of the system state from equilibrium can be arbitrary. We know that far from equilibrium the system remembers its initial state that is determined by the external impact from other systems surrounding the given one. However, there is a problem with how to describe the arbitrary interaction with any surrounding systems and the initial state of the system under consideration resulting from the interaction? It is quite obvious that in the general case this is impossible in principle. This problem is closely connected to the description of the thermodynamic evolution of real systems. We must understand that a non-equilibrium steady state is the result of the entire evolution of the system under constraints imposed. Therefore, at first we need to describe the temporal system evolution out of equilibrium including its initial state. If we want to describe the evolution stage before the final hydrodynamic one, we should have additional information about the features of specific processes which manifest themselves in the system at this stage of evolution. Microscopic information about this stage of the system evolution is not available, and macroscopic one in not sufficient, since most of the non-equilibrium effects at this stage belong to an intermediate scale between the macro- and microlevels. The special features of non-equilibrium processes were considered in Chap. 2. The description of collective and transient effects accompanying non-equilibrium processes, selforganization of turbulent structures, and their interaction can lead to instabilities of states, structural transitions, and the appearance of feedbacks. Such thin processes are studied by the control theory of adaptive systems. It is clear that this information will always be incomplete without the momentum and energy exchange between three scales: macro-, meso-, and microlevels which should be included in the system description out of equilibrium. All extreme principles such as Jaynes’s principle [35] can determine only the final state of the system but not the way to the goal of the system evolution. It means that we need other principles to describe the evolution path leading to the final state determined by the additional information about the evolution on the mesoscale. As for the entropy concept, it is also not defined far from equilibrium taking into account the non-equilibrium effects. So, we can conclude that the main problem of non-equilibrium statistical mechanics cannot be resolved in the framework of the approaches based on the generalization of classical methods and thermodynamic concepts implicitly related to the local equilibrium states. Since neither statistical distribution functions nor a set of macroscopic densities can describe processes far from equilibrium, we need a

92

3 Macroscopic Description in Terms of Non-Equilibrium …

principally new approach that being in the full correspondence with the most general results obtained in non-equilibrium statistical mechanics could include the processes on the mesoscopic scale.

3.10 The Disadvantages and New Opportunities to Close Transport Equations for High-Rate Processes In the second half of the last century, Zubarev [1] used the MEP principle to derive the most general distribution function describing processes far from equilibrium. Following Jaynes [35], he was the first who incorporated all the history of macroscopic fields in the system into the macroscopic constraints imposed on the system. According to the more recent papers [33, 34] in which the information aspects of the Jaynes approach using the MEP principle were considered, the information included in the imposed constraints determines the predictive ability of the mathematical model of the system evolution. The macroscopic information used by Zubarev in his integral constraints is the most full macroscopic information about the system. It means that his distribution function is the most general and can describe the macroscopic evolution of the system in the widest range of conditions far from equilibrium. Zubarev’s non-equilibrium distribution function results in the integral thermodynamic relationships between the macroscopic gradients and conjugated fluxes. Taking into account spatiotemporal correlations, the relationships generalize the linear thermodynamic relationships of irreversible processes to high-rate and high-gradient processes. The macroscopic transport equations closed by using the integral constitutive relationships are nonlinear, nonlocal in space, and retarding in time. The integro-differential equations of nonlocal hydrodynamics generalize the equations of classical hydrodynamics to non-equilibrium processes beyond the suitability of continuum mechanics. A characteristic feature of the new approach is the preservation in the generalized hydrodynamic equations of integral information about the system when describing its local properties [12–14]. The nonlocal effects are a payment for the incompleteness of macroscopic description of non-equilibrium processes. However, the generalized description is also incomplete because the spatiotemporal correlation functions in the nonlocal transport equations with memory are unknown nonlinear functionals of macroscopic gradients. No even nontrivial approximations have been obtained that allow us to close the nonlocal hydrodynamic equations far from local equilibrium. Their explicit form cannot be derived in the framework of statistical mechanics for real finite-size open systems out of equilibrium. Attempts to construct empirical models of integral kernels led to very rough models and did not allow satisfying the natural boundary conditions imposed on the system. This circumstance became an obstacle to using the nonlocal models in practical problems for several decades.

References

93

All attempts to introduce nonlocal hydrodynamics as a statistical–mechanical basis into the use of computational fluid dynamics and applied mechanics have not led to success until now due to too large a gap in the formulations and methods of solving problems. Only in recent years, works have appeared in which the correlation functions began to be simulated numerically using molecular dynamics on the basis of Zubarev’s method. Very interesting results were obtained for liquids with nanoparticles and flows in nanochannels [38, 39]. In particular, it was shown and experimentally confirmed that no traditional hydrodynamic models work on the nanoscale, while the interfacial interaction at the boundaries of particles and channel walls plays a decisive role. It must be noticed that these results have been obtained under quasi-stationary conditions, not for dynamic processes. Although the nonlocal memory transport models obtained by Zubarev are the only fundamentally new universal mathematical models obtained in science over the past two hundred years, they got no further development till the present. However, the very fact of the appearance of spatial correlations indicates the possibility of using them to describe the effects of collective interaction which can lead to the self-organization of new internal structures. These structures should bring into the system additional information about the system’s evolution far from equilibrium. This idea has been used in the new approach proposed in Chap. 5 of the book.

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15. Bergmann PG, Lebovitz JL (1955) New approach to nonequilibrium processes. Phys Rev 99(2):578–587 16. Lebovitz JL (1959) Stationary nonequilibrium gibbsian ensembles. Phys Rev 114(5):1192– 1202 17. Lebovitz JL, Shimony A (1962) Statistical mechanics of open systems. Phys Rev 128(4):1945– 1958 18. Zwanzig R (1965) Time-correlation functions in statistical mechanics. Annu Rev Phys Chem 16:67–102 19. Zwanzig R (1979) The concept of irreversibility in statistical mechanics. Pure Appl Chem 22(3–4):371–378 20. Zwanzig R (2001) Nonequilibrium statistical mechanics. Oxford Univ. Press, Oxford, New York 21. McLennan JA (1989) Introduction in nonequilibrium statistical mechanics. Prentice Hall, New Jersey 22. Eu BC (1998) Nonequilibrium statistical mechanics. Ensemble method. Kluwer, Boston, London 23. Kuzemsky AL (2007) Theory of transport processes and the method of nonequilibrium statistical operator. Int J Mod Phys B 21(17):2821–2949 24. Kuzemsky AL (2017) Statistical mechanics and the physics of many-particles model systems. World Sci., Singapore 25. Gibbs JW (1902) Elementary principles in statistical mechanics developed with especial reference to the rational foundations of thermodynamics. Dover, New York 26. Kubo R (1968) Thermodynamics. North-Holland, Amsterdam 27. Zwanzig R (1960) Ensemble method in the theory of irreversibility. J Chem Phys 33(5):1338– 1341 28. Maes C, Netocny K (2010) Rigorous meaning of McLennan ensembles. J Math Phys 51:015219 29. Zubarev DN (1970) Boundary conditions for statistical operators in the theory of nonequilibrium processes and quasiaverages. Theor Math Phys 3(2):276–286 30. Kuzemsky AL (2010) Bogoliubov’s vision: quasiaverages and broken symmetry to quantum protectorate and emergence. Int J Mod Phys B 24(8):835–935 31. Mitropolsky YA (1971) Averaging method in nonlinear mechanics. Moskow (in Russian) 32. Bogoliubov NN, Zubarev DN (1955) Method of the asymptotic approximation for the systems with rotating phase and its application to charged particles motion. Ukr Math J 7(1):5–17 33. Kuic D, Zupanovic P, Juretic D (2012) Macroscopic time evolution and MaxEnt inference for closed systems with Hamiltonian dynamics. Found Phys 42:319–339 34. Kuic D (2016) Predictive statistical mechanics and macroscopic time evolution: hydrodynamics and entropy production. Found Phys 46:891–914 35. Jaynes E (1979) The maximum entropy formalism. MIT, Cambridge 36. Poincaré H (1890) Sur le problème des trois corps et les équations de la dynamique. Acta Math 13:1–270 37. Zubarev DN, Bashkirov AU (1968) Statistical theory of brownian motion in a moving fluid in the presence of a temperature gradient. Physica 39:334–340 38. Rudyak VY, Minakov AV (2018) Thermophysical properties of nanofluids. Eur Phys J E 41:15 39. Rudyak VY, Belkin A (2015) Statistical mechanics of transport processes of fluids under confined conditions. Nanosyst: Phys, Chem, Math 6(3):366–377

Chapter 4

Thermodynamic Concepts Out of Equilibrium

Abstract Any theoretical approach should not lead to results that contradict the general laws of thermodynamics. However, all concepts of thermodynamics were formed under conditions close to thermodynamic equilibrium. The concept of local thermodynamic equilibrium was introduced for rather slow processes in weakly inhomogeneous systems that made it possible to use all relations of equilibrium thermodynamics locally, i.e. in the vicinity of each individual point of the system. For highspeed, high-gradient transport processes accompanied by all the features described in Chap. 2, fundamental difficulties arise with basic thermodynamic concepts which either lose their original meaning or become completely unusable far from local thermodynamic equilibrium. Such concepts require a deep rethinking and revision of their correct use. Moreover, such effects that do not occur near equilibrium require new concepts and approaches for their description. Therefore, in this chapter, we first briefly present the basic concepts and laws of classical thermodynamics, the foundations of linear thermodynamics of irreversible processes that help to solve the problem of closing the transport equations near local equilibrium, and then proceed to a critical analysis of the attempts to apply close-to-equilibrium representations to highly non-equilibrium processes. For this analysis, we need rigorous results obtained in non-equilibrium statistical thermodynamics described in Chap. 3. Moreover, due to the specific features of processes far from equilibrium, it turned out that in order to describe them, it is necessary to trace the entire thermodynamic evolution of the system under the conditions imposed. The complexity of real processes far from local equilibrium is so great that only synergetic approaches allow us to compress the information and transfer the description of evolution to the mesoscopic scale level. The order (or control) parameters related to the dynamic mesoscopic structures are not chosen arbitrarily but arise by themselves as a result of self-organization in the system due to nonlinearity, delay, and spatial correlations. The system’s temporal evolution far from local equilibrium should be described on the mesoscale where the internal dynamic structures evolve in a self-consistent way with the macroscopic dynamics of the system. The performed thermodynamic analysis leads us to the conclusion that we need a new interdisciplinary approach to describe such highly non-equilibrium processes as shock-wave processes in condensed media. We hope

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 T. A. Khantuleva, Mathematical Modeling of Shock-Wave Processes in Condensed Matter, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-981-19-2404-0_4

95

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that the approach presented in Chap. 5 permits us to get answers to some important questions of modern thermodynamics. Keywords Entropy production · Information entropy · Self-organization · Synergetics · Control parameters · Mesoscale evolution

4.1 Basic Concepts and Principles of Thermodynamics Thermodynamics emerged as a macroscopic, phenomenological theory of heat. In thermodynamics, a physical system is represented by a certain set of macroscopic measurable parameters. The connection between these parameters and the general laws to which they obey is deduced from the axioms treated as facts of experience [1–7]. Actually, thermodynamics is the theory of equilibrium states and therefore it would be more correct to call it thermostatics. However, the original name is generally accepted and used by most authors. At the present time, the development of thermodynamics is increasingly beginning to correspond to its name. All thermodynamic laws are obtained in statistical physics as some relations for statistical means of microscopic variables representing macroscopically measurable quantities. Thus, thermodynamics gives a necessary criterion for the correctness of any statistical theory built for macroscopic systems obeying the laws of thermodynamics. Choosing a microscopic (atomistic) model for such systems, we must endow it with such properties that the laws of thermodynamics would not be violated. Let us consider the general macroscopic characteristics of thermodynamic systems and the macroscopic parameters that represent them. Any thermodynamic system is concretized by specifying its main physical and chemical characteristics (total mass, chemical composition, etc.) and limiters that distinguish it from the surroundings (vessel walls, interfaces, external fields, etc.). Accordingly, the thermodynamic system is characterized by internal and external parameters. The parameters that can be set by external influences on the system by fixing external bodies or fields are called external parameters. In contrast to them, the internal parameters are determined by the state of the given system itself with the given external parameters. From a microscopic point of view, internal parameters are functions of all micro-parameters of the system that describe a given microscopic model. Which of the macroscopic parameters (volume, pressure, energy, mass, etc.) are attributed to external parameters and which to internal ones depends on the method of separating the thermodynamic system from its surroundings. Thermodynamic parameters are divided into intensive and extensive. Intensive parameters do not depend on the number of particles in the system and characterize the general state of the thermal motion of the body. These parameters are pressure, temperature, chemical potential, etc. Extensive or additive parameters at the same values as intensive parameters are proportional to the total mass or number of the particles in the system. This is energy, entropy, etc. Extensive parameters have a

4.1 Basic Concepts and Principles of Thermodynamics

97

certain macroscopic density and therefore can be represented as integrals of the corresponding densities over the system volume. By the nature of connections with external bodies or by the type of limiters that set thermodynamic systems apart from their surroundings, two main types of thermodynamic systems are distinguished: (i)

(ii)

adiabatically isolated systems separated from the surroundings by adiabatic walls. The energy of an adiabatically isolated system does not change at fixed external parameters but can change arbitrarily by the external bodies; systems that are in thermal contact with surrounding bodies separated by limiters. The energy of such a system can change at constant external parameters through direct interactions of the molecules of the system with the molecules of the surroundings.

Let us list the basic axioms of thermodynamics. (1)

The postulate of the existence of a state of thermodynamic equilibrium.

(2)

Any thermodynamic system under constant external conditions has a state of thermodynamic equilibrium in which its macroscopic parameters remain unchanged over time and from which the system cannot spontaneously get out. It is implied that a non-equilibrium thermodynamic system under constant external conditions ultimately passes into a state of thermodynamic equilibrium. The last statement can be viewed as an additional axiom that clarifies the concept of thermodynamic equilibrium. The postulate of additivity.

(3)

According to this postulate, the energy of a thermodynamic system is the sum of the energies of its macroscopic parts. Without the postulate of additivity, it would be impossible to separate the thermodynamic parameters into intensive and extensive ones and strictly define the concept of temperature. The postulate of the existence of temperature as a parameter that determines thermodynamic equilibrium with the surroundings. This postulate is sometimes considered the zero law of thermodynamics. According to this postulate, adiabatically isolated thermodynamic equilibrium systems form, upon thermal contact with other bodies, a common thermodynamic equilibrium system only if the temperatures of the initial systems are equal. On the basis of this postulate and the postulate of additivity, the temperature is introduced as a single intense parameter for all systems that determines their mutual equilibrium. The thermodynamical equilibrium state of an arbitrary system is completely determined by its external parameters and temperature since any thermodynamic equilibrium system can be considered as initially adiabatically isolated and then brought into thermal contact with an arbitrary body of a certain temperature. Consequently, all equilibrium internal parameters of a thermodynamic system are functions of external parameters and temperature.

98

(4)

4 Thermodynamic Concepts Out of Equilibrium

The law of energy conservation. This law applies to any macroscopic process and is interpreted as a universal law that expresses the conservation of motion in any of its forms. For thermodynamic systems, this law allows us to introduce the concept of the amount of heat as energy imparted to the system not through macroscopic work but through contact with external bodies through the boundaries.  Ak dak is an element If dE is an increment of the energy of the system, dW = k

of work performed by the system on external bodies, ak , Ak are generalized macroscopic coordinates and forces, and d Q is an elementary amount of heat transferred to the system by the surrounding bodies, then from the law of conservation of energy it follows d Q = dE + dW.

(5)

(4.1.1)

This is the first principle of thermodynamics. The second principle of thermodynamics. The second principle is based on Clausius’s postulate which states that heat cannot itself pass from a system with a lower temperature to a system with a higher temperature or Thomson’s postulate according to which it is impossible to build a perpetual motion machine of the second kind, i.e. a periodically operating machine that performs work only by cooling the heat reservoir below the temperature of the coldest part of the surroundings. Instead of all these postulates, one can introduce a set of axioms in which statements that are valid only for equilibrium states are separated from statements that are applicable to non-equilibrium systems. The most important consequence of the second law of thermodynamics is the existence of a new function of state, entropy S, determined for a quasi-static processes by the relation dQ = TdS

(4.1.2)

where T is the absolute temperature defined as a function of the empirically measured temperature and which is the integrating factor of the differential expression dE +

 k

Ak dak .

(4.1.3)

4.1 Basic Concepts and Principles of Thermodynamics

99

Thus, the first and second principles of thermodynamics make it possible to formulate the main differential relation from which all known relations for the state of thermodynamic equilibrium are obtained T d S = dE +



Ak dak .

(4.1.4)

k

The second principle implies the consequence T d S ≥ dE +



Ak dak .

(4.1.5)

k

The inequality sign refers to non-equilibrium states. The entropy increment in any adiabatic process cannot be negative d S ≥ 0.

(4.1.6)

The last relation is equivalent to the law of increasing entropy for adiabatic systems dS ≥ 0. dt

(6)

(4.1.7)

The content of the second principle of thermodynamics is to assert the existence of a function of state, entropy S, determined by (4.1.2) and the law of increasing entropy (4.1.7). Nernst’s postulate. The content of this postulate is the statement that for all condensed substances at zero absolute temperature, the entropy is equal to 0 S = 0, T = 0.

(4.1.8)

The third principle of thermodynamics follows from Nernst’s postulate. According to the principle, the absolute zero temperature cannot be reached in any finite process; it can only be asymptotically approached. The apparatus of thermodynamics establishes a close connection between thermal and caloric equations of state. Thermal equations relate generalized coordinates, forces, and temperature. Examples of thermal equations are the Clapeyron equation  for an ideal gas P V = RT and the Van der Waals equation for a real gas P + va2 (v − b) = kT where R = k N , v = V /N . Caloric equations relate the quantities measured calorimetrically as a certain amount of heat with temperature T and generalized mechanical parameters ak , Ak . Examples of caloric equations are the equations for the energy and heat capacity of an ideal gas E = 23 RT , cV = 23 R. The thermodynamic apparatus does not

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4 Thermodynamic Concepts Out of Equilibrium

contain theoretical results for establishing equations of state for specific thermodynamic systems. These equations are established empirically. Much later, some equations of state were derived from the “first principles” in statistical physics. Relationships between thermodynamic parameters can be obtained by the method of potentials developed by Gibbs. The simplest thermodynamic potential is internal energy presented as a function of entropy and external parameters E(S, ak ). In this case, (4.1.4) can be considered as the definition of the internal energy differential dE = T d S −



Ak dak =

k

 ∂E ∂E dak dS + ∂S ∂ak k

(4.1.9)

whence it follows that  T =

∂E ∂S



 , Ak = a

∂E ∂ak

 .

(4.1.10)

S

Equation (4.1.9) is caloric and (4.1.10) is thermal. By virtue of (4.1.10) 

∂ Ak ∂S

 a

  ∂T =− . ∂ak S

(4.1.11)

Another important thermodynamic potential is free energy  defined as  = E − ST .

(4.1.12)

Similar to (4.1.9), for the free energy differential we obtain d = −SdT −



Ak dak =

k

 ∂ ∂ dak dT + ∂T ∂ak k

(4.1.13)

hence follows     ∂ ∂ , Ak = − , S=− ∂T a ∂ak T

(4.1.14)

and therefore 

∂ Ak ∂T



 = a

∂S ∂ak

 . T

From (4.1.14) and (4.1.12), we obtain the Gibbs–Helmholtz equation

(4.1.15)

4.1 Basic Concepts and Principles of Thermodynamics

−T

101

∂ = E. ∂T

(4.1.16)

In this particular case, when the only generalized coordinate is volume a1 = V and the generalized force is pressure A1 = P, it is customary to use four potentials: internal energy E(S, V ), free energy (in the narrow sense) F(T , V ) = E − ST , thermodynamic potential (T , P) = F + T V , and enthalpy H (S, P) = E + P V H (S, P) = E + P V . The differentials of these potentials and their consequences have the form 

  ∂E , P=− ; (4.1.17) ∂V S V     ∂F ∂F d F = d(E − ST ) = −SdT − Pd V , S = − , P=− ; ∂T V ∂V T (4.1.18)     ∂ ∂ d = d(F + T V ) = −SdT − V d P, S = − , V =− ; ∂T V ∂P T (4.1.19)     ∂H ∂H d H (S, P) = d(E + P V ) = T d S + V d P, T = , V = . ∂S V ∂P S (4.1.20) dE = T d S − Pd V , T =

∂E ∂S



Potentials F,  are also called Helmholtz free energy and Gibbs free energy. Both of these potentials are free energy  and differ only in the choice of the external parameter: in the first case F(T , V ), it is the volume and in the second (T , P) it is the pressure. Other thermodynamic quantities can be selected as characteristic functions, for example, entropy S(E, V ) or volume V (E, S). The main differential relations (4.1.9) and (4.1.13) are generalized to the case of the systems with a changing chemical composition. If Nk , k = 1, ..., m are the numbers of molecules of various types that make up a thermodynamic system, then for such a system relation (4.1.9) can be written in the form dE = T d S −

 k

Ak dak +



μk d Nk

(4.1.21)

k

where μk are chemical potentials of system components. The relation (4.1.13) is generalized as d = −SdT − V d P + μd N .

(4.1.22)

In the case of a homogeneous one-component system, the differential expression for the thermodynamic potential (T , P) is

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4 Thermodynamic Concepts Out of Equilibrium

d = d(F + T V ) = −SdT − V d P.

(4.1.23)

By virtue of additivity, the potential  should linearly depend on the number of molecules N therefore from (4.1.23) for constants T = const, P = const, we obtain  = N μ.

(4.1.24)

Hence, the chemical potential has the meaning of the specific thermodynamic potential, i.e. this is the potential per molecule. In conclusion to this brief overview of the main provisions of classical thermodynamics, we present an expression for the simplest thermodynamic system, an ideal monatomic gas,  3 3 F = −RT ln V + ln T + ln b , E = RT . 2 2

(4.1.25)

This expression is obtained in statistical physics as a consequence of a certain atomistic model of matter.

4.2 Entropy Production in Transport Processes Here, we will consider only weakly non-equilibrium processes that occur near the state of thermodynamic equilibrium. In the state of complete thermodynamic equilibrium over the entire volume occupied by the medium, constant values of the thermodynamic parameters are maintained and the processes of mass, momentum, and energy transport are completely absent. The relationships of equilibrium thermodynamics describe the relations between thermodynamic parameters. If the degree of inhomogeneity (gradients of macroscopic density fields) and the velocity of the medium are small, we can assume that in a differentially small vicinity of each point of the system at each time moment there is a state of complete thermodynamic equilibrium which is characterized by a certain set of thermodynamic parameters. We can assume that the values of these parameters can be different at different points of the system and may change over time. Then, at a given point of the system at a given time moment, all the laws of equilibrium thermodynamics are fulfilled. So, the concept of local thermodynamic equilibrium makes it possible to use the results of equilibrium thermodynamics for inhomogeneous moving media. According to (4.1.5) and (4.1.6), the change in entropy in the system is due to a change in the internal energy and work done during compression (expansion). The change in entropy d S in the volume occupied by the system is due to both its inflow (outflow) from the outside through the boundary of the system volume d Se , and its occurrence inside the volume d Si : d S = d Se +d Si . The change in entropy d Se is due to reversible processes of entropy transfer along with the medium while the change

4.2 Entropy Production in Transport Processes

103

d Si accompanies irreversible transport processes and is called the entropy production. According to the second principle of thermodynamics, the entropy production in the system is always non-negative d Si ≥ 0.

(4.2.1)

In reversible processes d Si = 0, the transformation of internal energy into work and vice versa occur without losses and the system can return to its original state. In irreversible processes d Si > 0, a part of the mechanical energy is irretrievably lost due to dissipation, i.e. its conversion into internal heat energy. A non-equilibrium stationary state d S = 0 can be maintained only in open systems due to the release of constantly produced entropy from the system into the environment d Se = −d Si ≤ 0. For gaseous and liquid media near the state of local equilibrium, the internal energy is understood as thermal energy. In the local equilibrium state, we can introduce the specific and volumetric densities of internal energy E, ρ E, and entropy s, ρs (ρ is the mass density, υ = ρ −1 is specific volume) as we did in Sect. 1.1. The relationship (4.1.17) can be written locally for the densities as follows: T ds = d E + pdυ.

(4.2.2)

Let us substitute the specific entropy density s in the general transport equation (1.2.3) ∂ρs + ∇ · Jv s = σs . ∂t

(4.2.3)

Equation (4.2.3) is the entropy transport equation in Euler coordinates. The volumetric entropy flux density Js , similar to the momentum and energy fluxes, can be divided into a reversible part due to convective transport and a diffusion part which is responsible for the diffusive entropy transport at the molecular level: Js = ρvs +Js(d) . For liquid media, it is generally accepted that the diffusion part of the entropy flux is due only to the heat flux, i.e. the diffusion part of the internal energy flux, T Js(d) = J(d) E . We substitute the expression for the flux into Eq. (4.2.3) and, expanding the derivatives of the products, we obtain ρ

∂s ∂ρ +s + ρv · ∇s + s∇ · (ρv) + ∇ · Js(d) = σs . ∂t ∂t

Introducing the total derivative along the trajectory of the liquid particle and taking into account the continuity equation, we obtain the entropy transport equation in Lagrangian coordinates

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4 Thermodynamic Concepts Out of Equilibrium

ρ

ds + ∇ · Js(d) = σs . dt

(4.2.4)

The bulk density of the entropy source on the right represents the entropy production, an irreversible change in entropy as a result of non-equilibrium transport processes, which, by virtue of (4.2.1), is always non-negative σs ≥ 0. From the Eq. (4.2.2), we obtain ds dE d T = +p dt dt dt

  1 . ρ

(4.2.5)

Eliminating the total derivative ds/dt in (4.2.5) from Eq. (4.2.4), we obtain an expression for the local entropy production in the form σs =

ρ dE p dρ J(d) − +∇ · E . T dt Tρ dt T

Replacing the total derivatives from the internal energy transport equation (1.2.19) and the continuity equation (1.2.7), we obtain  1

p J(d) − p∇ · v + P ◦ ∇v + ∇ · q + ∇ · v + ∇ · E = T T T  1

q = P ◦ ∇v + ∇ · q − ∇ · . T T

σs =

Finally, we obtain an expression for the entropy production in transport processes in a continuous medium in the form of a sum of convolutions of flux terms P, q with gradients of velocity and inverse temperature  1 1 σs = P ◦ ∇v + q · ∇T ≥ 0. T T

(4.2.6)

From (4.2.6), it can be seen that in a spatially homogeneous medium, the entropy production turns to 0. If there are no temperature and velocity gradients in the medium, then there will be no irreversible diffusion fluxes of momentum and energy P = 0, q = 0. If σs = 0, then only reversible transport processes occur in the medium.

4.3 Linear Thermodynamics of Irreversible Processes

105

4.3 Linear Thermodynamics of Irreversible Processes Expression (4.2.6) for the entropy production is a special case of a more general expression in which the entropy production is determined by the sum of the convolutions of thermodynamic forces Xi (proportional to the gradients of the fields of hydrodynamic densities) and conjugate thermodynamic fluxes Ji for all transport processes occurring in the system [8–10] σs =



Ji ◦ Xi ≥ 0.

(4.3.1)

i

Expression (4.3.1) gives the formulation of the first postulate of linear thermodynamics of irreversible transport processes. The inhomogeneity of hydrodynamic fields creates density gradients, i.e. thermodynamic forces that generate fluxes associated with them. For example, the velocity vector gradient generates the viscous stress tensor and the temperature gradient generates the internal energy flux, i.e. the heat flux; in multi-component media concentration gradients generate the diffusion fluxes of the conjugate components of mixtures. However, it should be remembered that the division into forces and fluxes is conditional. Indeed, it can be assumed that, vice versa, inhomogeneities and gradients are created by transport processes, i.e. by fluxes. It is clear that the question of precedence does not make sense. Sometimes, a different terminology is used for the same concepts: they say, for example, about the response of the system to an external influence. If the external action is specified in the force characteristics (surface force, pressure), then the response will be the deformation of hydrodynamic fields. If the external action is specified in strain-rate characteristics, then the response will be the appearance of stress fields. Here, the densities of diffusive impulse and energy fluxes introduced in Chap. 1 were selected as thermodynamic fluxes. If the system is near the state of local thermodynamic equilibrium, then the system is weakly inhomogeneous, the gradients of the hydrodynamic fields, as well as the fluxes caused by them, are small. Considering them all small of the same order, the forces and fluxes can be related by linear relations Ji =



Li j X j .

(4.3.2)

j

Linear relationships between thermodynamic forces and fluxes are the essence of the second postulate of linear thermodynamics of irreversible transport processes. According to (4.3.2), all thermodynamic forces in the medium contribute to each flux. The transport coefficient matrix is called the Onsager matrix. For isotropic systems whose properties are independent of direction, the transport coefficient matrix is symmetric

106

4 Thermodynamic Concepts Out of Equilibrium

L i j = L ji . The off-diagonal matrix elements characterize the contributions of cross-effects, for example, the contributions of the temperature gradient to the viscous stress tensor and the velocity gradient to the heat flux. Under normal conditions, the contribution of cross-effects is negligible. Diagonal elements connect conjugated forces and flows. Linear relations (4.3.2) connect thermodynamic forces and fluxes of the same tensor dimension (Curie principle). However, for stationary systems, indirect conjugation of forces and fluxes is possible when any gradient due to the general balance changes all fluxes in the system regardless of their tensor dimension. Substituting relation (4.3.2) into the entropy production (4.3.1), we find that, in the framework of linear thermodynamics, an entropy production is a quadratic form with respect to the thermodynamic forces σs =



L i j X j ◦ Xi ≥ 0.

(4.3.3)

ji

Similarly, using linear relations (4.3.2), the entropy production can be expressed as a quadratic form with respect to fluxes. In the absence of cross-effects, the second principle of thermodynamics ensures the non-negativity of the diagonal elements of the transport coefficient matrix L ii ≥ 0 which determine the contributions of conjugate forces and fluxes to the entropy production in the system. The transport coefficients in the Onsager matrix have a physical meaning, in particular, non-negative diagonal elements characterize such dissipative properties of the medium as viscosity and thermal conductivity. As an example, consider the process of thermal conductivity, i.e. diffusion transport of internal energy in a thermal form in a stationary medium. In a one-component immobile medium, there are no cross-effects. In such a system, the only flux is a heat flux and the only thermodynamic force is the temperature gradient up to a factor. The real heat flux is conjugated with a reverse temperature gradient and is directed against the temperature gradient from hot to cold. Linear relation (4.3.2) in this case takes the form of the Fourier law of thermal conductivity q(r, t) =

κ ∇T (r, t), T2

(4.3.4)

where κ is the thermal conductivity of the medium (the diagonal element of the Onsager matrix). At not very large temperature differences, the thermal conductivity coefficient is considered a known constant of the medium. As the temperature

4.3 Linear Thermodynamics of Irreversible Processes

107

gradient increases, it begins to depend on the temperature, pressure, size, and geometry of the system. The latter effects are a sign that the used linear relationships in the form of Fourier’s law cease to be correct and the state of the medium is far from local equilibrium. The equation for the internal energy (heat) transport (1.2.19) in a stationary medium takes the form ρ

∂E =∇ ·q ∂t

(4.3.5)

Substituting the closing relation (4.3.4) into Eq. (4.3.5), as well as the thermal equation of state for thermal energy E = cV T , where cV is the heat capacity of the medium at constant volume, we obtain the equation of thermal conductivity (heat diffusion) ∂T κ ∇ 2 T. = ∂t ρcV T 2

(4.3.6)

The coefficient in Eq. (4.3.6) T 2 κρcV is called the temperature conductivity. With small differences in temperature and density, this coefficient is also approximately considered a constant coefficient. Then, the heat equation is a linear partial differential 2-order equation of parabolic type. All diffusion processes are rather slow, their typical times are long. For fast processes, the typical times are short and diffusion models become inadequate. They are also not suitable for describing the initial stage of slow processes. At short times, the finite propagation speed of disturbances in the medium plays an important role, while the parabolic equations correspond to infinitely large propagation velocities of disturbances. In the framework of continuum mechanics, the finite propagation speed of disturbances is taken into account only in the transport equations of a hyperbolic type according to which the mechanisms of momentum and energy transport include wave components. At large times, all wave effects can be neglected and the diffusion mechanism remains the predominant transport mechanism. Later, mathematical models of wave heat transport have been developed [11, 12]. As another example of the application of the closing relations of linear thermodynamics of irreversible transport processes, we consider the momentum transport in a medium near the state of local thermodynamic equilibrium. Here, the components of the mass velocity gradient tensor ∇v = ∇v(0) + ∇v(s) + ∇v(a) act as thermodynamic forces, and the components of the viscous stress tensor P = P(0) + P(s) + P(a) act as the conjugate thermodynamic fluxes. According to the second postulate of linear thermodynamics of irreversible transfer processes (4.3.2), each component of the viscous stress tensor is proportional to the corresponding  3  component of the velocity    Pii I, ∇v(0) = 13 ∇ · v I are gradient tensor. Their spherical parts P(0) = 13 1

linearly linked

108

4 Thermodynamic Concepts Out of Equilibrium 3 

Pii = η∇ · v.

(4.3.7)

1

The coefficient of proportionality η is called the coefficient of bulk or structural viscosity. This coefficient characterizes the contribution of internal friction to the stress tensor in the processes of uniform compression and expansion. Under normal conditions, this contribution is negligible but for inhomogeneous media with a complicated internal structure, it can become significant. The attenuation of acoustic (sound) waves due to their absorption by the medium is associated with the same effect. For shear motionsof a continuous medium, the symmetric components of the  tensors P(s) : Pii(s) = 21 Pi j + P ji − χ δi j are proportional to each other Pii(s) = 2μ

∂vi (s) . ∂r j

(4.3.8)

The coefficient μ is called the shear viscosity coefficient. The factor 2 in expression (4.3.8) appears such that, with a pure shear, for the deviatorial components of the tensor at i = j, the relation has its conventional form   ∂v j ∂vi Pii(s) = μ + . ∂r j ∂ri The shear viscosity coefficient μ characterizes the effects of internal or viscous friction when the medium moves. In a stationary medium, there will be no viscosity. Taking into account the shear components of the viscous stress tensor makes it possible to calculate the viscous friction on surfaces moving in liquid and gaseous media. In a similar way, for the anti-symmetric components of the viscous stress tensor, we obtain the relation   ∂v j ∂vi (a) − (4.3.9) Pii = ζ . ∂r j ∂ri The coefficient ζ is called the coefficient of rotational viscosity. The effects associated with rotational viscosity are often neglected. The reason why rotational effects are neglected is that they make a significant contribution to momentum transport in turbulent flows, while their description goes beyond the concept of a structureless medium. The appearance of large-scale vortex-wave structures in turbulent regimes violates the condition of applicability of continuum mechanics (1.3.5). All viscosity coefficients are diagonal elements of the Onsager transport coefficient matrix, and therefore they are all non-negative. Within the framework of linear

4.3 Linear Thermodynamics of Irreversible Processes

109

thermodynamics, the transport coefficients are not determined; they are considered given characteristics of the medium, i.e. empirical constants. However, it should be remembered that they are only within the limits of the applicability of the concept of a continuous medium. Thus, on the basis of the linear thermodynamics of irreversible transport processes, the closing relations for all components of the viscous stress tensor (4.3.7)–(4.3.9) which enter into the momentum transport equation (1.2.14) have been obtained. Substituting the closing relations (4.3.7)–(4.3.9) into the momentum transfer equation (1.2.14), we obtain the following vector equation: ρ

  dv = −∇ p + ∇(η∇ · v) + ∇ · 2μ∇v(s) + ρF. dt

(4.3.10)

In the one-dimensional case, when there is only one velocity component, we obtain the equation   2 dvx ∂ vx ∂p 4 . ρ =− + χ+ μ dt ∂x 3 ∂x2

(4.3.11)

If we neglect the viscosity in Eqs. (4.3.10) and (4.3.11), we obtain the Euler equations for an ideal fluid. To close the Euler equations for an incompressible fluid, it is only necessary to specify a pressure gradient. For a compressible medium, we need the continuity equation (1.2.7), the equilibrium equations of state for pressure and internal energy, as well as the internal energy transport equation (1.2.19) without the viscous stress and heat flux tensor. If the pressure gradient is assumed to be given, the Navier–Stokes equations (4.3.10) and (4.3.11) for an incompressible fluid are also closed. In general, they contain 4 unknowns in 4 equations. For a compressible medium, it is also necessary to use the equations of state that, as is commonly believed, remain valid near local equilibrium and the complete equation for the internal energy (1.2.19) that contains the heat conductivity coefficient of the medium. The Navier–Stokes equations for a viscous heat-conducting fluid underlie hydroaeromechanics. They describe irreversible processes of momentum and energy transport that insignificantly deviate the state of the system from local equilibrium. Such non-equilibrium processes are realized under the condition of the validity of the linear thermodynamics of irreversible transport processes that impose restrictions on thermodynamic forces and fluxes. The condition of the smallness of thermodynamic forces, i.e. gradients limits the degree of spatial inhomogeneity of the medium. Below, we will show that similar restrictions are imposed on the rates of transport processes. The condition of applicability of the concept of a continuous medium and linear thermodynamics, i.e. the smallness of thermodynamic forces and fluxes imposes a restriction not only on the local production of entropy but also on the integral entropy production in the entire system volume during the duration of these processes

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4 Thermodynamic Concepts Out of Equilibrium

T

S ≡

σs d V dt  S. 0

(4.3.12)

V

Here, S is the total entropy of the system, S is its change as a result of the transport processes in the system for a period T . It is clear from (4.3.12) that when the perturbation region propagates to the entire space and the perturbation duration is unlimited, the requirement that the state of the system be close to local equilibrium may be violated. If condition (4.3.12) is satisfied, then it turns out that the average rates of the transport process must be small. Let’s demonstrate this with a specific example. Consider the process of heat transport in a stationary and incompressible medium on the basis of Eq. (4.3.5) which we rewrite in the form ρ

∂E + ∇ · J E = 0. ∂t

(4.3.13)

The flow J E = κ∇T −1 is proportional to the gradient of the reciprocal temperature; its contribution to the local entropy production is 2  1 σs = J E ∇ · T −1 = κ ∇T −1 = (J E )2 ≥ 0. κ

(4.3.14)

Let the heat be transferred along the x-axis. Let us integrate Eq. (4.3.13) for onedimensional heat transport over the spatial domain V for which we choose a cylinder with a unit area base and a height along the x-axis equal to JE . Then from (4.3.13), we obtain an expression for the total thermal energy in the cylinder  = V ρ Ed V by means of the flux d = JE . Substituting this expression for the entropy production dt = J (4.3.14) with d E into (4.3.12), we can find the integral entropy production in dt the form T

S = 0

V

1 (JE )2 d V dt = κ

T 0

  1 1 d 2 d 1 1 d dt = d = (1 − 0 ). κ dt κ dt κ dt 0

(4.3.15) In formula (4.3.15), according to the mean value theorem, the average rate of change of the total energy in the system d appears. If d  1, the condition (4.3.12) can dt dt 0 be satisfied; even the total change in energy due to heat transport is finite 1− ∼ 1. 0

0 Without limiting the value 1− , by reducing the average speed d , one can obtain dt 0 an arbitrarily small change in entropy S in the system. Then it turns out that the restriction on the integral production of entropy in the system (4.3.12) imposes a

4.4 Revision of the Generally Accepted Thermodynamic Concepts …

111

restriction on the average transport rates and not on the change in the integral quantities themselves. Thus, very slow transport processes can always be described within the framework of continuum mechanics, while fast dynamic processes significantly deflect the system from local equilibrium.

4.4 Revision of the Generally Accepted Thermodynamic Concepts Out of Equilibrium Energy, entropy, entropy production, and information are the basic notions to describe a huge variety of phenomena both in physics and in animate nature [13–17]. The irreversibility in time of all processes in nature plays a principal role in thermodynamics [18–20]. The definition of the basic notions, the formulation of the second principle of thermodynamics, and its consequences on the admissible transport equations are open questions until now [21]. All the notions of thermodynamics were initially based on the concept of thermodynamic equilibrium. The first step away from equilibrium is the hypothesis of local thermodynamic equilibrium for non-equilibrium distributed systems. Assumption of the local equilibrium state permits to use the notions of equilibrium thermodynamics for the states out of equilibrium but does not allow one to describe irreversible processes. This is the base of classical thermodynamics. The linear thermodynamics of irreversible processes became the next step out of local equilibrium but not far from it [8, 10]. The problem of the proper description of irreversible processes is disputable up to now. The current situation in the treatment of irreversibility as noted in the review [21] is rather confused in its concepts and mathematical formulations. For example, the notion of entropy is used in a wide range of problems as an extensive quantity of physical systems, as a measure of information, choice, and uncertainty for different probability distributions [22], as a tool for statistical inference ([23], and in maximum entropy principle MEP). In all these quite different cases, they use just the same functional form of the entropy S=−



pk log pk .

k

However, for many complex systems that are history-dependent, non-ergodic, nonmultinomial, the different entropy concepts lead to the different entropy definitions. In statistical physics, the entropy function can be deduced directly from the probability distribution function. Boltzmann [24] proposed a statistical analogue of thermodynamic entropy linking the concept of entropy with molecular disorder or chaos. The irreversibility results from the assumption that information about individual molecular dynamics is forgotten after the collisions. This lack of memory considers being a source of irreversibility. In order for the randomization assumption to be reasonably applicable, the system should be large enough. Stabilization of the states

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4 Thermodynamic Concepts Out of Equilibrium

in the process of approaching equilibrium requires energy dissipation in the form of heat. Stabilization of the states out of equilibrium needs the input of energy from outside [25]. The numerous attempts [16, 26–33] to extend the notion of the entropy to nonequilibrium steady-state systems were usually based on the entropy definition close to the local equilibrium state. The question is put like this whether a function exists which achieves its extreme value in a stationary non-equilibrium state with the most probable distribution. In contrast to the close-to-equilibrium entropy concept, there is a variety of distinct entropies suitable for different non-equilibrium systems and models. Maes [34] has analyzed various forms of non-equilibrium entropies from a unified point of view and concluded that all the forms do not fit construction of non-equilibrium statistical thermodynamics. For several decades, artificial models have been constructed to determine entropy under non-equilibrium conditions; certain properties are attributed to these models that do not reflect the real physics of non-equilibrium processes. By using new entropy definitions or generalizing close-to-equilibrium concepts, attempts are made to describe the temporal evolution of a system far from equilibrium. However, without a rigorous statistical–mechanical basis, all known approaches do not allow one to really go far from the state of local equilibrium. For the construction of the probability distributions, maximization of the information entropy under the given constraints (principle of maximum entropy, MEP [23]) is used. According to MEP, only the information presented by the imposed constraints determines the statistical description of the system. At large time scales, the information about microscopic dynamics is incomplete and the information loss connected to the entropy production should be included in the description of macroscopic temporal evolution. Then the probability distributions become carriers of incomplete information. The paper [35] shows that maximization of the conditional information entropy leads to a loss of correlation between the initial phase space paths and final macroscopic states. The authors used MEP with hydrodynamic equations as constraints imposed on the system and obtained a probability distribution identical to the local equilibrium one. In Chap. 6, we show that by using MEP with the macroscopic constraints and developing in control theory speed-gradient principle (SGP) describing the way to the goal (entropy maximum), we get the same local equilibrium distribution as a stationary solution of the evolution equation resulting from the speed-gradient algorithm within Speed Gradient Principle (SGP). As noted in paper [36], the weakest point of the results obtained in statistical physics based on the traditional entropy form and the maximum entropy principle (MEP) is the dependence on the choice of the variables. The canonical distribution is obtained from MEP only for the canonical coordinates. The question arises, what variables are proper to describe non-equilibrium states? Is MEP the only tool of statistical physics that allows efficient derivations of previously known results or can it predict the behavior of a real system? Some answers to the questions are proposed in Chaps. 5,6, and 8. We consider that the proper variables should be defined by the same constraints as were used in the system description in accordance with MEP. The constraint imposed on the mean energy of the system is compatible with the use of

4.4 Revision of the Generally Accepted Thermodynamic Concepts …

113

canonical variables because it results in the equilibrium distribution. The farther from equilibrium, the more the detailed information should be included in the constraints imposed on the system. It means that MEP as a principle is as valid to describe a non-equilibrium state of the system as the imposed constraints are able to maintain this state. The objection to MEP is in reality the objection to the functional form of the entropy used out of equilibrium. Here, another question arises, what is to be maximized, what is to be minimized under different constraints? A whole variety of extremum principles is collected in paper [37]. The results obtained in non-equilibrium statistical thermodynamics [38] that became a basis of our approach were also based on MEP. In order to derive the most general non-equilibrium description, MEP was applied under the constraints with the most detailed admissible information included. Unfortunately, these general results are little known to a wide range of researchers and are still almost not used to describe real highly non-equilibrium processes. The traditional concept of “far away from equilibrium” processes as “far from the global equilibrium but not far from the local equilibrium” has been preserved to this day. This circumstance makes it difficult to perceive new approaches and understand the limits of their applicability. The main reason for the confusion, according to our opinion, lies in the outdated concept of non-equilibrium processes and misunderstanding of the physical nature of the processes far from local equilibrium. Nonequilibrium is not just a deviation from the equilibrium distribution function that can be described by several of its moments. Highly non-equilibrium processes are accompanied by a whole complex of the multi-scale momentum and energy exchange between different degrees of freedom and are qualitatively different compared to the near-equilibrium effects. Such a typical non-equilibrium phenomenon as turbulence requires fundamentally new approaches to its description. A brief overview of the special features of non-equilibrium processes based on the modern experimental data is presented in Chap. 2 of the paper. But since the problem of turbulence has not yet been solved, modified old approaches are often used. Under certain conditions, when the deviation from local equilibrium is not very large, this can be justified, but the extension of such approaches to high-rate processes is incorrect. A common misconception was that first one needs to define the non-equilibrium steady state and then describe the non-equilibrium process. However, a nonequilibrium steady state is established during the non-equilibrium process under the imposed constraints depending on the initial system state. Without the information about the system’s history, it is principally impossible to define both the nonequilibrium steady state and the notion of entropy. Unlike the close-to-equilibrium steady state in which the initial information is already forgotten, the steady state far from equilibrium should be defined by this information. Consider the main ideas behind the most commonly used macroscopic approaches to describe non-equilibrium processes. For the treatment of non-steady states of systems with rather significant gradients, the system is subdivided into smaller subsystems that can be considered homogeneous and yet macroscopic. A part of degrees of freedom that have not yet equilibrated and exert an influence on the overall behavior of the system can be included in an enlarged set of variables that obey the

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4 Thermodynamic Concepts Out of Equilibrium

laws of equilibrium thermodynamics like in the local equilibrium. In this case, the system relaxation goes through the additional variables that give a possibility to take into account such non-equilibrium effects as fluctuations, inertia, and activation processes. Therefore, this approach was applied to the systems of mesoscopic size and was even called “mesoscopic non-equilibrium thermodynamics” [39]. However, the choice of additional variables is arbitrary and also does not take into account the possibility of changing them during the process itself. Therefore, this approach is justified only for processes with diffusive transport mechanisms. In the more general case, spatial correlations appear on the mesoscale and form mesoscopic turbulent structures (which will be considered in the next chapters) that evolve over time depending on the system geometry, and initial and boundary conditions. Their size cannot be chosen as desired. Obviously, by using only the local equilibrium laws, it is impossible to describe the system’s temporal evolution far from local equilibrium. The approach based on the system division into small regions, as can be seen from the above, is connected to the theories with internal variables. These theories assume that there exist additional internal or hidden variables that complete the description of the local state. The choice of the internal variables added to the local equilibrium description is free. It may be somehow connected to the internal motions or structures but without any reference to the influence from outside. In Chap. 5, we show that in the framework of our approach the internal variables and the system discretization are entirely determined by the imposed boundary conditions and external influences. The recent trends in non-equilibrium thermodynamics beyond the local equilibrium hypothesis propose new statistical and phenomenological approaches involving nonlinear, nonlocal, and memory effects to describe high-rate and short-duration processes with large special gradients that are used in modern technologies, techniques, and miniaturized devices. The main ideas behind the macroscopic approaches to describe the processes far from equilibrium are presented in the book [40]. However, critical analysis of the approaches shows that they cannot describe the temporal evolution of the system from the initial state because they are based on the “rigid” mathematical models. According to Bogolyubov’s hypothesis, the system is gradually forgetting its initial state; the set of variables describing the system state is reducing during the temporal evolution. The behavior of the system due to the transport mechanisms alteration in multi-scale and multi-stage momentum and energy exchange is also transforming. It means that a mathematical model describing the system’s behavior during its evolution should change its type, i.e. be “flexible”. So, in order to extend thermodynamic description to really non-equilibrium processes, a principally new approach based on the rigorous ground of statistical mechanics is needed.

4.5 Thermodynamic Entropy, Information Entropy, and Information

115

4.5 Thermodynamic Entropy, Information Entropy, and Information In 1948, exploring the problem of the rational transmission of information through a noisy communication channel, Claude Shannon [22] proposed a new probabilistic approach to understanding communication and created the first mathematical theory of entropy. His ideas became the basis for the development of information theory that uses the concept of probability and ergodic theory to study the statistical characteristics of data. The basic concept of the information theory is the concept of entropy [17, 41]. Entropy is a measure of the uncertainty of a certain situation. The greater the uncertainty of the event space, the more information the message about its state contains. The lower the probability of an event, the more information is in the message about such an event. Shannon chose the entropy (a measure of uncertainty in the event space) in such a way that three conditions are met: • the entropy of a certain event with a probability equal to 1 is 0; • the entropy of two independent events is equal to the sum of the entropies of these events; • entropy is maximum if all events are equally probable. For independent random events X with n possible states, distributed with probabilities pi ≥ 0 (i = 1, … n), the Shannon information entropy is defined as follows: H (X ) = −

n 

pi log pi ,

i=1

n 

pi = 1.

(4.5.1)

i=1

The possible value of entropy is within 0 ≤ H (X ) ≤ logN .

(4.5.2)

The lower bound corresponds to the degenerate distribution. There is no uncertainty in the X values. The top bound corresponds to a uniform distribution when all n probabilities pi are equal to each other. If two random variables X and Y are somehow related to each other, then knowledge of one of them obviously reduces the uncertainty of the values of the other. The remaining uncertainty is estimated by conditional entropy. So, the conditional entropy of X under the condition of knowing Y is defined as H (X |Y = y) = −

n  i=1

p(xi |y) log p(xi |y) .

(4.5.3)

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4 Thermodynamic Concepts Out of Equilibrium

The mathematical expectation H (X |Y ) of a random variable H (X |Y = y) defined on (Y, p (y)) is called the conditional entropy of X with respect to Y H (X |Y ) = −



n 

p(y j )

j

  p(xi  y j ) log p(xi  y j ) .

(4.5.4)

i=1

For the conditional entropy, the following inequality (equality corresponds to independent variables) takes place: H (X ) ≥ H (X |Y ) .

(4.5.5)

The definitions for discrete random events (4.5.1) and (4.5.4) can be formally extended for continuous distributions given by the density of the probability distribution. The obtained functional given on the set of absolutely continuous probability distributions, a formal analogue of the concept of Shannon’s informational entropy for the case of a continuous random variable, was called differential entropy. For a continuous random variable ξ distributed over X ∈ R n (n < ∞), the differential entropy is defined as f (x) log f (x)d x.

H (ξ ) = −

(4.5.6)

X

The choice of the logarithm base in this formula (it must be greater than 1) determines the unit of measurement for the corresponding amount of information. So, in information theory, a binary logarithm is often used which corresponds to a unit of information bit and the functional is interpreted as the average information of a continuous source. In mathematical statistics, for reasons of convenience, the natural logarithm (the corresponding unit nat) is usually used. The differential entropy is interpreted as a measure of the uncertainty of a continuous distribution. It can be shown that the differential entropy takes on a maximum value on a uniform distribution. The conditional differential entropy for a quantity X at a given quantity Y = y is determined by the following formula: ∞ H (X |Y = y) = −

f X |Y (x) log f X |Y (x)d x.

(4.5.7)

−∞

Unconditional and conditional differential entropies can be both positive and negative values and can also be equal to infinity. This circumstance also indicates that

4.5 Thermodynamic Entropy, Information Entropy, and Information

117

differential entropy (conditional and unconditional) has a slightly different meaning than entropy that is always non-negative. Differential entropy is not invariant with respect to transformations of coordinates of a random variable and has no independent meaning. However, the difference between the differential entropies of two random variables distributed on one set is correct and the dimensionless value coincides with the difference in their entropies. Since the definition of Shannon’s entropy ceases to be directly applicable to random variables with a continuous probability distribution function, the appropriate value for maximization is the relative informational entropy. Unlike Shannon’s entropy, the relative entropy H has the advantage of being finite and well-defined for continuous x and invariant under coordinate transformations. For the differential entropy, inequality is valid that is similar to the entropy of a discrete source (4.5.5). As we can see, the definition of the information entropy coincides with the definition of thermodynamic entropy obtained on the basis of the canonical Gibbs distribution. In addition to its thermodynamic meaning associated with irreversible losses of mechanical energy for dissipation, the concept of entropy allows us to introduce another important concept of information. Information, along with matter and energy, is a primary concept and cannot be defined in a strict sense. In everyday life, information is usually understood as a set of information about the surrounding world which is an object of storage, transmission, and transformation. Signs and signals organized in sequence carry information by virtue of an unambiguous correspondence with objects and concepts of the real world, for example: objects and words denoting them. In communication theory, two definitions of Shannon’s concept of information are used. One of them coincides with the Boltzmann or Gibbs entropy and is actually a measure of uncertainty in the statistical description. The second is expressed through the difference between the values of the unconditional and conditional entropies. Specifying the second definition allows us to introduce a measure of information in open systems depending on the values of control parameters [42]. There are many definitions of this concept but quantitatively the amount of information can be determined through its relationship with entropy. For the systems that can reach an equilibrium state, the law of conservation of information and entropy holds [42]: S + I = Smax .

(4.5.8)

It follows from this law that in a state of thermodynamic equilibrium when the entropy reaches its maximum value S = Smax , the information I = 0. It means that near equilibrium in a medium that is considered continuum or structureless, its elements move chaotically. The maximum information Imax = Smax is achieved when S = 0. In thermodynamics, it is generally accepted that such a state is achieved at the maximum ordering of the system which is attainable only at zero absolute temperature T = 0. Orderliness means complete knowledge of the position and states of all elements of

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4 Thermodynamic Concepts Out of Equilibrium

the internal structure of the system. This is the level of reference for entropy which is defined in thermodynamics up to a constant. In fact, such a state is not attainable for macroscopic systems since even T = 0 in any physical system spontaneous fluctuations arise. The relationship between entropy and information is similar to the relationship between kinetic and potential energy with the difference that the latter can mutually transform into each other while the loss of information due to entropy, according to the second law of thermodynamics, is irreversible. The more orderly the system is, the more information it contains. The information carrier is the internal structure of the system. Any complex system is in this sense an information system. Informational aspects of complex systems are studied by information theory [22]. The base of the theory is the principle of maximum information according to which information should be processed, transmitted, and stored with the least loss. For this, it is necessary that all transport processes, including the transport of information, be accompanied by a minimum entropy production, and therefore the system should strive to preserve its internal structure. This principle is directly related to the living, most complex, systems since it is associated with the survival of the organism. Information is determined by the difference between unconditional and conditional entropies. It is a reduction in uncertainty due to “knowing something”. It is remarkable that information is symmetric, i.e.: I X Y = H (X ) − H (X |Y ) = H (Y ) − H (Y |X ) = IY X .

(4.5.9)

Information is always non-negative; it is zero when X and Y are independent; information is maximum and is equal to unconditional entropy when there is a one-to-one relationship between X and Y. Thus, unconditional entropy is the maximum information potentially contained in the system. However, despite the symmetry, the top bounds for I X Y and IY X are different when H (X ) > H (Y ): 0 ≤ I X Y ≤ H (X ), 0 ≤ IY X ≤ H (Y ). Shannon introduced the concept of mutual information about an event x that depends on the event y I (x, y) = log

p(x|y) . p(x)

(4.5.10)

Here, p(x|y) is a probability of an event x provided that the event y has occurred; p(x) is a probability of an event x before the event y has occurred. If there is a deterministic relationship between the events such as between stimulus and response of the system then p(x|y) = 1, I (x, y) = − log p(x) = Imax . A decrease in the conditional probability means the loss of information about the response of the system. For independent events p(x|y) = p(x), mutual information vanishes I (x, y) = 0. So, information is just a characteristic of the degree of dependence between some variables. This is an extremely general characteristic. The type of dependence can be absolutely any and, moreover, unknown to us. This does not interfere with

4.6 Local Entropy Production Near and Far from Equilibrium

119

calculating information quantitatively by comparing different types of dependencies among themselves, etc. The price for generality is only the impossibility, knowing the amount of information, to write an equation for the relationship of variables. We can also determine the joint entropy of X and Y by their two-dimensional distribution: H (X Y ) = H (X ) + H (Y |X ) = H (Y ) + H (X |Y ) .

(4.5.11)

Using joint entropy (4.5.11), we can write an expression for information I X Y = IY X = I in a symmetric form: I = H (X ) + H (Y ) − H (X Y ).

(4.5.12)

Intuitively, it is clear that the inclusion of a third variable can only increase the information. In this section, we have shown that the concepts of entropy and information are closely related. We already used the information entropy in Chap. 3 to derive local and non-equilibrium statistical distributions having incomplete information about states of macroscopic systems. In Sect. 4.8, we show the principle that allows us to use the concepts to describe the most probable states of macroscopic systems.

4.6 Local Entropy Production Near and Far from Equilibrium In the framework of linear thermodynamics of irreversible processes, the local entropy production is postulated by formula (4.3.1). Near local equilibrium, it is always non-negative according to the second law of thermodynamics. Far from equilibrium when nonlocal and memory effects take place, the entropy production has to be determined. Strictly, this can only be done on the basis of non-equilibrium statistical thermodynamics. The concept of entropy and entropy production is defined in papers [38, 43] within the method of a non-equilibrium statistical operator. Basing on the results presented in Chap. 3, it can be argued that the entropy of a non-equilibrium state is defined as the entropy of a quasi-equilibrium state at the same mean macroscopic densities as in a non-equilibrium state. According to (3.4.10), we obtain       t   s(r, t) = − ln f q ( p, r, r, t) = (r, t) + dr β r , t H p , r, r , r − μ(r , t) n (r , r, r )t ;        t (r, t) = ln d p dr exp − dr β r , t H p , r, r , r − μ(r , t)n (r , r, r )t .

(4.6.1)

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4 Thermodynamic Concepts Out of Equilibrium

As shown in [43], the local entropy (4.5.1) thus defined satisfies the transport equation (4.2.4)

 ∂ s(r, t) + div s(r, t)v(r, t) + β(r, t)q (r, t) = σs (r, t); ∂t σs (r, t) = q (r, t) · ∇ β(r, t) − β(r, t)P(r, t) · ∇v(r, t).

(4.6.2)

Far from equilibrium, the thermodynamic fluxes in the entropy production (4.6.2) are determined by the following integral relationships: t



P(r, t) = −

dr1 RP,P (r, r1 , t, t1 )β∇v(r1 , t1 );

dt1

−∞

t q(r, t) =

(4.6.3)

dr1 Rq,q (r, r1 , t, t1 ) ∇β(r1 , t1 ).

dt1 −∞

Here, σs (r, t) is the local entropy production that is the bulk density of entropy sources within the system. In the general case, the local entropy production for nonequilibrium processes is defined in paper [43] in the same form (4.6.2) as in linear thermodynamics (4.3.1). Close to equilibrium, when spatial nonlocality and retardation can be neglected, the local entropy production becomes the quadratic form of thermodynamic forces as in linear thermodynamics (4.3.3). σs (r, t) =



L ji X j (r, t) ◦ Xi (r, t) ≥ 0.

(4.6.4)

ji

In this case, the local entropy production is always non-negative and the entropy of an isolated system always increases. Indeed, if we use the relations of the linear thermodynamics of irreversible transport processes (4.3.2) or expressions (4.3.4), (4.3.7), and (4.3.8), we can show that the local entropy production will turn into a quadratic form (4.3.3) of the velocity and temperature gradients in which the transport coefficients on the main diagonal of the Onsager matrix are positive. In the general case, when the effects of spatial nonlocality and memory are taken into account in the expressions for dissipative fluxes (3.7.6), the local entropy production can change its sign and become negative σs (r, t) =

t  ji −∞

dt1

dr1 R ji (r, r1 , t, t1 )X j (r1 , t1 ) ◦ Xi (r, t)

≥0 . J ∗ above the elastic limit as the sum of the elastic and hydrodynamic components. J = (K +

4 4 ∂e G) e + (λ + μ) . 3 3 ∂t

Here K , G are elastic bulk and shear modules; λ, μ are bulk and shear viscosities. The model involves the action of both components at the same time while it is known that at the initial stage of loading, the medium shows only its elastic properties that gradually are replaced by plastic flow. Devoid of this deficiency, the Maxwell model for an elastic-viscoplastic relaxing medium explicitly introduces a dependence on a parameter that has the meaning of a stress relaxation time tr .  ∂J 4 ∂e J − J∗ = K+ G − . ∂t 3 ∂t tr ( J ) The equation can be easily integrated over time as a linear differential equation of the first order on the initial condition J (0) = 0. For the case of a constant relaxation time tr = const, its solution takes the form.

148

J=J

5 New Approach to Modeling Non-Equilibrium Processes

∗



t 1 − exp − tr



4 + (K + G) tr 3

t 0





dt  ∂e(x, t  ) t − t . exp − tr tr ∂t

At large times compared to the relaxation time t >> tr , in the limit of the complete relaxation tr → 0, the obtained solution results in a viscoplastic model of the medium. 4 ∂u ∂v ∂e J → J ∗ − (λ + μ) , =− , 3 ∂x ∂x ∂t 4 4 (λ + μ) = (K + G) tr , tr → 0. 3 3 In the case of frozen relaxation at short times t < t R 0 will be satisfied. Then using (6.6.6) and (6.6.5) we can show that the introduction of the slow time corresponds to the criterion of the system evolution by Glansdorf and Prigogine [5] ∂2 S(t; s(τ )) = −g[∇s Q(t; s(τ ))]2 ≤ 0. ∂τ ∂t

(6.6.7)

So, we can say that an extension of the SG principle for the thermodynamic evolution far from local equilibrium is justified by introducing the slow time describing the dynamic structure evolution at any rate for the processes without memory (Q ≥ 0). As to the high-rate processes with memory effects, there is a question: can the zone of negative entropy production restrain the evolution of the system and prevent it from reaching its goal? The question will be discussed in Chap. 9. In accordance with MEP, the goal functional for the system evolution under constraints imposed is written as follows:  drJ(r, t, s(t)) · X(r, t) +

Q(t, s(t)) =



λm m [J(r, t, s(t)), X(r, t)] −−−→ 0, t→∞

m

V

(6.6.8) where λm are Lagrangian multipliers and m is a number of constraints imposed on the system in the form of functionals. SG algorithm for the goal function with accounting the constraints imposed (6.6.8) is written in the same form ds(t) = −g∇ ds(t) (t, s(t)), dt dt ds(t) = −g∇s dt

(t, s(t)) =

 dr J(r, t; s(t)) · X(r, t) + V



d Q(t, s(t)). dt

λm m [∇s J(r, t, s(t)), X(r, t)].

m

(6.6.9) The Eqs. (6.6.9) describing the way of the structure evolution of the system differ from (6.6.5) due to the constraints imposed. The constraints do not allow the system to reach equilibrium. Since the constraints are able to maintain a stationary state of the

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6 Description of the Structure Evolution Using Methods of Control …

system out of equilibrium, the system can reach the final non-equilibrium stationary state. In this case, the integral entropy production in non-equilibrium stationary state can remain constant and nonzero. In the general case, the behavior of the entropy production out of equilibrium will be considered in the next section.

6.7 Entropy Production in a Stationary State Out of Equilibrium A stationary state can be established only in open system due to the entropy outflow into the environment. Near local equilibrium, the balance between the entropy production and its outflow, din S + dout S = 0, takes place at each time moment. The macroscopic constraints imposed on the system and maintaining the stationary state out of equilibrium entirely provide a constant exchange of entropy with the system surroundings. Far from equilibrium, new information arised from self-organization of turbulent structures interferes with this exchange and upsets the balance. The law of conservation of information and entropy [24] in the form (4.5.8) gives us a connection between the total entropy production and the information partially contained in the constraints imposed S + I = Smax .

(6.7.1)

Here S is the total entropy production defined by Eq. (6.6.2) and I is the total information determining the state deviation from equilibrium. It is known that the information included into the constraints imposed about the system out of local equilibrium is always incomplete. When self-organization arises in the system, the newly formed structures become carriers of information. If the constraints imposed on the system cannot maintain its steady state, the system begins to evolve. Then we can separate the total information in (6.7.1) into two parts: I (t) =  f I + I (s(t)), where  f I is the information given by the constraints imposed and I (s(t)) represents the information carried by the system structure and evolving during temporal evolution. As it follows from (6.6.8), the value of the integral entropy production is determined by the constraints imposed which contain the information  f I and by I (s(t)) which resulted from self-organization and continues to evolve. Then, the integral entropy production in the established stationary state in accordance with Eq. (6.6.8) can be rewritten in the form Q=

 d S +  f I + I (s(t)) = 0. dt

(6.7.2)

It follows from (6.7.2) that the balance between the entropy production and its outflow is broken due to internal structure evolution and the integral entropy production in

6.7 Entropy Production in a Stationary State Out of Equilibrium

195

a non-equilibrium stationary state depends on the rate of structural evolution of the system d S(s(t)) = dt

 dr J(r; s(t))νX(r) = −

 d  f I + I (s(t)) . dt

(6.7.3)

V

Equation (6.7.3) shows that far from equilibrium a stationary state is unstable due to ongoing structural evolution of the system even under the fixed constraints which are unable to retain internal processes. If the information I (s(t)) is gradually forgotten, the integral entropy production grows. Such situation takes place when temporal evolution of the system approaches to the final hydrodynamic stage and spatial correlations are attenuating in accordance with Bogolyubov’s hypothesis [12]. When new information occurs in the system due to self-organization of new internal structures, the integral entropy production can decrease making the system state more unstable. This is due to a change in the mechanisms of energy and momentum transport as a result of the transition to a turbulent mode of motion when dissipative effects give way to inertial ones which, in turn, give rise to retardation. When the system is slowly evolving only via the structural parameters, we say about quasi-stationary process. If the speed of the structural evolution is rather small at some stage, the stationary state can last for a certain finite period of time, for a certain lifetime. However, it will not last forever. According to the Prigogine theorem [5] in the framework of linear thermodynamics of irreversible processes, the entropy production in a stationary state close to local equilibrium has minimum value. There is still debate about whether the entropy production is minimum or maximum out of equilibrium. For different constraints, one can get both minimum and maximum entropy production in a stationary nonequilibrium state. Various principles proposed to determine the behavior of the entropy production have been analyzed and classified in paper [25]. In order to understand this issue to the end, it is necessary first to agree what kind of entropy production we are discussing and under what conditions. Close to local equilibrium, both local and integral entropy production are positive in a stationary state. Far from local equilibrium, local, integral, and total entropy production can behave differently [7]. Secondly, this is the most important thing, and none of the proposed principles in [25] is valid far from local equilibrium because all of them are based on a local albeit nonlinear description of the steady state of the system. Unlike the principles valid only not far from local equilibrium, we propose a new approach basing on the most general description derived in non-equilibrium statistical mechanics that allows us to describe the macroscopic system evolution beginning from the states far away from local equilibrium in full compliance with modern concepts and experimental data on highly non-equilibrium processes. Modeling structural evolution on the mesoscale and using the principles developed in the control theory of adaptive systems give us possibility to make quite definite conclusions about the thermodynamics of stationary states far from local equilibrium.

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6 Description of the Structure Evolution Using Methods of Control …

From Eq. (6.7.3) we see that to define any of them far from local equilibrium it is necessary to describe all temporal evolution of the system taking into account the evolution of its internal structure. Since a stationary state is establishing during temporal evolution in the direction to equilibrium, any stationary state is closer to equilibrium than all previous states of the system. It means that the total entropy production takes the maximum value allowed by the imposed constraints while the integral entropy production (the rate of the total one) tends to minimum in a stationary state maintained by the imposed constraints. From (6.7.1) it follows that the maximum value of the total entropy production in a stationary highly nonequilibrium state due to self-organization is always lower than close-to-equilibrium under the same constraints. SG algorithm providing the fastest way to the goal defines minimum integral entropy production in a stationary state far from local equilibrium. However, the dependence of the integral entropy production on the internal structure dynamics (6.7.3) makes a stationary state far from local equilibrium unstable in the general case. It means that in the framework of SG principle we cannot find = 0. The problem of the structure parameters in the stationary state expecting ds(t) dt stability far from local equilibrium is considered in Sect. 6.9. The set of differential Eqs. (6.6.5) needs initial conditions characterizing the internal structure induced by external impact. Accordingly to SG principle, the evolution path goes down from this point along the gradient of the hypersurface Q(s(t)) (6.6.2) constructed above a space of the parameters s(t). In the general case, the initial point is unknown, but in practice we can calculate one of the trajectory points from experimentally measured macroscopic characteristics, solving inverse problem with respect to the structure parameters. The approach was used in paper [23] for the elastic–plastic shock-induced wave evolution. It should be underlined that only the set of Eqs. (6.6.9) allows a closed formulation of problems on non-equilibrium processes. SG principle imbedded in the internal control mechanism of the non-equilibrium system is an engine of the system evolution. Transforming its internal structure, the system minimizes its irreversible losses and contains information about the history of the imposed external loading. The close-loops occurrence at the internal structure level generates new prospects of the external control, connected to development of new technologies, intellectual systems and prediction of catastrophic processes. So, proposed in the paper complex of “flexible” mathematical models based on nonlocal theory of non-equilibrium processes and SG principle describing the internal structure evolution of systems provides the effective theoretical approach to solve a wide range of the modern practical problems with the use of cybernetic physics methods.

6.8 Paths of Structural Evolution and Reduction of Irreversible …

197

6.8 Paths of Structural Evolution and Reduction of Irreversible Losses Due to Self-Organization In previous section we have shown that the integral entropy production under constraints described by the proposed model (6.6.9) decreases compared to its value described by the continuum mechanics model under the same constraints. This happens because far from equilibrium, self-organization occurs and a new mesostructure brings information into the system that lowers the level of its entropy. It is especially important to take into account synergistic effects [20, 26] when describing dynamic processes in condensed media where the effects of collective interaction are generated by the inertial properties of the medium that largely determine its macroscopic response to impulse influences. In Sect. 5.3 we have included the constraints imposed on the system and the self-organization of new dynamic (turbulent) structures in an open system into the modeling of its response to external influences. The self-organized structures are multiscale, and their spectra are determined by boundary and loading conditions and other limiters imposed on the system as a whole. Therefore, when describing the local properties of the medium, it is necessary to take into account the integral characteristics of the system, its sizes, geometry, and interaction with surroundings. Unlike traditional opinion that the friction in turbulent flow is always higher than in laminar one, now it is known that the situation is opposite: the entropy production in turbulent flow is lower than in laminar one under the same conditions [27]. According to Zubarev’s results [7] obtained in non-equilibrium statistical mechanics, the entropy production can even be negative due to synergistic processes of self-organization of new structures. This case will be considered in Chap. 8. The SG algorithm allows us to interpret the evolution of the system graphically. According to the SG algorithm, the integral entropy production under the constraints imposed (6.6.9) is represented as a hypersurface over the space of the control parameters. In order to reach the goal, path of the system evolution should go down the gradient of this surface. It is clear that until the phase point rolls down to the lowest point of the path, the system will be unstable. It is obvious that the region of local stability corresponds to a hole or a flat area of the entropy production surface. However, we should remember that feedbacks can change the shape of the surface. The path begins from the point defined by the known control parameters determining the sizes of mesoscopic structure of the system. As we have already told in Sect. 6.7, the initial point can be found from experimental data. Unlike the initial distribution function, the determination of the mesoscopic structure of the system is much more available. Since we have experimentally measured macroscopic characteristics, the structure parameters can be calculated by using the generalized macroscopic model with these parameters and solving inverse problem. The used experimental data connect the mathematical model with real state of the system. Next we consider several examples that will demonstrate to us the behavior of the integral entropy production in various situations and possible scenarios of the structure evolution.

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6 Description of the Structure Evolution Using Methods of Control …

In the Rayleigh’s problem presented in Sect. 6.3, the system temporal evolution is considered at the hydrodynamic stage only. Nonlocal generalization of the problem results in the integral expression for the shear stress component with the model spatial correlation function constructed in Chap. 5 ∞ P(y, t) = ν 0

  dy  π(y  − y − γ )2 ∂u . exp − ε ε2 ∂ y

(6.8.1)

Here y is coordinate in transverse directions relative to the plates, μ is shear viscosity, and u(y) is longitudinal mass velocity. As it was shown in paper [19], the nonlocal effects smooth the gradients and reduce the friction on the rigid plate. The generalized integral entropy production has the form ∞

∞ dyσ (y, t) =

0

dy 0

∞ =ν 0

∂u dy ∂y

∞ 0

∂u P(y, t) ∂y

  dy  π(y  − y − γ )2 ∂u = Q(t; ε, γ ). exp − ε ε2 ∂ y

(6.8.2)

In the case when we will take the SG algorithm in the form (6.3.4) corresponding to the hydrodynamic stage of evolution, we have the surface of the integral production as follows: 1 Q h (t; ε, γ ) = πt

∞ 0



y2 dy exp − 4t

 ∞ 0

   2  dy  π(y  − y − γ )2 y . exp − exp − 2 ε ε 4t (6.8.3)

For simplicity, we suppose the viscosity ν equal to 1. In Fig. 6.2 we can see that the surface (6.8.3) has maximum for the structureless medium (ε = 0, γ = 0) that tends to infinity when t → 0 and gradually reduces with time. The surface (6.8.4) smooths together with the stress component reducing. Figure 6.2 shows that any finite perturbations resulted in finite values of the nonlocal parameters ε, γ lead to their growing. According to SG principle, if the initial point of the evolutionary path is on the slope of the integral entropy production surface, the path should go down the hill along the gradient. It means that the turbulence will continue to develop. However, further the gradient in the form (6.3.4) will smooth and the surface will smooth too. The influence of such feedback will be considered in the last section of the chapter. In the Couette problem we have shear flow between two rigid boundary plates. For the steady flow of Newtonian fluid the shear stress between the plates is constant

6.8 Paths of Structural Evolution and Reduction of Irreversible …

199

Fig. 6.2 The shape of the integral entropy production surface near the hydrodynamic stage of the temporal evolution in the Rayleigh’s problem: a t = 1, b t = 2

P(y) = νdu/dy = const. Under the sticking conditions u(±h) = ±U linear velocity profile u(y) = (U/h)y is established. In dimensionless form one gets P(y) =

du = 1, u(±1) = ±1, u(y) = y. dy

(6.8.4)

High-rate shear in condensed medium cannot be described by Newtonian fluid model. High-rate interaction of the medium with rigid boundaries results in a complex of non-equilibrium effects that can make the velocity profile nonlinear and change shear stress. Out of equilibrium dimensionless model expression for the shear stress component takes a form h P(y) = −ν −h

  π  ∂u dy    π  exp − 2 (y − y + γ )2 + exp − 2 (y  − y − γ )2 2ε ε ε ∂ y h

−−−→ −ν ε,γ →0

−h

  ∂u ∂u = −ν . dy  δ | y  − y |  ∂y ∂y (6.8.5)

The correlation function constructed in Chap. 5 is modified (γ = β y [20]) to account for the given symmetry properties of the flow. The normalized parameters ε, γ describe nonlocal and boundary effects omitted in the Newtonian model.

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6 Description of the Structure Evolution Using Methods of Control …

For pure shear, when the shear stress P = εχ (ε, γ ) is constant, we get an integral– differential equation with respect to u(y) 1 −1

  π  ∂u dy    π  = χ (ε, γ ). exp − 2 (y − y + γ )2 + exp − 2 (y  − y − γ )2 2ε ε ε ∂ y (6.8.6)

From (6.8.6) we see that the nonlocal effects near the hydrodynamic stage of the flow evolution with the fixed gradient ∂u/∂ y = 1 decrease the shear stress. The surface of the integral entropy production near the hydrodynamic stage and far away from local equilibrium has different shape. Unlike the smoothed surface in Fig. 6.3a near local equilibrium, the surface in Fig. 6.3b out of local equilibrium has two peeks where the integral entropy production far from equilibrium is much higher than at the hydrodynamic stage. This is because near boundaries there are most non-equilibrium zones with large nonlocal parameters whereas their small values corresponding to continuum mechanics cannot describe the boundary zones. On the top of the surface for the hydrodynamic stage we have a flat zone where Newtonian medium model is valid. If the initial state of the flow is on the flat zone, the steady flow is stable. Far from local equilibrium, there are no stable zones on the surface. In the course of structural evolution, any highly nonequilibrium initial state generates local accelerations in the flow field that, in turn, induce pressure gradients along the flow. In result, the shear flow becomes unstable and the flow modes can change.

Fig. 6.3 Shapes of the integral entropy production surface: a near the hydrodynamic stage and b far from local equilibrium

6.8 Paths of Structural Evolution and Reduction of Irreversible …

∂ ∂y

1 −1

201

  π  ∂u dy    π  ∂u =a . exp − 2 (y − y + γ )2 + exp − 2 (y  − y − γ )2  2ε ε ε ∂y ∂t (6.8.7)

In both cases, large perturbations lead to the turbulence development. The turbulent structures decrease the integral entropy production because dissipation is replaced by inertial effects. However, far from equilibrium the turbulent structures can both grow and decrease depending on the initial state resulted from perturbation. Velocity profiles u(y) for decreasing structure size correspond to ones in accelerating flows and the growing structure sizes lead to the velocity profiles in flows with deceleration [28, 29] (Fig. 6.4). In open system, any subsequent impact can significantly change the shape of the surface (Fig. 6.5). In paper [30] it was shown that the integral entropy production in a plane jet calculated using continuum mechanics model is lower for laminar mode compared to the turbulent flow model but for the turbulent zone provides the lower value of the integral entropy production than continuum mechanics model (Fig. 6.6). So, any physical system changes its internal structure and mode of functioning in order to decrease its dissipation losses and retain information as much as possible.

Fig. 6.4 Evolution of the integral entropy production with nonlocal parameter

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6 Description of the Structure Evolution Using Methods of Control …

Fig. 6.5 The surface of the integral entropy production over the plane of the control parameters

Fig. 6.6 The integral entropy production during laminar-turbulent transition in plane jet

6.9 A New Look at the Problem of Stability of Non-Equilibrium Systems

203

6.9 A New Look at the Problem of Stability of Non-Equilibrium Systems Cybernetic methods allow a new point of view at the stability of non-equilibrium systems. Often non-equilibrium stationary state of macroscopic system simultaneously loses its stability, and it is difficult to find the reason of such behavior. In the framework of the nonlocal theory of non-equilibrium transport [20], the internal structure evolution can continue even in the stationary macroscopic state because the boundary conditions maintain not all degrees of freedom. As a result, a part of the structure parameters relaxes decreasing the entropy production rate. In Sect. 6.6 from Eq. (6.6.8) we have seen that a stationary state of the system can exist out of equilibrium only in the case when the imposed constraints can maintain it. The stationary boundary conditions that do not repel the system state far from local equilibrium are sufficient to maintain stationary process. However, as it is shown in non-equilibrium statistical mechanics [7], any description of macroscopic system far from local equilibrium is incomplete. Therefore, such constraints for macroscopic variables become unable to maintain the stable stationary state and it begins to evolve. It means that in the general case any constraints are incomplete and any state far from local equilibrium is unstable. Nevertheless, the non-equilibrium stationary state can be retained for a finite time interval. When the internal structure evolves rather slowly, macroscopic properties of the system can remain almost constant for some, maybe even quite a long time. The time of existence of slowly changing scale and type of the system structure defines an interval of the system stability. When the rate of the structure evolution suddenly grows, macroscopic properties of the system can essentially change. When due to the positive feedback the scale of macroscopic inhomogeneity coincides to the internal structure scale, the structure transformation occurs. In such threshold manner the system switches to another mode of evolution. Sometimes big changes may happen unexpectedly and be catastrophic. For example, it is a fracture of solid, non-equilibrium phase transformation, etc. The proposed nonlocal approach together with SG principle allows a direct calculation of the living time for the considered internal structure type under the imposed boundary conditions taking into account feedbacks in between the structure evolution and the dynamic of macroscopic fields. In the framework of the approach, the stability problem is reduced to the construction of the phase trajectory going down the surface defined by the goal functional from the initial phase point. When the phase point falls into a deepening or on a plane and cannot move further, the structure evolution, according to SG principle, goes at the constant speed ds/dt = const. Such mode of the system evolution can be called quasi-stationary. The structure evolution stops when ds/dt = 0. In the case the system has the internal structure of the constant scale and type that is usually considered in the theory of multiphase media. It must be underlined that the phase surface due to feedback with macroscopic dynamics changes during the evolution and the evolution path changes with it. The same changes occur in open systems under external action.

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The evolution speed and the living time of the quasi-stationary mode depend on the empiric coefficients g > 0 (matrix) in SG algorithm. Unknown in general case, the coefficient can be found from the experimental measurement of both macroscopic and structure dynamic characteristics. It is a complicated experimental problem because of the expensive measuring equipment and a large proportion of the fluctuation components in measured averaged values. It is clear that the more coefficients g, the less stable is the system. So, g are the more profound system characteristics than the conventional medium ones as elastic modules and transport coefficients.

6.10 Influence of Feedbacks on the Paths of the System Evolution and Prediction of the Final States As we have shown before, a macroscopic system evolves by means of its internal structure evolution on the mesoscale. Within control theory, the model of structural evolution is based on three main points: (1) accordingly to MEP, the goal function of the system evolution is defined by maximum of the total entropy production under constraints imposed; (2) the typical sizes of spatiotemporal correlations connected to the turbulent structure sizes or their rates are chosen as control parameters; (3) SG principle determines the fastest way to the goal of the system evolution. In the general case, the total entropy production of the system during temporal evolution is in stages: t Si (t; s(t)) ≡

dt 

0

t = 0

t −−−→ 0

dr σs (r, t  ; s(t))drdt  =

V



dr G(r, t  ) 

dt 





dr R(r, r , t  , t  ; s(t))G(r , t  ) −−−→ t→∞

V 



s(t)→0

t

dr (r, r ; s(t))G(r , t) −−−→

dr G(r, t) V

t  0

V

 dt

t→∞

dt 



t→∞

V

 dr kG2 (r, t)

dt 0

V

(6.10.1) If the initial state of the system is far from local equilibrium and characterized by finite-size correlations, then further according to Bogolyubov hypothesis [12], the correlations should decay during relaxation in the absence of external influence. At the initial stage of the system evolution far from local equilibrium when the main transport mechanism is a wave, spatial and temporal correlations cannot be separated. This stage takes place only for high-rate and short-duration processes. Closer to equilibrium but still out of local equilibrium, for quasi-stationary processes spatial nonlocal effects remain while retardation effects become negligible. At the

6.10 Influence of Feedbacks on the Paths of the System Evolution …

205

final hydrodynamic stage when the system approaches close to local equilibrium, all non-equilibrium effects disappear and the system evolution is described by linear thermodynamics of irreversible processes. In open systems, as it was shown in previous sections, the imposed constraints related to the exchange with surroundings slow down the process of evolution and can completely stop it if they are able to maintain a stationary state of the system out of equilibrium. As we have seen, it is possible only near local equilibrium since internal structure evolution can not be completely ceased far away from local equilibrium. Far from local equilibrium the constraints imposed on the system can give rise to selforganization of new dynamic structures in various forms: pulsations, wave-vortex and other turbulent structures. Such structures are unstable and evolve during relaxation or change due to external influence. If the constraints imposed are gradually weakened or forgotten over time, the turbulent structures disappear and the system becomes structureless as is commonly believed in continuum mechanics. Thus, when we want to describe a process far from local equilibrium, we should describe the evolution of turbulent structures under constraints containing the information about the system which is responsible for the structures. This is the main problem connected to the description of turbulence. Without inclusion of selforganization of turbulent structures and their evolution it is impossible to describe the system response to external impact which can bring the system out of local equilibrium. The evolution of the system takes place simultaneously at three scale levels micro, meso and macro. In response to mechanical perturbation of the system state, the energy exchange between them is directed from macro to micro level. Selforganization of the turbulent structures and their evolution appear on the mesoscale. Between the structure evolution on the mesoscale and macroscopic temporal evolution of the system feedbacks arise. Due to the feedbacks, even the system relaxation can become non-monotone. It is especially important to take into account synergistic effects [20, 26] when describing dynamic processes in condensed media where the effects of collective interaction are generated by the inertial properties of a dense medium that largely determine its macroscopic response to impulse loading. The macroscopic properties of the medium can be considered specified only near thermodynamic equilibrium. Therefore, it becomes clear that all attempts to use traditional macroscopic models with a set of empirical parameters, so called “rigid” models, will not have predictive ability. Analysis of the behavior of the integral entropy production in a wide range of conditions plays an important role in the developed approach since, striving to minimize its value, it determines the control mechanism that transfers the system from one state to another. Finally, by applying modern methods of the theory of adaptive control, it makes it possible to self-consistently describe the evolution of the internal structure. In the examples considered in the previous section, we have seen that the integral entropy production surface evolves during the temporal evolution. Due to feedback between macroscopic and mesoscopic evolutions the surface can change its shape and the evolution path can change its rate and even direction. The evolution

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6 Description of the Structure Evolution Using Methods of Control …

path essentially depends on the initial internal structure of the system and on the mode of external impact. If the path is known, it becomes possible to predict the future response of the system to impact that from the point of view of conventional approaches, based on “rigid” models, seems to be unpredictable. The presence of feedbacks when changes in macroscopic and structural variables are determined through each other gives rise to new dynamic structures and switches the system from one operating mode to another. It is shown in cybernetics [1] that a feedback system is much more resistant to external disturbances than systems with rigid program control. The numerical calculation of the velocity profiles of turbulent fluid flows in jets and channels based on nonlocal hydrodynamic models [30] showed that the “manual” control of the mesoparameters without using the governing equations based on SG method leads to a rapid “collapse” of the solution. In turn, the use of additional control in the form of the SG principle leads to the stabilization of the numerical solution. Various histories of acceleration of a fluid from a state of rest lead to various scenarios of the system evolution in the space of nonlocal parameters and can form significantly different velocity profiles. A change in the correlation characteristics in a medium according to the principle of minimizing the integral entropy production leads to the establishment of nonlinear (close to power-law) profiles [29]. Optimization of the mesoscale parameters of the system solves the problem of controlling high-rate processes in a condensed medium. The tested transition criterion makes it possible to reveal which of the modes will be implemented in practice. The numerical solution of the problem is very sensitive to the initial data and especially to the choice of the gain parameters in the SG algorithm. For large values of the gain parameters, when the “steps” of evolution are too large, the solution is usually unstable. By decreasing the evolution steps, we can violate an interaction of the medium with the boundaries of the system when the near-boundary zones and the flow core behave differently. Finally, nonphysical velocity profiles are obtained. In the general case, it can be assumed that the value of the parameters in SG algorithm is related to the properties of a particular medium but a careful study of both the physical meaning of these parameters and the development of appropriate numerical methods are required. Thus, taking into account the generation and evolution of the internal structure of the medium under highly non-equilibrium conditions, it becomes possible to create “flexible” mathematical models providing a uniform description of transient processes in a wide class of complicated phenomena. The proposed approach gives rise to new prospects for the development of new technologies, information systems, the study of natural multiscale processes, as well as with the prediction and prevention of catastrophic phenomena. Thus, internal control and self-organization are inherent elements of mathematical modeling processes far from local equilibrium since the closure of transport equations for high-rate and short-duration motions is impossible without them.

References

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References 1. Fradkov AL (2003) Cybernatic physics. Nauka, S. Petersburg (in Russian) 2. Fradkov AL (2007) Cybernetical physics: from control of chaos to quantum control. SpringerVerlag, Berlin 3. Fradkov AL (2017) RSTA. Horizons of cybernetical physics. Philos Trans Royal Soc A: Math Phys Eng Sci 375:20160223 4. Fradkov AL (2008) Speed-gradient entropy principle for nonstationary processes. Entropy 10:757–764 5. Glansdorff P, Prigogin I (1972) Thermodynamic theory of structure, stability and fluctuations. Wiley Interscience 6. Khantuleva TA (2013) Nonlocal theory of non-equilibrium transport processes. St. Petersburg State University Publ., St. Petersburg, 278p. (in Russian) 7. Zubarev DN (1974) Non-equilibrium statistical thermodynamics. Springer, Berlin 8. Rayleigh L (1911) On the motion of solid bodies in viscous liquids. Philos Mag 21:697–711 9. Landau LD, Lifshitz EM (2013) Fluid mechanics: course of theoretical physics, Vol 6, Kindle Edition, Pergamon Press. ISBN 0750627670, ISBN13: 9780750627672 10. Grinela M, Ottinger H (1997) Dynamics and thermodynamics of complex fluids, I-development of a general formalism. Phys Rev E 56:6620 11. Fradkov AL, Shalymov DS (2018) GENERIC and Speed-Gradient principle, IFACPapersOnLine (5th IFAC Conference on Analysis and Control of Chaotic Systems CHAOS 2018Eindhoven, The Netherlands, 30 October–1 November 51(33):121–126 12. Bogoliubov NN (1962) Problems of dynamic theory in statistical physics. Studies in statistical mechanics, North-Holland, Amsterdam, pp 1–118 13. Kuic D, Zupanovic P, Juretic D (2012) Macroscopic time evolution and MaxEnt inference for closed systems with hamiltonian dynamics. Found Phys 42:319–339 14. Jaynes E (1979) The maximum entropy formalism, MIY, Canbridge, MA 15. Jaynes ET (1980) The minimum entropy production principle. Ann Rev Phys Chem 31:579–601 16. Fradkov AL (2005) Application of cybernetic methods in physics. Physics- Uspekhi 48:103– 127 17. Fradkov AL, Shalymov DS (2015) Dynamics of non-stationary nonlimear processes that follow maximum differential entropy principle. Commun Nonlinear Sci Numer Simul 29:488–498 18. Fradkov AL, Krivtsov A (2011) Speed-gradient principle for description of transient dynamics in the systems obeying maximum entropy principle. In: Proceedings of AIP Conference 30th Workshop on Bayesian inference and maximum entropy methods in science and engineering, 1305, pp 399–406 19. Khantuleva TA, Shalymov DS (2017) RSTA modelling non-equilibrium thermodynamic systems from the speed-gradient principle. Philos Trans Royal Soc A: Math Phys Eng Sci 375:20160220 20. Khantuleva TA (2005) Self-organization at the mesolevel at high-rate deformation of condensed media. Khim Fiz 24(11):36–47 21. Khantuleva TA (2018) Thermodynamic evolution far from equilibrium. AIP Conf Proc 1959:100003-1–100003-4. https://doi.org/10.1063/1.5034750 22. Ravichandran G, Rosakis AJ, Hodovany J, Rosakis P (2002) On the convention of plastic work into heat during high-strain-rate deformation. In Shock compression of condensed matter-2001, Furnish MD, Thadhani NN, Horie YY (Eds). AIP Conference Proceedings-620. Melville. N.Y. pp 557–562 23. Fradkov AL, Khantuleva TA (2016) Cybernetic model of the shock induced wave evolution in solids. Procedia Struct Integr 2:994–1001 24. KlimontovichYL (1999) Entropy and information of open systems. Phys Usp. 42:375–384. https://doi.org/10.1070/PU1999v042n04ABEH000568 25. Stijn Bruers (2007) Classification and discussion of macroscopic entropy production principles. arXive:cond-mat/0604482v3, 2 May

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26. Haken H (2006) Information and self-organization. A macroscopic approach to complex systems. Springer, Berlin, Germany 27. Klimontovich YL (1990) Turbulent motion and the chaos structure. Nauka, Moscow 28. Khantuleva TA, Shalymov DS (2020) Nonlocal hydrodynamic modeling high-rate shear processes in condensed matter. J Phys Conf Ser 1560:012057 29. Schlihting H (1979) Boundary-layer theory. Mc Graw-Hill, New York 30. Nikulin IA (2005) Phd Thesis, S. Petersburg State University

Chapter 7

The Shock-Induced Planar Wave Propagation in Condensed Matter

Abstract Experimental and numerical study of shock-induced processes show that the accepted ideas of elastic-plastic transition within concepts of continuum mechanics cannot explain mechanical and physical properties of the shocked material related to defective structures occurring on a sub-continuum scale. New trend in mechanics called mesomechanics describes the deformation and destruction of the material as the multiscale process that defines a change in mechanisms of momentum and energy transport in a deformable solid depending on the strain-rate, boundary, and initial conditions. However, the physical processes on different scales associated with high-rate deformation are not well understood till present because experimental diagnostics does not allow us to obtain reliable data on deformation processes on the mesoscale in real time. Computer simulations as a tool for probing details of the deformation process also run into unresolved problems related to the choice of the closing relations embedded into the computer complex package or the unknown interaction laws between the mesostructure elements. It became clear that in order to develop predictive mathematical models, it is necessary to critically examine the fundamental postulates used to interpret shock compression phenomena and to develop first-principle approach capable to describe a whole complex of nonequilibrium processes in a deformable medium. In this chapter, presented in Chap. 5 approach, being suitable for any non-equilibrium conditions, is used to describe the processes of high-rate deformation of solids. The response of any condensed medium to external loading goes through all relaxation stages from elastic to hydrodynamic one. In contrast to the hydrodynamic limit, the processes occurring near the elastic limit have wave nature when spatial and temporal correlations cannot be separated. The specific feature of the response of a solid to shock loading is a very strong interaction of atoms with each other that corresponds to a very high degree of spatial correlation and memory about the initial state of the system. Shock on a solid breaks correlations into mesoscopic parts moving at different speeds like a wave packet. Similar to the generation of turbulence in liquids, strong shears cause the formation of rotational structures inside the propagating waveform. An irreversible part of the mesoscopic structures remain frozen into material after the wave when the solid state is restored. The synergistic formation of vortex-wave structures on the mesoscale defines the medium response to high-rate deformation (Sect. 7.4). Within © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 T. A. Khantuleva, Mathematical Modeling of Shock-Wave Processes in Condensed Matter, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-981-19-2404-0_7

209

210

7 The Shock-Induced Planar Wave Propagation in Condensed Matter

the framework of the developed approach, we proposed a new mathematical model of the propagating waveform generated by mesoscopic structures evolving over time. The integral formulation of the problem of the planar shock-induced wave propagation in a solid with the closing relationship for the relaxing stress tensor is given in Sects. 7.6 and 7.7. The explicit approximate solution to the problem obtained in Sect. 7.8 allowed us to decipher the information on the stress relaxation involved in the waveform evolution during its propagation along the material. The processing of experimental waveforms obtained at different distances from the impact surface for different materials in a wide range of impact velocities revealed the general laws of the relaxation processes taking into account the specificity of various materials (Sects. 7.9–7.11). In the first sections of this chapter, we provide a brief summary of the physical and thermodynamic properties of solids that are necessary to understand the new approach to describing shock-induced processes. More information can be found in books [1–6]. Keywords Shock-induced wave · Stress relaxation · Elastic–plastic transition · Waveform propagation · Mesoscale · Turbulent structure

7.1 Thermodynamic Properties of Solids All substances, regardless of their aggregation state, consist of particles (atoms, molecules, etc.) that interact with each other. The nature of their interaction determines the whole variety of physical and mechanical properties of real media. The main feature of metals in the solid state is a very strong interaction between atoms that attract when moving away and repel when approaching. The compression of solids requires tremendous pressure that is necessary to overcome the forces of the crystal lattice. Impact on a solid induces the pressure in it that has no thermal origin. The so-called cold pressure corresponds to a temperature equal to absolute zero. From the basic law of thermodynamics, T d S = d E + pd V , it follows that at absolute zero T = 0 the cold pressure is determined by the relation connecting it with cold or elastic energy pe = −∂ E e /∂ V .

(7.1.1)

The level of entropy in the case is considered constant and since entropy is determined up to a constant, as an initial level, we can choose a level with zero entropy, S = 0. Compression and expansion processes are reversible under such conditions. These are elastic processes determined only by the forces of interaction between atoms and the potential energy of their interaction. At the temperature T = 0 in the absence of external loading, a mechanical equilibrium between atoms is established (equilibrium of forces, not thermodynamic equilibrium!). It corresponds to the densest packing of particles located at a certain distance from each other. This is the most ordered state of the system called the

7.1 Thermodynamic Properties of Solids Fig. 7.1 Potential energy of crystal lattice E(V )

211

E

O

V0

V

Parabola Emin

crystal lattice that is typical for most metals in the solid phase. In this state, the cold energy E e (V ) has a minimum at specific volume V = V0 . Ideally, this state is characterized by maximum orderliness and, therefore, minimal entropy. That is why the level of entropy in this state can be taken as zero. In a real solid, the crystal lattice is imperfect, there are always packaging defects that lead to an increase in specific volume and a decrease in mass density. The dependence of cold potential energy E on the specific volume V is presented in Fig. 7.1. With increasing density ρ, the specific volume V decreases and the potential cold energy grows. With decreasing ρ, V increases, the cold energy also grows but only to a certain level due to the ∞ asymmetry of the shape of the potential, Q = − v0 Pe (V )d V . With this expansion, even at T = 0 condensed matter evaporates and behaves like a gas whose particles are not affected by the interaction forces. At a temperature T = 0, pressure and energy acquire thermal components due to thermal fluctuations p = pe + p T , E = E e + E T .

(7.1.2)

The thermal components increase the entropy level. This is valid at relatively low temperatures T < 105 ◦ C. The thermal energy is determined by the relationship E T = cV (T − T0 ) + E 0 , cV = 3N k,

(7.1.3)

where the specific heat at a constant volume cV is due only to harmonic vibrations of the lattice atoms. At low temperatures, atoms make small vibrations around the equilibrium position. When heated, the motion of atoms intensifies, the amplitude and energy of the oscillations increase. In an asymmetric potential well, the oscillations become anharmonic going beyond the parabolic part of the potential well. With increasing temperature, this effect is more pronounced. This can also happen with a strong shock. The increasing temperature leads to the thermal expansion of the condensed matter when

212

7 The Shock-Induced Planar Wave Propagation in Condensed Matter

the equilibrium position characterized by the minimum interaction energy shifts to the right. At very high temperatures, when the vibrations become chaotic, the solid behaves like a gas. The entropy level in this case reaches a maximum. At low temperatures and pressures, this state is not achieved but effects associated with anharmonic vibrations of atoms are manifested. Under normal conditions T ∼ 300 ◦ K due to thermal fluctuations, the volume expansion of solids, V > V0 , is about 1–2%.

7.2 Wave Processes in Crystal Lattice Oscillations of atoms due to spontaneous fluctuations always exist even at T = 0. At small amplitudes, the oscillations are harmonic. The interaction of atoms is transmitted in the form of waves. All atomic vibrations are a collection of waves of various lengths propagating in a crystal. In a bounded body, a stationary oscillation regime as a superposition of standing waves is established over time. All the properties of condensed matter such as heat capacity, thermal conductivity, and thermal expansion are explained from these positions. In the volume V there are N identical atoms with masses m that have 3N degrees of freedom. The motion of each atom is connected to the motion of its nearest neighbors in a lattice. The set of 3N coupled homogeneous equations has nontrivial solutions in the form of traveling waves only if the determinant of the set is equal to zero. This condition that interconnects the frequency and wavelength is called the dispersion relation. Considering this relation as an equation with respect to frequency, we obtain a cubic equation for the square of the frequency that has 3 real roots. These roots determine 3 modes of oscillations for each wave number: one longitudinal wave (oscillations in the direction of wave propagation) and two transverse waves with oscillations in the plane normal to the direction of wave propagation. Each normal oscillation, or mode, can be considered as a harmonic oscillator with a corresponding frequency. Let us consider for simplicity a one-dimensional linear chain of N atoms with mass m having only one degree of freedom (Fig. 7.2). The equation of motion of the nth atom is written under the assumption that a quasielastic force acts on it that is proportional to the relative displacements of neighboring atoms. This means that because of the short-range potential, direct interaction takes place only between the nearest neighbors which is transmitted along the chain. m

m a

Fig. 7.2 1D model of crystal lattice

m n-1

un

m

m n

n+1

m

7.2 Wave Processes in Crystal Lattice

m

213

  d 2un = β (u n+1 − u n ) − (u n − u n−1 ) . dt 2

(7.2.1)

Here u n is the displacement of the nth atom, β is the elastic constant, and a is the step of the crystal lattice. We got a set of N identical equations tied to each other. This means that all atoms oscillate according to one law. We seek normal vibration modes in the form u n = u 0 exp{i (kna − ωt)},

(7.2.2)

is the wave number inversely gde u 0 is the atomic displacement amplitude, k = 2π λ proportional to the wavelength λ, ω = 2π is the angular frequency inversely P proportional to the wave period p. The dispersion relation determines the dependence of the frequency ω on the wave number k, the same for each atom 

  β  ka  ω=2 sin . m 2

(7.2.3)

The difference between the oscillations of a discrete chain of atoms and the vibrations of a continuous string is the manifestation of the effects of the internal structure of the system in the form of wave dispersion that manifests itself in a nonlinear relationship between frequency and wave vector in the dispersion relation. It can be seen from the dispersion relation that the maximum frequency is reached for very short waves which length is defined by the structure of the atom chain  ωmax = 2

β π if k = , λ = 2a. m a

For long waves ak  π , 2a  λ, when the relationship  between frequency and

wave number is close to the linear dependence ω = a mβ k, the dispersion effects can be neglected. In wave mechanics, the phase speed of a wave C f is determined as ω 2 Cf = = k k



   β  ka  β , sin  −−→ a m 2 k→0 m

(7.2.4)

and the group speed C g is dω 2a Cg = = dk 2



   β  β ka  . cos  −−→ a  m 2 k→0 m

(7.2.5)

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7 The Shock-Induced Planar Wave Propagation in Condensed Matter

Fig. 7.3 Linear chain consisting of two kinds of atoms

The group speed occurs only in dispersive media when the internal structure of the medium affects the process of wave propagation. The group speed is always less than the phase one, like a wave packet speed. It can be seen from this that short-wave signals propagate more slowly than long-wave ones. Due to the spreading of wave packets, they also decay faster than long-length waves. In the long-wavelength limit when the dispersion disappears, both speeds coincide and become equal to the speed of sound propagation in the equilibrium system  C f = C g = C0 = a

β . m

In thermodynamics, the speed of sound is defined as follows C0 =

(7.2.6) 

∂P ∂ρ

. S,T

There are adiabatic sound speed at constant entropy S and isothermal sound speed at constant temperature T . Then for pressure near the equilibrium state, we obtain the expression p = (ρ − ρ0 )C02 . In a one-dimensional chain of two types of atoms, in contrast to a one-dimensional monoatomic chain (see Fig. 7.3), there two dispersion curves appear ω+ (k) and ω− (k). At small values of the wave number k  1, the two vibration frequencies behave differently: maximum frequency for the branch ω+ (k) is determined by the value of the coefficient of the quasi-elastic force β and the reduced mass of the atoms,

+

ω (k) =



1 1 2β + m M



+ (k), = ωmax

(7.2.7)

while the branch ω− (k) behaves similar to the case of the monoatomic chain,

ω− (k) = ka

β . 2(m + M)

(7.2.8)

In this case, the center of mass of the system during vibrations with these frequencies remains fixed while atoms of different types vibrate in antiphase. +  With increasing k, frequency decreases and tends to its lower boundary ωmin (k) = 2β m1 for the shortest possible wavelength λ = 4a. In this case, heavy atoms remain stationary and the wave propagates due to the displacement of light atoms. The group speed for the limiting frequency tends to zero.

7.3 Elastic Properties of Solids

215

Such vibrations can, for example, be excited in ionic crystals by electric soldering of a light wave. Therefore, the branch of oscillations was called optical. The optical branch of vibrations can arise not only as a result of the unequal masses of atoms. With equal masses, optical vibrations arise due to the difference in the distances between molecules (or between atoms within molecules) since this leads to a difference in the coefficients of elastic coupling between them. For the frequencies ω− (k) the oscillations occur in the same phase and have approximately the same amplitudes. This is typical for the acoustic branch for the monoatomic chain. The maximum possible frequency of acoustic vibrations does 

− (k) = 2β M1 . As the not depend on the mass of lighter atoms and is equal to ωmax masses of atoms m, M in the chain approach each other, the spectrum of acoustic and optical vibrations degenerates into the only acoustic branch. The spectrum of an infinite chain of atoms is continuous in the range from 0 + − + (k) and from ωmin (k) to ωmax (k). If the chain of atoms is limited, then the to ωmax spectrum of oscillations becomes discrete and as a result of reflection from the ends of the chain, traveling waves are replaced by standing ones. + − (k) to ωmin (k) The absence of a solution for frequencies in the range from ωmax indicates the presence of a forbidden zone. The atomic mass ratio determines both the band gap and the width of the optical branch. With a small difference in masses, the forbidden region turns out to be rather narrow and the ratio of the limiting frequencies of the optical branch approaches. In the case when it is much higher, the band gap will be wide and the frequencies of optical vibrations form a narrow region. It should be noted that for both the acoustic and optical branches, each longitudinal vibration in which the displacement of atoms occurs along the direction of propagation of the oscillations corresponds to two transverse ones (the displacement of atoms occurs in the orthogonal direction with respect to the direction of propagation). Thus, the character of atomic vibrations in a one-dimensional diatomic chain turns out to be much more complicated than in a monatomic one. Therefore, it should be expected that even more complex distributions of inhomogeneities in the medium can lead to fundamentally new types of motion of the particles of the condensed medium.

7.3 Elastic Properties of Solids The harmonic vibrations of the lattice atoms with a small amplitude determine the elastic properties of solids due to the cold components of energy and pressure. In the long wavelength limit, λ  2a, when the size of the lattice does not affect the propagation of waves, the medium can be considered as structureless or continuum. Then, under the condition 2a  λ  L (L is a distance traveled by the wave), transport processes in solids can be described in the framework of continuum mechanics. For the case of a one-dimensional motion of the medium along the axis x in a semi-space under plane compression, we have the equations of mass and momentum transport

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7 The Shock-Induced Planar Wave Propagation in Condensed Matter

(See Chap. 1) in the form ∂ρ ∂ + ρvx = 0, ∂t ∂x

(7.3.1)

∂ρvx ∂ (d) ) = 0. + (ρvx vx + Jvx ∂t ∂x

(7.3.2)

In the case of a weak external forces we can linearize these equations near the unperturbed state that is characterized by the mass density ρ = ρ0 , mass velocity (d) = K where K is a constant. Such a vx = 0, diffusive momentum flux density Jvx stationary state of the medium in the absence of transport processes corresponds to the model of an absolutely rigid body. The model of an elastic body is obtained if to introduce linear perturbations of these quantities of the same order of smallness (d) = K + p, where p is cold pressure (reversible ρ = ρ0 + ρ1 , vx = 0 + v, Jvx part of stress tensor) that arises under the force action on the solid. In the linear approximation with respect to the small additives, the Eqs. (7.3.1)–(7.3.2) take the form ∂ ρ1 ∂v + = 0, ∂t ρ0 ∂x

(7.3.3)

∂v ∂ p = 0. + ∂t ∂ x ρ0

(7.3.4)

We multiply Eq. (7.3.3) by the value C02 (C0 =



∂p ∂ρ

is adiabatic bulk velocity S

of sound in an equilibrium medium, not related to the thermal motion of atoms) and differentiate by x. Then we differentiate Eq. (7.3.4) with respect to t, subtract the (7.3.3) from (7.3.4), and get  2 ∂ 2v ∂2 p 2∂ v 2 ρ1 − C = − C . 0 0 ∂t 2 ∂x2 ∂ x∂t ρ0 ρ0

(7.3.5)

Equation (7.3.5) turns into the wave equation describing undamped oscillations of the medium provided that the right-hand side turns into 0. This condition is met by the equation of state of the medium for pressure p, p = ρ1 C02 .

(7.3.6)

The wave equation for mass velocity, 2 ∂ 2v 2∂ v − C = 0, 0 ∂t 2 ∂x2

(7.3.7)

7.3 Elastic Properties of Solids

217

v by using variables, ξ = t + Cx0 , ζ = t − Cx0 , is reduced to ∂ξ∂ ∂ζ = 0. Its solution x in opposite directions is the initial velocity waveforms propagating along the axis   x x v = f 1 t + C0 + f 2 t − C0 . For the wave traveling to the right along the axis x, 2

instead of (7.3.7), we can write the equation ∂v + C0 ∂∂vx = 0 which solution is f 2 . In ∂t this case the continuity Eq. (7.3.3) takes the form ∂ ρ1 ∂ v − = 0, ∂ζ ρ0 ∂ζ C0 whence the relationship determining deformation follows ρ1 v = = e  1. ρ0 C0

(7.3.8)

Here e = ex x is the only nonzero component of the strain tensor under the plane loading. Due to the weak compressibility of a solid because of the strong interaction of lattice atoms, the deformation is small in the field of elastic response of the medium. Taking into account (7.3.8), the equation of state (7.3.6) can be rewritten in the form p = ρ1 C02 = ρ0 C0 v = ρ0 C02 e,

(7.3.9)

(d) Then for the stress component we get Jvx = K + ρ0 C02 e = K (1 + e) where 2 ρ0 C0 = K is the bulk elastic modulus for all-around compression or expansion. It can be seen from (7.3.9) that the cold part of the stress tensor is proportional to the mass velocity of the medium or to the strain and not to the strain-rate as in Newtonian fluid. The cold pressure (7.3.9) is linked to the cold specific energy by the formula (7.1.1)

d E x = − px d



dρ1 dρ1 1 C 2 e2 = ρ1 C02 2 , E x = ρ1 C02 2 = 0 , ρ 2 ρ0 ρ0

(7.3.10)

It can be seen that the cold or elastic energy is a quadratic function of deformation and therefore of specific volume. This is the parabolic part of the potential well in which the vibrations of the lattice atoms are harmonic. If we take into account the contribution of the stress components corresponding to the cold shear that is described by the law i j = 2Gei j , i = j (G is shear modulus), then similarly to the one-dimensional fluid flow, we obtain  2 (s) 4 4G 2 2 p + x x = ρ0 C0 ex x + Gex x = ρ0 C0 + e = ρ0 Cl2 e, 3 3 3 ρ0

(7.3.11)

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7 The Shock-Induced Planar Wave Propagation in Condensed Matter

 where Cl = C02 + 43 ρG0 is longitudinal velocity of sound in the medium. The elastic modules K , G are determined by the properties of the crystal lattice of a solid. Small perturbations in a solid, as in any condensed medium, propagate in the form of elastic waves at the speed of sound which, unlike gas, is not related to the thermal velocity of particles in the medium but is determined by the ordered medium structure. In semi-space, in addition to longitudinal waves (the directions of the mass velocity in the wave and the wave propagation coincide), shear waves can also propagate (the mass velocities are directed normal to the direction of wave propagation). Elastic waves carry momentum in the medium. This is the wave mechanism of momentum and energy transport. Elastic waves do not transfer mass on a macroscopic scale because particles only vibrate with respect to a stationary position. Elastic wave is a reversible process. Entropy production at a wavelength λ is equal to zero

S =

λ

∂v p dx = ∂x

  ∂v 2 λ ρ0 C0 v d x = ρ0 C0 v /2 = 0. 0 ∂ x λ

In continuum mechanics, the elastic properties correspond to a medium model in which the components of the stress tensor linearly depend on the corresponding components of the strain tensor (in contrast to Newtonian model of a fluid where the viscous stress tensor is proportional to the strain-rate tensor, i.e., the mass velocity gradient). Unlike liquid and gas, a solid without external influence is a medium without pressure. In a liquid due to the redistribution of gravity, there is always hydrostatic pressure, in gas the thermal pressure associated with chaotic diffuse momentum transfer is added to it, and in a solid pressure appears only in response to external compression.

7.4 Plastic Deformation. Deficiencies of Continuum Mechanics If under external influence the stress exceeds the elastic limit and achieves yield stress of the material, the relaxation begins. The relaxation of the shear stress is associated with a change in the shape of the body without changing its volume. It means that the mass transport induced by the applied force becomes irreversible. Mass transport in a solid is called plastic flow or plastic deformation. In the framework of continuum mechanics, total deformation of a solid e under strong loading is separated into two parts e = eelastic + eplastic . Plastic deformation is usually described by hydrodynamic equations. Two hydrodynamic models are used to close macroscopic transport equations: model of ideal fluid and Newtonian model of viscous fluid. The ideal plasticity is based on the model of inviscid fluid. In this case plastic deformation proceeds without an increase in stress. In order to account the relaxation effect on the plastic properties of condensed matter, a viscous stress tensor,

7.4 Plastic Deformation. Deficiencies of Continuum Mechanics

219

  4 ∂vx 4 Px x = − χ + μ = χ + μ e˙x x , 3 ∂x 3 is used where e˙x x = e˙ is the strain-rate, and χ , μ are bulk and shear viscosities. The following relation establishes the connection between the viscosities and elastic modules 4 4 ρ0 C02 tr0 + Gtr G = χ + μ, 3 3

(7.4.1)

where tr 0 , tr G are the relaxation times of the bulk and shear stresses respectively. In a solid, the shear degrees of freedom relax much faster than the bulk ones tr G  tr 0 . If the bulk relaxation is frozen, the stress tensor corresponds to viscoelastic medium model 4 4 2 2 (s) 2 − Px(s) p + (s) x = ρ0 Cl ex x − Px x = ρ0 C 0 ex x + Gex x − μe˙ x x . 3 xx 3 3

(7.4.2)

This model describes uniaxial compression as a combination of elastic compression and shear relaxation at the same time instant. In the classical continuum mechanics, it means that the system state is close to the local thermodynamic equilibrium. However, the relaxation proceeds over time. During the relaxation transport mechanisms change. Transport mechanisms of elastic and plastic deformation are fundamentally different. Wave transport at the elastic stage of deformation is reversible whereas dissipative transport at the final plastic stage is irreversible. During the shear relaxation the elastic shear component decreases whereas the viscous component grows until at the time instant t = tr G they compensate each other. For example, when the shear relaxation is entirely completed, Gex x − μe˙x x = G(ex x − tr G e˙x x ) = 0, the plastic deformation is also finished and only elastic bulk wave is left. Behind the bulk wave an unperturbed state of the medium is established. This state, as well as the initial medium state, considers being uniform and equilibrium. It is through that due to the relaxation processes the responses of liquids and solids to applied forces depend on time. At sufficiently short times when the relaxation is considered frozen, liquids having insufficient time to flow behave elastically as if they were solid. Conversely, solids behave elastically at short times but they flow at sufficiently long times depending on the applied force. They behave as if they were liquid. Between the two extremes lies a transition zone where both hydrodynamic and elastic effects should take place. For a long time it was thought that plastic deformation was macroscopically uniform process [7]. In experiments, the recorded stress–strain curves are always smooth lines during the transition from the elastic to plastic regime. However, now we know that the plastic deformation is heterogeneous at all intermediate scales. The plastic deformation is not continuous process; it consists of series of steps during

220

7 The Shock-Induced Planar Wave Propagation in Condensed Matter

the relaxation which reduce the local stress slightly. Plastic deformation is not wave process and therefore cannot propagate [7, 8]. Accepted ideas of homogeneous, uniform states produced by shock compression are not generally correct and therefore models based on these simple ideas are not adequate. The obtained experimental and numerical evidence [8–31] shows departures from accepted ideas of elastic–plastic transition. The data are anomalous from the point of view of continuum mechanics. Mathematical treatments of planar shock loading of solids are usually based on variants of elastic–plastic model with the assumptions that the shocked material is homogeneous and that the stress and strain tensors can be separated into elastic and plastic components. The assumptions are generally far from adequate because the demarcation between the components depends on such loading conditions as strain-rate and shock duration. Specifically, the presented results indicate that mechanical and physical properties of the shocked material are determined by localized deformations occurring on a sub-continuum scale. At the initial stage of development of the theory of dislocations, it was believed that all the mechanical characteristics of crystalline solids can be described from the standpoint of the dynamics of dislocations moving in the potential barrier of the crystal lattice. However, numerous attempts to link the dislocation theory to the continuum mechanics failed. The reason was not only mathematical problems of the macroscopic description of the collective behavior of the shock-induced defects. It has become evident that our understanding of the internal processes during the plastic deformation is not true and is described by inadequate phenomenological models based on the concepts of the continuum mechanics [32–34]. Now the situation in the experimental and theoretical study of defective structures in deformable materials has radically changed. Structural studies of solid materials have revealed completely new types of defective structures for different deformation conditions. These include dislocation walls, localized shear bands, rotational structures, fragment boundaries, etc. The discovered structural formations belong to the so-called mesoscopic scale level (10−5 ÷ 10−3 cm). These formations play the role of independent elementary carriers of deformation of the material. New area of mechanics called “mesomechanics” [35, 36] describes the deformation and destruction of the material as the multiscale process responsible for the macroscopic properties of materials. According to modern concepts of the physics of strength and plasticity, there is a certain hierarchy of scale levels that defines a change in mechanisms of momentum and energy transport in a deformable solid, depending on the strain-rate, boundary and initial conditions, and the material properties. This means that the problem of an adequate description of deformation and fracture cannot be solved within the framework of a single-level approach. In-depth analysis of problems encountered in the study of the shock-induced processes and extensive bibliography are presented in [29]. The physical processes at different length scales associated with high-rate deformation are not well understood. The present state of experimental diagnostics does not allow us to obtain reliable data on deformation processes on a microscopic scale and especially in nanosecond regime. Recent advances in computer simulations were

7.5 Shock Wave as a Transient Highly Non-equilibrium Process

221

used as a tool for probing details of the deformation process. However, the obtained solutions must be compared with experimental data at different scales. Besides, the problem of the choice of the closing relations embedded into the computer complex package is still remained. Direct simulation of the particles motions requires the interaction laws between the internal structure elements on intermediate scales which are not known. So, the developed continuum models of plastic deformation and direct computer simulation of dynamic material response do not provide uniform descriptions of dynamic material response over a wide range of loading conditions. Despite considerable progress in modeling, there is no a first-principle approach capable to describe a whole complex of non-equilibrium processes in a deformable medium responsible for its dynamic macroscopic properties. To develop predictive continuum models, it is necessary to critically examine the fundamental postulates used to interpret shock compression phenomena in order to ensure that the correct mathematical framework is used to describe physical processes.

7.5 Shock Wave as a Transient Highly Non-equilibrium Process A shock wave is a steep compression wave that propagates at supersonic speed D relative to the medium ahead of it. The propagation speed of small perturbations in a solid is not equal to the thermal one, and is not connected with it in any way. If the gas behind the wave front accelerates to the speed of the wave itself, then the mass velocity in a solid is ten times less than the sound speedC. Here and below, we omit the subscript “l ” for the longitudinal sound speed Cl = C02 + 43 ρG0 . Therefore, for shock-induced waves in solids under compression up to 107–108 atm. Mach numbers are small M t R (shortduration loading). For the long-duration loading, the bulk part of degrees of freedom also relaxes and for the short-duration loading, a part of the shear degrees of freedom shows the elastic response. Therefore, the representation of the shear relaxation with the bulk degrees of freedom frozen as well as the division of the stress into elastic and plastic parts is incorrect in the general case. It means that the material response to the loading at different stages of the relaxation can radically change depending on the loading strain-rate, material properties and also on the size and shape of the sample. In addition, it was found that the process of establishing steady values of the sound speed in a medium requires a certain time which is much less macroscopic typical time but much more microscopic one. During this process, the propagation speed of the pulse can differ significantly from the equilibrium sound speed in the medium [58]. It means that the values of the elastic modules change in the transient zone with the contributions of the bulk and shear degrees of freedom. We never know the boundary between the elastic and plastic parts of the stress because it depends on the loading strain-rate and relaxation process. It means that it is impossible to determine which degrees of freedom are responsible for the shock-induced deformation. Therefore, it is incorrect to separate neither the contributions of the bulk and shear degrees of freedom nor the elastic and plastic components of the stress and deformation as we have done in the expression (7.6.5). So, the concept of the shock-induced waveform consisting of the elastic precursor and plastic front that was previously generally accepted cannot be considered adequate. Therefore, when describing dynamic processes with t R  tr , high-rate deformation, shock loading, we cannot separate the contributions of elastic, plastic and bulk, shear degrees of freedom in advance. The irreversible change of the material properties can be determined only after the waveform travelling in the form of new defective structures and partial dissipation into heat. Besides, the process of relaxation itself cannot be described only by the memory function MG (t, t ; tr G ) that determines attenuation of only temporal correlations during the wave propagation in the medium because spatial and temporal coordinates in a wave are linked together along the characteristics x = ±Ct. It means that it is

7.7 Formulation of the Problem of the Shock-Induced Wave Propagation …

231

necessary to take into account the wave nature of the momentum transport and include in the description of the system the evolution of its dynamic structure. Therefore, we should use the relationship for the longitudinal stress without separation of spatial and temporal correlations in the form

ω(t)

dt tr 0

(t)

dx ∂v R(t, t ; x, x ; tr , lr ) ; lr ∂x 0 0

  4G t, t < t R , Ct, t < , ω(t) =

(t) = Cl = C02 + , Ct ≥ , tR , t ≥ tR , 3 ρ0 (7.6.13)

J1 (x, t) = −

ρCl2 tr

In the next section we present the wave propagation problem formulation with the nonlocal and retarding constitutive relation (7.6.13) using the model correlation function developed in Chap. 5 that is capable to describe all stages of relaxation.

7.7 Formulation of the Problem of the Shock-Induced Wave Propagation in Condensed Matter The propagation of short pulses of moderate intensity in a condensed medium is induced by impact on the surface of the condensed medium at a speed V0  C that is much less the longitudinal sound speed in the medium. At macroscopic distances (∼10−3 m) from the surface of the impact, when the phase speed of the wave is established C = const, the total velocity of momentum transport can be correctly separated into the phase wave speed and mass velocity U = C +v under the condition v  C. In this case, mass transport takes place only inside the traveling wave. Consider a planar impact between two semi-spaces of condensed matter inducing two identical longitudinal waves that propagate along the axis x in opposite directions. In the case, we can consider only the leading front not taking into account the unloading process. It is known from experiments that the elastic precursor propagates at the speed of longitudinal sound. According to the generally accepted concept, when the elastic limit is exceeded, a two-wave structure forms that consists of both an elastic precursor and a plastic front rising behind it. Adopting such physical model of the shock-induced elastic–plastic wave, we put into it a completely different understanding of the relaxation and transient processes and use new mathematical model to describe it. We include the loading duration as an important characteristic of the process. The leading edge of the waveform propagates at the phase speed of longitudinal sound Cl but during the loading time interval it relaxes due to the spatiotemporal correlation broken by the shock. As a result, the material becomes less solid. Therefore, the so-called elastic precursor becomes not quite elastic. When the relaxation of the elastic precursor is completed, its amplitude remains constant. For each material,

232

7 The Shock-Induced Planar Wave Propagation in Condensed Matter

its own value for the height of the elastic precursor is set. The so-called plastic front is not quite plastic because it arises as a post-shock inertial effect but not as a hydrodynamic flow. After the shock, the solid state of the material is partially restored but a part of the shock-induced momentum is irreversibly lost. Therefore, the transporting mass plastic front gradually lags from the leading edge (precursor). Its speed is a group speed of the wave propagating in a medium with internal structure evidenced by the observed in experiments the mass velocity dispersion. We can consider the shock-induced waveform as a wave packet propagating along the material. Let us formulate the problem on the shock-induced waveform propagation on the base of the integral model (7.6.13). The wave propagation process is characterized by three relative scale parameters: (1) (2) (3)

the stress relaxation parameter τ = tr /t R determines the relaxation mode; the delay parameter θ = tm /t R characterizes the lag of the maximum of the plastic front from the elastic precursor; the relaxation length parameter ε = Ctr /L characterizes the nonlocal effects during the waveform traveling along the axis x (L is typical distance traveled by the wave).

All the introduced parameters should be included into the integral kernels of the model expression (7.6.13). In a wave process, the space and time variables are related to each other. Let us introduce the coordinate frame of reference associated with the elastic precursor that moves at a constant speed of longitudinal sound Cl . sound For simplicity, below we omit the subscript l and use C for longitudinal  speed. New dimensionless variables are following ζ = t1R t − Cx , ξ = Lx . The relationship between the typical scales of the problem ε/τ = Ct R /L determines the roles of the new variables for the short-duration loading. If the impact is short ε/τ  1, the nonlocal effects associated to the high-rate waveform change at the distance L can be neglected compared to its change inside the wave. In this case, the derivatives along x are small compared to the time derivatives ∂ζ∂  τε ∂ξ∂ . The separation of typical scales of the process is considered as the necessary condition for self-organization of new structures in the medium [59–61]. The mass transport Eq. (7.6.3) in the moving reference system can be written in dimensionless form as follows ∂ρ1 /ρ0 V0 ∂v V0 ε ∂v − + = 0. ∂ζ C ∂ζ C τ ∂ξ

(7.7.1)

Here, the mass velocity v is normalized by the shock velocity V0 . It can be seen from the continuity equation in (7.7.1) that in the first order in the small parameter V0 /C  1 and in the zero approximation in the parameter ε/τ  1, the deformation e is determined by the mass velocity of the medium e = ρ1 /ρ0 = (V0 /C)v.

(7.7.2)

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233

In the general case, beyond these assumptions, the concept of deformation introduced in classical continuum mechanics becomes incorrect. There is no correct definition of large deformation as well as is no need to introduce it when we can use mass velocity measured in real time in a wide range of loading conditions. The considerable part of difficulties connected to plastic deformation is explained by using obscure, misunderstood concept of deformation beyond the elastic limit. The mass velocity has definite physical meaning for any type motion and can be measured in real time. The momentum transport Eq. (7.6.3) is written in the new variables ∂v ∂ ε ∂

− + = 0. ∂ζ ∂ζ τ ∂ξ

(7.7.3)

Here, the integral expression for the normalized longitudinal stress component (7.6.10) = J1 /ρ0 C V0 in the new frame of reference takes a form

Ct

(ζ, ξ ; τ, θ, ε) = −τ

dξ



o

0

ω =

ω

dζ ζ (ζ, ζ ; τ, θ )



o

    ∂v ε ∂v   dζ ζ (ζ, ζ ; τ, θ )δ( ξ − ξ ) − + = τ ∂ζ τ ∂ξ 1



 ∂v ε ∂v − , ∂ζ τ ∂ξ

 ω(ζ ) =

ζ, 1,

ζ < 1, ζ ≥ 1. (7.7.4)

Here due to the condition of the scales separation, ε/τ  1, the derivatives along the x-axis and the spatial nonlocality in the integral kernel can be neglected. Substitution (7.7.4) without spatial derivatives into the momentum transport Eq. (7.7.3) results in an integral–differential equation containing dependence on only one wave variable ζ ∂v ∂ = ∂ζ ∂ζ

ω

dζ ζ (ζ, ζ ; τ, θ )

o

∂v . ∂ζ

(7.7.5)

In the case when the relaxation during loading is frozen, ζ (ζ, ζ ; τ, θ ) −−−→ 1, τ →∞

the Eq. (7.7.5) turns into the identity ⎛ ζ ⎞

∂v ∂ ⎝ ∂v dζ ⎠, = ∂ζ ∂ζ ∂ζ o

∂v ∂v = . ∂ζ ∂ζ

When the relaxation during the  is entirely completed during the long  loading loading, τ1 ζ (ζ, ζ ; τ, θ ) −−→ δ ζ − ζ , the Eq. (7.7.5) gives τ →0

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7 The Shock-Induced Planar Wave Propagation in Condensed Matter

 ∂v ∂ ∂v = τ , ∂ζ ∂ζ ∂ζ

∂ 2v ∂v = τ 2. ∂ζ ∂ζ

Integrating the last equation on the zero initial condition, we get the exponential plastic front rising without the elastic precursor. v=τ

  ∂v ζ , v(ζ ) = K exp . ∂ζ τ

(7.7.6)

The relaxation parameter τ and the parameter K are empirical. This shape of the plastic front is too simplified. Viscous-like flow within the shock is thought to be associated with the microscopic processes in the lattice. However, a real plastic front contains information not only about the viscous properties of a shock-loaded medium but also about structural processes on the mesoscale. These processes determine both the shape of the elastic–plastic waveform and its evolution during the front propagation. Basing on simple hydrodynamic models, it is not possible to predict shock wave risetimes in either fluids or solids over a range of shock loading conditions. It must be noted that the hydrodynamic model is valid only near local equilibrium as well as all models of continuum mechanics and therefore they cannot describe the highly non-equilibrium transition zone. Besides, at the final stage of relaxation, the values of the order ε/τ = Ct R /L can be rather large with small τ → 0 and the model Eq. (7.7.5) becomes incorrect. So, we can use the model Eq. (7.7.5) to describe the shock-induced waveform evolution during its propagation at the initial and transient stages without hydrodynamic mode. Our goal is to study transient highly non-equilibrium process whereas the final hydrodynamic stage of the process is described within continuum mechanics. Integrating (7.7.5) on the zero initial condition gives

ω v(ζ ; τ ) = o

dζ ζ (ζ, ζ ; τ, θ )

∂v . ∂ζ

(7.7.7)

According to the results obtained in non-equilibrium statistical mechanics, the integral operator with the temporal correlation function as an integral kernel is propagator or the operator of the system evolution. It means that when the operator with the parameters τ, θ acts on the function ∂v/∂ζ in the right-hand part of (7.7.7), it moves the waveform forward along the x-axis until its shape will correspond to the given τ, θ . This operator (propagator) defines the future waveform state with the mass velocity v(ζ ; τ, θ ) that differs from the velocity under the integral. The integral kernel describes the relaxation with typical relaxation time τ and retardation time θ. Because the Eq. (7.7.7) does not contain the dependence on spatial coordinate, this dependence can be included into the description of the waveform evolution during its propagation only through the parameters of the integral kernel. Each parameter τ must correspond to a point x, or ξ . A change in the parameters means the waveform evolution during its propagation from one point to another. Moreover, if to take the

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235

initial shock-induced waveform with finite acceleration ∂v/∂ζ and to substitute it under the integral in the right-hand of (7.7.7), then the integral operator (propagator) transforms it into a new waveform that will occur at a distance x from the shock surface corresponding to the parameters τ, θ . The Eq. (7.7.7) results in the nonlinear relationship between the normalized stress component J1 /ρ0 C V0 = (ζ ) and deformation e = (V0 /C)v(ζ ) via ∂v/∂ζ = (C/V0 )∂e/∂ζ

ω

(ζ ) = o

∂v dζ ζ (ζ, ζ ; τ, θ ) ω(ζ ) = ∂ζ





ζ, 1,

ζ < 1, ζ ≥ 1.

(7.7.8)

According to the relationship (7.7.8), the medium response to the shock-induced deformation with the rate ∂v/∂ζ can be both elastic and plastic depending on the initial waveform induced by shock loading and on dynamics of correlations governed by the integral kernel in the propagator. The relations (7.7.7) and (7.7.8) give

(ζ ) = v(ζ ). It means that the deformation with the rate (V0 /C)∂v(ζ )/∂ζ during the shock causes a stress ρ0 C V0 (ζ ; τ (ξ ), θ (ξ )) at the distance x = Lξ from the shock surface that, in turn, corresponds to the deformation (V0 /C)v(ζ ; τ (ξ ), θ (ξ )) determined for any point, ζ, ξ . Both the stress and the deformation appear inside the waveform that evolves during its propagation depending on loading conditions and initial medium properties. The shape of the waveform is determined by the initial loading pulse and its relaxation after the loading. For the shock of moderate intensity with t R  tr , τ  1, the state of the solid becomes unstable due to the broken correlations and motions on the mesoscale but final hydrodynamic stage of relaxation with parameter τ  1 is not achieved. The irreversible processes on the mesoscale result in various type inhomogeneities in the solid material. As the impact intensity increases, the material becomes less solid and more unstable. As experimental data show [48], the parameter τ decreases with the growth of the shock velocity and the solid medium response to high-rate shock can turn out close to hydrodynamic. This is possible to explain as follows. The initial long-range spatiotemporal correlations in solid are broken by the shock; the medium state right behind the shock is characterized by a finite value τ . Due to the relaxation, the medium state evolves during the wave propagation either in the direction to the hydrodynamic stage or to the solid state with new internal structure induced by the shock loading. The final state will depend on the whole complex of initial and loading conditions that govern the state evolution. So, the formulation of the problem of the shock-induced wave propagation in condensed matter is reduces to the integral-differential Eq. (7.7.8) describing the whole relaxation process during the wave propagation while the adopted conditions on the shock intensity V0 /C  1 and the scales separation ε/τ = Ct R /L  1 are

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7 The Shock-Induced Planar Wave Propagation in Condensed Matter

met. It must be noted that in the hydrodynamic limit at the final stage of the process, the last condition is not met. The diffusive transport processes are slow compared to the sound speed C0 . In the limit C0 → ∞, the spatiotemporal coordinates in the wave variable ζ = (t − x/C)t R are separated, ζ = t/t R , ξ = x/L. In the case, the momentum transport mechanism losses the wave character. Then the nonlocal effects becomes more important than the memory ones, τε = Ct R /L  1 . In order to close the formulation, it is necessary to determine the integral kernel ζ (ζ, ζ ; τ, θ ) using the modeling developed in Chap. 5 and to describe the evolution of the model relaxation parameters during the wave propagation using the Speed Gradient principle developed in the control theory and presented in Chap. 6. In the next section we first obtain a solution to the problem, compare it to experimental data and discuss the results.

7.8 Self-similar Quasi-Stationary Solution to the Problem Rewrite the integral-differential Eq. (7.7.8) with the model form of the integral kernel (5.9.1) depending on the variable ζ and the relaxation parameters τ, θ determined in Sect. 7.7

ω v= o



 2   π ζ − ζ − θ ∂v ζ, , ω(ζ ) = dζ exp − 1, τ2 ∂ζ

ζ < 1, ζ ≥ 1.

(7.8.1)

Iterative methods for solving such equations are developed in the theory of the special type operator sets described in Chap. 5. According to the results of nonequilibrium statistical mechanics, the integral operator in (7.8.1) is the evolution operator and its parameters evolve. We used the developed iterative procedure to describe the temporal evolution in the direction to final state. In the case of the waveform propagation, the iterative procedure should be deployed along the coordinate driving the evolution. It means that substitution of the waveform corresponding to a certain set of parameters τ0 , θ0 under the integral on the right results in a new waveform on the left corresponding to the parameters τ, θ in the integral kernel. The Eq. (7.8.1) is derived under the condition of the variable separation into fast temporal variable ζ and slow spatial variable ξ . The variable ζ defines the waveform induced by shock and the variable ξ governs the waveform evolution during its propagation in the medium. The dependence on the slow variable determining the distance ξ = x/L traveled by the wave after the shock remains in the Eq. (7.8.1) only through the model relaxation parameters τ (ξ ), θ (ξ ). Due to the integral kernel depending on the parameters τ (ξ ), θ (ξ ), the velocity on the left side of (7.8.1) v(ζ ; τ, θ ) differs from that under the integral on the right v(ζ ; τ0 , θ0 ) where the parameters τ0 , θ0 correspond to the initial waveform. It means that due to the relaxation parameters, the spatial nonlocality remains in (7.8.1) where the stress J1 = ρ0 Cv(ζ ; τ, θ ) and the plastic deformation are connected in different spatial points. This is a parametric

7.8 Self-similar Quasi-Stationary Solution to the Problem

237

spatial nonlocality. As to the memory effects, the stress is defined by the history of the strain-rate during the shock loading and the stress evolution is determined by the history of spatial correlations in the relaxation kernel. A fundamental difference from the approaches based on continuum mechanics where stress and strain are connected at the same spatiotemporal point, as in the case of an elastic wave or in a viscous flow, is that the plastic deformation is a process delayed in time and not localized in space. So, for elastic–plastic waves where the plastic deformation always retards from the elastic precursor, the conventional treatment of the shock-induced processes and of the plastic deformation concept should be revised. The integral operator in (7.8.1) describes the waveform evolution during its propagation in the medium. In order to get a solution to the integral-differential Eq. (7.8.1), an initial waveform is required. However, the initial profile of the mass velocity induced in the process of collision is unknown in the general case. Experimentally, it can be measured only at a certain distance from the surface of the collision. Most often the initial profile is taken rectangular assuming that the maximum pulse amplitude is reached instantly. Currently, it was established [50, 51] that the speed and time of the momentum transmission to a medium plays an important role in dynamic processes. First, it is necessary to understand what is the initial waveform induced by a shock. It is principally impossible to measure the initial waveform during the shock itself. In experiments, the waveform is recorded at a distance from the shock surface. By using the Eq. (7.8.1), we cannot describe a change of the medium state during the shock loading because the waveform changes only through the parameters τ (ξ ), θ (ξ ) which are constant on the waveform duration. Hence, we also do not know the medium state right after the momentum transmission during the shock. The values of the parameters τ (ξ ), θ (ξ ) near the shocked surface depend on the duration, intensity of the shock, and on the dynamic medium properties. In order to introduce the time and the rate of the momentum transmission to the medium, we assume that the impact is characterized by a certain averaged force per unit surface that during the loading time induces a constant acceleration, V0 /t R , or in the normalized form, ∂v/∂ζ = 1. Then we take the initial leading front as a straight line, v(ζ ) = ζ , that achieves its maximum at v(1) = 1.

Then, substituting the initial wave front ∂v/∂ζ = 1 under the integral in (7.8.1), we obtain an explicit self-similar solution to the Eq. (7.8.1) for the waveform in any point along its propagation [48]. In the case ω(ζ ) = ζ, ζ < 1, we get

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7 The Shock-Induced Planar Wave Propagation in Condensed Matter

v(ζ ; τ, θ ) =

√  √ π (ζ − θ ) πθ τ + erf erf , ζ ≤ 1. 2 τ τ

(7.8.2)

The solution (7.8.2) describes the waveform evolution during the loading. At the initial stage of the process t ≤ t R  tr , the parameter τ = tr /t R  1 is large. Taking into account ∂v/∂ζ = 1, in the limit τ → ∞ we get the initial wave front ζ = v(ζ ) without relaxation. √  τ 2 π ζ −θ θ

(ζ ; τ, θ ) −−−→ √ + = ζ = v. τ →∞ 2 τ τ π It means that at small times during the loading, the strain e = (V0 /C)v(ζ ) and the stress component J1 = ρ0 C V0 v(ζ ) are linearly connected in the same spatiotemporal point as in the elastic case. The relaxation has not begun and considered frozen. The smaller duration of the shock, the closer is the medium response of solids to the elastic one. At large τ , the medium response is always elastic. During long loading tr  t ≤ t R with small τ , the relaxation generates a flow of viscous liquid. In the limit τ → 0, the relaxation is entirely completed.

(ζ ; τ, θ ) −−→ τ (∂v/∂ζ ) → 0. τ →0

However, as we have shown above, the hydrodynamic stage is not achieved under the considered conditions. After the short-duration loading t R ≤ t  tr , the solution to the Eq. (7.8.1) is different √  √ π (ζ − θ ) π (1 − ζ + θ ) τ v(ζ ; τ, θ ) = (7.8.3) erf , ζ > 1. + erf 2 τ τ It describes the relaxation as post-shock effect that accompanies the wave propagation. At the initial stage of the post-shock relaxation, the waveform is close to the shock-induced form with finite τ that is rather large for the shocked solids and much less for liquids. The post-shock effect can be considered as an inertial effect of a dense medium. In the limit τ → ∞, the waveform preserves its amplitude.

(ζ ; τ, θ ) −−−→ 1. τ →∞

At large times t R  tr  t, after the relaxation is entirely completed both in space and time during the wave propagation, the post-shock effects disappear

(ζ ; τ, θ ) −−→ 0. τ →0

However, we must remember that the solution (7.8.3) does not describe closeto—equilibrium processes.

7.9 The Relaxation Model of the Shock-Induced Waveforms …

239

In the intermediate situations between the two limits for finite values of the parameters τ, θ , the obtained solution describes the medium response to shock loading including the relaxation of the elastic precursor and the post-shock amplitude recovery. The obtained solution (7.8.2)–(7.8.3) is quasi-stationary as far as the described waveform can evolve and propagate only through the model parameters τ (ξ ), θ (ξ ). However, such a process principally differs from quasi-stationary processes in the framework of continuum mechanics because the described waveform can evolve within transient zone.

7.9 The Relaxation Model of the Shock-Induced Waveforms During Propagation The obtained solution allows us to suggest that the impulse entering the target material during the impact and breaking the correlations forms a wave packet that immediately begins to propagate over the target. The impact transforms the initial elastic solid state with the parameters τ (0) → ∞, θ (0) → 0 into a state with the mesoscale structure characterized by finite values of the parameters τ (ξ > 0) and θ (ξ > 0). The wave packet propagates along the target and simultaneously relaxes. First the stress falls forming the so-called elastic precursor that rapidly relaxes. Its leading edge propagates at the speed of longitudinal sound but each following point gradually lags from the leading edge. The relaxation of the elastic precursor continues until its amplitude achieves the value of the dynamic elastic limit that can differ from its static value. Then the transfer of the impulse ends and if its amount is not enough for the occurrence of plastic flow, the state of the material begins to recover, the mesoscopic movements slow down, and the stress gradually reaches its initial value upon impact. As a result, the so-called plastic front is formed behind the precursor. The formed waveform can transfer mass over mesoscopic distances. This process is much slower than the relaxation of the elastic precursor and therefore this part of the waveform lags from it considerably. The time interval during which the plastic front rises is called risetime [17, 29]. One of the main questions posed by researchers of shock-induced processes was what material properties, internal structural effects, or loading conditions are responsible for the observed risetimes. Here we can say that these factors collectively contribute to the stress relaxation and its parameters. Mass transfer is an irreversible process by which a part of the mesoscopic structures behind the wave remains frozen in the material which changes its macroscopic properties. The new defective structure of the material absorbs a part of the kinetic energy transferred during the shock. As a result, the state after the passage of the wave front differs from the initial one and the amplitude of the stress in the wave does not reach the value of the shock stress [17, 48, 57]. The propagation speed of the plastic front is less than one of the elastic precursor. In the framework of conventional models [62] when only the shear relaxation forms the waveform, the propagation speed of the plastic front is equal to the volumetric sound speed.

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7 The Shock-Induced Planar Wave Propagation in Condensed Matter

Experimenters have long wondered whether the state produced immediately after shock transition is equilibrium [29]. Now thermodynamics allows us to answer the question. The shock-induced solid state is highly non-equilibrium but slow processes in weakly disturbed solid can be described by models of continuum mechanics. The shock loading make the state of the material less ordered, shifted to equilibrium, transient between solid and liquid. This state is not stable due to the relaxation and begins evolve either to equilibrium for very strong shock or back to solid state for the shock of moderate intensity. This issue requires research within the framework of the control theory of adaptive systems (Chap. 6). Now, see if the obtained solution describes the observed waveform behavior. Figure 7.4 shows the dependence of the waveform, given by the obtained solution (7.8.2)–(7.8.3), on the parameter θ which by its definition corresponds to the risetime. The leading edge of the elastic precursor before ζ = 1 is described by the solution (7.8.2). During the shock loading, the stress increases linearly until the transmission of the momentum to the medium is finished. At this time, the amplitude of the elastic precursor achieves its maximal value corresponding to the dynamic elastic limit that can significantly exceed the quasi-static limit. Due to the relaxation, it is decreasing with growing the delay parameter θ (ξ ) that defines the retardation of the restored part of the waveform from the elastic precursor. The plastic front starts to rise at ζ > 1 according to the solution (7.8.3). We can see (Fig. 7.5) that the amplitude of the elastic precursor relaxes with the decrease in the parameter τ = tr /t R , that is, in the decrease of the relaxation time tr and the growth of the loading duration t R . Elastic precursor stands out when τ is rather large and θ is not very large. We know that τ is large at the initial stage of the relaxation not far from the elastic limit and τ is small at the final stage of the relaxation when the medium response to the shock is close to hydrodynamic flow.

Fig. 7.4 The wavefronts at the relaxation parameters τ = 30 and θ = 10, 20, 30, 40 (from top to bottom)

7.9 The Relaxation Model of the Shock-Induced Waveforms …

241

Fig. 7.5 The wavefronts at the relaxation parameters θ = 15 and τ = 1, 3, 10, 30 (from bottom to top)

It must be noted that in all limiting cases described by the classical continuous models, the delay parameter θ does not play any role. By definition, θ defines the lag of the maximum  of the plastic front from the elastic precursor propagating at the velocity C = C02 + 43 ρG0 . We know that plastic front moves at the bulk sound speed C0 < C if the shear relaxation is completed. Then, we have the relationship ξ = C0 resulting in the linear dependence of the plastic front delay on the ξ /C+θ t R /L distance traveled by the wave  L C0 θ (ξ ) = ξ . 1− C0 t R C

(7.9.1)

However, when we do not separate shear and bulk relaxation, the speed C0 can differ from the bulk speed. It is important to indicate that the amplitude of the plastic front with large τ achieves the shock velocity but for small τ it decreases. Experimental data show that for not very high shock velocities, the wave amplitude is restored at ζ = 1 + θ . Further, the part of the initial waveform remains unchanged until the unloading wave arrives. At the intermediate stage of the relaxation, at some distance from the shock surface traveled by the wave during its propagation along the medium, the so-called twowave structure of the leading front is formed. For the shock loading, the risetime of the plastic front much exceeds the loading time (t R + tm )/t R = 1 + θ  1. In the case, the solution (7.8.3) governing the relaxation after the shock loading can be

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7 The Shock-Induced Planar Wave Propagation in Condensed Matter

approximated by the function 

 π (ζ − θ )2 v(ζ ; τ, θ ) −−−→ exp − . 1/θ →0 τ2

(7.9.2)

The same result is obtained with the traditional choice of the initial rectangular ∂v = δ(ζ ) at the forefront after its substitution in waveform with infinite acceleration ∂ζ (7.8.1). It means that the initial stage during shock loading described by the solution (7.8.2) is omitted at the intermediate stage of the process. The approximation (7.9.2) is much simpler to analyze and more visual to interpret the shock-induced waveforms. The maximum amplitude of the plastic front v(ζ ; τ, θ ) is achieved on the waveform plateau, ζ = θ , when the relaxation is completed. It is always equal to the shock velocity v(ζ = θ; τ, θ ) = 1 whereas the solution (7.8.3) √ π τ v(ζ = θ; τ, θ ) = vmax (τ ) = erf −−−→ 1 2 τ τ →∞

(7.9.3)

approaches to 1 only at large τ . Figure 7.6 gives a range of the parameters τ for which we can use the approximation (7.9.2). In the approximation (7.9.2), the amplitude of the elastic precursor is given by   √ 2  π Ae = v(ζ = 0; τ, θ ) = exp − θ . τ

Fig. 7.6 Dependence vmax (τ )

(7.9.4)

7.10 Decryption of the Information Recorded in the Experimentally …

243

In the process of the wave propagation from the shock surface, as shown by experimental data [59], the elastic precursor relaxes. According to (7.9.4), the amplitude of the elastic precursor decreases with the growth of the delay parameter θ and the decrease of the parameter τ . After the relaxation of the elastic precursor, its amplitude reaches a constant value. According to the expression (7.9.4), this means that the parameters τ, θ are linearly interconnected. We have already shown that the risetime (7.9.1) is linear function of ξ = x/L. Then τ is also linearly grows with the distance τ (ξ ) = θ (ξ )(π/|ln Ae |)1/2 . It means that for shock loading of solid materials the retardation of the plastic front can be considered as an inertial effect or a partial preservation of the initial waveform (memory effect). According to the hypothesis of attenuation of correlations, the system is gradually forgets its initial state and the memory effects decay with time. However, as we have seen above, the correlations can recover over time when the obtained during the shock energy is not enough to provide the hydrodynamic stage of the relaxation. In the case, the relaxation can be considered non-monotone. It is very interesting that such different concepts as memory, inertia and relaxation are closely related.

7.10 Decryption of the Information Recorded in the Experimentally Observed Waveforms Experimental diagnostics of short-duration processes in solids is based on the registration of the shock-induced mass velocity profiles on the back of the target [10, 11]. The shape of the mass velocity profiles turned out to be very sensitive to the relaxation and structural characteristics of a deformable material such as the density and mobility of dislocations, viscosity, initial and shock-induced mesoscopic structural inhomogeneities of the medium. In order to study the shock-wave behavior of materials, the method of recording the velocity profile during wave propagation in real time carried out using laser interferometers [63–66]. The high temporal and spatial resolution of interferometric methods made it possible to identify many features of the shock-wave behavior of materials including phase and structural transformations, spallation processes, etc. [20–27, 63–66]. Consider, for example, the mass velocity profile induced by the shock at a speed of 318 m/s that is recorded on the back side of a 5 mm thick metal target (Fig. 7.7). In order to compare the profile of the leading front with the model one given by the solution (7.8.2)–(7.8.3), it is necessary to determine the parameters τ, θ defining the waveform shape. Simplified representation (7.9.2) of the waveform makes it easy to determine the parameters τ, θ from the experimentally obtained velocity profiles at different distances from the impact surface. Two points on the profile are of the most important. First point defines the amplitude of the elastic precursor Ae that is achieved for the loading time t R . Second point

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7 The Shock-Induced Planar Wave Propagation in Condensed Matter

Fig. 7.7 Experimental mass velocity profile obtained at the impactor speed 318 m/s for the 5 mm target

U ms

time, ns

defines the amplitude of the plastic front vmax that lags behind the elastic precursor for a period tm . For our model of the leading front the period is the risetime. However, experimenters call the ratio of the wave amplitude to the maximum acceleration at the steady front the word risetime. From our point of view, this definition is not correct because each point of the front moves at different speed due to the wave packet spreading. By measuring the retardation of the plastic front tm and dividing it by the θ = tm /t R . By duration of the elastic precursor t R , we obtain the delay parameter  measuring the amplitude of the elastic precursor Ae = exp −π θ 2 /τ 2 and dividing it by the impact velocity, one can easily calculate the relaxation parameter τ from formula (7.6.4). Each pair of the parameters τ, θ corresponds to the velocity profile at a distance traveled by the wave as it propagates along the material. This means that the experimental measurement of the velocity profile induced by an impact on a solid material allows us to determine the change in the state of the material at different distances from the surface of the impact.

7.11 Comparison of the Model Waveforms with Experimental Data Now we can compare the model solution to the Eq. (7.8.1) with the experimental velocity profiles. Having the experimental values of the parameters τ, θ , we can substitute them into the solution (7.8.2)–(7.8.3) and get the velocity profile that can be compared to the experimental one. In Fig. 7.8 the comparison of the experimental front with the solution (7.8.2)–(7.8.3) is presented. In the middle part of the plastic wave front in Fig. 7.8 we can see a deviation from experimental curve on account of the model form of the correlation function. The experimental profile is steeper than the model one. Despite the deviation, we use Gaussian model of the correlation function because it allows us to obtain an explicit

7.11 Comparison of the Model Waveforms with Experimental Data

245

Fig. 7.8 The model profile (continuous line) and experimental dotted profile recorded on the back of 6 mm aluminum target; the impactor velocity is 190 m/s

solution and to analyze all the observed features of the evolution. If neces  waveform sary, it is always possible to modify the function Mζ ζ −ζτ −θ for numerical solution. Nevertheless, the behavior of the obtained solution is qualitatively corresponds to the known data on the relaxation process during elastic–plastic wave propagation. It must be noticed that the relaxation begins and finishes inside the waveform but we consider its parameters constant throughout the entire waveform. Therefore, the obtained parameters of the correlation model do not describe the real dynamics of correlations inside the waveform but define only the result of the elastic precursor relaxation during impact and the end of the material recovery after the shock. During the recovery, first the front becomes steeper on account for the smaller parameter τ then with the growing parameter τ , the front reaches a plateau (see Fig. 7.4 and 7.5). In order to describe all the details of the processes inside the waveform, it is necessary to take into account the dependences τ (ζ, ξ ), θ (ζ, ξ ) and to describe their evolution inside the waveform by methods presented in Chap. 6. This point will be discussed in the next chapter. If we have several velocity profiles recorded at different target thicknesses under the condition that the shock velocity was the same for each profile, we can trace the evolution of the waveforms during the wave propagation through the medium. As shown in [48], all experimental values of the parameters τ, θ obtained in this way and plotted on the phase plane {τ, θ} fell on two linear paths for two series of experiments on the shock loading at the impact velocity in between the interval 200 ÷ 400 m/s and different target thicknesses 3 ÷ 15 mm for two materials: 30XH4M

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7 The Shock-Induced Planar Wave Propagation in Condensed Matter

steel and D16 aluminum alloy. Both trajectories of the waveform evolution during its propagation are straight lines going under different angles to axis depending on the material elastic limit. With increasing thickness of the target, the phase point moves away from the origin, and with increasing velocity of the impact, it approaches the origin. We can see that during the wave propagation each phase point moves along the trajectory away from the coordinate origin. The experimental data confirm the −9 s = 0, 0084  1 under which the model profiles condition τε = CtLR ≈ 6000 m/s·14·10 10−2 m were obtained. All experimental profiles corresponding to the phase  points have almost steady amplitude of the elastic precursor Ae = exp −π θ 2 /τ 2 . The last condition implies only straight trajectories θ = kτ with coefficients k connected to the dynamic elastic limit. The evolution trajectories for aluminum alloy D-16 (θ = 0.84τ ) and steel 30XH4M (θ = 0.67τ ) are different. The rate of the plastic front retardation can be approximately evaluated from experimental data for all target thicknesses x as follows: C pl = x/((x/C) + tm ) ≈ 5000 m/s ≈ C0 . It means that all experimental data correspond to the quasi-stationary regime of the wave propagation when the plastic front moves at the constant velocity and the amplitude of the elastic precursor has a constant value. As phase points run away from the origin along its trajectories, the material after the loading tends to return to its initial solid state. It is interesting that straight trajectories correspond to the quasi-stationary wave propagation in rather wide range of shock velocities and distanced traveled by waves. It means that for each material within the range of conditions the value of the elastic precursor amplitude has already been established. During the wave propagation, the risetime grows with θ and the velocity profiles spread becoming less steep. This wave propagation cannot be called steady [55, 56] because waveforms change with the distance. We cannot say how long this mode of propagation will last and what a final state will be achieved. Problem of the waveforms evolution will be considered in the following chapters. Now, we can conclude that the proposed integral model of the shock-induced waveform provides the uniform description of both the relaxation of the elastic precursor and the formation of the plastic front during the wave propagation in condensed matter. All observed features of the waveforms propagation are described by dynamics of spatiotemporal correlations whose typical sizes are included in the waveform model as parameters. However, as shown in Chap. 5, these parameters are unknown functionals of the entire macroscopic history of the process, without the definition of which it is impossible to close the mathematical formulation of the problem. According to the obtained approximate solution of the problem, the evolution of the waveform during its propagation is completely described by the evolution of the model parameters. Therefore, it is impossible to describe the waveform propagation without describing the evolution of the parameters of the dynamic structure of the medium. In the next chapter, we will close the problem statement in a self-consistent way and propose a

References

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model for the evolution of the parameters τ, θ based on thermodynamic principles and methods of control theory of adaptive systems.

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25. Furnish MD, Trott WM, Mason J, Podsednik J, Reinhart WD, Hall C (2003) Assessing mesoscale material response via high resolution line-imaging VISAR. In: Furnish MD, Gupta YM, Forbes JW (eds) Shock compression of condensed matter. AIP Conference Proceedings, vol 706. Melville, N.Y, pp 1159–1163 26. Asay JR, Dwivedi SK, Gupta YM (2005) Mesoscale predictions of shock states in aluminium. Bull Am Phys Soc (Shock Compress) 50(5):122 27. Koskelo AC, Greenfield SR, Raisley DL, McClellan KJ, Byler DD, Dickerson RM, Luo SN, Swift DC, Tonk DL, Peralta PD (2008) Dynamics of the onset of damage in metals under shock loading. In: Proceedings of the AIP on Shock compression of condensed matter 2007, vol 955, pp 557–560 28. S. Case YH (2007) Discrete element simulation of shock wave propagation in polycrystalline copper. J. Mech Phys Solids 55:589–614 29. Asay JR, Chhabildas LC (2003) Paradigms and challenges. In: Horie Y, Davison L, Thadhani NN (eds) Shock wave research. high-pressure compression of solids VI: old paradigms and new challenges. Springer, pp 57–108 30. Lee J (2003) The universal role of turbulence in the propagation of strong shocks and detonation waves. In: Horie Y, Davison L, Thadhani NN (eds) High-pressure compression of solids VI: old paradigms and new challenges. Springer, pp 121–144 31. Mescheryakov YI (2003) Meso-macro energy exchange in shock deformed and fractured solids. In: Horie Y, Davison L, Thadhani NN (eds) High-pressure compression of solids VI: old paradigms and new challenges. Springer, pp 169–212 32. Panin VE, Egorushkin VE, Panon AV (2006) Physical mesomechanics of a deformed solid as a multilevel system. 1. Physical fundamentals of the multilevel approach. Phys Mesomech 9(3–4):9–20 33. Panin SV, Bykov AV, Grenke VV, Shakirov IV, Yussif SAK (2010) multiscale monitoring of localized plastic strain evolution stages in notched aluminum AA2024 alloy tension specimens by acoustic emission and television-optical techniques. Phys Mesomech 13(3–4):203–211 34. Makarov PV (2004) On the hierarchical nature of deformation and fracture of solids. Phys Mesomech 7(4):25–34 35. Physical Mesomechanics and computer-aided design of materials (1995). Panin VE (ed) Novosibirsk, Nauka, Siberian Publ. RAN 36. Panin VE (1998) Foundations of physical mesomechanics. Phys Mesomech 1:5–20 37. Taylor JW, Rice MH (1963) Elastic-plastic properties of iron. J Appl Phys 34:364 38. Taylor JW (1965) Dislocation dynamics and dynamic yielding. J Appl Phys 36(10):3146–3155 39. Khantuleva TA, Mescheryakov YuI (1999) Nonlocal theory of the high-strain-rate processes in a structured media. Int J Solids Struct 36:3105–3129 40. Khantuleva TA, Mescheryakov YI (1999) Kinetics and nonlocal hydrodynamics of mesostructure formation in dynamically deformed media. Phys Mesomech 2(5):5–16 41. Khantuleva TA (2000) Non-local theory of high-rate processes in structured media. In: Furnish MD, Chhabildas LD, Hixon RS (eds) CP505, shock compression of condensed matter-1999, APS 1-56396-923-8/00, pp 371–374 42. Khantuleva TA (2000) Non-local theory of high-rate straining followed by structure formations. J Phys. 4 France, vol 10. EDP Sciences, Les Ulis, pp 485–490 43. Khantuleva TA (2003) The shock wave as a nonequilibrium transport process. In: Horie Y, Davison L, Thadhani NN (eds) High-pressure compression of solids VI: old paradigms and new challenges. Springer, pp 215–254 44. Khantuleva TA (2005) Self-organization at the mesolevel at high-rate deformation of condensed media. Khim Fiz 24(11):36–47 45. Khantuleva TA, Serebryanskaya NA (2009) Relaxation of waves propagating in condensed media. Izvestiya VUZov, Fizika 52:165–171 46. Khantuleva TA, Meshcheryakov YI, Divakov AK (2010) Deciphering of experimental velocity profiles in non-stationary wave in framework of non-local model. In: Shock waves in condensed matter-2010, pp 275–278

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47. Khantuleva TA, Meshcheryakov YI (2010) Non-local model for shock-induced structureunstable state. In: Proceedings of the 1-st annual international conference proceedings of the royal society “From Static to Dynamic”, London. www.imperial.ac.uk/shockphysics/annual meeting 48. Meshcheryakov YI, Khantuleva TA (2015) Nonequilibrium processes in condensed media. Part 1. Experimental studies in light of nonlocal transport theory. Phys Mesomech 18(3):228–243 49. Mescheryakov YI, Khantuleva TA (2015) Nonlocal mechanics of nonequilibrium shock-wave processes. Mater Phys Mech 22(2):136–156 50. Morozov VA (1992) Specifics of loading, deformation and fracture of materials in submicrosecond and nanosecond ranges of pulse duration. In: Models of continuum mechanics. Proceedings of 14 International school on models of continuum mechanics, vol 1. Ghukovskii, p 324 51. Baizakov OD, Morozov VA, Sud’enkov YV (1993) Model of decay of elastic wave taking into account relaxation in surface region of shock loading. In: Gas dynamics and thermo-exchange. Dynamics of uniform and nonuniform media, vol 9. Leningrad, Publ. Leningrad University 52. Nesterenko VF (2001) Dynamics of heterogeneous materials. 2002. Springer, N.Y 53. Ravichandran G, Rosakis AJ, Hodovany J, Rosakis P (2003) On the convention of plastic work into heat during high-strain-rate deformation. In: Furnish MD, Thadhani NN, Horie Y-Y (eds) AIP Conference Proceedings of the Shock compression of condensed matter-2001, vol 620. Melville. N.Y, pp 557–562 54. Bever MB, Holt DL, Tichener AL (1973) Prog Mater Sci 17:5 55. Prieto FE, Renero C (1973) Steady shock profile in solids. J Appl Phys 44(9):4013–4401 56. Romero C, Prieto FE (1973) Steady shock profile in solids. J Appl Phys 44:4013–4016 57. Meyers MA (1977) A model for elastic precursor waves in the shock loading of polycrystalline metals. Mater Sci Eng 30(2):99–111 58. Sud’enkov YV, Pavlishin AI (2003) Nanosecond pressure pulses propagating at anomalously high velocities in metal foils. Tech Phys Lett 29(6):491–493 59. Haken H (1993) Advanced synergetics: instability hierarchies of self-organizing systems and devices. Springer, New York 60. Mikhailov AS, Loskutov AY (1991) Foundations of Synergetics II. Chaos and Noise, 2nd revised and enlarged edition, Springer Series in Synergetics. Springer, Berlin, Heidelberg 1996. ISBN 0-387-53448-2 61. Prigogine I, Nicolis G (1977) Self-organization. In: Non-equilibrium systems. Wiley 62. Razoryenov SV, Utkin AV, Fortov BE, Kanel GI (1996) Shock-wave phenomena in condensed matter. Yanus-K, Moscow 63. Barker LM, Hollenbuch RE (1965) Interferometric technique for measuring the dynamic mechanical properties of materials. Rev Sci Instrum 36(11):1617–1621 64. Barker LM (1967) Fine structure of compressive and release wave shapes in aluminum measured by the velocity interferometer technique. In: Symposium on high dynamic pressure, Paris, pp 369–382 65. Barker LM, Hollenbuch RE (1972) Laser interferometer for measuring of any reflecting surfaces. J Appl Phys 43(6): 4669–4679 66. Asay JR, Barker LM (1974) Interferometric measurement of shock-induced particle velocity and spatial variations of particle velocity. J Appl Phys 45:2540

Chapter 8

Evolution of Waveforms During Propagation in Solids

Abstract In Chap. 7, we have obtained the explicit solution to the problem of the shock-induced waveforms propagation that allows us to describe the medium response both during the loading and post-shock effects. As experiments show [1], the model parameters τ, θ depending on the distance traveled by the wave after impact evolve during the waveform propagation along the material. In Sect. 7.10, we have already traced the evolution of the parameters by using their experimental values obtained for the waveforms recorded at various distances from the impact surface (see Fig. 7.11). It is found out that all experimental points obtained by using the waveforms measured at different traveled distances fall down on the straight lines going at different angles for each material. In this chapter, the integral model developed in Chap. 7 is applied to describe the shock-induced waveform evolution during its propagation inside the material through the evolution of the model parameters by methods developed in cybernetic physics and considered in Chap. 6. The goal function for the waveform evolution is determined by the maximum of the total entropy production that points out the direction of the system evolution in accordance with MEP [2]. The speed-gradient (SG) algorithm determines the fastest evolutionary paths to the goal. A comparison of the theoretical paths obtained in this way with the experimental results for the quasi-stationary regime of the waveform propagation shows good agreement between them. The parameters τ, θ used in the integral model of the waveform, as it was found out, determine two limiting states inside the waveform. The observed decrease in the parameter τ during the so-called elastic precursor relaxation makes the material state less solid while the parameter θ defines the time interval needed for the restoration (at least partial) of the solid material state when the so-called plastic front reaches the plateau of the compression pulse. It means that the parameters evolve inside the waveform whereas the waveform evolution at a rather large distance from the impact surface is adequately described by the integral model with the constant parameters over the waveform duration. The described structure evolution inside the finite-duration waveform is an example of a process in which, as a result of the self-organization of new mesoscopic structures, the generalized integral entropy production after the unloading front can become negative.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 T. A. Khantuleva, Mathematical Modeling of Shock-Wave Processes in Condensed Matter, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-981-19-2404-0_8

251

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8 Evolution of Waveforms During Propagation in Solids

Keywords Shock-induced waveform · Stress relaxation · Temporal evolution · Speed-gradient principle · Integral entropy production

8.1 Waveforms Evolution Within the Integral Model According to SG principle, we should choose a goal function and control parameters for the waveform evolution. Unlike MEP [2] based on the maximization of the information entropy, we must use the total entropy production defined by Zubarev [3] and described in Chap. 3. The waveform evolution is entirely described by the evolution of the model parameters τ, θ along the x-axis. In order to compare the theoretical paths with the experimental ones, we must use the waveforms resulting from the shock of two semi-spaces filled with the same medium because the parameters τ, θ define only the forefront shape of the real waveform. All the processes capable to produce the entropy (transport of mass, momentum, and energy) take place only inside the waveform traveling along the medium. According to the law of thermodynamics applied to the loading process [4], the entropy change d S during the loading process is determined by irreversible losses during the mutual conversion of the deformation work J d(1/ρ) done by the stress J and specific internal energy E T d S = d E + J d(1/ρ).

(8.1.1)

It is generally accepted that d E is due to reversible energy exchange whereas d S determines dissipative losses of mechanical energy. However, as shown in Chaps. 4 and 5, we cannot separate the reversible and irreversible parts of thermodynamic fluxes and forces in advance because they change during the loading process. Since we want to consider the waveform evolution during its propagation along the x-axis, instead of describing temporal evolution, we can proceed to evolution as the waveform moves away from the impact surface. The expression (8.1.1) can be rewritten in terms of derivatives with respect to x dS dE d T = +J dx dx dx

  1 . ρ

(8.1.2)

  Going to the variable ζ = t1R t − Cx , ξ = Lx (see Sect. 7.7) in (8.1.2), and Ld = − τε ∂ζ∂ + ∂ξ∂ , we obtain that the main contributions to the energy conversion dx are given by the processes inside the waveform since τ/ε  1. Therefore, we can neglect the derivative ∂/∂ξ and use the obtained solution in Chap. 7 (7.8.2)– (7.8.3). In the case of the waveform propagation along the x-axis, the change of the deformation work by the stress J = ρ0 C V0 is J (ζ ; τ (ξ ), θ (ξ ))d(1/ρ) = V02 (ζ ; τ (ξ ), θ (ξ ))dv(ζ ; τ (ξ ), θ (ξ )) where the stress and strain are not split into elastic and plastic parts

8.1 Waveforms Evolution Within the Integral Model

T

253

dS ∂v dE − = V02 (ζ ; τ (ξ ), θ (ξ )) . dζ dζ ∂ζ

(8.1.3)

The normalized stress (ζ ) = v(ζ ) according to (7.7.8) has the model form (7.8.1)  ω(ζ )  2   π ζ − ζ  − θ (ξ ) ∂v  (ζ ; τ (ξ ), θ (ξ )) = exp − dζ . τ 2 (ξ ) ∂ζ 

(8.1.4)

0

Since far from equilibrium it is hardly possible to separate the dissipative loss from reversible changes [4] during loading, it would be appropriate to generalize the concept of the entropy production for the shock-induced wave propagation in solid materials. Then, the generalized local entropy production at the time moment ζ inside the waveform at the distance ξ = x/L is defined by the deformation work as a result of which both reversible and irreversible changes in the medium state can occur σ (ζ ; τ (ξ ), θ (ξ )) = (ζ ; τ (ξ ), θ (ξ ))

∂v (ζ ; τ (ξ ), θ (ξ )), ∂ζ

(8.1.5)

where the stress is the thermodynamic flux and ∂v/∂ζ corresponds to the thermodynamic force. However, we know from non-equilibrium statistical mechanics [3] that the local entropy production cannot define the goal of thermodynamic evolution. Whereas local entropy production can fluctuate during a highly non-equilibrium process, we use the integral entropy produced inside the waveform at any point x defined by the model parameters τ (ξ ), θ (ξ ). Integrating the relationship (8.1.4) over the waveform duration p, we get p

p σ (ζ, ξ )dζ =

0

dζ (ζ ; τ (ξ ), θ (ξ )) 0

∂v . ∂ζ

(8.1.6)

Integrating the thermodynamic relationship (8.1.3) over the waveform duration p and substituting it in the expression (8.1.5), we obtain the relationship p T S − E =

σ (ζ, ξ )dζ,

V02

(8.1.7)

0

where the reversible change in the internal energy E and irreversible change in entropy T S on the left-hand side are combined to define the generalized integral entropy production on the right side. It should be noticed that their combination corresponds to the change in free energy behind the wavefront T S − E = − .

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8 Evolution of Waveforms During Propagation in Solids

We can show that the generalization of the integral entropy production does not violate the second law of thermodynamics. Really, in the elastic limit when (ζ ; τ (ξ ), θ (ξ )) −−−→ v(ζ ), the integrand in (8.1.6) turns into a full differential of τ →∞ elastic energy p

p σ (ζ, ξ )dζ −−−→ τ →∞

0

0

∂v v 2 p dζ v = . ∂ζ 2 0

(8.1.8)

In the case of impact of semi-spaces p → ∞, the wave amplitude and the local entropy production become constants with time, σ (ξ ) −−−→ 1/2. For the finiteτ →∞ duration waveform in the elastic limit, the process is reversible, no mass transport is done, and the generalized entropy production is 0. Elastic waves propagate along material without influence on its properties. We see that reversible processes do not contribute to the final generalized integral production when the process under consideration is over. In the hydrodynamic limit when in correspondence with the linear thermodynamics of irreversible transport (ζ ; τ (ξ ), θ (ξ )) −−→ τ ∂v/∂ζ , the τ →0

generalized entropy production coincides with its conventional definition. The plastic flow carries mass and irreversibly deforms the material 

p



p

σ (ζ, ξ )dζ −−→ τ τ →0

0

0



∂v dζ ∂ζ

2 ≥ 0.

(8.1.9)

In accordance with MEP [2] (see Chaps. 4 and 6), the total entropy production during temporal evolution strives to reach its maximum value admissible under the imposed constraints. For the semi-space target, the wave propagation can be considered without any constraints. Maximization of the total entropy production (8.1.6) is chosen as a control goal. The total entropy production at the distance x tends to maximum when ξ → ∞ ξ dξ 0



p



ξ

σ (ζ, ξ )dζ = 0

dξ 0



p 0

∂v dζ ∂ζ

 ω(ζ )  2   π ζ − ζ  − θ (ξ  ) ∂v  dζ exp − . τ 2 (ξ  ) ∂ζ  0

(8.1.10) In the integral model of the shock-induced wave propagation presented in Chap. 7, the deformation work is done only by the force applied during the impulse transfer t R upon the impact that corresponds to the integration with upper limits ω(ζ ) = 1, p = 1 in (8.1.10). The rest part of the forefront is the restoration of the solid material state after the shock due to internal structure energy. In Fig. 8.1, we see that only the parts of the waveforms to the left of the gray line contribute to the entropy production. During the waveform propagation, the contributions decrease with the elastic precursor relaxation.

8.2 Speed-Gradient Principle for the Waveforms Evolution

255

Fig. 8.1 Elastic precursor relaxation at τ = 14, θ = 5 (line 3), 12 (line 2), 16 (line 1)

In Chap. 5, we have presented a new approach to describe transport processes far from local equilibrium in which the interrelation between the effective sizes of spatial correlations and the sizes of dynamic internal structure has been established [5]. In the case of the wave processes where spatial and temporal variables are combined, we can choose the temporal characteristics of the waveform to describe the evolution during propagation along the x-axis. Since the model parameters τ, θ control the waveform shape, we can choose the parameters τ, θ as the control ones. So, we have determined the total generalized entropy production inside the waveform and chosen its maximization as a goal of thermodynamic evolution. The model parameters τ, θ can be used as the control ones. Now, we are ready to describe the shock-induced wave evolution as it travels along the x-axis using control theory methods.

8.2 Speed-Gradient Principle for the Waveforms Evolution Among control algorithms describing the way to reach the goal, we use the most physically grounded speed-gradient (SG) algorithm designed for the control problems of continuous-time systems [6, 7]. The SG algorithm [6–8] provides the fastest way to the goal. In accordance with the SG principle, among all possible motions in the system, those are realized for which the input variables change in proportion to the gradient of the time derivative of the goal function. As the goal function of the system evolution, in Chap. 6, we have chosen the maximum of the total entropy produced in the

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8 Evolution of Waveforms During Propagation in Solids

system during the non-equilibrium process under the given loading conditions. In the general case, the rate of the parameters change included in the mathematical model of the system plays the role of control parameters. Since the model parameters in the proposed integral model are related to the duration of temporal correlations, such a choice of the goal function and control parameters is consistent with the hypothesis of N. N. Bogolyubov on the decay of space-time correlations during the relaxation process of the system [9]. The SG algorithm is written in two different forms: differential for wave-type processes and finite for slow diffusion-type processes. Since integral quantities such as total entropy production change more slowly than the waveform shape during its propagation, its evolution could be considered a slow process that can be described by equations of the SG algorithm in finite form. For the goal functional of the integral type (6.2.5), as it was noted in Sect. 6.2, the finite form of the SG algorithm coincides with the steepest descent algorithm. For the shock-induced waveform evolution, we have chosen the functional of the total entropy production (8.1.10) of the same type. According to the SG algorithm, the goal function should be differentiated by time. In our case, since evolution proceeds with distance ξ = x/L (for semi-space, the value L is conventional), we differentiate the goal functional by ξ . Then, the SG algorithm in a finite form is applied to the functional of the integral entropy production inside the waveform (8.1.6) instead of (8.1.10). So, we have the goal functional Q ξ Q(ξ ; τ, θ ) =

dξ 0



1 0

∂v dζ ∂ζ

1 0



 2  π ζ − ζ  − θ (ξ  ) ∂v dζ exp − ; 2  τ (ξ ) ∂ζ  

(8.2.1)

the rate of its change with the distance ξ dQ = dξ

1 0

∂v dζ ∂ζ

1 0



2    − θ (ξ ) π ζ − ζ ∂v dζ  exp − ; τ 2 (ξ ) ∂ζ 

(8.2.2)

the set of nonlinear differential equations corresponding to the finite form of the SG algorithm and determining the dependence of the parameters τ (ξ ), θ (ξ ) on the distance traveled by the wave from the impact surface dτ ∂ = −gτ dξ ∂τ



   ∂ dQ dθ dQ , . = −gθ dξ dξ ∂θ dξ

(8.2.3)

We take into account that the acceleration in the right sides (8.2.1)–(8.2.2) is ∂v = 1, ζ ∈ [0, 1] constant during the loading interval ∂ζ

8.2 Speed-Gradient Principle for the Waveforms Evolution

∂ ∂τ ∂ ∂θ



dQ dξ



dQ dξ



1 =

1 dζ

0



1 =

1 dζ

0

1 = 0

0

0

  2  π ζ − ζ  − θ (ξ ) ∂ dζ exp − ; ∂τ τ 2 (ξ ) 

∂ dζ exp = ∂θ 

1

1 dζ

0

0

257

(8.2.4)

  2  π ζ − ζ  − θ (ξ ) ∂ dζ exp − = ∂ζ  τ 2 (ξ ) 

  π (ζ − 1 − θ (ξ ))2 π (ζ − θ (ξ ))2 dζ exp − − exp − . τ 2 (ξ ) τ 2 (ξ ) (8.2.5)

The determination of the functions τ (ξ ), θ (ξ ) as solutions to the set of Eq. (8.2.3) requires setting the initial conditions. However, we do not know the initial values of the parameters resulting from the impact and defining the initial state of the given material. The waveform evolution begins just after the shock that breaks spatiotemporal correlations and considerably changes the material state. The gain parameters gτ , gθ characterize the rate of the waveform evolution. These quantities related to the slow evolution of the wave shape during its propagation have to depend on the properties of a specific material. Therefore, in a certain range of conditions, we can consider them empirical constants related to the stability of the material structure. The problem is how to determine these constants from the experiment. If one can record three waveforms in a non-stationary region separated by rather small time intervals, then one can find the evolution constants and not only fully reconstruct the entire evolution of the system in the past but also predict the future behavior of the system. In this case, we do not need the real initial conditions for the Eq. (8.2.3); it is sufficient to draw an integral curve through a given experimental point on the phase plane of the control parameters. In principle, when we have the initial conditions for the set of differential Eq. (8.2.3), the problem statement is complete. Taking the experimentally measured waveform at a certain distance x, we can find the corresponding values of the parameters τ (ξ ), θ (ξ ) related to the distance x and calculate the integrals (8.2.4)–(8.2.5). Solving the ordinary differential Eq. (8.2.3), one can draw the integral curves through the experimental values of the parameters τ (ξ ), θ (ξ ) and move along the curves either to higher or lower values of the distance x. In the next point x1 , new values of the parameters τ (ξ1 ), θ (ξ1 ) change the waveforms and therefore the integrals (8.2.4)–(8.2.5) change too. As a result, the integral curves change and we get the next point x2 with τ (ξ2 ), θ (ξ2 ), and the self-consistent procedure is repeated again and so on. The emergence of feedbacks when changes in the waveforms and the parameters related to the wave structures on the mesoscale are determined through each other can give rise to new dynamic structures and switch the system from one operating mode to another. The feedbacks make the numerical description of the waveform evolution

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8 Evolution of Waveforms During Propagation in Solids

very sensitive to the initial data and especially to the choice of the gain parameters in the SG algorithm. It can be expected that the values of the gain parameters in the SG algorithm are related to the properties of a particular material. However, a careful study of both the physical meaning of these parameters and the development of appropriate numerical methods requires significant efforts both on the part of experimenters and highly qualified specialists in computational mathematics. Various histories of acceleration during the loading lead to various scenarios of the waveform evolution in the space of the mesoscopic parameters which as a result can lead to various final states of the material. Besides, serious problems can arise due to the nonlinearity of the equations and corresponding multi-variance of the evolution. So, the proposed approach raises a whole series of new unsolved problems in various fields, the solution of which goes far beyond the scope of this book. However, it should be borne in mind that the “manual” control of the mesoscopic parameters without using the governing equations based on the SG algorithm leads to a rapid collapse of the numerical procedure. In turn, the use of internal control via feedbacks in the form of the SG principle leads to its stabilization.

8.3 The Waveform Evolution During Quasi-Stationary Wave Propagation In Sect. 7.11, we have considered the quasi-stationary regime of the shock-induced waveform propagation. An experimental study on shock loading of solid materials [10–14] allows us to analyze numerous waveforms recorded on targets of various thicknesses in a rather wide range of the shock velocities. However, we must take into account that in the experiments each shock and its results are unique. It is physically impossible to repeat a shock at exactly the same speed. We can analyze only the waveforms induced by shock at approximately the same speed. On the other hand, the mass velocity is recorded by an interferometer using a laser, the spot of which belongs to the meso-2 scale. In the vicinity of each point of the target backside, the measured mass velocity will be different. This averaging scale which is determined by the size of the laser beam spot smoothes the velocities on the meso-1 scale but such a scale is insufficient to equalize the velocity field on the target surface. Besides, the coordinates of two fundamentally important points on the waveforms (see Fig. 7.5) that determine the values of the control parameters of our model cannot be measured very accurately. So, processing experimental waveforms in order to establish the regularities of the change in the mesoscopic model parameters during the wave propagation turns out to be a very difficult task. In this regard, a very encouraging result is the very establishment of a quasi-stationary regime of the shock-induced wave propagation and the laws that characterize it. In this section, we want to prove that the obtained experimental regularities are in correspondence with our theoretical approach and SG principle.

8.3 The Waveform Evolution During Quasi-Stationary Wave Propagation

259

Within experiments used in the paper, it is impossible to load very thin targets. The smallest target thickness available for shock loading by using the given setup is about 3 mm. Later it was found out that the quasi-stationary regime starts approximately 2 mm from the impact surface. This regime is determined by the condition that during the further propagation of the waveform, the amplitude of the elastic precursor remains constant. The speeds of the elastic precursor and plastic front propagation Cl , C0 are set too. Since Cl > C0 , the plastic front lags behind the elastic precursor during the waveform propagation along the target. During the quasi-stationary wave propagation, all experimentally obtained values of the parameters τ, θ turn out to be linearly related in a wide range of shock velocities and target thicknesses. Moreover, all the experimental points τ, θ fall on one straight line outgoing from the origin of coordinates on the plane of the parameters τ, θ (Sect. 7.11). Such regularity was found in the location of points on straight lines: the experimental points for the shock-induced waveforms at higher velocities approach the origin of the phase plane and the points recorded on thicker targets are removed from the origin. It means that the phase point moves along the straight path away from the origin during the wave propagation inside the target. The growing values of the parameters τ, θ associated with the duration of wave packets on the meso-1 scale indicate a slow recovery of solid material correlations after the wave propagation. Since each material has its own elastic precursor amplitude, the evolutionary paths will be different for different materials. All of them will be straight lines starting from the origin on the phase plane of the parameters but going at different angles [1]. The observed quasi-stationary regime of the shock-induced wave propagation is fundamentally different from the so-called steady waves. Most of the theoretical models of shock-wavefronts describe stationary waves, the fronts of which move in a medium parallel to themselves without changing their shape [15]. Within the steady wave models, the risetime describing the duration of the medium state transition from elastic to plastic ones is fixed by the wave shape. Therefore, the steady wave models cannot describe the wave evolution in a rather wide range of loading conditions. Until now, there is no reliable experimental data to find out which processes on the mesoscale are responsible for the risetime values and how they can be used to predict the observed risetimes [16]. In paper [17], Lee claims that steady, one-dimensional models are not valid inside the shock-induced transition zone where turbulence and a strong departure from local equilibrium occur. Asay [16] doubts at all whether the so-called steady waves are really steady. Unlike a steady wavefront, the shape of which is stable during its propagation, the waveform described by the model (7.8.2)–(7.8.3) evolves during a quasi-stationary regime. As the model parameters τ, θ grow along the evolutionary path, the waveform spreads out. So, our notion of the shock-induced wave propagation is based on the concept of the wave packet that can transport mass, impulse, and energy at the group speed retarding from the phase speed in the medium with internal structure. It is well known that wave packets are spreading during their propagation in the medium with dispersion. Each interaction of the wave packet with the medium in homogeneities

260

8 Evolution of Waveforms During Propagation in Solids

increases the packet retardation. We believe that the observed shock-induced waveform on the meso-2 scale is the wave packet formed by interference of smaller packets on the meso-1 scale that arise as a result of the impact on the initially inhomogeneous medium. We think that the so-called elastic precursor corresponds to the stress relaxation after the shock that partially breaks spatiotemporal correlations into meso-1 parts. As a result, the material state with the remaining short correlations becomes less elastic and more plastic. This part of the waveform propagates slowly compared to the rest elastic one and gradually lags behind highlighting the elastic precursor. And the so-called plastic front corresponds to the partial recovery of the initial correlations that make the material state more elastic on the plateau of the compression pulse. Such an idea is closer to the point of view of Gilman [18, 19] who believes that the names “elastic precursor” and “plastic front” do not correspond to the physics of the processes of their formation. However, we use these names to make them easier to read because they are generally accepted assuming that new terminology is unlikely to help understand new ideas. Now, we apply the SG principle presented in Chap. 6 to describe the quasistationary regime of the shock-induced waveform propagation within the integral model developed in Chap. 7. According to the SG principle, the waveform evolution is determined by the minimization of the integral entropy production for the loading ∂v = 1 is considered constant time (8.2.2). Within the integral model, the acceleration ∂ζ for the loading interval ζ ∈ [0, 1]. Since the quasi-stationary regimeis determined  of the constancy of the elastic precursor Ae = exp −π θ 2 /τ 2 = by the   condition exp −πk 2 = const, the integral must also be constant dQ = dξ

1

1 dζ

0

0



2   π ζ − ζ  − θ (ξ ) dζ exp − = const. τ 2 (ξ ) 

(8.3.1)

Then, the integrals (8.2.4)–(8.2.5) turn to 0 and the SG algorithm (8.2.3) gives dθ dτ = 0, = 0. dξ dξ

(8.3.2)

We have got that the model parameters do not evolve although experimental data suggest otherwise. This result is obtained using the approximate solution (7.8.2)– (7.8.3) that describes only the shape of the wave at the given parameters characterizing the beginning and end of the non-monotonic stress relaxation inside the waveform. Therefore, it is not surprising that, within the framework of this model solution, the parameters are fixed inside the waveform. In order to describe the waveform evolution, it is necessary to come back to the thermodynamic relationship (8.1.2) and rewrite it via the derivatives with respect to the variable ξ T

∂v dE dS − = V02 (ζ ; τ (ξ ), θ (ξ )) . dξ dξ ∂ξ

(8.3.3)

8.3 The Waveform Evolution During Quasi-Stationary Wave Propagation

261

Integration of (8.3.3) over the waveform duration p gives p

∂ ∂ξ

p (T S − E)dζ =

0

(ζ ; τ (ξ ), θ (ξ ))

V02 0

∂v dζ. ∂ξ

(8.3.4)

In this case, the goal of the waveform evolution during propagation is the minimization of the integral entropy production in the form dQ = dξ

θ (ζ ; τ (ξ ), θ (ξ )) 0

∂v dζ , ∂ξ

(8.3.5)

where the integration goes over the risetime θ in the opposite direction because the waveform propagates to the left. Now we can calculate the integral entropy  production(8.3.5) using an approximate solution (7.9.2) for (ζ ) = v(ζ ) = exp − τπ2 (ζ − θ )2 that is valid for rather large risetimes θ . θ



θ

0

∂v v dζ = ∂ξ

θ



σ (ζ, ξ )dζ = 0

= 0

θ + 0

θ 0

∂ v2 dζ = ∂ξ 2

θ 0

 1 ∂ 2π exp − 2 (ζ − θ )2 dζ = 2 ∂ξ τ



1 ∂ 2π dθ exp − 2 (ζ − θ )2 dζ 2 ∂θ τ dξ  1 ∂ 2π dτ 2 exp − 2 (ζ − θ ) dζ . 2 ∂τ τ dξ

For the goal function (8.3.5), the control parameters are in the finite form are

(8.3.6)

dθ dτ , . The SG equations dξ dξ

dθ ∂(d Q/dξ ) dτ ∂(d Q/dξ ) = −gθ ; = −gτ . dξ ∂(dθ/dξ ) dξ ∂(dτ/dξ )

(8.3.7)

Then, we can calculate the following integrals: ∂(d Q/dξ ) = ∂(dθ/dξ )

θ 0

 1 ∂ 2π exp − 2 (ζ − θ )2 dζ = 2 ∂θ τ θ

=− 0

  1 ∂ 2π θ 2 2π 1 2 1 − exp − 2 exp − 2 (ζ − θ ) dζ = ; 2 ∂ζ τ 2 τ (8.3.8)

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8 Evolution of Waveforms During Propagation in Solids

  1 ∂ 2π 2π θ 2 1θ exp − 2 (ζ − θ )2 dζ = − exp − 2 2 ∂τ τ 2τ τ 0 √ 2π θ 1 (8.3.9) . + √ erf τ 4 2

∂(d Q/dξ ) = ∂(dτ/dξ )

θ

We can show that both integrals are non-negative. Since axes are in opposite directions, both parameters can only grow in accordance with the SG principle

 1 dθ 2π θ 2 dτ = gθ 1 − exp − 2 ≥ 0. ≥ 0; dξ 2 τ dξ We know that in the quasi-stationary regime, the ratio τθ = k = const [1]. From the integrals (8.3.8)–(8.3.9), it follows that in the quasi-stationary regime of the shock-induced waveform propagation, the rate of the waveform evolution is constant. However, the scenarios of evolution for different materials can be different depending on the value of the constant k. When the evolution goes along the straight path, both parameters grow dθ dτ =k . dξ dξ

(8.3.10)

On the phase plane of the parameters τ (ξ ), θ (ξ ), the evolution path is governed by the equation of a straight line θ (ξ ) = kτ (ξ ). For each material, the quasi-stationary paths differ by the coefficient k (k = 0.84 for aluminum; k = 0.67 for steel). Since the quasi-stationary regime of the waveform propagation is not valid close to the impact surface, the extension of the straight paths to the origin is incorrect. We believe that just after the shock the parameter τ rapidly decreases with the elastic precursor relaxation and only then it begins to grow. In Fig. 8.2, we can see that in a narrow zone near the impact surface where the elastic precursor relaxes, the parameter τ first falls while the risetime grows. When the evolutionary path goes to the straight line, the material state begins to recover after shock. We have no available experimental waveforms to trace how long the quasistationary regime can last and what can occur during the further propagation of the wave. The wave model proposed in Chap. 7 also can become invalid at large distances from the impact surface where the dissipative processes play the main role.

8.4 Paths of the Waveform Evolution and Experimental Results

263

Fig. 8.2 The initial stage of the waveform evolution corresponding to the elastic precursor relaxation

8.4 Paths of the Waveform Evolution and Experimental Results The SG algorithm [6–8] allows one to interpret the evolution of the system graphically [20]. In the general case, the integral entropy production of the system under constraints imposed defines a certain evolving hypersurface over the phase space of control parameters. According to the SG principle, minimization of the integral entropy production of the system as the goal of evolution is achieved when the phase point descends down the hypersurface. SG algorithm defines the fastest descent path. It is clear that until the phase point determining the system state goes down, the system will be unstable. The lowest point of the path defined the final state of the system. This state is stable if the surface gradient in the point is zero. However, when the point is on the top of a certain amount, the state will be meta-stable since any fluctuation can remove the system from this state. Now we want to present the shock-induced waveform evolution in the quasistationary regime of propagation in accordance with the SG principle as a path on the surface of the integral entropy production. We have seen that the surface of the integral entropy production for the quasi-stationary regime in the form (8.3.1) is the plane over the plane of the control parameters τ, θ . Therefore, the system does not evolve as shown (8.3.2). We can  on the surface constructed for the  show that approximate value d Q/dξ ≈ exp −π θ 2 /τ 2 for rather large θ (see Sect. 7.9), the surface is really plane. The view  of this surface is shown in Fig. 8.3. The condition Ae = exp −π θ 2 /τ 2 = exp −π k 2 = const determining the quasi-stationary regime of the waveform propagation  givesus the plane whose height above the level of the parameters τ, θ plane is exp −π k 2 . The intersection of this plane with the surface is a straight line. The greater the height of the plane, the more

264

8 Evolution of Waveforms During Propagation in Solids

Fig. 8.3 The surface of the integral entropy production during the elastic precursor

the straight line will turn from us. It means that for different k and different materials, the straight lines on the plane will go at different angles as in [1]. So, we have got the evolutionary paths but it is not sufficient to describe the evolution because on the plane the phase point cannot move. Then we use the integral entropy in the form (8.3.5) first without the condition Ae =  production   exp −π θ 2 /τ 2 = exp −π k 2 = const in the zone near the impact surface where the elastic precursor exceeds its quasi-stationary value due to the decreasing correlation scale τ . As a result, near the origin of the phase plane, the surface of the integral entropy production rises and a slight slope appears. Later, the evolution proceeds in the quasi-stationary regime along the straight path [21]. The projection of this path onto the phase plane coincides with the experimental line constructed from the points taken from the recorded shock-induced waveforms at various distances from the impact surface (Fig. 8.4). In the first sections of this chapter, we considered the waveforms induced by the impact of two semi-spaces. Such a model allows us to describe only the forefront of the wave while the whole waveform is infinite. All real waveforms are finiteduration. They are formed both by loading and unloading. In the next sections, we will consider how the finite-duration waveforms are induced and how they evolve propagating along the target material.

8.5 Modeling Finite-Duration Waveforms

265

Fig. 8.4 The projection of the evolutionary path onto the phase plane in the quasi-stationary regime of the waveform propagation (a); two different materials (b)

8.5 Modeling Finite-Duration Waveforms In the previous Chaps. 6–7, we considered the case of the impact between two semispaces when only the loading front propagates in the material and no unloading occurs. In real shock-induced processes, the striker and the target have a finite length and the propagating waveforms have a finite duration. Let us first consider how in the process of interaction between the two bodies of the striker and the target, a finite-duration waveform arises. According to the conventional model of the process, the shock induces two waves simultaneously: one wave runs in the target, and another propagates in the opposite direction inside the striker. It is known that in a symmetric collision of two bodies, the momentum is equally distributed between them and the mass velocity in the wave propagating in the target is equal to half the speed of the striker. When the wave in the target reaches its free surface on the backside, the mass velocity doubles and becomes equal to the striker’s velocity. On the back surface of the target, the mass velocity is registered by laser in real time as the forefront of the waveform. The wave inside the striker reaches its back surface and is reflected from it and comes back into the target as an unloading wave. The striker adheres to the target due to inertia during the time equal to 2lstr /Cstr where lstr is the length of the striker and Cstr is the longitudinal sound velocity corresponding to the striker’s material. When it bounces off the target, the unloading begins forming a trailing edge of the waveform. Thereby, the finiteduration waveforms have a forefront induced by loading and a trailing edge formed by unloading. There is a plateau between the top of the forefront and the unload beginning. The duration of the plateau is determined by the striker’s length lstr and the risetime of the forefront. The forefront consists of the so-called elastic precursor and plastic front. The trailing edge is believed to consist of the same parts while the elastic precursor on the back of the waveform has twice the amplitude of the elastic precursor at the forefront.

266

8 Evolution of Waveforms During Propagation in Solids

According to the conventional concept of the shock-induced waveform [18, 19, 22], the elastic precursor propagates at the longitudinal sound speed Cl whereas the plastic front propagates at the speed of volumetric sound C0 because only shear degrees of freedom relax. It is usually believed that in the case of a short striker, the top of the trailing edge moves at the speed of longitudinal sound Cl . As a result, during the propagation of the waveform in the target material, the plastic front inside the waveform of fixed duration constantly lags behind approaching the trailing edge. This effect is called “hydrodynamic attenuation”. In reality, such an ideal model based on the close-to-equilibrium concepts of continuum mechanics is not entirely realized beyond the elastic limit. Although experimental studies reproduce the main parts of the waveforms, the mechanisms of their formation are misunderstood. We apply the integral model presented in Chap. 7 to explain both formation of the experimentally observed waveforms and the main regularities of their evolution during propagation inside the material. Within our model solution (7.8.2)–(7.8.3), all changes in the waveform during its propagation through the target material occur due to the evolution of the parameters τ (ξ ), θ (ξ ). In Fig. 8.5, we can see the waveforms induced by the striker of the fixed length at various values of the relaxation and retardation parameters. The waveforms are constructed for the normalized longitudinal stress J1 /ρ0 C V0 ≡ (ζ ; τ (ξ ), θ (ξ )) = v(ζ ; τ (ξ ), θ (ξ )) (recall that V0 is the shock velocity) in accordance with the obtained approximate solution (7.8.2)–(7.8.3). 1

0.5

0

0.5

1

0

2

4

6

8

ζ N1

Fig. 8.5 The waveforms at the parameters θ = 0.5, τ = 0.1, 0.5, 1, 2, 5, 10, 100 (from down to top)

8.5 Modeling Finite-Duration Waveforms

267

Similar waveforms are observed in experiments on shock loading of moderate intensity. Their special feature is the formation of a characteristic kink at the forefront of the wave called “elastic precursor”. Within our model, the kink arises at the time instant when the loading force ceases to act and the aftereffect begins at the . model parameters τ ∼ θ ∼ 10 = 102 . Outside of this range of conditions, the elastic precursor does not occur. For example, at small values of the delay parameter, as shown in Fig. 8.5, these waveforms look different. We see that unloading superimposed on an already ongoing relaxation causes a tensile wave; near local equilibrium (τ ∼ θ 1 indicates the time moment when the unloading begins. Unlike the case of the quasi-stationary wave propagation when the values of the parameters are rather large and assumed to be constant for the waveform duration, inside the finite-duration waveform, the parameters τ, θ can evolve. First, due to the energy obtained by the medium upon impact, a high stress arises in a thin near-surface layer of the target material that immediately begins to relax. In the process of the elastic precursor relaxation, fragmentation of spatiotemporal correlations occurs corresponding to a decrease in the parameter τ , and the formation of the wave forefront begins. Figure 8.14 shows the part of the generalized integral entropy production surface that is responsible for the elastic precursor relaxation. The gradient descent from the hill induced by the shock energy goes with decreasing τ and increasing θ which corresponds to the initial part of the evolutionary path with a turn marked in Fig. 8.2 with a pink dotted line. Further, the path governed by the SG algorithm dτ ∂ = −gτ dξ ∂τ



 dQ , dξ

∂ dθ = −gθ dξ ∂θ



dQ dξ

 (8.8.2)

turns into the well where the parameters reach their limiting values. On the path to the well, correlations grow and due to the self-organization, the material is partially restored. So, we see that inside the finite-duration waveform, the value of the parameter τ responsible for a scale of spatiotemporal correlations and turbulent structures changes inside the waveform from a lower value at the top of the elastic precursor up to a higher value at the pulse plateau. It means that our correlation model is not accurate.

8.8 Evolution of the Waveforms During Their Propagation

277

Fig. 8.14 The shape of the generalized integral entropy production surface (8.8.1) near the shockinduced hill

To correct it, we could use the correlation function of the asymmetric form with two parameters τ . In Fig. 8.15, we can see that it is possible to pass from the shape for τ = 20, θ = 12 to the shape for τ = 40, θ = 12 and get the shape with τ = 20 for ζ ≤ θ and τ = 40 for ζ ≥ θ . However, it is not always necessary since the waveform evolution during the wave propagation along the medium can be entirely described by parameters τ, θ . In this case, we use another goal function in the form d Q(τ, θ ) = dξ

θ 0

∂v dζ (ζ ; τ (ξ ), θ (ξ )) − ∂ξ +

d+θ

dζ − (ζ ; τ (ξ ), θ (ξ ))

d

∂v ∂ξ

(8.8.3)

and the SG algorithm as in d(d Q/dξ ) dθ d(d Q/dξ ) dτ , . = −gτ = −gθ dξ d(dτ/dξ ) dξ d(dθ/dξ )

(8.8.4)

As shown in [1], the experimentally obtained values of the parameters τ, θ grow with the distance from the impact surface in the quasi-stationary mode of the waveform propagation. We can calculate all integrals in the algorithm but the result will strongly depend on the backfront shape. In the general case, the shock-induced stress relaxes both in the target and in the striker. As a result, the backfront can significantly

278

8 Evolution of Waveforms During Propagation in Solids

Fig. 8.15 The passage to the correlation function with two typical values of the parameter τ

differ from the forefront. The regularities of its behavior are very poorly studied. In Fig. 8.16, we see that for a long striker, the entropy well moves away from the hill. Most often, the experimenters were interested in the spall strength of the material that is determined, as shown in Fig. 8.6, by the residual stresses at the trailing edge before the arrival of reflected waves. With respect to the spall strength, it is known Fig. 8.16 The part of the surface (8.8.1) for a long striker

8.8 Evolution of the Waveforms During Their Propagation

279

that its value first rapidly falls, and then its almost constant value is established. It means that the scenarios of the shock-induced waveform evolution are almost the same both for finite-duration waveforms and for semi-spaces at a rather large distance from the impact surface where the quasi-stationary regime of the wave propagation is set. If we have two experimentally recorded waveforms at different distances from the shocked target surface of a given material (ζ ; τ (ξ1 ), θ (ξ1 )), (ζ ; τ (ξ2 ), θ (ξ2 )), we can define the gain parameters for the material gτ , gθ characterizing the rate of the waveform evolution during its propagation in the quasi-stationary regime. As a result, it becomes possible to unambiguously predict the waveform shape at any distance traveled by the wave in the future and restore the past. Until now, we have described the free evolution of the waveform without any constraints imposed. However, all experimental waveforms are recorded on the backside of the target which is the system boundary. In the framework of MEP (see Sects. 4.6, 6.6, and 6.9), all the constraints imposed on the system evolution should be included into the goal function. In our case, the constraint can provide that the evolutionary path will pass through the given experimental point, τ (ξ1 ), θ (ξ1 ). Within the SG principle, the generalized integral entropy production surface is given by the conditional functional d Q(τ, θ ) = dξ

θ 0

∂v dζ (ζ ; τ (ξ ), θ (ξ )) − ∂ξ +

⎡

+λ⎣v(ζ )| L −

ω o

d+θ

dζ − (ζ ; τ (ξ ), θ (ξ ))

d

⎤ 2   π ζ − ζ − θ (ξ ) ∂v ⎦ 1 dζ  exp − ; τ 2 (ξ1 ) ∂ζ  



∂v + ∂ξ (8.8.5)

λ is Lagrange multiplier. SG algorithm in the finite form defines an evolutionary path on the surface (8.8.5) d(d Q/dξ ) dθ d(d Q/dξ ) dτ , . = −gτ = −gθ dξ d(dτ/dξ ) dξ d(dθ/dξ ) The equations can be rewritten in the integral form ξ τ (ξ ) = τ (ξ1 ) − gτ ξ1

d Q(τ, θ ) dξ , θ (ξ ) = θ (ξ1 ) − gθ dτ

ξ dξ ξ1

d Q(τ, θ ) . (8.8.6) dθ

Taking into account the opposite direction of the axes ζ, ξ and the relief of the surface (8.8.5), the parameters τ, θ should increase with gradient descent over the surface d Q(τ, θ )/dξ from the point τ (ξ1 ), θ (ξ1 ) and decrease when lifting. In the general case, different limiting states are possible depending on the initial state of the target material, the impact strain-rate, and its duration. After the quasi-stationary

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8 Evolution of Waveforms During Propagation in Solids

wave propagation along a finite-size target, the limiting state is the solid with a new internal structure induced inside a propagating waveform. When the stress falls abruptly during unloading, a spall may occur. When the correlation scale during the elastic precursor relaxation becomes too small, the medium response is the plastic flow that will stop close to the impact surface of the target. So, we have got the completed mathematical formulation of the problem of the shock-induced waveform evolution during its propagation along the target material. For the quasi-stationary wave propagation this formulation results in the evolution paths coinciding with experimental data [21].

References 1. Meshcheryakov YI, Khantuleva TA (2015) Nonequilibrium processes in condensed media. Part 1. Experimental studies in light of nonlocal transport theory. Phys Mesomech 18(3):228–243 2. Jaynes E (1979) The maximum entropy formalism. MIT, Cambridge 3. Zubarev DN (1974) Non-equilibrium statistical thermodynamics. Springer, Berlin 4. Khantuleva TA, Meshcheryakov YI (2016) Nonequilibrium processes in condensed media. Part 2. Structural instability induced by shock loading. Phys Mesomech 19(1):69–76 5. Khantuleva TA (2013) Nonlocal theory of non-equilibrium transport processes. St. Petersburg State University Publ, St. Petersburg (in Russian) 6. Fradkov AL (2007) Cybernetical physics: from control of chaos to quantum control. Springer, Berlin 7. Fradkov AL (2017) Horizons of cybernetical physics. Phil Trans R Soc A 375:20160223 8. Fradkov AL (2008) Speed-gradient entropy principle for nonstationary processes. Entropy 10:757–764 9. Bogoliubov NN (1962) Problems of dynamic theory in statistical physics. In: Studies in statistical mechanics. North-Holland, Amsterdam, pp 1–118 10. Meshcheryakov YI, Atroshenko SA (1992) Multiscale rotations in dynamically deformed solids. Int J Solids Struct 29:2761 11. Meshcheryakov YI, Divakov AK (1994) Multiscale kinetics and strain-rate dependence of materials. Dymat J 1(1):271–287 12. Meshcheryakov YI, Divakov AK, Zhigacheva NI (2004) Shock-induced structural transition and dynamic strength of solids. Int J Solids Struct 41:2349–2362 13. Meshcheryakov YI, Divakov AK, Zhigacheva NI, Makarevich IP, Barakhtin BK (2008) Dynamic structures in shock-loaded copper. Phys Rev B 78:64301–64316 14. Meshcheryakov YI, Divakov AK, Zhigacheva NI, Barakhtin BK (2013) Regimes of interscale momentum exchange in shock deformed solids. Int J Impact Eng 57:99–107 15. Prieto FE, Renero C (1973) Steady shock profile in solids. J Appl Phys 44(9):4013–4016 16. Asay JR, Chhabildas LC (2003) Paradigms and challenges. In: Horie Y, Davison L, Thadhani NN (eds) Shock wave research. High-pressure compression of solids VI: old paradigms and new challenges. Springer, pp 57–108 17. Lee J (2003) The Universal Role of Turbulence in the propagation of strong shocks and detonation waves. In: Horie Y, Davison L, Thadhani NN (eds) High-pressure compression of solids VI: old paradigms and new challenges. Springer, pp 121–144 18. Gilman JJ (2002) Mechanical states of solids. In: Furnish MD, Thadhani NN, Horie Y-Y (eds) Shock compression of condensed matter-2001. AIP Conference on Proceedings, vol 620. Melville, N. Y, pp 36–41 19. Gilman JJ (2003) Response of condensed matter to impact. In: Horie Y, Davison L, Thadhani NN (eds) High pressure shock compression of solids VI. Old paradigms and new challenges. Springer, pp 279–296

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20. Khantuleva TA, Shalymov DS (2017) Modelling non-equilibrium thermodynamic systems from the Speed-Gradient principle. Phil Trans R Soc A: Math Phys Eng Sci 375:20160220 21. Fradkov AL, Khantuleva TA (2016) Cybernetic model of the shock induced wave evolution in solids. Proc Struct Integr 2:994–1001 22. Swegle JW, Grady DE (1985) Shock velocity and the prediction of shock-wave times. J Appl Phys 58(2):692–699 23. Meyers MA (1977) A model for elastic precursor waves in the shock loading of polycrystalline metals. Mater Sci Eng 30(2):99–111 24. Meshcheryakov YI (2021) Multiscale mechanics of shock wave processes. Springer Nature Singapore 25. Kanel GI, Razorenov SV, Fortov VE (2004) Shock-wave Phenomena and the properties of condensed matter. Springer, New York 26. Ravichandran G. Rosakis AJ, Hodovany J, Rosakis P (2003) On the convention of plastic work into heat during high-strain-rate deformation. In: Furnish MD, Thadhani NN, Horie Y-Y (eds) Shock compression of condensed matter-2001. AIP Conference on Proceedings, vol 620. Melville. N.Y, pp 557–562 27. Bever HB, Holt DL, Titchener AL (1973) The stored energy of cold work. Pergamon Press, London

Chapter 9

Abnormal Loss or Growth of the Wave Amplitude

Abstract In Chap. 8, we considered the shock-induced waveforms shape of which is determined by the integral model constructed in Chap. 7. The model parameters are related to the spatiotemporal correlations dynamics inside the waveform. Within the approach proposed in Chap. 5 to describe processes far from local equilibrium, the correlation scales can be considered the dynamic structure sizes. The elements of this dynamic structure are carriers of mass, momentum, and energy in the wave transport mechanism as particles but what they are was not clear. To model the dynamics of the spatiotemporal correlations in shock-induced waveforms, it is necessary to reveal the physical nature of the shock-induced structure on the mesoscale. In the chapter, we will show that the experimentally recorded waveform is the result of the superposition of the moving wave packets that, in turn, can be considered the mesoparticles (Sects. 9.1 and 9.2). The wave packets originated by the shock-induced wave in the medium with dispersion are moving at different velocities. Their interaction can considerably enhance the velocity inhomogeneity and induce strong shears leading to the rotational modes occurrence. The formed turbulent structures partially remain frozen into the material after the unloading front passing. On the basis of the integral model of the waveform in Sect. 9.3, we were able to explain from the standpoint of the dynamics of correlations how the behavior of experimentally measurable quantities such as the dispersion of the mass velocity and the velocity defect on the plateau of the compression pulse is associated with the processes of structure formation. In Sect. 9.7, we show that the self-organization of turbulent structures is an example of the process that is accompanied by the negative integral entropy production that was predicted in non-equilibrium statistical mechanics (see Chap. 4). Unlike turbulence in liquids where dissipation gives the greatest contribution to the entropy production, the inertial properties of the solid material play a critical role in the transition to turbulence during high-rate deformation of the solid material. In Sect. 9.8, we show that the effects arising at the mesoscale under shock loading of solid materials have many similarities with quantum effects. Keywords Wave packet · Dispersion · Interference · Self-organization · Mesoparticle · Turbulence · Quantum effects

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 T. A. Khantuleva, Mathematical Modeling of Shock-Wave Processes in Condensed Matter, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-981-19-2404-0_9

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9.1 Mass Velocity Dispersion and Interference of Shock Waves on the Mesoscale Within the framework of traditional concepts of continuum mechanics, it was believed that in a plane collision, all particles of the medium at the same distance from the impact surface move with the same velocity. In reality, the deformable medium is inhomogeneous; a material always has various defects of the crystal lattice (impurities, dislocations, and internal boundaries) in a rather wide range of intermediate scales between micro and macro levels. As a result, different parts of the medium receive different velocities upon impact, which leads to a scatter in velocities on the mesoscale. The physical nature of these mesovolumes formed during high-rate deformation is unclear. Modern experimental tools do not allow visualizing processes on the time scale of shock loading. The observed defects of various natures that have arisen in the material after loading (group of dislocations, shear bands, and rotations) do not make it possible to determine the type of mesoparticle during the stress relaxation. In order to reveal the physical nature of the processes on the mesoscales, it is necessary to understand what happens during shock loading and what happens after the force stops acting. The special feature of the shock loading is such a short time interval during which the force is applied to the impact surface that the discrete atomic structure of the solid material immediately scatters the induced pressure wave. It is known that the difference between the oscillations of a discrete chain of atoms and the vibrations of a continuous string lies in the manifestation of the effects of the internal structure of the system in the form of wave dispersion. The presence of dispersion leads to a nonlinear relationship between frequency and wave vector and affects the process of wave propagation. The interference of the waves in a dispersive medium forms a space–time limited wave formation called a wave packet [1]. In Sec. 2.7, we have already considered the concept of the wave packet seeing the internal structure effects during highly non-equilibrium processes. We will now briefly recall some of the properties of the wave packets to compare them with evolving waveforms during their propagation in the material. The superposition of harmonic waves with a small scatter in wavenumbers δk/k0  1 has the form. 1 (x, t) = √ 2π k−k0 →∞



ei (kx−ω(k)t) A(k − k0 )dk.

(9.1.1)

The function A(k − k0 ) −−−−−→ 0 defines the wave amplitude. We introduce the notation k = k0 + k  in the form (8.7.1) and get a wave packet with the amplitude B(x − v0 t).  1   ei(k x−[ω(k0 +k )−ω(k0 )]t) A(k  )dk  (x, t) = ei (k0 x−ω(k0 )t) √ 2π i(k0 x−ω0 t) = B(x − v0 t)e . (9.1.2)

9.1 Mass Velocity Dispersion and Interference of Shock Waves on the Mesoscale

285

Expanding the function ω(k0 + k  ) into a Taylor series in a small parameter k  , in the linear approximation, we obtain the velocity v(k0 ) = v0 at which the main maximum of the packet travels at a speed different from the phase one. ω(k0 + k  ) − ω(k0 ) ≈

dω    k k = v(k0 )k . dk 0

  is called group velocity. Due to the dispersive propThe velocity v(k0 ) = dω dk k0 erties of the medium, wave packets propagate at a group speed that is always less than the phase one. In the linear approximation, the packet does not spread out. The spatial extent of the packet is determined by the spread in wavenumbers δx ∼ 1/δk. The next term in the expansion takes into account the dependence of the group velocity on the wavenumber.  dω   1 d 2 ω  2 1 dv(k)  2 ω(k0 + k ) − ω(k0 ) ≈ = v(k0 )k  + k0 k + k0 k k k  2 dk 2 dk 2 dk 0 

Accounting for this dependency leads to a spreading of the package during its propagation. (x, t) = e

i (k0 x−ω(k0 )t)

1 √ 2π



eik (x−v0 t+ 2 dk2 |k0 k 

1 d2 ω



t)

f (k  )dk  .

(9.1.3)

As a result, short-wave signals propagate more slowly and decay faster than longwave ones. In the long-wavelength limit, when the dispersion disappears, both speeds coincide and become equal to the speed of sound propagation in a equilibrium system. In the long-wavelength limit λ >> 2a, when the size of the lattice does not affect the propagation of waves, the medium can be considered as a continuum. In quantum mechanics [1, 2], the problem of the wave packets scattering on the inhomogeneities of the potential is solved. It was found that the transmitted and reflected wave packets exit after scattering with a certain delay. The times and lengths of the delay are determined by the wave packet pulses and the shape of the potential. The delay of the reflected packet corresponds to the penetration depth of the wave into the potential barrier. In addition, due to the dispersion, wave packets spread out on the inhomogeneities of the potential. It is known that the distribution density of the particle coordinate for a minimizing wave packet is a Gaussian or normal distribution. Only the Gaussian distribution function determines the minimum value of the coordinate and momentum uncertainty xp. These special properties of the Gaussian function make it useful for many cases. Moreover, the Gaussian function is directly related to the wave functions of the harmonic oscillator. Therefore, the Gaussian wave functions play an important role in quantum theory. For a Gaussian wave packet with the maximum in the point x0 moving at the group velocity v0 , the function B in the expression (9.1.2) has a form.

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B(x) = 

  (x − x0 − v0 t)2 , exp − 2σ 2 (t) 2π σ 2 (t) 1

(9.1.4)

where the packet width is determined by the variance of this distribution.   h2t 2 σ 2 (t) = σ02 1 + . 4m 2 σ04

(9.1.5)

In the linear approximation in k, the packet retains its shape and size, and its variance is constant σ (t) = σ0 . Quadratic term k 2 determines the spreading of the wave packet over time. The Gaussian distribution persists over time while packet width grows. In this case, the greater the particle mass m, the slower the spreading of the wave packet. Moreover, the stronger the initial package is concentrated, the faster it will spread. If we divide the numerator and denominator in the exponent in (9.1.4) by v0 , we get the wave packet in time representation. B(t) = 

  (x/v0 − x0 /v0 − t)2 exp − . 2σ 2 (t)/v02 2π σ 2 (t)/v02 1

Here x0 /v0 is the time when the packet amplitude reaches its maximum moving at the group velocity v0 and t − x/v0 is the wave variable connected to the point x0 . The transition to a frame of reference moving with phase velocity C requires the introduction of a time shift (individual for each point) which also takes into account the phase shifts due to interaction with potential inhomogeneities of the material during the wave propagation t − x/v0 − α/v0 = ζ − ζ  . As a result, we get the wave packet in the form. B(t) = 

  (ζ − ζ  − θ )2 exp − , 2σ 2 (t)/v02 2π σ 2 (t)/v02 1

(9.1.6)

which, up to a normalization factor, coincides with the form of the model correlation function constructed in Chap. 7 based on the approach developed in Chap. 5. All possible time shifts ζ  will be integrated into the resulting waveform. The obtained match is not accidental, and it has a deep meaning. Further, we will see that very important consequences follow from this analogy. Until now, the nature of the so-called mesoparticles that transfer mass, momentum, and energy in a dynamically deformable solid material was not understood. Now we can assume that this transport is carried out by the wave packets that are obtained as a result of the fragmentation of spatiotemporal correlations upon impact and which move like mesoparticles. However, due to the spreading of the wave packets, it is impossible to identify them completely with particles. They can be considered as some quasi-particles for some time such as risetime in the shock-induced waveform.

9.1 Mass Velocity Dispersion and Interference of Shock Waves on the Mesoscale

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So, we can see that a shock on a solid can be considered as the scattering of initially elastic waves by the inhomogeneities of the material structure after passing through a potential barrier with the subsequent formation of wave packets. After loading, the packets continue to propagate by inertia breaking up the space–time correlations and decreasing the order of the crystal lattice. Under conditions of high-rate deformation, the long-range order of the interatomic interaction is destroyed and only the shortrange order is retained. These retained by correlations mesovolumes are able to move as whole particles regardless of what processes occur inside them and can transport mass, momentum, and energy in the waveform. These mesovolumes do not form the material internal structure; they are rather dynamic field formations propagating along the medium. A combination of such properties is possessed by a single physical object, the wave packet. In turn, the wave packets can interact with each other and be scattered by larger inhomogeneities of the medium. The motion of the wave packets is possible only in the time interval that corresponds to the duration of the shockinduced waveform propagating in the medium. Outside the wave, the velocities of the wave packets equalize, the boundaries between them disappear, and the material becomes a continuous medium. The interaction of the wave packets can lead to a significant increase in the mass velocity inhomogeneity on the mesoscale due to arising resonance effects. Each mesopacket induces a spherical wave in the medium. Since the transverse size of the mesopacket is small compared to the wavelength which is of the order of the stress relaxation length in the medium, waves induced by a large number of the mesopackets can interfere with each other. If the phase difference, the spread in sizes, and velocities of mesopackets as wave sources are small, an interference pattern with pronounced maxima is formed where the velocity of the mesopackets can increase several times. It is these maxima of the velocity inhomogeneities that are the sources of local instabilities which generate new structures in the condensed medium. It is known from optics that an increase in the size of the wave source (transverse dimensions of mesopackets) and the degree of monochromaticity (scatter in phases and velocities) leads to blurring of the interference pattern and decreasing in the maxima of the wave field. Therefore, on the macroscale, such effects are not observed. In Fig. 9.1, we can see the wave structures on the mesoscale formed at the initial stage of their propagation as a result of the plane collision of solids with inhomogeneities on the surface at a speed of 300 m/s obtained by smoothed particle simulations. A similar situation can take place for the shock-induced waveforms which partially reflect from internal and external boundaries in the material and partially penetrate inter-grain and interphase boundaries [3]. Depending on the ratio of wavelengths and distances between potential inhomogeneities, the interference of incident and reflected wave packets can give rise to the resonance effects which form waves with large amplitudes. Propagating at different speeds, the mass fluxes inside the wave packets can generate high-rate shifts which, in turn, induce rotational degrees of freedom [4]. The self-organization of vortex-wave structures associated to the phenomenon of turbulence is experimentally observed in solids under shock loading [5–7].

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9 Abnormal Loss or Growth of the Wave Amplitude

Fig. 9.1 Formation of wave structures on the mesoscale at plane collision of solids with inhomogeneities on the impact surface at a speed of 300 m/s

All types of inhomogeneities including turbulent structures (pulsations, mesoflows, mesoshears, and rotations) arise inside the waveform and absorb a part of kinetic energy not due to dissipation (diffusion is too slow for a shock duration) but due to dispersion and self-organization of new internal structure [8–10]. The turbulent structures are only partially irreversible and only a part of the dynamic structures remain frozen into the material after passing the waveform. Figures 9.2 and 9.3 show both shear and vortex structures in shocked materials. So, the experimentally observed mass velocity dispersion in the shock-induced waveforms is closely related to the formation of wave packets on the mesoscale in

Fig. 9.2 Shock-induced mesoshear in a solid

9.2 The Shock-Induced Waveform as a Wave Packet

289

Fig. 9.3 Shock-induced rotational structures on the mesoscale

a real material and indicates the initial stage of the post-shock material structure formation.

9.2 The Shock-Induced Waveform as a Wave Packet Experimental studies of the shock-induced waveform propagation in solid materials [5, 6, 11–13] indicate the presence of such typical features of these processes that are similar to the properties of wave packets. In the experiments, such wave packets were detected simultaneously at two scale levels: meso-1 and meso-2. The mass velocity  dispersion v 2 − v 2 = D 2 (ζ ) is recorded in real time inside the waveform on the backside of the metal target. Being recorded on a much smaller scale compared to the waveform itself, the mass velocity dispersion can be considered as scatter in the velocities of wave packets on the mesoscale-1 [2]. In turn, the waveform is recorded at a spot compared to the diameter of the laser beam on the target backside and can be related to the mesoscale-2 [14, 15]. We can explain the mechanism of the observed waveforms generation as follows. When the shock breaks the space–time correlations in the solid, its parts of meso1 scale are moving at different velocities as the wave packets. Due to the chaotic component of their motion, a part of the shock energy is losing. This process accompanied by the stress decreasing is observed experimentally as relaxation of the elastic precursor. At this stage, the material response acquires plastic properties that bring it closer to the hydrodynamic response. However, for the shock of moderate intensity, the hydrodynamic stage of relaxation is not reached. Due to the inertia of the

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9 Abnormal Loss or Growth of the Wave Amplitude

medium, the post-shock effects allow the stress to accumulate for some time and reverse the relaxation process. At this stage, the chaotically moving wave packets begin to interact with each other. At each interaction, the packet loses a part of the pulse and decelerates; the boundaries of the wave packets begin to blur. The wave packets speeds level out in some places and their size grows. Their motions become more organized; the longitudinal components grow while the transverse ones decrease. It is in the middle of the plastic front that the mass velocity dispersion experimentally recorded in real time on the mesoscale-1 reaches its maximum. This stage corresponds to the so-called risetime when the plastic front rises until it reaches a plateau of the compressive pulse where the material is at least partially restored to its original solid state. As a result, the wave packets on the mesoscale-1 gradually form a much larger packet on the mesoscale-2 that is the experimentally observed shock-induced waveform. During the formation of the wave packet on the mesoscale-2, the amplitudes and delays of all wave packets on the mesoscale-1 are summed up. In paper [10], we have shown that in a result of interference, the waves induced by short-duration pulses in the inertial medium with sufficiently long relaxation and delay times can be added up, greatly increasing the mass velocity and stress. Due to such resonance effect, the resulting wave packet, as can be seen in Figs. 9.4 and 9.5, transports mass and momentum in quantities significantly exceeding the capabilities of one total pulse at the same strain-rate. This effect can occur in dense inertial media under short loads when the loading duration is much shorter than the relaxation time of the medium (rather large values of the parameters τ, θ ). Then, with the successive action of several short pulses, it

Fig. 9.4 Superposition of two pulses with unit amplitude and strain-rate and with different intervals between them 

9.2 The Shock-Induced Waveform as a Wave Packet

291

Fig. 9.5 Superposition of 4 pulses with unit amplitude and strain-rate and with different intervals between them 

is possible to achieve a local increase in the maximum stress value in the resulting waveform by several times. In practice, such stress increase has been used in tools such as jackhammers and saws for a long time without a rigorous theoretical base. Then, the experimentally observed waveform on the mesoscale-2 is a result of the superposition of the wave packets on the mesoscale-1 v(ζ ) =

n

i

ω

(ζ + ii ; τi , θi )  o





2  π ζ − ζ  − θ (ξ ) ∂v . (9.2.1) dζ exp − τ 2 (ξ ) ∂ζ  

During the elastic precursor relaxation, the parameters τi characterizing the correlation scale of the mesopackets-1 sharply fall while θi slowly grow. The intervals between the mesopackets-1 are very small. In the resulting waveform model on the right side of the expression (9.2.1), the parameter τ corresponds to the minimum reached during the elastic precursor relaxation. During the risetime of the plastic front, both parameters τi , θi grow until the parameter θi reaches its maximum on a plateau θ . The intervals between the mesopackets-1 are due to delays caused by their interaction with each other, which intensifies with increasing mass velocity at the wavefront. The increase in the intervals between the wave packets on the mesoscale1 during the risetime can lead to a significant increase in mass velocity variance recorded experimentally in real time. We see that the stress relaxation inside the waveform on the mesoscale-2 is non-monotonic (Figs. 8.7, 8.8, 8.9 and 8.10). Although both the parameters τ, θ evolve inside the waveform, the mathematical model of the waveform constructed in Chap. 7 uses only their limiting values inside the waveform at a fixed distance traveled by the wave from the impact surface. During

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9 Abnormal Loss or Growth of the Wave Amplitude

the waveform propagation along the target material, it evolves due to the change in the model parameters τ, θ depending on the traveled distance. Therefore, our model describes the shape of the plastic front only qualitatively retaining only its most important points such as the elastic precursor amplitude and the plateau beginning after the plastic front risetime. However, rather small deviations from the recorded waveform shape are observed between the limiting points. So, the used model of the shock-induced waveform adequately describes both the elastic precursor relaxation and the risetime evolution during the waveform propagation along the target material. The processes responsible for the plastic front shape and the material recovery after the shock are very diverse, complex, and little studied. We know that for the shock of moderate intensity at a distance of a little more than 2 mm, the elastic precursor amplitude remains constant during the waveform propagation while the risetime grows at a constant rate. In the Sects. 7.11 and 8.3, this process is observed as an establishment of the quasi-stationary regime of the shock-induced waveform propagation. In this case, it has been shown that the correlation parameter τ linearly related to the risetime θ also grows during the waveform propagation. It means that the material state is gradually restored with the distance traveled by the waveform. The growing risetime can be explained by the waveform interaction and their scatter by the material structure defects which increase their delays and scatter in velocities on the mesoscale-2. As the wave packets in quantum mechanics [1, 2], the experimentally observed waveforms on the mesoscale-2 [14] spread out in the process of their propagation due to an increase in the delay of individual packets when a waveform passes through an inhomogeneous material (Fig. 9.6). In the quasi-stationary regime of the waveform propagation, the spreading is going at the elastic precursor amplitude fixed due to the growing spatiotemporal correlations when the material state is restored after the shock. The experimentally observed scatter in the amplitudes of the waveforms on the mesoscale-2 increases with the impact velocity as in paper [16]. Fig. 9.6 The waveform spreading during propagation from the impact surface at the distance x1 (line 3), x2 > x1 (line 2), x3 > x2 > x1 (line 1).

9.3 Behavior of the Mass Velocity Dispersion and the Waveform Amplitude Loss

293

9.3 Behavior of the Mass Velocity Dispersion and the Waveform Amplitude Loss The experimentally observed waveforms propagating in the quasi-stationary mode are adequately described by the model developed in Chap. 7. After the shock, the material is almost entirely restored and the waveform amplitude is close to the initial one. In this case, the mass velocity dispersion is observed only during the risetime of the forefront and has maximum in its middle where the forefront is the steepest [13–15]. Experimental studies on shock loading of solid materials have shown that the strain-rate and variation of the mass velocity grow synchronously in the middle of the wavefront [13, 17]. In Fig. 9.7, the behavior of mass velocity and its variance is presented. This allows us to establish the law of proportionality between the variation in mass velocity and the strain-rate. D∼

de . dt

(9.3.1)

Since the variation of the mass velocity D describes the average amplitude of the pulsations of the mass velocity on the mesoscale-1 and the strain-rate describes the deformation inside the waveform on the mesoscale-2, the relationship (9.3.1)

Fig. 9.7 Mass velocity variance D (blue dotted line) in the middle of the forefront and inside the waveform propagating in the quasi-stationary regime (red line)

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9 Abnormal Loss or Growth of the Wave Amplitude

establishes a connection between the two mesoscales. It is worth noting that a similar effect has been well known for turbulent fluid flows [18]. However, experiments [13, 14, 17] show that a quite different situation can arise with an increase in the impact speed in a threshold manner. When the mass velocity dispersion does not decay during risetime and remains almost constant on the pulse plateau, the waveform amplitude on the mesoscale-2 drops dramatically but the velocity inhomogeneity of the wave field on mesoscale-1 increases greatly. In such situations, the mass velocity variation increases at the forefront of the waveform until the plateau and falls to zero only at the falling edge of the compression pulse. It means that most part of the kinetic energy remains on the mesoscale-1 and the material state is not restored after the shock. As a result, the material is in an unstable state and unable to maintain the stress induced by impact. A considerable part of kinetic energy is going to the pulsations on mesoscale-1 due to the impulse loss on the mesoscale-2. This situation is presented in Fig. 9.8. In the case of shock loading, the loss of the kinetic energy on the mesoscale-2 due to the mass velocity loss on the waveform plateau was called the mass velocity defect. Its determination is based on an independent measurement of the free surface velocity U f and the projectile velocity during its collision with the target V0 . In a symmetric collision, the mass velocity is equal to half the striker velocity but when the wave reaches the free surface, its amplitude doubles. If the material state is entirely

Fig. 9.8 Mass velocity variance D (blue dotted line) inside the waveform .propagating with the mass velocity defect (difference between red and gray lines)

9.4 Dependence of the Waveform Amplitude Loss on the Impact Velocity

295

restored on the waveform plateau, the waveform amplitude should be equal to the striker velocity U f = V0 . As experiments show, the mass velocity amplitude is close to the striker velocity when the waveform propagates in the quasi-stationary regime. However, in general, this is not the case. Only the propagation of elastic waves is completely reversible. Some part of the kinetic energy is always lost for the interaction of the wave packets through which the energy exchange between different scales is carried out. In this case, the mass velocity defect is determined as the difference between the striker velocity and the maximum value of the free surface velocity on the compression pulse plateau U = V0 − U f . The large value of this defect indicates that the state of the material structure inside the propagating waveform is unstable and a new material structure may remain in the trail after passing a wave. The magnitude of the velocity defect depends on many factors including the initial state of the material internal structure, the strain-rate, the aptitude of the material to structural and phase transformations, etc. The paper [17] presents numerous examples of the mass velocity behavior in experiments with different materials in a wide range of loading conditions. Thus, registration of both the mass velocity inside the waveform on the mesoscale2 and simultaneously its variations in real time gives information about the character of the energy exchange between the mesoscale-1 and the mesoscale-2 at different stages of high-rate deformation. A quantitative characteristic of the energy exchange is the defect of the mass velocity at the plateau of the compression pulse. It occurs due to the loss of momentum and energy during structural transformations of the material under high-rate deformation as a result of which the initial state of the material is not restored. In this case, after the passage of the wave, the material acquires a new internal structure and new mechanical properties. So, together with the dynamic yield point, the threshold for structural instability, and spall strength, the mass velocity defect is the most important characteristic of the dynamic deformation and fracture of materials [17].

9.4 Dependence of the Waveform Amplitude Loss on the Impact Velocity Experimental studies on shock loading of solid materials [11–13, 17] have revealed the dependence of the free surface velocity on the pulse plateau U f on the impact velocity U f = (V0 ). It turned out that at velocities below a certain critical value V0 < V0 ∗, the velocity U f is close to the impact velocity V0 slowly deviating from it with its increasing. At the impact velocity V0 > V0 ∗, the dependence U f = (V0 ) changes radically. The value of the impact velocity V0 ∗ at which the break in the dependence U f = (V0 ) occurs corresponds to the threshold of the structural transition in the given material. Above the impact velocity V0 > V0 ∗, the mass velocity

296 Fig. 9.9 The dependence of the free surface velocity U f on the impact velocity V0

9 Abnormal Loss or Growth of the Wave Amplitude

Uf

V0*

V0

defect begins to grow sharply, which indicates an increase in energy consumption for self-organization of the shock-induced mesostructure in the material. The dependence U f = (V0 ) is shown schematically in Fig. 9.9. The experimentally obtained dependencies U f = (V0 ) for real materials can be found in the paper [17]. The observed threshold change in the pulse amplitude at the plateau leads not only to a sharp increase in the mass velocity defect U = V0 − U f but simultaneously to an increase in the variation in the mass velocity D on the mesoscale-1 which is also recorded in real time. There is a transition from the behavior scenario of the mass velocity variance in Fig. 9.7 to the scenario in Fig. 9.8. It means that with an increase in the impact intensity, the shock-induced chaotic movement of the wave packets on the mesoscale-1 sharply increases which does not have time to transform into directional movement on the pulse plateau for the duration of the waveform on the mesoscale-2 and to completely recover the material solid state. As a result, high-rate deformation of a solid becomes more irreversible since a part of kinetic energy transported by the waveform remains in its trail frozen into the material in the form of various shear and rotational structures. In this regard, it should be noted that the threshold growth of the mass velocity defect correlates not only with an increase in dispersion but also with a change in the spall strength of the given material. The spall strength W of the material is determined as the difference between the maximum value of the free surface velocity at the plateau of the compression pulse and the first minimum at its trailing edge. In Fig. 9.10, we can see that when the impact velocity exceeds the critical one V0 ∗, the spall strength begins to drop noticeably. We see that the mass velocity defect is almost unchanged with increasing impact velocity until its critical value V0 ∗ is reached. With an increase in the impact velocity above the critical value V0 > V0 ∗, the growth rate of the mass velocity defect U sharply increases. In this case, due to the chaotic movement of the wave packets on the mesoscale-1, the dispersion responsible for the velocity pulsations increases sharply. Then, a drop in the spall strength W with an increase in the impact velocity becomes quite expected since the material does not recover after a strong impact and many defects appear in its structure on both mesoscales.

9.5 Multi-scale Momentum and Energy Exchange in Wave Processes

297

Fig. 9.10 Behavior of the mass velocity defect U (red line), the mass velocity variance D (blue dotted line), and the spall strength W (green line)

V0*

V0

Experimental studies have shown that both the mass velocity defect and the rest of the measured characteristics of the deformation process of a given material, even at the same impact velocity, change with the distance traveled by the waveform from the impact surface. This fact confirms the influence of the waveform evolution during their propagation which is described in Chap. 8. Thus, the experimental data on shock loading indicate that the macroscopic response of a material to shock loading is influenced by both the initial state of the material structure and the characteristics of the loading process. The observed regularities in the behavior of such characteristics as the mass velocity defect, velocity variance, and spall strength cannot be explained within conventional models of continuum mechanics. The very occurrence of such a loss of momentum that is not determined by the dissipation of kinetic energy into a thermal form [8, 9] seems anomalous if the processes on the mesoscale are not taken into account. So, it is necessary to describe multi-scale processes of exchange of momentum and energy inside the waveform accompanying its propagation in the medium. In the next section, we will try to do it in the framework of our approach.

9.5 Multi-scale Momentum and Energy Exchange in Wave Processes Within the framework of continuum mechanics, it is believed that 90–95% of the work of plastic deformation is converted into heat. The transformation of mechanical energy into a thermal form is carried out by dissipative processes based on the diffusion momentum transport. However, experiments [8, 9] show that depending on the strain-rate and its duration, only 35–50% of the work of dynamic deformation is converted into heat. The rest of this work is stored in the material as latent energy in the form of structural defects, cracks, localized shear bands, and other structural inhomogeneities [18]. The fact is that in the fast short-duration processes, diffusion does not have time to completely transform the directed motion of the medium on the macroscale into the chaotic motion of the medium particles on the microscale.

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9 Abnormal Loss or Growth of the Wave Amplitude

The energy obtained by the medium during shock loading has time only to reach some intermediate level between macro and microscales. Wave transport plays the main role in short-duration processes. As we have already shown, propagating along the medium with dispersion, waves form wave packets that can transport mass, momentum, and energy. As a result of interference of the wave packets, larger packets and other formations on the mesoscale may arise. In experiments on shock loading of solid materials [11–13, 17, 23], the waveform characteristics are recorded at two mesoscale levels at once. Mass velocity pulsations are recorded in real time in the form of mass velocity dispersion D 2 = v 2 − v 2 . They are defined by the chaotic motions of the wave packets on the mesoscale-1. The larger wave packets formed due to the interference between wave packets on the mesoscale-1 are the waveform recorded on the mesoscale-2. It is important to note that interference is observed between mesopackets at both the meso-1 and meso-2 levels. In the general case, a hierarchy of scale levels is formed through which the kinetic energy of the impactor transforms into rotational–shear wave structures on the mesoscales and partially dissipates into the thermal component of the internal energy of the material. In this case, the average velocity of deterministic motion at a higher level may be reduced by chaotic pulsation modes at a lower level due to multi-scale and multi-stage energy exchange between different-scale impulse carriers. Consider the energy exchange between two scale levels, mesoscale-1 and mesoscale-2. However, if we determine volumetric internal energy density by the ∂v (ζ ) are given traditional transport equation where the stress and the strain-rate VC0 ∂ζ at the same instant ζ and point ξ . ∂v ∂E = (ζ, ξ ) (ζ, ξ ), ∂ζ ∂ζ

(9.5.1)

then, taking into account the relationship obtained in Chap. 7 for the stress tensor

(ζ ; τ (ξ ), θ (ξ )) = v(ζ ; τ (ξ ), θ (ξ )), it turns out that the energy coincides with elastic energy E = v 2 /2, and the process of all its transformations within the shockinduced waveform is entirely reversible. However, experimental studies on shock loading [11–13, 17] suggest otherwise. It means that the concept of energy as of the other thermodynamic quantities must be redefined under the non-equilibrium conditions. First, the work of the deformation forces is performed only during loading while the response of the medium due to inertia is formed significantly after the impact and propagates in the medium in the form of a wave. For the shock-induced waveform propagation, the stress and strain-rate are not related to the same spatial point and time instant. The strain-rate during the loading at the impact surface generates the stress after the loading inside the waveform at any distance from the impact surface. Secondly, the energy between different degrees of freedom is distributed unevenly and the kinetic energy of vibrations is not equal to the potential one. In this case, instead of (9.5.1), the internal energy at any instant during and after the shock is

9.5 Multi-scale Momentum and Energy Exchange in Wave Processes

299

defined by the deformation work as follows: ζ E=

(ζ ; τ (ξ ), θ (ξ )) 0

∂v0 dζ, ∂ζ

(9.5.2)

where the acceleration during the shock loading is ∂v0 /∂ζ = 1, 0 ≤ ζ ≤ 1 and ∂v0 /∂ζ = 0, ζ > 1. Then, the internal energy is not a full differential and not thermodynamic potential. ζ

1

(ζ ; τ (ξ ), θ (ξ ))dζ, 0 ≤ ζ ≤ 1 and E =

E= 0

(ζ ; τ (ξ ), θ (ξ ))dζ, ζ ≥ 1. 0

(9.5.3) To determine the kinetic energy, we need to know the total energy and subtract the potential one. Since the internal energy is not a thermodynamic potential far from local equilibrium, we can determine full mechanical energy as a full differential. ∂E ∂v0 = , E = v0 = v(ζ, ξ )v0 (ζ, ξ = 0), ∂ζ ∂ζ

(9.5.4)

where v(ζ, ξ )v0 (ζ, ξ = 0) is the two-point correlation function. Then we get the full mechanical energy as the sum of potential and kinetic energies. ζ E= 0

∂v0

(ζ ; τ (ξ ), θ (ξ )) dζ + ∂ζ

ζ v0 (ζ ) 0

∂ dζ. ∂ζ

(9.5.5)

Individually, these energies, not being total differentials, cannot be determined correctly. It is easy to show that these energies are not equal to each other and lose their generally accepted meaning. Near local equilibrium when spatial correlations contract to one point, we return to elastic energy.  E p = Ek = 0

ζ

v(ζ )

∂v v2 dζ → , 0 ≤ ζ ≤ 1. ∂ζ 2

(9.5.6)

Now we can calculate the total mechanical energy received by the medium under loading and its evolution for the waveform duration based on the definition. E(ζ, ξ ) = v0 (ζ, ξ = 0) (ζ ; τ (ξ ), θ (ξ )).

(9.5.7)

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9 Abnormal Loss or Growth of the Wave Amplitude

As a result of the impact, the material acquires energy E(ζ = 1, ξ = 0) = v0 (ζ = 1, ξ = 0) (1; τ (0), θ (0)) = v02 (1, 0) = 1. Then, the energy drops during the elastic precursor relaxation and grows on the plastic front until it reaches the constant value E(1 + θ; ξ ) = v0 (1 + θ; 0) (1 + θ; τ (ξ ), θ (ξ )) = (1 + θ ; τ (ξ ), θ (ξ )) on the pulse plateau where ζ = 1 + θ and v0 (1 + θ; 0) = 1. During the stress relaxation, the energy from the mesoscale-2 goes down to the mesoscale-1 forming interacting wave packets. During the risetime, the process goes back from mesoscale-1 to mesoscale2. If the material state is restored after the shock, the value (1+θ; τ (ξ ), θ (ξ )) = 1. It means that all the energy transformations with the transitions from mesoscale-2 to mesoscale-1 and back were reversible inside the waveform. However, in experiments [11–13, 17] with an increase in the impact velocity, the mass velocity defect U = 1 − (1 + θ ; τ (ξ ), θ (ξ )) ≥ 0 is recorded on the waveform plateau. The appearance of the velocity defect on the pulse plateau indicates that the reverse process of restoring the initial material state is not completed. A part of the full mechanical energy remained on the mesoscale-1 in the form of chaotic pulsations. 1 − E(1 + θ ) = 3n D 2 /2.

(9.5.8)

Here, 3D 2 /2 is the kinetic energy of a moving wave packet and n is the number of wave packets on the mesoscale -1. For the chaotic movement, the kinetic energy is defined correctly. Unloading material in the state with the defect leads to irreversible structure formation after the waveform passing. As a result, the material state becomes less solid and the spall strength falls with an increase in the mass velocity defect as shown in Fig. 9.10. In Fig. 9.11, we can see various multi-scale turbulent structures formed in the shock-induced crack inside the spall zone in the metal target. These structures resemble a twisted jet of a dense viscous medium flowing from a source in the

Fig. 9.11 Multi-scale turbulent structures in the crack inside the spall zoneinduced by the shock on the metal target

9.6 The Shock-Induced Structure Instability

301

nose of a crack under pressure. It can be seen that the jet consists of smaller, also twisted streams. So, we have made sure that both the variation of the mass velocity recorded in real time and the velocity defect on the pulse plateau are indeed important information characteristics of non-stationary wave processes induced by a short pulse action of moderate intensity. The rapid growth of the mass velocity defect with an increase in the intensity of impact indicates a growing possibility of a catastrophic change in the mechanical properties of the material up to its destruction. In the section, we have considered the energy exchange between the two mesoscales. However, just the same exchange takes place between any different scales far from local thermodynamic equilibrium.

9.6 The Shock-Induced Structure Instability One of the most urgent problems of high-rate deformation of solid materials is the study of the development of plastic flow instabilities which lead to local structural transformations such as localized shear bands, rotations, and phase transitions [17, 18]. High-rate deformation is a heterogeneous multi-scale process at different stages of which different scales and different mechanisms of mass, momentum, and energy transport are involved. Sometimes, the transition from one stage to the next one occurs gradually and sometimes unexpectedly. It is very difficult to describe such a transient process, but even more difficult to predict the probability of its catastrophic scenario. The nucleation and development of any structural inhomogeneity begin with the occurrence of a local instability of plastic deformation. The entire volume of a dynamically deformable material cannot pass into a structurally unstable state because it takes a lot of energy. The size of the zones of local instability corresponds to the mesoscopic sizes of structural inhomogeneities. Therefore, both experimental research and theoretical modeling of the nucleation and development of structural instability should also be carried out on the mesoscale. Macroscopic medium response to shock loading averages mesoscopic effects approaching the medium state to local equilibrium and making it more stable. Unlike many macroscopic experimental studies, the mass velocity waveforms recorded on the free surface of the target with a laser interferometer in experiments on uniaxial deformation [17] reflect the response of one mesoscopic structural element. Therefore, the experimental data obtained in this way really make it possible to study transport processes directly on the mesoscale. Probing different points on the same target surface under the same loading conditions gives different waveforms with different mass velocity defects U and velocity variance D behavior. Evaluations of the results obtained show that the mass velocity variance D defines the dispersion of mass and momentum transport on the mesoscale1 while the mass velocity defects U are a variation of the velocity on the mesoscale2. An increase in pulsations on the mesoscale-1 can lead to instability of the waveform on the mesoscale-2, and the growth of pulsations on the mesoscale-2 can provoke

302

9 Abnormal Loss or Growth of the Wave Amplitude

instability of macroscopic material response up to destruction or disintegration of the material. Hence, both characteristics are necessary to study how the local instabilities occur during high-rate deformation of solid materials. We know that both the mass velocity defect and the velocity dispersion on the mesoscale-1 increase with the impact speed. The experimental fact [14] is that both waveform parameters τ, θ introduced in the integral model in Chap. 7 decrease with increasing impact speed gradually reaching some almost constant values. It means that the higher the impact intensity, the stronger the correlations are fragmented and the less solid the material becomes, and its recovery, although incomplete, occurs faster. The constant values of the waveform parameters correspond to the establishment of the constant amplitude of the elastic precursor when the quasi-stationary mode of the wave propagation is established. It was noted that a structurally unstable state of a material on the mesoscale-2 develops when the rate of increase in the velocity variation D on the mesoscale-1 exceeds the increase in mass velocity on the plastic front [15, 17]. However, when and under what conditions this can happen is unknown. When the material state on the pulse plateau of the waveform on the mesoscale-2 is not entirely restored, the motion of the wave packets on the mesoscale-1 can become turbulent [19, 20]. Numerous shears between the wave packets cause rotations of mesoscopic fragments of crystal lattice which make the material state irreversible. In this case, inertial effects play an important role when turbulence occurs. As is known, the generation of turbulent pulsations in both liquids and solids is largely provoked by the initial structural inhomogeneities of the medium [21, 22]. The shock-induced relaxation zone is highly turbulent and unstable [7]. Unloading wave superimposed on such an unstable state significantly enhances the velocity inhomogeneity of motions on the mesoscales giving rise to structures of different scales that remain frozen in the medium after the wave passing. This means that the lost energy from the mesoscale-2 goes not to the kinetic energy of wave packets on the mesoscale-1 but is stored in the new material structure (Fig. 9.12). The lack of experimental data for different thicknesses at the same projectile velocity does not allow answering the question of at what distance from the impact surface the transition to turbulence occurs. It is only known that large-scale pulsations arise at small target thicknesses at relatively high impact velocities whereas a quasistationary regime is established with increasing target thickness. The transient zone between the two relaxation processes where the evolutionary path makes a turn in Fig. 8.2 is the most unstable. It is in this zone that active control of the loading process is possible to obtain the desired structures. Figure 9.13 shows that superimposed small pulses just after the shock change the integral entropy production surface and the direction of evolutionary paths. Thus, if we describe the evolution of a system subjected to a sufficiently intense external influence to bring the system into the structurally unstable state, we can define possible evolutionary paths based on the SG principle as shown in Chaps. 6 and 8. It is the problem of internal control. Then, it is necessary to formulate the problem of external control with its own goal compatible with the given possibilities. If we assume that a certain sequence of weak external influences with its own control

9.6 The Shock-Induced Structure Instability

303

Fig. 9.12 Traces of turbulent structures in a metal after the passage of a shock-induced wave of moderate intensity

Fig. 9.13 The evolutionary paths during the structure formation on the mesoscale-2: without external influence and with additional small pulses

parameters can help in achieving the set goal, then these influences will enter the conditional goal functional as imposed constraints that can turn the system evolution in the desired direction. Problems that combine internal and external control are the most difficult in control theory but also the most interesting from a practical point of view. Methods for solving these problems would allow the development

304

9 Abnormal Loss or Growth of the Wave Amplitude

of completely new technologies for obtaining materials with desired properties. We hope that in the future it will be possible.

9.7 Self-organization of Turbulent Structures in the Entropy Well In order to obtain materials with the required internal structure, one must first be able to determine what structures are obtained as a result of self-organization upon impact on a solid material. In Sect. 8.7, we have shown that self-organization occurs when the evolutionary path leads the material state after the shock to the entropy well. Our vision of the processes on the mesoscales is based on the concept of wave packets. High-rate deformation of the solid material during the shock induces the elastic wave that propagates in a heterogeneous medium. Due to dispersion in the medium, wave packets are formed that can transport mass, momentum, and energy at a group speed that is less than the phase one. Moving at different speeds, the wave packets retain the short-range order of the material structure but destroy the longrange one. Shattered spatiotemporal correlations make the state of the material structurally unstable. In this state, elastic stress immediately starts to relax quickly until the elastic precursor reaches a certain constant level corresponding to the dynamic elastic limit for a given material. However, after that, the relaxation is not completed. Each wave packet when interacting with inhomogeneities of the material structure acquires a phase delay and at the same time rapidly spreads when moving. As a result, the phase difference between individual packets disappears, their motion becomes coherent, and their amplitudes are summed up forming one larger packet that we identify with the waveform on the mesoscale-2. It means that for the impact of moderate intensity, the relaxation continues in a non-monotonous way, the stress grows on the plastic front, and the material solid state restores. This process can be partially reversible partially not depending on the material state on the compression pulse plateau. Even in the ideal case when the state of the material after impact is completely restored in a semi-infinite waveform, its propagation along the target is an irreversible process due to the loss of kinetic energy when its leading front spreads. As to the finite-duration waveform, the material state after its propagation always changes because the material internal structure transforms. When the unloading wave comes, two flows of the wave packets move toward each other and their interaction increases the velocity heterogeneity on the mesoscale-1. As a result, a part of them loses momentum and remains in the trace of the wave in the form of new defects on the mesoscale-2. This process is irreversible but the entropy produced by the waveform is negative due to the new information about the wave propagation retained by the material. Figure 8.17 in the previous chapter shows the entropy well where the waveform evolution leads. All real processes leave behind traces in the environment to one degree or another.

9.7 Self-organization of Turbulent Structures in the Entropy Well

305

A strong shock imparts such a large impulse to a part of the material that its structure is completely and irreversibly destroyed. Depending on the strain-rate and pulse duration, fracture occurs at different stages of wave relaxation. At small distances from the collision surface, correlations retain rather large fragments of the material while with an increase in the duration of the input pulse, the structures become smaller, and melting sites of the material appear [23]. In Chap. 5, we have noticed that, as in quantum mechanics, the size spectrum of spatiotemporal correlations for the system of finite size should be discrete. This effect is determined by the influence of the imposed boundary conditions in the finite-size system. The size spectrum evolves over time due to the rest free degrees of freedom in accordance with the Jaynes principle [24] in the direction to more stable states available under constraints imposed. In Chap. 6, we have shown that the self-organization of turbulent structures decreases the integral entropy production in fluids and liquids. In liquid media, a significant part of the loss of kinetic energy falls on dissipation since their initial state is much less ordered and, therefore, more equilibrium than the solid state. Therefore, despite the decrease in dissipative losses due to structure formation, the integral entropy production in high-speed liquid flows remains nonnegative [25]. Such processes are quasi-stationary when the temporal evolution of the system state is slow compared to gradients inside it. The self-organization of turbulent structures defines the transition from laminar flow mode to the turbulent one. For the solid materials whose initial states are much more ordered than liquid states, the situation is different. Unlike fluids, mass and momentum transport in solids is carried out in the waveform while the diffusion rate is too low compared to the sound speed. Only in dynamically deformed solids, as shown in Sect. 8.7, the integral entropy production can become negative [26]. In Chap. 4, we have shown that this does not contradict the second law of thermodynamics. This effect is explained by the fact that during unloading that is superimposed on the stress state of the material in the propagating waveform, two fluxes of the wave packets collide moving toward each other. As a result, the wave packets lose their kinetic energy and remain frozen into the material in the trace of the waveform. In Chap. 8, we have seen that at this stage of the evolution of the material state inside the waveform, the phase point characterizing the state of the system falls into the entropy well. What does it mean? The negative values of the generalized integral entropy production, as shown in Sect. 8.7, correspond to the free energy losses due to boundaries formation between new structural elements. In this case, the lost kinetic energy goes to potential lattice energy. Then, falling into the entropy well means the wave packets on the mesoscale are retained by the potential well. Since in Sect. 9.1 we have shown that the wave packets on the mesoscale-1 can be considered as quasiparticles on the mesoscale-2, we get a complete analogy with the quantum problem of confining a particle by a potential well [1, 2]. In quantum mechanics, the problem of the moving particle or quasi-particle in a potential well is solved [1, 2]. It was proven that a well deep enough should contain discrete energy levels. Even in a shallow well, as shown, there is at least one discrete energy level. It means that a new internal structure formed in the waveform trace

306

9 Abnormal Loss or Growth of the Wave Amplitude

on the mesoscale-2 is ordered. The new information that this ordered structure is carrying, as shown in Sect. 4.5, reduces the entropy production. Thus, we have demonstrated that falling of a phase point into the entropy well during the temporal waveform evolution means the self-organization of a new internal structure of the material as a result of shock-induced waveform propagation. In Chap. 5, we have shown how the problem of determining the size spectrum of the dynamic structure of a system under quasi-static loading using imposed boundary conditions can be formulated. In this case, a mathematical apparatus was used developed on the basis of functional analysis for nonlinear operator systems of a special type. In paper [27], it was tested on the example of a quasi-stationary shear flow of a liquid between two parallel planes. It was shown that far from local equilibrium, already in the first approximations, boundary layers and moment structures are obtained due to which the velocity profiles become nonlinear and the flow itself is unsteady. How to formulate the spectrum problem for the wave packets on the mesoscale is unclear. Like in quantum mechanics, it is possible to get the energy spectrum knowing the shape of the entropy well. However, what information can be extracted from this data regarding the type and size of the formed structural elements? To answer the question, we need special studies and experimental data. Otherwise, quite a new problem formulation is necessary. In any case, additional experimental information is required. In order to determine the sizes and types of the new structure, it is necessary to describe the dynamics of wave packets on the mesoscale-1 within the framework of quantum mechanics. The process of the wave packet scattering by potential inhomogeneities can be considered as the interference of incident and multiple reflected wave packets from the original defective structure of the material. In this case, depending on the ratio of the wave packet lengths and distances between defects, interference can both enhance and weaken the velocity inhomogeneity within the waveform. Since the wave packets arise as a result of the action of short pulses of moderate intensity, the parameters of these pulses and, first of all, their duration are the decisive factor for the self-organization on the mesoscale. Only the control of the external action of short impulses superimposed on the system that is relaxing according to the internal control will allow one to control the processes of structure formation.

9.8 Quantum Effects on the Mesoscale One of the unsolved problems of modern physics is the identification of regularities to which the transport processes are subjected at an intermediate scale level between the deterministic behavior of a macroscopic system and the probabilistic one at the micro level of elementary particles. Macroscopic size systems containing a huge number of elementary particles obey the laws of classical physics. The neglect of the effects of the microscopic structure of the system in continuum mechanics allows one to consider continuous distributions of mass, momentum, and energy over the

9.8 Quantum Effects on the Mesoscale

307

system volume. The elementary particles interact and exchange discrete quanta of momentum and energy in accordance with the laws of quantum mechanics. The result of the interaction can be predicted only with a certain probability. Modern studies of shock-wave processes in the condensed matter [3–6] have shown that high-rate and short-duration processes are also not described by models of continuum mechanics. Far from local equilibrium, the system response always lags behind the impact from outside because of the finite propagation speed of disturbances in the system and depends on the interaction effects between the internal structural elements. Highly non-equilibrium processes are often accompanied by the self-organization of new dynamic structures that can give rise to oscillations, instabilities, and the formation of feedbacks between the structure effects and macroscopic properties of the system. As a result, the local properties of the system are determined by its integral time-evolving state under external influence. The dynamics of correlations in the system makes the system behavior unstable and poorly predictable. The self-organization of vortex-wave structures associated to the phenomenon of turbulence is experimentally observed in solids under shock loading. We have shown that the self-organization in a shocked solid is accompanied by negative integral entropy production which corresponds to the loss of free energy for the formation of the new internal structure on the mesoscale. Such a loss of energy and momentum leads to their lag behind the waveform in which they arose and to their transformation from dynamic structures to stationary ones when they remain frozen in the material behind the outgoing wave. The zone of self-organization on the integral production surface constructed over the control parameters plane corresponds to the well where the evolutionary path of the system leads. Then we can say that this is a potential well that captured mesoparticles formed by wave packets interaction. In quantum mechanics, a deep well should contain discrete energy levels. Comparison of the physical nature of the processes far from local equilibrium with quantum mechanics principles shows that some of them take place on the mesoscale between macro and micro description levels despite the huge difference in the mathematical description and representations [28]. Below we list some provisions in common with quantum mechanics. (1) (2) (3) (4) (5) (6) (7)

Any external influence on the system that deflects its state far from equilibrium leads to a change in the structure and properties of the system. Wave processes are the main transport mechanism. Discrete carriers of mass, impulse, and energy are wave packets. Any characteristics on the mesoscale can be predicted with a certain degree of probability. The potential well can hold a mesoparticle (wave packet) that has lost a significant part of its kinetic energy. The internal structure characteristics on the mesoscale have discrete size and energy spectra. Mathematical apparatus of functional analysis is used to solve inverse problems: to seek internal structure from integral system characteristics.

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It means that to predict the mechanical properties of condensed matter after shock loading, to obtain materials with desired internal structure, and to control the processes of their manufacture, we need to adapt some of the results obtained in quantum mechanics to solve similar problems on the mesoscale. It is necessary to develop more general theoretical approaches which require more experimental data on the processes on the mesoscale. In the book, we could only catch a glance beyond conventional concepts and approaches and see a huge new field awaiting its researchers. This is just the beginning.

References 1. Bohm D (1952) Quantum theory. Prentice Hall Inc., N.-Y 2. Ivanov MG (2012) How to understand quantum mechanics. Moscow-Izhevsk (in Russian) 3. Meshcheryakov YI, Atroshenko SA (1992) Dynamic rotations in crystals. Izvestiya Vuzov Fizika 4: 103−122 (in Russian) 4. Liu C, Yan Y, Lu P (2014) Physics of turbulence generation and sustenance in a boundary layer. Comput Fluids 102(10):353−384 5. Meshcheryakov YI Atroshenko SA (1992) Multiscale rotations in dynamically deformed solids. Int J Solids Struct 29:2761 6. Meshcheryakov YI, Divakov AK, Zhigacheva NI, Makarevich IP, Barakhtin BK (2008) Dynamic structures in shock-loaded copper. Phys Rev B 78:64301–64316 7. Lee J (2003) The universal role of turbulence in the propagation of strong shocks and detonation waves. In: Horie Y, Davison L, Thadhani NN (eds) High-pressure compression of solids VI: old paradigms and new challenges. Springer, pp 121–144 8. Ravichandran G, Rosakis AJ, Hodovany J, Rosakis P (2003) On the convention of plastic work into heat during high-strain-rate deformation. In: Furnish MD, Thadhani NN, Horie Y-Y (eds) Shock compression of condensed matter-2001. AIP conferences proceedings, vol 620. Melville, NY, pp 557–562 9. Bever HB, Holt DL, Titchener AL (1973) The stored energy of cold work. Pergamon Press, London 10. Khantuleva TA, Meshcheryakov YI (2017) Mesoscale plastic flow instability in a solid under high-rate deformation. Phys Mesomech 20(4):417–424 11. Meshcheryakov YI, Divakov AK (1994) Multiscale kinetics and strain-rate dependence of materials. Dymat J 1(1):271–287 12. Meshcheryakov YI, Divakov AK, Zhigacheva NI (2004) Shock-induced structural transition and dynamic strength of solids. Int J Solids Struct 41:2349–2362 13. Meshcheryakov YI, Divakov AK, Zhigacheva NI, Barakhtin BK (2013) Regimes of interscale momentum exchange in shock deformed solids. Int J Impact Eng 57:99–107 14. Meshcheryakov YI, Khantuleva TA (2015) Nonequilibrium processes in condensed media. Part 1. Experimental studies in light of nonlocal transport theory. Phys Mesomech 18(3):228–243 15. Khantuleva TA, Meshcheryakov YI (2016) Nonequilibrium processes in condensed media. Part 2. Structural instability induced by shock loading. Phys Mesomech 19(1):69–76 16. Case S, Horie Y (2007) Discrete element simulation of shock wave propagation in polycrystalline copper. J Mech Phys Solids 55:589–614 17. Meshcheryakov YI (2021) Multiscale mechanics of shock wave processes. Springer nature. Singapore 18. Panin VE (1998) Foundations of physical mesomechanics. Phys Mesomech 1:5–20 19. Yano K, Horie Y (1999) Discrete-element modeling of shock compression of polycrystalline copper. Phys Rev B 59(21):13672–21368

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20. Koskelo AC, Byler DD, Diskerson RM, Luo SM, Swift DC, Tonk DL, Peralta PD (2007) Dynamics of the onset of damage in metals under shock loading. In: Shock compression of condenced matter. In: Proceedings of AIP, vol 955–208, pp 557–560 21. Grady DE, Kipp ME (1987) The growth of unstable thermoplastic shear with application to steady–wave shock compression in solids. J Mech Phys Solids 35(1):95–119 22. Faisst H, Eckhardt B (2004) Sensitive dependence on initial conditions in transition to turbulemce in pipe flow. J Fluid Mech 594:343–352 23. Meshcheryakov YuI, Zhigacheva NI, Konovalov GV, Divakov AK, Morozov VA (2021) Formation of multi-scale structure under shock loading of solids. Lett J Tech Phys 47(7):6–9 (in Russian) 24. Jaynes E (1979) The maximum entropy formalism. MIT, Cambridge 25. Khantuleva TA, Shalymov DS (2017) Modelling non-equilibrium thermodynamic systems from the Speed-Gradient principle. Phil Trans Royal Soc A: Math Phys Eng Sci 375:20160220 26. Zubarev DN (1974) Non-equilibrium statistical thermodynamics. Springer, Berlin 27. Khantuleva T, Shalymov D (2020) Nonlocal hydrodynamic modeling high-rate shear processes in condensed matter. J Phys Conference Series 1560(1):012057. https://doi.org/10.1088/17426596/1560/1/012057 28. Khantuleva TA, Kats VM (2020) Quantum effects on the mesoscale. Particles 3(3):562-575. https://doi.org/10.3390/particles3030038

Chapter 10

The Stress–Strain Relationships for the Continuous Stationary Loading

Abstract Until now, mechanics do not fully understand the specific features of dynamic processes, assuming that the presence of a time derivative in a mathematical model is sufficient for the process to be considered dynamic. Therefore, they often tried to use the data obtained under quasi-static conditions to describe shock-wave processes. Naturally, this led to errors and contradictions and raised questions about what physical processes initiated by shock effects are (Asay, Chhabildas, Paradigms and challenges in shock wave research. In: Horie Y, Davison L, Thadhani NN (eds) High-pressure compression of solids VI: old paradigms and new challenges. Springer, pp 57–108, (2003), [1]). In Sect. 10.2, we want to show that the dependence of the shear stress on plastic strain for uniaxial planar loading cannot be found using experimental relationships for the simple tension–compression of thin rods (Sect. 10.1) recalculated to semi-space. In Sect. 10.3, based on the solution to the problem of the planar shock-induced waveform propagation obtained in Chap. 7, we derive the stress–strain relationship for continuous loading at the constant strain-rate. Then in Sect. 10.4, we show that it is incorrect to separate stress and strain both into elastic and plastic and into bulk and shear parts in advance. It is possible to determine whether the process is reversible or irreversible only after its end by calculating the resulted integral entropy production. In Sect. 10.5, the surface of the integral entropy production is constructed over the plane of the parameters that bind the typical times of relaxation, delay, and loading duration. It was found that the long-term deformation of condensed matter is a dissipative process, while a short impact, depending on its intensity, can be either reversible or irreversible (Sect. 10.6). Experimental research on shock loading of metals [2–5] that revealed relationships between the quantities describing the shock-induced mass and momentum transport elucidates specific features of material response to the shock loading that characterize dynamic loading unlike quasi-static one. In Sect. 10.7, the fundamental difference between shock-induced and quasi-static processes is considered. We show that the elastic precursor can form only under short-duration loading due to the delay of post-shock effects. Prediction of final states and possibilities to control them are discussed. Keywords Stress–strain relationship · Dynamic and quasi-static loading · Shock and long loading · Plastic deformation · Elastic–plastic transition © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 T. A. Khantuleva, Mathematical Modeling of Shock-Wave Processes in Condensed Matter, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-981-19-2404-0_10

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10 The Stress–Strain Relationships for the Continuous Stationary Loading

10.1 Stress–Strain Relationships for Continuous Quasi-Static Loading Under normal conditions, it is believed that plastic deformation of solids is due to the relaxation of shear degrees of freedom that relax much faster than the bulk ones. From Chap. 7, we know that both plastic deformation and elastic–plastic transition in real solid materials cannot be described in the framework of continuum mechanics. Usually, the relationship between stress and strain is obtained from experiments. Since the bulk relaxation is considered frozen, experimental data on the uniaxial tension of thin rods is used to study the plastic properties of real materials. Tension testing is a fundamental test in which a sample is subjected to a controlled tension by gradually applying load until failure. The stress–strain relationship can be determined by measuring the deformation during the load. The values of stress and strain determined from the tensile test can be plotted as a stress–strain curve (Fig. 10.1). Such mechanical properties as ultimate tensile strength, breaking strength, maximum elongation, and reduction in area can be directly measured via a tensile test. From these measurements, the following properties can also be determined: Young’s modulus, Poisson’s ratio, yield strength, and strain hardening characteristics. Uniaxial tensile testing is most commonly used for obtaining the mechanical characteristics of isotropic materials [6–11]. There are several points of interest in the diagram above: • P: This is the proportionality limit that represents the maximum value of stress at which the stress–strain curve is linear. • E: This is the elastic limit that represents the maximum value of stress at which there is no permanent set. Even though the curve is not linear between the proportionality limit and the elastic limit, the material is still considered elastic in this region, and if the load is removed at or below this point, the specimen will return to its original length. Fig. 10.1 The stress–strain diagram. Adapted from Mechanical Properties of Materials, by AM Kirkby, 2015, Retrieved from mechanicalc.com

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313

• Y: This is the yield point that represents the value of stress above which the strain will begin to increase rapidly. The stress at the yield point is called the yield strength. • U: This point corresponds to the ultimate strength which is the maximum value of stress on the stress–strain diagram. The ultimate strength is also referred to as the tensile strength. After reaching the ultimate stress, specimens of ductile materials will exhibit necking in which the cross-sectional area in a localized region of the specimen reduces significantly. In brittle materials, the ultimate tensile strength is close to the yield point whereas in ductile materials, the ultimate tensile strength can be higher. • F: This is the fracture point or the break point that is the point at which the material fails and separates into two pieces. Up to the elastic limit, the strain in the material is also elastic and will be recovered when the load is removed so that the material returns to its original length. However, if the material is loaded beyond the elastic limit, then there will be permanent deformation in the material that is referred to as a plastic strain. If the load is removed at the indicated point (σ, ε), the stress and strain in the material will follow the unloading line as shown in Fig. 10.2. Some materials break very sharply without plastic deformation. It is called a brittle failure. Others, more ductile materials, including most metals, experience some plastic deformation and possibly necking before fracture. A ductile material can withstand large strains even after it has begun to yield whereas a brittle material can withstand little or no plastic strain. The figure below shows representative stress– strain curves for a ductile material and a brittle material (Fig. 10.3). In the figure above, the ductile material can be seen to strain significantly before the fracture point, F. There is a long region between the yield at point Y and the Fig. 10.2 Elastic and plastic strain. Adapted from Mechanical Properties of Materials, by AM Kirkby, 2015, Retrieved from mechanicalc.com

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10 The Stress–Strain Relationships for the Continuous Stationary Loading

Fig. 10.3 The stress–strain diagrams for ductile (left) and brittle (right) materials. Adapted from Mechanical Properties of Materials, by AM Kirkby, 2015, Retrieved from mechanicalc.com

ultimate strength at point U where the material is strain hardening. There is also a long region between the ultimate strength at point U and the fracture point F in which the cross-sectional area of the material is decreasing rapidly and necking is occurring. The brittle material in the figure above can be seen to break shortly after the yield point. Additionally, the ultimate strength is coincident with the fracture point. In this case, no necking occurs. When force is applied to a material, the material deforms and stores potential energy, just like a spring. The strain energy (i.e. the amount of potential energy stored due to the deformation) is equal to the work expended in deforming the material. The total strain energy corresponds to the area under the load deflection curve. The elastic strain energy can be recovered, so if the deformation remains within the elastic limit, then all of the strain energy can be recovered. Because the area under the stress–strain curve for the ductile material above is larger than the area under the stress–strain curve for the brittle material, the ductile material can absorb much more strain energy before it breaks and has a higher modulus of toughness (Fig. 10.4). Stress–strain curves for materials are commonly needed; however, without representative test data, it is necessary to come up with an approximation of the curve. The Ramberg–Osgood [9] equation can be used to approximate the stress–strain curve for a material knowing only the yield strength, ultimate strength, elastic modulus, and percent elongation of the material. The stress–strain curve in the plastic region can be approximated by the expression. σ = H enp ,

10.2 The Stress–Strain Relationship for Continuous Planar Loading ...

315

Fig. 10.4 The strain energy. Adapted from Mechanical Properties of Materials, by AM Kirkby, 2015, Retrieved from mechanicalc.com

where εp is the plastic strain, H is the strength coefficient with the same units as stress, and n is the strain hardening exponent and is unitless. In this case, the total deformation takes the form. e = eel + e pl =

 σ 1/n σ . + E H

A relationship was proposed by Ramberg and Osgood that is frequently used to approximate the stress–strain curve for a material. This relationship is exponential and is used to describe the plastic strain in a material. All the results presented above are obtained for uniaxial stress in thin rods. There are no similar results for plane loading with uniaxial strain. These processes are quite different and it was impossible to use the data on mechanical material properties in experiments on the shock loading of solid materials. In the next section, we consider the ways how to use the data obtained from the uniaxial tensile testing for the case of planar loading and analyze their suitability for the shock loading processes.

10.2 The Stress–Strain Relationship for Continuous Planar Loading with Accounting Only Shear Relaxation The problem of longitudinal elastic–plastic deformation in semi-space under planar loading was considered already long ago. In one of the first papers [12], D. Wood had got a solution to the problem supposing that.

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10 The Stress–Strain Relationships for the Continuous Stationary Loading

(1) (2) (3) (4)

deformations are divided into elastic and plastic components; elastic components are determined by Hook’s law; plastic deformation does not change the material density; the relationships between stress and plastic deformations do not depend on the strain-rate.

Taking into account the assumptions above, the stress–strain relationship was derived in a form J = ρ0 C02 eel + σ (e pl ). The first term corresponds to the elastic volume compression and the second one arises due to the shear resistance during plastic deformation. The dependence of the shear stress on plastic strain can be found using experimental relationships for the simple tension–compression of thin rods recalculated to semi-space. The algorithm of the recalculation was presented in the work. In order to analyze the suitability of Wood’s approach, we can construct the stress– strain relationship for uniaxial strain under planar loading of solids using the solution constructed in Chap. 7 and based on the integral model. First, we will consider the case when the contributions of the bulk and shear degrees of freedom can be separated in accordance with their modules. Under normal conditions, it is believed that plastic deformation of solids is due to the relaxation of shear degrees of freedom that relax much faster than bulk degrees of freedom tr G  tr0 . Considering the bulk relaxation frozen, in accordance with (7.6.7), the longitudinal stress component with the shear relaxation is described by the integral equation

J1 (x, t) =

ρ1 C02

4 − Gtr G 3

ω(t) 0

dt  ∂v MG (t, t  ; tr G ) , ω(t) = tr G ∂x



t, t < t R , tR , t ≥ tR , (10.2.1)

where ρ1 C02 describes the elastic compression and the integral term describes the and shear relaxation with typical time tr G . Taking into account that ∂∂vx = − ∂e ∂t e = ρ1 /ρ0 , we can rewrite (10.2.1) for continuous loading as follows: J1 (x, t) =

ρ0 C02 e

4 + Gtr G 3

t 0

dt  ∂e MG (t, t  ; tr G )  . tr G ∂t

(10.2.2)

  In the wave variables ζ = t1R t − Cx , ξ = Lx introduced in Chap. 7, on the condition ∂ζ∂  τε ∂ξ∂ , the normalized stress T = J1 /ρ0 C02 takes the form

10.2 The Stress–Strain Relationship for Continuous Planar Loading ...

ζ

C2 C2 T(ζ ; τG ) = 02 e + G2 C C

dζ  ζ (ζ, ζ  ; τG )

o

317

∂e , ∂ζ 

(10.2.3)

where K + 43 G = ρ0 (C02 + C G2 ) = ρ0 C 2 and τG = tr G /t. All variables with the dimension of time are set in conventional units. At the initial stage of the loading when t  tr G , the shear relaxation is also frozen tr G →∞

ζ (ζ, ζ  ; τG ) −−−−→ 1 and the shear response is elastic τ →∞

T(ζ ; τG ) −−−→ ζ

C2 C02 e + G2 e = e. 2 C C

(10.2.4)

During the stage when the relaxation is frozen, the momentum transport is reversible and not accompanied by dissipation. For long loading when t R  tr G , the shear relaxation results in the model of viscous–elastic medium τG →0

T(ζ ; τG ) −−−→

C G2 ∂e C02 C02 e + τ e. → G C2 C 2 ∂ζ C2

(10.2.5)

From (10.2.5), it is clear that in the limiting cases, the separation of the relaxation scales of bulk and shear degrees of freedom is equivalent to the stress and strain separation into elastic and plastic parts. As it was shown in Chaps. 5 and 7, the additive combination of the elastic and hydrodynamic medium responses to describe transient processes is suitable only for low strain-rates. The transient zone between the two limiting cases is usually neglected in theoretical models. By using the model (10.2.3), we can describe the transient process when t ∼ tr G . The momentum transport Eq. (7.7.3) with the stress (10.2.3) takes the form ⎡ ⎤ ω C G2 ∂e ∂e ⎦ ∂ ⎣ C02   e+ 2 dζ ζ (ζ, ζ ; τG )  . (10.2.6) = ∂ζ ∂ζ C 2 C ∂ζ o

Taking into account the identity C02 + C G2 = C 2 , the Eq. (10.2.5) is reduced to the equation identical to (7.7.7) but for the shear deformation e. Its integration gives the equation for the shear relaxing part of the medium response to continuous plane loading ζ TG (ζ ; τG ) = o

dζ  ζ (ζ, ζ  ; τG )

∂e . ∂ζ 

(10.2.7)

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10 The Stress–Strain Relationships for the Continuous Stationary Loading

For the correlation model determining the integral kernel in (10.2.6), we have the Gaussian form developed in Chap. 5 and applied in Chap. 7

 2 π ζ − ζ  − θG

ζ (ζ, ζ ; τG , θG ) = exp − . τG2 

In this case, the Eq. (10.2.7) is identical to the Eq. (7.8.1) that for the constant acceleration ∂v/∂ζ = 1 has the approximate solution (7.8.2). Using the solution, we can construct the stress–strain relationship for the case of continuous plane loading at the constant strain-rate ∂e/∂ζ = const when e = ζ ∂e/∂ζ in the explicit form √  √  C G2 τG ∂e π (e − θG ) π θG C02 T(e; τG , θG ) = 2 e + 2 + erf erf . C C 2 ∂t τG τG

(10.2.8)

For the constant values of the parameters τG , θG , the shear stress component is presented in Fig. 10.5.

Τ

e Fig. 10.5 The shear stress relaxation for various relaxation parameters τG at θG = 0.5

10.2 The Stress–Strain Relationship for Continuous Planar Loading ... Fig. 10.6 The stress–strain diagram for plane loading at the constant strain-rate and accounting only the shear relaxation

319

e

Τ(e)

C02 e C2

e

In accordance with the relationship (10.2.8), the stress–strain diagram for continuous plane loading with the constant strain-rate and accounting only the shear relaxation is presented in Fig. 10.6. However, it is necessary to take into account that the parameters τG = tr G /t, θG = tmG /t are decreasing over time. It means that the shear stress component will decrease during relaxation tending to 0 in the limit τG → 0. In Fig. 10.6, the decrease is shown in a red dotted line. The stress–strain relationship (10.2.8) is obtained under the condition that the relaxation of bulk degrees of freedom is frozen while all stages of the shear relaxation are taken into account. The separation of the contributions of bulk and shear degrees of freedom is not equivalent to the division of stress and strain into elastic and plastic parts. The relationship (10.2.8) is much more general and can describe the transition from elastic medium response to plastic under the adopted conditions. Therefore, this relationship allows us to analyze the validity of the assumptions made in Wood’s work [12]. First of all, the division of the deformation into elastic and plastic parts (assumption 1) is incorrect for transient processes because their contributions change over time. As we have already shown that elastic modules are not constant far from equilibrium (Chap. 7), besides, as experiments show, they grow with strain-rate (assumption 2). The division is only justified in the near-equilibrium case τG  1 when the stress– strain relationship takes the form (10.2.5) for long quasi-static loading. Assumption 3 can take place only in the limiting case τG → 0 of ideal plasticity. And finally, assumption 4 about the independence of the stress–strain relationship on the strainrate is valid also only for the ideal plasticity because only the elastic part meets the condition. The relationship (10.2.8) shows that in dynamic transient processes the strain-rate plays a decisive role during the loading. All mechanical models of the elastic and deformable solid are based on near-equilibrium concepts suitable only for quasi-static processes. High-strain-rate dynamic loading induces such multi-scale mechanisms of mass–momentum transport and energy exchange that lead to the self-organization

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10 The Stress–Strain Relationships for the Continuous Stationary Loading

of turbulent internal structure while at low strain-rates these effects cannot arise. In paper [13], modeling of high-rate shear processes in condensed media is considered. For quasi-static continuous loading at low strain-rates, the medium response keeps up with the loading, and the loading history does not directly influence it. That is why the neglecting of post-effects can be justified at low strain-rates. So, it was demonstrated that the adopted assumptions about the stress–strain division into elastic and plastic parts neglecting the strain-rate are not adequate for the dynamic loading.

10.3 The Stress–strain Relationship for Continuous Plane High-Rate Loading We can see that in the case when only the shear degrees of freedom can relax, the C2 stress–strain curve cannot intersect the line T(e) = C02 e. Approaching asymptotically the line, the stress continues to grow indefinitely. This is the case of loading a semispace with a semi-space. For the real system of finite size, it is impossible; the material C2 either will flow or fail. In the general case, both straight lines T(e) = e, T(e) = C02 e can change their angles of inclination with the strain-rate increasing due to change in the speeds C, C0 . Moreover, for the long-duration loading, the bulk part of the degrees of freedom also relaxes and at the initial stage of loading, a part of the shear degrees of freedom shows the elastic response. Therefore, the concept of the shear relaxation under the bulk degrees of freedom frozen is incorrect. In the general case, it is impossible to separate not only elastic and plastic parts of deformation but also contributions of different degrees of freedom in advance. The material response to the loading at different stages of the relaxation can radically change depending on the loading strain-rate, material properties, and also on the size and shape of the sample. The approximate solution to the problem of the shock-induced waveform propagation (7.8.2) obtained in Chap. 7 describes the medium response during the loading interval t R at the constant acceleration ∂v/∂ζ = 1 √   √ π (ζ − θ ) πθ τ ∂v erf , ζ ≤ 1. + erf

(ζ ; τ, θ ) = 2 ∂ζ τ τ

(10.3.1)

In this expression, all quantities with the dimension of time are referred to as the loading duration t R . In particular, the relaxation and delay parameters directly depend on the duration t R τ = tr /t R , θ = tm /t R . For the shock loading, the interval t R is very short. In the general case, the loading duration can be arbitrary if the averaged acceleration (or strain-rate) during the loading is considered constant. It means that we can use the solution (10.3.1) to describe the medium response to any continuous loading taking into account the parameters changing.

10.3 The Stress–strain Relationship for Continuous Plane High-Rate Loading

321

At the initial stage of the process t ≤ t R  tr when the parameter is large τ = tr /t R >> 1, the solution (10.3.1) gives v(ζ ) ≈ ζ . In this case, the strain e = (V0 /C)v(ζ ) and the stress component J1 = ρ0 C V0 v(ζ ) are linearly connected in the same spatiotemporal point. It means that at small times during the loading, the response of any medium is elastic independent of the strain-rate. However, the duration of the elastic stage according to the solution (10.3.1) increases with the strain-rate. During long loading tr  t ≤ t R when the values of the relaxation parameter τ  1 are small, the medium response becomes hydrodynamic. The stress is proportional to the strain-rate as in the flow of viscous liquid. τ →0

J1 = ρ0 C V0 v(ζ ) −−→ ρ0 C V0 τ

∂e ∂v ∂e = ρ0 C 2 τ =μ . ∂ζ ∂ζ ∂ζ

(10.3.2)

During the transition region between the two limits, the elastic medium response is gradually replaced by the hydrodynamic one. Their combination is changing over time. Neither the elastic modules nor the transport coefficients describe the relaxation process. The elastic modules are changing during relaxation with increasing dependence on the strain-rate. In Chap. 7, we have shown that after the loading stops t R ≤ t  tr , the stress relaxation is not completed yet. The post-effects are described by the solution √  √  π (ζ − θ ) π (1 − ζ + θ ) τ ∂v

(ζ ; τ, θ ) = erf , ζ > 1. + erf 2 ∂ζ τ τ

(10.3.3)

In this case, the loading duration t R is fixed and the parameters of the solution (10.2.3) are constant. For the large τ >> 1, the post-relaxation is considered frozen and the impactinduced stress remains in the medium after the loading indefinitely long. It means that the medium has non-attenuating memory. τ →∞

J1 = ρ0 C V0 v(ζ ) −−−→ ζ ρ0 C V0 .

(10.3.4)

For the small τ  1, the post-relaxation entirely disappears. The medium forgets the history of its loading. τ →0

J1 = ρ0 C V0 v(ζ ) −−→ ρ0 C V0 τ

∂v → 0. ∂ζ

At the intermediate stage of the relaxation, the medium memory is attenuated in accordance with Bogolyubov’s hypothesis. So, the elastic–plastic transition during continuous loading can be described by the solution (10.3.1). Rewritten in terms of deformation, as we have done in (10.2.3), the relationship (10.3.1) takes a form

322

10 The Stress–Strain Relationships for the Continuous Stationary Loading

T(e; τ, θ ) =

√   √ π (e − θ ) πθ τ ∂e + erf erf . 2 ∂t τ τ

(10.3.5)

At the initial stage of the loading when t  tr , the medium response is elastic τ →∞

T(ζ ; τ, θ ) −−−→ ζ e.

(10.3.6)

During the stage when the relaxation is frozen, the momentum transport is reversible and not accompanied by dissipation. For long loading when t R >> tr , the medium response is a hydrodynamic flow of viscous liquid τ →0

T(ζ ; τ ) −−→ τ

∂e . ∂ζ

(10.3.7)

The hydrodynamic description of plastic behavior of the medium under the loading is valid only near equilibrium within continuum mechanics. The transient zone is far from equilibrium, where multi-scale and multi-stage momentum and energy exchange processes lead to the self-organization of various types of shear and turbulent structures. Such effects cannot arise during quasi-static loading. As we have already seen, the stress–strain relationship (10.3.5) describes all stages of relaxation from elastic one up to hydrodynamic medium response. However, we should remember that the stress in Fig. 10.5 is presented for the constant values of the parameters τ, θ whereas both of them are decreasing during the loading. It means that the relaxation is non-monotone, and the stress will not remain constant and will fall down as the loading duration increases as shown in Fig. 10.7. The model stress–strain relationship (10.3.5) qualitatively describes the medium response to the high-rate loading. In order to describe the properties of a specific material, we should describe the temporal evolution of the parameters τ, θ as in Chap. 8, knowing at least two of their values out of the elastic zone from the experiment. The model parameters evolution determines the stress relaxation inside a sample of a given material to a large extent. It is obvious that the relaxation time should depend on the sample width. The stress relaxation in a thin sample will complete faster than in the thick one. There is no reliable experimental data on plane high-rate continuous loading of materials. It is found out that in rather thin plates, the stress drops very quickly [14]. However, in most practical cases, at high loading rates, after a while the material is destroyed. In order to predict the destruction, it is necessary to control the stress redistribution inside the sample of a given size and shape via the interference of wave packets on the mesoscale (Chap. 9) during high-rate loading. It is an extremely complicated problem. The obtained stress–strain relationship (10.3.5) can only indicate the influence of different factors which are related to the material destruction.

10.4 Reversible and Irreversible Loading–unloading Processes

323

Fig. 10.7 Possible stress–strain curves for the plane dynamic loading; the stress drop with decreasing the relaxation parameters is shown by dotted lines

It should be remembered that this relation was obtained on the basis of an approximate solution to the problem of the propagation of a non-steady wave, provided that the velocity of the mass of matter carried by the wave is small compared to the sound speed. This means that it is incorrect to describe rather long high-rate loading using the relation (10.3.5). The problem of high strain-rates and large deformations cannot be solved within only one scale modeling. It is a multi-disciplinary problem where macroscopic modeling of mechanical medium response to high-rate loading is inextricably linked with multi-scale wave packets dynamic at the intermediate between macro and micro levels.

10.4 Reversible and Irreversible Loading–Unloading Processes The solution (10.2.1) describes the case of two interacting semi-spaces. For the real systems, the loading duration t R is always finite. Then, in any case, it is necessary to take into account the medium response to impact both during the loading and posteffects. We have already known that in general it is impossible to separate reversible

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10 The Stress–Strain Relationships for the Continuous Stationary Loading

and irreversible processes until they are not completed. Therefore, we cannot separate elastic and plastic parts of stress or strain in advance. Now we can consider the loading–unloading process in the medium in the entire range of parameters between the two limits and determine how the process becomes irreversible. In order to do this, we need first to load the medium assuming that the average loading force causes a constant acceleration ∂v/∂ζ = 1. Then at the moment ζ = 2, we unload the medium applying the force in the opposite direction, respectively, with the acceleration ∂v/∂ζ = −1 [15]. As before, all variables with the dimension of time are referred to t R . The loading time interval is described by the solution (10.4.1) √   √ π (ζ − θ ) πθ τ ∂v

1 (ζ ; τ, θ ) = erf , + erf 2 ∂ζ τ τ

0 ≤ ζ ≤ 1.

(10.4.1)

When the unloading follows just after the loading stops, it is necessary to take into account the post-loading effects. Then the stress over the unloading time interval is a combination of the post-loading effect described by the solution (10.3.3) and (10.4.1) with opposite sign

2 (ζ ; τ, θ ) =

√  √  π (ζ − θ ) π (1 − ζ + θ ) τ ∂v erf , + erf 2 ∂ζ τ τ

1 ≤ ζ ≤ 2. (10.4.2)

However, after unloading, the stress continues to relax. In this case, both the effect of loading and unloading must be taken into account. √  √  π (ζ − θ ) π (1 − ζ + θ ) τ ∂v erf − + erf 2 ∂ζ τ τ √   √ π (ζ − 2 − θ ) πθ τ ∂v + erf erf , 2 ≤ ζ ≤ 3; − 2 ∂ζ τ τ (10.4.3) √  √  π (ζ − θ ) π (1 − ζ + θ ) τ ∂v

4 (ζ ; τ, θ ) = erf − + erf 2 ∂ζ τ τ √  √  π (ζ − 2 − θ ) π (3 − ζ + θ ) τ ∂v erf , ζ ≥ 3. + erf − 2 ∂ζ τ τ

3 (ζ ; τ, θ ) =

As a result, we get a general scheme for loading–unloading processes of various durations depending on the values of the relaxation parameters τ, θ . Its large values correspond to short-duration processes while its small values determine quasi-static processes. Indeed, for large values of the relaxation parameter τ >> 1 and low values of the retardation parameters θ, the induced stress J1 = ρ0 C V0 v(ζ ) is proportional to the strain e = (V0 /C)v(ζ ) both during loading and unloading. Large values of

10.4 Reversible and Irreversible Loading–unloading Processes

325

the parameter τ = tr /t R mean that the loading duration is short compared to the relaxation time. In the elastic limit τ → ∞, the relaxation is frozen and we get a reversible loading–unloading process without post-effects τ →∞

1 (ζ ; τ ; 0) −−−→ ζ τ →∞

2 (ζ ; τ, θ ) −−−→ 1 ·

∂v ≈ v, ∂ζ ∂v , ∂ζ

τ →∞

3 (ζ ; τ ; 0) −−−→ (3 − ζ )

ζ ≤ 1; 1 ≤ ζ ≤ 2;

∂v , 2 ≤ ζ ≤ 3; ∂ζ

τ →∞

4 (ζ ; τ ; 0) −−−→ 0,

ζ >> 3.

For low values of the relaxation time or long-duration loading, plastic deformation takes place. We can say that there is an elastic–plastic transition. The final state is not plastic flow but the stress relaxation occurred due to the irreversible structure formation on the mesoscale. The observed formation of local inhomogeneities at the mesoscopic scale level such as the localized shear bands, microcracks, traces of microflows, and rotational structures significantly change the macroscopic properties of the medium. In the opposite case for τ  1 and the retardation parameters θ  1, we are close to the hydrodynamic limit τ →0

1 (ζ ; τ ; 0) −−→ τ

∂v , ∂ζ

τ →0

2 (ζ ; τ, θ ) −−→ 0 , τ →0

3 (ζ ; τ ; 0) −−→ −τ τ →0

∂v , ∂ζ

4 (ζ ; τ ; 0) −−→ 0,

ζ ≤ 1; 1 ≤ ζ ≤ 2; 2 ≤ ζ ≤ 3; ζ >> 3.

In both limiting cases, the post-effects are absent: in the elastic limit, it is due to approaching mechanical equilibrium of internal forces and the hydrodynamic one takes place close to the local equilibrium thermodynamic state. However, the model (10.4.1) is not quite correct in the hydrodynamic case where the wave transport mechanism disappears and only quasi-stationary processes take place. In the transient zone τ ∼ 1 and θ ∼ τ , the waveforms evolve due to the posteffects. The duration of the post-effects grows with the retardation parameter θ . In this case, the waveform represents a wave packet formed due to the dispersive properties of defective material after the high-rate loading. The wave packets propagate at a group speed that is always lower than the phase one. The obtained experimental data

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10 The Stress–Strain Relationships for the Continuous Stationary Loading

show that the waveforms spread out [16] during their propagation like wave packets in quantum mechanics. We see that at low values of the relaxation and delay parameters, long continuous loading induces elastic–plastic transition. The stress remains constant whereas the force continues to act on the medium. Such medium response to long-duration loading at low strain-rates can be described by the model of viscous liquid in the framework of continuum mechanics. This is the final hydrodynamic stage of the system relaxation near local thermodynamic equilibrium. However, in the transition zone that is neither elastic nor plastic, the continuum mechanics models become invalid. Spatiotemporal sizes of the zone depend on the material properties, strain-rate, and loading duration. The higher the strain-rate, the more the transient zone is. At very high strain-rates, the classical hydrodynamic regime of plastic deformation may not be achieved. In Fig. 10.8, the trapezium with base 3 is a completely elastic reversible process when the load is completely removed by unloading that is equal to the load with the opposite sign. The trapezium corresponds to the limiting case τ → ∞. Any decrease in the parameters τ leads to irreversible mass transport which is responsible for plastic effects. Figure 10.9 shows that with an increase in the delay θ that can be caused by the inertia of the condensed medium, the maximum stresses are reached already during unloading after which a zone of significant negative stresses is formed. In this case, the elastic precursor disappears and the plateau of the compressive pulse erodes.

Fig. 10.8 The stress behavior for various values of the relaxation parameter τ and the delay parameter θ = 1

10.5 Entropy Production Surfaces for Various Duration Loading ...

327

Fig. 10.9 The stress behavior for various values of the relaxation parameter τ and the delay parameter θ = 0.1

So, we see that the model of elastic solid is valid at large τ and low θ whereas the model of viscous or ideal liquid is valid at low values of both τ and θ. The most deviation of the system response to loading is observed in the intermediate region τ ∼ 1 with inertial effects increasing. This transient zone is highly nonequilibrium where the models of continuum mechanics become unsuitable. Unlike the quasi-static processes characterized by low strain-rates, the dynamic processes just fall into the transient zone. Therefore, the use of conventional continuum models to describe such processes is incorrect.

10.5 Entropy Production Surfaces for Various Duration Loading and Possible Evolutionary Paths As it is shown in the previous section, the medium response of the material to loading is entirely defined by the strain-rate and the relaxation and delay parameters τ, θ of the integral model developed in Chap. 7. In order to determine whether the considered process is reversible or irreversible, it is necessary to calculate the integral entropy production and construct the surface of its values above the plane of the parameters τ, θ . For the loading–unloading processes, the induced stresses are described by the expressions (104.1)−(10.4.3). The work on the deformation of the material is performed only at the time intervals [0,1] and [2,3]. According to the results of

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10 The Stress–Strain Relationships for the Continuous Stationary Loading

Chap. 8, the integral entropy production during the loading–unloading process is defined as follows: dQ (τ, θ ) ≡ (τ, θ ) = dx

1 0

∂v

1 (ζ ; τ, θ ) dζ − ∂ζ

3

3 (ζ ; τ, θ ) 2

∂v dζ . ∂ζ

(10.5.1)

Here, we take into account that the acceleration during loading is constant ∂v/∂ζ = 1, and during unloading, it changes its sign. The shape of the surface (τ, θ ) is presented in Fig. 10.10. In Fig. 10.10, we see that near the origin where the parameter values are small, a hill rises. It indicates the dissipation zone where the deformation work transforms into heat. The loading–unloading processes characterized by the parameters from this zone are quasi-static. Near the hill, the surface descends into the entropy well with negative values of the integral entropy production. As we have shown in Sect. 9.7, this is the self-organization zone where irreversible structures remain after the loading–unloading process. This zone appears only due to the unloading process. Under continuous loading of semi-space, the surface will grow infinitely as shown in Fig. 10.11. For the loading–unloading processes, the most part of the surface (τ, θ ) is the plane with zero level of the entropy production (Fig. 10.12).

Fig. 10.10 The shape of the integral entropy production surface (τ, θ ) over the model parameters plane

10.5 Entropy Production Surfaces for Various Duration Loading ...

Fig. 10.11 The integral entropy production surface for the semi-space loading

Fig. 10.12 The large-scale surface (τ, θ ) view

329

330

10 The Stress–Strain Relationships for the Continuous Stationary Loading

It means that the medium response to the loading–unloading process with very large values of the parameters is reversible and exists in the medium without a direct relation to its source. In this zone, elastic or acoustic long waves propagate. All of the above zones, in which processes of a completely different nature occur, are interconnected in the evolution of the medium state after the end of loading– unloading.

10.6 Probable Evolutionary Paths and Final States In Chap. 9, we revealed the physical nature of the shock-induced processes on the mesoscale. The mesoscopic processes define both the waveforms generation and their evolution during propagation along the material described in Chap. 8. The shock induces a very short elastic wave of large amplitude. The induced stress causes the entropy growth but cannot exist long. Just during the shock, the wave disperses on the internal structure of the medium and decomposes into wave packets on the mesoscale-1. The stress rapidly falls. The interaction between the spreading wave packets results in the waveforms on the mesoscale-2. Due to the chaotic component of the wave packets motion, the waveform loses a part of its impulse and contributes to the entropy production. The mesoscopic processes very quickly form a hill on the surface of the entropy production inside the waveform, the zone of positive entropy production. The waveform state on the top of the hill is unstable. From here, the waveform evolution during its propagation along the medium begins. In accordance with MEP and SG principles, the evolutionary path should go down (Fig. 10.13). But what direction does the waveform choose? Depending on the medium initial state, loading duration, and strain-rate, the evolutionary paths and final states will be different. Any fluctuation can change the direction of the evolution. The descent from the hill to the origin can lead to a hydrodynamic scenario and result in a localized melting site. When the unloading wave arrives, the descent into the entropy well results in turbulent structures along the waveform path. This is the zone of self-organization where the entropy production inside the waveform is negative (Fig. 10.14). Since only a part of the wave packets energy is absorbed by the structure formation on the mesoscale-1, the waveform continues to propagate in the medium. We know that when the waveform passes the distance a little more than 2 mm, the quasistationary mode of its propagation is established. In this case, the evolutionary path goes along a straight line from the origin to the zone of large parameters τ, θ . The waveform spreading and so-called hydrodynamic attenuation of its amplitude transform it into an elastic wave with a large wavelength. This is the reversible long-wave zone. When the evolutionary path goes down the surface of the integral entropy production, it is possible to change its direction or even to stop the evolution with the help of additional perturbations imposed on the system during its relaxation. In Fig. 10.15, we can see that a series of successive small disturbances delays the evolution of the

10.6 Probable Evolutionary Paths and Final States

Fig. 10.13 The descent from the hill to the origin Fig. 10.14 The descent into the entropy well

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Fig. 10.15 Evolutionary paths on the entropy production surface with perturbations

system. The additional external control of small impulses can make it possible to achieve the desired scenario of the evolution of the structure of the system in order to obtain a material with predetermined properties.

10.7 Fundamental Difference Between Shock and Quasi-Static Loading All basic concepts in the mechanics of elastic and deformable solids were introduced in the framework of continuum mechanics which describes deformation processes without taking into account the internal structure of a material and determines only its macroscopic (integral) plastic and strength characteristics. The mathematical apparatus of continuum mechanics adequately describes the macroscopic response of the material to sufficiently slow loading that does not lead to large plastic deformations. Such conditions determine small deflections of the system state from thermodynamic equilibrium. However, the change in the material internal structure such as the appearance of microcracks, localized shears, and other structural inhomogeneities

10.7 Fundamental Difference Between Shock and Quasi-Static Loading

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as a result of its intensive loading affects its strength. There is an urgent need to study the mechanisms of the defective structure formation and their influence on the material properties. In contrast to continuous mechanics, the theory of dislocations describes the microscopic behavior of a deformable medium and makes it possible to reveal the physical nature of elementary structural inhomogeneities arising in the process of deformation. Dislocations have a charge and interact with each other in a nonlocal manner. Their redistribution causes collective effects that give rise to structures on the mesoscale. However, attempts to connect it with macroscopic medium properties failed. It turned out that at different stages of deformation, the different physical mechanisms of transport of mass, momentum, and energy are involved on different scales. In addition, the classical interpretation of the dynamics of dislocations does not take into account the effects of collective interaction that lead to the appearance of large-scale structures. In particular, in the framework of the theory of dislocations, the process of relaxation of the elastic precursor has not found its explanation [17]. And the reason for this is easy to understand if we take into account that an elastic precursor occurs only when a shock loading is applied to a dense inert medium and is accompanied by long-lasting post-effects. This is a purely dynamic process that cannot occur under quasi-static loading. We can see that the elastic precursor separating the medium response to direct loading and post-effects stands out only as a result of the shock loading whereas under continuous action, no break in the front occurs (Fig. 10.16). Figure 10.17

Fig. 10.16 The difference between the shock-induced waveform and continuous loading

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Fig. 10.17 Formation of the elastic precursor upon shock loading at rather large values of the parameters τ, θ

shows that the elastic precursor appears when the relaxation and delay parameters are large and τ ≥ θ . It must be noticed that in Fig. 10.16 the strain-rate during the continuous loading is approximately 25 times lower compared to the shock loading at the same maximum amplitude. According to the solution obtained in Chap. 7, the elastic precursor reaches its maximum at the time instance when the loading force stops. Then the plastic front rises as a post-shock effect during which the original material state restores after shock. The so-called two-wavefront is not forming during the slow continuous loading at the constant strain-rate. In this case, the elastic precursor cannot stand out and be separated from the plastic front. Moreover, the experimentally observed plateau on the top of the compression pulse is not a plastic flow. On the opposite, this part of the shock-induced waveform corresponds to the restored (though may be partially) original solid material state. The main difference between quasi-static and dynamic processes is determined by the degree of deviation of the medium state from thermodynamic equilibrium. Any dynamic process is highly non-equilibrium that is characterized by large gradients of macroscopic fields and a high rate of their change. The finite propagation speed of disturbances in the medium excites the wave mechanism of the transport of mass,

References

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momentum, and energy. As a result, the material response to high-rate loading comes with a delay and may last after the end of the action of the disturbing force. Dispersion of waves in a inhomogeneous medium and their interference in the presence of a delay can form large-scale vortex-wave structures due to the inertial properties of the medium. As we have shown in Chap. 9, the only physical objects that can be carriers of such properties in solids are wave packets formed by the wave propagation in an inhomogeneous medium. The wave packets move, spread, and interact in a wide range of scales between macro and micro where little-studied processes take place and approaches to their description have not been developed. The dynamics of the wave packets on the mesoscale is the link between macroscopic properties of the deformable solid material and its microstructure. These processes can cause large stress gradients, sharp internal boundaries between elastic and plastic states, and the formation of turbulent structures. A part of them remains frozen in the deformed material as combinations of rotations and shear bands on the mesoscale. The processes on the mesoscale responsible for the high-rate deformation of solid materials occur much earlier than the transition to macroscopic plasticity takes place. The deformation work on the macroscale, as shown in paper [18], does not have time to convert into thermal energy on the microscale because the diffusion transport responsible for the dissipation of energy is too slow. The energy remains on the mesoscale in the form of moving wave packets inside the impact-induced waveform. Unlike quasi-static loading, the time and the dependence on the strain-rate cannot be excluded from the stress–strain relationship for dynamic processes. At high strainrates, the delay and the post-shock effects largely determine the response of the material to shock loading.

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