Continuum Thermodynamics and Constitutive Theory 3030439887, 9783030439880

Table of contents :
Preface
Acknowledgements
Contents
1 Introduction
1.1 Different Approaches to Constitutive Theory
1.2 The Role of Constitutive Theory
References
2 Some Elements of Continuum Mechanics
2.1 Basic Notions
2.2 Kinematics and Motion
2.3 Index Notation
2.4 Generalized Integral Theorems of Gauss and Stokes Type
2.5 Time Derivatives
2.6 Different Measures of Deformation
Reference
3 Balance Equations
3.1 Introduction
3.2 Global Balance Equations for Single-Component Systems
3.3 Local Balance Equations for Single-Component Systems
3.3.1 Mass
3.3.2 Momentum
3.3.3 Angular Momentum
3.3.4 Spin
3.3.5 Total Energy
3.3.6 Internal Energy
3.3.7 Balance of Entropy
3.4 Partial Balance Equations and Equations for the Mixture
3.4.1 Partial Balance of Mass
3.4.2 Partial Balance of Momentum
3.4.3 Partial Balance of Energy
3.5 Maxwell's Equations as Conservation Equations
3.6 Summary
References
4 Some Elements of Thermodynamics
4.1 Thermostatics
4.2 Different Formulations of the Second Law of Thermodynamics
4.3 Gibbs' Equation
References
5 State Spaces and Constitutive Equations
5.1 Introduction
5.2 Properties of the State Space
5.2.1 Small and Large State Spaces
5.2.2 State Space and Thermodynamic Process
5.3 Examples of State Spaces
5.3.1 Global Systems
5.3.2 Examples of State Spaces in Field Formulation
5.3.3 Examples of Constitutive Equations on Small State Spaces
5.3.4 Shape Memory Alloys
5.4 State Space with Internal Variables
5.5 The State Space of Extended Thermodynamics
5.6 Summary
References
6 Thermodynamics of Irreversible Processes (TIP)
6.1 Introduction
6.2 Example 1: A Viscous, Heat-Conducting Fluid
6.3 Example 2: A Heat-Conducting Mixture of Different Chemical Components
6.4 Example 3: Thermoelectricity
References
7 Thermodynamics of Irreversible Processes with Internal Variables
7.1 Introduction
7.2 Example 1: Liquid Crystals
7.2.1 Some Properties of Liquid Crystals
7.2.2 Exploitation of the Dissipation Inequality
7.3 Example 2: Colloid Suspensions
7.3.1 Special Case: Stationary Couette Flow
7.4 Example 3: A Stress-Strain Relation for a Material with After Effects
7.5 Example 4: Irreversible Thermodynamics of Flexible Fibers
7.5.1 Deformation of a Fiber
7.5.2 Balance Equations
7.5.3 Entropy
7.5.4 Constitutive Equations
7.5.5 An Illustrating Example
7.5.6 The Case of a Non-isotropic Orientation Distribution of Fibers
7.6 Example 5: Systems with Internal Mechanical Parts and Analogy …
7.6.1 The Second Law of Thermodynamics
7.6.2 Equations of Motion for Different Mechanical Equipments
7.6.3 Thermodynamical Models
7.7 Example 6: A Viscous, Incompressible Fluid with Two Internal Variables
7.8 Gyarmati's Wave Approach
7.9 Summary of the Method of Irreversible Thermodynamics with Internal Variables
References
8 Rational Thermodynamics
8.1 Introduction
8.2 Structure of the Balance Equations
8.3 Principles Restricting Constitutive Functions
8.3.1 Objectivity or Material Fame Indifference
8.3.2 Material Symmetry
8.4 Representation Theorems for Isotropic Materials
8.4.1 Example: State Space Including One Symmetric Tensor and One Vector
8.4.2 Physical Examples
8.4.3 Example: Reduction of the Constitutive Equation for Linear Elastic Materials
8.5 The Second Law of Thermodynamics
8.5.1 Class of Materials
8.5.2 No-Reversible Direction Axiom
8.5.3 Liu Technique
8.6 Example 1: A Simple Heat-Conducting Viscous Fluid
8.6.1 The Balance Equations on the State Space
8.6.2 The Restrictions on Constitutive Functions and the Entropy Production
8.6.3 Exploitation of Objectivity and Equilibrium Conditions
8.6.4 Equilibrium Conditions
8.6.5 Discussion
8.7 Example 2: A Heat-Conducting Fluid with a Scalar Internal Variable
8.8 Example 3: Material Damage—An Exploitation of Liu Equations …
8.8.1 Balance Equations and Equation of Motion for the Internal Variable
8.8.2 Exploitation of the Second Law According to Liu
8.8.3 Exploitation of the Residual Inequality
8.8.4 Summary
8.9 Example 4: Liquid Crystals of Biaxial Particles or Suspensions of Flexible Fibers
8.9.1 The Choice of the Variables
8.9.2 Exploitation of the Dissipation Inequality
8.9.3 The Entropy Production
8.10 Outlook: Derivatives of Balance Equations as Additional Constraints
8.11 Summary of Rational Thermodynamics' Exploitation of the Dissipation Inequality
References
9 Extended Thermodynamics
9.1 Motivation
9.2 Extended Irreversible Thermodynamics
9.2.1 Example 1: A Heat-Conducting Fluid
9.2.2 Example 2: Rheological Models
9.2.3 Other Examples
9.3 Rational Extended Thermodynamics
9.3.1 The Formal Structure of Rational Extended Thermodynamics
9.3.2 Example: Viscous Heat-Conducting Body
9.4 Summary
References
10 Shock Fronts and Hyperbolic Systems of Differential Equations
10.1 Introduction
10.2 Hyperbolic Differential Equations and the Method of Characteristics
10.2.1 Example: Wave Equation
10.2.2 Well-Posed Initial-Boundary-Value Problems
10.3 The Burgers Equation
10.3.1 Characteristics of the Burgers Equation and Well-Posed Cauchy Problems
10.3.2 Examples of Initial Conditions
10.4 Discontinuous Solutions from Continuous Initial Data
10.5 The Hugoniot Equation as Jump Condition at Field Discontinuities
10.5.1 Weak Solutions
10.5.2 The Hugoniot Condition
10.5.3 The Hugoniot Condition Applied to the Burgers Equation
10.6 Two Uniqueness Theorems in the Case of Diverging Characteristics
10.6.1 The Lax Criterion
10.6.2 The Entropy Condition
10.6.3 Application of the Entropy Criterion to the Burgers Equation
10.7 A Physical Example
10.8 Summary
References
11 A Short Survey of Thermodynamics of Material Surfaces
11.1 Introduction
11.2 Definition of the Surface Fields
11.3 Geometry of the Surface
11.4 Surface Balance Equations
11.5 An Example: The Measurement of Surface Tension
11.6 Summary
References
12 Outlook: Mesoscopic Theory of Complex Materials
12.1 Introduction: Complex Materials
12.1.1 Examples of Internal Structure
12.2 Mesoscopic Concept
12.3 Mesoscopic Balance Equations
12.4 Example: Mesoscopic Theory of Uniaxial Liquid Crystals
12.4.1 Mesoscopic Balance Equations
12.4.2 Macroscopic Balance Equations
12.4.3 Macroscopic Constitutive Quantities
12.4.4 Order Parameters
12.4.5 Differential Equation for the Distribution Function and for the Alignment Tensors
12.4.6 Summary
12.4.7 A Constitutive Equation from the Mesoscopic Background: An Example
12.5 Summary of the Mesoscopic Theory
References
Index

Citation preview

Christina Papenfuß

Continuum Thermodynamics and Constitutive Theory

Continuum Thermodynamics and Constitutive Theory

Christina Papenfuß

Continuum Thermodynamics and Constitutive Theory

123

Christina Papenfuß FB 2 Hochschule für Technik und Wirtschaft Berlin, Germany

ISBN 978-3-030-43988-0 ISBN 978-3-030-43989-7 https://doi.org/10.1007/978-3-030-43989-7

(eBook)

© Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

About 15 years ago, I planned a lecture on continuum thermodynamics for engineering students of the third or fourth year. Special emphasis should be given on constitutive theory and the role of the second law of thermodynamics. At that time, I could not find any textbook giving an overview of the different “schools” of thermodynamic constitutive theory. In the lecture, I wanted to give an overview for the beginner with a critical view on the advantages, possibilities, presuppositions, and limitations of the different approaches. The methods should be illustrated by simple examples. The manuscript for that lecture was the starting point of the present book. Chapters 2 and 3 form a compact introduction to continuum theory, especially kinematics and balance equations. Chapter 4 is devoted to the notion of the thermodynamic state space, the set of variables in the domain of constitutive mappings. Different “schools” of thermodynamic constitutive theory have a different philosophy of choosing this state space. The choice of the state space is the first criterion for classifying the different theories. Chapter 5 deals with thermodynamics, especially the second law and the notion of thermodynamic equilibrium. Beginning with Chap. 6, we start with the exploitation of the second law of thermodynamics with respect to constitutive relations. In Chap. 6, the classical method of Thermodynamics of Irreversible Processes (TIP) is introduced. With the hypothesis of local equilibrium and a classical Gibbs relation, this theory is as close as possible to thermostatics. But in contrast to equilibrium, there are thermodynamic fluxes, like a heat flux. The constitutive relations for the fluxes are derived from the entropy production. This procedure leads to the Fourier’s law for the heat flux, Fick’s law for the diffusion flux, and Newton’s law for the viscous stress tensor. In Chap. 7, internal variables are introduced in addition to the equilibrium variables within the frame of TIP. This allows to model a more diverse material behavior, including non-Newtonian fluids of different kinds and other materials with an internal structure. For example, liquid crystals, colloid suspensions, fiber suspensions, and rheological models are considered here. Rational thermodynamics (see Chap. 8) aims to treat continuum thermodynamics in an axiomatic way. Constitutive functions are restricted by the second law of thermodynamics, material symmetry, and the principle of objectivity. The method of exploiting these restrictions is discussed in detail v

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and applied to various examples. The last two subsections of the chapter are devoted to two topics of current research: In addition to the balance equations, the differentiated equations may be considered in the exploitation of the dissipation inequality. This leads to less restrictive results on constitutive functions than the classical procedure. This is demonstrated in a simple example. Finally, an example with so-called dual internal variables is considered. This extension of the single internal variable formalism allows one to derive a hyperbolic evolution equation (a wave-type equation) for internal variables. This way, non-dissipative and dissipative processes may be treated in a unified way. The balance equations together with the classical constitutive relations of Fourier, Newton, and Fick lead to parabolic differential equations for the wanted fields. This type of differential equation allows for the infinite speed of propagation of disturbances. This unsatisfactory result was the starting point of Extended Thermodynamics (ET), dealt with in Chap. 9—the branch of ET based on rational thermodynamics as well as the branch based on irreversible thermodynamics. The propagation of disturbances can be studied easily on the “toy example” of the Burgers equation, as one can see in Chap. 10. Especially, the propagation of shock fronts is investigated, and the role of the second law in the formation of shock fronts is discussed. The shock fronts in Chap. 10 are treated as surfaces of discontinuity of field quantities without any structure perpendicular to the surface. Chapter 11 is an outlook on material surfaces, equipped with surface densities of mass, momentum, and energy and with fluxes within the surface. For the exploitation of the second law with respect to surface constitutive quantities, the reader is referred to the literature. Chapter 12 is an outlook on the so-called mesoscopic theory of complex materials. The idea of this approach is to enlarge the domain of all field quantities instead of introducing internal variables. The macroscopic field quantities are obtained by an averaging procedure with a distribution function. The example of liquid crystals is treated in detail. This presentation of thermodynamic constitutive theory is far from being complete. Variational formulations and General Equation for the Non-Equilibrium Reversible–Irreversible Coupling (GENERIC) have been left out. In almost all examples, the material is in the fluid state—especially large deformations of solids are not considered at all. Microscopic background theories are out of scope and relativistic as well as quantum effects are not considered. The intention of the book is to give an overview of the wide field of continuum thermodynamics and constitutive theory and to compare the different methods of exploiting the second law of thermodynamics. Chapters 2–8 may be the basis for an undergraduate course for engineers and physicists. The examples are kept simple and have been selected mainly for pedagogical reasons. For an extensive collection of interesting examples of modern physics and constitutive theory, the reader is referred f.i. to the book [1] or the review article [2]. Berlin, Germany

Christina Papenfuß

Preface

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References 1. D. Jou, J. Casas-Vazquez, G. Lebon. Extended Irreversible Thermodynamics. Springer-Verlag, Berlin, Heidelberg, New York (1993) 2. V.A. Vito Antonio Cimmelli, D. Jou, T. Ruggeri, P. Ván, Entropy principle and recent results in non-equilibrium theories. Entropy 16, 1756–1807 (2014)

Acknowledgements

It is my pleasure to thank all those who have contributed to the success of this book: Prof. W. Muschik, who awakened my interest in thermodynamics and continuum physics and who has accompanied my scientific path from my diploma thesis until today with many suggestions and discussions; Prof. G. Brunk, who gave the impulse for my lecture on thermodynamic material theory during my time at the Institute of Mechanics at the TU Berlin; Prof. V. Popov, who supported me until today to offer a lecture on Thermodynamic Constitutive Theory at the TU Berlin, whereby the concept has developed into this book; all my colleagues and students who, with numerous discussions and comments, have contributed to this book; and finally, Dr. M. Hildebrandt, who read this book with much interest and patience before its final completion and who pointed out mistakes and parts that might be incomprehensible for the reader.

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Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Global Balance Equations for Single-Component Systems . 3.3 Local Balance Equations for Single-Component Systems . 3.3.1 Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Internal Energy . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.7 Balance of Entropy . . . . . . . . . . . . . . . . . . . . . . . 3.4 Partial Balance Equations and Equations for the Mixture . 3.4.1 Partial Balance of Mass . . . . . . . . . . . . . . . . . . . 3.4.2 Partial Balance of Momentum . . . . . . . . . . . . . . . 3.4.3 Partial Balance of Energy . . . . . . . . . . . . . . . . . .

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Different Approaches to Constitutive Theory 1.2 The Role of Constitutive Theory . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Some Elements of Continuum Mechanics . . . . . . . . . . . 2.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Kinematics and Motion . . . . . . . . . . . . . . . . . . . . . 2.3 Index Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Generalized Integral Theorems of Gauss and Stokes 2.5 Time Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Different Measures of Deformation . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.5 Maxwell’s Equations as Conservation Equations . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Some Elements of Thermodynamics . . . . . . . . 4.1 Thermostatics . . . . . . . . . . . . . . . . . . . . . 4.2 Different Formulations of the Second Law of Thermodynamics . . . . . . . . . . . . . . . . 4.3 Gibbs’ Equation . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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State Spaces and Constitutive Equations . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Properties of the State Space . . . . . . . . . . . . . . . . . . . . 5.2.1 Small and Large State Spaces . . . . . . . . . . . . . 5.2.2 State Space and Thermodynamic Process . . . . . 5.3 Examples of State Spaces . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Global Systems . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Examples of State Spaces in Field Formulation 5.3.3 Examples of Constitutive Equations on Small State Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Shape Memory Alloys . . . . . . . . . . . . . . . . . . 5.4 State Space with Internal Variables . . . . . . . . . . . . . . . 5.5 The State Space of Extended Thermodynamics . . . . . . . 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Thermodynamics of Irreversible Processes (TIP) . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Example 1: A Viscous, Heat-Conducting Fluid . . . . . 6.3 Example 2: A Heat-Conducting Mixture of Different Chemical Components . . . . . . . . . . . . . . . . . . . . . . 6.4 Example 3: Thermoelectricity . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Thermodynamics of Irreversible Processes with Internal Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Example 1: Liquid Crystals . . . . . . . . . . . . . . . . . . . 7.2.1 Some Properties of Liquid Crystals . . . . . . . 7.2.2 Exploitation of the Dissipation Inequality . . 7.3 Example 2: Colloid Suspensions . . . . . . . . . . . . . . . 7.3.1 Special Case: Stationary Couette Flow . . . . .

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Example 3: A Stress-Strain Relation for a Material with After Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Example 4: Irreversible Thermodynamics of Flexible Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Deformation of a Fiber . . . . . . . . . . . . . . . . . . . 7.5.2 Balance Equations . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4 Constitutive Equations . . . . . . . . . . . . . . . . . . . 7.5.5 An Illustrating Example . . . . . . . . . . . . . . . . . . 7.5.6 The Case of a Non-isotropic Orientation Distribution of Fibers . . . . . . . . . . . . . . . . . . . . 7.6 Example 5: Systems with Internal Mechanical Parts and Analogy to Rheological Models . . . . . . . . . . . . . . . 7.6.1 The Second Law of Thermodynamics . . . . . . . . 7.6.2 Equations of Motion for Different Mechanical Equipments . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.3 Thermodynamical Models . . . . . . . . . . . . . . . . . 7.7 Example 6: A Viscous, Incompressible Fluid with Two Internal Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Gyarmati’s Wave Approach . . . . . . . . . . . . . . . . . . . . . . 7.9 Summary of the Method of Irreversible Thermodynamics with Internal Variables . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Rational Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Structure of the Balance Equations . . . . . . . . . . . . . . . . . 8.3 Principles Restricting Constitutive Functions . . . . . . . . . . 8.3.1 Objectivity or Material Fame Indifference . . . . . . 8.3.2 Material Symmetry . . . . . . . . . . . . . . . . . . . . . . . 8.4 Representation Theorems for Isotropic Materials . . . . . . . . 8.4.1 Example: State Space Including One Symmetric Tensor and One Vector . . . . . . . . . . . . . . . . . . . . 8.4.2 Physical Examples . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Example: Reduction of the Constitutive Equation for Linear Elastic Materials . . . . . . . . . . . . . . . . . 8.5 The Second Law of Thermodynamics . . . . . . . . . . . . . . . 8.5.1 Class of Materials . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 No-Reversible Direction Axiom . . . . . . . . . . . . . 8.5.3 Liu Technique . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Example 1: A Simple Heat-Conducting Viscous Fluid . . . 8.6.1 The Balance Equations on the State Space . . . . . . 8.6.2 The Restrictions on Constitutive Functions and the Entropy Production . . . . . . . . . . . . . . . . .

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8.6.3

Exploitation of Objectivity and Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.4 Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . 8.6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Example 2: A Heat-Conducting Fluid with a Scalar Internal Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Example 3: Material Damage—An Exploitation of Liu Equations and of the Entropy Production . . . . . . . . . . . . . 8.8.1 Balance Equations and Equation of Motion for the Internal Variable . . . . . . . . . . . . . . . . . . . 8.8.2 Exploitation of the Second Law According to Liu 8.8.3 Exploitation of the Residual Inequality . . . . . . . . 8.8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Example 4: Liquid Crystals of Biaxial Particles or Suspensions of Flexible Fibers . . . . . . . . . . . . . . . . . . 8.9.1 The Choice of the Variables . . . . . . . . . . . . . . . . 8.9.2 Exploitation of the Dissipation Inequality . . . . . . 8.9.3 The Entropy Production . . . . . . . . . . . . . . . . . . . 8.10 Outlook: Derivatives of Balance Equations as Additional Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11 Summary of Rational Thermodynamics’ Exploitation of the Dissipation Inequality . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Extended Thermodynamics . . . . . . . . . . . . . . . . . . . . . . 9.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Extended Irreversible Thermodynamics . . . . . . . . . 9.2.1 Example 1: A Heat-Conducting Fluid . . . . 9.2.2 Example 2: Rheological Models . . . . . . . . 9.2.3 Other Examples . . . . . . . . . . . . . . . . . . . . 9.3 Rational Extended Thermodynamics . . . . . . . . . . . 9.3.1 The Formal Structure of Rational Extended Thermodynamics . . . . . . . . . . . . . . . . . . . 9.3.2 Example: Viscous Heat-Conducting Body . 9.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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171 176 176 177

10 Shock Fronts and Hyperbolic Systems of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Hyperbolic Differential Equations and the Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Example: Wave Equation . . . . . . . . . . . . . . . 10.2.2 Well-Posed Initial-Boundary-Value Problems .

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10.3 The Burgers Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Characteristics of the Burgers Equation and Well-Posed Cauchy Problems . . . . . . . . . . . 10.3.2 Examples of Initial Conditions . . . . . . . . . . . . . 10.4 Discontinuous Solutions from Continuous Initial Data . . . 10.5 The Hugoniot Equation as Jump Condition at Field Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 The Hugoniot Condition . . . . . . . . . . . . . . . . . . 10.5.3 The Hugoniot Condition Applied to the Burgers Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Two Uniqueness Theorems in the Case of Diverging Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.1 The Lax Criterion . . . . . . . . . . . . . . . . . . . . . . . 10.6.2 The Entropy Condition . . . . . . . . . . . . . . . . . . . 10.6.3 Application of the Entropy Criterion to the Burgers Equation . . . . . . . . . . . . . . . . . . 10.7 A Physical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . 184 . . . . . 184 . . . . . 186 . . . . . 187 . . . . . 188 . . . . . 188 . . . . . 189 . . . . . 191 . . . . . 192 . . . . . 192 . . . . . 193 . . . .

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199 199 200 201 205 213 215 216

12 Outlook: Mesoscopic Theory of Complex Materials . . . . . . . . . 12.1 Introduction: Complex Materials . . . . . . . . . . . . . . . . . . . . 12.1.1 Examples of Internal Structure . . . . . . . . . . . . . . . 12.2 Mesoscopic Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Mesoscopic Balance Equations . . . . . . . . . . . . . . . . . . . . . 12.4 Example: Mesoscopic Theory of Uniaxial Liquid Crystals . . 12.4.1 Mesoscopic Balance Equations . . . . . . . . . . . . . . . 12.4.2 Macroscopic Balance Equations . . . . . . . . . . . . . . 12.4.3 Macroscopic Constitutive Quantities . . . . . . . . . . . 12.4.4 Order Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.5 Differential Equation for the Distribution Function and for the Alignment Tensors . . . . . . . . . . . . . . .

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219 219 219 221 223 224 226 227 228 229

11 A Short Survey of Thermodynamics of Material Surfaces 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Definition of the Surface Fields . . . . . . . . . . . . . . . . . 11.3 Geometry of the Surface . . . . . . . . . . . . . . . . . . . . . . 11.4 Surface Balance Equations . . . . . . . . . . . . . . . . . . . . 11.5 An Example: The Measurement of Surface Tension . . 11.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12.4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.7 A Constitutive Equation from the Mesoscopic Background: An Example . . . . . . . . . . . . . . . 12.5 Summary of the Mesoscopic Theory . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Chapter 1

Introduction

Abstract The subject of the present book is the different aspects of constitutive theory in phenomenological continuum thermodynamics. Some references to other statistical approaches to the constitutive theory are given.

1.1 Different Approaches to Constitutive Theory There exist many different approaches toward a constitutive theory. We are concerned here only with methods that somehow involve the thermodynamic aspect. We have to distinguish between macroscopic (phenomenological) theories and theories with a more refined background. The macroscopic quantities of the phenomenological theories are directly measurable, whereas all other theories need some statistical averaging procedure. A wide field of statistical theories is based on the phase space, which is spanned by the particle positions and momenta and, for non-spherical particles, also by orientations and angular momenta. A phase space distribution function is introduced that contains information about the positions and momenta of all, say, 1023 particles. This information is not available in experiments. Macroscopic quantities have to be calculated by integrating over particle positions and velocities. The dynamics in this phase space approach is given in terms of an equation of motion for the phase space distribution function. From this dynamics on the phase space, the phenomenological equations have to be recovered. For the averaging procedure, a distribution function is needed. In the kinetic theory based on the Boltzmann equation, this is a one-particle distribution function. From this Boltzmann equation, the phenomenological equations of continuum theory [1–3] and the equations of extended thermodynamics [4, 5] can be derived. However, the Boltzmann equation is restricted to dilute gases, where interactions between atoms or molecules are rare and the motions of particles are independent from each other. In dense gases or liquids, the motion of one particle is influenced by the surrounding particles, and a one-particle distribution function is not sufficient to describe the state of the system. A derivation of the phenomenological equations of classical and extended thermodynamics from the level of many-particle distribution functions is much more complicated [6]. The Boltzmann equation is © Springer Nature Switzerland AG 2020 C. Papenfuß, Continuum Thermodynamics and Constitutive Theory, https://doi.org/10.1007/978-3-030-43989-7_1

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1 Introduction

replaced by an infinite coupled set of equations of motion for the one-, two-, three-, ..., N-particle distribution functions, the BBGKY hierarchy [7–10]. From this theory, as well as from the Boltzmann theory, expressions for constitutive quantities in terms of interaction forces or potentials between molecules are obtained. Another approach starting from the molecular level is molecular dynamics. Newton’s equation of motion is solved for the system of N interacting particles [11–16] with particle numbers N up to 106 , practically limited by the available computing time. The crucial point in all these microscopic models is the assumption about the inter-particle interaction. In the present book, we do not deal with the microscopic background, but we are concerned with phenomenological theories. The present book arose from a two-semester course, given regularly at the Technische Universität Berlin between 1999 and 2005 for graduate students, as well as different one-semester courses given since 2007. The book is intended as a textbook for such a course, as well as a source of material for anybody, who wants to get an overview of the large field of thermodynamic constitutive theory.

1.2 The Role of Constitutive Theory Continuum mechanics undertakes to determine the wanted fields. In classical theories for simple materials, the wanted fields are the five fields of mass density, internal energy density, and the three components of material velocity. Complex materials and polar continua need more than these classical five fields for their description. If the material consists of non-spherical particles, the spin density (density of internal angular momentum) is an additional wanted field. Complex materials have an internal structure that can change under the action of external fields. In order to deal with the constitutive behavior of complex materials, additional (macroscopic) variables are needed, to take into account the internal structure. Examples of such complex materials, which we will consider in later chapters, are liquid crystals, solids with microcracks, protein solutions, fiber suspensions, and polymer melts. There are different approaches toward a constitutive theory of complex materials. We will treat theories with internal variables and the so-called mesoscopic theory. The complexity of the material not only enforces the introduction of additional fields, but also aims to deal with fast phenomena. In the so-called extended thermodynamics, fluxes such as the heat flux and the stress tensor, or even higher order fluxes, are considered as additional wanted fields. The reason for including these additional variables into the set of wanted fields is to avoid the problem of infinite speeds of propagation of disturbances in classical thermodynamics (see Chap. 9). The balance equations and additional differential equations for the internal variables form the set of differential equations for the wanted fields. However, they are not a closed system of equations of motion because additional field quantities, apart from the wanted fields, appear in the equations. Constitutive equations are needed in

1.2 The Role of Constitutive Theory Fig. 1.1 The balance equations and the constitutive equations are the set of field equations to be solved. Republished with permission of [Elsevier], from [17]; permission conveyed through Copyright Clearance Center, Inc.

3

material under consideration

order to create a closed system of these equations. From the physical point of view, it is clear that the balance equations alone cannot determine the evolution of the wanted fields because the balance equations are the same for any material. Constitutive equations are the place where a different material behavior enters the theory. Inserting them into the balance equations, we obtain differential equations whose order depends on the chosen constitutive equations and, therefore, on the material under consideration. Consequently, we have to distinguish between the general balance equations that are valid for arbitrary materials and those balances containing inserted special constitutive equations that are called balances on the state space (because the domain of the constitutive equations is called a state space). Constitutive equations can be formulated only after the domain of the constitutive mappings, the state space, has been fixed (see Fig. 1.1). After this choice, the constitutive mappings are not arbitrary, but they are restricted by physical principles such as objectivity and material symmetry, and by the second law of thermodynamics. The second law, on the other hand, does not determine the constitutive equations completely. After taking into account the second law, there is still some freedom for different forms of constitutive equations. This is expected because constitutive behavior differs for different materials with the same set of variables, compatible with the second law of thermodynamics, i.e., observed in nature. There are different ways to derive these restrictions on constitutive equations from the second law. We will deal with these approaches mainly in historical order. Each chapter will start with an introduction to the ideas, methods, and assumptions of the

4

1 Introduction

respective theories. Afterward, we will discuss several examples for the application of each method. Between the different “schools” of thermodynamic constitutive theory, a lot of “fight” arose in the literature, sometimes with a rather polemic character. For some argumentation of the “Rational school” against “Irreversible Thermodynamics School”, see f.i. [18]. Our aim here is neither to discuss these different lines of argumentation nor to give preference to any of the schools of thermodynamics, but the assumptions and possibilities of each method are presented.

References 1. H. Grad, Principles of the kinetic theory of gases, in Handbuch der Physik XII, ed. by S. Flügge (Springer, Berlin, 1958) 2. S. Chapman, T.G. Cowling, The Mathematical Theory of Nonuniform Gases (Cambridge University Press, Cambridge, 1970) 3. L.E. Reichl, A Modern Course in Statistical Physics. Edward Arnold Ltd 4. D. Jou, J. Casas-Vazquez, G. Lebon, Extended Irreversible Thermodynamics (Springer, Berlin, Heidelberg, New York, 1993) 5. I. Müller, T. Ruggeri, Extended Thermodynamics, vol. 37 (Springer Tracts in Natural Philosophy, Berlin, Heidelber, New York, 1993) 6. W. Dreyer. Molekulare Erweiterte Thermodynamik Realer Gase. Habilitationsschrift, Technische Universität Berlin 7. M. Born, H.S. Green, Proc. Roy. Soc. Lond. A 188, 10 (1946) 8. M. Born, H.S. Green, Proc. Roy. Soc. Lond. A 190, 455 (1947) 9. N. Bogoljuboff, J. Phys. USSR 10, 265 (1946) 10. J.G. Kirkwood, J. Chem. Phys. 14, 180 (1946) 11. D. Baalss, S. Hess, Nonequilibrium molecular-dynamics studies on the anisotropic viscosity of perfectly aligned nematic liquid crystals. Phys. Rev. Lett. 57, 86 (1986) 12. S. Hess, D. Frenkel, M.P. Allen, On the anisotropy of diffusion in nematic liquid crystals: test of a modified transformation model via molecular dynamics. Mol. Phys. 74(4), 765–774 (1991) 13. H. Sollich, D. Baalss, S. Hess, Anisotropy of the viscosity of a nematic discotic liquid crystal via non-equilibrium molecular dynamics. Mol. Cryst. Liq. Cryst. 168, 189–195 (1989) 14. M. Kröger, S. Hess, Viscous Behavior of Polymeric Liquids Investigated by NEMD-Simulations and by a Fokker-Planck Equation (Elsevier, Amsterdam, 1992), p. 422 15. M. Kröger, W. Loose, S. Hess, Structural changes and rheology of polymer melts via nonequilibrium molecular dynamics. J. Rheol. 37, 1057–1080 (1993) 16. S. Hess, W. Loose, Slip flow and slip boundary coefficient of a dense fluid via nonequilibrium molecular dynamics. Phys. A 162, 138–144 (1989) 17. W. Muschik, C. Papenfuss, H. Ehrentraut, A sketch of continuum thermodynamics. J. NonNewton. Fluid Mech. 96, 255–290 (2001) 18. C. Truesdell, Six Lectures on Modern Natural Philosophy (Springer, New York, Wien, Berlin, 1966), pp. 49–59

Chapter 2

Some Elements of Continuum Mechanics

Abstract The Eulerian and the Lagrangian description of continuum mechanics are introduced as well as the material time derivative and the deformation gradient. As mathematical tools for the following chapters, integral theorems and Reynolds’ transport theorem are given.

2.1 Basic Notions Definition: A region G ⊂ R3 of the three-dimensional Euclidean space with boundary ∂G is called a system if: 1. G is simply connected and closed, and 2. ∂G has a piecewise continuous unit normal vector. This is necessary in order to apply the integral theorems of Gauss and Stokes. The complement of G is called the environment of the system. The system is divided into continuum elements. These material elements are large enough to contain so many particles (atoms or molecules) that statistical averaging applies and yet the microscopic structure of matter does not become visible. This means that the linear dimension of the continuum element is much larger than the inter-particle distance. On the other hand, the linear dimension of the continuum element is much smaller than the length scale of interest in a macroscopic theory. Both assumptions together restrict the range of applicability of continuum theory.

2.2 Kinematics and Motion Definition: An observer or a frame of reference is a set of instruments allowing an event to be assigned a set of coordinates: position and time (x, t). The position x in a continuum means the position of the center of mass of the surrounding continuum element. A change of the observer is a change of the set of measuring instruments. © Springer Nature Switzerland AG 2020 C. Papenfuß, Continuum Thermodynamics and Constitutive Theory, https://doi.org/10.1007/978-3-030-43989-7_2

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Fig. 2.1 The configuration mapping between the coordinate at initial time and the coordinate of the same material element in the actual configuration

In the following, we will deal only with non-relativistic theories. Then the timescale for all observers is the same, i.e., the times they measure for the same event can differ only by an unimportant additive constant. This description of events relative to the frame of an observer is called space fixed or Eulerian description . Another possibility is the material or Lagrangian description. A coordinate X is introduced that somehow “labels” the different continuum elements. For convenience, this coordinate X is identified with the position X(·) of the continuum element at an initial time t = 0, i.e., in the reference configuration. The configuration mapping, χt at time t, is the mapping between the material coordinates and the space fixed coordinates (see Fig. 2.1): χt : X −→ x(t) , x(t) = χt (X) = χ (X, t).

(2.1)

The configuration mapping is injective because material elements, being at different positions at an initial time, are at different positions for all later times. This mapping is called motion, if 1. χ is continuous with continuous inverse, i.e., a topological mapping, and 2. the set of mappings for different times is differentiable with respect to time. The first point means that neighboring continuum elements remain neighboring for all times, and vice versa neighboring elements were not separated at former times. This excludes macroscopic cracking of the material as well as “gluing” together. With this presupposition, the Jacobi determinant of the mapping  := det

∂(x1 , x2 , x3 ) = 0 ∂(X 1 , X 2 , X 3 )

(2.2)

exists and is non-zero due to the injectivity of the configuration mapping. The corresponding tensor is the deformation gradient: F=

∂x . ∂X

(2.3)

It is a mapping between material coordinates and space fixed coordinates. In continuum physics, we deal with the material in a region G. In the Eulerian description, this region is time independent. In the Lagrangian description, the region is moving together with the material, i.e., the boundary ∂G is moving with the material

2.2 Kinematics and Motion

7

velocity v. In this case, the region G contains always the same material elements and is called a body . It is also possible to consider a region with an arbitrary time dependence.

2.3 Index Notation We will denote the components of a tensor with respect to a Cartesian basis by Latin indices, with summation convention over indices appearing two times. In general, we will use a Cartesian basis, and we will not distinguish between covariant and contravariant components. If the basis is a space fixed one, i.e., the coordinate system of the space fixed observer, we will use small indices. If the basis is material fixed, i.e., the coordinate system of an observer fixed to the material element, we will use capital indices. With these conventions, the components of the deformation gradient are: ∂ xi . (2.4) Fi K = ∂ XK In invariant notation, scalar products are denoted by a · (dot). Scalar products between second- and higher order tensors are understood in such a way that in the corresponding index notation summation is over the indices that are closest together, for instance: (a · A)k = ai Aik

(2.5)

A : B = Aik Bki .

(2.6)

2.4 Generalized Integral Theorems of Gauss and Stokes Type A general version of the Gauss integral theorem, which includes the classical theorems of Gauss and Stokes as special cases, reads ([1] page 18): 

 ∇ ∗ a(x)dV = G\F

∂G

 n ∗ a(x)d A +

n ∗ |[a(x)]|d A.

(2.7)

F

In this version, the fields, defined in the region G, may be discontinuous at a surface of discontinuity F. |[a]| is the difference of the values of the field quantity a on both sides of the surface of discontinuity: |[a]| = a 1 − a 2 ,

(2.8)

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2 Some Elements of Continuum Mechanics

where the index 1 refers to the side where the positive surface unit normal n points, and the index 2 refers to the opposite side. The symbol ∗ in (2.9) summarizes three different possible operations: multiplication with a scalar field, scalar product with a vectorial or tensorial field, or vector (wedge) product with a vector field. Especially, the theorem of Gauss will be important: 

 ∇ · a(x)dV = G\F

∂G

 n · a(x)d A +

n · |[a(x)]|d A

(2.9)

F

for a vector field a.

2.5 Time Derivatives Field quantities can be expressed in terms of material coordinates (X, t) or in terms ˆ of the coordinates of a space fixed observer (x, t). In the following, we will skip ˆ only by their domains. the symbol “ ˆ ” and distinguish the mappings  and  In the first case, the space variable is not time dependent, and the time derivative of the field quantity, the material time derivative , is: ∂(X, t) d(X, t) ˙ =: (X, t) = . dt ∂t

(2.10)

In the second case, the position variable (the position of the material element) is time dependent, and the total time derivative of the field quantity is: d(x(t), t) ∂(x(t), t) ∂(x(t), t) ˙ =: (x(t), t) = · x˙ + . dt ∂t  ∂ x     convective part

x˙ (t) =

dx dt

=

dχ(X,t) dt

(2.11)

local time derivative

=: v(x, t) is the material velocity.

2.6 Different Measures of Deformation Let us denote the displacement of a material element from position X to position x with u: X + u = x. (2.12) The gradient with respect to the material coordinate X gives: ∂u ∂x ∂X + = ∂X ∂X ∂X

2.6 Different Measures of Deformation

δ+

9

∂x ∂u = =F ∂X ∂X

(2.13)

with the unit tensor δ. For the right-hand side, we have already introduced the deformation gradient ∂∂Xx = F. In addition, we introduce the displacement gradient ∂u =H ∂X

(2.14)

with F=

∂(X + u) ∂x = ∂X ∂X =δ+H.

(2.15)

The deformation gradient has a polar decomposition F = R·U

(2.16)

R · R T = R T · R = δ.

(2.17)

with an orthogonal tensor R:

U describes the stretching of the material element and is therefore denoted as (right) stretch tensor (because the above decomposition is the right polar decomposition). The orthogonal tensor R accounts for the rotation of the material element. A measure of deformation that is independent of the rotation is the Cauchy strain tensor C: C = FT · F T    ∂u ∂u = δ+ · δ+ ∂X ∂X  T  T ∂u ∂u ∂u ∂u =δ+ + + · ∂X ∂X ∂X ∂X = δ + H T + H + H T · H.

(2.18)

In the geometrical linear theory, it is assumed that the displacement gradient is small and nonlinear terms in the displacement gradient can be neglected in the deformation measure. This can be interpreted as a linearization of kinematical relations and has to be distinguished from a linearization of constitutive equations. Applying geometrical linear theory to the Cauchy strain, we obtain: C =δ+

∂u + ∂X



∂u ∂X

T .

(2.19)

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2 Some Elements of Continuum Mechanics

Another measure of deformation (not including the rigid body motions) can be defined:  

∂u T 1 ∂u + = (2.20) 2 ∂X ∂X with C = δ + 2.

(2.21)

Within the geometrical linear theory,  is the symmetric part of the displacement gradient. The question of the dependence of the constitutive function stress tensor on deformation is discussed in Sect. 8.4.3.

Reference 1. J. Behne, W. Muschik, M. Päsler, Ringvorlesungen zur Theoretischen Physik: Theorie der Elektrizität (Vieweg, Braunschweig, 1971)

Chapter 3

Balance Equations

Abstract The extensive quantities of a global system may change due to a flux over the boundary or due to production and supply within the volume, resulting in balance equations for the conserved quantities of mass, momentum, angular momentum, and energy. We start with the global balance equations and derive the local form in regular points of the continuum (where the field quantities are continuously differentiable) as well as at surfaces of discontinuity. In addition to one-component systems, we consider mixtures of different chemical components.

3.1 Introduction We will start with the formulation of the global balance equations for systems consisting of one single chemical component. From these equations, we will derive the local formulation in regular points of the continuum, as well as at surfaces of discontinuity, using Reynolds’ transport theorem. In the last part, we will consider mixtures of different chemical components with chemical reactions. For textbooks on continuum mechanics, see for instance [1–3].

3.2 Global Balance Equations for Single-Component Systems Phenomenological thermodynamics is concerned with so-called Schottky systems ˙ power [4]. A Schottky system G interacts with its surroundings via heat exchange Q, exchange W˙ , and material exchange n˙ e = (n˙ e1 , n˙ e2 , . . . , n˙ eα )T through a partition ∂G (see 3.1). The different components of the row of external change of mole numbers n˙ e correspond to the different chemical components. Extensive quantities are those, which are proportional to the mass of the system, like momentum or energy. In contrast, temperature is an intensive quantity. The global balance equations represent the mathematical formulation of the fact that the amount of an extensive quantity in a volume can change by a flux through the boundary of the system, by production, and by supply (in contrast to production, supply can be © Springer Nature Switzerland AG 2020 C. Papenfuß, Continuum Thermodynamics and Constitutive Theory, https://doi.org/10.1007/978-3-030-43989-7_3

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Fig. 3.1 A Schottky system G exchanges heat, power, and material with its surroundings

suppressed by an appropriate partition). The global balance equation for an arbitrary extensive quantity with specific density ψ, flux J ψ , production density π ψ , supply density σ ψ , and mass density  reads:    d (x, t)ψ(x, t)dV = − J ψ (x, t) · da + (σ ψ (x, t) + π ψ (x, t))dV, dt G(t) ∂G(t) G(t)

(3.1) where da = nda and n is the outward unit normal vector to the surface ∂G. As an example, we consider the balance of mass. Then, we have ψ ≡ 1 and σ 1 ≡ 0, π 1 ≡ 0 which expresses the conservation of mass: d dt



d (x, t)dV = M G (t) = − dt G(t)

 ∂G(t)

J 1 (x, t) · da.

(3.2)

There are different possible choices for the region G: (a) G is called a body if the total mass M G within G is conserved. The body moves with the matter. For a body, we have M G = const. and d G M (t) = 0 = − dt

 ∂G(t)

J 1 (x, t) · da.

(3.3)

(b) G can be an arbitrary region moving with velocity w(x, t) (see Fig. 3.2). Then we have  d G J 1 (x, t) · da. (3.4) 0 = M (t) = − dt ∂G(t) The mass flux through the boundary is given by the convective flux: J 1 =  (v − w) ,

(3.5)

where v is the material velocity. If w = v, the region is a body, and the mass flux through ∂G is zero.

3.2 Global Balance Equations for Single-Component Systems δ

Fig. 3.2 The mapping velocity w is the velocity of the boundary of the region G(t). In general, it is different from the velocity of material elements

13 δ

v

Fig. 3.3 Balance of mass for a tube

(c) If G is fixed in space (w(x, t) = 0), we have d G M (t) = − dt

 ∂G(t)

v(x, t) · da.

(3.6)

As an illustrating example, we consider a tube with cross sections A1 and A2 . The material velocity is parallel to the rotation symmetry axis, i.e., perpendicular to the cross sections. The tube is impermeable (Fig. 3.3). Consequently, M G (t) =

d dt







(x, t)dV = G

1 v1 da − A1

2 v2 da = 1 v¯1 A1 − 2 v¯2 A2 A2

(3.7) with average velocities v¯1 and v¯2 and mass densities 1 and 2 in the cross sections. v¯ A is the mass flux through the cross section. Balance of Momentum, Angular Momentum, and Energy Explicitly, we have the following set of global balance equations if the region G is chosen as a body: Balance of Mass

d G d M = dt dt

 (x, t)dV = 0.

(3.8)

G(t)

M G is the total mass in G. Balance of Momentum The balance of momentum is Newton’s law:  d G d I = (x, t)v(x, t)dV = F r es . dt dt G(t)

(3.9)

14

3 Balance Equations

I G is the total momentum in G, and F r es is the sum of all forces acting on G, which can be divided into surface forces and volume forces:    d (x, t)v(x, t)dV = (x, t) f (x, t)dV + t(x, t) · d A dt G(t) G(t) ∂G(t) (3.10) with stress tensor t and specific volume force g. Balance of Angular Momentum d G J = M. dt

(3.11)

J G is the total angular momentum, and M is the total torque, including the torque resulting from forces. The torque can be split into surface and volume terms, and the angular momentum is the sum of the moment of momentum and an internal angular momentum with density s:  d G d J = (x × (x, t)v(x, t) + s) dV dt dt G(t)  = (x × (x, t) f (x, t) + m(x, t)) dV G(t)  + (x × t(x, t) + (x, t)) · d A

(3.12)

∂G(t)

with volume density of torques m and surface density of torques (couple stresses) . Balance of Energy d G E = − Q˙ + P. dt

(3.13)

P is the power of allin the case of continuous fields forces and torques acting on G, and Q˙ is heat exchange with the surroundings. It is counted as positive if heat is flowing out of the system. For internal energy U G , we have d G U = − Q˙ + P i , dt

(3.14)

where P − P i is the power transformed into a change of kinetic energy. The total energy is decomposed into kinetic energy of translational motion, kinetic energy of rotation, and internal energy, resulting in the balance of energy

3.2 Global Balance Equations for Single-Component Systems

15

   d G d (x, t) v(x, t) · v(x, t) + s(x, t) · θ −1 · s(x, t) + e(x, t) dV E = dt dt G(t)     (x, t)g · v(x, t) + (x, t)m(x, t) · θ −1 · s(x, t) dV q(x, t) · da + =− ∂G(t) G(t)    v(x, t) · t(x, t) + θ −1 · s(x, t) · (x, t) · da + ∂G(t)

(3.15) where the right-hand side is the power of volume forces and torques as well as the power of stresses and couple stresses on the surface ∂G(t). In order to evaluate the time derivative (integrand and region G(t) time dependent), we need the following theorem: Proposition (Reynolds’ transport theorem) [5]: d dt



(x, t)dV  ∂ (x, t) + ∇ · [w(x, t)(x, t)] dV = G(t) ∂t  |[(x, t) {u(x, t) − w(x, t)}]| · da. − 



G(t)

(3.16)

F(t)

Here, (x, t) is a density (an arbitrary field quantity), w(x, t) is the already introduced field of the mapping velocity belonging to G(t), and u(x, t) is the velocity of an interface F(t) in G(t) at which discontinuities may appear. Outside of F(t) all fields are differentiable. The surface element da of F(t) is directed from the side marked by 1 to that marked by 2 . For a derivation of Reynolds’ transport theorem in the case of continuous fields (no surface of discontinuity F). A proof of the more general form of the transport theorem used here later in the derivation of the jump conditions can be found in [6]. The jump conditions will be discussed later in connection with shock waves and related phenomena in Chap. 10. Up to here, it has been assumed that field quantities can be discontinuous, but the values of the fields are finite at each point. Then, the interface is called inert. If the possibility is considered that field quantities behave like the δ-distribution at a surface of discontinuity, the jump conditions are modified. An example of such a “δ-like” localized field quantity is the electric surface charge density at the surface of an electrically conducting body. Other examples are the surface excess quantities introduced by Gibbs [7] in the description of phase boundaries. In a thin region around the phase boundary, the values of the bulk quantities change drastically. Extrapolating the bulk fields to the phase boundary results in jumps of these extrapolated bulk fields (see Fig. 3.4). This jump is assigned to the surface as a (singular) surface density.

16

3 Balance Equations

singular sur face Ψ

-

Ψs =

Ψ −Ψ

Ψ

+ Ψ

Ψs =

+ Ψ −Ψ

xs

G-

G+

x

Fig. 3.4 Definition of the surface fields: + and − are the extrapolated bulk solutions, and is the solution in the presence of the surface (located at xs ). The difference between the extrapolated fields ± and the real value of the field quantity close to the surface defines the surface quantity

s

One can derive a surface balance equation for this surface density [5, 8]. The general form of such a surface balance is shown in Chap. 11. The reader interested in surface and transition zone phenomena can find examples of the above treatment in [9] (chemically reacting multiphase mixtures), [8] (liquid crystals), [10] (general description, thermodynamics of irreversible processes), and [11, 12] (general description, biological membranes). Applying to (3.1) Reynolds’ transport theorem to the time derivative of the integral in (3.1) and the generalized version of Gauss’ theorem (2.9) for a region with a surface of discontinuity F(t) of the fluxes, we obtain an equation where all expressions are volume integrals. If we now identify (x, t) ≡ (x, t)ψ(x, t) (ψ(x, t): specific density, i.e., quantity per unit mass) in (3.16), then (3.1) results in 

 G(t)

  ∂ [(x, t)ψ(x, t)] + ∇ · w(x, t)(x, t)ψ(x, t) dV ∂t   ψ =− ∇ · J (x, t)dV + (σ ψ + π ψ )(x, t)dV G(t)



G(t)



{ − J ψ (x, t) + (x, t)ψ(x, t){u(x, t) − w(x, t)} · n f }da. (3.17)

+ F(t)

Here, u is the velocity of the surface F of discontinuity with unit normal vector n f . Since in this equation all differentiations are under the integrals by the help of Reynolds’ transport theorem and Gauss’ theorem, we can now derive local balance equations.

3.3 Local Balance Equations for Single-Component Systems

17

3.3 Local Balance Equations for Single-Component Systems Since (3.17) is valid for an arbitrary region G(t) including an arbitrary interface F(t), the following local balances are valid:  ∂ (ψ) + ∇ · wψ + J ψ − σ ψ − π ψ = 0, ∂t



ψ(u − w) − J ψ · n f = 0.

(3.18) (3.19)

These are the balances and the jump conditions at interfaces of a one-component system which we will now discuss in more detail. The flux J ψ can be split into a non-convective part J ψ,nonc and the convective part  (v − w) ψ: J ψ = J ψ,nonc +  (v − w) ψ = J ψ,nonc + J 1 ψ.

(3.20)

The convective part vanishes if G(t) is a body (v ≡ w).

3.3.1 Mass We obtain from (3.18) and (3.19) the local balance of mass and its jump condition if we consider the mass density: φ = ρ with specific density ψ = 1.



∂  + ∇ · (v) = 0, (u − v) · n f = 0. ∂t

(3.21)

3.3.2 Momentum The momentum density (x, t)v(x, t) is defined in such a way that its integral over G is the momentum in G. The balance of momentum reads: ψ ≡ v,

J v ≡ −t  + J 1 v, σ v ≡  f , π v ≡ 0.

(3.22)

According to (3.22), the non-convective part of the flux of momentum is the negative transposed Cauchy stress tensor t. For the proof that this flux can be written in such a way with a second-order tensor t, see for instance [3]. The supply of momentum is the external force density (x, t) f (x, t), and no production of momentum appears. Consequently, (3.18) and (3.19) result by (3.22) in:

18

3 Balance Equations

 ∂ (v) + ∇ · vv − t −  f = 0, ∂t



v(u − v) + t · n f = 0.

(3.23) (3.24)

3.3.3 Angular Momentum The specific angular momentum is the sum of the moment of momentum and the specific spin s(x, t) (internal angular momentum, for instance, due to rotations of elongated particles in liquid crystals): S(x, t) := x × v(x, t) + s(x, t).

(3.25)

We have ψ ≡ S,

J S ≡ −[x × t] −  + J 1 s, σ S ≡ [x × f + m], π S ≡ 0 .

(3.26)

Additional to the angular momentum caused by the stress tensor [x × t] (x, t), we have a couple stress field (x, t) acting on the specific spin s(x, t). The supply of angular momentum is the momentum of the external force density and an additional external couple force g(x, t). The production terms of angular momentum are zero. Consequently, (3.18) and (3.19) result by (3.26) in: ∂ (S) + ∇ · (vS − (x × t) −  ) − (x × f + m) = 0, ∂t



S(u − v) + (x × t) +  · n f = 0.

(3.27) (3.28)

3.3.4 Spin If we multiply x×(3.23) and subtract this from (3.27), we obtain the spin balance from which its jump condition follows: ∂ (s) + ∇ · (vs −  ) −  : t − m = 0, ∂t



s(u − v) +  · n f = 0.

(3.29) (3.30)

Here,  is the Levi-Civita tensor by which the antisymmetric part of the stress tensor is introduced into the spin balance (3.29). The field  : t(x, t) is the spin production, and m(x, t) the field of spin supply density.

3.3 Local Balance Equations for Single-Component Systems

19

3.3.5 Total Energy The total specific energy density is defined by: u(x, t) :=

1 1 2 v (x, t) + s(x, t) · −1 (x, t) · s(x, t) + e(x, t). 2 2

(3.31)

Here (x, t) is a moment of inertia tensor field which describes the connection between spin and rotational energy, and e(x, t) is the field of internal energy density. To state the balance of total energy, we identify: ψ ≡ u, J e ≡ −t · v −  · −1 · s + q + J 1 e, σ e ≡ ( f · v + m · −1 · s + r ), e π ≡ 0.

(3.32) (3.33) (3.34)

Here q(x, t) is the field of the heat flux density, and r (x, t) is the supply by energy absorption. Consequently, the balance of total energy can be expressed:  ∂ (u) + ∇ · vu − t  · v −  · −1 · s + q ∂t − f · v − m · −1 · s − r = 0,

(3.35)

and the jump condition of the total energy is:



u(u − v) + t  · v +  · −1 · s − q · n f = 0.

(3.36)

3.3.6 Internal Energy Subtracting v·(3.23) and (3.29)·−1 · s, we obtain from (3.35) the balance of internal energy: ∂ (e) + ∇ · (ve + q) − ∇v : t − ∇(−1 · s) : ∂t − r + +(−1 · s) ·  : t = 0.

(3.37)

The jump condition of the internal energy follows by (3.19) from the comparison of (3.37) with (3.18):



e(u − v) − q · n f = 0,

(3.38)

under the presupposition that there is no singular energy production at the surface of discontinuity.

20

3 Balance Equations

Let us summarize the balance equations in regular points of the continuum for a material without internal angular momentum: Mass ∂ + ∇ · (v) = 0 ∂t

(3.39)

Momentum  ∂ ((x, t)v(x, t)) + ∇ · −t T (x, t) + (x, t)v(x, t)v(x, t) ∂t −(x, t) f (x, t) = 0

(3.40)

Internal Energy ∂ ((x, t)e(x, t)) + ∇ · (q(x, t) + v(x, t)(x, t)e(x, t)) ∂t = r (x, t) + t(x, t) : (∇v(x, t))

(3.41)

These are the differential equations for the five wanted fields of mass density , material velocity v, and internal energy density e. If there is an internal angular momentum density present in the material, the balance of moment of momentum has to be considered, in addition, and the balance of energy is modified (see above). The balances are valid for any material. They are linear differential equations of first order in time and space. In order to make a closed system of equations from these, constitutive equations for the fields of stress tensor t, heat flux q, and energy absorption density r are needed. The order of the differential equations can change after constitutive equations are inserted, but they are still linear in the highest derivatives, i.e., they are quasilinear. Finally, the balance equations can be written in a more compact form using the balance of mass and the definition of the material time derivative. The balance of mass is multiplied with the field ψ and subtracted from the general balance equation. + v · ∇ψ =  dψ , we obtain the general balance equation: With  ∂ψ ∂t dt (x, t)

d ψ(x, t) + ∇ · J ψ,nonc (x, t) − σ ψ (x, t) − ψ (x, t) = 0. dt

(3.42)

In this form, the flux is the non-convective one. Explicitly, we have: Mass d + ∇ · v = 0 dt

(3.43)

3.3 Local Balance Equations for Single-Component Systems

21

Momentum d (v(x, t)) − ∇ · t T (x, t) − (x, t) f (x, t) = 0 dt

(3.44)

de(x, t) + ∇ · q(x, t) = r (x, t) + t(x, t) : (∇v(x, t)) dt

(3.45)

(x, t)

Internal Energy (x, t)

Remark: In the formulation of the balance equations, the notion of temperature does not appear. The thermodynamical quantities, in addition to purely mechanical ones, are the internal energy density and the heat flux density. If the temperature is introduced, it is a variable in the domain of the constitutive mappings, an element of the state space (see next chapter).

3.3.7 Balance of Entropy The different formulations of continuum thermodynamics assume that, to a region G, there can be assigned an additional extensive quantity, entropy, for which a balance equation holds. A corresponding density can be introduced, and the local form of the balance equation can be derived. The specific entropy density is denoted with η, the non-convective entropy flux with φ, entropy supply with z, and the entropy production density with σ . For the entropy, we have the balance equation: ∂ ((x, t)η(x, t)) + ∇ · (φ(x, t) + (x, t)η(x, t)v(x, t)) − z(x, t) = σ (x, t) ∂t

(3.46)

d η(x, t) + ∇ · φ(x, t) − z(x, t) = σ (x, t) dt

(3.47)

(x, t)

Integration of the above equation over the system volume leads to the global balance of entropy: dS ˙ − Z = . − (3.48) dt ˙ is the entropy exchange over the boundary with the S is the system entropy,  environment, Z is the entropy supply, and  is the entropy production. The role of the entropy in the formulation of the second law of thermodynamics will be discussed in Chap. 5.

22

3 Balance Equations

3.4 Partial Balance Equations and Equations for the Mixture In this subsection, we shall deal with mixtures consisting of different chemical components that can undergo chemical reactions [13–15]. The different chemical components are numbered by an index ν. Let us consider an example of a chemical reaction: (3.49) 4  Fe +3 O2 −→ 2 Fe2 O3 .



  ν=1

ν=2

ν=3

The balance equations for the different chemical components are called partial balance equations, and the densities of the different chemical components are partial densities. From the chemical reaction equation, it is obvious that there is no conservation of mass of a particular chemical component. Also, momentum or energy is not conserved for a single chemical component, but only for the mixture as a whole. The axiom of mixture theory states: The balance equations for the mixture are the same as the balance equations for a single-component system. From the standpoint of the balance equations, the mixture cannot be distinguished from a single-component system. In this subsection, we assume that all field quantities are continuous, i.e., no surface of discontinuity exists.

3.4.1 Partial Balance of Mass For the mass density of a particular chemical component, there maybe production due to chemical reactions, but no supply and no non-convective flux. Therefore, the partial balance of mass of chemical component ν reads: ∂ ν + ∇ · (ν v ν ) = Pνchem . ∂t

(3.50)

Pνchem is the production of mass of component ν due to chemical reactions. The mass densities of the different chemical components are additive: 

ν = .

(3.51)

ν

Taking the sum of (3.50) over the chemical components, we have:   ∂  ν +∇ · Pνchem . (ν v ν ) = ∂t ν ν ν

  

(3.52)

3.4 Partial Balance Equations and Equations for the Mixture

23

Due to (3.51), this is the balance of mass. Comparing with (3.39), we obtain for the production terms:  Pνchem = 0 (3.53) ν

and for the flux terms (the terms under the divergence): 

ν v ν = v,

(3.54)

ν

which states the additivity of partial momenta. The material velocity of the mixture is the weighted sum of the velocities of the components: v=

 ν ν



vν .

(3.55)

3.4.2 Partial Balance of Momentum For the momentum of chemical component ν, there are the flux term and the production term due to external forces analogous to the equation for a single component and an additional production term due to chemical reactions:  ∂ (ν v ν ) + ∇ · ν v ν v ν − tν T =  fν + P momentum . (3.56) ν ∂t P momentum is the production density because of chemical reactions. ν Comparing (3.40) and (3.56), we have:  ν

ν f ν + P momentum =  f, ν

and vv − t T =

(3.57)

 ν v ν v ν − t νT = ν

 ν vv + ν δv ν δv ν − t νT , =

(3.58)

ν

where we have defined: δv ν = v ν − v. It follows: t=



(t ν − ν δv ν δv ν ) .

ν

The stress tensors of the chemical components are not additive.

(3.59)

(3.60)

24

3 Balance Equations

3.4.3 Partial Balance of Energy The balance of energy can be treated the same way as the balance of mass and the balance of momentum, i.e., by stating the equations for the components with a production due to chemical reactions and comparing the sum of the equations with the total balance of energy. We will give here only the result in the case of a simple fluid (s = 0). The internal energy e of the mixture: e=



(eν + ν (v − v ν ) · (v − v ν ))

(3.61)

ν

as well as the heat flux q: q=

 ν

qν +

ν v (v − v ν ) · (v − v ν ) + v ν · v ν (v − v ν ) + (e − eν ) v ν 2

− (t ν − t) · v ν − t · (v − v ν )) .

(3.62)

Thus e and q are not additive.

3.5 Maxwell’s Equations as Conservation Equations If the material is interacting with electromagnetic fields, we have additional wanted fields. Maxwell’s equations are the equations of motion for the electromagnetic fields E (electric field) and B (magnetic induction). In regular points, i.e., in points where the fields are continuously differentiable, they read: ∇ · D = ρe ∇·B=0

(3.63) (3.64)

∇×H = J+

(3.65)

∂D ∂t ∂B ∇×E=− , ∂t

(3.66)

where D is the dielectric displacement, H is the magnetic field, ρ e is the density of (free) electric charges, and J is the current of free electric charges. The sources ρ e and J are given from outside, and D and H are connected with E and B by constitutive laws:

3.5 Maxwell’s Equations as Conservation Equations

D = ε0 E + P M 1 B− , H= μ0 μ0

25

(3.67) (3.68)

where P is the electric polarization, M is the magnetization, and 0 , μ0 are the susceptibility and permittivity of the vacuum. Polarization and magnetization are constitutive quantities. At surfaces of discontinuity of the fields, jump conditions can be derived similar to those discussed in Sect. 3.3 (see also Chap. 10). Maxwell’s equations have to be solved together with the balance equations after inserting constitutive relations. However, Maxwell’s equations are not of balance type, i.e., they do not have the form of a general local balance equation (3.18). On the other hand, exploiting the second law of thermodynamics with respect to constitutive equations, all equations of motion have to be taken into account. There, the fact that Maxwell’s equations are not of balance type causes problems. However, from Maxwell’s equations, two different forms of charge conservation equations can be derived. The form appropriate for the exploitation of the dissipation inequality is the balance of total electric charges, including polarization charges. In this form, the charge density and the flux are constitutive quantities. We will discuss the derivation of this conservation equation here. Maxwell’s equations can be rewritten as: ˇ = ρ tot ∇·D ∇·B=0 ˇ ∂D ∇ × Hˇ = J tot + ∂t ∂B ∇×E=− , ∂t

(3.69) (3.70) (3.71) (3.72)

introducing the quantities: ˇ = D− P D M . Hˇ = H + μ0

(3.73) (3.74)

Then the source terms in (3.69) to (3.72) are calculated from Maxwell’s equations with (3.68): ˇ = ∇ · D − ∇ · P = ρe − ∇ · P ρ tot = ∇ · D

(3.75)

26

3 Balance Equations

and ∂D 1 = ∇ × H + ∇ × M/μ0 = ∇ × Hˇ ∇×M+ μ0 ∂t ˇ ∂D ∂D ∂P = J tot + = J tot + − (3.76) ∂t ∂t ∂t 1 ∂P + = J+ ∇ × M. (3.77) ∂t μ0

J+

⇒

J tot

The form of Maxwell’s equations (3.69) to (3.72) with these new quantities is the same as (3.63) to (3.66). However, in (3.69) to (3.72), the sources ρ tot and J tot are constitutive quantities, in contrast to ρ e and J, which are given from outside. ρ tot and J tot are the density and the current not only of free electric charges but also of ˇ and Hˇ are related to E and B by Maxwell’s ether polarization charges. The fields D relations, i.e., they are related the same way as in vacuum: ˇ = ε0 E D 1 B. Hˇ = μ0

(3.78) (3.79)

These are no constitutive quantities. Note that some confusion arises in the literature because (3.69) to (3.72) are also called Maxwell’s equations [16]. For the density of free electric charges, we have the continuity equation: ∂ρ e + ∇ · J = 0. ∂t

(3.80)

However, this form of the conservation equation is not appropriate for the exploitation of the second law of thermodynamics, because the free electric charges ρ e and the current J are given from outside. They are no constitutive functions. From (3.80), a continuity equation follows for the total charge density: ∂ρ tot + ∇ · J tot = 0 ∂t

(3.81)

because ∂ρ tot ∂P ∂P 1 ∂ρ e + ∇ · J tot = −∇ · +∇ · J +∇ · + ∇ · (∇ × M) = 0. ∂t ∂t ∂t ∂t μ0 (3.82) The field ρ tot and the flux J tot are material-dependent quantities because the polarization charges ∇ · P are constitutive quantities, defined on the state space.

3.5 Maxwell’s Equations as Conservation Equations

27

Equations (3.80) and (3.81) are balance-type equations derived from Maxwell’s equations for the electromagnetic fields. This balance-type form is suitable to be taken into account together with the balance equations of mass, momentum, energy, and angular momentum in an exploitation of the dissipation inequality according to Liu. This is not possible directly with Maxwell’s equations. In addition, there are contributions of the electromagnetic field to the total momentum, angular momentum, and energy. We will not deal with an example of such a treatment of dynamic electromagnetic phenomena here, but examples can be found in [8, 17]. Also, in irreversible thermodynamics, the treatment of electromagnetic phenomena, not restricting ourselves to quasi-statics, requires a balance-like structure of the equations (see, for example, [18, 19]).

3.6 Summary We started from the global balance equations for the extensive quantities mass, momentum, and energy. These equations express the fact that the amount of these quantities contained in a region G(t) can change in time by production or supply within G or by a flux over the boundary of G. The time derivative of the volume integral over the time-dependent region G(t) has been transformed using Reynolds’ transport theorem, and Gauss’ theorem has been applied to the flux term. The resulting form of the global balance equation is appropriate to derive the local balance equations. The local balance equations are the differential equations that determine the time evolution of the wanted fields. They are not a closed set of equations but require the addition of constitutive equations. These are discussed in the following chapters. For textbooks on continuum mechanics, see [1–3, 16, 20] We have started with balance equations for single-component systems. In the case of mixtures of different chemical components undergoing chemical reactions, partial balance equations for the components have been formulated. For a particular chemical component, mass, momentum, and energy are not conserved. It is an axiom (empirical fact) that the balance equations of the whole mixture look the same as the balance equations of a single-component system. Comparing the partial balance equations and the equations of the mixture, it follows that the constitutive quantities are not additive. Stress tensor, heat flux, and internal energy of the mixture are not the sum over the different chemical components of these quantities. Finally, we derived a form of conservation equation of electric charge with constitutive quantities. This balance-type equation can be taken into account in the exploitation of the second law of thermodynamics in the presence of electromagnetic fields.

28

3 Balance Equations

References 1. J. Altenbach, H. Altenbach, Einführung in die Kontinuumsmechanik. Studienbücher (TeubnerVerlag, Stuttgart, 1994) 2. A. Bertram, Axiomatische Einführung in die Kontinuumsmechanik (B.I. Wissenschaftsverlag, Mannheim, Wien, Zürich, 1989) 3. E. Becker, W. Bürger, Kontinuumsmechanik (Teubner Studienbücher, Stuttgart, 1975) 4. W. Schottky, Thermodynamik (Springer, Berlin, 1929) 5. T. Alts, K. Hutter, I: surface balance laws and their interpretation in terms of three dimensional balance laws averaged over the phase change boundary layer, II: Thermodynamics, III: thermostatics and its consequences, IV: on thermostatic stability and well-posedness. J. Non-Equilib. Thermodyn. 13, 221–301 (1988) 6. W. Muschik, Fundamentals of nonequilibrium thermodynamics, in Non-equilibrium Thermodynamics with Applications to Solids, ed. by W. Muschik. CISM Courses and Lectures, vol. 336 (Springer, Wien, New York, 1993), pp. 1–63 7. J.W. Gibbs, Collected Works (Yale University Press, New Haven, 1948) 8. C. Papenfuss, Contribution to a Continuum Theory of Two Dimensional Liquid Crystals (Wissenschaft- und Technik Verlag, Berlin, 1995) 9. R. Prud’homme, Fluides hétérogènes et réactifs: écoulements et transferts, Lecture Notes in Physics (Springer, Berlin, Heidelberg, New York, 1988) 10. D. Ronis, D. Bedeaux, I. Oppenheim, On the derivation of dynamical equations for a system with an interface I: General theory. Phys. A 90A, 487–506 (1978) 11. D. Ronis, I. Oppenheim, On the derivation of dynamical equations for a system with an interface: II The gas-liquid interface. Physica 117A, 317–354 (1983) 12. A. Grauel, Feldtheoretische Beschreibung von Grenzflächen (Springer, Berlin, New York, London, 1989) 13. W.H. Müller, W. Muschik, Bilanzgleichungen offener mehrkomponentiger Systeme I Massenund Impulsbilanzen. J. Non-Equilib. Thermodyn. 8, 29–46 (1983) 14. W. Muschik, W.H. Müller, Bilanzgleichungen offener mehrkomponentiger Systeme II. Energieund Entropiebilanz. J. Non-Equilib. Thermodyn. 8, 47–66 (1983) 15. W. Muschik, C. Papenfuss, H. Ehrentraut, Concepts of Continuum Thermodynamics (Kielce University of Technology, Technische Universität, Berlin, 1996). ISBN 83-905132-7-7 16. I. Müller, Thermodynamics (Pitman Advanced Publishing Program, Boston, 1985) 17. I. Müller, Grundzüge der Thermodynamik (Springer, Berlin, 1994) 18. J. Verhas, C. Papenfuss, W. Muschik, A simplified thermodynamic theory for biaxial nematics using Gyarmati’s principle. Mol. Cryst. Liq. Cryst. 215, 313–321 (1992) 19. C. Papenfuss, J. Verhas, W. Muschik, A simplified thermodynamic theory for biaxial nematics. Z. Naturforsch. 50a, 795–804 (1994) 20. K. Wilma´nski, Thermomechanics of Continua (Springer, Berlin, 1998)

Chapter 4

Some Elements of Thermodynamics

Abstract The zeroth, first, and second law of thermodynamics are fundamental axioms. For the second law of thermodynamics, there exist different formulations, f.e. the non-existence of a perpetuum mobile of the second kind, Clausius formulation or the statement, that the entropy production is non-negative in each volume element at any time.

4.1 Thermostatics In this section, we shall deal mainly with global systems. Definition: Time-independent states of isolated systems are called states of equilibrium. Definition: A partition is called adiabatic, if it does not let any heat flux pass through it. A system is called thermal homogeneous, if it does not contain any adiabatic partition in its interior. Zeroth law of thermodynamics: The state space of thermal homogeneous systems in equilibrium is (4.1) Z ∗ = {a, n, ∗}, where a is the row containing the work variables, n is the row of mole numbers of the different chemical components, and * is exactly one thermodynamical variable. This variable can be the internal energy U or the thermostatic temperature T . The work variables are defined in such a way that the work differential reads: dW =



Aα daα ,

(4.2)

α

with so-called affinities Aα . This means that equilibrium states of thermal homogeneous systems are uniquely characterized by the mole numbers n ν , the work variables aα , and one thermal variable. © Springer Nature Switzerland AG 2020 C. Papenfuß, Continuum Thermodynamics and Constitutive Theory, https://doi.org/10.1007/978-3-030-43989-7_4

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4 Some Elements of Thermodynamics

Fig. 4.1 Equilibrium states form a subspace of the non-equilibrium state space. z(·) is a thermodynamic process. A projection P ∗ from the non-equilibrium space onto the equilibrium subspace induces a trajectory z ∗ (·) in equilibrium subspace (an accompanying process)

Z

z( ) P*

thermodynamic process

z*( )

.

.

A

B equilibrium subspace

The dimension of any non-equilibrium state space Z is higher than the dimension of the equilibrium state space Z ∗ . Therefore, there exist projections P ∗ from the non-equilibrium state space onto the equilibrium subspace (see Fig. 4.1). Definition: A trajectory z ∗ in the equilibrium subspace induced by a projection P ∗ (i.e., a mapping with (P ∗ )2 = P ∗ ) of a process z(•): P ∗ z(t) = z ∗ (t) P ∗ (a, n, . . . , ∗, a, ˙ n, ˙ . . . )(t) = (a, n, ∗)(t) is called an accompanying process. Embedding axiom: Extensions of any quantity M to non-equilibrium have to be compatible with the definition in equilibrium. Therefore, for any path z in state space with end points A and B in equilibrium subspace, it holds for the integrals along the different paths: 

B

z A

eq eq ˙ M(t)dt = MB − M A = z∗



B

M˙ ∗ (t)dt.

(4.3)

A

The equilibrium state space (4.1) is the state space of thermostatics. In a field formulation of thermostatics, the state space includes the corresponding fields of the work variables, the mole numbers, and one thermal variable: ∗ = {a(x, t), n(x, t), ∗(x, t)}. Zfield

(4.4)

No gradients are present in this state space. A possible choice of the thermal variable is the thermostatic temperature.

4.1 Thermostatics

31

In the case of a single-component system, we have only one mole number n. If no external fields are present, the only work variable of a fluid is the volume, and we have for the state space of a discrete system: Z ∗ = {V, n, T }.

(4.5)

The First Law of Thermodynamics is the balance of energy d G E = − Q˙ + P dt

(4.6)

˙ with power P and heat exchange Q.

4.2 Different Formulations of the Second Law of Thermodynamics There exist many different formulations of the second law [1, 2], which are not all equivalent. This is due to the fact that there is no unique and natural generalization of the notion of entropy defined in equilibrium to non-equilibrium [3]. Similar problems occur with the definition of non-equilibrium temperature [4, 5]. 1. The eldest formulation of the second law is a verbal one: There is no perpetuum mobile of the second kind. A perpetuum mobile of the second kind is a machine taking energy from a reservoir and transforming this energy completely into mechanical work, without changes in the environment. It is an empirical fact that such a machine cannot be constructed. The next formulation is the one given by Clausius. It deals with the whole system (not a field formulation), and it is global in time in the sense that a whole cyclic process is considered. The time integral is one over the cyclic process, i.e., after the process, the system is in its initial state again (but there are changes in the environment). 2. This formulation of the second law of thermodynamics can be shown to follow  ˙ dt ≤ 0 for a closed system and from the first formulation: Q(t) T (t)    Q(t)  ˙ e − ν μν n˙ ν dt ≤ 0 for a system exchanging n eν moles of chemical comT (t) ponent ν per unit time with its environment. μν is the chemical potential of component ν in the surrounding reservoir. In both cases, the system is in contact with a reservoir, which by definition is always in equilibrium. The temperature T is the (equilibrium) temperature of the reservoir. Q˙ is the heat exchange rate from the reservoir to the system. An example of a cyclic process considered in this formulation is sketched in Fig. 4.2.

32

4 Some Elements of Thermodynamics

3. The next formulation of the second law of thermodynamics is global in time and in space, too: In an isolated system, the equilibrium entropy increases in time, with the system going from an equilibrium state A to a new equilibrium state B. This spontaneous process in an isolated system is the result of a removing of internal hindrances: eq

eq

SB > S A .

(4.7)

This formulation avoids the problem of a definition of entropy in non-equilibrium. 4. The next formulation is related to the one above, but here a spontaneous process from a non-equilibrium state to an equilibrium state in an isolated system is considered: In an isolated system relaxing from a non-equilibrium state C to an equilibrium state D, the entropy increases in time: eq

S D > SC .

(4.8)

5. A formulation local in time states: In an isolated system, the (global) entropy S does not decrease in time: ˙ ≥ 0 ∀t. S(t)

(4.9)

Such a formulation local in time seems to be more restrictive than the above formulations global in time. However, it can be shown that they are equivalent [1, 6]. 6. Finally, the most restrictive formulation is local in time and space. The formulations 1 to 5 follow from this one, but they are not equivalent. This local formulation is the one that “fits” to the continuum theory, and it is the one which will be used later to exploit the second law of thermodynamics: Dissipation inequality: The entropy production is non-negative at each position and each instant: (x, t)

d η(x, t) + ∇ · φ(x, t) − z(x, t) = σ(x, t) ≥ 0 ∀(x, t). dt

(4.10)

The entropy production is zero only in equilibrium states.

4.3 Gibbs’ Equation The so-called Gibbs relation arises from the balance of entropy if a special constitutive ˙ namely, the expression equation for the entropy exchange with the surroundings , ˙ ˙ = Q˙ , from thermostatics for a heat exchange Q with a heat bath of temperature T ,  T

4.3 Gibbs’ Equation

33

Fig. 4.2 In this example of a cyclic process, the system is 1. in contact with a heat reservoir of temperature T1 , 2. with a reservoir of temperature T2 , and 3. with a heat reservoir of temperature T3 . In the first two steps, heat is flowing from the system to the heat baths, and in the last step, heat is flowing back to the system such that the energy of the system after Step 3 is the same as before Step 1

is inserted. It results in: Global formulation dS ˙ = − dt dS 1 − Q˙ =  dt T 1  ˙G dS i − E − P =  = 0, dt T

(4.11)

where entropy production  = 0 in thermostatics and the global balance of internal energy have been inserted. In examples, the respective expression for the internal power P i of the forces and moments acting on the system has to be inserted. Field formulation The analogous derivation in field formulation gives 

1 de 1 dη = + pi . dt T dt T

(4.12)

In the examples, we have to insert an expression for the power density pi of the for a change of the system acting on the environment (for instance, pi = − p2 d dt density against a pressure p).

34

4 Some Elements of Thermodynamics

References 1. W. Muschik, Formulations of the second law–recent developments. J. Phys. Chem. Solids 49, 709 (1988) 2. W. Muschik, Aspects of Non-Equilibrium Thermodynamics (World Scientific, Singapore, 1990) 3. K. Hutter, The foundation of thermodynamics, its basic postulates and implications. A review of modern thermodynamics. Acta Mechanica 27, 1–54 (1977) 4. W. Muschik, Empirical foundation and axiomatic treatment of non-equilibrium temperature. Arch. Rat. Mech. Anal. 66, 379 (1977) 5. W. Muschik, G. Brunk, A concept of non-equilibrium temperature. Int. J. Eng. Sci 15, 377–389 (1977) 6. W. Muschik, J. Nonequilib. Thermodyn. 4, 277 (1979)

Chapter 5

State Spaces and Constitutive Equations

Abstract Constitutive quantities depend on position and time implicitly through the state variables. The domain of the constitutive functions is the state space. A non-equilibrium state space may include gradients and time derivatives of the wanted fields, or internal variables, or fluxes like heat flux and stress tensor. Thermodynamic constitutive theories are classified according to the choice of the state space.

5.1 Introduction We have seen in Chap. 1 that constitutive equations are needed in order to make a closed system of differential equations out of the balance equations. Constitutive quantities depend on the state of the system and on the material. The state space is the domain of the constitutive mappings M, which themselves are material dependent. Introducing the constitutive equations into the balance equations, we end up with a closed system of differential equations for the wanted fields. Due to the fact that the constitutive equations are different for different materials, the final set of differential equations for the wanted fields is material dependent. We will mainly deal with the field formulation of continuum thermodynamics. However, constitutive equations for the global system can be treated too, and we will do so in a few examples. The state space Z includes, at a minimum, the wanted fields, i.e., in the simplest case of a five-field theory, the fields of mass density, material velocity, and internal energy density. In general, in non-equilibrium, more variables are needed in order to describe material behavior properly. These additional variables can be gradients and/or time derivatives of the wanted fields, or other quantities that are not directly related to the wanted fields. We will study a lot of examples in this chapter and in the following chapters. The class of variables in the state space can be used to classify different thermodynamic theories (see below). The choice of the state space is a very crucial point because all results of constitutive theory depend on it. This choice depends on the material considered, on the kind of experimental behavior

© Springer Nature Switzerland AG 2020 C. Papenfuß, Continuum Thermodynamics and Constitutive Theory, https://doi.org/10.1007/978-3-030-43989-7_5

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36

5 State Spaces and Constitutive Equations

to be described, and also on the “philosophy” of the branch of non-equilibrium thermodynamics. Before classifying the different branches of thermodynamics, we will discuss some properties of state spaces.

5.2 Properties of the State Space The concept of state space is local in space and time. Field quantities are taken at one point, at one moment, without considering their time evolution and space dependence. The state space includes independent variables. At one point, the values of the fields and the corresponding gradients are independent variables. For instance, one can have different fields of mass density, having the same value at one point, but different gradients at that point (see Fig. 5.1). This independence of the values of the fields and the gradients is true only locally. The mass densities 1 and 2 (Fig. 5.1, [1]) cannot coincide at all points. Analogously, field quantities and their time derivatives are independent at one instant of time. The distinction between small and large state spaces (to be treated in the next subsection) is a distinction between constitutive mappings being functions or being functionals, respectively.

5.2.1 Small and Large State Spaces The elements of the state space, the state variables, are denoted by z. Let Z denote the state space. Material properties at an instant of time can either depend on the state z ∈ Z at that instant of time only or on the whole process the system has undergone. In the second case, constitutive quantities depend also on states the system went through in earlier times. Then, the state space is called a small state space, whereas we have the definition [1].

Fig. 5.1 For different fields of mass density, the values coincide in one point, but the gradients are different at that point. Republished with permission of [Elsevier], from [1]; permission conveyed through Copyright Clearance Center, Inc.

ρ

ρ1(.) ρ2(. ) ρ1(.) = ρ2(. ) ρ1(.) = ρ2(. )

x

5.2 Properties of the State Space

37

A state space is called large if material properties M are defined by mappings local in time. M : z(t) → M(t) ∀t. (5.1) For constitutive mappings nonlocal in time, we introduce the following. Definition [1]: For a fixed time t and real s ≥ 0: z t (s) := z(t − s), s ∈ [τ , 0]

(5.2)

is called the history of the process z(·) between t − τ and t. Definition [1]: A state space is called small if material properties M are defined by mappings on process histories: M : z t (·) → M(t) ∀t.

(5.3)

In the following chapters, we will deal with large state spaces. However, a lot of examples of constitutive theory with small state spaces can be found in the literature, see, for example, [2–5].

5.2.2 State Space and Thermodynamic Process Definition: A path in state space is called a thermodynamic process zˆ : zˆ : [t1 , t2 ] → Z t → zˆ (t).

(5.4)

A thermodynamic process is a solution of the balance equations. After the state space has been chosen, the differentiations of constitutive quantities in the balance equations have to be carried out using the chain rule. This is shown here for an example with the state space Z = {, v, e, ∇, ∇v, ∇e, } ˙ ∇ · q (, v, , ∇, ∇v, ∇, ) ˙ = ∂q ∂q ∂q ∂q · ∇ + : ∇v + · ∇e + : ∇ + = ∂ ∂v ∂e ∂∇ ∂q ∂q ∂q + : ·∇∇v + : ∇∇ + · ∇ . ˙ ∂∇v ∂∇ ∂ ρ˙

(5.5)

For a given state space, constitutive functions can be stated in systematic way using representation theorems [6] (see Sect. 8.3.2).

38

5 State Spaces and Constitutive Equations

5.3 Examples of State Spaces This part shall give an idea of possible choices of state spaces and constitutive equations on them.

5.3.1 Global Systems The following variables are necessary to describe the state of a global system: • Mole numbers n for the chemical composition (The notation with a formal vector n means a row of mole numbers of the different components.); • Work variables a for mechanical forces, geometry, and electromagnetic fields; • Thermodynamical variables z like internal energy, temperature, and possibly other variables, for instance, time derivatives.   A state of the system zˆ := a, n, z ∈ Z is an element of the state space Z of discrete systems. This state space is a non-equilibrium state space because more than one thermal variable is included (see Sect. 4.1 in the preceeding chapter for a definition of equilibrium). Examples of constitutive equations are the caloric equation of state U (V, n, T ) and the thermal equation of state p(V, n, T ). For an ideal gas, for instance, we have: p=

n RT , V

(5.6)

J where R is a constant (R = 8.31 mol ). If the ideal gas is monatomic, the internal K energy is given by: 3 (5.7) U = n RT. 2

For non-ideal gases, other thermal equations of state have to be used such as the well-known Van der Waals equation: p(V, T, n) =

RT an 2 − V V2 −b n

(5.8)

with material-dependent constants a and b. A more general form is a virial expansion with material-dependent virial coefficients Bi (T ):  p(V, T, n) = RT

 n 2  n 3 n + B3 + ··· B1 + B2 V V V

 .

(5.9)

5.3 Examples of State Spaces

39

5.3.2 Examples of State Spaces in Field Formulation A suitable state space to describe heat conduction includes density, temperature, and the gradient of temperature: Z = {, T, ∇T }. (5.10) A simple constitutive equation for the heat flux is Fourier’s law: q = −κ(T, )∇T

(5.11)

with heat conductivity κ. For a viscous fluid in a flow field (position-dependent velocity field), constitutive quantities depend on the velocity gradient, and a suitable state space is Z = {, T, ∇v}.

(5.12)

The constitutive equation for the stress tensor under the assumption of a linear dependenotes the symmetric traceless part dence on the velocity gradient reads: where T 1 1 of a tensor ( A := 2 ( A + A ) − 3 trace( A)δ), μ and λ are viscosity coefficients, the shear viscosity and volume viscosity, respectively, and p is the pressure. Here and in the following, δ denotes the unit tensor. For dielectric materials, we chose the state space: Z = {, T, E}.

(5.13)

For the so-called linear materials, the dielectric displacement D is a linear function of the electric field E. For anisotropic materials, the constitutive function permeability  is a tensorial quantity: D = (, T ) · E. (5.14)

5.3.3 Examples of Constitutive Equations on Small State Spaces Now, we will consider some examples of history-dependent constitutive mappings. If the dependence of the constitutive functional on the state space variable z is linear, it can be written as  t m(t − θ)z(θ)dθ (5.15) M(t) = 0

with an integral kernel m, depending on the material. Ferroelectric Materials

40

5 State Spaces and Constitutive Equations

In ferroelectric materials, the dependence of the polarization P on the electric field E is nonlocal in time. The polarization depends not only on the value of E, but also on the electric field at earlier times:  t χ(t − θ) · E(θ)dθ, (5.16) P(t) = 0

with susceptibility tensor χ. Consequently, the polarization shows after effects. Measuring polarization and electric field, one observes a hysteresis: The polarization corresponding to a given value of the electric field is different for decreasing and increasing electric field, respectively (see Fig. 5.2). The same effect is observed for the dependence of the magnetization on the magnetic field in the case of ferromagnetic materials (Fig. 5.2).

5.3.4 Shape Memory Alloys Shape memory alloys show an interesting behavior upon application of mechanical deformation and changing temperature: applying a mechanical load achieves a deformation as is true for many other materials. However, for shape memory alloys, the old shape can be recovered on heating up. This is shown in Fig. 5.3 for a wire. The behavior of the shape memory alloy can be explained by a temperaturedependent load-deformation diagram (see Fig. 5.4) [7]. At all temperatures, there is a hysteresis in the load-deformation diagram, i.e., it is reasonable to consider a small state space. The corresponding constitutive equation is a functional on the process history. At low temperature T1 (see Fig. 5.4a), the behavior is quasi-plastic: Applying a load to the undeformed state, one starts with zero deformation on an elastic line. On this line, one can move reversibly, increasing and decreasing the load, as long as the point P1 is not reached. At the point P1 , plastic flow starts; there is a deformation without increasing the load. At point P2 , a second elastic

Fig. 5.2 In ferroelectric and ferromagnetic materials, a hysteresis is observed

5.3 Examples of State Spaces

41

Fig. 5.3 Applying a load to the wire at 20◦ C causes the wire to deform. Heating it up to 60◦ C causes it to recover its old shape

Fig. 5.4 Load versus deformation-diagrams of a shape memory alloy at different temperatures

line starts. If the load is decreased after the point P2 was reached, the process goes along the second elastic line. At point P3 , the state of zero load is reached. It shows a finite deformation. This explains why the wire in Fig. 5.3 is deformed at 20◦ C. At the higher temperature T2 (see Fig. 5.4b), the behavior is still quasi-plastic, but the plastic flow starts at a smaller load. Increasing the temperature to T3 (see Fig. 5.4c), the behavior changes qualitatively. It is called pseudo-elastic because, for small loads, the material remains on the first elastic line without hysteresis. At this temperature, a state without deformation corresponds to zero load. Applying a load at this temperature and decreasing the load again causes the shape memory alloy to recover the shape it had before the load was applied. At the higher temperature T4 , the behavior is qualitatively the same, but the first elastic line extends to higher loads. A model has been developed [7] explaining this temperature-dependent mechanical

42

5 State Spaces and Constitutive Equations

behavior of shape memory alloys. It is based on the assumption that changes in the temperature between T2 and T3 cause a phase transition between different crystal modifications of the alloy to occur.

5.4 State Space with Internal Variables For practical purposes, constitutive functionals are not very convenient, and one would like to deal with large state spaces and constitutive functions. To create a large state space can be achieved including additional internal variables α in state space. These are independent variables in non-equilibrium. In equilibrium, they cannot be independent because the dimension of the equilibrium subspace is given by the zeroth law (see Sect. 4.1 in the preceeding chapter). Internal variables are dependent in equilibrium, but for given values of the independent equilibrium variables, the internal variables can have different values, i.e., in some points, there is no unique mapping between the equilibrium variables and the internal variables. This is shown in Fig. 5.5. Such a non-unique value of the internal variable occurs at a phase transition point. Phase transitions will be discussed in Sect. 7.2.2.2. For the internal variables, additional equations of motion (rate equations) are needed. The relaxation of the internal variables cannot be influenced directly by the environment, but their relaxation is governed by internal processes. This is the reason for calling them “internal variables”. Sometimes, internal variables are introduced formally without interpretation. A model of the internal structure of the material is needed to give the internal variables a physical meaning. Some examples of internal variables are: 1. in the example of the shape memory alloys: the mole fraction of the different crystal modifications; 2. the order parameters introduced in Landau theory of phase transitions [8] (for example, alignment tensors in liquid crystal theory, or the fraction of the supraconducting phase at the phase transition to supra-conduction);

Fig. 5.5 The equilibrium states form the lower dimensional equilibrium subspace in non-equilibrium state space. If the non-equilibrium variables include internal variables, there can correspond eq eq different values α1 and α2 of the internal variable to the same value of the equilibrium variables

internal variable

α

equilibrium subspace

α eq 2 e eq

α1

ρ

5.4 State Space with Internal Variables

43

3. a variable related to the shape of the deformable particles in suspensions of colloids and micro-emulsions; 4. damage parameters in the theory of plasticity, fatigue, and material failure; 5. the orientation of microscopic magnets in the theory of ferromagnetism. Examples of the application of internal variables can be found in [9–16].

5.5 The State Space of Extended Thermodynamics The state space of so-called extended thermodynamics includes, in addition to the classical fields of mass density, momentum density, and energy density, (nonconvective) fluxes. Usually, these are the heat current and the stress tensor [17–20]. To treat these fluxes as additional wanted fields, additional equations of motion for them are needed. Within a purely macroscopic theory, these additional equations have to be postulated. A derivation of these additional equations of motion is possible from the background of kinetic theory (Boltzmann theory) in the case of dilute gases [17, 18] or, with a more sophisticated microscopic background, even for dense gases and fluids [21]. In order to describe phenomena far from equilibrium, even higher order fluxes have been included in state space, and equations of motion for them have been derived from kinetic theory [18, 22].

Table 5.1 Typical choices of large state spaces in the different branches of thermodynamical constitutive theory. Sometimes, the equilibrium temperature is denoted by  (in thermostatics and Thermodynamics of Irreversible Processes (TIP)) in order to distinguish it from the non-equilibrium temperature T Thermostatics

Internal variables

TIP

Rational thermodynamics

State space

{n, a, T } or {n, a, e}

{n, a, T, ∇n, ∇a, ∇T, αk , . . . }

{n, a, T }

{n, a, T, e, ∇e, . . . , n, ˙ a, ˙ T˙ } {n, a, e, q, t}

Additional assumptions

Equilibrium

Rate equations for the internal variables

Local Equilibrium, linear constitutive laws, φ = q/T

s, φ, σ: primitive concepts

Advantage

Avoids history-dependent constitutive equations

Extended dynamics

Balance equations for q, t

Finite speeds of propagation

44

5 State Spaces and Constitutive Equations

5.6 Summary The different choices of state spaces presented in this chapter are summarized in Table 5.1. The corresponding constitutive theories will be discussed in other chapters.

References 1. W. Muschik, C. Papenfuss, H. Ehrentraut, A sketch of continuum thermodynamics. J. NonNewton. Fluid Mech. 96, 255–290 (2001) 2. G. Brunk. Recent developments in nonequilibrium thermodynamics. Lecture Notes in Physics, vol. 199 (Springer, Berlin, Heidelberg, New York, Tokyo, 1984), pp. 90–119 3. G. Brunk, On energy exchange between multidimensional fluctuating systems in equilibrium, in Proceedings of the Meeting ‘Escuela de Termodinámica de Bellaterra’ (1885), pp. 383–386 4. G. Brunk. Entropieproduktion und Gleichgewichtsschwankungen für eine Klasse nichtlinearer Materialgleichungen vom Nachwirkungstyp. ZAMM, 61(T), 80–82 (1981) 5. G. Brunk, Thermodynamic properties of the JOSEPHSON superconduction tunnel junction and related consequences for equilibrium fluctuations. J. Non-Equilib. Thermodyn. 5, 339– 360 (1980) 6. G.F. Smith, On isotropic integrity bases. Arch. Rat. Mech. Anal. 18, 282–292 (1965) 7. I. Müller, Grundzüge der Thermodynamik (Springer, Berlin, Heidelberg, 1994) 8. L.D. Landau, E.M. Lifschitz. Lehrbuch der Theoretischen Physik, vol. V Statistische Physik, 6th edn. (Akademie-Verlag, Berlin, 1984) Translated into German 9. V. Ciancio. On rheological equations and heat dissipation in anisotropic viscoanelastic media. Atti Acc. Sc. Lett. ed Arti di Palermo 1, 3–16 (1981) 10. G.A. Maugin, R. Drouot, Internal variables and the thermodynamics of macromolecule solutions. Int. J. Engn. Sci. 21, 705–724 (1983) 11. J.L. Ericksen, A thermodynamic view of order parameters for liquid crystals, in Orienting Polymers, Proceedings Minneapolis 1983, ed. by J.L. Ericksen (Springer, Berlin, New York, Tokio, 1983), pp. 27–36 12. S. Hess. Irreversible thermodynamics of nonequilibrium alignment phenomena in molecular liquids and in liquid crystals. Z. Naturforsch. 30a, 728–733 (1975) 13. W. Muschik, Internal variables in non-equilibrium thermodynamics. J. Non-Equilib. Thermodyn. 15, 127–137 (1990) 14. G.A. Maugin, W. Muschik, Thermodynamics with internal variables, part I. General concepts. J. Non-Equilib. Thermodyn. 19, 217–249 (1994) 15. G.A. Maugin, W. Muschik, Thermodynamics with internal variables, part II. Appl. J. NonEquilib. Thermodyn. 19, 250–289 (1994) 16. L. Longa, D. Monselesan, H.-R. Trebin, An extension of the Landau-Ginzburg-de Gennes theory for liquid crystals. Liq. Cryst. 2(6), 769–796 (1987) 17. D. Jou, J. Casas-Vazquez, G. Lebon, Extended Irreversible Thermodynamics (Springer, Berlin, Heidelberg, New York, 1993) 18. I. Müller, T. Ruggeri, Extended Thermodynamics, vol. 37 (Springer Tracts in Natural Philosophy, Berlin, Heidelber, New York, 1993) 19. I. Müller, Extended thermodynamics of classical and degenerate gases. Arch. Rat. Mech. Anal. 83, 286–332 (1983) 20. D. Jou, J. Casas-Vasquez, G. Lebon, Extended irreversible thermodynamics. Rep. Prog. Phys. 51, 1105–1179 (1988) 21. W. Dreyer, Molekulare Erweiterte Thermodynamik Realer Gase (Habilitationsschrift, Technische Universität Berlin) 22. K. Ikenberry, C. Truesdell, On the pressures and the flux of energy in a gas according to Maxwell’s kinetic theory. J. Rational. Anal. 5 (1956)

Chapter 6

Thermodynamics of Irreversible Processes (TIP)

Abstract Classical Thermodynamics of Irreversible Processes (TIP) applies different concepts known from thermostatics to non-equilibrium situations (not too far from equilibrium): According to the local equilibrium assumption, entropy density and internal energy density depend on the equilibrium variables in the continuum element only. In contrast to equilibrium, there are fluxes between the different continuum elements. A Gibbs relation for the differential of the entropy holds. The entropy flux is heat flux divided by temperature. An expression for the entropy production is derived, and thermodynamic fluxes and forces are identified. Linear relations between the fluxes and forces are the constitutive relations. The Curie principle restricts the cross-coupling effects, and Casimir-Onsager relations reduce the number of independent coefficients. The second law determines the sign of material coefficients. The examples of a viscous, heat-conducting fluid, a mixture of diffusing chemical components, and a thermoelement are treated. Fourier’s law of heat conduction, a Newtonian viscous stress tensor, and Fick’s law of diffusion are found.

6.1 Introduction Irreversible thermodynamics is a non-equilibrium theory that uses insofar as is possible, the concepts of thermostatics. Many examples [1, 2] are in field formulation, but the approach can be applied to global systems as well. The following assumptions are made [3–11]: • The state space of irreversible thermodynamics is the equilibrium subspace: Z(x, t) = {, T, a}(x, t),

(6.1)

consisting of the fields of mass density, thermostatic temperature, and work variables. This presupposition, in which the equilibrium subspace is sufficient for describing constitutive properties in non-equilibrium, is called the hypothesis of local equilibrium . The values of these fields differ from volume element to volume element, thus describing a non-equilibrium situation. Gradients or time derivatives are not included in the state space, i.e., constitutive quantities are assumed not to © Springer Nature Switzerland AG 2020 C. Papenfuß, Continuum Thermodynamics and Constitutive Theory, https://doi.org/10.1007/978-3-030-43989-7_6

45

46

6 Thermodynamics of Irreversible Processes (TIP)

depend on these gradients. In this approach, the quantities defined on the abovementioned state space are internal energy and entropy density. Equations for the fluxes like heat flux and stress tensor are derived by exploiting the dissipation inequality, and they turn out to depend on gradients. Therefore, in TIP, energy and entropy, on one hand, and the fluxes, on the other hand, are a priori not assumed to depend on the same set of variables. This contradicts the principle of equipresence, stated as an axiom in rational thermodynamics (see Chap. 8). However, for special choices of the state space in rational thermodynamics, it can be shown by exploiting the dissipation inequality that this presupposition of TIP is fulfilled. • For one-component systems, it is presupposed that the entropy transport is only caused by heat transport; that means the entropy flux density is given by: 1 q. T

=

(6.2)

For materials consisting of several chemical components, the above constitutive assumption has to be modified to: =

1  1 q− μν J ν , T T ν

(6.3)

where the sum is over the different chemical components, and μν are the respective chemical potentials. • Also, in non-equilibrium, the time derivative of the entropy density satisfies the equilibrium Gibbs equation: 1 de p d dη = − 2 , dt T dt  T dt

d ∂ := + v·∇ dt ∂t

(6.4)

for single-component systems, or 1 de p d 1  dν dη = − 2 − μν dt T dt  T dt T ν dt

(6.5)

for systems of several chemical components. Here, the change of the partial mass densities contributes to the change of the entropy. The Gibbs equation, together with the balance equations for mass and internal energy, allows an expression for the entropy production that has the form of a sum of products of “fluxes and forces”, Ji and X i to be derived: σ =



Ji X i ≥ 0.

i

Examples will be given in the following sections.

(6.6)

6.1 Introduction

47

• It is assumed that the fluxes and the forces are connected by linear constitutive laws:  L ik X k . (6.7) Ji = k

Equation (6.7) shows that, in general, a thermodynamic flux is caused not only by the corresponding force of the same index, but also by all other forces due to cross-coupling effects. The thermodynamic fluxes and forces can be scalar quantities or tensors of any order. The indices in (6.7) run over the scalar quantities, the components of vectors, and all components of second- and higher order tensors. The coefficients L ik are constitutive quantities. The number of independent constitutive coefficients is reduced by two principles: • The Curie principle states: In isotropic materials, there is no coupling between fluxes and forces of different tensorial orders, i.e., scalar fluxes depend only on all scalar forces, vectorial fluxes only on all vectorial forces, and so on. With the assumption of linear constitutive relations, this Curie principle can be proved by representation theorems (see Sect. 8.3.2) • Casimir-Onsager reciprocal relations hold: L ik (B, ω . . . ) = i k L ki (−B, −ω . . . ).

(6.8)

Here, the quantities (B, ω . . . ) have to transform in such a way that the force density is even under reversal of motion (B is the magnetic induction; ω is the angular velocity). Reversal of motion means that the system passes through the same states as in the original process, but with the sequence reversed in time. The k are determined by the transformation properties of the thermodynamical forces under reversal of motion: X k (B, ω . . . ) = k X k (−B, −ω . . . ).

(6.9)

It is an observation that, in nature, quantities are either even or odd, i.e., they do not change under reversal of motion or change sign in the reversed process compared to the original one. Examples for so-called even quantities with  = +1 are: – temperature gradient, – concentration gradient, and – electric field. Examples for so-called odd quantities with  = −1 are: – magnetic induction, – velocity gradient,

48

6 Thermodynamics of Irreversible Processes (TIP)

– electric current, – heat current, and – time derivatives of concentrations or any other even quantities. A derivation of Onsager-Casimir reciprocal relations from a statistical background can be found in [12] and, on a phenomenological basis, in [13, 14]. Because of linear constitutive equations (6.7), the entropy production (6.6) is a quadratic form of the forces. The second law of thermodynamics (6.6) requires that L ik is a positive definite matrix. Many examples applying the method of thermodynamics of irreversible processes can be found in [1]. In the book by De Groot and Mazur, as well as in the present book, a large state space is presupposed, and all constitutive relations are local in time. The existence of symmetry relations of the Casimir-Onsager type has been proved in the case of small state spaces, too, without using any assumptions or arguments on the microscopic level, see f.i. [15–18]. This has been applied to electric networks of linear two poles and to viscoelastic materials. The introduction of the Casimir-Onsager symmetry relations, which were proved using microscopic statistical arguments was one of the points of criticism on irreversible thermodynamics by Truesdell. Another point of criticism of the rational thermodynamics school was the assumption of local equilibrium in irreversible thermodynamics and the fact that fluxes are taken as time derivatives of state variables [19].

6.2 Example 1: A Viscous, Heat-Conducting Fluid The state space of TIP for a single-component system is given by the equilibrium variables Z = {e, }. We have assumed that we are dealing with a simple fluid without internal angular momentum. The entropy density is defined on this state space, and the derivatives are carried out according to the chain rule: ∂ d η(e, ) := η(e, ) + v · ∇η(e, ) dt ∂t   ∂η ∂e ∂η ∂ ∂η ∂η = + +v· ∇e + ∇ ∂e ∂t ∂ ∂t ∂e ∂ ∂η d ∂η de + = ∂e dt ∂ dt   p d 1 de − 2 . = T dt  dt

(6.10)

The last equal sign is due to Gibbs’ equation. The balance of entropy with φ = z = Tr reads:

1 T

q,

6.2 Example 1: A Viscous, Heat-Conducting Fluid

49

dη +∇ ·φ−z = dt dη r 1 1 = + ∇ · q − 2 (∇T ) · q − . dt T T T

σ =

(6.11)

The time derivative of the entropy density is transformed using (6.10): σ =

 T



de p d − 2 dt  dt

 +

1 1 r ∇ · q − 2 (∇T ) · q − . T T T

(6.12)

The divergence of the heat flux density q and the material time derivative of the internal energy density are eliminated by means of the balance of internal energy: 

de + ∇ · q = t : ∇v + r. dt

(6.13)

For the time derivative of the mass density  the balance of mass d + ∇ · v = 0 dt

(6.14)

is used. The result for the entropy production is: σ =

1 p 2 1 t : ∇v +  ∇ · v − 2 (∇T ) · q. T T 2 T

(6.15)

Assuming that our material is composed of spherical particles (there is no internal angular momentum), it can be shown that the stress tensor is symmetric [20]. Decomposing the stress tensor into its symmetric traceless part and the isotropic part t = t + 13 trace(t)δ the entropy production reads: 1 1 σ = t :  ∇v + T T  F1  J1



 1 1 p + trace(t) ∇ · v − 2 (∇T ) · q ,    3 T   F2    J3

(6.16)

F3

J2

which has the form of a sum of products of three fluxes and three forces: one product of second-order tensors, one product of scalars, and one product of vectors. The constitutive equations in the form of linear relations between fluxes and forces are: J3 = L 33 F3 :

q = −κ∇T

(6.17)

J1 = L 11 F1 :

t = 2μ ∇v

(6.18)

1 p + trace(t) = λ∇ · v. 3

(6.19)

J2 = L 22 F2 :

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6 Thermodynamics of Irreversible Processes (TIP)

These are Fourier’s law of heat conduction and the viscous stress tensor of a Newtonian fluid. ∂η and 13 the trace of the stress tensor The sum of the equilibrium pressure p = −2 ∂ is called dynamic pressure pdyn = p + 13 trace(t). The dynamic pressure vanishes for vanishing divergence of the velocity field ∇ · v = 0 ⇒ p = − 13 trace(t), as well as the symmetric traceless part of the stress tensor: ∇v = 0 ⇒ t = 0. Especially in equilibrium we have: pdyn |eq = 0 and t |eq = 0. μ is the shear viscosity, λ is the bulk viscosity, and κ is the heat conductivity. Writing down these constitutive equations, it has been used that (at least for isotropic media) there is no cross-coupling between scalars, vectors, and tensors, according to the Curie principle (see the introduction to this chapter). This means that the vectorial flux q is connected only with the vectorial force ∇T and so on for the scalar flux and the tensorial flux. According to the second law of thermodynamics, the matrix of coefficients between fluxes and forces is positive definite. It follows: L 11 ≥ 0,

⇒

L 11 T = 2μ ≥ 0,

L 22 ≥ 0, L 33 ≥ 0 L 33 L 22 T = λ ≥ 0, = κ ≥ 0. T2

(6.20) (6.21)

Viscosity coefficients and the heat conductivity are positive.

6.3 Example 2: A Heat-Conducting Mixture of Different Chemical Components We assume that there is diffusion of different chemical components, but no chemical reactions, and there is heat conduction, but no velocity field is present (v ≡ 0). A suitable state space to describe such a fluid is: Z = {T, c1 , . . . , cn , }.

(6.22)

The concentrations of the different chemical components cν are defined as cν = ν , and therefore, ν cν = 1. Gibbs’ fundamental equation for a mixture at rest (v ≡ 0 ⇒ d = 0) reads: dt T

n  dη de dcν = − |T . μν dt dt dt ν=1

(6.23)

The time derivative of the concentrations cν of the components has to be taken at constant temperature. μν are the chemical potentials. The time derivative of the internal energy density is eliminated by means of the balance of internal energy:

6.3 Example 2: A Heat-Conducting Mixture of Different Chemical Components



de + ∇ · q = t : ∇v + r = r, dt

51

(6.24)

where the first term on the right-hand side is zero because of the presuppositions v ≡ 0. In addition, radiation absorption r is neglected. The time derivatives of different concentrations are given by the partial mass balances, where the total and the partial time derivatives are the same due to v ≡ 0. The diffusion flux is introduced: J ν = ν v ν − ν v = ν v ν .

(6.25)

It vanishes if all components have the same material velocity. The partial balance of mass can be written with our presuppositions dν ∂ν + ∇ · (ν v ν ) = + ∇ · J ν = 0. ∂t dt The mass balance may be written in terms of concentrations cν = that

∂ ∂t

= 0: 

(6.26) ν , 

∂cν 1/dν = = −∇ · J ν + Pνchem = −∇ · J ν , ∂t dt

where we use

(6.27)

where the productions Pνchem are zero because of the absence of chemical reactions. The last form of the balance equation is also valid in case of non-zero material velocity. The result for the time derivative of the entropy density is: T

n  dη = −∇ · q + μν ∇ · J ν + r . dt ν=1

(6.28)

The sum of all diffusion fluxes J ν is zero: n  ν=1

Jν =

n 

(ν v ν − ν v) = v − v = 0.

(6.29)

ν=1

It is assumed that there is a contribution due to heat flux and a contribution due to diffusion of chemical components to the entropy flux of a multicomponent system: φ=

n 1  1 q− μν J ν . T T ν=1

(6.30)

With this constitutive equation in the balance of entropy, and with (6.28), one has:

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6 Thermodynamics of Irreversible Processes (TIP)

dη +∇ ·φ dt 

n n  1 μν q 1  =− ∇·q+ μν J ν . ∇ · Jν + ∇ · − ∇ · T T T T ν=1 ν=1

σ =

(6.31)

One of the partial mass fluxes can be eliminated by means of (6.29): n 

Jν = 0

⇒

Jn = −

n−1 

Jν

(6.32)

n−1 1 1  σ = − 2 q˜ · ∇T − J ν · ∇ (μν − μn ) |T , T T ν=1

(6.33)

ν=1

ν=1

where we have defined: q˜ := q −

n 

μν J ν .

(6.34)

ν=1

At constant temperature, the chemical potentials depend on the n − 1 independent concentrations only. Therefore, ∇ (μν − μn ) |T =

n−1 n−1   ∂ μνi ∇ci , (μν − μn ) · ∇ci =: ∂ci i=1 i=1

(6.35)

where μνi abbreviates the partial derivatives of the chemical potentials. The result for the entropy production is again a sum of products of fluxes and forces:

1 ∇T − σ = − 2 q˜ ·  T    force flux

n−1 . 1  μνi J ν · ∇ci  T i,ν=1    force flux

Linear constitutive relations read:

(6.36)

6.3 Example 2: A Heat-Conducting Mixture of Different Chemical Components

53

n−1  1 ˜ q = −L ∇T − L qm μm j ∇c j qq T2 m, j=1

(6.37)

n−1  1 L km μm j ∇c j . J k = −L kq ∇T − T m, j=1

(6.38)

The index k is the number of the chemical component with díffusion flux J k . L qq is the coefficient of heat conductivity, and L ik are diffusion coefficients. A heat flux as a consequence of a concentration gradient is called Dufour effect with corresponding coefficients L qk . The reciprocal effect of a mass flux due to a temperature gradient is thermal diffusion with coefficients L iq . Both effects are socalled cross-coupling effects, where a thermodynamical force induces not only the corresponding flux, but also other fluxes. Since all forces ∇T and ∇ci are even, functions under inversion of motion Onsager reciprocal relations hold: L qk = L kq ∀k.

(6.39)

Let us consider the important and simpler case of two chemical components. In this case, there is only one independent diffusion flux, and we have: 1 q˜ = −L qq ∇T − L q1 μ1 ∇c1 T2 1 J 1 = −L 1q ∇T − L 11 μ1 ∇c1 , T

(6.40) (6.41)

with reciprocity relation L q1 = L 1q

(6.42)

between the Dufour coefficient L q1 and the thermal diffusion coefficients L 1q . This relation was known from experiments before thermodynamics of irreversible processes was developed, a fact, which favored this method.

6.4 Example 3: Thermoelectricity This is an example not in field formulation, but considering the global system. We deal with the following setup: Two wires of different metals are connected in points A and B, which are in heat reservoirs of temperatures T A = T and TB = T + T . The total system consisting of the heat reservoirs, the wires, and the capacitor (see Fig. 6.1) is isolated.

54

6 Thermodynamics of Irreversible Processes (TIP)

Fig. 6.1 The principle of a thermoelectric device

V 3

The total balance equations of energy and of entropy of the whole system, consisting of the two reservoirs and the capacitor, read: dU − Q˙ tot = 0 dt dS ˙ tot = σ. − dt

(6.43) (6.44)

In addition, we assume that the heat capacity of the electric capacitor vanishes, and 3 therefore, its internal energy does not change: dU = 0. Due to the isolation of the dt whole system, the total fluxes vanish: ˙ tot = 0 Q˙ tot = 0, 

(6.45)

and therefore, the balance of energy simplifies: dU A dU B dU = + =0 dt dt dt

⇒

dU A dU B =− . dt dt

(6.46)

Gibbs’ fundamental equation for heat bath 1, heat bath 2, and the capacitor read (according to Sect. 4.3), respectively: dS A 1 dU A = dt T A dt dS B 1 dU B = dt TB dt dS3 V =− I dt T3

(6.47) (6.48) (6.49)

because in the heat reservoirs there is only heat exchange, and in the capacitor there is only electric work. −V I is the power of the electric work at the capacitor (I : electric current, V : Voltage at the capacitor). Therefore, we have for the entropy production of the whole system σ := σ tot :

6.4 Example 3: Thermoelectricity

σ = = = = =

55

dS tot dS A dS B dS3 = + + dt dt dt dt 1 dU A 1 dU B V + − I T A dt TB dt T3 dU B 1 dU B 1 V − + − I T dt T + T dt T3 V

T dU B − I − 2 T dt T

T ˙ V − 2 QB − I T T

(6.50)

if T  T and T3 ≈ T (lowest order of the Taylor expansion). The last expression gives the entropy production as a sum of products of fluxes and forces:

T V I − 2 Q˙ 2 . σ = −  T T   flux1    flux force1

force2

(6.51)

2

Heat flow and electric current are the causes of entropy production. The linear relations between thermodynamic forces and fluxes are: V

T − L 12 2 T T V

T Q˙ 2 = −L 21 − L 22 2 . T T I = −L 11

(6.52) (6.53)

The coefficients L 12 and L 21 account for the cross-coupling effects. Both forces − T T2 and − VT are even under inversion of motion, and therefore, the reciprocity relations are Onsagerian ones: (6.54) L 12 = L 21 . Two special cases are important: 1. There is a temperature difference between the two connecting points A and B ( T = 0), but there is no electric current I = 0. In this case, it follows from (6.52): L 12 V =− . (6.55)

T T L 11 The fact that the temperature difference causes a voltage between the points A and B is called the Seebeck effect. It is applied for temperature measurement. Conversely, a voltage applied to the capacitor causes a temperature difference, and this effect is applied in thermo-elements and in special refrigerators. The ratio in (6.55) is the thermoelectric force.

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6 Thermodynamics of Irreversible Processes (TIP)

2. There is no temperature difference between A and B, but a voltage V is applied. Then, we have from (6.52) and (6.53): V T V Q˙ 2 = −L 21 . T I = −L 11

(6.56) (6.57)

The cross-coupling effect, the fact that the applied voltage causes not only an electric current but also a heat flux, is called Peltier effect. It is applied in Peltier elements for cooling purposes. The ratio of the heat flux between points A and B and the electric current is a (material dependent) constant, the Peltier constant . It is the flux of heat at points A and B per electric current in the isothermal state: =

L 21 Q˙ = . I L 11

(6.58)

Due to the Onsager reciprocal relations, the thermoelectric force and the Peltier constant are connected. For any material, it holds: L 21 L 12 V = . = =− T T L 11 T L 11

T

(6.59)

This relation is known as the second relation of Thomson and was known before the derivation from the linear force-flux relations of irreversible thermodynamics.

References 1. S.R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics (North-Holland Publishing, Amsterdam, 1963) 2. J. Meixner, H.G. Reik, Thermodynamik der Irreversiblen Prozesse, in Handbuch der Physik, vol. III./2 (Springer, Berlin, 1959), p. 413 3. C. Garrod, J. Hurley, Symmetry relations for the conductivity tensor. Phys. Rev. A 27, 1487– 1490 (1983) 4. P. Glansdorff, I. Prigogine, Themodynamic Theory of Structure, Stability and Fluctuations (Wiley-Interscience, London, 1971) 5. J. Hurley, C. Garrod, Generalization of the Onsager reciprocity theorem. Phys. Rev. Lett. 48, 1575–1577 (1982) 6. S. Machlup, L. Onsager, Fluctuations and irreversible processes II. Systems with kinetic energy. Phys. Rev. 91, 1512–1515 (1953) 7. L. Onsager, Reciprocal relations in irreversible processes I. Phys. Rev. 37, 405–426 (1931) 8. L. Onsager, Reciprocal relations in irreversible processes II. Phys. Rev. 38, 2265–2279 (1931) 9. L. Onsager, S. Maclup, Fluctuations and irreversible processes. Phys. Rev. 91, 1505–1512 (1953) 10. I. Prigogine, Introduction to Thermodynamics of Irreversible Processes (Interscience, New York, 1961)

References

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11. G.F. Smith, R.S. Rivlin, The anisotropic tensors. Quart. Appl. Math. 15, 308–314 (1957) 12. F. Reif, Statistische Physik und Theorie der Wärme, 2nd edn. (Walter de Gruyter, Berlin, New York, 1985) 13. W. Muschik, A phenomenological foundation of non-linear Onsager-Casimir reciprocity relations. J. Non-Equilib. Thermodyn. (2), 109 (1977) 14. W. Muschik, A phenomenological foundation of non-linear Onsager-Casimir-reciprocal relations. Periodica Polytechnica Ser. Chem. Eng. 42, 85–96 (1998) 15. G. Brunk, Recent Developments in Nonequilibrium Thermodynamics, vol. 199 of Lecture Notes in Physics (Springer, Berlin, Heidelberg, New York, Tokyo, 1984), pp. 90–119 16. G. Brunk, On energy exchange between multidimensional fluctuating systems in equilibrium, in Proceedings of the meeting ‘Escuela de Termodinámica de Bellaterra’ (1885), pp. 383–386 17. G. Brunk, Entropieproduktion und Gleichgewichtsschwankungen für eine Klasse nichtlinearer Materialgleichungen vom Nachwirkungstyp. ZAMM 61(T), 80–82 (1981) 18. G. Brunk, Thermodynamic properties of the JOSEPHSON superconduction tunnel junction and related consequences for equilibrium fluctuations. J. Non-Equilib. Thermodyn. 5, 339– 360 (1980) 19. C. Truesdell, Six Lectures on Modern Natural Philosophy (Springer, New York, Wien, Berlin, 1966), pp. 49–59 20. A. Bertram, Axiomatische Einführung in die Kontinuumsmechanik (B.I. Wissenschaftsverlag, Mannheim, Wien, Zürich, 1989)

Chapter 7

Thermodynamics of Irreversible Processes with Internal Variables

Abstract Introducing internal variables as additional independent quantities in the continuum element, a much richer constitutive behavior can be described than with classical thermodynamics of irreversible processes. The first example is liquid crystals, where the phase transition is obtained as a special case of the evolution equation for the internal variable and flow birefringence can be explained. The second example is colloid suspensions, where the theory predicts a rich non-Newtonian flow behavior. A solid material with after effects and suspensions of flexible fibers are treated as well. Finally, mechanical model systems with damping are modeled as thermodynamic systems with internal variables.

7.1 Introduction The classical Thermodynamics of Irreversible Processes (TIP) can deal only with a restricted number of different material behaviors, due to the a priori choice of the state space. If, for instance, the fluid consists of only one chemical component, and there are no electromagnetic fields present, there is only one possibility: The state space is Z = {, T } or an equivalent one, and the material is the viscous heat-conducting liquid of Example 1 in Chap. 6, i.e., the constitutive equations are completely determined. Since there are more complicated materials showing different effects (see examples in the following), additional variables have to be introduced into the state space, and within the framework of TIP these are internal variables. A reason to introduce internal variables can also be to avoid a description in terms of history functionals in case of history-dependent materials, i.e., to go over from small to large state spaces. These additional variables can be scalars, vectors, or tensors, and they have the following properties: 1. The internal variables are independent in non-equilibrium but become dependent in equilibrium. For a one-component fluid without external fields, there are only two independent equilibrium variables: for instance, density and temperature. However, corresponding to a certain value of density and temperature, there is not necessarily a unique value of the internal variable (see 5.5). This happens at

© Springer Nature Switzerland AG 2020 C. Papenfuß, Continuum Thermodynamics and Constitutive Theory, https://doi.org/10.1007/978-3-030-43989-7_7

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7 Thermodynamics of Irreversible Processes with Internal Variables

a point of phase transition if the value of the internal variable is different in the two phases. 2. The dynamics of the internal variables cannot be controlled directly from outside. By changes in the vicinity of the system, only density and temperature can be controlled, which indirectly causes changes in the value of the internal variables. 3. For the internal variables, equations of motion are needed in addition to the balance equations. These equations are material dependent. Within the TIP, they are obtained from the exploitation of the entropy production. The internal variables can be used without giving them a priori a physical meaning [1]. However, in most of the examples discussed in the following, we will use internal variables having a physical interpretation from the beginning. For the definition of internal variables, see [2–4], and for examples of the application of internal variables see [1, 5–21]. Textbooks on irreversible thermodynamics with internal variables include [1, 20, 22–25].

7.2 Example 1: Liquid Crystals 7.2.1 Some Properties of Liquid Crystals Liquid crystals were discovered by Reinitzer1 more than one hundred years ago. Since then the fascinating properties of this “fourth state of matter”, which combines the properties of the fluid with those of the solid state, have been a subject of active research. On one hand, liquid crystalline phases behave like fluids, as they do not have a well-defined shape but flow like highly viscous liquids. On the other hand, they are anisotropic, i.e., material properties depend on the orientation of the sample relative to the measuring device. For example, the electrical conductivity depends on the orientation of the material. The same is true for the dielectric constant. This anisotropy of the dielectric constant leads to optical anisotropy (see Fig. 7.1). This optical anisotropy led to the discovery of liquid crystals under the polarizing microscope, and it is also the property that is used in the most important application of liquid crystals: optical devices (liquid crystal displays, LCD).

7.2.1.1

Liquid Crystalline Phases

Liquid crystals consist of form-anisotropic molecules. They can be prolate (elongated) or oblate (disk shaped). In the following, it will always be assumed that the 1 These unusual substances were mentioned for the first time in a letter from the botanist F. Reinitzer

to the physicist O. Lehmann in 1888. This letter is published in [26]. O. Lehmann was the first one to investigate the physical properties of liquid crystals, especially the morphology of textures under the polarizing microscope.

7.2 Example 1: Liquid Crystals Fig. 7.1 In the liquid crystalline state, different dielectric constants are measured parallel to the preferred particle orientation and perpendicular to the preferred particle orientation. Republished with permission of Springer, from [27]; permission conveyed through Copyright Clearance Center, Inc.

61

Example: anisotropic index of refraction

liquid crystal

simple liquid

effective molecular shape is uniaxial, so that the orientation of the molecule can be described by one direction, the microscopic director n. Liquid crystals exhibit a variety of ordered phases between the solid crystalline phase and the isotropic liquid phase. The phase transitions are induced by temperature changes. Heated to above the clearing temperature, the material becomes an isotropic liquid. In most cases, the nematic phase occurs upon cooling to below the clearing point; in some cases, other liquid crystalline phases occur. In the nematic phase, as well as in the isotropic phase, there is no ordering of the centers of mass of the particles. In the nematic phase, in contrast to the isotropic phase, the molecular orientations show long-range ordering. In addition to this orientational ordering, a one-dimensional positional ordering occurs in the smectic phases. The mass density is modulated periodically, i.e., the centers of mass form layers. The microscopic director can be parallel to or tilted with respect to the normal vector to the smectic layers. In the first case, the liquid crystal is in the smectic A phase; in the second case, it is in the smectic C phase. Some liquid crystalline phases are sketched in Fig. 7.2. Under special boundary conditions, or the action of electromagnetic fields, it is possible that the liquid crystal is biaxial, i.e., material properties are different in all three perpendicular directions. However, in most cases, there is rotation symmetry around one axis, called macroscopic director d. Then, the phase is called uniaxial. This macroscopic property has to be distinguished from the rotation symmetry of the (microscopic) particles, which is presupposed here in any case. Macroscopic definition of the alignment tensor We define the second-order alignment tensor in such a way that: 1. It vanishes in the high-temperature phase (the isotropic phase). 2. It is non-zero in the low-temperature phase (the nematic phase). 3. It is a dimensionless quantity.

62

7 Thermodynamics of Irreversible Processes with Internal Variables

Fig. 7.2 Some liquid crystalline phases. Republished with permission of Springer, from [27]; permission conveyed through Copyright Clearance Center, Inc.

isotropic phase

nematic phase

smectic A phase

scmectic C phase

These are the properties of an order parameter in the Landau theory of phase transitions, where the alignment tensor has been introduced as order parameter. A definition of the alignment tensor with the above properties is: a=

e − 1 3

1 3

trace(e )δ

trace(e )

(7.1)

with the dielectric tensor e (D = e · E). In principle, analogous definitions could be given by any other anisotropic property of the liquid crystalline state. However, the dielectric susceptibility is directly related to optical properties and is, therefore, the most important property of liquid crystals. If the material is (macroscopically) uniaxial, with symmetry axis denoted by d from symmetry arguments, the alignment tensor is of the form: a = S dd

(7.2)

with a scalar quantity Maier-Saupe-order parameter S and unit vector d.

7.2.1.2

Anisotropic Viscosity

Liquid crystals are not only optically anisotropic. The dielectric constant, measured with static electric fields, is also a tensorial property. Also in the measurement of a viscosity coefficient, the anisotropic nature of liquid crystals shows up. To illustrate this, consider a simple flow geometry, a Couette flow. The velocity field is in the x-direction, depending linearly on the y-coordinate (a constant gradient):

7.2 Example 1: Liquid Crystals

63

∇v = γey ex

(7.3)

with shear rate γ. We define the viscosity coefficient as the ratio between the respect tive stress tensor component and the shear rate γyx . Let us assume that the material has uniaxial symmetry even in the flow field, and the scalar order parameter S does not depend on position. There are different possibilities for the preferred orientation, i.e., the macroscopic director d, namely d can be parallel to the flow field, d can be parallel to the gradient, d can be perpendicular to both the flow direction and the gradient, or d can be in the plane of the velocity and the gradient under an angle of 45◦ with the velocity. In all these different geometries, different viscosity coefficients are measured. These are the different Miesowich viscosities.

7.2.1.3

Franck Elastic Energy

In several experimental situations, the orientational order is not homogeneous, but there is a gradient of the alignment tensor. This can be enforced by boundary conditions or by external fields. There are even liquid crystalline phases, the so-called blue phases, which are characterized by a periodic lattice of alignment tensor gradient. The energy density with inhomogeneous alignment is higher than with a homogeneous orientation. The energy difference is called elastic energy, although it is not the elastic energy of a solid but of an orientational order (Fig. 7.3). A representation theorem for the elastic energy density, depending on the alignment tensor gradient up to second order in the alignment tensor gradient, reads [28]: ee = κ1 (∇ · a)2 + κ2 (∇ × a)2 .

(7.4)

In the special case of a uniaxial distribution function, i.e., a = S dd , and for a homogenous scalar order parameter ∇S = 0, this assumes the form:

Fig. 7.3 A simple shear experiment: Couette geometry. Republished with permission of Springer, from [27]; permission conveyed through Copyright Clearance Center, Inc.

γ=

∂ vx ∂y

y

x

64

7 Thermodynamics of Irreversible Processes with Internal Variables

ee = k1 (∇ · d)2 + k2 (d · ∇ × d)2 + k3 (d × (∇ × d))2 .

(7.5)

All terms in (7.5) satisfy the so-called head-tail symmetry: d ↔ −d is a symmetry operation. Expression (7.5) is the Franck elastic energy. The first term is the energy of a pure splay deformation, the second one the energy of a pure bend deformation, and the third one the elastic energy of a pure twist deformation.

7.2.2 Exploitation of the Dissipation Inequality The alignment tensor is an internal variable in the sense of the above definition. It is an independent variable in non-equilibrium. In equilibrium, its value is determined by the equilibrium variables density and temperature in the simple case of a one-component system without external fields. For the following exploitation of the dissipation inequality with the methods of irreversible thermodynamics [29–31], the alignment tensor, but not its gradient, is included in the set of variables. The alignment tensor may vary from continuum element to continuum element, but this does not influence constitutive properties. This assumption can be looked at as a version of the local equilibrium hypothesis, generalized to internal variables (Fig. 7.4). For the entropy density η and the internal energy density e, the constitutive assumption of an additive decomposition into a part depending on the equilibrium variables and an alignment-tensor-dependent part is made: η = η0 (0 , ) + ηa (a) e = 0 (a = 0) + a (a).

(7.6) (7.7)

For the alignment-tensor-independent parts, the Gibbs equation in the usual form holds:

splay

bend twist

Fig. 7.4 Splay, twist, and bend deformation of the director field

7.2 Example 1: Liquid Crystals

65

1 d0 p d dη0 = − 2 . dt T dt  T dt With the usual assumptions of TIP  = entropy:

q T

(7.8)

and z = Tr , it results from the balance of

dη +∇ ·−z dt dη0 dηa 1 1 = + +q·∇ + ∇ ·q−z dt dt T T  d0 p d dηa q 1 = − 2 + − 2 · ∇T + ∇ · q − z T dt  T dt dt T T    de da p d dηa q r 1 = − − + − 2 · ∇T + ∇ · q − . T dt dt T dt dt T T T σ=

(7.9) (7.10) (7.11) (7.12)

The balance of internal energy for a medium with internal angular momentum: 

ds de = −∇ · q + t : ∇v + r −  · θ −1 · s, dt dt

(7.13)

the balance of mass, and the balance of internal angular momentum with the simplifying assumptions of vanishing couple stress and couple force: 

ds = − : t dt

(7.14)

are inserted: σ=

+

1 da q dηa −  − 2 · ∇T dt T dt T

 1 p 1 t : ∇v + s · θ −1 · ( : t) + ∇ · v T T T 

= 

 1 da dηa da 1 − : + q · − 2 ∇T   da T da dt T     J2

 J f 1

f1

1 + T 

2

  1 1 p + trace(t) ∇ · v+ t :   ∇v   3 T  

 f3 f4 J3

J4

(7.15)

66

7 Thermodynamics of Irreversible Processes with Internal Variables

+

  1 antisym  t : (∇v)antisym +  · θ −1 · s .

 T  

(7.16)

f5

J5

Next, the linear constitutive relations between the fluxes J1 . . . J5 and the forces f1 . . . f5 are expressed. It is assumed that the anisotropy of the liquid crystal is given explicitly by the dependence of internal energy and entropy on the alignment tensor, but otherwise material coefficients are scalars. Then, the Curie principle applies: there is no coupling between fluxes and forces of different tensorial orders, and no coupling between symmetric and antisymmetric tensors. With these assumptions, the flux-force relations read:   1 da dηa da = L11  − (7.17) + L14 ∇v dt da T da

q=−

1 T

1 L22 ∇T T2

(7.18)

  1 p + trace(t) = L33 ∇ · v 3

1 t = L41  T



1 da dηa − da T da

(7.19)

 + L44 ∇v

   1 antisym t = L55 (∇v)antisym +  · θ −1 · s . T

(7.20)

(7.21)

Equation (7.17) is the equation of motion for the internal variable, the alignment tensor. It is of the form of a pure relaxation equation without a flux term (i.e., a divergence). In the following, the expression in the bracket a − T ηa = fa is abbreviated as the alignment-tensor-dependent part of the free energy density fa . The constitutive equation (7.18) is the classical Fourier equation with heat conductivity κ = T12 L22 . From (7.19), it follows that, for vanishing flow field, p = − 13 trace(t). The remaining two equations are the constitutive relations for the symmetric traceless part of the stress tensor t , and for the antisymmetric part of the stress tensor t antisym . In order to exploit further (7.17) and (7.20), expressions for the alignment tensor dependence of ηa and a are needed. We will make ansatzes involving terms up to fourth and second orders, respectively. The different orders may be motivated by microscopic considerations. It turns out that the entropic contribution is more important than the

7.2 Example 1: Liquid Crystals

67

energetic contribution, which justifies the truncation after the second-order term in the energy density. 1 1 ηa (a) = − A0 a : a + B trace (a · a · a) 2 3 1 1 2 − C1 (a : a) − C2 trace (a · a · a · a) 4 4 1 a (a) = − a : a. 2

(7.22) (7.23)

The coefficients A0 , B, C1 , C2 , and  are material-dependent parameters, which are assumed to be constant, and, especially independent of temperature. Here, the Cayleigh-Hamilton [32] theorem could be used to transform the expression a · a · a · a, because this is not an independent invariant. However, the above form is the most practical one. The derivations are carried out: dηa = −A0 a + Ba · a − C1 a : aa − C2 a · a · a da da = −a. da

(7.24) (7.25)

With these ansatzes from (7.17), the relaxation equation da 1 dfa = −L11  +L14 ∇v dt T da = L11  −A(T )a + Ba · a − C1 a : aa − C2 a · a · a + L14 ∇v , results, with A(T ) = A0 −

1 . T

(7.26)

(7.27)

For the symmetric traceless part of the stress tensor, we obtain the constitutive equation: t = −L41 

dfa +L44 T ∇v da

= L41 T  −A(T )a + Ba · a − C1 a : aa − C2 a · a · a

+ L44 T ∇v . (7.28)

Let us consider now some consequences and special cases of these equations.

68

7.2.2.1

7 Thermodynamics of Irreversible Processes with Internal Variables

Equation of Motion for the Alignment Tensor Without Flow Field

For vanishing velocity field: v ≡ 0 → ∇v ≡ 0

(7.29)

the relaxation equation for the alignment tensor simplifies to: da 1 dfa = −L11  dt T da = L11  −A(T )a + Ba · a − C1 a : aa − C2 a · a · a .

(7.30)

The right-hand side of this equation is proportional to the derivative of a potential, the free energy density fa . In other words, for vanishing velocity field, the time derivative of the alignment tensor is governed by a potential. For a non-vanishing velocity gradient, such a derivation from a potential is possible only in very special flow geometries but not in general [33].

7.2.2.2

Comparison to Landau Theory of Phase transitions

The equation of motion for the alignment tensor without a flow field (7.30) can be interpreted as the equation of motion for the order parameter in the dynamical Landau theory. Landau theory was developed to deal with phase transitions, originally with phase transitions in ferromagnetic materials. It has been applied to various kinds of phase transitions, for instance, the transition nematic/isotropic phase in liquid crystals [28, 34–38], other transitions between liquid crystalline phases [35, 39, 40], the transition to the superfluid phase of liquid helium, and the transition to the superconducting phase [41]. We will restrict ourselves here to the case where no spatial gradients of the order parameter are present. In general, it is assumed that there exists an order parameter that is non-zero in one phase (usually the lower temperature phase) and zero in the other phase. The free energy density f is assumed to depend on this order parameter in an analytical way, and the series expansion of the free energy density, with respect to the order parameter, is truncated at some order. The equation of motion for the order parameter O is assumed to be of the form: df dO =C , dt dO

(7.31)

(with a constant C) which leads to the equilibrium condition: df = 0, dO

(7.32)

7.2 Example 1: Liquid Crystals

69

the necessary condition for an extremum of the free energy. The equation of motion for the alignment tensor (7.30) derived with the methods of TIP is of the form of (7.31). The corresponding equilibrium condition is: dfa =0 da

(7.33)

with the alignment-tensor-dependent free energy density: −

1 1 1 1 fa (a) = − A(T )a : a + B trace (a · a · a) − C1 (a : a)2 T 2 3 4 1 − C2 trace (a · a · a · a) . 4

(7.34)

From this, it follows that in equilibrium A(T )a − B a · a +C1 a : aa + C2 a · a · a = 0.

(7.35)

The number of extremum values of the free energy expression depends on the value of the coefficient A(T ), which changes with temperature. This will be discussed here for the case of a uniaxial phase (the experimentally most important case) with axis of rotation symmetry d. Then the products of the alignment tensor a = S dd = S dd − 13 δ with d · d = 1 can be calculated, and it follows for the equilibrium condition (7.33): 1 2 1 A(T )S dd − BS 2 dd + C1 S 3 dd + C2 S 3 dd = 0, 3 3 3 and it follows

2 1 1 A(T )S − BS 2 + C1 S 3 + C2 S 3 = 0. 3 3 3

(7.36)

(7.37)

At high temperatures T , there exists only one minimum of the free energy density at S = 0, the isotropic phase (see Fig. 7.5). On lowering the temperature, there occurs a second minimum at temperature T = Tc∗ , at which the value of the free energy is higher than at the isotropic minimum. This second minimum is metastable, and the corresponding ordered phase can be obtained as a metastable phase by overheating. At the temperature Tc , both minima have the same value of the free energy. At this temperature, the clearing temperature, there occurs the phase transition. The order parameter jumps between zero (isotropic phase) and a finite value (ordered phase). As the variable S is discontinuous at the phase transition, it is a first-order transition. On lowering the temperature further, the second minimum becomes the absolute minimum and the liquid crystalline phase is the stable one. The isotropic phase (the minimum at order parameter zero) becomes unstable at temperature T = T ∗ . Up to this temperature, the isotropic phase can be obtained by supercooling as a metastable phase.

70

7 Thermodynamics of Irreversible Processes with Internal Variables

Fig. 7.5 The free energy density as a function of the scalar order parameter for different temperatures. Republished with permission of Springer, from [27]; permission conveyed through Copyright Clearance Center, Inc.

7.2.2.3

Flow Birefringence

Let us assume a stationary state in a flow field ( ∇v = 0). Assume that for vanishing velocity gradient the liquid is isotropic (i.e., the temperature is higher than the clearing temperature) and that for ||∇v|| small the deviation from the isotropic phase is small, i.e., ||a|| is small. With this assumption, the constitutive equation for the entropy density will be truncated after the second-order term. With this truncation and the condition of stationarity: da =0 (7.38) dt equation (7.26) yields: 0 = L14 ∇v −A(T )L11 a

(7.39)

or, for the alignment tensor in such a stationary state: a=

L14 1 ∇v . L11 A(T )

(7.40)

As the alignment tensor was defined proportional to the anisotropic part of the dielectric constant (see (7.1)), there should be a measurable dielectric anisotropy under

7.2 Example 1: Liquid Crystals

71

the action of the flow field. An anisotropy of the dielectric tensor for electric ac fields at optical frequencies is observed as optical birefringence. This was confirmed experimentally [30, 42]. A related phenomenon is the observed acoustically induced birefringence. There the acoustic wave induces a periodic velocity gradient, which again causes a periodic dielectric tensor field. This can be observed by refraction.

7.2.2.4

Orientational Order Influencing the Viscosity

Let us consider the constitutive equation for the symmetric traceless part of the stress tensor, (7.20). In general, it gives a nonlinear dependence of the stress tensor on the alignment tensor. If we are dealing with the material at a temperature where it is isotropic in the absence of a flow field, we can assume that ||a|| is small and truncate the expansion of the free energy after the second-order term in the alignment tensor. Then, the constitutive relation t = −L41 A(T )a + L44 ∇v

(7.41)

results. In the case of a stationary state, we can insert the solution (7.40) for the alignment tensor. The resulting expression for the stress tensor:   L41 t= − L14 + L44 ∇v L11

(7.42)

is a linear function of the velocity gradient, and the factor is the viscosity in the usual L14 due to the flow-induced notation. This viscosity is modified by the term − LL41 11 orientational ordering. If the temperature is low enough that the material is in the liquid crystalline state without a flow field, then the coupling of velocity gradient and alignment tensor in a stationary state leads to reorientation of the preferred axis (flow alignment), to an increase of the order parameter, and eventually to the induction of phase biaxiality (the loss of a rotation symmetry axis).

7.3 Example 2: Colloid Suspensions Colloid suspensions consist of a solid component of particle sizes large compared to atomic dimensions but small compared to macroscopic dimensions in a solvent. Some examples of colloid suspensions are: suspensions of DNA threads (typical diameter a few micrometer), suspensions of elastic spheres, for instance, of polystyrol (applied, for example, in ferrofluids [43, 44]), suspensions of proteins in water (which are very important in biological cells), and paints are suspensions of (anorganic) particles in

72

7 Thermodynamics of Irreversible Processes with Internal Variables

a solvent. Related to suspensions are emulsions like fat micelles in water (f.i. milk). For reference, see, for instance, [18–20]. We are dealing now [19] with a mixture of two different components, denoted by indices ν = 1 for the solvent and ν = 2 for the solid particles. The following assumptions are made: 1. The solvent is incompressible. 2. The solid particles behave elastically with specific elastic energy density 21 cν  · , where  is the strain  = FT · F, and cν is the mass fraction of the solid component times elastic modulus E. 3. The deformation gradient of the solid particles is an internal variable. 4. For the solvent the classical Gibbs equation holds. 5. For the solid component the differential of the entropy is: 2

dη2 ρ2 de2 cν  d = − : dt T dt T dt

(7.43)

with the last contribution due to elastic deformation of the solid component. 6. The temperature is constant: ∇T = 0. 7. The stress tensor is symmetric, i.e., no internal angular momentum has to be taken into account. The exploitation of the entropy inequality gives the following expression for the entropy production: dη +∇ ·−z dt q r dη1 dη2 = 1 + 2 +∇ · − dt dt T T p1 d1 2 de2  q r 1 de1 d 1 − − cν  : − 2 ·   = + ∇T + ∇ · q − T dt 1 T   dt T dt T dt T T T σ=

=0

=0

=

1 r  r  de d + ∇ · q − − cν  : − T dt T T T dt T  d 1 . = t : ∇v − cν  : T T dt (7.44)

In the last equation, the total balance of internal energy (3.45) of the suspension has been used, assuming that the suspension behaves fluid like with an internal variable. t is the stress tensor of the mixture. Finally, this stress tensor can be decomposed into the symmetric traceless part and the isotropic part (according to the last assumption the antisymmetric part is zero): 1 t = t + trace(t)δ. 3

(7.45)

7.3 Example 2: Colloid Suspensions

73

However, for the incompressible solvent, it holds ∇ · v = 0 and t : ∇v = t : ∇v

(7.46)

Also, the time derivative of the deformation gradient can be rewritten, introducing the so-called Jaumann derivative, or co-rotational derivative˚ :   d d : =: +·ω−ω· dt dt =:  :˚ , (7.47) which holds for any antisymmetric second-order tensor ω and especially for the antisymmetric part of the velocity gradient ω = ∇v(antisym) . The Jaumann derivative is the time derivative seen by an observer moving and rotating with the fluid flow. In the framework of thermodynamics of irreversible processes, it is an axiom that the entropy production should be formulated in terms of co-rotational time derivatives, because local equilibrium is valid for an observer moving and co-rotating with the fluid. With these transformations, the expression for the entropy production reads: σ=

1 t : ∇v −cν  :˚  . T

(7.48)

The linear constitutive relations between the two second-order symmetric tensorial fluxes and forces are: 1 t = L11 ∇v −L12 cν  T

(7.49)

1 ˚  = L21 ∇v −L22 cν . T

(7.50)

Positivity of the entropy production requires: L11 ≥ 0, L22 ≥ 0.

(7.51)

One of the forces, ∇v , is odd under time reversal (because the velocity has the opposite sign for the reversed motion), and the other force,  is even under time reversal. (As the entropy production itself is odd, the reversed sign under inversion of motion must occur for the corresponding fluxes: the stress tensor t is an even function, and˚  is an odd function under inversion of motion.) Consequently, we have Casimir-Onsager reciprocal relations for the cross-coupling coefficients: L12 = −L21 .

(7.52)

74

7 Thermodynamics of Irreversible Processes with Internal Variables

This relation is inserted into (7.50), and by (7.49) the internal variable, the deformation gradient,  is eliminated: t −L11 ∇v . cν L12

1

=−T

(7.53)

This expression together with the Jaumann derivative of it (with  = const., T = const.) ˚ 1˚ t − L11 ∇v T ˚ =− (7.54) cν L12 is inserted into the second linear constitutive relation (7.50) to give the differential equation for the stress tensor: 1 ˚ t + t =T T cν L22    =:τ1



 L11 L22 L212 L11 ˚ + ∇v + ∇v L12 L22 2 cν L22  

 =:2μτd

=:2μ

˚ t +τ1 t = 2μ



(7.55)

˚ ∇v +τd ∇v

 .

(7.56)

In this final equation, the internal variable  has been eliminated. Neglecting the ˚ ˚ time derivatives ∇v = 0 and t = 0 yields the Stokes constitutive equation for the stress tensor: t = 2μ ∇v (7.57) with the viscosity coefficient μ. Some consequences of the modification of this equation will be discussed in the next section.

7.3.1 Special Case: Stationary Couette Flow We will apply now the equations obtained from the irreversible thermodynamics treatment to a special flow geometry, a simple Couette flow [45]. The velocity is in the y-direction with a constant gradient κ in the x-direction. Assume that the flow is stationary, i.e., all partial time derivatives vanish. We assume that the spatial dependence is on the x-coordinate only, i.e., ∂(·) ∂(x)

∇(·) = 0 0

(7.58)

7.3 Example 2: Colloid Suspensions

75

and v · ∇(·) = 0. Therefore, we have d(·) ∂(·) = + v · ∇(·) = 0, dt ∂t

(7.59)

the material time derivatives vanish. In Cartesian components, we have: vx = 0, vy = γx, vz = 0. ⎞ 010 ((∇v)) = γ ⎝ 0 0 0 ⎠ , 000

(7.60)

⎛

⎛ 01 γ⎝ 10 (( ∇v )) = 2 00

⎞ 0 0⎠ 0

⎛ ⎞ 0 10   γ ((ω)) = (∇v)antisym = ⎝ −1 0 0 ⎠ , 2 0 00 ∇ · v = 0.

(7.61)

(7.62)

(7.63)

(7.64)

Due to the assumed stationarity, we have: d∇v =0 dt dt = 0. dt

(7.65) (7.66)

We have denoted the component representation with respect to the chosen coordinate system with double brackets (( )). For the Jaumann derivatives, it follows: ˚ ∇v = ∇v ·ω − ω· ∇v

(7.67)

˚ t = t · ω − ω · t,

(7.68)

76

7 Thermodynamics of Irreversible Processes with Internal Variables

and in components we have: ⎛ ⎞ −1 0 0 γ2 ⎝ ˚ 0 1 0⎠. (( ∇v )) = 2 0 00

(7.69)

We introduce the Cartesian components of the stress tensor (t denotes the invariant object, the tensor, whereas ((t)) is the representation of the tensor in components with respect to a chosen coordinate system.) ⎞ ⎛ ⎞ t11 t12 t13 t11 t12 t13 ((t)) = ⎝ t21 t22 t23 ⎠ = ⎝ t12 t22 t23 ⎠ . t31 t32 t33 t13 t23 t33 ⎛

(7.70)

For the coordinate representation of˚ t it results in: ⎞ ⎞ ⎛ ⎛ −t t 0 t t t γ ⎝ 12 11 ⎠ γ ⎝ 12 22 23 ⎠ −t11 −t12 −t13 . (7.71) −t22 t12 0 − ((˚ t)) = ((t · ω − ω · t)) = 2 −t t 0 2 0 0 0 23 13 Finally, we have to take the traceless parts of t and ˚ t and insert the component representations into (7.56). It results in a coupled system of algebraic equations for the components of the stress tensor. The solution of these equations is: ⎛

⎞ σ¯ τ 0 (( t )) = ⎝ τ −σ¯ 0 ⎠ 0 0 0

(7.72)

with the shear stress τ and the normal stress σ, ¯ depending on the shear rate γ: τ=

μγ + τ1 τd μγ 3 1 + τ12 γ 2

(7.73)

σ¯ = −μτd γ 2 + τ1 γ

μγ + τ1 τd μγ 3 , 1 + τ12 γ 2

(7.74)

where the relaxation times τ1 and τd depend on the concentration of solid particles. Shear stress and normal stress as functions of the shear rate may be displayed for different fractions of the solid component [19]. For certain values of the mass fraction cν = 2 of the solid particles, there is a region of shear rates where the shear stress decreases with increasing shear rate and at higher shear rates reaches a plateau, finally increasing slightly again. This can be interpreted as plastic flow.

7.4 Example 3: A Stress-Strain Relation for a Material with After Effects

77

7.4 Example 3: A Stress-Strain Relation for a Material with After Effects The experiment we want to describe in this section is a uniaxial tension applied normal to the surface of a test specimen. The system will be described in terms of global quantities, not in a field formulation. The tension is denoted by σ, ˜ to be distinguished from the entropy production σ. The length of the specimen changes 0 . from l0 to l, and the strain is defined as  = l−l l0 We denote the internal energy of the system with U and the entropy with S. It is assumed that in addition to the entropy (equivalent to internal energy or temperature) and strain we have one scalar internal variable ξ in the state space: Z = {S, , ξ}.

(7.75)

The thermal variable S and the work variable  are the independent variables in equilibrium. Internal energy U is a constitutive function defined on this set of variables. This leads to the total time derivative: ∂U dS ∂U d ∂U dξ dU = + + . dt ∂S dt ∂ dt ∂ξ dt

(7.76)

On the other hand, we postulate a generalized Gibbs’ equation for the time derivative of the entropy: dS 1 dU σ˜ d A d ξ = − + (7.77) dt T dt T dt T dt which can be solved for the derivative of internal energy: dS d dξ dU =T + σ˜ −A dt dt dt dt

(7.78)

The coefficients can be identified as: ∂U ∂S ∂U σ˜ = ∂ ∂U . A=− ∂ξ T=

(7.79) (7.80) (7.81)

For the time derivative of the internal energy, we have the first law of thermodynamics: dU ˙ + σ˜ d . = −Q dt dt

(7.82)

78

7 Thermodynamics of Irreversible Processes with Internal Variables

˙ is the heat exchange rate (energy per time) with the surroundings (but it is not the Q time derivative of some energy function). This first law, (7.82), and the generalized Gibbs’ equation (7.77) are inserted into the balance of entropy (with entropy pro˙ and entropy supply assumed to be zero). It is duction , entropy exchange rate , ˙ Q ˙ assumed  = T : dS dt ˙ 1 dU σ˜ d A d ξ Q − + = + T T dt T dt T dt A dξ . = T dt ˙ + =

(7.83)

is identified as the thermodynamic force, and dξ is the corresponding flux. The dt linear force-flux relation is the equation of motion for the internal variable: A T

A dξ =L , dt T

(7.84)

with a material coefficient L. The constitutive equation for the internal energy is assumed to be of the form: U = U00 + U10  + U01 ξ +

 1 U20 2 + 2U11 ξ + U02 ξ 2 , 2

(7.85)

which is the most general second-order expression in the variables  and ξ. The material coefficients Uij may still be functions of the thermal variable (here: entropy). Due to the equilibrium conditions: ∂U ( = 0, ξ = 0) = 0 ∂ ∂U ( = 0, ξ = 0) = 0 ∂ξ

(7.86) (7.87)

we have U10 = 0 and U01 = 0. Equations (7.85) and (7.80) result in σ˜ = U20  + U11 ξ

(7.88)

−A = U11  + U02 ξ .

(7.89)

With (7.89) the linear force-flux relation results in: L dξ = − (U11  + U02 ξ) dt T

(7.90)

7.4 Example 3: A Stress-Strain Relation for a Material with After Effects

79

Between (7.90) and (7.88), the internal variable ξ is eliminated by differentiating ˙ respectively, and inserting equation (7.88) with respect to time, solving for ξ and ξ, for ξ and ξ˙ in (7.90). It results in the following differential equation between stress and strain: (7.91) σ˜ + τe σ˙˜ = E0 ( + τσ ˙) , where the abbreviations: T 1 L U02

(7.92)

U20 T 2 L U02 U20 − U11

(7.93)

2 2 U11 U02 U20 − U11 = U02 U02

(7.94)

τe = τσ = E0 = U20 −

have the interpretations of relaxation times, and elastic constant, respectively. In rheology, a material with a stress-strain relation of the form equation (7.91) is called a Kelvin body. A special case is the so-called Voigt body, which is obtained if τe = 0, and has the stress-strain relation: σ˜ = E0 ( + τσ ˙) .

(7.95)

The general solution of this equation for a given time-dependent stress σ(t) ˜ is: 1 E0 τσ

(t) =



t

0

  (t − s) σ(s)ds. ˜ exp − τσ

(7.96)

This shows that not only the stress applied at the present instant of time is relevant for the strain, but also stresses applied at all earlier times. An analogous solution, involving not only the applied stress, but also its time derivative under the integral is obtained for the Kelvin body. Let us consider finally the solution of (7.91) for a special time dependence of the applied stress:  σ(t) ˜ =

0 for t ≤ 0 σ∞ for t > 0.

(7.97)

The solution (7.96) reads for this stress function: (t) =

σ∞ E0 τσ



t 0

e−

(t−s) τσ

ds =

t σ∞ 1 − e − τσ for t > 0 E0 0 for t ≤ 0.

(7.98) (7.99)

80

7 Thermodynamics of Irreversible Processes with Internal Variables

Fig. 7.6 The strain in a Kelvin body as a function of time for a stress, jumping at time t = 0 from zero to a finite value σ∞

This solution is shown in Fig. 7.6. The graph shows clearly the after effect: The strain does not jump instantaneously to the value σE∞0 , corresponding to the applied stress at positive times, but reaches this value only asymptotically.

7.5 Example 4: Irreversible Thermodynamics of Flexible Fibers Other examples of a system with internal degrees of freedom are suspensions of flexible fibers [20, 46–49], republished with permission of Elsevier, from [49]; permission conveyed through Copyright Clearance Center, Inc. The internal variable is related to the deformation of elastic microparticles. For example, proteins, certain saccharides, and DNA molecules are well described by the above model. The particles are like elastic slender bars and may be twisted and bent. In our model, a possible flow-induced anisotropy is neglected, and the orientation distribution of the fibers is presumed isotropic. The purpose here is to investigate the consequences of the flexibility of the particles. The deformation of a particle (bend and twist) stores energy, so influencing the macroscopic mechanical properties of the fluid.

7.5.1 Deformation of a Fiber In our model, the fibers are assumed to be straight if not loaded. Then, one can choose a coordinate s along this fiber orientation and an orthogonal tensor U(s) describing the distortion of the fiber. The macroscopic angular distortion tensor defined by ϕ := U T ·

dU ds

(7.100)

7.5 Example 4: Irreversible Thermodynamics of Flexible Fibers

81

takes into account the local deformation of the flexible fibers and is the average over all fiber orientations in the continuum element. s is the local coordinate along the fiber, and x is the position of the continuum element. s is only introduced in order to describe the local fiber deformation. The tensor ϕ is obviously skew-symmetric as U is orthogonal, U T · U = δ,

(7.101)

dU dU T · U + UT · = 0, ds ds

(7.102)

where δ is the unit tensor and

i.e.,



dU U · ds T

T + UT ·

dU = 0. ds

(7.103)

This way, we introduce the angular distortion vector (the vector invariant of the angular distortion tensor) as ϕ × δ = ϕ. (7.104) Only in this case we will denote the vector by the symbol arrow in order to distinguish it from the tensor ϕ. Let n denote the unit vector tangential to the undeformed fiber. The scalar product of ϕ and n gives the twist: t = ϕ · n, (7.105) and the component of ϕ perpendicular to n is the bend: b = ϕ − n(n · ϕ).

(7.106)

An illustrating example will be shown. As the fibers are elastic, two material coefficients, torsion stiffness μt and bending stiffness μb , have to be introduced. The torque acting in a cross section of the fiber is given by (7.107) τ = μt n(n · ϕ) + μb [ϕ − n(n · ϕ)]. The elastic energy stored per unit length of a fiber is given by ue =

μt μb (ϕ · n)2 + (ϕ2 − (ϕ · n)2 ). 2 2

(7.108)

The fibers are deformed by flow and they continuously relax. The angular distortion of an individual fiber ϕ depends on the local orientation of the fiber,

82

7 Thermodynamics of Irreversible Processes with Internal Variables

ϕ = ϕ(n).

(7.109)

The function is odd, ϕ(−n) = −ϕ(n), according to the definition of ϕ (7.100), as turning the fiber (changing ϕ(n) to ϕ(−n)) gives a −sign in the line element ds. An obvious approximation for the function ϕ(n) is the first term of its expansion by spherical harmonics; ϕ(n) = α · n, (7.110) where α is a second-order pseudotensor as ϕ is an axial vector. Verhas [50] elaborated the theory based on this linear approximation and the theory gives only one relaxation time, while the relaxation times for twist and bend are expected to differ. As it will be shown in this section, the above shortcoming is ceased if the approximation (7.110) is improved with a third-order term (with a traceless tensor β): ϕ(n) = α · n + n(n · β · n),

(7.111)

which can be decomposed as ϕ(n) = α0 n + αa · n + αd · n − n(n · αd · n) + n(n · T · n),

(7.112)

where α0 = 1/3trα, αa is the skew-symmetric part of α, αd is the deviatoric part of α, and T is the deviatoric part of the sum of the tensors α and β. The latter form shows the twist and bend separately; the first and the last terms are due to twist, while the others are due to bend. Inserting the approximation equation (7.111), the elastic energy of a fiber per unit fiber length reads ue (n) =

μt μb (α0 + n · T · n)2 + (αa · n + αd · n − n(n · αd · n))2 , 2 2

(7.113)

the average of which over all possible fiber orientations is  1 ue (n)d2 n = 4π S 2     μb 1 a a 1 d d 2 μt α0 2 + T:T + α :α + α :α = 2 15 2 3 5

u¯ e =

(7.114)

if uniform distribution of the fiber orientations is supposed, i.e., the fiber orientations are distributed isotropically. A more refined description introducing a non-uniform distribution of orientations is discussed in chapter 7.5.6. If the total length of the fibers in a unit volume is denoted by l, we obtain und = lue

(7.115)

7.5 Example 4: Irreversible Thermodynamics of Flexible Fibers

83

for the part of the specific internal energy stored in the deformation of the fibers, and not being dissipated. From the formula, one can see that this not dissipated part of the energy is a homogeneous quadratic function of the scalar α0 , and the tensors αa , αd , and T.

7.5.1.1

Two Illustrating Examples

In both examples, the straight fiber is along the x-direction. The deformation is shown in Fig. 7.7. In the first example, the deformed fiber is in the x-y-plane, and the bending axis is the z-axis. In this example, we have ⎛ ⎞ 1 n = ⎝0⎠. 0

(7.116)

The rotation matrix U, mapping the fiber segment in the undistorted state to the one in the distorted state, reads ⎛ ⎞ cos α − sin α 0 U = ⎝ sin α cos α 0 ⎠ , (7.117) 0 0 1 ⎛ − sin α − cos α dU = ⎝ cos α − sin α ds 0 0 and

Fig. 7.7 a Bend deformation, b twist deformation of a fiber

⎞ 0 dα 0⎠ , 0 ds

(7.118)

84

7 Thermodynamics of Irreversible Processes with Internal Variables

⎛ ⎞ ⎛ ⎞ 0 −1 0 0 dα dα dU ϕ := U T · = ⎝1 0 0⎠ , ϕ =⎝ 0 ⎠ . ds 0 0 0 ds −1 ds

(7.119)

The deformation is pure bend: ϕ · n = 0, and consequently the twist t is zero, and the bend b = ϕ. In the second example, the straight fiber is again along the x-direction. The rotation axis is now the x-axis. ⎛ ⎞ 1 0 0 U = ⎝ 0 cos β − sin β ⎠ , (7.120) 0 sin β cos β ⎛

⎞ 0 0 0 dβ dU = ⎝ 0 − sin β − cos β ⎠ , ds 0 cos β − sin β ds

(7.121)

and ⎛

⎞ ⎛ ⎞ 00 0 −1 dβ dβ dU ϕ := U T · = ⎝ 0 0 −1 ⎠ , ϕ=⎝ 0 ⎠ . ds ds ds 01 0 0

(7.122)

The deformation is pure twist: ϕ·n =−

dβ =t ds

(7.123)

the twist, and the bend b = 0.

7.5.2 Balance Equations The balance equation for mass is dρ + ρ∇ · v = 0, dt

(7.124)

which, for volume-preserving motions (incompressible liquids), reduces to ∇ · v = 0.

(7.125)

v is the material velocity of the fluid-fiber mixture, and ρ is the mass density. The derivative dtd is the material (or total) time derivative

7.5 Example 4: Irreversible Thermodynamics of Flexible Fibers

∂ d = + (v · ∇). dt ∂t

85

(7.126)

The balance equation of linear momentum reads ρ

dv = ρf + ∇ · t T, dt

(7.127)

where t is Cauchy’s stress tensor, and f is the body force per unit mass. Although the internal angular momentum and the torque will be neglected later, we write down the balance for the angular momentum in general [20, 51]. ρ

ds = 2w(t) + ρg + ∇ · . dt

(7.128)

Here, s is the internal angular momentum per unit mass, g is the torque exerted on the material by external fields, and  is the couple stress tensor. The vector w(t) stands for the vector invariant of Cauchy’s stress tensor. In the following, we suppose that the average rotation of the fibers (in addition to thermal fluctuations) is coupled to the macroscopic mechanical motion. Consequently, the internal angular momentum (due to thermal motion) can be neglected compared to the moment of momentum (due to macroscopic motion). Moreover, the rotational energy of internal angular momentum need not be regarded. If we also neglect the torque, which is reasonable as long as no electromagnetic fields are present, (7.128) reduces to 2w(t) + ∇ ·  = 0.

(7.129)

The balance equation of internal energy is of basic importance for constitutive modeling. For the total energy, we can write   v2 )dV = − q · dA + v · t · dA + 2    + ω  ·  · dA + ρv · f dV + ρω m · m dV .

d dt



ρ(u +

(7.130)

The left-hand side is the change of the total (internal plus kinetic) energy of the part of the material regarded. Kinetic energy due to particle rotations has already been neglected. The terms on the right-hand side are heat flow, and power of stress, of couple stress, of forces, and of couple forces. The vectors ω  and ωm are the angular velocities of what the couple stress and the torque act on, respectively. Now, we assume that both ω m and ω  are equal to the macroscopic angular velocity of the volume element, which is equal to the vorticity of the flow field. This is a classical result of Doi and Edwards [52]. It has been shown in [53] that the angular velocity of a fiber is equal to the vorticity of the flow field, even in a second gradient theory. (7.131) ω m = ω  = ω = ∇ × v.

86

7 Thermodynamics of Irreversible Processes with Internal Variables

Making use of the other balance equations and of Gauss’ theorem, we get ρ

du + ∇ · q =˚ d:t d + :∇ω, dt

(7.132)

where ˚ d and t d stand for the symmetric part of the velocity gradient and of Cauchy’s stress tensor, respectively.

7.5.3 Entropy The independent state variables in the domain of the entropy are the internal energy, the pseudoscalar α0 and the tensors αa , αd and T, the latter taking into account the state of the deformation of the fibers. Assume that the entropy depends on the dissipated energy only,  1  η = η e u − und , ρ

(7.133)

where und is the non-dissipated energy (stored in the deformation of the fibers). From here and from (7.114) and (7.115), we get    lμb  1 a a 1 d d  lμt  2 2 α0 − T:T − α :α + α :α . η = ηe u − 2ρ 15 2ρ 3 5

(7.134)

The balance equation for the entropy, for vanishing entropy supply, is ρ

dη + ∇ · φ = σ ≥ 0. dt

(7.135)

The entropy flux is supposed to obey the constitutive equation for local equilibrium systems 1 ∂s 1 where = . (7.136) φ= q T T ∂u Making use of the entropy function in (7.134) and the balance equation of the internal energy, we obtain the expression of the entropy production density

7.5 Example 4: Irreversible Thermodynamics of Flexible Fibers

σ=

 1 ˚ d 2lμt ˚ T:T − d:t + :∇ω − lμt α0 α˙0 − T 15  lμb a a lμb d ˚d 1 ˚− α :α α :α − q · ∇T . − 3 5 T

87

(7.137)

˚d are the co-rotational time derivatives of the tensors T, αa , ˚a , and α Here, ˚ T, α and αd , respectively. The introduction of co-rotational derivatives is here arbitrary, because it holds T :˚ T = T : T˙

(7.138)

˚a = αa : α˙a αa : α

(7.139)

˚d = αd : α˙d . αd : α

(7.140)

It is motivated by the fact that the local equilibrium assumption is reasonable for an observer moving with the fluid, and therefore fluxes and forces should be expressed in terms of co-rotational time derivatives, i.e., the time derivatives in a co-moving frame: dT ˚ T= + T · ω − ω · T, dt

˚a = α

d dαa ˚d = dα + αd · ω − ω · αd . + αa · ω − ω · αa , α dt dt

The tensors  and ∇ω are decomposed into their symmetric and antisymmetric parts: (7.141) :∇ω = s :(∇ω)s + a :(∇ω)a . It results in the entropy production density Tσ =˚ d:t s + s :(∇ω)s + a :(∇ω)a − lμt α0 α˙0 2lμt ˚ lμb a lμ ˚d − q · 1 ∇T . ˚a − b αd :α T:T − α :α − 15 3 5 T

(7.142)

The first term on the right-hand side refers to affine deformations, the second and the third to distortions, the next terms to the relaxations of the distortions of the fibers while the last term is due to heat propagation. The set of process rates is   ˚d ; ˚ ˚a ; α˙0 q; t d ; d ; α T; a ; α

88

7 Thermodynamics of Irreversible Processes with Internal Variables

and the set of corresponding forces is 

−

lμb d 2lμt 1 ∇T ;˚ d; (∇ω)d ; − α ; − T; T 5 15  lμb a α ; −lμt α0 ; . (∇ω)a ; − 3

7.5.4 Constitutive Equations In an isotropic material, there is no coupling between different tensorial orders of fluxes and forces, and there is no coupling between the pseudotensors and the tensor ˚ d. With the above thermodynamic fluxes and forces, Onsager’s linear laws read [24, 54–61] 1 q = − Lq ∇T = −κ∇T , T

(7.143)

t d = 2η˚ d.

(7.144)

The heat conduction equation is the classical Fourier law with heat conductivity κ. The symmetric part of the stress tensor is the viscous stress with viscosity η. It is not influenced by the fiber distortion. t T, d = Ld00 (∇ω)d − Ld01 lμ5b αd − Ld02 2lμ 15

˚d = Ld (∇ω)d − Ld lμb αd − Ld 2lμt T α 10 11 5 12 15 t ˚ T = Ld20 (∇ω)d − Ld21 lμ5b αd − Ld22 2lμ T 15

a = La00 (∇ω)a − La01 lμ3b αa , ˚a = La10 (∇ω)a − La11 α

(7.145) (7.146)

lμb a α , 3

d α0 = −L0 lμt α0 . dt

(7.147)

7.5 Example 4: Irreversible Thermodynamics of Flexible Fibers

89

The couple stresses  are not only due to the gradient of angular velocity, but also due to fiber deformation (see (7.145 and 7.146)). The fact that the couple stresses are non-zero causes an antisymmetric part of the stress tensor (see 7.129), and the fiber suspension is a micropolar continuum. The relaxation of twist and bend deformations of the fibers is caused by the gradient of angular velocity, as well as by the fiber deformation (see (7.145), (7.146), (7.147)). The bend and the twist deformations have different relaxation times because the material coefficient Ld11 and Ld12 on one hand and Ld21 and Ld22 on the other hand are independent of each other. For the fiber co-rotating with fluid, we have ω = ∇ × v, and ∇ω is a second-order gradient term in terms of the material velocity. In this sense, only a second-order gradient theory leads to a coupling between flow v and fiber deformation, as it has been pointed out already from a different point of view in [53]. The pseudoscalar α0 and the tensors αd , and T are related to the distortion of the fibers, so we suppose they are of α-type, i.e., not changing under time inversion. On the other hand, ω is of β-type, i.e., changing sign under time inversion. Consequently, the Onsager-Casimir reciprocal relations are Ld10 = −Ld01 ,

Ld20 = −Ld02 ,

Ld12 = Ld21 ,

La10 = −La01 .

(7.148)

The inequalities λ > 0, Ld22 ≥ 0,

η > 0, Ld00 ≥ 0,

Ld11 ≥ 0,

La00 ≥ 0, La11 ≥ 0, Ld11 Ld22 ≥ (Ld12 )2

L011 ≥ 0 (7.149)

are the consequences of the second law of thermodynamics. According to Casimir’s reciprocal relations, the second law gives no restrictions on the values of Ld01 and Ld20 . The general theory does not imply any further restriction on the value of the coefficients. Equation (7.147) shows that the pseudoscalar α0 tends to zero even if it were non-zero some time ago and it cannot be excited at later times, because there is no coupling to other variables. Therefore, after some relaxation time, α0 is always zero. Because the trace of  does not appear in the balance of energy and in the entropy production it is not relevant, we conclude that the distinction between the symmetric and the deviatoric parts of the tensor  is not necessary, i.e., s = d . There is no coupling between the symmetric part of the velocity gradient (the affine deformation of the fluid), and the distortion of the fibers because ˚ d is the only secondorder polar tensor among the thermodynamic forces. If we had regarded the affine

90

7 Thermodynamics of Irreversible Processes with Internal Variables

deformation of particles in the liquid, equation (7.144) should have turned to the relation known in viscoelasticity [1, 19]. We will see explicitly in an example that the stress tensor may have a skewsymmetric part in a fiber suspension. The equations of motion are those of a micropolar continuum, because couple stresses  are non-zero. Moreover, the theory based on the approximation in (7.112) takes into account the fact that the twist and the bend of the fibers relax with different relaxation times. The authors expect that the improvement of the approximation (7.112) to higher orders does not change the equations because of the nature of the isotropic tensors. To decide if it is really so needs further investigations.

7.5.5 An Illustrating Example In this example, we assume that torsion does not play a role, because torsion stiffness is infinitely small. In this case, (7.145) for the couple stresses simplifies to d = Ld00 (∇ω)d − Ld01

⇒

lμb d α 5

˚d = Ld (∇ω) ˚d ; ˚ d − Ld lμb α  00 01 5

(7.150)

(7.151)

inserting (7.150) again, we end up with ˚d = αd + β(∇ω)d + γ (∇ω) ˚ d 

(7.152)

with the abbreviations lμb d L 5 11  lμb  d d L11 L00 − Ld01 Ld10 β= 5 γ = Ld00 . α=−

(7.153) (7.154) (7.155)

We consider the Poisseuille flow between parallel plates with distance h: ⎞ ⎛ v0  yh − y2 h2 ⎠, 0 v=⎝ 0

(7.156)

7.5 Example 4: Irreversible Thermodynamics of Flexible Fibers

91

where v0 is time independent. Vorticity and gradient of vorticity are calculated as ⎞ ⎛ 0 −1 0 v 0 ω = (∇v)a = 2 (h − 2y) ⎝ 1 0 0 ⎠ 2h 0 0 0 ⎛ ⎞ 0 v0 ω = 2 (h − 2y) ⎝ 0 ⎠ 2h −1

(7.157)

⎛

⎞ 000 v0 (∇ω)d = 2 ⎝ 0 0 1 ⎠ 2h 010

(7.158)

˚ d = (∇ω)d · ω − ω · (∇ω)d (∇ω) ⎛ ⎞ 001 v02 = 4 (h − 2y) ⎝ 0 0 0 ⎠ . 4h 100

(7.159)

The symmetric traceless part of the couple stresses is of the form ⎛

⎞ ab c ⎠. e d = ⎝ b d c e −a − d We are interested in a stationary solutions for the couple stresses ⎛ ⎞ 2b −a + d e v ˚d = 0 (h − 2y) ⎝ d − a −2b −c ⎠ .  2h2 e −c 0

(7.160)

d dt

= 0, and

(7.161)

With these matrices, (7.152) has the solution ⎛

⎞ 00c d = ⎝ 0 0 e ⎠ ce0 e= c=

v03 (h − 2y)2

γ α

− 4βv0 h4

2v02 h2 (h − 2y)2 + 8α2 h6

v0 v03 (h − 2y)2 γ − 4βαv0 h4 +β 2 2h 2v02 h2 (h − 2y)2 + 8α2 h6

(7.162)

(7.163) (7.164)

92

7 Thermodynamics of Irreversible Processes with Internal Variables

and ⎛

⎞ 0 ∇ · d = ⎝ 0 ⎠ e, y

(7.165)

is non-zero, which shows that the antisymmetric part of the stress tensor is non-zero, because ∇ ·  = w(t).

(7.166)

The couple stresses vanish for vanishing flow, c(v0 = 0) = 0, e(v0 = 0) = 0. This example shows that the fiber suspension is a micropolar fluid, even if the fiber orientations are distributed isotropically. Bending of the fibers is the only reason for an antisymmetric part of the stress tensor here.

7.5.6 The Case of a Non-isotropic Orientation Distribution of Fibers In the case of a non-isotropic distribution of fiber orientations, we introduce an ˆ t) of fiber orientations n. ˆ The internal energy orientation distribution function f (x, n, density is the average over all fiber orientations: 

 u¯ e =

S2

ˆ t) f (x, n,

 lμt lμb a ˆ 2+ ˆ nˆ · αd · n)) ˆ (α0 + nˆ · T · n) (α · nˆ + αd · nˆ − n( d2 nˆ 2 2

lμt 2 α0 + 2α0 A : T + TT :: A(4) + = 2

lμb a (α · αa ) : A + 2(αa · αd ) : A + (αd · αd ) : A − (αd : A)2 . 2

The moments of the orientation distribution function  ˆ t) nˆ nˆ d2 nˆ , A= f (x, n, S2  ˆ t) nˆ nˆ nˆ nˆ d2 nˆ A(4) = f (x, n,

(7.167)

(7.168)

S2

are a measure of the degree of orientational order. They are denoted as orientation tensors. A closure relation for the fourth moment is needed, and it is assumed A(4) = AA.

(7.169)

The average of the internal energy over all orientations depends on the orientation tensor A, which is the additional independent variable now:

7.5 Example 4: Irreversible Thermodynamics of Flexible Fibers

93

 lμt  2 α0 + 2α0 A : T + TT :: AA + 2  lμb  a a a d (α · α ) : A + 2(α · α ) : A + (αd · αd ) : A − (αd : A)2 (7.170) 2 u¯ e (x, t) =

with the co-rotational time derivative ˚ u¯ e = ˚ α0 lμt (α0 + A : T) + ˚ T : Alμt (α0 + A : T)  lμb a  a ˚ : α · A + A · αa + 2αd · A α + 2 ˚d : A · αa − αd : AA + +lμb α    1 a a a d d d ˚ α · α + α · α − α : Aα A : lμt (α0 + A : T) T + lμb . 2 (7.171) The expression (7.171) is inserted into the entropy production σ=

  1 1 1 ˚ d u¯ e d:t + :∇ω − q · ∇T − ˚ T T T

(7.172)

with the assumption of incompressibility ∇ · v = 0.

(7.173)

The result for the entropy production is linear in the thermodynamic fluxes and forces and leads to the following linear constitutive equations: q=−

M11 ∇T = −κ∇T . T

(7.174)

The heat conduction equation is the classical Fourier equation. There is no influence of fiber orientation or deformation. ˚ α0 = −M22 lμt (α0 + A : T) .

(7.175)

Now, the differential equation for the scalar deformation variable α0 is not a pure relaxation equation, because the right-hand side depends on A : T, which may have any sign and may cause a growth of α0 . d − M34 t d = 2η˚



lμt (α0 + A : T) T d lμb a a a d d d + α · α + lμb α · α − lμb α : Aα 2

(7.176)

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7 Thermodynamics of Irreversible Processes with Internal Variables

˚ A = M43˚ d − M44



lμt (α0 + A : T) T d lμb a + α · αa + lμb αa · αd − lμb αd : Aαd . 2

(7.177)

The orientational order of the fibers and the viscous flow are coupled. The effect of flow alignment (the parallel orientation of particles in a flow field) is well known in other materials like polymer solutions or liquid crystals [62–64], but also for fiber suspensions [65–69]. It leads to a modified constitutive equation for the viscous stress tensor and to the effect of shear thinning. In our case of deformable fibers, the stress tensor also depends on fiber deformation, as it was found also in [70, 71].  d d = M55 (∇ω)d − M56 2lμt A · αa − αd : AA −M57 lμt (α0 + A : T) A

(7.178)

˚d = M65 (∇ω)d − M66 2lμt A · αa − αd : AAd α −M67 lμt (α0 + A : T) A

(7.179)

 d ˚ T = M75 (∇ω)d − M76 2lμt A · αa − αd : AA −M77 lμt (α0 + A : T) A

(7.180)

a lμb  a α · A + A · αa + 2αd · A 2

(7.181)

a lμb  a α · A + A · αa + 2αd · A . 2

(7.182)

a = M88 (∇ω)a − M89

˚a = M98 (∇ω)a − M99 α

The relaxation of the different measures of deformation, as well as the couple stresses, depends on the orientational order of the fibers (the orientation tensor). With respect to the orientation tensor, the equations are nonlinear.

7.6 Example 5: Systems with Internal Mechanical Parts and Analogy …

95

7.6 Example 5: Systems with Internal Mechanical Parts and Analogy to Rheological Models In this section, we deal with several examples of a thermodynamical description of systems with dissipative mechanical parts. We will deal with global systems, not with a field formulation, and we use internal variables without defining their physical meaning. We will show that by introducing different numbers and tensorial orders of internal variables, many different equations of motion can result. Finally, we will discuss the analogy of the derived equations of motion with the differential equations in different rheological models. From the mechanical model can be derived a relation between the force and the velocity. The mechanical model is said to correspond to a rheological model if, by replacing the force by the stress and the velocity by the strain, the rheological constitutive equation is obtained. In the literature, the method of irreversible thermodynamics with internal variables has been applied successfully to rheology [1]: Viscoelasticity, nonlinear generalized Newtonian flow, streaming birefringence, and normal stress effects have been described. We will deal with a comparison of mechanical models and thermodynamical models for systems relevant for the constitutive theory in rheology. The method of internal variables has given a rather general way for finding models of drag in several kinds of fluid media. The equations of non-steady drag force are derived from the ideas of non-equilibrium thermodynamics. The first law of thermodynamics is the global balance equation of energy for the system consisting of the mechanical part and the surrounding fluid: dU ˙ + P = −Q ˙ + F · v, = −Q dt

(7.183)

˙ is the heat supply, P is the power, F where U is the total energy of the system, Q is the force acting on the particle or the mechanical equipment in the interior of the system, and v is the velocity of that part of the equipment the force is acting upon (see examples below). The right-hand side is the sum of the heat supply from the surroundings and the power of the forces.

7.6.1 The Second Law of Thermodynamics The global balance of entropy for vanishing entropy supply reads: dS ˙ =  ≥ 0, + dt

(7.184)

and the second law of thermodynamics is expressed by the inequality. The entropy ˙ are constitutive quantities, depending on the material S and the entropy exchange  properties of the system, and the relevant variables have to be chosen. One relevant

96

7 Thermodynamics of Irreversible Processes with Internal Variables

variable is the internal energy, and this is the only one in thermostatics if the volume is assumed to be constant. In order to describe non-equilibrium processes, the domain of the constitutive mappings has to include more variables, becoming dependent in equilibrium. Here, we will use dynamical variables ξ1 , ξ2 , . . . , ξn without specifying their physical meaning in the beginning. The internal variables are somehow connected to the motion of the internal parts of the system. With this choice of variables, the entropy is a function S(U, ξ1 , ξ2 , . . . , ξn ). As the entropy of a system has a maximum in adiabatic equilibrium, according to Morse’s lemma [72], the state variables ξ1 , ξ2 , . . . , ξn can be chosen such that the entropy function is: 1 1 1 S = S(U − ξ12 − ξ22 − · · · − ξn2 ). 2 2 2

(7.185)

In general, such an expression for the entropy, without mixed terms of the internal variables and without higher order terms, is obtained by a variable transformation. This is always possible for variables without a specified interpretation. In addition, we assume that also in non-equilibrium the inverse temperature is the derivative of the entropy: 1 = S , (7.186) T which in equilibrium reduces to

∂S 1 = , T ∂U

(7.187)

and for the entropy supply we assume the constitutive relation: ˙ = 

∂S ˙ Q. ∂U

(7.188)

The entropy production is determined by applying he chain rule   dS dU dξ1 dξ2 dξn = S − ξ1 − ξ2 − · · · − ξn dt dt dt dt dt   1 dU dξ1 dξ2 dξn − ξ1 − ξ2 − · · · − ξn = T dt dt dt dt

(7.189)

and inserting the balance of energy into the entropy inequality: T  = F · v − ξ1

dξ1 dξ2 dξn − ξ2 − · · · − ξn . dt dt dt

(7.190)

The entropy production (7.190) has the form of a sum of products of fluxes and forces, and linear constitutive relations between fluxes and forces are assumed. The dynamical variables can be divided into even and odd (α-type and β-type) variables,

7.6 Example 5: Systems with Internal Mechanical Parts and Analogy …

97

not changing sign or changing sign, respectively, under time reversal. Depending on the transformation properties of the variables, Onsager reciprocal relations or Casimir reciprocal relations [73] hold. The internal variables can be scalar, vectorial, or tensorial quantities. The different choices of the set of relevant dynamical variables lead to different thermodynamical models and to different sets of equations of motion.

7.6.2 Equations of Motion for Different Mechanical Equipments 7.6.2.1

A Rigid Sphere Moving in a Compressible Ideal Fluid

The relation between the force F and the velocity v for a sphere moving slowly in an ideal (non-viscous) fluid is known to be [74, 75]: 

  2  d F 2π 3 3 c dF +F m+ + R 3 4πRc2 dt 2 R dt    2π 3 d v R d2 v , R  + = m+ 3 dt c dt 2

(7.191)

where m is the mass of the sphere, R is its radius,  is the density of the fluid, and c is the velocity of sound in it. We are interested in a thermodynamical model with internal variables giving the same relation. This will be given in Sect. 7.6.3. In the following examples, we consider only one-dimensional motions, and therefore force and velocity are scalar quantities.

7.6.2.2

A Mass Connected to a Vessel with a Piston and a Force Acting on the Piston

The motion is assumed to be one-dimensional, see Fig. 7.8. The force F is acting on the piston. The force causes a velocity ˙l of the piston in the fluid of viscosity μ, and the same force is acting on the particle with particle velocity x˙ : F = μ˙l F = m¨x.

(7.192)

The velocity of the point P is v = ˙l + x˙ . Combining these equations, we end up with a relation between force and velocity involving first time derivatives: v˙ =

F˙ F + . m μ

(7.193)

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7 Thermodynamics of Irreversible Processes with Internal Variables

Fig. 7.8 1. A mass point connected to a vessel with a piston and a force acting on the piston and 2. A mass point connected to a vessel with a piston and a spring in series. Forces acting on the different parts

7.6.2.3

1.

2.

m

m

F P x

l 0

m

A Mass Connected to a Vessel with a Piston and a Spring in Series

For the forces acting on the piston, on the spring, and on the mass, we have, respectively: (7.194) Fp = μ˙l, Fs = kL, Fm = m¨x. L is the length change of the (Hookean) spring. In a stationary state, the external force at point P is for the equipment in series (see Fig. 7.8) F = Fp = Fs = Fm .

(7.195)

˙ and we get for the time derivative of the The velocity of point P is v = x˙ + ˙l + L, velocity: F˙ F¨ F (7.196) v˙ = x¨ + ¨l + L¨ = + + m μ k 7.6.2.4

A Mass Point Connected to a Vessel with a Piston and a Spring in Parallel

In this equipment (see Fig. 7.9), the relation between the forces in a stationary state is F = Fp + Fs + Fm . (7.197) The ends of the spring and the piston are connected rigidly and therefore ˙l = L˙ and ˙ For ¨l we have v = x˙ + ˙l = x˙ + L.

7.6 Example 5: Systems with Internal Mechanical Parts and Analogy …

99

Fig. 7.9 A mass point connected to a vessel with a piston and a spring in parallel 3 3 ˙ ˙ ˙ ˙ ˙ ˙ ¨l = Fp = F − Fs − Fm = F − k L˙ − m d x = F − k (v − x˙ ) − m d x . μ μ μ μ μ dt 3 μ μ μ dt 3 (7.198) With v¨ = x¨˙ + ¨˙l and x¨ = Fmm we end up with

v¨ +

k F˙ F¨ k v˙ = F+ + . μ mμ m μ

(7.199)

7.6.3 Thermodynamical Models 7.6.3.1

A Rigid Sphere Moving in a Compressible Ideal Fluid

We try to model the system with one vectorial α-type (even) and one vectorial βtype (odd) variable. The idea behind this is that these internal variables are related to the absolute value of the velocity (odd variable) and the pressure distribution (even variable) around the sphere. The entropy production reads: T = F · v − β ·

dα dβ −α· . dt dt

(7.200)

Linear relations between forces and fluxes are: v = L11 F + L12 β + L13 −

dα , dt

dβ dα = −L12 F + L22 β + L23 , dt dt

− α = −L13 F + L23 β + L33

dα , dt

(7.201) (7.202) (7.203)

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7 Thermodynamics of Irreversible Processes with Internal Variables

where we already introduced Casimir reciprocity relations between the α-type (even) variable F and the β-type (odd) variables β and dα dt L12 = −L21 L13 = −L31 ,

(7.204)

and the Onsager reciprocity relation between the β-type variables β and L23 = L32 .

dα : dt

(7.205)

The second law is expressed by the inequalities: L11 ≥ 0; L22 ≥ 0; L33 ≥ 0; L22 L33 ≥ L223 .

(7.206)

We know that a stationary motion is possible with F = 0, and for such a special state equation (7.202) is exploited. We conclude that L22 = 0 if in this stationary state β stat = 0. From (7.206) follows that L23 = 0. A solution of (7.202) is: β = L12 I where

dI =F dt

(7.207)

dα , dt

(7.208)

and, therefore, from (7.201) and (7.203): v − L212 I = L11 F + L13 − α = −L13 F + L33

dα . dt

(7.209)

These equations are differentiated with respect to time: dv dF d2 α − L212 F = L11 + L13 2 dt dt dt dF d2 α dα = −L13 + L33 2 . − dt dt dt

(7.210) (7.211)

d2 α dt 2

from the second equation is inserted into the first one. Together with (7.208), it yields: 1 dF dv − L212 F = L11 − dt dt L33

 v − L212 I − L11 F + L213

dF dt

 .

(7.212)

Finally, differentiation with respect to t with I˙ = F gives the relation between the velocity and the force:

7.6 Example 5: Systems with Internal Mechanical Parts and Analogy …

    L213 d2 F L11 dF L212 2 L11 − + L12 + F + L33 dt 2 L33 dt L33 1 dv d2 v = + 2, L33 dt dt

101

(7.213)

which is of the form of the mechanical Stokes law equation (7.191). Next, we study a system with one β-type internal variable, i.e., with one dynamic degree of freedom β. The entropy production is: T = F · v − β ·

dβ . dt

(7.214)

The Onsager equations are convenient in the form: dβ dt dβ −β = L21 v + L22 . dt F = L11 v + L12

(7.215)

The inequalities: L11 ≥ 0, L22 ≥ 0, L11 L22 − L12 L21 ≥ 0

(7.216)

is an α-type variable (even under are the consequences of the second law. As dβ dt inversion of motion) while v is of β-type (odd), a reciprocal relation of Casimir type holds: (7.217) L21 = −L12 , which makes the last inequality in (7.216) trivial. With (7.215), the entropy production is: dβ dt     2 dβ dβ dβ dβ · v + L21 v + L22 · = L11 v2 + L22 = L11 v + L12 ,(7.218) dt dt dt dt T = F · v − β ·

where it has been used the Casimir reciprocity L12 = −L21 . Consequently, the crosscouplings do not contribute to the entropy production. The limit case with L22 = 0 is of particular interest. Eliminating β, we get: F = L11 v + L12

dβ , dt

which, in the reversible limit (L11 = 0⇒  = 0), reduces to

(7.219)

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7 Thermodynamics of Irreversible Processes with Internal Variables

F = L12

dβ dv dv = L12 (−L21 ) = L212 , dt dt dt

(7.220)

i.e., to Newton’s second law for a particle with mass m: m = L212 ,

(7.221)

β = L12 v.

(7.222)

1 1 s = s(u − β 2 ) = s(u − mv2 ), 2 2

(7.223)

and the internal variable:

The entropy function is:

meaning that it depends on the difference between the total energy u and the kinetic energy, i.e., on the internal energy e = u− 21 mv2 . 7.6.3.2

A Mass Point Connected to a Vessel with a Piston and a Force Acting on the Piston

In the following subsection, the internal variables are scalar quantities because we consider only one-dimensional motions. We try to model this system with one α-type (even) and one β-type (odd) internal variable. The resulting equation of motion is the same as in the first example, except for the scalar nature of α and β. Comparing (7.213) to (7.193), we identify the coefficients: L11 −

L213 1 1 = 0, L212 L33 + L11 = , L212 = , L33 = 0 L33 μ m 1 1 ⇒ L13 = √ , L11 = . μ k

(7.224) (7.225)

The resulting Onsager equations show that the α-type variable is not relevant: v=

7.6.3.3

1 1 1 dβ F + √ β, − = − √ F, α = 0. μ dt m m

(7.226)

A Mass Point Connected to a Vessel with a Piston and a Spring in Series

We try to model also this system by one α-type and one β-type variable. A comparison of (7.213) to (7.196) gives for the coefficients

7.6 Example 5: Systems with Internal Mechanical Parts and Analogy …

L11 L33 − L13 =

1 1 1 , L212 L33 + L11 = , L33 = 0, L12 = √ k μ m 1 1 ⇒ L13 = √ , L11 = . μ k

103

(7.227) (7.228)

The resulting Onsager equations are: 1 1 dα 1 F+√ β+√ μ m k dt dβ 1 1 − = − √ F, α = √ F. dt m k v=

7.6.3.4

(7.229) (7.230)

A Mass Point Connected to a Vessel with a Piston and a Spring in Parallel

The same thermodynamical model with one α-type and one β-type variable gives comparing (7.199) and (7.213): L33 =

μ 1 μ , L212 = , L212 L33 + L11 = k m km 1 L11 L33 + L213 = k

=⇒ L11 = 0, L33 =

7.6.3.5

μ 1 1 , L12 = √ , L13 = √ . k m k

(7.231) (7.232)

(7.233)

Summary

We have shown that the systems described above, containing a mechanical part in their interior, can be treated with the methods of irreversible thermodynamics with appropriate internal variables. Another example where it would be interesting to apply this method is the dynamic measurement of the mass of a porous material. There, the mass is not constant due to the varying amount of adsorbed matter [76]. In a thermodynamic description, the internal variable would be related to the mass of the adsorbed matter. Examples 2–4 are interesting as mechanical analogies to rheological models for non-Newtonian liquids as well as for solids, which are not simple linear elastic materials. A mechanical model is said to be analogous to the rheological model if the rheological constitutive equation is obtained by replacing the force by the stress and the velocity by the velocity gradient for liquids, or the strain for solids, respectively, and the constants in the mechanical equation by material coefficients

104

7 Thermodynamics of Irreversible Processes with Internal Variables

[1]. So, for instance, in (7.226) replacing F by t and v by the strain  the constitutive equation ˙t t ∇ v˙ = + (7.234) m μ is obtained. Example 3, (7.196), results in the rheological equation ∇ v˙ =

˙t ¨t t + + . m μ k

(7.235)

7.7 Example 6: A Viscous, Incompressible Fluid with Two Internal Variables The state space for this example is given by the equilibrium variables and two scalar internal variables α, β: Z = {e, , α, β}. The fluid is assumed incompressible, i.e., d = 0 and consequently ∇ · v = 0. dt For the entropy flux we suppose φ=

r 1 q, z = . T T

(7.236)

For the dependency of the entropy density on the equilibrium variables, the classical Gibbs relation is assumed, and the balance equations of entropy and internal energy are inserted: d η(e, , α, β) = dt ∂η dα ∂η dβ ∂η de ∂η d + + + = = ∂e dt ∂ dt ∂α dt ∂β dt   1 de p d ∂η dα ∂η dβ = − 2 + + = T dt  dt ∂α dt ∂β dt ∂η dα ∂η dβ 1 de + + = T dt ∂α dt ∂β dt due to the incompressibility of the fluid. For the entropy production we find dη +∇ ·φ−z = dt dη r 1 1 =  + ∇ · q − 2 (∇T ) · q − = dt T T  T  ∂η dα ∂η dβ 1 1 r 1 de + + + ∇ · q − 2 (∇T ) · q − = = T dt ∂α dt ∂β dt T T T

σ=

(7.237)

7.7 Example 6: A Viscous, Incompressible Fluid with Two Internal Variables

 ∂η dβ ∂η dα + − ∂α dt ∂β dt   ∂η dβ ∂η dα + − = ∂α dt ∂β dt 

=

1 (∇T ) · q + T2 1 (∇T ) · q + T2

1 t : ∇v = T 1 t : ∇v T

105

(7.238)

and with a quadratic dependence of the entropy on the internal variables η(e, , α, β) = η0 (e, ) + a(e, )α2 + b(e, )αβ + c(e, )β 2

(7.239)

we end up with σ = (2aα + bβ)

dβ 1 1 dα + (bα + 2cβ) − 2 (∇T ) · q + t : ∇v .(7.240) dt dt T T

There are two scalar pairs of thermodynamic fluxes and forces, one vectorial pair and one pair of second-order tensors. Consequently, in an isotropic material, the only cross-coupling effect is between the two scalar internal variables: dα = l11 (2aα + bβ) + l12 (bα + 2cβ) = L11 α + L12 β dt dβ = l21 (2aα + bβ) + l22 (bα + 2cβ) = L21 α + L22 β dt 1 q = −L33 2 (∇T ) T t = L44 ∇v .

(7.241) (7.242) (7.243)

(7.244)

The constitutive equation for the heat flux is the classical Fourier equation, and the traceless part of the stress tensor is the viscous stress of a Newtonian fluid. In the equations for the time derivatives of the internal variables, we have introduced the abbreviations L11 = 2l11 a + l12 b, L12 = l11 b + 2l12 c, L21 = 2l21 a + l22 b and L22 = l21 b + 2l22 c. α˙ = L11 α + L12 β β˙ = L21 α + L22 β

(7.245) (7.246)

α¨ = L11 α˙ + L12 β˙ = L11 α˙ + L12 (L21 α + L22 β)   L22 = L11 α˙ + L12 L21 α + (α˙ − L11 α) L12 = α˙ (L11 + L22 ) + α (L12 L21 − L22 L11 ) .

(7.247)

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7 Thermodynamics of Irreversible Processes with Internal Variables

With the abbreviations L11 + L22 = A L12 L21 − L22 L11 = B,

(7.248) (7.249)

we end up with the second-order differential equation α¨ = Aα˙ + Bα.

(7.250)

The solution of this differential equation is of exponential form or an oscillating one, depending on the values of the material coefficients:   ⎧   A2 A2 A ⎪ t 4 −Bt + C e − 4 −Bt ⎪ 2 C , e e ⎪ 1 2 ⎪ ⎨ A α(t) = e 2 t (C  1 + C2t),   ⎪ ⎪ A ⎪ A2 ⎪ e 2 t C1 cos ⎩ B − t + C2 sin B− 4

if  if 2 A t , if 4

A2 4

− B > 0;

2

A 4

− B = 0;

2

− B < 0.

A 4

(7.251) This very simple example shows that the balance equations and relaxation equations for the internal variables, all being first order in time, may lead to higher order differential equations after eliminating one of the internal variables.

7.8 Gyarmati’s Wave Approach More than five decades ago, Gyarmati stated [77] that the entropy of a thermodynamic system out of equilibrium can depend on the process rates in addition to the equilibrium variables. The so-called wave approach of Gyarmati introduces the thermodynamic fluxes as additional variables in the state space [78, 79]. They are internal variables because their value cannot be controlled directly from outside, and they become dependent in equilibrium, namely, their value is zero. However, they are not related to the internal structure of the medium but to dynamical processes. The aim of introducing these variables is to avoid the paradox of infinite propagation speeds. This is the same motivation as in extended thermodynamics (see Chap. 9). This wave approach is similar to the branch of extended thermodynamics developed from irreversible thermodynamics [80]. Historically, the wave approach of Gyarmati was the first solution to the problem of infinite propagation speeds, appearing in classical thermodynamics (see Sect. 9.1). Gyarmati derived wave-type equations (hyperbolic equations, see Chap. 10) governing the classical transport processes such as heat conduction, multicomponent diffusion, and thermodiffusion.

7.9 Summary of the Method of Irreversible Thermodynamics with Internal Variables

107

7.9 Summary of the Method of Irreversible Thermodynamics with Internal Variables The different steps in the exploitation of the dissipation inequality in field formulation are summarized as follows: 1. The set of variables the entropy density depends upon has to be chosen. These are internal energy density, mass density, and internal variables. In many cases, a physical interpretation of the internal variables is given, but variables without interpretation can be used too. In Gyarmati’s wave approach, the entropy density is assumed to depend also on the process rates. 2. In the balance of entropy, assumptions for the entropy flux φ = Tq and for the entropy supply z = Tr are made. 3. The balance of internal energy is inserted for the heat flux. 4. If necessary, other equations of motion are inserted. These can include, for instance, the balance of angular momentum or Maxwell’s equations for the electromagnetic fields. 5. The resulting expression for the entropy production is of the form of a sum of products of so-called thermodynamic fluxes and forces. Linear relations between the fluxes and forces are assumed. According to the Curie principle, there is no coupling of different tensorial orders in isotropic media. 6. From the resulting set of Onsager equations (linear force-flux relations), the internal variables are eliminated. Differential equations in time result for the constitutive quantities. In some cases, the dynamical equations for the internal variables are of interest, too. We have exploited the dissipation inequality according to this method in the examples of liquid crystals, colloid suspensions, fiber suspensions, and dissipative mechanical systems with analogies to rheological models. Generally, relaxation equations for the internal variables are the result. After eliminating the internal variables from the set of Onsager equations, we obtain differential equations for the constitutive quantities, which can account for history-dependent material behavior. In the case of Gyarmati’s approach, wave equations for the wanted fields are derived [77]. This solves the problem of infinite propagation velocities, otherwise, present in classical thermodynamics. A related approach solving this problem is extended thermodynamics. In Chap. 9, we will also sketch the problem of infinite velocities in connection with the classical parabolic heat conduction equation. Thermodynamics of irreversible processes with internal variables has been applied to many other systems and phenomena, such as electric conduction [81], electric and magnetic polarization [82–84], heat conduction and radiation [7–9, 79, 80, 85], viscoelastic and plastic deformations [6, 10–12, 14, 18, 19, 25, 73, 78, 86–90], and liquid crystals [16].

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References 1. J. Verhas, Thermodynamics and Rheology. Society for the Unity of Science and Technology, Budapest, Budapest, 2017. Edition almost completely equivalent to the print edition jointly published by Akadémiai Kiadó, Budapest (ISBN 963-05-7389-X) and Kluwer Academic Publishers, Dordrecht (ISBN 0-7923-4251-8) (1997) 2. G.A. Maugin, W. Muschik, Thermodynamics with internal variables, part I. General concepts. J. Non-Equilib. Thermodyn. 19, 217–249 (1994) 3. G.A. Maugin, W. Muschik, Thermodynamics with internal variables, part II. Applications. J. Non-Equilib. Thermodyn. 19, 250–289 (1994) 4. W. Muschik, Internal variables in non-equilibrium thermodynamics. J. Non-Equilib. Thermodyn. 15, 127–137 (1990) 5. G.A. Maugin, R. Drouot, Internal variables and the thermodynamics of macromolecule solutions. Int. J. Eng. Sci. 21, 705–724 (1983) 6. V. Ciancio, E. Turrisi, G.A. Kluitenberg, On the propagation of linear longitudinal acoustic waves in isotropic media with shear and volume viscosity and a tensorial internal variable. I. General formalism. Physica 125A, 640–652 (1984) 7. V. Ciancio, J. Verhás, On heat conduction in media with isotropic microstructure. Atti. Academia Peloritania dei Pericolanti (Messina) 68, 41–53 (1990) 8. V. Ciancio, J. Verhás, A thermodynamic theory for radiating heat transfer. J. Non-Equilib. Thermodyn. 17, 33–43 (1990) 9. V. Ciancio, J. Verhás, On thermal wawes and radiating heat transfer. Acta Phys. Hung. 69, 69 (1990) 10. G.A. Kluitenberg, On the thermodynamics of viscosity and plasticity. Physica 29, 633–652 (1963) 11. G.A. Kluitenberg, On heat dissipation due to irreversible mechanical phenomena in continuous media. Physica 35, 177–192 (1967) 12. G.A. Kluitenberg, E. Turrisi, V. Ciancio, On the propagation of linear transverse acoustic waves in isotropic media with mechanical relaxation phenomena due to viscosity and a tensorial internal variable. I. General formalism. Physica 110A, 361–372 (1982) 13. L. Restuccia, G.A. Kluitenberg, Hidden vectorial variables as splitting operators for the polarization vector in the thermodynamic theory of dielectric relaxation. J. Non-Equilib. Thermodyn. 15, 335–346 (1990) 14. E. Turrisi, V. Ciancio, G.A. Kluitenberg, On the propagation of linear transverse acoustic waves in isotropic media with mechanical relaxation phenomena due to viscosity and a tensorial internal variable. II. Some cases of special interest (Poynting-Thomson, Jeffrey, Maxwell, Kelvin-Voigt, Hooke and Newton media). Physica 116A, 594–603 (1982) 15. J. Verhás, A thermodynamic approach to viscoelasticity and plasticity. Acta Mech. 53, 125– 139 (1984) 16. J. Verhás, Irreversible thermodynamics of nematic liquid crystals. Acta Phys. Hung. 55, 275– 291 (1984) 17. V. Ciancio, On rheological equations and heat dissipation in anisotropic viscoanelastic media. Atti Acc. Sc. Lett. ed Arti di Palermo 1, 3–16 (1981) 18. J. Verhas, A thermodynamic approach to viscoanelasticity and plasticity. Acta Mech. 53, 125–139 (1984) 19. J. Verhas, Irreversible thermodynamics for rheological properties of colloids. Int. J. Heat Mass Transfer 30, 1001–1006 (1987) 20. J. Verhas, The thermodynamic theory of non-Newtonian flows. J. Non-Equilib. Thermodyn. 18, 311 (1993) 21. J.R. Rice, Inelastic constitutive relations for solids: an internal-variable. J. Mech. Phys. Solids 19, 433–455 (1971) 22. G.D.C. Kuiken, Thermodynamics of Irreversible Processes—Applications to Diffusion and Rheology (Wiley, Chichester, New York, Brisbane, 1996)

References

109

23. J. Meixner, H.G. Reik, Thermodynamik der Irreversiblen Prozesse, in Handbuch der Physik, vol. III./2 (Springer, Berlin, 1959), p. 413 24. I. Prigogine, Introduction to Thermodynamics of Irreversible Processes (Interscience, New York, 1961) 25. K.C. Valanis, Irreversible Thermodynamics of Continuous Media (Springer, Wien, 1971) 26. H. Stegemeyer, H. Kelker, 100-jähriges Jubiläum: Flüssigkristalle-Blaue Phasen. Nachr. Chem. Tech. Lab. 36(4), 360–364 (1988) 27. C. Papenfuss, Applied Wave Mathematics—Selected Topics in Solids, Fluids, and Mathematical Methods, chapter Dynamics of Internal Variables from the Mesoscopic Background for the Example of Liquid Crystals and Ferrofluids (Springer, Berlin Heidelberg, 2009), pp. 89–125 28. L. Longa, D. Monselesan, H.-R. Trebin, An extension of the Landau-Ginzburg-de Gennes theory for liquid crystals. Liq. Cryst. 2(6), 769–796 (1987) 29. S. Hess, Irreversible thermodynamics of nonequilibrium alignment phenomena in molecular liquids and in liquid crystals. Z. Naturforsch. 30a, 728–733 (1975) 30. S. Hess, Irreversible thermodynamics of nonequilibrium alignment phenomena in molecular liquids and in liquid crystals ii. Z. Naturforsch. 30a, 1224–1232 (1975) 31. I. Pardowitz, S. Hess, On the theory of irreversible processes in molecular liquids and liquid crystals, nonequilibrium phenomena associated with the second and fourth rank alignmenttensors. Physica 100A, 540–562 (1980) 32. W. Gröbner, Matrizenrechnung, no. 103 (B.I.-Hochschultaschenbücher. Bibliographisches Institut, Mannheim, Wien, Zürich, 1966) 33. H. See, M. Doi, R. Larson, J. Chem. Phys. 92, 792 (1990) 34. G. Vertogen, W.H. de Jeu, Thermotropic Liquid Crystals, Fundamentals, no. 45 in Chemical Physics (Springer, Berlin, Heidelberg, 1988) 35. P.G. de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1974) 36. P.-G. De Gennes, J. Prost, The Physics of Liquid Crystals. Monographs on Physics, 2nd edn. (Clarendon Press, Oxford, 1995) 37. E.G. Virga, Variational Theories for Liquid Crystals (Chapman and Hall, London, 1994) 38. P.-G. de Gennes, Simple Views on Condensed Matter. Series in Modern Condensed Matter Physics, vol. 4 (World Scientific, Singapore, New Jersey, London, Hong Kong, 1992) 39. H. Grebel, R.M. Hornreich, S. Shtrikman, Landau theory of cholesteric blue phases: the role of higher harmonics. Phys. Rev. A 30, 3264 (1984) 40. H. Grebel, R.M. Hornreich, S. Shtrikman, Landau theory of cholesteric blue phases. Phys. Rev. A 28, 1114 (1983) 41. L.D. Landau, V.L. Ginzburg, K theorii sverkhrovodimosti. Zh. Eksp. Teor. Fiz. 20, 1064 (1950). English translation: on the theory of superconductivity, in Collected papers of L. D. Landau, ed. by D. ter Haar (Pergamon, Oxford, 1965), pp. 626–633 42. S. Hess, Birefringence caused by the diffusion of macromolecules or colloidal particles. Physica 74, 277–293 (1974) 43. P. Ilg, M. Kröger, Magnetization dynamics, rheology, and an effective description of ferromagnetic units in dilute suspension. Phys. Rev. E 66(2) (2002). Art. no. 021501 44. P. Ilg, M. Kröger, S. Hess, Magnetoviscosity and orientational order parameters of dilute ferrofluids. J. Chem. Phys. 116(20), 9078–9088 (2002) 45. J. Verhas, C. Papenfuss, W. Muschik, A simplified thermodynamic theory for biaxial nematics using Gyarmati’s Principle. Mol. Cryst. Liq. Cryst. 215, 313–321 (1992) 46. T. Nishimura, Orientation behaviour of fibers in suspension flow through a branching channel. J. Non-Newton. Fluid Mech. 73(3), 279–288 (1997) 47. J. Azaiez, Investigation of the abrupt contraction flow of fiber suspensions in polymeric fluids. J Non-Newton. Fluid Mech. 73(3), 289–316 (1997) 48. M. Vincent, Fibre orientation calculation in injection moulding of reinforced thermoplastics. J. Non-Newton. Fluid Mech. 73(3), 317–326 (1997) 49. C. Papenfuss, J. Verhas, Constitutive theory of fiber suspensions. J. Non-Newton. Fluid Mech. 253, 27–35 (2018)

110

7 Thermodynamics of Irreversible Processes with Internal Variables

50. J. Verhás, Thermodynamic theory for couple stress, in Second Workshop on Dissipation in Physical Systems (Borkow, Poland, 1997) 51. V.K. Stokes, Theory of Fluids with Microstructure (Springer, Berlin, 1984) 52. M. Doi, S.F. Edwards, The Theory of Polymer Dynamics (Clarendon, Oxford, 1986) 53. E. Abisset-Chavanne, J. Férec, G. Ausias, E. Cueto, F. Chinesta, R. Keunings, A secondgradient theory of dilute suspensions of flexible rods in a newtonian fluid. Arch. Computat. Methods Eng. 22, 511–527 (2015) 54. C. Garrod, J. Hurley, Symmetry relations for the conductivity tensor. Phys. Rev A 27, 1487– 1490 (1983) 55. P. Glansdorff, I. Prigogine, Themodynamic Theory of Structure. Stability and Fluctuations (Wiley-Interscience, London, 1971) 56. J. Hurley, C. Garrod, Generalization of the Onsager reciprocity theorem. Phys. Rev. Lett. 48, 1575–1577 (1982) 57. S. Machlup, L. Onsager, Fluctuations and irreversible processes II. Systems with kinetic energy. Phys. Rev. 91, 1512–1515 (1953) 58. L. Onsager, Reciprocal relations in irreversible processes I. Phys. Rev. 37, 405–426 (1931) 59. L. Onsager, Reciprocal relations in irreversible processes II. Phys. Rev. 38, 2265–2279 (1931) 60. L. Onsager, S. Maclup, Fluctuations and irreversible processes. Phys. Rev. 91, 1505–1512 (1953) 61. G.F. Smith, R.S. Rivlin, The anisotropic tensors. Quart. Appl. Math. 15, 308–314 (1957) 62. M. Kröger, Rheologie und Struktur von Polymerschmelzen. Ph.D. thesis, Technische Universität Berlin, W und T Verlag, Berlin, 1995 63. H. Ehrentraut, S. Hess, Viscosity coefficients of partially aligned nematic and nematic discotic liquid crystals. Phys. Rev. E 51(3), 2203–2212 (1995) 64. G. Rienacker, M. Kröger, S. Hess, Chaotic and regular shear-induced orientational dynamics of nematic liquid crystals. Physica A 315, 537–568 (2002) 65. S. Halelfadl, P. Estellao, B. Aladag, N. Doner, T. Marao, Viscosity of carbon nanotubes waterbased nanofluids: Influence of concentration and temperature. Int. J. Therm. Sci. 71, 111–117 (2013) 66. Z. Fan, S.G. Advani, Rheology of multiwall carbon nanotube suspensions. J. Rheol. 51(4), 585–604 (2007) 67. Z. Fan, S.G. Advani, Characterization of orientation state of carbon nanotubes in shear flow. Polymer 46(14), 5232–5240 (2005) 68. A. B. Sulong, J. Park, Alignment of multi-walled carbon nanotubes in a polyethylene matrix by extrusion shear flow: mechanical properties enhancement. J. Compos. Mater. 45(8), 931–941 (2011) 69. F. Folgar, C.L. Tucker, Orientation behavior of fibers in concentrated suspensions. J. Reinf. Plast. Compos. 3, 98–119 (1984) 70. M. Keshtkar, M.C. Heuzey, P.J. Carreau, Rheological behavior of fiber-filled model suspensions: effect of fiber flexibility. J. Rheol. 53(3), 631–650 (2009) 71. M. Keshtkar, M.C. Heuzey, P.J. Carreau, M. Rajabian, C. Dubois, Rheological properties and microstructural evolution of semi-flexible fiber suspensions under shear flow. J. Rheol. 54(2), 197–222 (2010) 72. M. Morse, Relation between the critical points of a real function of n independent variables. Trans. Am. Math. Soc. 27, 345–396 (1925) 73. S.R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics (North-Holland Publishing, Amsterdam, 1963) 74. H. Lamb, Hydrodynamics, 6th edn. (Cambridge University Press, Cambridge, 1963) 75. L.D. Landau, F.M. Lifshitz, Course of Theoretical Physics, vol. 6 (Pergamon Press, Oxford, 1959) 76. J.U. Keller, Theory of measurement of gas adsorption equilibria by rotational oscillations, in 25th International Conference on Vacuum Microbalance Techniques (University of Siegen, Siegen, Germany, 1993)

References

111

77. I. Gyarmati, On the wave approach of thermodynamics and some problems of non-linear theories. Non-Equilib. Thermodyn. 2, 233–260 (1977) 78. I. Gyarmati, Non-Equilibrium Thermodynamics (Springer, 1970) 79. D. Fekete, A systematic application of Gyarmati’s wave theory of thermodynamics to thermal waves in solids. Phys. Stat. Sol. (b) 105, 161–174 (1981) 80. L.S. Garcia-Colin, R.F. Rodriguez, On the relationship between extended thermodynamics and the wave approach to thermodynamics. J. Non-Equilib. Thermodyn. 13, 81–94 (1988) 81. G.A. Kluitenberg, On dielectric and magnetic relaxation phenomena and non-equilibrium thermodynamics. Physica 68, 75–92 (1973) 82. G.A. Kluitenberg, On dielectric and magnetic relaxation phenomena and vectorial internal degrees of freedom in thermodynamics. Physica 87A, 302–330 (1977) 83. G.A. Kluitenberg, On vectorial internal variables and dielectric and magnetic relaxation phenomena. Physica 109A, 91–122 (1981) 84. G.A. Kluitenberg, V. Ciancio, On linear dynamical equations of state for isotropic media. I. General formalism. Physica 93A, 273–286 (1978) 85. D. Fekete, Application of the fundamental principle of dissipative processes to Gyarmati’s wave theory. Russ. J. Phys. Chem. 57, 2700–2703 (1983) (in russian) 86. V. Ciancio, G.A. Kluitenberg, On linear dynamical equations of state for isotropic media. II. Some cases of special interest. Physica 99A, 592–600 (1979) 87. S.R. De Groot, Thermodynamics of Irreversible Processes (North-Holland Publ. Co., Amsterdam, 1951) 88. D. Jou, J.M. Rubi, J. Casas-Vazquez, A generalized Gibbs equation for second order fluids. J. Phys. A 12, 2515–2520 (1979) 89. I. Müller, Thermodynamics (Pitman Publ. Co., London, 1985) 90. B. Nyíri, Thermodynamical derivation of equations of motion for multicomponent fluids. Acta Phys. Hung. 60, 245–260 (1986)

Chapter 8

Rational Thermodynamics

Abstract Rational thermodynamics starts out with a few axioms: Constitutive functions fulfill the principle of equipresence, i.e., depend all on the same set of variables, symmetry relations, the requirements of objectivity, and the second law of thermodynamics. The second law of thermodynamics restricts the class of materials. These restrictions are derived by the method of Liu. In the first example, a simple, heatconducting, viscous fluid, this method leads exactly to the assumptions and predictions of irreversible thermodynamics. In other examples, including material damage and fiber suspensions, the state space includes an internal variable. If the differentiated balance equations are considered as additional constraints, the restrictions on constitutive functions become weaker.

8.1 Introduction Here, we want to deal only with the branch of the theory that uses large state spaces. In our examples, the relation between entropy flux and heat flux will be universal, i.e., no special form of constitutive relation for φ is assumed. Usually in rational thermodynamics, temperature and entropy are primitive concepts, and mainly the theory is in field formulation. The presuppositions in this main branch of rational thermodynamics can be summarized as: • The existence of fields of entropy density η(x, t), entropy flux density φ(x, t), entropy supply z(x, t), and entropy production σ(x, t) fulfilling the dissipation inequality (local in time and space) (x, t)

dη(x, t) + ∇ · φ(x, t) − z(x, t) = σ(x, t) ≥ 0 ∀(x, t) dt

(8.1)

is assumed. • Entropy and temperature are primitive concepts, meaning that nothing is said about how to measure these quantities. A priori knowledge of the physical meaning of entropy and temperature is presupposed. • There are balance equations for the wanted fields. © Springer Nature Switzerland AG 2020 C. Papenfuß, Continuum Thermodynamics and Constitutive Theory, https://doi.org/10.1007/978-3-030-43989-7_8

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In classical theory, these are the five fields of mass density, momentum density, and energy density. If internal variables are included in state space, additional rate equations for these variables are needed. If electromagnetic fields are present, Maxwell’s equations are the equations of motion for these. Maxwell’s equations themselves do not have the form of balance equations. In the general scheme in Sect. 8.2, the structure of balance equations, leading to a quasilinear system of field equations, is used. The method of Liu [1] to exploit the dissipation inequality is based on the structure of balance equations. Therefore, we have reformulated Maxwell’s equations in Sect. 3.5 as balance equations for the charge density and the magnetic flux. For an application in the exploitation of the dissipation inequality, we refer to the literature [2, 3]. In all cases, there are more unknown field quantities than there are balance equations, and additional constitutive equations are needed. Constitutive equations express the different behaviors of different materials. Constitutive equations are treated in a systematic manner in rational thermodynamics. The constitutive quantities are defined as mappings on the state space. The principle of equipresence states that all constitutive quantities depend on the same set of variables. This dependence is restricted by the second law of thermodynamics. In addition, material symmetry has to be taken into account [4]. The most general form of constitutive equations taking into account the material symmetry is given by representation theorems on the state space [5–7]. The balance equations (and rate equations and Maxwell’s equations), together with the constitutive equations, are the set of field equations to be solved. For given initial and boundary conditions, a solution is called a thermodynamic process. However, the border between rational thermodynamics and other non-equilibrium thermodynamic theories is not sharp, and a non-equilibrium temperature definition that can principally be verified experimentally can be found in [8, 9]. This branch of thermodynamics, using operational definitions of temperature and entropy, is sometimes called non-classical thermodynamics. For articles, reviews, and textbooks on rational thermodynamics see, for instance, [4, 10–13].

8.2 Structure of the Balance Equations Before carrying out the differentiations of the constitutive functions, balance equations are linear partial differential equations of first order in time and in space. Therefore, applying the chain rule to the constitutive quantities gives partial derivatives that are one order higher than the order of the derivatives included in state space. The resulting equations are linear in these highest derivatives. Let us consider an example: Assume that the state space includes (, T, ∇T ). After applying the chain rule, the balance of internal energy becomes:

8.2 Structure of the Balance Equations

115

de +∇ ·q dt   ∂e ∂e ˙ ∂e ˙ ) + ∂q · ∇ + ∂q · ∇T + ∂q : ∇∇T = ˙ + T+ · (∇T ∂ ∂T ∂∇T ∂ ∂T ∂∇T = t : ∇v + r. 

(8.2) According to the classification of partial differential equations, the resulting set of equations is a quasilinear system of partial differential equations. Formally, this set of equations for the wanted fields u A with the fluxes f A (constitutive quantities, defined on the state space) with variables z B , and productions and supplies π A can be written as du A + ∇ · f A = πA dt du A  ∂ f A + · ∇z B = π A . dt ∂z B B

(8.3) (8.4)

Here, we assumed that the wanted fields are included in the state space, but a similar formulation is possible if they are state space functions, too. In this general form, we do not distinguish between the different tensor orders of the wanted fields u A . The summation is over the different state space variables. The order of the resulting differential equations is determined by the order of the derivatives in the state space.

8.3 Principles Restricting Constitutive Functions 8.3.1 Objectivity or Material Fame Indifference This part deals with the question of how different observers see the material behavior of the same material. This question has to be distinguished from the question of how the material behaves in different states of motion [14–16]. Some confusion arises in the literature due to a mixing up of these two different situations. In non-relativistic theories, changing the observer is described as a rule by a Euclidean transformation of space-time described in general by a time-dependent proper orthogonal transformation Q(t) and by a time-dependent translation c(t) (see, for instance, [12]). Objectivity or material frame indifference is the fact that different observers (moving with respect to each other arbitrarily) observe the same form of constitutive function. This fact does not say anything about the transformation properties of the value of the constitutive quantities. The transformation property of the value depends on how the variables in the domain of the constitutive mappings transform under a change of the observer. The form of the constitutive function is the same for different

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observers as it is a material property, and the same material is observed in different frames. So, if one observer finds, for instance, a quadratic dependence of a material property on some variable, the same will be true for any other observer. The transformation properties of the state space elements have to be concluded from measurements in different frames. It is possible that these fields transform like scalars, vectors, and tensors of different orders; it is also possible that they transform completely differently. It is observed that mass density and internal energy are of scalar behavior. That means, they are invariant for all observers; gradients of scalars are vectors (tensors of first order). For other quantities, the tensorial or vectorial behavior is only an approximation for small observer velocities, but in usual experimental situations this is a reasonable assumption. In detail we have:  −→ ,

 −→ Q · ,

∇ −→ Q · ∇,

∇v −→ Q· ∇v ·Q ,

e −→ e,

(8.5)

q −→ Q · q,

(8.6)

∇e −→ Q · ∇e,

(8.7)

t −→ Q · t · Q .

(8.8)

Here, the −→ denotes changing of the observer. These transformation properties of the different physical quantities have to be concluded from experimental observations. The above transformation properties will be used in Sect. 8.6.

8.3.2 Material Symmetry The positions of material elements in the reference configuration form the material manifold. Material symmetry deals with properties on the material manifold. The position variable in this consideration is the material coordinate X. We will consider transformations of the material in the reference configuration that are volume preserving. The volume elements of the continuum are then rotated or volume-preserving deformed. Let Z be the state space, where all gradients are formulated as derivatives with respect to the material coordinate. Definition: A transformation U is a symmetry transformation, if all constitutive properties M have the same value before and after the transformation:

8.3 Principles Restricting Constitutive Functions

M(U(Z)) = M(Z).

117

(8.9)

The material does not “notice” a symmetry transformation. For example, for cubic crystal lattices, rotations of π2 around any of the cube lines or of 2π around one of the four space diagonals of the cubic elementary cell are a 3 symmetry transformation. The different crystal lattices can be classified according to their symmetry transformations. This defines the different crystal classes. According to Noll, solids and liquids can be distinguished as follows: 1. In solids, there can exist only symmetry transformations, which are rotations. 2. In liquids, all volume-preserving transformations are symmetry transformations. The second point, however, is not so clear for liquids with an internal variable, where the internal structure can change under the transformation. If this classification of liquids holds, one can show that the state space includes only the density and not the deformation gradient. The symmetry transformations form a group in the mathematical sense because: 1. The identity, the transformation which does not change anything, is a symmetry transformation. 2. The inverse element to each transformation is the back transformation, also a symmetry transformation. 3. Applying two symmetry transformations, one after the other, gives another symmetry transformation. However, the group of symmetry transformations is not commutative. The order of the applied transformations is relevant. For isotropic solids, the whole orthogonal group S O(3) is the symmetry group. All rotations are symmetry transformations. If the material is non-chiral, also inversions are symmetry transformations. For any rotation Q and any constitutive mapping M on the state space Z, it holds that: M(Q(Z)) = M(Z).

(8.10)

8.4 Representation Theorems for Isotropic Materials The aim of this section is to derive the most general form of constitutive equation compatible with material symmetry, if the state space is chosen. If the material (solid or liquid) is isotropic, all rotations Q are symmetry transformations and (8.10) holds. We will deal here only with this symmetry class. Representation theorems for other crystal classes can be found in [5]. Now, in addition to symmetry transformations on the material manifold, we introduce observer transformations. These transform the coordinate systems of different observers, one into another, but refer to the same material coordinate system.

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An arbitrary rotation Q can be applied on one hand as symmetry transformation on the material manifold and on the other hand as an observer transformation. For an observer transformation, constitutive quantities transform like tensors of different orders: for instance, internal energy density as a scalar, heat flux density as a vector, and the Cauchy stress tensor as a second-order tensor. Suppose that we have a state space of the form: Z = {T,

∂xi ∂T , , . . . }, ∂XA ∂XA

(8.11)

where gradients are formulated as derivatives with respect to the material coordinate X. The components of vectors with respect to a Cartesian coordinate system in the reference configuration are denoted with capital indices: for instance, X A . Small indices denote components of vectors and tensors with respect to a coordinate system in the actual configuration. Let Q act as an observer transformation, transforming the components of the coordinate x in the actual configuration. The tensor property of constitutive functions leads to the equations: ∂x j ∂T ∂xi ∂T , , . . . ) = e(T, Q i j , ,...) ∂XA ∂XA ∂XA ∂XA ∂x j ∂T ∂xi ∂T qi (T, , , . . . ) = Q ji q j (T, Q i j , ,...) ∂XA ∂XA ∂XA ∂XA ∂x j ∂T ∂xi ∂T ti j (T, , , . . . ) = Q ki tkl (T, Q i j , , . . . )Q l j . ∂XA ∂XA ∂XA ∂XA e(T,

(8.12) (8.13) (8.14)

On the other hand, we can apply the same rotation as a transformation of the material on the material manifold. Due to the fact that the material is supposed to be isotropic, a rotation is a symmetry transformation, and it holds: ∂xi ∂T ∂xi ∂T , , . . . ) = e(T, Q B A, Q B A, . . . ) ∂XA ∂XA ∂XB ∂XB ∂xi ∂T ∂xi ∂T qi (T, , , . . . ) = qi (T, Q B A, Q B A, . . . ) ∂XA ∂XA ∂XB ∂XB ∂xi ∂T ∂xi ∂T ti j (T, , , . . . ) = ti j (T, Q B A, Q B A , . . . ). ∂XA ∂XA ∂XB ∂XB e(T,

(8.15) (8.16) (8.17)

Setting equal the right-hand sides of the respective equations of (8.12) and (8.15), (8.13) and (8.16), as well as (8.14) and (8.17), we obtain:

8.4 Representation Theorems for Isotropic Materials

119

∂x j ∂T ∂xi ∂T , , . . . ) = e(T, Q B A, Q B A , . . . ) (8.18) ∂XA ∂XA ∂XB ∂XB ∂x j ∂T ∂xi ∂T Q ji q j (T, Q i j , , . . . ) = qi (T, Q B A, Q B A , . . . ) (8.19) ∂XA ∂XA ∂XB ∂XB ∂x j ∂T ∂xi ∂T Q ki tkl (T, Q i j , , . . . )Q l j = ti j (T, Q B A, Q B A , . . . ), (8.20) ∂XA ∂XA ∂XB ∂XB e(T, Q i j

or ∂x j ∂T ∂xi ∂T , , . . . ) = e(T, Q B A, Q B A , . . . ) (8.21) ∂XA ∂XA ∂XB ∂XB ∂x j ∂T ∂xi ∂T Q T · q(T, Q i j , , . . . ) = q(T, Q B A, Q B A , . . . ) (8.22) ∂XA ∂XA ∂XB ∂XB ∂x j ∂T ∂xi ∂T Q T · t(T, Q i j , , . . . ) · Q = t(T, Q B A, Q B A , . . . ), (8.23) ∂XA ∂XA ∂XB ∂XB e(T, Q i j

which must hold for any ∂∂xX iA and any ∂∂T , especially for ∂∂xX iB Q TB A and XA Inserting this in the argument of the constitutive functions, together with Q TB A Q AC = δ BC

∂T ∂XB

Q TB A .

(8.24)

(because the rotation Q is an orthogonal transformation), yields: ∂x j T ∂T T ∂xi ∂T Q B A, Q B A , . . . ) = e(T, , , . . . ) (8.25) ∂XB ∂XB ∂XA ∂XA ∂x j T ∂T T ∂xi ∂T Q T · q(T, Q i j Q , Q , . . . ) = q(T, , , . . . ) (8.26) ∂XB BA ∂XB BA ∂XA ∂XA ∂x j T ∂T T ∂xi ∂T Q T · t(T, Q i j Q B A, Q B A , . . . ) · Q = t(T, , , . . . ). (8.27) ∂XB ∂XB ∂XA ∂XA e(T, Q i j

Finally, we introduce the deformation gradient F = ∂∂Xx . Note that the index of the first component of the deformation gradient refers to a coordinate system in the actual configuration, whereas the second index refers to the coordinate system in the reference configuration. The resulting equations, valid for any isotropic material, read: e(T, Q · F · Q T , (∇T ) · Q T , . . . ) = e(T, F, ∇T, . . . )

(8.28)

Q · q(T, Q · F · Q , (∇T ) · Q , . . . ) = q(T, F, ∇T, . . . ) Q · t(T, Q · F · Q T , (∇T ) · Q T , . . . ) · Q = t(T, F, ∇T, . . . ).

(8.29) (8.30)

T

T

T

T

Equations (8.28) to (8.30) are the conditions to be satisfied by scalars, vectors, and second-order tensors, respectively, in an isotropic medium. An isotropic representation theorem is the most general form of constitutive equation in accordance with

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these conditions. This form of constitutive equation of course depends on the number of scalar, vectorial, and tensorial quantities in the domain of the constitutive mapping, i.e., on the state space. For a given number of vectors and tensors of different orders, the representation theorems can be found in tables, for instance, in [5, 6, 17]. Second-order tensors have to be decomposed into symmetric and antisymmetric parts. Equivalently, for the antisymmetric part a vector can be introduced by a = A : , where  is the totally antisymmetric third-order tensor.

8.4.1 Example: State Space Including One Symmetric Tensor and One Vector The state space in this example includes one symmetric tensor S and one vector v: Z = {S, v}.

(8.31)

Let us consider the representation theorem for an arbitrary scalar quantity. Scalar invariants from the vector v: v·v

(8.32)

because only the length of a vector does not change under rotation. Scalar invariants from the symmetric second-order tensor S: Three main invariants, I1 , I2 , and I3 , of a second-order tensor can be defined by the characteristic equation: det(S + λδ) = λ3 + I1 λ2 + I2 λ + I3 .

(8.33)

The main invariants I1 , I2 , and I3 transform like scalars under rotations of the coordinate system because neither the eigenvalues λ nor the determinant changes under such a rotation, and therefore the characteristic equation is invariant: det( Q · S · Q T + λ Q · δ · Q T ) = det( Q · (S + λδ) · Q T ) = det(( Q T · Q) · (S + λδ)) = det(S + λδ).

(8.34)

There can be built only three independent invariants from the components of a symmetric second-order tensor because, in the main axis system, the tensor has only three non-zero components, the three eigenvalues, and the invariants cannot depend on the coordinate system. A possible choice are the three invariants I1 , I2 , and I3 . Another possible choice is the traces trS, trS2 , and trS3 , which are independent and invariant under rotation, like any trace of a tensor.

8.4 Representation Theorems for Isotropic Materials

121

Scalar invariants from the vector v and the symmetric second-order tensor S: These following terms are scalar invariants v · S · v, v · S · S · v, v · S3 · v, . . . v · Sn · v,

(8.35)

which can be proved by applying a rotation to these expressions. However, the invariants v · S3 · v, . . . v · Sn · v are not independent as a consequence of the CayleighHamilton theorem, which states that the tensor S satisfies its characteristic equation: S3 + I1 S2 + I2 S + I3 δ = 0.

(8.36)

Solving this equation for S3 shows that S3 , and by multiplication also higher powers, can be expressed in terms of S2 , S, and the invariants I1 , I2 , and I3 . In summary, a set of independent invariants consists of the terms: trace(S), trace(S · S), trace(S · S · S), v · v, v · S · v, v · S · S · v.

(8.37)

A constitutive function is an arbitrary function of these invariants, which only in special cases is a linear function. The representation theorem for an arbitrary vectorial constitutive function R is derived from the representation theorem for a scalar quantity, including an additional vector c in the set of variables. It can be shown that the invariants, which can be constructed from one symmetric second-order tensor S and two vectors v and c, are: v · v, v · S · v, v · S · S · v, trace(S), trace(S · S), trace(S · S · S), c · c, c · S · c, c · S · S · c, c · v, c · S · v, c · S · S · v.

(8.38)

With the help of the vector c, we can construct the scalar quantity c · R, for which the invariants are given by the representation theorem (8.38). On the other hand, the quantity c · R is linear in the arbitrary vector c and can include only invariants linear in this vector: c · R = αc · v + βc · S · v + γc · S · S · v,

(8.39)

where the coefficients α, β, and γ may depend on all scalar invariants (8.37). Since (8.39) is valid for arbitrary c, it must hold: R = αv + β S · v + γ S · S · v.

(8.40)

This is the representation theorem for a vectorial constitutive quantity R on the chosen state space. The representation theorem for a second-order tensorial constitutive function is derived analogously with the help of an arbitrary second-order tensor. For a symmetric second-order tensor T , the theorem yields:

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8 Rational Thermodynamics

T = aδ + bS + cS · S + dvv +e (v S · v)symm + f (v S · S · v)symm ,

(8.41)

where again the coefficients a, b, c, d, e, and f may depend on all scalar invariants (8.37).

8.4.2 Physical Examples 1. In order to describe heat conduction, the state space has to include the temperature gradient in addition to the equilibrium variables. If we want to deal with fast phenomena, the time derivative of temperature is included, too: Z = {T, T˙ , , ∇T }.

(8.42)

The representation theorem for the constitutive quantities internal energy density, heat flux, and stress tensor reads: e = e(T, T˙ , , (∇T ) · (∇T )) q = κ(T, T˙ , , (∇T ) · (∇T ))∇T t = − p(T, T˙ , , (∇T ) · (∇T ))δ

+τ (T, T˙ , , (∇T ) · (∇T ))∇T ∇T.

(8.43) (8.44) (8.45)

There is nothing supposed about linearity. The functions e(·), κ(·), p(·), and τ (·) can be arbitrary functions. 2. A fluid with heat conduction and viscous flow, including fast temperature changes, is described well in terms of the state space: Z = {, T, T˙ , ∇T, (∇v)symm }.

(8.46)

With this set of variables, the representation theorem for isotropic tensor functions reads:  e = e T, T˙ , , (∇T ) · (∇T ), ∇ · v, (∇v)symm : (∇v)symm , trace ((∇v)symm · (∇v)symm · (∇v)symm ) , (∇T ) · (∇v)symm · (∇T ), (8.47) (∇T ) · (∇v)symm · (∇v)symm · (∇T )) q = −κ(◦)∇T + q1 (◦)(∇v)symm · ∇T + q2 (◦)(∇v)symm · (∇v)symm · ∇T

(8.48)

t = − p(◦)δ + 2μ(◦)(∇v)symm + t2 (◦)(∇v)symm · (∇v)symm + t3 (◦)(∇T )(∇T ) + t4 (◦) ((∇T ) · (∇v)symm (∇T ))symm + t5 (◦) ((∇T ) · (∇v)symm · (∇v)symm (∇T ))symm ,

(8.49)

8.4 Representation Theorems for Isotropic Materials

123

where we introduced the abbreviation:  (◦) = T, T˙ , , (∇T ) · (∇T ), ∇ · v, (∇v)symm : (∇v)symm , trace ((∇v)symm · (∇v)symm · (∇v)symm ) , (∇T ) · (∇v)symm · (∇T ) , (∇T ) · (∇v)symm · (∇v)symm · (∇T )) . (8.50)

8.4.3 Example: Reduction of the Constitutive Equation for Linear Elastic Materials A material is called elastic if the dependence of the (Cauchy) stress tensor t on the configuration mapping is only through the deformation gradient F, i.e., through the locally linearized configuration mapping χ. In contrast, a material is called viscoelastic if the stress tensor depends in addition on the time derivative of the deformation gradient. A material is called plastic if there is a residual strain after the unloading. For the deformation gradient, we have the (unique) polar decomposition: F = R · U = V · R,

(8.51)

R · RT = RT · R = δ

(8.52)

with an orthogonal tensor R:

and U · U = FT · F V · V = F · FT .

(8.53) (8.54)

U · U is denoted as right Cauchy-Green tensor and V · V as left Cauchy-Green tensor. Proposition: For an isotropic elastic material, the stress tensor depends only on the left Cauchy-Green tensor, instead of the deformation gradient. Proof: Due to the supposed isotropy, the symmetry group contains all orthogonal transformations Q: t (F · Q) = t (F). (8.55) As the tensor R (and consequently R T ) in the polar decomposition (8.51) is orthogonal, it can be inserted for Q, and we obtain: t (F) = t (F · R T ) = t (V · R · R T ) = t (V ).

(8.56)

124

8.4.3.1

8 Rational Thermodynamics

Different Measures of Deformation and Geometrical Linearization

We define the Green tensor as: G :=

 1 T F · F−δ . 2

(8.57)

It vanishes in case of a rigid body motion. The displacement u is defined as: u = x − X,

(8.58)

∂x ∂u = − δ =: δ F. ∂X ∂X

(8.59)

and the displacement gradient is:

Geometrical linearization: Neglecting all nonlinear terms in the deformation measure, i.e., restricting to small deformations, we have the following relations between the different measures of deformation:  1 (δ + δ F)T · (δ + δ F) − δ 2   1 1 T = δ F + δ F + δ FT · δ F = δ F + δ FT , 2 2 G=

(8.60)

V 2 = F · F T = (δ + δ F) · (δ + δ F)T    2 1 T T T δF + δF = δF + δF + δF · δF + δ = δ + 2 = (δ + G)2 .

(8.61)

As this is an equation for symmetric tensors, we can take the square root: V = δ + G,

(8.62)

which shows the equivalence of the different measures of deformation in the linear case. This linearization in the deformation measure is denoted as geometrical linearization, and it concerns the variables in the domain of the constitutive functions only. It has to be distinguished from the so-called physical linearization, which means that

8.4 Representation Theorems for Isotropic Materials

125

constitutive functions (for instance, the stress tensor) are assumed to depend only in a linear way on the variables (especially the measure of deformation) in their domain.

8.4.3.2

Representation Theorem for the Stress Tensor and Physical Linearization

Let us consider the stress tensor as a constitutive function on the state space: Z = {ρ, T, V }.

(8.63)

The representation theorem for an isotropic material reads: t = φ0 δ + φ 1 V + φ2 V 2 ,

(8.64)

where φ0 , φ1 , and φ2 are coefficients depending on the scalar invariants of the secondorder tensor V (see Sect. 8.4) and on the scalar variables density and temperature. After geometrical linearization, the left Cauchy-Green tensor V can be replaced by V = δ + G: (8.65) t = φ0 δ + φ1 (δ + G) + φ2 (δ + G)2 . Assuming physical linearity, the term (δ + G)2 = δ + 2G + G 2 is replaced by δ + 2G, and the representation theorem for the stress tensor can be summarized as: t = 0 δ + 1 G

(8.66)

0 = φ0 + φ1 + φ2 , 1 = φ1 + 2φ2 .

(8.67)

with

In order to have a linear homogeneous constitutive function t, the scalar coefficient 1 must be independent of G, and the scalar coefficient 0 must be linear in G. From the list of scalar invariants of a second-order tensor (see Sect. 8.4.1), there is only one scalar invariant linear in a second-order tensor, namely, the trace of the tensor. Consequently, the coefficients have the following form: 0 = λ L trace(G), 1 = 2G

(8.68)

with constant Lame coefficients λ L and G. In summary, applying the requirements of isotropic material symmetry, geometric and physical linearity, and the representation theorem, we have derived the stressstrain relation of the Hook material: t = λtrace(G)δ + 2μG.

(8.69)

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8 Rational Thermodynamics

8.5 The Second Law of Thermodynamics In this section, we reason why the second law of thermodynamics implies restrictions on constitutive functions at all. Essential to this argument is the fact that the balance equations, after the chain rule is applied to the constitutive functions, form a system of equations linear in the so-called higher derivatives, i.e., the derivatives not included in the state space but one order higher. These higher derivatives will be put together in a row y. Then the system of balance equations and the dissipation inequality can be written symbolically as: A· y = C

(8.70)

B· y≥D

(8.71)

with a Matrix A, rows B and C, and a scalar function D. All these quantities are functions of the state- space variables.

8.5.1 Class of Materials We are not interested in a special constitutive equation, but we are looking for a class of materials that is defined as follows: Definition: All constitutive equations defined on the chosen large state space satisfying the balance equations and the dissipation inequality determine the class of materials. In principle, there are two possibilities excluding each other to find this class of materials: 1. There exist (mathematical) solutions of the balance equations that do not satisfy the dissipation inequality. These solutions are excluded from the set of solutions observed in nature by the second law, i.e., the second law “forbids” certain processes. Formally, this means: For fixed A, C, B, and D the dissipation inequality excludes certain y, i.e., certain process directions in state space (represented by the higher derivatives) are not allowed. 2. The constitutive functions are such that there exist only solutions of the balance equations satisfying the dissipation inequality. In this case, the second law restricts the “allowed” constitutive equations and does not restrict thermodynamic processes. Formally in this case, the A, C, B, and D have to be determined so that all process directions are possible, which means that, whatever y may be, the dissipation inequality is always satisfied. Without additional physical tools, there is no possibility to decide which of these cases is the correct one. Consequently, we need an amendment to the second law [18] for deciding how to exploit the dissipation inequality with respect to the balance

8.5 The Second Law of Thermodynamics

127

equations. This amendment to the second law is the no-reversible direction axiom, which we will now discuss.

8.5.2 No-Reversible Direction Axiom According to possibilities 1 and 2, we have to look for a criterion to decide whether constitutive properties can exclude process directions in state space, or whether constitutive properties are restricted in such a way that all process directions in state space are allowed. The no-reversible direction axiom enforces case 2. That means that, for all materials and to each fixed process direction in state space, initial conditions are possible so that the process runs along the prescribed process direction. A detailed discussion of that axiom is rather sophisticated [18], but here we will give a shortened version sufficient for clarifying the problem. We consider an arbitrary material at a fixed non-equilibrium state Z (x, t) (fixed position, fixed time, local). The coefficients in the balance equations (8.70) and the dissipation inequality (8.71) are A(Z ), C(Z ), B(Z ), D(Z ).

(8.72)

We now presuppose that case 1 may be possible. Assumption: At arbitrary, but fixed Z there are allowed and forbidden directions in state space (see Fig. 8.1). May y1 be an allowed process direction, as in (8.73). Then, A · y1 = C,

Fig. 8.1 At an arbitrary but fixed state Z , there are directional derivatives representing directions that are allowed (→) or forbidden (− − − −). “Allowed” means in agreement with the dissipation inequality; “forbidden” means in contradiction to it. Figure taken from [19]

B · y1 ≥ D

(8.73)

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8 Rational Thermodynamics

is valid. If y2 is a forbidden process direction, we have: A · y2 = C,

B · y2 < D.

(8.74)

We now prove the following: Proposition: A consequence of (8.73) and (8.74) is the statement: There exists a reversible process direction αy1 + (1 − α)y2 for each arbitrary, but fixed, state. The real number α satisfies: 0 < α :=

D − B · y2 < 1. B · (y1 − y2 )

(8.75)

Proof: Multiplying (8.73) with α > 0 and (8.74) with β > 0, we get by addition of both: . A · (αy1 + βy2 ) = (α + β)C, α + β = 1, . B · [αy1 + (1 − α)y2 ] = D.

(8.76) (8.77)

. By the demand α + β = 1, the process direction αy1 + (1 − α)y2 in (8.76) satisfies the balance equations. α is determined by (8.77), from which immediately (8.75) follows. The inequalities in (8.75) are induced by (8.73) and (8.74). From (8.77) we  conclude that αy1 + (1 − α)y2 is a reversible process direction. By this proposition, we proved the existence of at least one reversible process direction if, at an arbitrary non-equilibrium state Z , both kinds of allowed and forbidden process directions exist. By arguments of continuity, we conclude the existence, not only of a reversible direction, but also of a piece of a reversible trajectory. Because we are in non-equilibrium, we have to exclude such reversible trajectories by the no-reversible direction. Axiom [20]: Except in equilibrium subspace, reversible process directions in state space do not exist. A consequence of this axiom is the fact that, in non-equilibrium, all process directions are either allowed or forbidden, but no non-equilibrium state exists with both kinds of process directions because then, according to (8.75), a reversible process direction can be constructed. By the no-reversible direction axiom and the proposition (8.75), we have proved the following proposition: Proposition: If Z is no trap (a state that cannot be left by any process), the conclusion: ∧y:

A(Z ) · y = C(Z ) −→ B(Z ) · y ≥ D(Z )

(8.78)

is valid; that means all solutions of the balance equations satisfy the dissipation inequality. By this proposition, restrictions for the A, C, B, and D result. These restrictions characterize the class of materials we are looking for.

8.5 The Second Law of Thermodynamics

129

8.5.3 Liu Technique We start out with Liu’s Proposition [1, 21, 22]. By use of the inclusion (8.78), the following statement is valid: In large state spaces, constitutive equations satisfy the Liu relations: B(Z ) = λ(Z ) · A(Z ),

(8.79)

λ(Z ) · C(Z ) ≥ D(Z ).

(8.80)

Here the state function λ is only unique if A has its maximal rank. The entropy production density: σ := λ · C − D ≥ 0 (8.81) is independent of the process direction, i.e., it depends only on the state space element and not on higher derivatives. The set of equations (8.79) will be denoted as “Liu equations”, and equation (8.80) is the residual inequality. The Liu equations are as many equations as there are higher derivatives (elements in the row y). These are more equations than there are unknown factors λ, which is the same number as the number of balance equations. The equations remaining after eliminating the unknowns λ from the Liu equations are the restrictions on constitutive functions. The determination of these restrictions on constitutive functions is the main aim of the procedure. In addition, after eliminating the multipliers λ from the residual inequality, an expression for the entropy production is obtained. In the examples, it turns out that this expression is of the form of a sum of products where one factor can be denoted as a thermodynamic force and the other one as a thermodynamic flux. In this way, we have reached the starting point of the exploitation method of irreversible thermodynamics. Linear relations between forces and fluxes can be used now as lowest order approximation away from equilibrium. However, in contrast to irreversible thermodynamics, we have here derived restrictions on constitutive functions (from Liu equations) before using linear forceflux relations, and these restrictions have to be taken into account in addition to the force-flux relations. Finally, let us remark that in examples the Liu equations can be derived without writing out the balance equations on the state space in matrix form. For this purpose, the derivatives of constitutive functions are carried out according to the chain rule. In the resulting expression, the higher derivatives are put as factors in front of brackets. The Liu equations are obtained by setting the brackets behind the higher derivatives equal to zero. The remaining equation contains no higher derivatives. It is the residual inequality. A modified technique, which is not as general as the Liu technique, is the ColemanNoll technique. On the other hand, the Coleman-Noll technique is easier to handle in examples than the Liu procedure, and it has been applied widely (see for example [23–27]). In this technique [23, 28], the inclusion (8.78) is enforced by:

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8 Rational Thermodynamics

B(Z ) = 0

∧

D(Z ) ≤ 0

(8.82)

A(Z ) and C(Z ) are not restricted.

(8.83)

For a comparison, see [29]. Remark: The exploitation of the dissipation inequality by the Liu procedure gives the most general form of constitutive equations compatible with the second law of thermodynamics. These different possible constitutive equations form a whole class of materials. However, the result of this exploitation depends on the choice of the state space, which is always the first step in the procedure.

8.6 Example 1: A Simple Heat-Conducting Viscous Fluid A simple example is now considered, see also [19]. We choose a large state space consisting of mass density, internal energy density, their gradients, and the velocity gradient: (8.84) Z := (, e, ∇, ∇e, (∇v)symm ). For constitutive functions on this domain, we have: t(x, t) = t((x, t), e(x, t), ∇(x, t), ∇e(x, t), (∇v)symm (x, t)), q(x, t) = q((x, t), e(x, t), ∇(x, t), ∇e(x, t), (∇v)symm (x, t)),

(8.85) (8.86)

where we did not introduce new symbols for the mappings. We assume here that there is no internal spin, and consequently the stress tensor is symmetric: t(x, t) = t  (x, t).

(8.87)

8.6.1 The Balance Equations on the State Space In the balance equations for momentum (3.40) and energy (3.35), we find the expressions ∇ · t and ∇ · q, which we obtain from (8.85) and (8.86) by using the chain rule. Thus, (3.40) and (3.35) result in: ∂t ∂t ∂t · ∇ − · ∇e − : ∇∇ ∂ ∂e ∂∇ ∂t ∂t − : ∇∇e − : ·∇(∇v)sym −  f = 0, ∂∇e ∂(∇v)sym ∂q ∂q ∂q · ∇ + · ∇e + : ∇∇ e˙ + ∂ ∂e ∂∇ ˙v −

(8.88) (8.89)

8.6 Example 1: A Simple Heat-Conducting Viscous Fluid

+

131

∂q ∂q : ∇∇e + : ·∇(∇v)sym − t : (∇v)sym − r = 0. ∂∇e ∂(∇v)sym

(8.90)

The dissipation inequality results in: 

 ∂η ∂η ∂η ∂η ∂η sym ˙ ˙ ˙ ˙ + e˙ + · (∇) + · (∇e) +  : ((∇v) ) ∂ ∂e ∂∇ ∂∇e ∂(∇v)sym ∂ ∂ ∂ · ∇ + · ∇e + : ∇∇ + ∂ ∂e ∂∇ ∂ ∂ r : ∇∇e + ≥ 0. (8.91) + : ·∇(∇v)sym − sym ∂∇e ∂(∇v)  The three balance equations for mass, momentum (8.88), internal energy (8.90), and the dissipation inequality (8.91) can be written in matrix form: A · y = C,

B · y ≥D,

(8.92)

if we introduce the so-called higher derivatives that are the derivatives of the state space variables (8.84): ˙ (∇e), ˙ ((∇v) ˙ sym ), ∇∇, ∇∇e, ∇(∇v)sym ) ˙ e, ˙ v˙ , (∇), y = (,

(8.93)

( denotes the transposed matrix), then the matrices A and C become: ⎛

⎞ 10 0 00 0 0 0 0 ∂t ∂t ∂t ⎜ ⎟ A = ⎝ 0 0 δ 0 0 O − ∂∇ − ∂∇e − ∂(∇v)sym ⎠ , ∂q ∂q ∂q 0  0 0 0 0 ∂∇ ∂∇e ∂(∇v)sym

(8.94)

⎛

⎞ −δ : ∇v ∂t ∂t ⎜ ⎟  f + ∂ · ∇ + ∂e · ∇e C=⎝ ⎠. ∂q · ∇ − · ∇e r + t : (∇v)sym − ∂q ∂ ∂e

(8.95)

In case of vectorial and tensorial elements, scalar products between them have to be taken. The matrices B and D forming the dissipation inequality (8.92)2 are:

∂η ∂η ∂η  ∂η 0  ∂∇  ∂∇e  ∂(∇v) B =  ∂η sym ∂ ∂e D=

r 

−

∂ ∂

· ∇ −

∂ ∂e

∂ ∂ ∂ ∂∇ ∂∇e ∂(∇v)sym

 · ∇e .



,

(8.96) (8.97)

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8 Rational Thermodynamics

This simple example demonstrates that due to the linearity of the balance equations and of the dissipation inequality in the derivatives ˙ and ∇, we get (after having applied the chain rule) linear equations (8.92) in the higher derivatives y. A, C, B, and D are state functions defined on the state space (in the example on (8.84)), and y is beyond the state space variables. As we can see in the special example, A, C, B, and D contain derivatives of the constitutive functions with respect to the state variables. Consequently, they depend on the constitutive equations, which are not yet fixed.

8.6.2 The Restrictions on Constitutive Functions and the Entropy Production For the example, we calculated the coefficients of the balance equation and of the dissipation inequality (8.94) to (8.97). According to Liu’s proposition, these coefficients have to satisfy Liu relations (8.79) and (8.80). Because A has three rows, we obtain for: (8.98) λ = (λ1 , λ2 , λ3 ). Then (8.79) results in nine equations: ∂η , ∂ ∂η  , ∂e 0, ∂η ,  ∂∇ ∂η ,  ∂∇e ∂η  , ∂(∇v)sym ∂ , ∂∇ ∂ , ∂∇e ∂ . ∂(∇v)sym

λ1 =  λ3 = λ2 = 0= 0= 0= ∂q ∂t + λ3 = ∂∇ ∂∇ ∂q ∂t + λ3 = λ2 · ∂∇e ∂∇e ∂q ∂t λ2 · + λ3 = ∂(∇v)sym ∂(∇v)sym λ2 ·

By (8.95) and (8.97), the dissipation inequality (8.80) results in:

(8.99) (8.100) (8.101) (8.102) (8.103) (8.104) (8.105) (8.106) (8.107)

8.6 Example 1: A Simple Heat-Conducting Viscous Fluid

133

  ∂t ∂t · ∇ + · ∇e − λ1 δ : ∇v + λ2 ·  f + ∂ ∂e   ∂q ∂q · ∇ − · ∇e +λ3 r + t : (∇v)sym − ∂ ∂e ∂ ∂ r − · ∇ − · ∇e. ≥  ∂ ∂e

(8.108)

We now exploit the Liu relations. According to the chosen state space (8.84), we obtain from (8.102) to (8.104) that the specific entropy depends only on mass density and on specific internal energy: η = η(, e). (8.109) From (8.99) and (8.100) follows immediately: λ1 = λ1 (, e), λ3 = λ3 (, e).

(8.110)

Consequently, the quantities η, λ1 , and λ3 depend only on equilibrium variables also in non-equilibrium. This fact is connected to the assumption used in classical irreversible thermodynamics: the hypothesis of local equilibrium. It states that the variables the entropy depends upon are the equilibrium ones. The fluxes, however, depend on gradients of these equilibrium variables (which are thermodynamic forces). A dependence of the entropy density on the equilibrium variables mass density and internal energy density only has been shown here by Liu procedure. However, this conclusion is possible only where the state space includes first-order gradients of the equilibrium variables solely, and no higher order gradients. According to (8.101) and (8.110), (8.105) to (8.107) result in: ∂ ( − λ3 q) = 0, ∂(∇v)sym (8.111) from which follows immediately, because of the chosen state space (8.84): ∂ ( − λ3 q) = 0, ∂∇

∂ ( − λ3 q) = 0, ∂∇e

 − λ3 q = K (, e).

(8.112)

8.6.3 Exploitation of Objectivity and Equilibrium Conditions We now exploit the transformation properties of constitutive functions under a change of the observer. For (8.85) and (8.86), we have: Q · t · Q = t(, e, Q · ∇, Q · ∇e, Q · (∇v)sym · Q ), Q · q = q(, e, Q · ∇, Q · ∇e, Q · (∇v)

sym



· Q ).

(8.113) (8.114)

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8 Rational Thermodynamics

Analogously, we obtain for the entropy flux density: Q ·  = (, e, Q · ∇, Q · ∇e, Q · (∇v)sym · Q ).

(8.115)

We now have all tools for exploiting (8.112) by the principle of material frame indifference. By the use of (8.114), (8.115), and (8.110), for the original observer (Q = δ) (8.112) we can state: (, e, ∇, ∇e, (∇v)sym ) − λ3 (, e)q(, e, ∇, ∇e, (∇v)sym ) = K (, e). (8.116) Because of the principle of material frame indifference, the mappings , λ3 , q, and K are observer invariant. Therefore, (8.112) can be expressed for the changed observer: (, e, Q · ∇, Q · ∇e, Q · (∇v)sym · Qtop ) −λ3 (, e)q(, e, Q · ∇, Q · ∇e, Q · (∇v)sym · Q ) = K (, e),

(8.117)

or according to (8.112): Q ·  − λ3 Q · q = K (, e) = Q · K (, e), ∧Q.

(8.118)

Due to the fact that this equation is valid for all Q (for all observers), we obtain by use of (8.112): K (, e) = 0 (8.119) and therefore (, e, ∇, ∇e, (∇v)sym ) = λ3 (, e)q(, e, ∇, ∇e, (∇v)sym ).

(8.120)

We now have to exploit the dissipation inequality (8.108) which, according to (8.101), can be written in the form: 

 ∂ ∂q − − (λ1 δ − λ3 t) : (∇v) − λ3 · ∇ ∂ ∂     ∂ ∂q 1 − · ∇e + λ3 − r ≥ 0. − λ3 ∂e ∂e  sym

We have: λ3

∂ ∂ ∂λ3 ∂q ∂λ3 − = = −q . (λ3 q − ) − q ∂ ∂ ∂ ∂ ∂

(8.121)

(8.122)

Here the last equation stems from (8.119). Because t was presupposed to be symmetric, we obtain: λ1 δ − λ3 t =: F(, e, ∇, ∇e, (∇v)sym ) = F .

(8.123)

8.6 Example 1: A Simple Heat-Conducting Viscous Fluid

135

Therefore, in the double scalar product in (8.121), only the symmetric part of the velocity gradient contributes. Consequently, we obtain by (8.122) and, by the analogous equation for e: σ := − (λ1 δ − λ3 t) : (∇v)

sym

  1 r ≥0 + q · ∇λ3 (, e) + λ3 − 

(8.124)

(where ( )sym denotes the symmetric part). For continuing the exploitation of the dissipation inequality, we need the equilibrium conditions, which will be discussed in the next section.

8.6.4 Equilibrium Conditions The preconceived opinion that in equilibrium all quantities are time independent and there are no gradients, turns quickly out to be wrong, as a simple example demonstrates. If there is a gravitational field, we have a gradient of mass density also in equilibrium, and if an observer is freely falling in this gravitational field, he sees that the mass density is changing in time, although the system is in equilibrium. Consequently, time independence and vanishing gradients cannot be used as equilibrium conditions if no additional conditions are taken into account. First of all, we want to formulate equilibrium conditions with respect to a special observer. Having done so, we can transform them to arbitrary observers. Proposition [30]: Let f be a function of the variables X. If X · f (X) ≥ 0 and f is continuous at X = 0, then f (0) = 0 follows. Equilibrium Conditions: A system is said to be in equilibrium if there exists an observer  † , so that the velocity field is constant, all material time derivatives, all non-convective fluxes, energy supply, and entropy production are zero:  v ≡ c = const, †

d dt

† ≡ 0,

(8.125)

q † ≡ 0,

† ≡ 0,

(8.126)

r ≡ 0,

σ ≡ 0.

(8.127)

†

†

The equilibrium conditions with regard to the dissipation inequality (8.124) are: ((∇v)sym )† ≡ 0 −→ ((∇v)sym ) eq ≡ 0, q eq ≡ 0, r eq ≡ 0.

(8.128)

Due to the above proposition, we see from (8.124) and from the equilibrium conditions the following relations: eq

eq

λ1 δ − λ3 t eq ≡ 0,

(∇λ3 )eq ≡ 0,

eq

λ3 −

 eq 1 ≡ 0. T

(8.129)

136

8 Rational Thermodynamics

As according to (8.110) the λ-s depend only on the equilibrium variables  and e, we can omit the mark eq for them. With λ3 = T1 , we have shown in this example that the constitutive relation  = Tq for the entropy flux holds, as is generally assumed in irreversible thermodynamics. In the next subsection, we discuss the considered example in more detail.

8.6.5 Discussion Introducing the large state space (8.84) Z := (, e, ∇, ∇e, (∇v)sym ),

(8.130)

we obtain by Liu technique a specific entropy (8.109) that is independent of the gradients, even in non-equilibrium: η = η(, e).

(8.131)

According to (8.119), we obtain for the entropy flux density: (, e, ∇, ∇e, (∇v)sym ) = λ3 (, e)q(, e, ∇, ∇e, (∇v)sym ).

(8.132)

The form of the non-equilibrium stress tensor follows from (8.129)1 t(Z) = − p(, e)δ + tdyn (Z), with the static pressure: p(, e) := −

λ1  λ3

(8.133)

(8.134)

and with the dissipative part of the pressure tensor: tdyn (Z) = t(Z) + p(, e)δ,

tdyn eq = 0.

(8.135)

Because of (8.110)2 , (8.129)2 results in: ∂λ3 ∂λ3 ∇ + ∇e|eq = 0. ∂ ∂e

(8.136)

If now the internal energy is uniform in equilibrium, the same will follow for the mass density by (8.136): (∇e)eq = 0 −→ (∇)eq = 0.

(8.137)

8.6 Example 1: A Simple Heat-Conducting Viscous Fluid

137

From (8.129)3 , we obtain for the equilibrium temperature: 1 1 (, e) = |eq (, e) = λ3 . T T

(8.138)

By (8.133), (8.134), and (8.138), the entropy production density (8.124) results in: σ=

1 1 dyn t (Z) : (∇v)sym + q(Z) · ∇ ≥ 0. T T

(8.139)

The entropy balance equation becomes, by (8.133) and (8.119): η˙ + ∇ ·

1 r q 1 = tdyn : (∇v)sym + q · ∇ + . T T T T

(8.140)

Inserting here the balance equations (3.43) to (3.45), we obtain: η˙ =

p  e˙ − , ˙ T T

(8.141)

which is consistent with (8.99) and (8.100). This result (8.141) is the well-known Gibbs equation, proved here in non-equilibrium by the Liu procedure for a special choice of the state space.

8.7 Example 2: A Heat-Conducting Fluid with a Scalar Internal Variable In this section, we will treat again the example of a fluid with one internal variable ξ, already treated with the method of irreversible thermodynamics in Sect. 7.6.3. The notation is here for one scalar variable ξ, which can be identified with α or β in Sect. 7.6.3. Analogously, one can deal with a vectorial or tensorial internal variable with the same kind of results. In this case, our results here compare to Sect. 7.2, where the second-order alignment tensor was the internal variable. The state space is chosen to include the equilibrium variables density and temperature, the internal variable, its time derivative, and the temperature gradient in order to account for heat conduction: ˙ ∇T }. Z = {, T, ξ, ξ,

(8.142)

We suppose that the internal variable obeys a relaxation equation of the form: dξ = G(, T, ξ). dt

(8.143)

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8 Rational Thermodynamics

˙ because G is a constitutive function, assumed not to depend on ∇T , and not on ξ, the evolution equation for ξ is assumed to be a first-order linear differential equation. We suppose that no internal angular momentum is present, i.e., the stress tensor is symmetric. In addition, we assume that there is no other entropy supply than by heat supply, i.e., z = Tr . Finally, we suppose that constitutive functions do not depend on velocity. Therefore, we have not included velocity in the state space. However, in principle, velocity could be a state space element, and therefore derivatives of the velocity are higher derivatives (see also the previous example). With these presuppositions, the entropy inequality as the local formulation of the second law of thermodynamics is exploited according to the method of Liu (see Sect. 8.5.3):   d r dη + ∇ · φ − + λ1 + ∇ · v dt T dt   dv +λ2 ·  − ∇ · t −  f dt   de +λ3  + ∇ · q − t : ∇v − r dt   dξ − G(, T, ξ) +λ4 dt ∂η ˙ ∂η ∂η ∂η d∇T ∂η +  ξ˙ +  ξ¨ =  ˙ +  T + ∂ ∂T ∂∇T dt ∂ξ ∂ ξ˙ σ=

∂φ ∂φ ∂φ ∂φ ∂φ · ∇ ξ˙ · ∇ + · ∇T + : ∇∇T + · ∇ξ + ∂ ∂T ∂∇T ∂ξ ∂ ξ˙   d + ∇ · v +λ1 dt  ∂t ∂t ∂t dv · ∇ − · ∇T − : ∇∇T +λ2 ·  − dt ∂ ∂T ∂∇T  ∂t ∂t · ∇ξ − − · ∇ ξ˙ −  f ∂ξ ∂ ξ˙    ∂e d ∂e dT ∂e d∇T ∂e dξ ∂e dξ˙ + + · + + +λ3  · ∂ dt ∂T dt ∂∇T dt ∂ξ dt ∂ ξ˙ dt +

+

∂q ∂q ∂q · ∇ + · ∇T + : ∇∇T ∂ ∂T ∂∇T  dq + · ∇ξ − t : ∇v − r dξ   dξ − G(, T, ξ) ≥ 0. +λ4 dt (8.144)

8.7 Example 2: A Heat-Conducting Fluid with a Scalar Internal Variable

139

This expression for the entropy production involves the higher derivatives , ˙ T˙ , ¨ ∇, ∇∇T , ∇ξ, ∇ ξ, ˙ v˙ , ∇v. The factors in front of the higher derivatives ˙ = d∇T , ξ, ∇T dt appear in the Liu equations:

˙ :

∂e ∂η + λ1 + λ3  =0 ∂ ∂ ∂e ∂η + λ3  =0 T˙ :  ∂T ∂T ∂η ∂e ξ¨ :  + λ3  =0 ˙ ∂ξ ∂ ξ˙ 

∂η ∂e d∇T :  + λ3  dt ∂∇T ∂∇T ∂φ ∂q ∂t ∇ : − λ2 · + λ3 ∂ ∂ ∂ ∂φ ∂q ∂t − λ2 · + λ3 ∇∇T : ∂∇T ∂∇T ∂∇T ∂q ∂φ ∂t − λ2 · + λ3 ∇ξ : ∂ξ ∂ξ ∂ξ ∂φ ∂q ∂ t ∇ ξ˙ : − λ2 · + λ3 ∂ ξ˙ ∂ ξ˙ ∂ ξ˙ ∇v :

(8.145) (8.146) (8.147)

=0

(8.148)

=0

(8.149)

=0

(8.150)

=0

(8.151)

=0

(8.152)

v˙ : λ2 = 0 λ1 1 − λ3 t = 0.

(8.153) (8.154)

From (8.153) it follows immediately: λ2 = 0.

(8.155)

For example, (8.146) can be used to calculate λ3 : ∂η

λ3 = − ∂T . ∂e

(8.156)

∂T

In Equilibrium, it follows λ3 = − T1 , and analogously to example 1, λ3 is identified generally. Then we have from (8.145): ∂η ∂e − λ3  ∂ ∂ ∂η ∂f  ∂e − = = T ∂ ∂ T ∂ λ1 = −

(8.157)

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8 Rational Thermodynamics

with the definition of the specific free energy density f := e − T η. After inserting the multipliers λ1 , λ2 , and λ3 , the remaining Liu equations are the restrictions on the constitutive functions, imposed by the second law of thermodynamics: ∂η 1 ∂e − =0 ˙ T ∂ ξ˙ ∂ξ ∂η 1 ∂e − =0 ∂∇T T ∂∇T ∂φ 1 ∂q − =0 ∂ T ∂ 1 ∂q ∂φ − =0 ∂∇T T ∂∇T ∂φ 1 ∂q − =0 ∂ξ T ∂ξ 1 ∂q ∂φ − =0 ˙ T ∂ ξ˙ ∂ξ λ1 δ t= λ3

(8.158) (8.159) (8.160) (8.161) (8.162) (8.163) (8.164)

or shortly with the definition of the free energy density and the abbreviation k = φ − T1 q: ∂f ˙ ∇T } = 0 where z 1 ∈ {ξ, ∂z 1

∂k ˙ = 0 where z 2 ∈ {, ∇T, ξ, ξ} ∂z 2 ∂f δ. t = − ∂

(8.165) (8.166) (8.167)

From the second law of thermodynamics, we have shown in (8.165) that the free energy density f is not a function of all state space variables, but only of the equilibrium variables  and T and of the internal variable ξ: f (, T, ξ).

(8.168)

This is the same as the local equilibrium assumption made in irreversible thermodynamics. However, here it is a conclusion from the dissipation inequality for a special choice of the state space. From (8.166), it follows that the difference φ − T1 q may only be a function of temperature (or a constant). The result (8.166) together with a representation theorem for isotropic material shows (in analogy to the previous

8.7 Example 2: A Heat-Conducting Fluid with a Scalar Internal Variable

141

example) that the vectorial constitutive function k depends only on temperature. A vector depending solely on a scalar variable can only be a constant. Since this constant vanishes in equilibrium (q = 0, φ = 0), it follows: k =φ−

q = 0. T

(8.169)

Equation (8.167) shows that in our example the stress tensor is isotropic, i.e., there is only a scalar pressure. This would be different for a tensorial internal variable. All conclusions drawn by applying the Liu procedure depend on the choice of the state space in the beginning. The residual inequality assumes the form:   ∂e ∂η ˙ ∂φ ∂q · ∇T + λ3  ξ˙ + · ∇T ξ+ ∂ξ ∂T ∂ξ ∂T   dξ −G +λ4 dt    =0   1 ∂q 1 ∂f ˙ ∂φ − · ∇T ≥ 0. −  + ξ T ∂ξ ∂T T ∂T      

σ=

=

production

of

internal

variable

heat

(8.170)

conduction

The entropy production is a function of the state space variables and does not contain higher derivatives. This fact is generally valid. The last bracket in equation (8.170) is zero because of the relaxation equation for the internal variable. As the difference φ − T1 q is zero, the entropy production simplifies to: σ=−

∂f ˙ 1 ξ − 2 q · ∇T ≥ 0. T ∂ξ T

(8.171)

This is exactly the same expression we obtained from the irreversible thermodynamics treatment of a material with one scalar internal variable in Sect. 7.6.

8.8 Example 3: Material Damage—An Exploitation of Liu Equations and of the Entropy Production As an example of an internal variable, we introduce a tensorial damage parameter together with its relaxation equation.

142

8 Rational Thermodynamics

8.8.1 Balance Equations and Equation of Motion for the Internal Variable As we are dealing with a solid, we will formulate the balance equations in terms of the deformation variable. We will restrict ourself to small deformations. Consequently, strain  1 ∇u + (∇u)T (8.172) = 2 is introduced as measure of deformation, and the balance equations are formulated in terms of the actual coordinate x(t). The material is described as non-micropolar, i.e., there is no internal angular momentum and the balance of angular momentum is not independent of the balance of momentum. The remaining balance equations are as follows: Balance of mass: ˙ +  trace ˙ = 0 (8.173) with mass density . The balance of mass simply plays the role of a constraint between the mass density and the strain. Balance of momentum: v˙ − ∇ · t −  f = 0. (8.174) f is the specific density of volume forces, and t denotes the Cauchy stress tensor, a constitutive function. Balance of internal energy: e˙ + ∇ · q − t : ˙ = 0 .

(8.175)

e is the internal energy density, q is the heat flux density, and the remaining term is the (internal) power of the stress. It is supposed that there is no radiation absorption. Internal energy and heat flux are constitutive functions. In addition, we introduce the second-order damage tensor D as an internal variable with the equation of motion dD =G (8.176) dt with a constitutive function G. The second law of thermodynamics is expressed by the dissipation inequality: σ = η˙ + ∇ ·  ≥ 0

(8.177)

with specific entropy density η, entropy flux , and entropy production σ. The dissipation inequality is exploited according to Liu together with the balances of mass, momentum and internal energy, as well as the equation of motion for the

8.8 Example 3: Material Damage—An Exploitation of Liu Equations …

143

damage tensor. The corresponding multipliers are denoted by  , v , e , and  D . These are functions of the state space variables.

8.8.2 Exploitation of the Second Law According to Liu The second law of thermodynamics is fulfilled by any thermodynamic process, if and only if the following inequality is fulfilled: η˙ + ∇ ·  + (Balance of mass) 

+v · (Balance of momentum) +e (Balance of internal energy) + D (Relaxation of damage tensor) ≥ 0.

(8.178)

The implications of this inequality on constitutive functions are exploited in the following. We assume that for material behavior strain and strain gradient, temperature and temperature gradient and the damage tensor together with its gradient are relevant. They are included in the domain of constitutive functions, the state space Z. Z = {T, , D, ∇T, ∇, ∇ D}.

(8.179)

Constitutive quantities depend on position and time through the space and time dependence of the field quantities in the state space Z. The time and space derivatives of all constitutive functions are carried out according to the chain rule. Application of the chain rule to all constitutive functions in the inequality (8.178) leads to an expression linear in the so-called higher derivatives not included in the state space. These are in our case: ˙ d ∇T, d ∇, d ∇ D, ∇∇T, ∇∇, ∇∇ D, v. ˙ T˙ , ˙, D, dt dt dt

(8.180)

Corresponding to each higher derivative, there is an equation restricting constitutive functions. From this set of equations, the quantities  D , v , and e are eliminated. The results are most conveniently written in terms of the specific free energy density f := e − T η and the extra entropy flux k =  − Tq . It is found that ∂η

e = − ∂T ∂e ∂T

(8.181)

144

8 Rational Thermodynamics

and e is set equal to the negative inverse temperature: 1 e = − . T

(8.182)

This is in agreement with thermostatics. From the equation corresponding to the higher derivative v˙ , we calculate: v˙ :

v = 0 ,

(8.183)

which is taken into account in the following. ˙ we calculate: From the equation corresponding to the higher derivative D, ˙ : D

D =

1 ∂f . T ∂D

(8.184)

The remaining restrictions on constitutive functions from the second law are: t=−

1∂f T ∂

∂f = 0 z i ∈ {∇T, ∇, ∇ D} ∂z i ∂k = 0 u i ∈ {∇T, ∇, ∇ D}. ∂u i

(8.185) (8.186) (8.187)

In summary, the exploitation of the second law with the balance equations as constraints has restricted the dependence of the extra entropy flux and of the free energy density to the following variables: k(T, , D) .

(8.188)

Thus, the extra entropy flux does not depend on gradients. The dependence of the free energy density is then f (T, , D). (8.189) The free energy density also does not depend on gradients. In addition to the above restrictions on constitutive functions, the method of Liu results in an expression for the entropy production, which is a function of the state space variables:  σ=

 1 ∂f ∂k .. ∂k ∂k − 2 q · ∇T + ·∇D + : G ≥ 0. (8.190) .∇ + ∂T T ∂ ∂D ∂D

This inequality is exploited now further making use of the fact the entropy production is of the form

8.8 Example 3: Material Damage—An Exploitation of Liu Equations …

σ=



vi f i (v j ) ≥ 0 ,

145

(8.191)

i

which can be interpreted as a sum of products of fluxes and forces. One can show that for continuous functions f i (v j ) it follows that f i is a homogeneous function vi = 0 ⇒ f i = 0.

(8.192)

This fact will be exploited in the following.

8.8.3 Exploitation of the Residual Inequality In order to make it more clearly readable for the following exploitation, we write the entropy production in the form of a formal scalar product between rows. It is understood that in the lines involving vectors or tensors the scalar product (contraction over all pairs of indices in components) has to be taken: ⎛ ⎜ σ=⎜ ⎝

∂k ∂T

−

∂k ∂ ∂k ∂D ∂f ∂D

1 q T2

⎞ ⎛

⎞ ∇T ⎟ ⎜ ∇ ⎟ ⎟·⎜ ⎟ ⎠ ⎝∇D⎠. G

(8.193)

Under the presupposition that the dependence of the constitutive functions on the state space variables is continuous, it follows according to (8.192) that each constitutive quantity in the row at the left-hand side is a homogeneous function of the state space variable in the row at the right-hand side in the same line. This fact is exploited especially for the second and third lines: ∂k = f 1 (Z)∇ , ∂

(8.194)

where f 1 (Z) is up to now an arbitrary function of the state space variables. On the other hand, k does not depend on ∇ as a consequence of the dissipation inequality ∂k is also independent of (see (8.188)) and therefore, the left-hand side of (8.194), ∂ ∇. Consequently, the only possibility to fulfill (8.194) is: f1 = 0

⇒

∂k = 0. ∂

The extra entropy flux does not depend on the strain.

(8.195)

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8 Rational Thermodynamics

The exploitation of the third line of (8.193) shows analogously that k does not depend on the damage tensor: ∂k = f 2 (Z)∇ D , ∂D

(8.196)

where f 2 (Z) is up to now an arbitrary function of the state space variables. On the other hand, k does not depend on ∇ D according to the Liu procedure (see (8.188)) ∂k is also independent of ∇ D, and the and therefore, the left-hand side of (8.196), ∂D only way to fulfill (8.196) is: f 2 (Z) = 0

⇒

∂k = 0. ∂D

(8.197)

The exploitation of the line above in the entropy production does not restrict the extra entropy flux k, because of the unknown constitutive function of heat flux q in the corresponding equation. Concerning the dependence of the extra entropy flux k on temperature, we cannot draw any conclusion from the entropy production.

8.8.4 Summary Exploiting the dissipation inequality according to Liu, we have obtained restrictions on the free energy and on the extra entropy flux, but the production of damage is not restricted. The second part of the exploitation of the second law of thermodynamics, namely, the conclusions drawn from the entropy production, has lead to additional restrictions. We found that the extra entropy flux k depends solely on temperature. In summary, we have the following dependence of constitutive functions: k(T )

(8.198)

f (T, , D).

(8.199)

If there is no material anisotropy present (in addition to the relevance of a tensorial damage parameter), there exists no vector function k depending on a scalar, the temperature, solely. We conclude from (8.198) that the extra entropy flux vanishes, k = 0, i.e., we have the classical relation between entropy flux and heat flux: =

q . T

(8.200)

The only cause for the entropy flux is the heat flux. There is no contribution from the damage parameter.

8.8 Example 3: Material Damage—An Exploitation of Liu Equations …

147

Finally, there are two contributions to the entropy production: heat conduction and production of damage σ=−

∂f 1 q · ∇T + : G ≥ 0. T2 ∂D

(8.201)

8.9 Example 4: Liquid Crystals of Biaxial Particles or Suspensions of Flexible Fibers Originally published in [31] Copyright by Creative Commons License.

8.9.1 The Choice of the Variables In this example (see [31]), we want to deal with anisotropic materials, which are biaxial. In such materials, constitutive properties are different in three pairwise orthogonal directions. Examples for such materials are most crystals, and biaxial liquid crystals (for more details about uniaxial and biaxial liquid crystals, see Chap. 12). In all cases, the orientation of the material is relevant and this can be described by the orientations of three material fixed axes, given by unit vectors l, m, n. These axes can be chosen such that they are pairwise orthogonal. In general, the axes change their orientation in time, keeping the orthogonality. Therefore, for all times the following constraints have to be taken into account: l ·l =1 m·m=1 n·n=1 l · n = 0 l · m = 0 m · n = 0.

(8.202) (8.203)

We want to deal with materials, where the orientations are not homogeneous in space, i.e., the gradients ∇l, ∇m, and ∇n are relevant. The normalization leads to the following constraints: l · (∇l)T = 0, m · (∇m)T = 0, n · (∇n)T = 0.

(8.204)

˙ m, ˙ and In addition, the orientations may depend on time, and the time derivatives l, n˙ are elements of the state space. The normalization leads to: l·l =1⇒

d (l · l) = 2l˙ · l = 0 dt

d ˙ ·m=0 (m · m) = 2 m dt d n · n = 1 ⇒ (n · n) = 2n˙ · n = 0. dt

m·m=1⇒

(8.205) (8.206) (8.207)

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8 Rational Thermodynamics

In summary, our state space Z contains the equilibrium variables density, and temperature, velocity and velocity gradient, the orientation axes, their gradients, and time derivatives: ˙ m, ˙ n, ˙ ∇l, ∇m, ∇n} Z = {, T, v, ∇v, l, m, n, l,

(8.208)

with the abovementioned constraints to be taken into account. As the choice of state space variables with constraints is not very convenient, we will introduce now another equivalent choice. Instead of three orientations which are ortho-normalized, it is equivalent to introduce the mapping Q, which maps a reference system with ortho-normalized vectors l 0 , m0 , n0 to the vectors l, m, n. Q is a rotation, i.e., it is an orthogonal tensor Q T · Q = δ. Therefore, it has three independent components, the same number as the vectors l, m, n with the constraints. The spatial dependence of the orientations is described by the gradient ∇ Q, which has nine independent components. Due to this number of independent components, the following mapping to a second-order tensor O is a one-to-one mapping (which can also be checked directly from this definition): O=

 1 (∇ Q) · Q T :  2

(8.209)

with the totally antisymmetric third-order tensor . The system of three orientation axes moves like a rigid body, because the lengths and angles are fixed. Therefore, we can describe the time dependence of the orientations in the following way by an angular velocity vector : ∂Q =  × Q. ∂t

(8.210)

Exploiting the derivatives of the constitutive functions according to the chain rule, we obtain higher derivatives. Among others, there occurs the time derivative of the tensor O. This time derivative can be expressed by other derivatives, which are partly higher derivatives, and partly elements of the state space: Proposition:  T dO 1 ∂v ∂ QT T = (∇) + · · Q : dt 2 ∂x ∂x or in components:

dOik ∂i ∂ Q r s ∂vm 1 = + irl Q ls . dt ∂xk 2 ∂xm ∂xk

Proof: In components we have:

(8.211)

(8.212)

8.9 Example 4: Liquid Crystals of Biaxial Particles or Suspensions of Flexible …

149

  1 dOik d ∂ Qr s = − irl Q ls = dt 2 dt ∂xk     d ∂ Qr s 1 ∂ Q r s dQ ls Q ls + = − irl 2 dt ∂xk ∂xk dt    ∂ ∂ Qr s ∂ 1 Q ls + vm − irl 2 ∂t ∂xm ∂xk  ∂ Qr s + lkm k Q ms = ∂xk     ∂ ∂ Qr s ∂vm ∂ Q r s ∂ Qr s 1 − Q ls + vm − irl 2 ∂xk ∂t ∂xm ∂xk ∂xm  ∂ Qr s + lkm k Q ms = ∂xk   ∂ dQ r s ∂vm ∂ Q r s 1 Q ls − − irl 2 ∂xk dt ∂xk ∂xm  ∂ Qr s + lkm k Q ms = ∂xk    ∂vm ∂ Q r s ∂  1 Q ls − irl r op o Q ps − 2 ∂xk ∂xk ∂xm

 ∂ Qr s + lkm k Q ms = ∂xk       ∂ Q ps 1 ∂vm ∂ Q r s ∂o + irl Q ls δio δlp − δi p δlo Q ps + o 2 ∂xk ∂xk ∂xk ∂xm 1 ∂ Qr s − (δik δr m − δim δr k ) k Q ms = 2 ∂xk   ∂vm ∂ Q r s ∂l 1 ∂i Q ls Q ls − Q is Q ls + irl Q ls = 2 ∂xk ∂xk ∂xk ∂xm ∂i ∂vm ∂ Q r s 1 + irl Q ls . (8.213) ∂xk 2 ∂xk ∂xm i The first term ∂ is a higher derivative, whereas the another term contains only ∂xk the state-space variables ∇v, Q, and ∇ Q, which can be expressed in terms of O. The choice of variables in the state space, and therefore the exploitation of the dissipation inequality, is similar for the example of suspensions of flexible fibers (see also Sect. 7.5). There the second-order tensor Q describes the local fiber orientation. The second-order tensor O accounts for the local deformation of the fibers.  is the time derivative of the orientation tensor Q.

150

8 Rational Thermodynamics

8.9.2 Exploitation of the Dissipation Inequality We will exploit now the dissipation inequality according to Liu with the following state space: Z = {, T, v, ∇v, Q, O, },

(8.214)

where all constitutive quantities are assumed not to depend on v. As the materials discussed here are micropolar media, we have to take into account the balance of internal angular momentum in addition to the balances of mass, momentum, and energy. The following inequality has to be exploited:   r dη  d + ∇ · v  +∇ ·φ− +λ dt T dt   dv p T +λ  − ∇ · t −  f dt   de u T T +λ  + ∇ · q − t : ∇v − r −  : ∇ + ( : t ) ·  dt   ds s T +λ ·  −  : t − ∇ ·  − m ≥ 0. dt

(8.215)

After exploiting the differentiations of the constitutive functions according to the chain rule, it results an inequality linear in the following higher derivatives: d(∇v) ˙ ˙ , O, , ∇, ∇T, ∇∇v, ∇ O, ∇. T˙ , , ˙ v˙ , dt

(8.216)

These higher derivatives are not all independent, see (8.212), which is a constraint ˙ Q, ∇v, ∇ Q, and ∇. Q, ∇v and ∇ Q are state space variables, whereas between O, ∇ is not included in the state space. This gradient shows up in the list of higher derivatives. After inserting this constraint, we can write down the Liu equations, corresponding to the different higher derivatives: ˙ : T˙ : ˙ : ∇v ∇ :

∂η ∂e ∂s + λu  + λ + λs · =0 ∂ ∂ ∂ ∂η ∂e ∂s  + λu  + λs · =0 ∂T ∂T ∂T ∂η ∂e ∂s + λu  + λs · =0  ∂∇v ∂∇v ∂∇v ∂η ∂ ∂tT  + − λp · + ∂O ∂ ∂ 

(8.217) (8.218) (8.219)

8.9 Example 4: Liquid Crystals of Biaxial Particles or Suspensions of Flexible …

    ∂e ∂T ∂s ∂q u T + +λ  + + =0 λ ·  ∂O ∂ ∂O ∂ ˙ :  ∂η + λu  ∂e + λs · ∂s = 0  ∂ ∂ ∂ T T ∂q ∂ t ∂ ∂ − λp · + λs · + λu =0 ∇ : ∂ ∂ ∂ ∂ ∂ ∂q ∂tT ∂T ∇T : − λp · + λs · + λu =0 ∂T ∂T ∂T ∂T ∂q ∂tT ∂T ∂ − λp · + λs · + λu =0 ∇∇v : ∂∇v ∂∇v ∂∇v ∂∇v ∂ ∂q ∂tT ∂T ∇O : − λp · + λs · + λu =0 ∂O ∂O ∂O ∂O p v˙ : λ = 0. s

151

(8.220) (8.221) (8.222) (8.223) (8.224) (8.225) (8.226)

From this set of equations, the multipliers λ , λ p , λs , and λu can be calculated. Between the specific spin density and the orientation change velocity , the relation: s =· (8.227) holds.  is the moment of inertia tensor. For biaxial liquid crystals, it is reasonable to assume this tensor to be constant. In case of suspensions of flexible fibers, this is only an approximation, because in principle this tensor changes if the fibers are deformed. We assume here that these deformations are small and the variation of  can be neglected. Then, the specific spin density s depends only on the orientation change velocity , and all other partial derivatives vanish. In this case we have: ∂s = ∂

(8.228)

and the equations for the multipliers simplify to: ∂η

⇒

Equation (8.218)

λu = − ∂T =− ∂e ∂T

Equation (8.217) Equation (8.221)

∂η

⇒

1 T

1 ∂e ∂η λ = − +  ∂ T ∂   ∂ η − T1 e s · −1 λ =− ∂ λ p = 0.

⇒

(8.229) (8.230) (8.231) (8.232)

1 For the derivative ∂T ∂e we insert T . This relation is known in equilibrium. In non∂T equilibrium, it needs some additional argumentation, see Example 1 or [4].

152

8 Rational Thermodynamics

The remaining (8.219), (8.220), (8.222), (8.223), (8.224), and (8.225) give, after inserting the multipliers, the restrictions on constitutive functions. This will be shown here only under the assumption of a constant moment of inertia tensor. We have:   ∂ η − T1 e =0 ∂∇v    ∂ η − T1 e ∂ 1 ∂q =T  + − ∂O ∂ T ∂    T ∂ η − T1 e ∂ − · −1 · ∂ ∂   ∂ η − T1 e 1 ∂q ∂T ∂ − = · −1 · ∂ T ∂ ∂ ∂   ∂ η − T1 e ∂ 1 ∂q ∂T − = · −1 · ∂T T ∂T ∂T  ∂  T ∂ η − T1 e ∂ 1 ∂q ∂ − = · −1 · ∂∇v T ∂∇v ∂∇v  ∂  T ∂ η − T1 e 1 ∂q ∂ ∂ − = · −1 · . ∂O T ∂O ∂ ∂O

(8.233)

(8.234) (8.235)

(8.236) (8.237) (8.238)

Introducing the free energy density f := e − T η and the difference k =  − T1 q, the resulting restrictions on the constitutive functions can be written as: ∂f =0 ∂∇v ∂f ∂k ∂f ∂T  = − +T + · −1 · ∂O ∂ ∂ ∂ T ∂k 1 ∂f ∂ · −1 · =− for u i ∈ {, T, ∇v, O}. ∂u i T ∂ ∂u i

(8.239) (8.240) (8.241)

Equation (8.241) shows that the difference k =  − T1 q is surely non-zero if the free energy density depends on the orientation change velocity ( = 0), and if the couple stresses  depend on any of the variables , T , ∇v, O. In this case, the very frequently made constitutive assumption (see, for instance, irreversible thermodynamics) of the entropy flux  being heat flux divided by temperature is not fulfilled. Equation (8.240) is a differential equation for the couple stress. It reduces to an algebraic equation if the free energy density does not depend on . In this case, the ∂k . couple stresses can be calculated as  = − ∂∂ Of + T ∂

8.9 Example 4: Liquid Crystals of Biaxial Particles or Suspensions of Flexible …

153

8.9.3 The Entropy Production The residual inequality is built up by all terms in (8.215), which contain no higher derivatives, but only state space functions. The constraint equation (8.212) has been inserted, and z = Tr . We will deal here only with the case of a constant moment of inertia tensor. For the multipliers, equations (8.229) to (8.232) are inserted: ∂ ∂ Q ∂η  ∂η : Q˙ + · ∇v · (∇ Q) · Q T · :  + ·: ∂Q 2 ∂O ∂Q ∂x   T ∂Q ∂ s  ·: − g −  : t +λ ∇ · v − λ · − ∂Q ∂x     ∂e ∂e : Q˙ − · ∇v · (∇ Q) · Q T · :  + ( : t T ) · s · −1 +λu  ∂Q 2 ∂O  ∂q ∂Q T + ·: − t : (∇v) ∂Q ∂x σ=

∂η  ∂η ∂ ∂ Q : Q˙ + · ∇v · (∇ Q) · Q T · :  + ·: ∂Q 2 ∂O ∂Q ∂x   T 1 ∂f ∂Q 1 ∂f ∂ ∇ ·v− · −1 · − ·: − g −  : t + 2 T ∂ T ∂ ∂Q ∂x    ∂e  ∂e 1  : Q˙ + · ∇v · (∇ Q) · Q T · :  + ( : t T ) · s · −1 − T ∂Q 2 ∂O  ∂Q ∂q ·: − t T : (∇v) + ∂Q ∂x  ∂f :× Q =− T ∂Q    =

change

of

orientation

 ∂f · ∇v · (∇ Q) · Q T · :  − 2T ∂ O    coupling

between

+

viscous

flow

and

∂Q ∂k ·: ∂Q ∂x    transport

of

orientational

orientation

  ∂Q 1 ∂f ∂T · −1 · − ·: − T ∂ ∂Q ∂x    coupling

between

change

of

orientation

and

order

1 ∂f ∇ ·v + 2 T ∂   

gradient

viscous

of

flow

orientation

154

8 Rational Thermodynamics

   1  ( : t T ) · s · −1 − t T : (∇v) −    T deviation

of

particle

angular

velocity

from

angular

velocity

of

(8.242) . the

continuum

element

One could start now an exploitation of the entropy production in the sense of irreversible thermodynamics, identifying forces and fluxes, and writing down linear relations between them. For a comparison of the present example to a rational thermodynamics treatment of fiber reinforced solids, see [32], and for a treatment of particle reinforced rubber see [33].

8.10 Outlook: Derivatives of Balance Equations as Additional Constraints As an example, we will consider a material with an internal variable in the state space. After the classical exploitation, we will present an extended procedure with more general results. Classical procedure: We chose the state space Z = {, T, ξ, ∇ξ, ∇T }.

(8.243)

The balance of entropy together with the classical balance equations and the equation of motion for the internal variable results in the inequality η˙ + ∇ ·  + λ1 (˙ − ∇ · v) + λ2 · (˙v − ∇ · t −  f )   +λ3 (e˙ − ∇ · q − t : ∇v) + λ4 ξ˙ − G ≥ 0.

(8.244)

After applying the chain rule, we obtain an expression linear in the higher derivatives ˙ ∇ξ, ˙ ∇, ∇∇T , ∇∇ξ, v˙ , ∇v. The corresponding Liu equations ˙ = d∇T , ξ, , ˙ T˙ , ∇T dt read: ˙ : T˙ :

 

∂e ∂η + λ1 + λ3  =0 ∂ ∂

∂η ∂e + λ3  =0 ∂T ∂T

⇒

λ1 = ∂η

⇒

∂f T ∂

(8.245)

1 T

(8.246)

λ3 = − ∂T =− ∂e ∂T

∂η ∂e ∂f + λ3  + λ4 = 0 ⇒ λ4 = ∂ξ ∂ξ T ∂ξ d∇T ∂e ∂η ∂f :  + λ3  =0 ⇒ =0 dt ∂∇T ∂∇T ∂∇T ξ˙ :



(8.247) (8.248)

8.10 Outlook: Derivatives of Balance Equations as Additional Constraints

∂f =0 ∂∇ξ v˙ : λ2 = 0 ∂φ ∂k ∂q ∇ : + λ3 =0 ⇒ =0 ∂ ∂ ∂ ∂φ ∂k ∂q ∇∇T : + λ3 =0 ⇒ =0 ∂∇T ∂∇T ∂∇T ∂φ ∂q ∂k ∇∇ξ : + λ3 =0 ⇒ =0 ∂∇ξ ∂∇ξ ∂∇ξ ∇v : λ1 1 − λ3 t = 0. d∇ξ : dt



∂η ∂e + λ3  =0 ∂∇ξ ∂∇ξ

⇒

155

(8.249) (8.250) (8.251) (8.252) (8.253) (8.254)

The extra entropy flux k = φ − Tq = k(T ) = 0 as in classical irreversible thermodynamics. The free energy density f = e − T η = f (, T, ξ) is depending only on the equilibrium variables and on the internal variable, but not on the gradient of the internal variable. The same result for the free energy density is obtained, if the state space is extended by the second-order gradient or higher order gradients of the internal variable. On the other hand, the successful approach of the Ginzburg-Landau theory is based on a free energy, depending on an internal variable and the gradient of it. Such a free energy density is thermodynamically consistent if the derivatives of balance equations are taken into account as constraints as well in the Liu procedure. We will demonstrate this in our simple example. Extended Liu procedure: The state space is the same as in the classical procedure: Z = {, T, ξ, ∇ξ, ∇T }.

(8.255)

A thermodynamic process fulfills the entropy inequality, the balance equations, and consequently also the derivatives of balance equations. As an example, we take into account the gradient of the equation of motion for the internal variable in addition to the other constraints. η˙ + ∇ ·  + λ1 (˙ − ∇ · v) + λ2 · (˙v − ∇ · t −  f )     +λ3 (e˙ − ∇ · q − t : ∇v) + λ4 ξ˙ − G + λ5 ∇ ξ˙ − ∇G ≥ 0.

(8.256)

The material time derivative and the gradient do not commute:

∇ ξ˙ = ∇



 ∂∇ξ ∂ξ ˙ + (∇v) · ∇ξ. + v · ∇ξ = + v · ∇∇ξ + (∇v) · ∇ξ = ∇ξ ∂t ∂t

156

8 Rational Thermodynamics

The Liu equations read ˙ : T˙ :

 

∂e ∂η + λ1 + λ3  =0 ∂ ∂

∂η ∂e + λ3  =0 ∂T ∂T

⇒

λ1 = ∂η

⇒

∂f (8.257) T ∂

λ3 = − ∂T =− ∂e ∂T

1 (8.258) T

∂e ∂f ∂η + λ3  + λ4 = 0 ⇒ λ4 = ξ˙ :  ∂ξ ∂ξ T ∂ξ ∂η ∂f d∇T ∂e :  + λ3  =0 ⇒ =0 dt ∂∇T ∂∇T ∂∇T d∇ξ ∂e  ∂f ∂η :  + λ3  + λ5 = 0 ⇒ λ5 = dt ∂∇ξ ∂∇ξ T ∂∇ξ v˙ : λ2 = 0 ∂φ ∂k ∂q ∂G ∂G ∇ : + λ3 + λ5 =0 ⇒ = −λ5 ∂ ∂ ∂ ∂ ∂ ∂φ ∂k ∂q ∂G ∂G ∇∇T : + λ3 + λ5 =0 ⇒ = −λ5 ∂∇T ∂∇T ∂∇T ∂∇T ∂∇T ∂φ ∂q ∂G ∂G ∂k ∇∇ξ : + λ3 + λ5 =0 ⇒ = −λ5 ∂∇ξ ∂∇ξ ∂∇ξ ∂∇ξ ∂∇ξ ∇v : λ1 1 − λ3 t + λ5 ∇ξ = 0.

(8.259) (8.260) (8.261) (8.262) (8.263) (8.264) (8.265) (8.266)

The restrictions on constitutive functions resulting from this extended procedure are weaker than the results of the classical Liu procedure. Especially, the free energy density may depend on the gradient of the internal variable (see (8.261)), and the Ginzburg-Landau theory can be recovered. The extra entropy flux may be a function of the whole set of state space variables, and cannot be concluded to be zero—heat flux and entropy flux are independent constitutive functions. One can show in general: 1. The more constraints (derivatives of balance equations) are taken into account in the Liu procedure, the weaker are the restrictions on constitutive functions. 2. Higher derivatives of balance equations are constraints as well. Up to a maximal order of derivation, they have to be taken into account. Derivatives of the balance equations higher than this maximal order have no influence in the Liu procedure. The maximal order of differentiation to be taken into account depends on the chosen state space. For more considerations on differentiated balance equations in the Liu procedure, see [34–37]. Another interesting example [38–40] is the so-called dual internal variables, which allow a unified treatment of dissipative processes and non-dissipative processes (wave-type equations).

8.11 Summary of Rational Thermodynamics’ Exploitation of the Dissipation Inequality

157

8.11 Summary of Rational Thermodynamics’ Exploitation of the Dissipation Inequality Summary of the steps according to the method of Liu: 1. All balance equations, and if necessary other equations of motion, have to be formulated in such a way that the right-hand side is zero. These equations are the balance of mass, of momentum, of energy, and (for micropolar media) of angular momentum. The additional equations of motion can be, for instance, Maxwell’s equations for the electromagnetic fields, or equations of motion for internal variables. 2. The dissipation inequality is formulated, and the left-hand sides of the balance equations and other equations of motion are multiplied, with (undetermined) state functions λi . 3. The set of variables relevant for the constitutive functions, i.e., the state space, has to be chosen. All constitutive functions depend on this same set of variables. 4. The derivatives of constitutive functions have to be carried out according to the chain rule. 5. The resulting expression is linear in the higher derivatives not included in the state space. The higher derivatives are put in front of brackets. 6. Setting the brackets equal to zero gives the so-called Liu equations. These are as many equations as there are higher derivatives. 7. The multipliers λi are calculated from the Liu equations. There are as many multipliers as there are balance equations (and other equations of motion). The number of these multipliers is less than the number of Liu equations. The remaining Liu equations, not used for calculating the multipliers, are the restrictions on constitutive functions imposed by the second law. In these equations, the multipliers have to be substituted. 8. With the Liu equations, the resulting expression for the entropy production does not contain any higher derivatives. It is a state function. 9. The expression for the entropy production, the residual inequality, can be exploited for equilibrium conditions. 10. The simpler method of Coleman and Noll [23] is a special case of the procedure by Liu.

References 1. I.S. Liu, Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Rat. Mech. Anal. 46, 131–148 (1972) 2. C. Papenfuss, Contribution to a Continuum Theory of Two Dimensional Liquid Crystals (Wissenschaft- und Technik Verlag, Berlin, 1995) 3. C. Papenfuss, Liquid crystalline surface tension and radius dependence of the internal pressure in liquid crystalline bubbles and droplets. Mol. Cryst. Liq. Cryst. 367, 2999–3006 (2001)

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4. 5. 6. 7.

I. Müller, Thermodynamics (Pitman Publ. Co., London, 1985) G.F. Smith, On isotropic integrity base. Arch. Rat. Mech. Anal. 18, 282 (1965) G.F. Smith, R.S. Rivlin, The anisotropic tensors. Quart. Appl. Math. 15, 308–314 (1957) C.C. Wang, A new representation theorem for isotropic functions. Arch. Rat. Mech. Anal. 36, 166 (1970) W. Muschik, Empirical foundation and axiomatic treatment of non-equilibrium temperature. Arch. Rat. Mech. Anal. 66, 379 (1977) W. Muschik, G. Brunk, A concept of non-equilibrium temperature. Int. J. Eng. Sci 15, 377–389 (1977) K. Hutter, The foundation of thermodynamics, its basic postulates and implications. A review of modern thermodynamics. Acta Mechanica 27, 1–54 (1977) Marsden and Hughes, Mathematical Foundations of Elasticity (Practice Hall, Inc., Englewood Cliffs, New Jersey, 1983) C. Truesdell, W. Noll, Non-Linear Field Theories of Mechanics, Encyclopedia of Physics, vol. III/3, Sect. 98 (Springer, Berlin, etc., 1965) I. Müller, Grundzüge der Thermodynamik (Springer, Berlin, Heidelberg, 1994) W. Muschik, Is the heat flux density really non-objective? a glance back, 40 years later. Continuum Mech. Thermodyn. 24, 333–337 (2012) Muschik. Objectivity and frame indifference of acceleration-sensitive materials—50th anniversary of JTAM. J. Theor. Appl. Mech. 50(3), 807–817 (2012) W. Muschik, L. Restuccia, Systematic remarks on objectivity and frame-indifference, liquid crystal theory as an example. Arch. Appl. Mech. 78–95, 837 (2008) A.J.M. Spencer, Theory of invariants, in Continuum Physics, vol. 1, ed. by A.C. Eringen (Academic Press, New York and London, 1971), pp. 239–353 W. Muschik, H. Ehrentraut, An amendment to the second law. J. Non-Equilib. Thermodyn. 21, 175 (1996) W. Muschik, C. Papenfuss, H. Ehrentraut, A sketch of continuum thermodynamics. J. NonNewtonian Fluid Mech. 96, 255–290 (2001) W. Muschik, in Proceedings of the International Conference on Nonlinear Mechanics (Science Press, Beijing, Shanghai, 1985) I.S. Liu, I. Müller, Extended thermodynamics of classical and degenerate gases. Arch. Rat. Mech. Anal. 83, 285 (1983) W. Muschik, Fundamental remarks on evaluating dissipation inequalities, in Lecture Notes in Physics, vol. 199, ed. by J. Casas-Vazquez, D. Jou, G. Lebon (Springer, 1984), pp. 388–397 B.D. Colemann, W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rat. Mech. Anal. 13, 167–168 (1963) S. Blenk, H. Ehrentraut, W. Muschik, Macroscopic constitutive equations for liquid crystals induced by their mesoscopic orientation distribution. Int. J. Eng. Sci. 30(9), 1127–1143 (1992) B.D. Coleman, On thermodynamics, strain impulses and viscoelasticity. Arch. Rat. Mech. Anal. 17, 230–254 (1964) B.D. Coleman, M.E. Gurtin, Thermodynamics with internal state variables. J. Chem. Phys. 47(2), 597–613 (1967) B.D. Coleman, D.R. Owen, On the thermodynamics of elastic-plastic materials with temperature-dependent moduli and yield stress. Arch. Rat. Mech. Anal. 70 (1979) W. Muschik, R. Ellinghaus, Zur Auswertung von Dissipationsungleichungen. Z. angew. Math. Mech. (ZAMM) 68, 232–233 (1988) V. Triani, C. Papenfuss, V.A. Cimmelli, Exploitation of the second law: Coleman-noll and liu procedure in comparison. J. Non-Equilib. Thermodyn. 33(1), 47–60 (2008) W. Muschik, Recent developments in nonequilibrium thermodynamics, in Lecture Notes in Physics, vol. 199 (Springer, Berlin, 1984) C. Papenfuss, Thermodynamic restrictions on constitutive functions for fiber suspensions. AAPP (Atti della Accademia Peloritana dei Pericolanti) 97(1), A4 (2019) T. Alts, Thermodynamics of thermoelastic bodies with kinematic constraints. Fibre reinforced materials. Arch. Rat. Mech. Anal. 61, 253–289 (1976)

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

References

159

33. R.A. Schapery, A micromechanical model for linear viscoelastic behavior of particle-reinforced rubber with distributed damage. Eng. Fract. Mech. 25, 845–867 (1986) 34. P. Ván, Exploiting the second law in weakly non-local continuum physics. Period. Polytech. Ser. Mech. Eng. 49(1), 79–94 (2005) 35. P. Rogolino, V.A. Cimmelli, Differential consequences of balance laws in extended irreversible thermodynamics of rigid heat conductors. Proc. R. Soc. A 475, 20180482, 01 (2019) 36. V.A. Cimmelli, An extension of liu procedure in weakly nonlocal thermodynamics. J. Math. Phys. 48(113510), 1–13 (2007) 37. V.A. Cimmelli, A. Sellitto, V. Triani, A generalized coleman-noll procedure for the exploitation of the entropy principle. Proc. R. Soc. A 466, 911–925 (2010) 38. P. Van, A. Berezovski, J. Engelbrecht, Internal variables and dynamic degrees of freedom. J. Non-Equilib. Thermodyn. 33(3), 235–254 (2008) 39. P. Van, A. Berezovski, Microinertia and internal variables. J. Non-Equilib. Thermodyn. 28(4), 1027–1037 (2016) 40. A. Berezovski, P. Van, Internal Variables in Thermoelasticity. Solid Mechanics and its Applications, 1st edn. (Springer International Publishing AG, Cham, Switzerland), p. 6 (2017)

Chapter 9

Extended Thermodynamics

Abstract Fourier’s equation of heat conduction in the balance of internal energy leads to a parabolic differential equation for the temperature and therefore to infinite speeds of propagation of disturbances. Extended thermodynamics gives a solution to this problem on one hand, and on the other hand it is suitable for a treatment of processes at short timescales or high frequencies. Both branches of extended thermodynamics are sketched: One branch generalizing the method of irreversible thermodynamics and the other branch based on rational thermodynamics. The second branch is based on the Liu procedure, and it is intimately connected to the mathematical theory of hyperbolic systems of differential equations.

9.1 Motivation In a linear approximation, we have Fourier’s law of heat conduction: q(, T, ∇T ) = −κ∇T,

(9.1)

with the additional assumption κ = const. and (, T, ∇T ) = e0 (, T ).

(9.2)

The balance of energy for an incompressible fluid ( =const.) at rest (v = 0) without energy supply reads: ∂e + ∇ · q = 0. (9.3)  ∂t With the constitutive equations in linear approximation, we end up with (κ = const.): 

∂e ∂T − κ∇ · (∇T ) = 0. ∂T ∂t

© Springer Nature Switzerland AG 2020 C. Papenfuß, Continuum Thermodynamics and Constitutive Theory, https://doi.org/10.1007/978-3-030-43989-7_9

(9.4)

161

162

9 Extended Thermodynamics

The final equation is the heat conduction equation: c T˙ − κT = 0.

(9.5)

∂e is the heat capacity. This is a parabolic differential equation. The problem is c = ∂T that parabolic equations allow for infinite speeds of propagation of disturbances. It is easy to check that the following temperature field is a solution of the parabolic heat conduction equation with the initial condition T ( y, t = 0) = T0 ( y)

1 T (x, t) = (4π Dt)3/2



    x−y 2 3 T0 ( y) exp − d y, 4Dt −∞ ∞

where D=

κ , c

(9.6)

(9.7)

and in the case of a δ-shaped initial disturbance from constant temperature T0 :    x 2 1 + T0 . exp − T (x, t) = (4π Dt)3/2 4Dt

(9.8)

The temperature distribution at times t > 0 is different from the constant value T0 at all positions x. This is so for arbitrarily small times, i.e., the temperature disturbance has spread with infinite velocity. Giving a “δ-shaped” initial temperature distribution at time t0 for the heat conduction equation (9.5), the temperature has a constant value T0 everywhere except at x = 0 (see Fig. 9.1). For times t2 > t1 > t0 , the temperature disturbance T − T0 is greater than zero everywhere. The disturbance has spread from x = 0 to any position in an arbitrarily small time interval. This infinite propagation velocity of disturbances is not observed in nature and contradicts the principle of relativity. This paradox was the motivation for the development of extended thermodynamics. The problem with infinite speeds of propagation is due to the parabolic type of the heat conduction equation (see (9.5) and, for the classification, [1]), roughly speaking because it is of first order in time

Fig. 9.1 For an initially δ-shaped temperature distribution, the disturbance spreads with infinite speed

T

t=0 t1 >0 t 2>t1>0 x

9.1 Motivation

163

and of second order in the space derivatives. The same type of parabolic equation occurs, for instance, when the expression for the viscous stress tensor of a Newtonian fluid t = μ ∇v + ν∇ · vδ is inserted into the balance of momentum. The resulting Navier-Stokes equations are parabolic equations allowing for infinite propagation velocities of disturbances. The mathematical classification of partial differential equations can be found in [2]. We will give here only the characterization of a special case of hyperbolic systems later in this chapter. Infinite propagation velocities do not occur for solutions of hyperbolic differential equations. This fact is exploited systematically in the branch of extended thermodynamics developed from rational thermodynamics (see Sect. 9.3). Historically, extended irreversible thermodynamics (see Sect. 9.2) was first, and the oldest approach to this problem stems from Cattaneo. His argumentation from elementary kinetic theory was the following: Consider a volume element of linear dimension of the mean free path of the particles, i.e., collisions of particles within the volume element can be neglected. Suppose that the average (thermal) energy of particles at the upper boundary of the volume element is higher than the average energy at the lower boundary. Then particles flying downward carry a higher energy than those flying upward. This way, the thermal energy is flowing opposite to the energy gradient, with a flux proportional to the gradient. As the temperature is proportional to the average thermal (kinetic) energy of the particles, this leads to the classical Fourier law of heat conduction: q = −κ∇T . Cattaneo [3] argued that the particles need a finite time, the mean time of free flight, for the path between the upper and the lower boundaries. Therefore, the temperature gradient at earlier time is relevant. This is approximated by the first-order Taylor series with respect to time:

˙ ) . q = −κ ∇T − τ (∇T

(9.9)

Introducing this equation into the simplified energy balance (9.3) leads to the partial differential equation: C



∂T ˙ ) =0 + ∇ · −κ ∇T − τ (∇T ∂t

(9.10)



−1 for the temperature. If κ is a constant, and with the formal expansion 1 − τ ∂t∂ = 1 + τ ∂t∂ , we end up with the Cattaneo equation of heat conduction:   ∂ ∂T C 1+τ = κT ∂t ∂t ∂T ∂2 T κ + τ 2 = T. ∂t ∂t C

(9.11)

It is of the form of a telegrapher’s equation, i.e., it is a wave equation with damping. The solutions are traveling waves with damping, which have a finite speed of propagation.

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9 Extended Thermodynamics

9.2 Extended Irreversible Thermodynamics The development of extended thermodynamics started with extended thermodynamics of irreversible processes [4, 5]; for an overview over the early history of extended thermodynamics, see [6]. In Extended Irreversible Thermodynamics (EIT), the assumption of local equilibrium is no longer made, but the entropy density may depend on the stress tensor t and the heat flux q, in addition to depending on the equilibrium variables. The assumptions are as follows: 1. There is a local dissipation inequality with constitutive functions entropy density η and entropy flux , and a non-negative entropy production. 2. Stress tensor and heat flux are included in the domain of the entropy density as independent variables. These fluxes are included in order to deal with the exchange between different volume elements. No gradients or time derivatives are included in the set of variables. 3. The entropy density is a concave function of all its variables, i.e., the matrix of the second derivatives is negative definite. The exploitation of the dissipation inequality is analogous to the procedure in classical irreversible thermodynamics: 1. The total differential of the entropy density with respect to all its variables (a generalized Gibbs equation [7]) is inserted into the balance of entropy. 2. For the entropy supply, it is assumed z = Tr . For the entropy flux, in some cases, the assumption  = Tq is made, but sometimes this is generalized. 3. The other balance equations are inserted and, for the entropy production, an expression in the form of a sum of products of fluxes and forces is derived. 4. Linear relations between fluxes and forces are assumed.

9.2.1 Example 1: A Heat-Conducting Fluid Let us consider the example of a heat-conducting simple fluid, neglecting viscous flow: Z = {, e, q}. (9.12) The exploitation of the dissipation inequality is analogous to the procedure in classical TIP. It is presupposed that a generalized Gibbs equation, involving also the heat flux density, holds:   1 de p d dq dη . (9.13) = − 2 + 2aT q · dt T dt  dt dt

9.2 Extended Irreversible Thermodynamics

With the assumption φ =

q T

165

, the dissipation inequality reads:

r dη +∇ ·φ− dt T r dη ∇ · q 1 = + − 2 (∇T ) · q − T T T  dt p d dq 1 1  de r − 2 + 2aT q · + ∇ · q − 2 (∇T ) · q − = T dt  dt dt T T T   dq 1 1 t : ∇v + p∇ · v + 2aT q · = − (∇T ) · q T dt T ⎛   1 1 ⎜ ⎜ t :  ∇v + p + trace(t) ∇ = ⎜  · v T ⎝ 3    flux force force flux ⎞ σ=

⎟ 1 dq ⎟ − (∇T ) ⎟ , + q · 2aT  ⎠ dt T   flux  force 

(9.14)

where first Gibbs’ equation and then the balance of energy  de + ∇ · q = t : ∇v + r dt = −∇ · v have been used. The final expression for the and the balance of mass d dt entropy production is again a sum of products of fluxes and forces. Linear constitutive equations between the fluxes and forces read: t = 2μ ∇v

(9.15)

1 p + trace(t) = λ∇ · v 3

(9.16)

˙ q = −κ∇T − αq.

(9.17)

There is no coupling between scalars, vectors, and tensors. The last equation (9.17) ˙ . leads to Cattaneo’s equation of heat conduction if q˙ is approximated by − κτ ∇T α In this example, we have made use of a generalized Gibbs equation, including the heat flux, but the part of the entropy differential concerning the equilibrium variables is the classical Gibbs equation. The assumptions can be made less restrictive in the sense that we do not assume such an equation for the entropy differential, but the total differential of the entropy density is exploited in general. Then the pre-factor of the internal energy differential is a generalized non-equilibrium temperature, depending on the heat flux, too. For a detailed review and discussion on the definitions of nonequilibrium temperature, see [8, 9]. The classical Gibbs equation with equilibrium temperature is recovered as the lowest order (in terms of the heat flux) approximation.

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9 Extended Thermodynamics

This more general version, also including viscous flow, has been treated in [10]. The differences between non-equilibrium temperature and equilibrium temperature on one hand, and non-equilibrium pressure and equilibrium pressure on the other hand, turn out to be of second order in the variables heat flux and pressure tensor.

9.2.2 Example 2: Rheological Models In this example, we are interested in flow- and deformation-related effects on a viscoelastic body. We restrict ourselves to small deformations (i.e., all measures of deformation are equivalent) and to isotropic material that is not micropolar. In addition, the considerations are limited to lowest order in the non-equilibrium variables: here, the inelastic part of the pressure tensor. The outcome of the exploitation of the dissipation inequality will be the so-called Pointing-Thomson model of a viscoelastic body. We are not interested here in thermal effects. Therefore, heat flux is not included in the state space. In addition to the symmetric deformation gradient  = 21

∇u + (∇u)T , which is the Green tensor G within the geometrically linearized theory (see 8.54), and internal energy density (the equilibrium variables), the pressure tensor is included in the state space. The choice of the pressure tensor P = −t instead of the stress tensor t is only a matter of taste. The pressure tensor is decomposed into an elastic part P  and an inelastic part P  with: P = P  + P  P  = −2G + λ L δ.

(9.18) (9.19)

G and λ L are LAMÉ coefficients, and  = 13 trace . Due to the fact that the material is not micropolar, the pressure tensor is symmetric, and we have the decomposition into irreducible parts: P  = p  δ+ P  .

(9.20)

The inelastic part is treated as an independent variable in the sense of extended thermodynamics. In summary, the state space includes the variables: Z = {e, , P  , p  }.

(9.21)

Internal energy density and deformation gradient form the equilibrium subspace.

9.2 Extended Irreversible Thermodynamics

167

With this choice of the state space, the total differential of the entropy density is: dη ∂η de ∂η d ∂η ∂η d p  d P  = + : + + : dt ∂e dt ∂ dt dt ∂ p  dt ∂ P  =

1 de P  d c d p  C d P  + : − : − . T dt T dt T dt T dt

(9.22)

For the derivatives with respect to the equilibrium variables e and , the relations known from the (classical) Gibbs relation for solids have been inserted, and we have introduced the abbreviations: C = −T

∂η ∂ P 

, c = −T

∂η . ∂ p 

(9.23)

The time derivative of the internal energy is eliminated by means of the balance of internal energy: here, for a solid, for the case of vanishing heat flux and energy supply: d de . (9.24)  = −P : dt dt P : d is the work of the stresses, and P is the total pressure, consisting of the elastic dt and the inelastic contribution: (9.25) P = P  + P  . For the time derivative of the entropy, the result can be expressed: dη 1 d P  d c d p  C d P  =− P: + : − : − dt T dt T dt T dt T dt 1  d P  d 1  d P : − P : + : =− T dt T dt T dt − =− =−

c d p  C d P  : − T dt T dt

c d p  C d P  1  d P : − : − T dt T dt T dt

1  d c d p  1 C d P  d  − p − : − P  : T dt T dt T dt T dt

with =

1 trace. 3

(9.26)

(9.27)

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9 Extended Thermodynamics

Finally, we have decomposed the first term into symmetric traceless parts and the trace. For the time derivative of the entropy density, the balance of entropy is inserted where, in our case, entropy supply is assumed to be zero. For the entropy flux, it holds  = 0 to linear order in the deformation gradient  and in the non-equilibrium variable P because the representation theorem on the chosen state space (9.21) does not give any vector invariant that is linear in the second-order tensor variables. With 

dη = −∇ ·  + σ + z = σ, dt

(9.28)

we end up with the expression for the entropy production: ⎛ ⎞   1 ⎝  d  d p d P d ⎠. σ=− + p  + C : +c P : T dt dt dt dt

(9.29)

Due to the definitions of C and c as derivatives of the entropy density C = −T ∂η and c = −T ∂∂ηp , they are zero in equilibrium (which is necessary for ∂ P 

the maximum of the entropy in equilibrium). Therefore, to the lowest order in the nonequilibrium variables we have the representation (with p  |eq = 0 and P  |eq = 0): C = α P  , c = β p  .

(9.30)

The result for the entropy production reads: ⎛



⎞

d  d P ⎠ +α T σ = − P  ⎝ dt dt J1    

f1

 d p   d +β −p .  dt dt    J2

(9.31)

f2

Linear constitutive relations between forces and fluxes are: d P  d  +α dt dt  d p d − bp  = +β . dt dt

− a P  =

(9.32) (9.33)

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169

The resulting differential equations for the symmetric traceless part of the deformation gradient and for the trace, respectively, read: − 2ηˆ

d  d P  = τ + P  dt dt

(9.34)

d d p  = τˆ + p  dt dt

(9.35)

−ξ with the abbreviations:

1 α , τ = 2ηα ˆ = 2a a β 1 ξ = , τˆ = ξβ = . b b

ηˆ =

(9.36)

τ and τˆ have the dimension of time and are interpreted as relaxation times. Finally, we introduce again the total pressure tensor P = P  + P  . The equation for the symmetric traceless part is transformed to: d  d = τ − 2ηˆ dt dt −2ηˆ

 P − P





+ P − P

d P d  d  = τ + 2Gτ + P +2G  dt dt dt d P τ + P = −2G dt

with τσ =



d   +τσ dt

ηˆ + τ . G

(9.37)

 (9.38)

(9.39)

Analogously, we obtain for the trace: τˆ with

  dp d + p = −3K  + τˆσ dt dt

2 ξ + τˆ . K = λ L + G, τˆσ = 3 3K

(9.40)

(9.41)

The differential equations (9.38) and (9.40) for the constitutive quantity pressure tensor represent the Pointing-Thomson model of a viscoelastic body. They include as special cases:

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9 Extended Thermodynamics

1. The Kelvin-Voigt model for relaxation time τ = 0:  P = −2G

d   +τσ dt

 .

(9.42)

2. The Maxwell model for Young modulus G = 0: τ

d P d  + P = −2ηˆ . dt dt

(9.43)

9.2.3 Other Examples Extended irreversible thermodynamics has been applied to a large variety of systems: Viscoelastic fluids, where it has to be taken into account that these are mixtures of a solvent and a high molecular weight component (the concentrations are additional variables) [11, 12], as well as polymer melts and solutions [10, 13–17]. For an overview of the literature in the field, with hundreds of references, see [5] and [18]; see also [10, 19] and for a historical overview, see [6]. A comparison of the theoretical results with experimental results is possible, for instance, investigating the scattering of light or sound waves see [1, 20]. The propagating wave is a (small) perturbation of the equilibrium state, and the dynamics of the perturbation is governed by the EIT equations of motion. For solving these equations, the method of Laplace transformation is usually applied. An application to second sound in rigid solids can be found in [21]. Heat conduction equations beyond the Fourier law are a topic of intensive modern research. Contributions of EIT in this direction can be found f.e. in [22], and for an overview from the viewpoint of extended irreversible thermodynamics, see [23].

9.3 Rational Extended Thermodynamics In extended thermodynamics fluxes, in the simplest case, heat flux density and stress tensor (or the trace and the symmetric traceless part of the stress tensor) are included in state space, but derivatives of the state space variables are not. In rational extended thermodynamics, additional balance-type equations for the fluxes are introduced [1, 24]. In purely phenomenological theory, the validity of such balance equations for the fluxes is an assumption. Such balance equations can be derived from kinetic theory, i.e., from a level closer to the microscopic (molecular) level. However, this derivation usually starts from Boltzmann’s equation [1, 10, 25– 27], which is valid only for dilute gases. A derivation of balance-type equations from

9.3 Rational Extended Thermodynamics

171

the full set of microscopic Hamilton equations is also possible (see [28]) and leads in general to the BBGKY hierarchy of coupled equations [29–32]. In order to describe phenomena far from equilibrium, even higher order fluxes have been included in state space, and equations of motion for them have been derived from kinetic theory [1, 20, 33]. The steps in this branch of extended thermodynamics can be summarized as follows: 1. The set of variables for the constitutive quantities, the state space, is chosen to include the equilibrium variables and at least the non-convective fluxes of momentum and energy, the stress tensor, and heat flux. Higher order fluxes may be included, but no time or space derivatives are included in the state space. 2. All wanted fields are supposed to obey balance-type differential equations. For the extensive quantities, these are the usual balance equations. For the fluxes, this is an additional assumption. 3. There exists a constitutive quantity entropy obeying a balance equation with a non-negative production and a constitutive quantity entropy flux. In addition, it is supposed that entropy density is a concave function with respect to all its variables. With respect to the non-equilibrium variables, this can be interpreted as a stability requirement for equilibrium states, but with respect to the equilibrium variables, it is an assumption. 4. If the state space is chosen as described in the first point, the dissipation inequality is exploited according to Liu [34] (see Chap. 8). It can be shown that the resulting restrictions on constitutive functions result in a general mathematical structure of the theory: namely, in a set of hyperbolic partial differential equations with a convex extension. This fact and the properties of such a system will be discussed in the next section.

9.3.1 The Formal Structure of Rational Extended Thermodynamics It is assumed that all wanted fields obey balance-type equations. We will denote the wanted fields, by u A , where A is the index numbering the different wanted fields. For vectorial or tensorial wanted fields, the components are counted separately. In the following, we will use a summation convention over repeated capital indices, meaning summation over all indices, index values running from 1 to a fixed number M. For a discussion of the mathematical structure of rational extended thermodynamics, see also [24]. The equations of motion for the wanted fields are of balance type and read: ∂u A + ∇ · f A (u B ) = π A . ∂t

(9.44)

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9 Extended Thermodynamics

The flux f A of the quantity u A contains the convective and the non-convective part. π A is the production and supply density of the wanted field u A . The flux f A is a constitutive quantity, defined on the state space Z. The state space in extended thermodynamics contains all wanted fields u B and no other variables. Carrying out the derivative of the constitutive quantity according to the chain rule equation (9.44) results in the following quasilinear first-order partial differential equation: ∂fA ∂u A + · ∇u B = π A . ∂t ∂u B

(9.45)

This is the general form of balance equation on the state space in extended thermodynamics. There are M such balance equations, as many as there are wanted fields, which is the same number as the number of state space variables. The dissipation inequality is the local formulation of the second law of thermodynamics. Entropy density and entropy flux are constitutive quantities on the state space Z: ∂ ∂η ∂u A + · ∇u B = σ ≥ 0. (9.46) ∂u A ∂t ∂u B With the set of balance equations (9.45), the dissipation inequality is exploited according to Liu, i.e., the M balance equations are multiplied by M quantities  A and subtracted from the dissipation inequality: ∂η ∂u A ∂ + · ∇u B ∂u A ∂t ∂u B   ∂u A ∂fA + − A · ∇u B − π A = σ ≥ 0. ∂t ∂u B

(9.47)

Due to the fact that no derivatives are included in the state space, all derivatives are higher order ones. Therefore, the factors in front of the time derivatives ∂u∂tA and the factors in front of the space derivatives ∇u A vanish: ∂η ∂u A : − A = 0 ∂t ∂u A ∂fB ∂ ∇u A : − B = 0. ∂u A ∂u A

(9.48) (9.49)

On the other hand, with the choice of variables Z = {u A }, the total differentials of the constitutive functions are: ∂η du A ∂u A ∂ d = du A , ∂u A dη =

(9.50) (9.51)

9.3 Rational Extended Thermodynamics

173

which gives, together with (9.48) and (9.49): dη =  A du A ∂fB d =  B du A =  B d f B . ∂u A

(9.52) (9.53)

After inserting (9.48) and (9.49) into the dissipation inequality (9.47), it remains the residual inequality, i.e., the entropy production:  A π A = σ ≥ 0.

(9.54)

∂ A ∂2η = < 0, ∂u B ∂u B ∂u A

(9.55)

From (9.48), it follows:

because η was assumed to be a concave function of all its variables. This shows that A the derivative ∂ is nowhere zero. Therefore, the transformation u A →  A is an ∂u B allowed coordinate transformation. Corresponding to the coordinate transformation, the constitutive functions η and  are Legendre transformed: η( ˜ A ) = −η(u A ) +  A u A ˜ A ) = −(u A ) +  A f A . (

(9.56) (9.57)

The Legendre transform η˜ of the concave function η is a convex function. The total differentials of the constitutive functions in terms of the new variables  A are: dη( ˜ A ) = −dη(u A ) + d( A u A ) = − A du A + d A u A +  A du A = d A u A (9.58)

˜ A ) = −d(u A ) + d( A f A ) = − A d f A + d A f A +  A d f A d( = d A f A , (9.59) and the coefficients of the differentials can be expressed as: ∂ η˜ ∂ A ˜ ∂ = . ∂ A

uA =

(9.60)

fA

(9.61)

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9 Extended Thermodynamics

In terms of the new variables  A , the system of field equations reads: ∂fA ∂u A ∂ B + · ∇ B = π A ( B ). ∂ B ∂t ∂ B

(9.62)

With this variable transformation, the system of field equations results in a system of partial differential equations of the form: B AB

∂ B + A AB · ∇ B = π A ( B ), ∂t

(9.63)

where the matrices A and B are defined as: ∂u A ∂ 2 η˜ = ∂ B ∂ B ∂ A ˜ ∂fA ∂2 = = . ∂ B ∂ B ∂ A

B AB = A AB

(9.64) (9.65)

Definition: A system of quasilinear partial differential equations for the wanted fields A ∂ B B AB (9.66) + A AB · ∇ B = π A ( B ) ∂t is called a symmetric hyperbolic system with a convex extension, if: 1. 2. 3. 4.

The matrix B AB is symmetric and positive definite. The matrices AiAB , where i is the component index i = 1, 2, 3, are symmetric.  A · π A ≥ 0. There exists an additional balance equation with a convex density η˜ (here in terms of the new variables  A ) and a non-negative production σ in regular points, as well as at surfaces of discontinuity σs : ∂ η˜ ˜ =σ≥0 +∇ · ∂t ˜ · n = σs ≥ 0. −s|[η]| ˜ + |[]|

(9.67) (9.68)

s = w ⊥ is the normal component of the surface velocity. Such systems of quasilinear hyperbolic equations with a convex extension have the following properties [35–37]:

9.3 Rational Extended Thermodynamics

175

1. For an initial-boundary-value problem, with Cauchy data that are not characteristic (see Chap. 10) and fulfill a very weak requirement on the regularity of the functions, there exists a solution and the solution is unique. The regularity requirement on the Cauchy data is that initial functions are elements of the Sobolev space H 4 (R4 ) (see, for instance, [2, 38]), meaning that the function and its derivatives up to the fourth order are square integrable. This means that, for a very general class of initial data, there exists a unique solution of the field equations. 2. The solution depends continuously on the initial data, i.e., a small variation in the initial function leads only to a small variation in the solution of the field equations. 3. In finite volumes, the solution remains finite. 4. Perturbations spread only with finite velocities. The fact that the balance equations and the dissipation inequality together form a system of symmetric hyperbolic equations in extended thermodynamics is used in literature as an argument in favor of this branch of thermodynamics. From a physical point of view, these properties are the properties to be expected from solutions of balance equations. Solutions to initial value problems should exist for arbitrary initial data. Smooth dependence of solutions on the initial functions is usually observed. However, such a smooth dependence does not exist in deterministic chaotic systems. There, very similar initial data can lead to completely different solutions at later times. The last point in the list of properties is the solution to the problem of infinite propagation velocities present in classical thermodynamics, which was the starting point of extended thermodynamics. However, the argument cannot be reversed. There exist other types of field equations having solutions with finite propagation velocities. Examples are nonlinear field equations having soliton solutions (see examples in [39–41]). With the exploitation of the dissipation inequality, and the transformation of the balance equations to the multipliers  A as new variables, we can show now easily that the field equations for the new variables form a symmetric hyperbolic system with a convex extension: 1. The matrix B AB is symmetric due to (9.64) and the Schwarz theorem: B AB =

∂ 2 η˜ ∂ 2 η˜ = = BB A . ∂ B ∂ A ∂ A ∂ B

(9.69)

The matrix is positive definite, because η˜ is a convex function. 2. With the same argument as above, the matrix A AB is symmetric because it is a mixed second derivative: A AB =

˜ ˜ ∂2 ∂2 = = AB A . ∂ B ∂ A ∂ A ∂ B

(9.70)

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9 Extended Thermodynamics

3.  A π A ≥ 0 holds because of the residual inequality. 4. The entropy density (as a function of the multipliers) plays the role of the convex density, which has positive productions in regular points as well as at surfaces of discontinuity, according to the second law of thermodynamics.

9.3.2 Example: Viscous Heat-Conducting Body For the state space: Z = {, e, v, t, q}

(9.71)

the balance equations are: d + ∇ · v = 0 dt dv  + ∇ · tT =  f dt 

de + ∇ · q = r + t : ∇v dt dq +∇ · L = G dt dt + ∇ · M = N. dt

(9.72) (9.73) (9.74) (9.75) (9.76)

The fluxes L and M, as well as the productions G and N, are constitutive quantities. The need of additional constitutive equations is not avoided if t and q are considered as wanted fields; however, it is often assumed for the higher order fluxes L and M and productions that constitutive equations of lowest non-trivial order in the nonequilibrium variables stress tensor and heat flux density are sufficient. For the exploitation of the dissipation inequality and applications, for instance, concerning the wave speed of light and sound in this viscous, heat-conducting body, the reader is referred to the literature [1, 27, 42].

9.4 Summary In summary, the state space of extended thermodynamics includes fluxes (at least heat flux and stress tensor) but no gradients or time derivatives of variables. It can be shown that, with constitutive functions defined on this state space, the balance equations form a hyperbolic system of partial differential equations [1]. Such systems of differential equations enforce finite speeds of propagation of disturbances. With the additional assumption of a concave entropy density, as a function of the wanted fields (or a convex density as a function of the multipliers  A ) one can show

9.4 Summary

177

from the structure of the resulting equations that the field equations in extended thermodynamics form a symmetric hyperbolic set of equations with a convex extension. Such systems of differential equations have the desirable properties of finite speeds of propagation only, finite solutions in finite volumes, and smooth dependence of the solution on initial conditions. In Extended Irreversible Thermodynamics (EIT), the assumption of local equilibrium is no longer made, but the entropy density may depend on the stress tensor t and the heat flux q, in addition to depending on the equilibrium variables. The assumptions are as follows: 1. There is a local dissipation inequality with constitutive functions entropy density η and entropy flux , and a non-negative entropy production. 2. Stress tensor and heat flux are included in the domain of the entropy density as independent variables. These fluxes are included in order to deal with the exchange between different volume elements. No gradients or time derivatives are included in the set of variables. 3. The entropy density is a concave function of all its variables, i.e., the matrix of the second derivatives is negative definite. The exploitation of the dissipation inequality is analogous to the procedure in classical irreversible thermodynamics. This leads to relaxation equations for heat flux and stress tensor.

References 1. I. Müller, T. Ruggeri, Extended Thermodynamics, vol. 37 (Springer Tracts in Natural Philosophy, Berlin, Heidelber, New York, 1993) 2. R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. II (Interscience Publishers, New York, London, 1962) 3. C. Cattaneo, C.R. Acad. Sci. Paris 247, 431 (1958) 4. I. Müller, Zum Paradoxon der Wärmeleitungstheorie. Zeitschrift fuer Physik 198, 329 (1967) 5. D. Jou, J. Casas-Vazquez, G. Lebon, Extended irreversible thermodynamics: an overview of recent bibliography. J. Non-Equilb. Thermodyn. 17, 383–396 (1992) 6. G. Lebon, D. Jou, Early history of extended irreversible thermodynamics (1953–1983): an exploration beyond local equilibrium and classical transport theory. Eur. Phys. J. H 40(2), 205–240 (2015) 7. D. Jou, J.M. Rubi, J. Casas-Vazquez, A generalized Gibbs equation for second order fluids. J. Phys. A 12, 2515–2520 (1979) 8. J. Casas-Vazquez, D. Jou, Temperature in non-equilibrium states: a review of open problems and current proposals. Rep. Prog. Phys. 66, 1937–2023 (2003) 9. D. Jou, L. Restuccia, Caloric and entropic temperatures in non-equilibrium steady states. Phys. A 460, 246–253 (2016) 10. D. Jou, J. Casas-Vazquez, G. Lebon, Extended Irreversible Thermodynamics (Springer, Berlin, Heidelberg, New York, 1993) 11. P. Goldstein, L.S. Garcia-Colin, A thermodynamic basis for transport phenomena in viscoelastic fluids. J. Chem. Phys. 99, 3913 (1993) 12. P. Goldstein, L.S. Garcia-Colin, Transport processes in a viscoelastic binary mixture. J. NonEquilib. Thermodyn. 19, 170–183 (1994)

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13. D. Jou, J. Casas-Vazquez, M. Criado-Sancho, Polymer solutions under flow: phase separation and polymer degradation. Adv. Polym. Sci. 120, 205–266 (1995) 14. M. Criado-Sancho, D. Jou, J. Casas-Vazquez, On the spinodal line of polymer solutions under shear. J. Non-Equilib. Thermodyn. 18, 103–120 (1992) 15. J. Casas-Vazquez, M. Criado-Sancho, D. Jou, Dynamical and thermodynamical approaches to phase separation in polymer solutions under flow. Europhys. Lett. 23, 469–474 (1993) 16. M. Criado-Sancho, D. Jou, J. Casas-Vazquez, Nonequilibrium thermodynamics and the degradation of polymers under shear flow. J. Non-Equilib. Thermodyn. 19, 137–152 (1994) 17. G. Lebon, D. Jou, J. Casas-Vazquez, Nonequilibrium entropy and the second law of thermodynamics: a simple illustration. Int. J. Thermophys. 14, 671–683 (1993) 18. D. Jou, J. Casas-Vazquez, G. Lebon, Recent bibliography on extended irreversible thermodynamics and related topics (1992–1995). J. Non-Equilb. Thermodyn. 21, 103–121 (1996) 19. D. Jou, J. Casas-Vasquez, G. Lebon, Extended irreversible thermodynamics. Rep. Prog. Phys. 51, 1105–1179 (1988) 20. W. Weiss, Zur Hierarchie der Erweiterten Thermodynamik (Thesis. TU Berlin, Berlin, 1990) 21. G. Lebon, T. Ruggieri, A. Valenti, Extended thermodynamics revisited: renormalized flux variables and second sound in rigid solids. J. Phys. C 20, 025223 (2008) 22. D. Jou, V.A. Cimmelli, Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: an overview. Commun. Appl. Ind. Math. 7(2), 196–222 (2016) 23. G. Lebon, Heat conduction at micro and nanoscales: a review through the prism of extended irreversible thermodynamics. J. NonEquilibrium Thermodyn. 39, 35–59 (2014) 24. T. Ruggeri, M. Sugiyama, Rational Extended Thermodynamics Beyond the Monatomic Gas, 1st edn. (Springer International Publishing AG, Cham, Switzerland, 2015) 25. H. Grad, Principles of the kinetic theory of gases, in Handbuch der Physik XII, ed. by S. Flügge (Springer, Berlin, 1958) 26. S. Chapman, T.G. Cowling, The Mathematical Theory of Nonuniform Gases (Cambridge University Press, Cambridge, 1970) 27. I. Müller, Extended thermodynamics of classical and degenerate gases. Arch. Rat. Mech. Anal. 83, 286–332 (1983) 28. W. Dreyer, Molekulare Erweiterte Thermodynamik Realer Gase. Habilitationsschrift, Technische Universität Berlin 29. M. Born, H.S. Green, Proc. Roy. Soc. Lond. A 188, 10 (1946) 30. M. Born, H.S. Green, Proc. Roy. Soc. Lond. A 190, 455 (1947) 31. N. Bogoljuboff, J. Phys. USSR 10, 265 (1946) 32. J.G. Kirkwood, J. Chem. Phys. 14, 180 (1946) 33. K. Ikenberry, C. Truesdell, On the pressures and the flux of energy in a gas according to maxwell’s kinetic theory. J. Rational. Anal. 5, 1–54 (1956) 34. I.S. Liu, Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Rat. Mech. Anal. 46, 131–148 (1972) 35. M.G. Boillat, Sur l’ existance et la recherche d’équations de conservation supplémentaires pour les systèmes hyperboliques. C. R. Acad. Sc. Paris, 278 A, 909–913 (1974) 36. K.O. Friedrichs, P.D. Lax, Systems of conservation equations with a convex extension. Proc. Natl. Acad. Sci. USA 68, 1686–1688 (1971) 37. P.D. Lax, Hyperbolic Systems of Conservation Laws and The Mathematical Theory of Shock Waves (Society for Industrial and Applied Mathematics, Bristol, 1973) 38. H.T. Triebel, Höhere Analysis (VEB Deutscher Verlag der Wissenschaften, Berlin, 1972) 39. R.S. Johnson, Water waves and Korteweg-de Vries equations. J. Fluid Mech. 97(4), 701–719 (1980) 40. P. Lax, Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. XXI, 467–490 (1968) 41. G.A. Maugin, in Nonlinear Waves in Solids, ed. by A. Jeffrey, Ju. Engelbrecht (Springer, Berlin, New York, Wien, 1994), p. 109 42. I. Müller, Thermodynamics (Pitman Publishing Corporation, London, 1985)

Chapter 10

Shock Fronts and Hyperbolic Systems of Differential Equations

Abstract Rational extended thermodynamics leads to symmetric hyperbolic systems of differential equations for the wanted fields. A simple model equation with these properties in one dimension is the Burgers equation. Properties of solutions of such equations, especially the evolution and propagation of shock fronts, are studied on the example of this Burgers equation.

10.1 Introduction Exploiting the dissipation inequality, we have used up to now the local balance equations valid in regular points of the continuum, i.e., we assumed the field quantities to be continuously differentiable. However, there are many physical examples, where fields are discontinuous: phase boundaries, shock fronts evolving as discontinuities of the mass density in gases, domain walls in ferromagnetic systems, or grain boundaries in polycrystals. In the first part of this chapter, we introduce the method of characteristics for solving hyperbolic partial differential equations. Then, we consider one single example of a partial differential equation, the one-dimensional Burgers equation. This example equation allows the study of effects such as the evolution of discontinuous solutions for continuous initial conditions, the problem of non-uniqueness of solutions, and a mathematical criterion for the “selection” of a unique solution. This criterion is intimately connected to the notion of symmetric hyperbolic systems of partial differential equations with a convex extension (see Sect. 9.3.1).

10.2 Hyperbolic Differential Equations and the Method of Characteristics Generally, a differential equation is denoted as quasilinear if it is linear in the highest derivatives. Here and in the following sections about the Burgers equation, we will deal only with one first-order quasilinear, one-dimensional partial differential © Springer Nature Switzerland AG 2020 C. Papenfuß, Continuum Thermodynamics and Constitutive Theory, https://doi.org/10.1007/978-3-030-43989-7_10

179

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10 Shock Fronts and Hyperbolic Systems of Differential Equations

equation for a wanted field u(x, t) depending on time and one space variable. The general form of such an equation is: a(u, x, t)

∂u ∂u + b(u, x, t) = c(u, x, t), ∂t ∂x

(10.1)

where, in the general nonlinear case, the coefficients a and b are functions of the wanted field u. However, in case of a quasilinear equation, the coefficients do not and ∂∂ux , and the equation is linear in these depend on the (highest) derivatives ∂u ∂t quantities. In addition, the value of u on some line in the x − t-plane has to be given. This line in the x-t-plane, together with the values of u given on it, is the Cauchy data. Special cases are as follows: 1. Initial value problem: The values of u for all x are given at time t = 0: u(x, t = 0). The line in the x-t-plane is the x-axis. 2. Boundary-value problem: For a certain value x = x0 , the value of u is given for all times u(x0 , t). Other mixed cases of initial-boundary values are possible. The solution of (10.1), together with Cauchy data, forms a surface in the space spanned by the variables x, t, u as sketched in Fig. 10.1. For the solution u(x, t), there can be given a first-order Taylor expansion around the Cauchy data u 0 = u(x0 , t0 ): u(x, t) − u 0 =

∂u ∂u (x0 , t0 )(t − t0 ) + (x0 , t0 )(x − x0 ), ∂t ∂x

(10.2)

which can be written with the usual scalar product in R3 as: ⎞ ⎞ ⎛ ∂u (x0 , t0 ) t − t0 ∂t ⎝ x − x0 ⎠ · ⎝ ∂u (x0 , t0 ) ⎠ = 0. ∂x u(x, t) − u 0 −1 ⎛

(10.3)

As the vector (t − t0 , x − x0 , u(x, t) − u 0 )T is tangential to the surface of solu T tions, (10.3) shows that the vector ∂u (x0 , t0 ), ∂∂ux (x0 , t0 ), −1 is normal to that ∂t surface. On the other hand, the differential equation (10.1) in the point (u 0 , x0 , t0 ) can be written in the form: ⎞ ⎛ ⎞ ⎛ ∂u (x0 , t0 ) a(u 0 , x0 , t0 ) ∂t ⎝ b(u 0 , x0 , t0 ) ⎠ · ⎝ ∂u (x0 , t0 ) ⎠ = 0. (10.4) ∂x c(u 0 , x0 , t0 ) −1 As the second vector in this scalar product is normal to the surface of solutions, the first one must be a tangential vector. Therefore, it can be written in the form:

10.2 Hyperbolic Differential Equations and the Method of Characteristics

181

Fig. 10.1 The solutions of the differential equation as a function of position and time form a hypersurface. This surface can be parametrized by σ and s

dt (s; σ ) ds dx(s; σ ) b(u, s, σ ) = ds du(s; σ ) . c(u, s, σ ) = ds a(u, s, σ ) =

(10.5) (10.6) (10.7)

Now the partial differential equation has been transformed to a set of ordinary differential equations in terms of the variable s, the equations depending on the parameter σ . The first two equations have to be solved to give the characteristic curves in the x-t-plane. In general, this is coupled to solving the last equation in order to obtain the solution u(s, σ ) along the characteristic curves, where s is the parameter running on a curve that lies on the solution surface. This curve is the integral curve to the vector field (a, b, c). The parameter σ determines the “starting point” of the characteristic curve on the line of Cauchy data. For different values of σ , we obtain different integral curves. Each of these integral curves is a characteristic (see Fig. 10.1).

10.2.1 Example: Wave Equation The wave equation in one dimension: 2 ∂ 2u 2∂ u − c =0 ∂t 2 ∂x2

for the wanted field u(x, t) is fulfilled if either:

(10.8)

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10 Shock Fronts and Hyperbolic Systems of Differential Equations

Fig. 10.2 The solution of the wave equation: The solution u is moving with constant shape along the characteristics

or

∂u ∂u +c =0 ∂t ∂x

(10.9)

∂u ∂u −c = 0. ∂t ∂x

(10.10)

Let us consider the first of these first-order differential equations. We identify the coefficients in the general first-order differential equation of Sect. 10.2 as a(u, x, t) = 1, b(u, x, t) = c, c(u, x, t) = 0. This leads to the following set of ordinary differential equations: dt (s; σ ) =1 ds dx(s; σ ) =c ds du(s; σ ) = 0. ds

(10.11) (10.12) (10.13)

The last equation shows that the solution u is constant along the characteristic curve. The first and the second equation can be integrated to: t = s + t0 , x = cs + σ = ct + σ.

(10.14)

Here, it has been set already t0 = 0, which is only the choice of the zero point of the parameter on the characteristic to be on the line of Cauchy data. σ is the value of x for t = 0. This shows that the characteristic curves in the x-t-plane are independent of the initial data. They are straight lines with slope 1c . An initial function u(σ ) is transported constantly along these characteristic curves (see Fig. 10.2).

10.2.2 Well-Posed Initial-Boundary-Value Problems If the Cauchy problem is well posed, the differential equation allows to be calculated the value of the wanted field u in a neighborhood of the Cauchy surface (see the

10.2 Hyperbolic Differential Equations and the Method of Characteristics

183

method of characteristics, Sect. 10.2). On the other hand, if Cauchy data are given on a characteristic curve, these Cauchy data contradict in general the differential equation because the differential operator determines the value of the wanted field along the characteristics. The data on the characteristic cannot be arbitrary. The characteristic curves have to intersect the Cauchy surface in order for the wanted field to be calculated along the characteristics. Otherwise, the Cauchy problem is denoted as characteristic. Definition: The Cauchy problem is well posed if the Cauchy data are not characteristic for the differential operator. This can be checked by the following criterion: Theorem: The Cauchy problem, with Cauchy data given on a line in the x-t-plane with unit normal ν with components ν1 and ν2 , is characteristic if and only if the principal part of the differential operator, evaluated with the components of ν instead of the components of the ∇-operator, is zero. The principal part of a differential operator is the part containing the highest derivatives. In our case of quasilinear first-order differential equations with two variables, we have the principal part of the differential operator: L p (∇, x, t, u) = a(u, x, t)∇1 + b(u, x, t)∇2 ,

(10.15)

with the abbreviations ∇1 = ∂t∂ and ∇2 = ∂∂x . The criterion for the Cauchy problem to be characteristic is: a(u, x, t)ν1 + b(u, x, t)ν2 = 0.

(10.16)

 ∂t ∂ x T A tangential vector to the line where Cauchy data are given is ∂σ , ∂σ . Conse∂x ∂t and − ∂σ . quently, a normal vector to the line of Cauchy data has the components ∂σ With (10.16), this leads to the condition: a(u, x, t)

∂x ∂t − b(u, x, t) |x0 ,t0 = 0 ∂σ ∂σ

(10.17)

for the Cauchy data to be characteristic. The question whether an initial-boundary-value problem is well posed can also be considered from a more formal point of view: The method of characteristics (see Sect. 10.2) introduces a coordinate change from the variables x, t to the variables s, σ . This coordinate change is allowed if the Jacobi matrix is regular, i.e., its determinant is non-zero:   ∂t ∂ x a(u, x, t) b(u, x, t) ∂s | |x0 ,t0 = det det ∂s x0 ,t0 ∂t ∂x ∂t ∂ x ∂σ ∂σ

∂σ

∂σ

∂t ∂x − b(u, x, t) |x0 ,t0 = 0. = a(u, x, t) ∂σ ∂σ

(10.18)

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10 Shock Fronts and Hyperbolic Systems of Differential Equations

This is exactly the criterion (10.17) for the surface to be not characteristic. In physical problems, we usually encounter initial value problems. In such an initial value problem, the Cauchy data are given on a line parallel to the x-axis in the x-t plane. In the case of the wave equation, we have seen that the characteristic curves are straight lines in the x-t-plane with slope 1c , independent of the Cauchy data. For this differential equation, the x-axis is not characteristic, and an initial value problem is well posed.

10.3 The Burgers Equation The following first-order partial differential equation: ∂u ∂u +u = π(u) ∂t ∂x

(10.19)

for the wanted field u(x, t), depending on one space variable x and the time t, is denoted in the literature as Burgers’ equation [1]. It can be rewritten in the form of a balance equation:   ∂ 21 u 2 ∂u ∂u ∂ f (u) + = + = π(u) (10.20) ∂t ∂x ∂t ∂x with the constitutive equation for the flux term f (u) = 21 u 2 and a production π(u). In the sense of thermodynamics, this can be interpreted as constitutive functions depending on the state variable, the wanted field u.

10.3.1 Characteristics of the Burgers Equation and Well-Posed Cauchy Problems If a partial differential equation is hyperbolic, like the Burgers equation (10.19) (for a classification of differential equations, see Sect. 9.3.1; symmetry properties of the matrices do not make sense here in the case of one equation for one variable), then there exists always a coordinate transformation to a new coordinate s such that, in terms of the coordinate s, the original partial differential equation transforms to a system of ordinary differential equations with a parametric dependence on a coordinate σ (see Sect. 10.2). The coordinate s is the characteristic coordinate. In our case, we have originally the coordinates x and t. The characteristic coordinate is a certain combination of the coordinates x and t, which we will determine in the following. The curves in the x-t-plane where σ is constant are the characteristics of the considered partial differential equation.

10.3 The Burgers Equation

185

In order to find the characteristic coordinate, assume the wanted field u to be a function u(x(s), t (s); σ ). We want the left-hand side of the differential equation to be a total differential. The chain rule gives: ∂u ∂t ∂u ∂ x du = + = π. ds ∂t ∂s ∂ x ∂s

(10.21)

Comparing this with Burgers’ equation (10.19), we find that this is the case if: ∂t =1 ∂s

⇒ t = s + s0 = s (s0 = 0) ∂x =u ∂s

⇒

∂x = u. ∂t

(10.22) (10.23)

The last equation gives the slope of the characteristic curve ∂∂tx , which depends on the actual value of u. Along the characteristic curve, we have the ordinary differential equation for u: du = π. (10.24) ds In case of the homogenous Burgers equation, π = 0, the solution u of this equation is constant along the characteristic curves. The value on the characteristic curve depends on the initial condition. This set of ordinary differential equations found here is exactly the same set of differential equations to be found from general considerations of a first-order partial differential equation in Sect. 10.2: In the general first-order quasilinear differential equation considered in Sect. 10.2, we have to identify for the homogeneous Burgers equation: a(u, x, t) = 1 b(u, x, t) = u c(u, x, t) = 0,

(10.25) (10.26) (10.27)

and the corresponding set of ordinary differential equations is: dt =1 ds dx =u ds du = 0. ds

(10.28) (10.29) (10.30)

For finite Cauchy data u(x0 , t0 ), the characteristic curves are straight lines in the x-t-plane with a non-zero slope. The x-axis is not a characteristic. Therefore, an initial value problem is well posed in the case of the Burgers equation.

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10 Shock Fronts and Hyperbolic Systems of Differential Equations

10.3.2 Examples of Initial Conditions Let us consider now two examples of initial conditions for the homogeneous Burgers equation (10.31): ∂u ∂u +u =0 (10.31) ∂t ∂x given at time t0 = 0 on the whole line x ∈ R: 1. ⎧ ⎨ u 1 ∈ R for x < 0 u 2 for x > 0 u(x, 0) = ⎩ with u 2 ∈ R < u 1

(10.32)

⎧ ⎨ u 1 ∈ R for x < 0 u 2 for x > 0 u(x, 0) = . ⎩ with u 2 ∈ R > u 1

(10.33)

2.

In both cases, the initial function u(x, 0) is discontinuous. The corresponding characteristic curves in the x-t-plane are shown in Fig. 10.3. As we have seen in general for the homogeneous Burgers equation, the characteristics are straight lines with the slope determined by the initial value of u. In the case of the first initial function (10.32), the slope of the characteristics starting at negative x-values u11 is smaller than the slope of the characteristics starting at positive x-values u12 . Therefore, the two sets of characteristics are intersecting. In the region of intersecting characteristics, the question arises about the value of u because along one characteristic the value is constantly u 1 and along the other intersecting characteristic it is constantly u 2 . The answer to this question will be given by the Hugoniot equation in Sect. 10.5. In the case of the second initial condition (10.33), the slope of the characteristics for negative values of x is larger than the slope of the characteristics for positive x-values. The characteristics are diverging. Therefore, there exists a region in the xt-plane that is not reached by characteristics. There the value of u is not determined by the initial condition. This problem will be analyzed in Sect. 10.6. In the second example with the initial condition (10.33), the characteristics do not intersect and there is a (triangular) region in the x-t-plane whose points are not connected by characteristics with the initial line t = 0, x ∈ R. For all other points (x, t), the value of u(x, t) is equal to the initial value u 1 or u 2 , respectively. The value of u depends on which point on the initial line is the starting point of the corresponding characteristic. For the homogeneous Burgers equation, the value of u is constant along the characteristic, as can be seen from (10.30). For the points in the x-t-plane that are connected with the initial surface by characteristics, there exists a solution to the initial value problem.

10.4 Discontinuous Solutions from Continuous Initial Data

187

Fig. 10.3 For some choice of initial data, characteristics are intersecting. For another choice of initial data, characteristics are diverging, i.e., there is a region in the x-t-plane that is not reached by characteristics

10.4 Discontinuous Solutions from Continuous Initial Data The fact that characteristics can intersect arises not only for discontinuous initial functions, but also for continuous ones. For simplicity, let us consider an example with a continuous, but not continuously differentiable initial function. However, the effect also occurs for smooth initial values. As an example, we consider: u(x, t = 0) =

⎧ ⎨

1 for x < 0 1 − x for 0 ≤ x ≤ 1 ⎩ 0 for x > 1

(10.34)

(see Fig. 10.4). The characteristics for x > 1 are vertical lines. The characteristics starting at x < 0 have slope 1, and the characteristics starting in between have continuously varying slopes. There is an intersection of characteristics, not at time equal to zero, but occurring for the first time at some finite time tc . The right-hand side of Fig. 10.5 shows the other case: The characteristics for the initial distribution shown on the right-hand side of Fig. 10.4. In this case, the characteristics are diverging; there is no intersection in the positive time direction.

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10 Shock Fronts and Hyperbolic Systems of Differential Equations

Fig. 10.4 The solution of Burgers’ equation evolving in time for two different initial conditions

Fig. 10.5 The characteristics corresponding to the initial conditions of Fig. 10.4

10.5 The Hugoniot Equation as Jump Condition at Field Discontinuities In the region of intersecting characteristics, there are two possible values of the wanted field u consistent with the initial condition and the characteristic equation: the one coming from the left-hand side characteristic and the one coming from the right-hand side characteristic. The Hugoniot equation to be proved below determines the line in the x-t-plane that separates the two regions, the value of u being 1 in one region and 0 in the other region in our example.

10.5.1 Weak Solutions At points of discontinuity, the solution u cannot be a solution of the differential equation in the classical sense because it is not differentiable. Let us consider the differential equation: ∂ f 0 (u(x, t)) ∂ f 1 (u(x, t)) + = π, ∂t ∂x

(10.35)

10.5 The Hugoniot Equation as Jump Condition at Field Discontinuities

189

which is the general form of a balance-type differential equation in one space dimension. In case of a single wanted field and one spatial variable, it is always possible to write a quasilinear first-order partial differential equation in such a form. In case of a system of differential equations, integrability conditions have to be fulfilled to ensure such a balance-type form. u is called a weak solution of the differential equation (10.35) if:  ∂φ(x, t) ∂φ(x, t) − f 0 (u(x, t)) − f 1 (u(x, t)) − π φ(x, t) dxdt = 0 ∂t ∂x G⊂R2 (10.36) for any test function φ(x, t) being an element of a suitable set of test functions to be chosen; here: φ ∈ C0∞ (G), (10.37)

the set of analytical functions with a compact support within the region G, where the balance equation is considered. Due to the compact support of the test functions, they are zero at the boundary of region G, where the balance equation (10.35) holds.

10.5.2 The Hugoniot Condition Let us consider now a region G ∈ Rx × Rt that is separated into two subregions G + and G − by a surface of discontinuity  (see Fig. 10.6). In the interior of each subregion u(x, t) is a classical solution. Starting from the balance equation (10.35), multiplying by a test function, integrating over the regions G + and G − , respectively, and partial integration gives:

  ( f 0 (u(x, t))φ(x, t)) νt+ + ( f 1 (u(x, t))φ(x, t)) νx+ dl

 ∂φ(x, t) ∂φ(x, t) − f 0 (u(x, t)) + − f 1 (u(x, t)) − π φ(x, t) dxdt = 0 ∂t ∂x G+ (10.38) ∂G +

Fig. 10.6 A region divided by a surface of discontinuity  into two subregions G + and G −

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10 Shock Fronts and Hyperbolic Systems of Differential Equations

  ( f 0 (u(x, t))φ(x, t)) νt− + ( f 1 (u(x, t))φ(x, t)) νx− dl ∂G −

 ∂φ(x, t) ∂φ(x, t) − f 0 (u(x, t)) − f 1 (u(x, t)) − π φ(x, t) dxdt = 0. + ∂t ∂x G− (10.39) Here ν + is the outward unit normal vector to the boundary ∂G + of the region G + , with components νt+ and νx+ . Analogously, ν − is the outward unit normal vector to the boundary ∂G − of the region G − , with components νt− and νx− . dl is the line element on the boundary. The boundary ∂G + consists of the parts G + 0 and , which is the surface of discontinuity (see Fig. 10.6), and the boundary ∂G − consists of the parts G − 0 and . As the support of the test functions φ is within the region G, − + the boundary terms on G + 0 and G 0 vanish. The outward unit normal for G on the surface of discontinuity is oriented opposite to the outward unit normal for G − : νt+ = −νt− =: νt νx+

=

−νx−

=: νx .

(10.40) (10.41)

On the other hand, we require u to be a weak solution: 

∂φ(x, t) ∂φ(x, t) − f 0 (u(x, t)) − f 1 (u(x, t)) − π φ(x, t) dxdt = 0 ∂t ∂x G+ (10.42)

 ∂φ(x, t) ∂φ(x, t) − f 0 (u(x, t)) − f 1 (u(x, t)) − π φ(x, t) dxdt = 0. ∂t ∂x G− (10.43)

Adding (10.38) and (10.39), and subtracting (10.42) and (10.43) by using (10.40) and (10.41), gives the wanted jump condition in integral form:



f 0 (u + (x, t))νt + f 1 (u + (x, t))νx  − f 0 (u − (x, t))νt − f 1 (u − (x, t))νx φ(x, t)dxdt = 0 



(|[ f 0 (u(x, t))]|νt + |[ f 1 (u(x, t))]|νx ) φ(x, t)dxdt = 0,

(10.44)

where u + (x, t) is the value of the wanted field at the surface of discontinuity  taken at the side of G + , and u − (x, t) is the value at  at the −-side. The bracket |[...]| denotes the jump at the surface of discontinuity: |[u]| := u + − u − .

(10.45)

10.5 The Hugoniot Equation as Jump Condition at Field Discontinuities

191

The integral jump condition (10.44) holds for any subset of the surface of discontinuity  and any test function φ. Therefore, the integrand must be zero, and it follows the local form of the Hugoniot equation: |[ f 0 (u(x, t))]|νt + |[ f 1 (u(x, t))]|νx = 0.

(10.46)

In case of three spatial variables (x → x), the component νx has to be replaced by the (spatial) normal vector ν x . The time component of the normal vector ν t is the surface velocity s. This can be seen by giving the surface a representation in implicit form (x, t) = 0. The , ∂ )T . With surface normal is proportional to the gradient of : (νt , νx )T ∼ ( ∂ ∂t ∂x ∂  = x − c · t, the change of  in time ∂t is the negative surface velocity s ((νt , νx ) = (−s, 1)), and the local Hugoniot condition can be written as: s|[ f 0 (u(x, t))]| − |[ f 1 (u(x, t))]| = 0.

(10.47)

Weak solutions with a surface of discontinuity are called shocks if at the surface of discontinuity the Hugoniot condition is fulfilled.

10.5.3 The Hugoniot Condition Applied to the Burgers Equation We have written the Burgers equation in the form of a balance equation: ∂ 1 u2 ∂u + 2 = π. ∂t ∂x

(10.48)

The corresponding jump condition at surfaces of discontinuity reads: 1 s|[u]| − |[u 2 ]| = 0. 2

(10.49)

This equation determines the velocity s of the surface of discontinuity in terms of the jump of the fields. For the initial conditions (10.32) and (10.33), the +-region is the positive x-axis, and the −-region is the negative x-axis. For initial condition (10.32) with u − = 1, u + = 0, we have: s=

1 |[u 2 ]| 1 02 − 12 1 = = . 2 |[u]| 2 0−1 2

For initial condition (10.33) with u − = 0, u + = 1, we have:

(10.50)

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10 Shock Fronts and Hyperbolic Systems of Differential Equations

s=

1 12 − 02 1 1 |[u 2 ]| = = . 2 |[u]| 2 1−0 2

(10.51)

In both cases, the line of discontinuity in the x-t-plane is a straight line with slope 21 .

10.6 Two Uniqueness Theorems in the Case of Diverging Characteristics In the case of the initial condition (10.33) with diverging characteristics, we can construct two solutions of the homogeneous Burgers equation: 1. A classical (continuous) solution: u 1 (x, t) =

⎧ ⎨ ⎩

x t

0 for x < 0 for 0 < x < t , 1 for x > t

(10.52)

which fulfills the differential equation everywhere, except for the fact, that it is not differentiable in the points x = 0 and x = t, but it is continuous there. 2. A discontinuous solution:  0 for x < ts = 21 t u 2 (x, t) = , (10.53) 1 for x > ts = 21 t which is a classical solution where it is differentiable and fulfills the Hugoniot equation at the discontinuity. Therefore, both solutions are solutions in the weak sense. In the following two sections, two criteria selecting the solution that occurs in nature are given.

10.6.1 The Lax Criterion The Lax criterion [2] is the following statement: A discontinuity occurs only in the case of characteristic curves intersecting in the positive time direction. In our example, it follows that the solution is the continuous one and no shock occurs. The Lax criterion can be stated as: The discontinuity occurs only if this is caused inevitably by the initial conditions.

10.6 Two Uniqueness Theorems in the Case of Diverging Characteristics

193

10.6.2 The Entropy Condition The entropy condition is another criterion selecting a unique solution among the two solutions in the example in Sect. 10.6. In general, the Lax criterion and the entropy criterion are not equivalent, but the Lax criterion is more restrictive [3]. Only in the limiting case of small jumps at the surface of discontinuity, it can be shown that the Lax criterion and the entropy criterion are equivalent. Consider the following balance-type equation: ∂ f (u) ∂u + = 0. ∂t ∂x

(10.54)

Assume that there exists a concave function U (u) of the wanted field u, which also fulfills a conservation equation in regular points of the continuum: ∂U (u) + ∇ · F(u) = 0. ∂t

(10.55)

Definition: The field u fulfills the entropy condition if, at surfaces of discontinuity (with surface velocity s, given by the Hugoniot equation), it holds: − s|[U ]| + |[F]| · n ≥ 0.

(10.56)

Remarks: 1. The inequality (10.56) can be interpreted as the quantity U increasing at the surface of discontinuity. 2. Mathematically, the existence of an arbitrary concave quantity, being conserved in regular points and increasing at surfaces of discontinuity, is required. Physically, the entropy density can be taken as the quantity U with the inequality (10.56) interpreted as entropy inequality. This implies the assumption that the entropy density is a concave function in all its variables. 3. For one partial differential equation and one wanted field u, there exists always the concave function U = − 21 u 2 , for which the conservation equation follows: ∂(− 21 u 2 ) ∂ F(u) + = 0, ∂t ∂x ∂ F(u) ∂ f (u) = . with − u ∂u ∂u

(10.57) (10.58)

For systems of more partial differential equations and more wanted fields, the construction of an additional conservation equation requires some integrability conditions to be fulfilled.

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10 Shock Fronts and Hyperbolic Systems of Differential Equations

10.6.3 Application of the Entropy Criterion to the Burgers Equation Multiplying Burgers equation with (−u), we can derive the following additional conservation equation, valid in regular points of the continuum:     ∂ − 13 u 3 ∂ − 21 u 2 + = 0. ∂t ∂x

(10.59)

The concave quantity, which is conserved in regular points, is U (u) = − 21 u 2 . The entropy condition at surfaces of discontinuity reads: − s|[U (u)]| + |[F(u)]| ≥ 0       1   1 s  u 2  −  u 3  ≥ 0. 2 3

(10.60)

If a weak solution of the Burgers equation satisfies the inequality (10.60), it is called an allowed shock solution. The entropy criterion states that only this kind of solutions occurs in nature. For the two different initial conditions, we have:

10.6.3.1

Case 1: Intersecting Characteristic Curves

In the example (10.34): u(x, t = 0) =

⎧ ⎨

1 for x < 0 1 − x for t ≤ x ≤ 1 ⎩ 0 for x > 1

(10.61)

we have found the solution u(x, t) =

⎧ ⎨ ⎩

1−x 1−t

1 for x < t for 0 ≤ x ≤ 1 for t < tc = 1 , 0 for x > 1

(10.62)

which is a classical one for times smaller than a critical time and, after the critical time, a discontinuous (weak) solution with the velocity of the line of discontinuity s = 21 given by the Hugoniot condition:  u(x, t) =

1 for x < ts + s = 21 t + 0 for x > ts + s = 21 t +

1 2 1 2

.

(10.63)

It is easily seen that the discontinuous solution fulfills the entropy condition:

10.6 Two Uniqueness Theorems in the Case of Diverging Characteristics

      1   1 s  u 2  −  u 3  2 3   1 1 2 1 1 1 = u R − u 2L + − u 3R + u 3L 2 2 2 3 3   1 1 1 1 2 1 2 1 0 − 1 + − 03 + 13 = > 0, = 2 2 2 3 3 12

195

(10.64)

where u R denotes the solution on the right-hand side and u L the solution on the left-hand side. This solution satisfies the entropy inequality. Therefore, the discontinuity occurs after the time tc .

10.6.3.2

Case 2: Non-intersecting Characteristic Curves

Consider the initial condition from the beginning of Sect. 10.6: u(x, t = 0) =

⎧ ⎨

0 for x < 0 x for 0 ≤ x ≤ 1 . ⎩ 1 for x > 1

(10.65)

The corresponding characteristic curves are diverging. For later times, there exists a continuous classical solution and a discontinuous weak solution. Applying the entropy condition to the discontinuous solution, we find:       1   1 s  u 2  −  u 3  2 3   1 1 2 1 1 1 = u R − u 2L + − u 3R + u 3L 2 2 2 3 3   1 1 1 1 2 1 2 1 1 − 0 + − 13 + 03 = − < 0, = 2 2 2 3 3 12

(10.66)

which does not fulfill the entropy condition. Therefore, in this case, the solution to the initial value problem is the continuous classical one. A discontinuity occurs only in the case of intersecting characteristics.

10.7 A Physical Example The flux term in the Burgers equation is of the form of the convective flux in the balance of momentum. The balance of momentum

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10 Shock Fronts and Hyperbolic Systems of Differential Equations

  ∂ ( v) + ∇ · v v − t − f = 0 ∂t

(10.67)

is considered in the special case of an incompressible fluid (i.e., ∇ · v = 0 and = const.) and vanishing stresses and forces. In one space dimension, we end up with ∂ ( v) + v · ∇v + v∇ · v = 0 ∂t ∂ v + v · ∇v = 0 , ∂t

(10.68) (10.69)

and in one dimension: ∂v ∂v +v =0, ∂t ∂x

(10.70)

which is Burgers’ equation.

10.8 Summary In the beginning of this section, we have shown how the method of characteristics allows a partial differential equation to be transformed into a system of ordinary differential equations. For one first-order partial differential equation, this is always possible, supposing that the Cauchy data are not characteristic. For higher order partial differential equations or systems of partial differential equations, the method of characteristics for solving the differential equation(s) works only if the equations form a hyperbolic system, which means that certain conditions must be fulfilled by the differential equations. The classification of systems of partial differential equations can be found in [4]. Especially interesting are so-called symmetric hyperbolic systems with a convex extension (see [2, 5–8]). In Chap. 9, we show that rational extended thermodynamics always leads to such systems of differential equations and, therefore, to finite speeds of propagation of disturbances. In the last section, we have treated a special balance-type differential equation in one dimension, the Burgers equation. With the method of characteristics, we have investigated the behavior of weak solutions of this model differential equation that are discontinuous. We have shown how such a discontinuity can evolve from the initial data. For the velocity of the surface of discontinuity, we have derived the Hugoniot condition. Finally, for certain initial conditions, there exists more than one weak solution of the Burgers equation. The entropy condition is the criterion ruling out which of these solutions occurs in nature. It turns out that a discontinuity occurs only if characteristics are intersecting in the positive time direction. For an application of a generalized Burgers equation, see [9].

References

197

References 1. J.M. Burgers, Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. Verhandelingen der Koninklijke Nederlandse Akademie van Wetenschappen 17(2), 1–53 (1939) 2. P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (Society for Industrial and Applied Mathematics, Bristol, 1973) 3. W. Dreyer, S. Seelecke, Entropy and causality as criteria for the existence of shock waves in low temperature heat conduction. Contin. Mech. Thermodyn. 4, 23–36 (1992) 4. R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. II (Interscience Publishers, New York, London, 1962) 5. M.G. Boillat, Sur l’ existance et la recherche d’équations de conservation supplémentaires pour les systèmes hyperboliques. C. R. Acad. Sc. Paris 278 A, 909–913 (1974) 6. K.O. Friedrichs, P.D. Lax, Systems of conservation equations with a convex extension. Proc. Nat. Acad. Sci. USA 68, 1686–1688 (1971) 7. P. Lax, Shock waves and entropy, in Proceedings of a Symposium Conducted by the Mathematics Research Center, Contributions to Nonlinear Functional Analysis, ed. by Zarantonello (Academic Press, The University of Wisconsin, 1971), pp. 603–634 8. T. Ruggeri, Symmetric-hyperbolic system of conservative equations for a viscous heat conducting fluid. Acta Mech. 77(3), 167–183 (1983) (Review Article) 9. V. Ciancio, L. Restuccia, The generalized Burgers equation in viscoanelastic media with memory. Phys. A 142, 309–320 (1987)

Chapter 11

A Short Survey of Thermodynamics of Material Surfaces

Abstract In this chapter, densities of physical quantities, like mass, momentum, and energy are attributed to the surface, and there are fluxes within the surface. The curvature of the surface enters into the balance equations for the surface quantities. The surface quantities are coupled to the surrounding bulk medium by jump terms of the bulk fields. These jump terms are the only terms present in the case of an idealized mathematical surface, where they fulfill the Hugoniot jump condition (see the preceding chapter).

11.1 Introduction In Chap. 10, when deriving the Hugoniot jump condition, we have allowed for discontinuities of the field quantities but it is supposed that the values of all fields remain finite. This implies that integrals of field quantities over infinitely thin layers vanish. However, there are many situations, where physical quantities are localized on thin surfaces: in polarizable media in the electric field, there exist surface charges, in membranes (for example, cell membranes and other biological membranes, see, for instance, [1–3], or rubber balloons [4]), and thin films (for instance, LangmuirBlodget films, or adsorption layers on the surface of catalysts) matter is concentrated in a layer that is sometimes only a few molecular diameters thick. The surface layer with its influence on the orientation of liquid crystal molecules is of great importance in liquid crystal devices [5–8]. These systems are modeled as mathematical surfaces without any thickness. The volume densities of the field quantities are of δ-type, localized at the surface. The volume density of mass, for instance, reads: (x, t) = 0 (t)δ(x s (t) − x),

(11.1)

where x s is a point on the surface. The position of the surface can depend on time. The δ-distribution is normalized:  (11.2) δ(x s − x)d 3 x = 1.

© Springer Nature Switzerland AG 2020 C. Papenfuß, Continuum Thermodynamics and Constitutive Theory, https://doi.org/10.1007/978-3-030-43989-7_11

199

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11 A Short Survey of Thermodynamics of Material Surfaces

Other examples modeled the same way are phase boundaries, for example, the boundary between ice and water [9] or the boundary between the isotropic and the nematic phase. In these cases, the surface quantities are a mathematical model for a spatial distribution of the corresponding field with gradients that are orders of magnitude larger perpendicular to a surface s than in directions tangential to the surface. Surface mass is assigned to the surface to balance the difference between the extrapolated bulk mass and the mass density in the real system (see Sect. 11.2). Other physical surface quantities are introduced analogously. The value of the surface quantities depends on the definition of the surface position. This concept of surface quantities traces back to the work of Gibbs [10], and has been applied to phase boundaries in simple liquids (see, for instance, [11–14]), and also to gas-liquid and solid-liquid interfaces [15, 16]. For a description of the Gibbs method of excess quantities, see also [17]. In the book by Rusanov [17], you also find a huge number of applications of the theory of phase boundaries and films, for instance, phase equilibria, adsorption phenomena, temperature dependence of surface tension, and chemical reactions involving phase boundaries. Throughout the book by Rusanov, the variables are the equilibrium variables, temperature, pressure, and chemical composition, as it is assumed for local equilibrium states. The method applied there is irreversible thermodynamics. Numerous other applications of irreversible thermodynamics methods to the surface of discontinuity as a separate thermodynamic system can be found f.i. in [18, 19] In general, the description in terms of a mathematical surface of discontinuity of the bulk fields is an approximation, where no structure perpendicular to the surface is taken into account. However, we will present here only the balance equations, but we will not exploit the dissipation inequality on the surface. Generally, in this chapter, there are not only surface densities but also production and supply densities localized on the surface and fluxes tangential to the surface.

11.2 Definition of the Surface Fields Let  be a quantity defined in the bulk, for which we have an equation of motion: ∂ ˇ = F[] ∂t

(11.3)

ˇ with some functional F[]. In the +-region and in the −-region of the bulk without an adjacent surface of discontinuity (see Fig. 11.2), we have: ∂ ± = Fˇ ± [ ± ], ∂t

(11.4)

i.e.,  + and  − are the solutions of the differential (11.3) if no phase boundary is present. Now we define the surface fields s as:

11.2 Definition of the Surface Fields

201

singular sur face Ψ

-

Ψs =

Ψ −Ψ

Ψ

+ Ψ

G-

Ψs =

+ Ψ −Ψ

xs

G+

x

Fig. 11.1 Definition of the surface fields:  + and  − are the extrapolated bulk solutions,  is the solution in the presence of the surface (located at xs ). The difference between  ± and  defines the surface quantity s

s (r, t) = (r, t) −  − (r, t)− (r, t) −  + (r, t)+ (r, t).

(11.5)

(r, t) is the field in the presence of the surface, whereas  + and  − are the extrapolated bulk solutions. + and − are the Heaviside step functions for the +- and −-region, respectively. This way, s is defined not only on the singular surface but also in a surface region and in the bulk. This is important, if one wants to consider surface structure perpendicular to the surface [15], [11]. We will need the definition of the surface fields (11.5) only on the singular surface. For the mass density, this definition coincides with the definition of the excess mass introduced by Gibbs [10]. The value of the surface quantities depends on the definition of the surface position. The definition of the surface quantities is sketched in Fig. 11.1. The definition (11.5) has been introduced to consider the structure of a surface region. For this purpose, a multipole expansion of the surface excess densities s around the δ-distribution on the surface is written down, and equations of motion for the multipole moments are derived [11, 15]. The exact equations of motion are approximated by neglecting higher order multipoles. The equations for the higher order multipoles result in boundary conditions for the bulk equations. The equations of motion for the lowest order moments are the equations of motion for the surface quantities. These are the same equations as we will discuss in this chapter.

11.3 Geometry of the Surface The considered surface divides the bulk into the +-region and the −-region. In general, the surface is curved, and therefore we use a curvilinear system of surface coordinates and a Cartesian system in R3 (see Fig. 11.2b)). The distinction between these two coordinate systems is most clearly seen in index notation, which is therefore

202

11 A Short Survey of Thermodynamics of Material Surfaces

a)

e surface

s

bulk + bulk −

e b) C

τ1

τ2

1 ξ = const .

ξ 2 = const .

xs

Fig. 11.2 a The surface ∫ divides the bulk into the +-region and the −-region. b Definition of the quantities characterizing the surface geometry

used throughout this chapter. Tensor components referring to the curvilinear surface coordinate system will be denoted by Greek indices, and components with respect to the Cartesian coordinate system by Latin indices. A superscript “⊥” denotes a vector or tensor projected orthogonal to the surface. Summation convention applies to Latin and Greek indices. e is the unit normal vector to the surface, pointing toward the +-region. τ 1 and τ 2 are the tangential vectors to the surface coordinate lines ξ 2 (x) = constant, ξ 1 (x) = constant (see Fig. 11.2b): ταk =

∂χk k ∈ {1, 2, 3}, α ∈ {1, 2}. ∂ξ α

(11.6)

χ(ξ α ) = 0 (α ∈ {1, 2}) is the representation of the surface in R3 . The notation used here is essentially the same as in [9]. An intrinsic characterization of surface geometry is given by the metric tensor g with gαβ = ταk τβk . The second fundamental form (curvature tensor) is denoted with b with: k = −ekβ ταk = −ekβ ταk , (11.7) bαβ = ek τα,β and the mean curvature is: KM =

1 tr(b). 2

(11.8)

11.3 Geometry of the Surface

203

The covariant derivative of the Levi-Civita connection is denoted by “;” and μ

φα;β = φα,β − αβ φμ ,

(11.9)

μ

where αβ are the Christoffel symbols: σ μν =

 1 σα  k k k k k k k τμ,ν τα + τμk τα,ν . + τν,μ ταk + τνk τα,μ − τμ,α τνk − τμk τν,α g 2

(11.10)

The covariant derivatives of co- and contravariant vectors and tensors are given by: μ

vα;β = vα,β − αβ vμ v α;β

=

Tαβ;γ = αβ T ;γ

=

α μ + βμ v μ Tαβ,γ − αγ Tμβ

(11.11)

v α,β

T αβ ,γ

+

α γμ T μβ

(11.12) μ βγ Tαμ

(11.13)

β γμ T αμ .

(11.14)

− +

The balance equations will be given not only in components with respect to coordinates (surface coordinates and Cartesian coordinates in R3 ) but also in invariant notation. The scalar product of vectors a, b ∈ R3 will be denoted by a · b. Scalar products of vectors tangential to the surface ∫ (at , bt ∈ Tx s s) are given by: at ∗ bt = a α gαβ bβ .

(11.15)

In invariant notation, the covariant derivative will be denoted by \∇. The velocity of the surface is defined by:       dχk ξ α  , t , t ∂χk ∂ξ α  , t ∂χk (ξ α , t) = α + = w = dt ∂ξ ∂t ∂t = ταk w α + w ⊥ ek k

with the tangential vectors: ταk =

∂χk . ∂ξ α

(11.16) (11.17)

(11.18)

The vector w corresponds to the scalar surface velocity s in the case of one space dimension in Chap. 10. Capital letters (for instance  ) refer to the reference configuration, and small letters (for instance ξ α ) to the actual configuration at time t: ξ is a position on the surface in the actual configuration, and  is the corresponding point in the reference configuration. Now the component of the velocity w tangential to the surface is chosen as the tangential component of the material velocity on the surface of discontinuity. This is a special choice of the surface coordinate system, i.e., of the parametrization. Then,

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11 A Short Survey of Thermodynamics of Material Surfaces

the first term of the sum in (11.16) that is the tangential component of the material velocity is the tangential component of the velocity w, and the second term can be identified with the normal component: ∂ξ α ( , t) , ∂t ∂χl (ξ α , t) l w⊥ = e. ∂t wα =

(11.19) (11.20)

In the following, the surface coordinate system is always chosen this way: surface coordinate lines are particle trajectories, i.e., within the surface, we refer to a material coordinate system. From here, the time derivatives of the tangential and normal vectors at a fixed actual surface coordinate can be expressed by the surface velocity: ∂ k ∂ ∂χk ∂ ∂χk τα = = ∂t ∂t ∂ξ α ∂ξ α ∂t   ∂ = α w ⊥ ek = w ⊥,α ek + w ⊥ ek,α = w ⊥,α ek − w ⊥ bαβ τ kβ , (11.21) ∂ξ where the definition of the second fundamental form (11.7) and (11.20) have been used. Due to the fact that the tangential vectors are orthogonal to the normal vector, this also gives the time derivative of the unit normal vector: ταk ek = 0, therefore: ταk

∂ek ∂τ k = −ek α . ∂t ∂t

(11.22)

(11.23)

As the time derivative of the normal vector has a tangential component only, we have (with τ kα ταl = δ kl ) ∂el ∂τ l ∂ek = τ kα ταl = −τ kα el α = −w ⊥,α τ kα . ∂t ∂t ∂t

(11.24)

In the last identity, (11.21) has been used. Also, the time derivatives of gradients of the basis vectors of the tangential space and of the unit normal vector are related to the velocity: k ∂e,β

∂t

 =

∂ek ∂t

 ,β

  ⊥ kα = −w ⊥,α τ kα ,β = −w ⊥,αβ τ kα − w,α τ ,β ,

(11.25)

11.3 Geometry of the Surface k ∂τα,β

∂t

205

 ⊥ k  = w,α e − w ⊥ bαγ τ kγ ,β ⊥ ⊥ k k = w,αβ ek + w,α e,β − w ⊥,β bαγ τγk − w ⊥ bαγ,β τ kγ + w ⊥ bαγ τγ,β .

(11.26)

The relations (11.21) to (11.26) show that for certain choices of the state space not all variables are independent of each other.

11.4 Surface Balance Equations The global balance equation for an additive quantity s within the time-dependent subregion G(t) of the surface reads [9, 14, 20]:  ∂ d ψs (ξ α , t)da s = dt ∂t G(t)     =− φαs h α ds − |[φk + ψ v k − w k ]|ek da C(s) G(t)  + (πs + σs ) da,

(11.27)

G(t)

ψs is the corresponding surface density (quantity per unit area; in terms of a volume density, it is a “δ”-like quantity, localized at the surface), φs is the flux through the boundary curve C(s) (see Fig. 11.3), and πs and σs are surface production and supply density. h is the unit normal vector in the tangential plane to the surface ∫ and perpendicular to the curve C(s). φ is a non-convective bulk flux of the density ψ defined in the bulk. The brackets |[ ]| denote the difference between the limiting values φ+ and φ− of the bulk fields on both sides of the surface, as introduced already in Chap. 10 in the context of the Hugoniot equation: |[φ]| := φ+ − φ− .

(11.28)

The discontinuity of the bulk fields contributes to the balances of the surface quantities as an additional production. As the surface is in general non-material (v = w), there is also a convective flux contribution |[ψ(v k − w k )]|ek due to the discontinuity (see (11.27)). Using the transport theorem on the curved surface [9]: d dt





 ψs da = G(t)

G(t)

 ∂ψs α ⊥ + (ψs w );α − 2ψs K M w da ∂t

(11.29)

and Gauss’ theorem in curvilinear coordinates with the covariant derivative \∇ · s = αs ;α :

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11 A Short Survey of Thermodynamics of Material Surfaces

Fig. 11.3 Definition of the quantities in the surface balance: a πs and σs are production density and supply density of ψs within the surface, φs is the non-convective flux tangential to the surface. b The bulk flux is discontinuous at the surface s

h

a) . C πs

φs

σs

bulk +

b) surface

φ+

s

bulk − φ-

 C(s)

φαs h α ds

 = G(t)

φαs;α da

(11.30)

with the unit vector h in the surface and normal to the curve C(s); pointing outward from the region G, one obtains the local balance equation:     ∂ψs − 2K M w ⊥ ψs + φαs + ψs w α ;α = −|[φk + ψ v k − w k ]|ek + πs + σs ∂t (11.31) from the global one. Now the quantities in this abstract balance equation have to be identified with physical quantities. We will show here the balance equations of a micropolar medium, i.e., we allow for an internal angular momentum. This occurs, for example, due to particle rotations in liquid crystals. One could encounter also such an internal angular momentum due to rotations of electric dipoles in dipolar media. Then, the quantities ψs have to be identified with either mass, momentum, energy, or angular momentum. The respective identifications of the fluxes are given in Table 11.1. In contrast to a simple fluid, we have couple forces m, couple stresses , and a spin density s (Table 11.2). The resulting balance equations are: Balance of Mass ∂ s − 2s K M w ⊥ + (s w α );α = −]|(v m − w m )|[em , ∂t

(11.32)

11.4 Surface Balance Equations

207

Table 11.1 Identification of the quantities in the abstract balance with physical quantities Balance of ψs φs πs σs ψ φ Mass Momentum Angular momentum Energy

s s w s (x s × w +ss ) s (es + 1/2w · w + 2θ1eff ss · ss )

0 −t s −x s × t s −s qs − w · ts 1 − θeff ss · s

0 0 0 0

0 s f s s (x s × f s +ms ) s ( f s · w 1 + θeff ss · ms )

 v (x × v +s) (e+ 1/2v · v+ 1 2θe f f s · s)

0 −t −x × t − q−v·t 1 − θeff s·

Table 11.2 List of the symbols used in the balance equations Symbol Physical quantity s v ts t f ss s ms s  es e qs q θeff

Surface mass density Material velocity in the bulk Surface stress tensor Stress tensor defined in the bulk Acceleration due to external fields Spin per unit mass defined on the surface Surface couple stress tensor Surface couple forces Spin density in the bulk Bulk couple stress tensor Surface specific internal energy Specific internal energy in the bulk Surface heat flux Bulk heat flux Effective moment of inertia of the particles

Balance of Momentum ∂ (s w k ) − 2s w k K M w ⊥ + (s w k w α − tskα );α ∂t = −]| − t km + v k (v m − w m )|[em + s f k , Balance of Angular Momentum    ∂   klm l m s  xs w + ssk − 2K M w ⊥ s klm xsl w m + ssk ∂t

(11.33)

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11 A Short Survey of Thermodynamics of Material Surfaces

  α klm l m  − klm xsl tsmα + kα xs w + ssk ;α s − w s     = |[klm xsl t mn + kn +  klm xsl w m + ssk v n − w n ]|en   +s klm xsl f m + m ks .

(11.34)

x s is the position vector of a surface element, and  is the totally antisymmetric tensor of third order. Balance of Energy ∂ ∂t

   1 m m 1 1 m m s es + w w + s s 2 2 θeff s s   1 m m 1 m m ⊥ −s 2K M w es + w w + s s 2 2θeff s s     1 k kα 1 m m 1 m m α k kα + q s − w ts − s  + s es + w w + s s wα θeff s s 2 2θeff s s ;α 1 = |[q k − v m t mk − s m mk θeff    1 m m 1 m m  k v − w k ]|ek + e + v v + s s 2 2θeff   1 k k k k (11.35) +s f s w + s m + rs . θeff s s

θeff is an effective specific moment of inertia, obtained by averaging over particles. It has been introduced in order to write the rotational part of the kinetic energy in terms of the specific spin only instead of the spin density and a rotational velocity. Into all these balances, the curvature enters explicitly by the mean curvature K M and implicitly by the covariant derivative. The term including the mean curvature is due to the fact that the surface area changes by mapping the curved surface with a velocity field perpendicular to the surface (see Fig. 11.4). The balance equations of the surface quantities are coupled to those of the bulk quantities through the jump terms |[ϕ]| := ϕ+ − ϕ− , ϕ+ and ϕ− being the limiting values of the bulk fields on both sides of the surface. The discontinuity of the bulk fields plays the role of an additional production term in the surface balances.

Fig. 11.4 Mapping the curved surface ∫ with w ⊥ , the surface area changes

w

⊥

w

⊥

11.4 Surface Balance Equations

209

The vector product between the balance of momentum and the position vector is subtracted from the balance of angular momentum. We end up with the balance of spin. The vector product between the balance of momentum and the position vector on the surface x s reads: ∂  klm l m  s  xs w − 2s klm xsl w k K M w ⊥ ∂t + (s klm xsl w m w α − klm xsl tsmα );α   = |[klm xsl t mn + v m (v n − w n ) ]|en + s f m klm xsl − klm

l ∂xsl mα klm ∂x s t −  s w m w α . s ξα ∂ξ α

(11.36)

Subtracting this from the balance of angular momentum with: ∂xsl m α w w = klm ταl w m w α = kαβ w α w β + kα⊥ w ⊥ w α = kα⊥ w ⊥ w α , ∂ξ α (11.37) we obtain the balance of spin: klm

∂ (s ssk ) − 2K M w ⊥ s ssk + (s ssk w α − kα s );α ∂t = |[km − s k (v m − w m )]|em + s m ks + klm ταl tsmα + kα⊥ s w ⊥ w α . (11.38) Similarly, the balance of internal energy results from the balance of energy, the balance of momentum, and the balance of angular momentum:  k  ss ∂ α (es s ) − 2K M w ⊥ s es + qs;α − wsk ;α tskα − kα + (s es w α );α ∂t θeff ;α s   1 k 1 k = −|[q m − (v k − w k )t km + s − ss km + e(v m − w m ) θeff θeff  k + (v − w k )(v k − w k )(v m − w m ) 2  (s k − ssk )(s k − ssk )(v m − w m )]|em + 2θeff 1 k 1 +s rs − klm ταl tsmα s − kα⊥ w ⊥ w α s s k . (11.39) θeff s θeff s In invariant notation, the balance equations are (an index t denotes a vector projected onto the tangential plane to the surface, T denotes the transposed of a secondorder tensor, T r∗ is the trace with the surface metric, ∗ is the tangential scalar product with the surface metric, and \∇ is the covariant derivative):

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Balance of Mass ∂ s − 2s K M w · e + \∇ ∗ (s wt ) = −|[(v − w)]| · e, ∂t

(11.40)

Balance of Momentum   ∂ (s w) − 2s K M ww · e + \∇ ∗ s wwt − tstT = |[t − v(v − w)]| · e + s f , ∂t (11.41) Balance of Spin   ∂ (s ss ) − 2K M s ss w · e + \∇ ∗ s wt ss − stT ∂t = |[ − s(v − w)]| · e + s ms + T r∗ (τ t × t st ) + s (w × e) w · e, (11.42) Balance of Internal Energy ∂ (es s ) − 2K M w · es es ∂t       1 \∇ ∗ q st − T r∗ \∇ws ∗ t sT − T r∗ \∇ ss ∗ sT + \∇ ∗ (s es w) θeff 1 = −|[q − (v − w) · t + (s − ss ) ·  + e(v − w) θeff   + (v − w) · (v − w)(v − w) + (s − ss ) · (s − ss )(v − w)]| · e 2 2θeff 1 1 +s rs − T r∗ (τ t × t s ) · ss − w · e (s × w t ) · e. (11.43) θeff θeff The balances of momentum and of spin can be decomposed into components tangential and perpendicular to the surface. As an example, this is shown for the balance of momentum. Using (11.7), (11.21), and (11.24), the decompositions are:  ∂  ⊥ k ∂w k = w e + w γ τγk ∂t ∂t ∂τγk ∂w γ k ∂ek ∂w ⊥ k e + w⊥ + τγ + w γ = ∂t ∂t ∂t ∂t γ   ∂w ∂w ⊥ k e − w ⊥ g μν w ⊥,ν τμk + τγk + w γ w ⊥,γ ek − w ⊥ bγβ τβk = ∂t ∂t  ⊥  ∂w k γ ⊥ + w w,γ =e ∂t   ∂w β k ⊥ βν ⊥ γ ⊥ β (11.44) − w w bγ , +τβ −w g w ,ν + ∂t

11.4 Surface Balance Equations

211

  k w;α = w ⊥ ek + w γ τγk ;α

γ

k = w ⊥,α ek + w ⊥ ek;α + w ;α τγk + w γ τγ;α    k  k γ μ + w γ τγ,α = w ⊥,α ek + w ⊥ e;α + τγk w γ,α + w μ αμ − τμk γα

= w ⊥,α ek − w ⊥ bαβ τβk + τγk w γ,α + w γ ek bγα     = ek w ⊥,α + w γ bγα + τβk −w ⊥ bαβ + w β,α . Transforming \∇w the identity

k = −bαβ τβk e;α

(11.45)

(11.46)

has been used. Finally, for the divergence of the stress tensor we have:   kα = ts⊥α ek + tsγα τγk ;α ts;α

γα

k ⊥α k k γα k = ts⊥α ;α e + ts e;α + ts ;α τγ + ts τγ;α   ⊥α α ek − ts⊥α bαβ τβk = ts ,α + ts⊥μ αμ  γα  γ μα α γμ ν + ts ,α + αμ ts + αμ ts τγk + tsγα bγα ek − γα τνk tsγα   γα = ek ts⊥α ,α + ts bγα   +τγk −ts⊥α bαγ + tsγα,α .

(11.47)

We get the balance equation of the tangential component of momentum:   ∂s w β + s −w ⊥ g βν w ⊥,ν − w γ w ⊥ bγβ − 2s K M w ⊥ w β ∂t + s,α w α w β + s w α,α w β   + s w α −w ⊥ bαβ + w β,α + ts⊥α bαβ − tsβα,α + |[−t lm + vl (v m − w m )]|em τ lβ − τ lβ s f l = 0,

(11.48)

and of the normal component of momentum ∂s w ⊥ + s w γ w ⊥,γ − 2s w ⊥ w ⊥ K M ∂t + s,α w ⊥ w α + s w ⊥ w α,α     γα + s w α w ⊥,α + w γ bγα − ts⊥α ,α + ts bγα + |[−t km + v k (v m − w m )]|em ek − s f k ek = 0.

(11.49)

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A similar procedure results in the balance equation of the tangential component of the spin. Using the balance of mass, we finally get the shortest form of the balance equations. Tangential Component of the Balance of Momentum

s

  ∂w β = −s −w ⊥ g βν w ⊥,ν − w γ w ⊥ bγβ ∂t   (11.50) βα −s w α −w ⊥ bαβ + w β,α − ts⊥α bαβ + ts ,α +|[−t lm + (vl − wl )(v m − w m )]|em τ lβ + τ lβ s f l = 0.

Normal Component of the Balance of Momentum

s

    ∂w ⊥ γα = −s w γ w ⊥,γ + s w α w ⊥,α + w γ bγα + ts⊥α ,α + ts bγα (11.51) ∂t −|[−t km + (v k − w k )(v m − w m )]|em ek − s f k ek = 0.

Tangential Component of the Balance of Spin

s

β   ∂ss γ βα β = −s −ss⊥ g βν w ⊥,ν − ss w ⊥ bγβ − ⊥α s bα + s ,α ∂t   β +s w α ss⊥ bαβ + ss ,α − |[−lm + (s l − ssl )(v m − w m )]|em τ lβ β

β

+τ kβ klm ταl tsmα + s m s + s  α⊥ w ⊥ w α = 0. (11.52) Normal Component of the Balance of Spin

s

    ∂ss⊥ γ γ γα = −s ss w ⊥,γ − s w α ss⊥,α + ss bγα + ⊥α s ,α + s bγα ∂t (11.53) −|[−km + (s k − ssk )(v m − w m )]|em ek − s m ks ek +ek klm ταl tsmα = 0,

11.4 Surface Balance Equations

213

and the balance of Internal Energy s

   ∂es βα  α = −qs,α + ts −w ⊥ bαβ + w β,α + ts⊥α w ⊥,α + w β bαβ ∂t    1 βα  ⊥ β + s −s bα + s β,α + ts⊥α w ⊥,α + w β bαβ − s es,α w α θeff 1 k 1 k km −|[q m − (v k − w k )t km + ( s − s ) + e(v m − w m ) θeff θeff s  + (v k − w k )(v k − w k )(v m − w m ) 2 (s k − ssk )(s k − ssk )(v m − w m )]|em − s rs + 2θe f f 1 k s k −klm ταl tsmα s − kα⊥ w ⊥ w α s . θe f f s θeff s (11.54)

These are the equations of motion for the fields of surface mass density, surface energy, surface momentum, and surface spin. Constitutive equations are necessary to close the system of differential equations. The geometry of the surface is determined by the geometry at an initial time and the mapping velocity w. In this sense, the balance of surface momentum is the equation of motion for the surface geometry. The mean curvature does not enter into (11.50)–(11.54) in contrast to (11.48)– (11.53). Equations (11.48)–(11.53) are balance equations for quantities per unit area, for instance, s es is the internal energy per unit area. For quantities per unit area, the change of surface area on a moving surface is important. This change of surface area results in the mean curvature terms. Specific quantities, i.e., quantities per unit mass are unaffected by a change in surface area. Therefore, the mean curvature does not occur in (11.50)–(11.54).

11.5 An Example: The Measurement of Surface Tension Surface tension is introduced in order to explain the observed mechanical behavior of liquid-gas interfaces, namely, the fact that they support tangential forces. Consequences are that gas bubbles in a liquid are spherical and that insects can walk on a water surface. A measurement of surface tension is used to characterize the interaction of a pair of materials. As a consequence of the curvature dependence, the pressure in nano-porous systems depends on the pore size and shape, and it is a question, how to define the pressure in such systems [21]. Capillary rise experiments are performed in order to measure the surface tension. A capillary is immersed in the liquid in a vessel (see Fig. 11.5). Depending on the material, the level of the liquid in the capillary is higher or lower than the level in the vessel. In the first case, the meniscus in the capillary is convex; in the second

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11 A Short Survey of Thermodynamics of Material Surfaces

case, it is concave. The relevant equation to describe this experiment is the normal component of the balance of momentum (11.51). In equilibrium (v = 0, w = 0), it reduces to: γα km k m k k ts⊥α ,α + ts bγα + |[t ]|e e − s f e = 0.

(11.55)

The following constitutive assumptions (the assumptions of a simple liquid) are usually made: 1. There are no surface forces perpendicular to the surface: ts⊥α = 0.

(11.56)

2. The stress tensor on both sides of the surface, i.e., in the liquid and in air, is isotropic: t = pδ. (11.57) 3. The surface stress tensor is an isotropic surface tensor: tsγα =

γ γα g , 2

(11.58)

and the scalar coefficient γ is called surface tension. If in addition we define the position of the singular surface such that s = 0 (which is always possible), (11.55) reduces to: γ α b + |[ p]| = 0. 2 α

(11.59)

In equilibrium, the pressure difference between liquid and air |[ p]| is balanced by the gravitational force: gh( − 0 ) , (11.60) |[ p]| = − cos(θ) where 0 is the density of the air and θ is the contact angle (see Fig. 11.5). Finally, it is assumed that the shape of the meniscus in the capillary is a section of a sphere. Then, the mean curvature is given by: bαα = 2K M =

2 , R

(11.61)

where R is the radius of curvature. Thus, we finally end up with the well-known formula for the surface tension: γ = gh( − 0 )

R . cos(θ)

(11.62)

11.5 An Example: The Measurement of Surface Tension

215

Fig. 11.5 In a capillary rise experiment, the rise of the liquid column h and the contact angle θ are measured

r

θ

h

liquid

If the same experiment is performed with a micropolar medium like a liquid crystal, the normal stresses do not vanish, i.e., the surface supports normal forces. With a certain choice of the state space, it can be shown that the tangential part of the surface stress tensor is still isotropic but the normal stresses are given in terms of derivatives of the free energy with respect to surface geometrical quantities and with respect to an internal variable alignment tensor. For the exploitation of the surface dissipation inequality, we refer to the literature [22, 23]. Assuming isotropy of the bulk stress tensor in the absence of a velocity gradient, the normal component of the balance of momentum reduces to: ts⊥α;α +

γ α b + |[ p]| = 0. 2 α

(11.63)

Inserting (11.60) for the pressure jump |[ p]| (11.63) shows that the surface tension γ cannot be simply determined from the height of the liquid column h and the contact angle  because of the divergence of the normal stresses entering in (11.63).

11.6 Summary There are certain examples where field quantities are not only discontinuous, but show localization at a surface. For these surface densities, balance equations can be derived. Compared to the balance equations for the volume densities in the bulk, there are two differences: 1. If the surface geometry changes, the area of a surface element may change (imagine, for instance, surface densities defined on a balloon that is blown up). This fact shows up in the balance equations as a term proportional to the mean curvature.

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2. The surface, where field quantities are localized, is embedded in a bulk medium. The flux of the bulk quantities may be discontinuous at the surface. This jump of the bulk flux shows up as a production term in the surface balance equations. The balance of momentum in the stationary case is the equation needed for the interpretation of capillary rise experiments to determine the surface tension of a fluid by measuring its contact angle and its height in a capillary. From an exploitation of the surface dissipation inequality, it can be shown that for micropolar media the constitutive equation for the surface stress is more complicated than for simple liquids. Consequently, the balance of momentum is modified by an additional normal stress term.

References 1. W. Harbich, W. Helfrich, The swelling of egg lecitin in water. Chem. Phys. Lipids 36, 39–63 (1984) 2. W. Helfrich, Size distributions of vesicles: the role of the effective rigidity of membranes. J. Physique 47, 321–329 (1986) 3. G. Beblik, R.-M. Servuss, W. Helfrich, Layer bending rigidity of some synthetic lecitins. J. Physique 46, 1773–1778 (1985) 4. W. Dreyer, On thermodynamics and kinetic theory of ideal rubber membranes. J. Appl. Mech. 48, 345–350 (1981) 5. H. Schadt, W. Helfrich, Voltage-dependent optical activity of a twisted nematic liquid crystal. Appl. Phys. Lett. 18(4), 127–128 (1971). Feb 6. R. Hirning, W. Funk, H.-R. Trebin, M. Schmidt, H. Schmiedel, Threshold behaviour and electro-optical properties of twisted nematic layers with weak anchoring in the tild and twist angle. J. Appl. Phys. 70(8), 4211–4216 (1991) 7. J.D. Lee, A.C. Eringen, Boundary effects of orientation of nematic liquid crystals. J. Chem. Phys. 55(9), 4509–4512 (1971) 8. J.D. Lee, A.C. Eringen, Alignment of nematic liquid crystals. J. Chem. Phys. 55(9), 4504–4508 (1971) 9. T. Alts, K. Hutter, I: Surface balance laws and their interpretation in terms of three dimensional balance laws averaged over the phase change boundary layer, II: Thermodynamics, III: Thermostatics and its consequences, IV: On thermostatic stability and well-posedness. J. Non-Equilib. Thermodyn. 13, 221–301 (1988) 10. J.W. Gibbs, Collected Works (Yale University Press, New Haven, 1948) 11. D. Ronis, D. Bedeaux, I. Oppenheim, On the derivation of dynamical equations for a system with an interface: I General theory. Phys. A 90A, 487–506 (1978) 12. A. Grauel, Thermodynamics of an interfacial fluid membrane. Phys. A 103A, 468–520 (1980) 13. D. Ronis, Statistical mechanics of systems nonlinearly displaced from equilibrium I. Physica 99A, 403–434 (1979) 14. A. Grauel, Feldtheoretische Beschreibung von Grenzflächen (Springer, Berlin, New York, London, 1989) 15. D. Ronis, I. Oppenheim, On the derivation of dynamical equations for a system with an interface: II The gas-liquid interface. Physica 117A, 317–354 (1983) 16. D. Ronis, On the derivation of dynamical equations for a system with an interface: III The solid-liquid interface. Physica 121A, 1–37 (1983) 17. A.I. Rusanov, Phasengleichgewichte und Grenzflächenerscheinungen (Akademie Verlag, Berlin, 1978)

References

217

18. K. Glavatskiy, D. Bedeaux, Non-equilibrium thermodynamics for surfaces; square gradient theory. Eur. Phys. J. Special Top. 222, 161–175 (2013) 19. R. Rurali, L. Colombo, X. Cartoixa, O. Wilhelmsen, T.T. Trinh, D. Bedeaux, S. Kjelstrup, Heat transport through a solid-solid junction the interface as an autonomous thermodynamic system. Phys. Chem. Chem. Phys. 18, 13741–13745 (2016) 20. M.E. Gurtin, On thermodynamical laws for the motion of a phase interface. J. Appl. Math. Phys. (ZAMP) 42 (1991) 21. O. Galteland, D. Bedeaux, B. Hafskjold, S. Kjelstrup, Pressures inside a nano-porous medium the case of a single phase fluid. Front. Phys. 7(60), 1–10 (2019) 22. C. Papenfuss, Contribution to a Continuum Theory of Two Dimensional Liquid Crystals (Wissenschaft- und Technik Verlag, Berlin, 1995) 23. C. Papenfuss, Liquid crystalline surface tension and radius dependence of the internal pressure in liquid crystalline bubbles and droplets. Mol. Cryst. Liq. Cryst. 367, 2999–3006 (2001)

Chapter 12

Outlook: Mesoscopic Theory of Complex Materials

Abstract The mesoscopic concept introduces field quantities defined on an enlarged domain, the mesoscopic space, which takes into account the internal structure of the complex material. In addition, a mesoscopic distribution function is introduced as a statistical element. Some examples are mentioned, and the mesoscopic theory of liquid crystals is presented in more detail. Balance equations on the mesoscopic level are presented, and constitutive equations are discussed. Order parameters, the alignment tensors, are introduced, based on the orientation distribution function. The second-order alignment tensor is the internal variable, introduced in Chap. 7.

12.1 Introduction: Complex Materials Let us call substances consisting of spherical particles with interactions depending only on the inter-particle distance simple materials. However, most of the interesting and practically important materials have a more complicated internal structure. We call these materials with internal degrees of freedom complex materials. Several examples of such complex materials have been investigated in the preceding chapters. We want to sketch now an alternative way to take into account an internal structure of the material within a continuum theory.

12.1.1 Examples of Internal Structure Steel Steel consists of micro-crystallites of different orientations of the crystal axes. The crystal axes change from micro-crystallite to micro-crystallite. Under the action of a mechanical load, the micro-crystallites can be reoriented. The resulting macroscopic properties are history-dependent material properties and after effects.

© Springer Nature Switzerland AG 2020 C. Papenfuß, Continuum Thermodynamics and Constitutive Theory, https://doi.org/10.1007/978-3-030-43989-7_12

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12 Outlook: Mesoscopic Theory of Complex Materials

Shape Memory Alloys These materials show a very spectacular hysteresis effect already discussed in Sect. 5.3.4: If the original shape is deformed at low temperature, and the material is then heated above a critical temperature, the original undeformed shape is recovered. There are several applications of these shape memory alloys, such as switches controlled by temperature and even medical applications [1]. The underlying internal structure is a microcrystalline material with different orientations of the microcrystallites. In addition, the material can undergo a phase transition between martensite and austenite. The phase transition at the critical temperature, and reorientation of crystal axes under mechanical load, can explain the different stress-strain diagrams at different temperatures resulting in the hysteresis effect. Polymer Melts and Solutions Polymer molecules are long chains of organic subunits. Polymer melts or polymer solutions with a low-molecular-weight solvent show non-Newtonian flow behavior [2–4]. Let us consider, as an example, a plane Couette flow (see Fig. 7.3) with the velocity in the x-direction, the gradient in the y-direction, and the shear rate defined x . In a Newtonian fluid, the stress tensor component tx y depends linearly by γ = ∂v ∂y on the shear rate. In polymer melts and solutions, however, we observe a nonlinear dependence, i.e., shear thinning or shear thickening, respectively (see Fig. 12.1). This non-Newtonian behavior can be explained by assuming stretching and reorientation of the polymer chains in the flow field. The theoretical description of the internal degrees of freedom connected with the conformation of the polymer chain is possible on different levels of accuracy (see Fig. 12.2). The simplest description is in terms of an end-to-end vector connecting the two ends of the molecule. More information is included in the conformation tensor. The tensor ellipsoid of the conformation tensor is the smallest ellipsoid including the polymer chain. The most elevated description is in terms of the positions and orientations of the N subunits of the polymer molecule.

shear thickening

shear thickening Newtonian fluid

Newtonian fluid shear thinning

Fig. 12.1 Non-Newtonian fluid in a shear flow

shear thinning

12.1 Introduction: Complex Materials

221

isotropic

end-to-end vector

end-to-end vectors in a volume element

in the flow field

orientation distribution

anisotropic

Fig. 12.2 Polymer chains are stretched and elongated in the flow field. The orientation distribution of end-to-end vectors becomes anisotropic

Liquid Crystals The liquid crystalline state shows properties of the liquid phase as well as those of the solid phase [5]. For instance, liquid crystals show fluid-like non-Newtonian flow behavior. On the other hand, they are anisotropic like solids, showing birefringence, anisotropic electric and heat conductivity, and so on ([5, 6] and see Sect. 7.2). Thermotropic liquid crystals consist of rigid non-spherical particles. In the uniaxial case, the particles are rotation symmetric. The axis of rotation symmetry is called the microscopic director n. The molecules can be rod-like or disk-like. In all liquid crystalline phases, there exists an orientational order of the microscopic directors. Liquid crystals consisting of rigid particles without rotation symmetry are called liquid crystals of biaxial molecules. In this case, the orientation of at least one of the molecular axes shows a long-range order in the liquid crystalline phase.

12.2 Mesoscopic Concept There are two principally different possibilities for dealing with complex materials within continuum thermodynamics: The first way is to introduce additional fields that depend on position and time. These fields can be internal variables [7–9], order or damage parameters [5, 10], Cosserat triads [11], directors [12, 13], and alignment and conformation tensors [14–17]. The other way is a so-called mesoscopic theory.

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12 Outlook: Mesoscopic Theory of Complex Materials

Table 12.1 The mesoscopic variable and the manifold M for different complex materials. Republished with permission of Springer, from [18]; permission conveyed through Copyright Clearance Center, Inc. Material Internal structure Manifold M Steel Shape memory alloys Polymer solutions Liquid crystals of uniaxial molecules Liquid crystals of biaxial molecules

Orientation of crystal axes of micro-crystallites Fraction of different phases and orientation of crystal axes Orientation and length of end-to-end vector Orientation of particle = microscopic director Orientation of three molecular axes

S O(3) [0, 1] × S O(3) S 2 × [0, r ] S2 S O(3)

The idea is to enlarge the domain of the field quantities. The new mesoscopic fields are defined on the space R3x × Rt × M. The manifold M is given by the set of values the internal degree of freedom can take. Therefore, the choice of M depends on the complex material under consideration (examples see below), but if the internal structure is known, the nature of M is clear. We will see later that the manifold M should be such that differentiation and integration are possible on it. We call this way of dealing with the internal structure of complex materials a mesoscopic concept because it includes more information than a macroscopic theory on R3x × Rt , but no interactions on the molecular level are considered like in a microscopic approach. The mesoscopic level is between the microscopic and the macroscopic levels. The domain of the mesoscopic field quantities R3x × Rt × M is called mesoscopic space. In Table 12.1, the physical meaning of the additional mesoscopic variable and of the manifold M are given for the examples considered in the previous section. The orientation of a triad of crystal axes or molecular axes is determined by a rotation matrix ∈ S O(3) mapping the actual triad to a reference system. Beyond the use of additional variables m, the mesoscopic concept introduces a statistical element, the so-called Mesoscopic Distribution Function (MDF) f (m, x, t) generated by the different values of the mesoscopic variable of the particles (or subunits) in a volume element: f (m, x, t) ≡ f (·), (·) ≡ (m, x, t) ∈ M × R3 × R1 .

(12.1)

The MDF describes the distribution of m in a volume element around x at time t, and, therefore, it is normalized:  f (m, x, t) dM = 1. (12.2)

12.2 Mesoscopic Concept

223

Now the fields such as mass density and momentum density are defined on the mesoscopic space. For distinguishing these fields from the macroscopic ones, we add the word “mesoscopic”, but we will use the same symbol as for the corresponding macroscopic field. Consequently, the mesoscopic mass density is defined by: (·) := (x, t) f (·) .

(12.3)

Here, (x, t) is the macroscopic mass density. By use of (12.2), we obtain:  (x, t) =

(m, x, t) dM.

(12.4)

This equation shows that the system can be formally treated as a mixture of components (see Sect. 3.4) having the partial density (·) [19, 20]. Here, the “component index” m is a continuous one. We will come to this point again in connection with the balance of mass. Other mesoscopic fields defined on the mesoscopic space are the mesoscopic material velocity v(·) of the particles belonging to the mesoscopic variable m at time t in a volume element around x, the external mesoscopic acceleration f (·), the mesoscopic stress tensor t(·), and the mesoscopic heat flux density q(·). Macroscopic quantities are obtained from mesoscopic ones as averages, with the MDF as probability density:  A(x, t) =

M

A(·) f (·)dm.

(12.5)

This again shows that the complex material can be seen as a mixture of components with different values of the mesoscopic variable. The summation over the indices of different chemical components is replaced by integration over the continuous variable m.

12.3 Mesoscopic Balance Equations Let G denote a region in R3 × M and X the mesoscopic density of an extensive quantity. Then, the global quantity in the region G changes due to a flux over the boundary of G and due to production and supply within G: d dt



 Xd xdm = 3

G

∂G

 φ X (·)da +

 X (·)d3 xdm.

(12.6)

G

A generalized Reynolds transport theorem in the mesoscopic space [21] is used to transform the time derivative, and Gauss’ theorem is applied to the boundary integral. This is the point where it is necessary to be able to integrate and to differentiate on the

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12 Outlook: Mesoscopic Theory of Complex Materials

manifold M. Then, in regular points of the continuum, we get the local mesoscopic balance [22] ∂ X(·) + ∇x · [v(·)X(·) − S(·)] + ∇m · [u(·)X(·) − R(·)] = (·). ∂t

(12.7)

Here S(·) and R(·) are fluxes of the quantity X(·) in position space and in mesoscopic space, respectively. (·) is the production. The independent field u(·), defined on the mesoscopic space, describes the change in time of the set of mesoscopic variables. With respect to m, the mesoscopic change velocity u(·) is the analog to the mesoscopic material velocity v(·) referring to x. If a molecule is characterized by (m, x, t), then for t → +0 it is characterized by (m + u(·)t, x + v(·)t, t + t). Besides the usual gradient, also the gradient with respect to the set of mesoscopic variables appears as an additional flux term on M in all balance equations. In general, there is a convective and a non-convective flux on M. Entropy Balance Besides these mesoscopic balances, we have a balance of entropy. The production of mesoscopic entropy is not necessarily positive for each component, just as, analogously, the partial entropy production of a chemical component in a mixture may be negative. Only the sum over all components, or the integral over the mesoscopic variable, respectively, is positive definite. The set of balance equations is not a closed system of equations but requires constitutive equations for mesoscopic quantities. The domain of the constitutive mappings is the state space: here, a mesoscopic one. There are the possibilities that the mesoscopic state space includes only mesoscopic quantities, or that it includes mesoscopic and macroscopic quantities. An example of the second kind is discussed for liquid crystals in [20, 23–25]. The balance equations to be considered in any case are the balances of mass, momentum, and energy. Depending on the internal degrees of freedom, additional balance equations have to be taken into account. In the example of liquid crystals, this will be the balance of internal angular momentum connected to rotations of the molecules.

12.4 Example: Mesoscopic Theory of Uniaxial Liquid Crystals An example for m is the microscopic director n in mesoscopic liquid crystal theory [20, 22, 26–28]. This microscopic director is defined as a unit vector pointing in the temporary direction of a needle-shaped rigid particle, or, if the particle is of a plane shape, the microscopic director is perpendicular to the particle. As the microscopic director is defined on a molecular level, it is not a macroscopic field describing the ”mean orientation” but, rather, represents a mesoscopic variable which spans the

12.4 Example: Mesoscopic Theory of Uniaxial Liquid Crystals

a)

Rotation symmetric molecules

microscopic director

microscopic directors

225

b)

microscopic director

orientation distr ibution function

Fig. 12.3 The Orientation Distribution Function (ODF) a in the uniaxial and b in the biaxial liquid crystalline phases. In all liquid crystalline phases, the ODF is anisotropic. Republished with permission of Springer, from [18]; permission conveyed through Copyright Clearance Center, Inc.

two-dimensional unit sphere S 2 . The unit sphere is the manifold M in this example. The MDF in (12.1) describes the orientation of the molecules in a volume element. In the case of liquid crystals, the MDF is called Orientation Distribution Function (ODF). It is a distribution function on the unit sphere. It is always observed a so-called head-tail symmetry, i.e., one orientation and the reversed one are equally probable: f (x, t, n) = f (x, t, −n).

(12.8)

Consequently, an inversion symmetry with respect to the origin of the graph of the ODF on the unit sphere is observed (see Fig. 12.3). The ODF allows identification of the different phases. In the isotropic phase, all particle orientations are equally probable, and the orientation distribution function is isotropic, i.e., a homogeneous function on the unit sphere S 2 . The other extreme is the totally ordered phase, where all particle orientations are the same. The corresponding distribution function has a non-zero value only for this single orientation, i.e., it is δ-shaped. Due to thermal motion, this totally ordered phase does not occur at nonzero temperature. There is partial ordering of orientations, and the corresponding distribution functions show some concentration around a preferred orientation. There are two possibilities: that the ODF is rotation symmetric around an axis d, or that there is no such rotation symmetry. In the first case, the phase is called uniaxial; in the second case, it is called biaxial, see Fig. (12.3). In most cases, nematic liquid crystalline phases are observed to be uniaxial. Biaxiality can be induced by the action of crossed electric and magnetic fields [29], by the action of a flow field [30], or by boundary conditions. This symmetry of the phase has to be distinguished from the symmetry of the particles, which are assumed here in all cases to be rotation symmetric. For applications of the mesoscopic concept to liquid crystals, see [19, 23, 24, 26, 31–36].

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12 Outlook: Mesoscopic Theory of Complex Materials

12.4.1 Mesoscopic Balance Equations The derivative with respect to the mesoscopic variable microscopic director n is denoted by \∇n . It is the covariant derivative on the unit sphere, and the orientation change velocity u = n˙ is tangential to the unit sphere. We obtain the special balances by special identifications of the abstract quantities X(·), (·), S(·), R(·) in (12.7). These balances are expressed [26]: Mass

∂ (·) + ∇x · {(·)v(·)} + ∇n · {(·)u(·)} = 0. ∂t

(12.9)

Momentum ∂ [(·)v(·)] + ∇x ∂t + ∇n

  · v(·)(·)v(·) − t  (·)   · u(·)(·)v(·) − T  (·) (·) f (·).

(12.10)

Here f (·) is the external acceleration, t  (·) is the transposed stress tensor, and T (·) is the transposed stress tensor on the unit sphere S 2 , i.e., the non-convective momentum flux on the unit sphere. 

Angular Momentum In addition, we have a balance of angular momentum, which is independent of the balance of momentum, due to rotations of the particles, which are not point like: S(·) := x × v(·) + s(·), ∂ [(·)S(·)] + ∇x ∂t + ∇n

(12.11)

  · v(·)(·)S(·) − (x × t(·)) −  (·)   · u(·)(·)S(·) − (x × T (·)) − W  (·)

= (·)x × f (·) + (·)m.

(12.12)

Here, s is the vector of the specific spin (internal angular momentum due to particle rotations), m the vector of volume torque density, the second-order tensor  is the surface torque (the non-convective flux of angular momentum in position space), and W is the analog to  with respect to the orientation variable n. All these quantities are mesoscopic ones, depending on position, time, and orientation.

12.4 Example: Mesoscopic Theory of Uniaxial Liquid Crystals

227

Total Energy etot (·) :=

1 2 1 v (·) + s(·) · −1 · s(·) + e(·), 2 2

(12.13)

∂ [(·)etot (·)] ∂t 

 + ∇x · v(·)(·)etot (·) − t(·) · v(·) − (·) · −1 · s(·) + q(·)   + ∇n · u(·)(·)etot (·) − T (·) · v(·) − W(·) · −1 · s(·) + Q(·) = (·) f (·) · v(·) + (·)m(·) · −1 · s(·) + r (·).

(12.14)

Here is r the absorption supply,  is the moment of inertia tensor of the particles, q is the heat flux density, and Q is the heat flux density on M. Here again, all quantities are mesoscopic ones. Summary • Compared to macroscopic balance equations, the mesoscopic equations include additional flux terms in orientation space. • Compared to simple materials, complex materials require consideration of the balance of angular momentum. This is clear from the internal degree of freedom, the rotations of particles.

12.4.2 Macroscopic Balance Equations For the extensive quantities of mass density, momentum density, and spin density, the macroscopic quantities are integrals of the mesoscopic ones over the unit sphere:  (·)d2 n (x, t) = 2 S  (·)v(·)d2 n (x, t)v(x, t) = S2  (·)s(·)d2 n. (x, t)s(x, t) =

(12.15) (12.16) (12.17)

S2

Therefore, we obtain the macroscopic balance equations by integrating the mesoscopic ones over all orientations. For the balance of mass, we have:  S2

∂ (·)d2 n + ∂t



 S2

∇x · (v(·)(·)) d2 n +

S2

∇n · (u(·)(·)) d2 n

(12.18)

228

12 Outlook: Mesoscopic Theory of Complex Materials

is equal to ∂ ∂t



 S2

(·)d2 n + ∇x

leads to

 S2

· (v(·)(·)) d2 n +

S2

∇n · (u(·)(·)) d2 n

∂ (x, t) + ∇x · (v(x, t)(x, t)) = 0. ∂t

(12.19)

(12.20)

The last term in (12.19) is zero due to Gauss’ theorem on the unit sphere and the fact that S 2 is a closed surface. Accordingly, there is no boundary term on S 2 : 

 S2

∇n (u)d2 n =

∂ S2

u · ndl = 0.

(12.21)

The mesoscopic balance of mass integrated over the unit sphere shows again the analogy to mixture theory (see also [20]) summing up the equations for the different components of the mixture (here the different orientations) gives the balance equation of the mixture. The right-hand side is zero, in analogy to the fact that the sum over the components of productions due to chemical reactions is zero. In the case of mesoscopic theory of liquid crystals, chemical reactions are replaced by rotations of particles (the flux in orientation space). Analogously to the macroscopic balance of mass, other macroscopic balance equations of momentum and angular momentum are obtained. Along with the balance of energy, these are the balance equations of a micropolar continuum [37–41].

12.4.3 Macroscopic Constitutive Quantities From the integrated mesoscopic equations, we can identify the macroscopic constitutive quantities in terms of mesoscopic ones. For example, for the stress tensor and the internal energy, we find:  t(x, t) =

S2

(t(·) + (x, t) (v(x, t)v(x, t) − v(·)v(·))) d2 n

(12.22)

and e(x, t) = e(·) +

 1 1 v(·)2  − v(x, t)2 + θu(·)2  − s(x, t)2 , (12.23) 2 θ

where   is the abbreviation for S 2 f (·) d n . Similar relations can be derived for the heat flux and for the energy supply. These relations between the mesoscopic constitutive quantities and the corresponding macroscopic ones are analogous to the relations between the constitutive

12.4 Example: Mesoscopic Theory of Uniaxial Liquid Crystals

229

equations in the partial balance equations and the macroscopic ones in mixture theory (see Sect. 3.4). Summary For constitutive quantities, the macroscopic quantities are in general not simply averages of the corresponding mesoscopic ones. Instead, deviations of the mesoscopic velocity from the macroscopic one and of the mesoscopic angular velocity from the macroscopic one, make additional contributions.

12.4.4 Order Parameters We will give now a definition of the alignment tensor based on the Orientation Distribution Function (ODF). This alignment tensor is the internal variable introduced before in Sect. 7.2 for a purely macroscopic example. Using the microscopic director n as a mesoscopic variable, we can introduce the family of the macroscopic fields of order parameters, which is defined by different moments of the ODF:  f (·) nn d2 n, (12.24) a(x, t) := S2  f (·) nnnn d2 n, (12.25) a(4) (x, t) := S2  f (·) n . . . n d2 n, etc. (12.26) a(6) (x, t) := S2

6 times

These are tensors of successive order. Only the even-order tensors are non-zero, due to the inversion symmetry of the orientation distribution function ( f (x, −n, t) = f (x, n, t)). The symbol denotes the traceless symmetric part of the tensor in its argument [42]. If the ODF is rotation symmetric with axis of rotation symmetry d, the alignment tensors can be written as: a(k) = S (k) d . . . . . . d

(12.27)

k times with scalar order parameters S (k) , and a unit vector d. These fields of order parameters describe macroscopically the mesoscopic state of the system introduced by n. Consequently, these fields are the link between the mesoscopic background description of the liquid crystal and its description by additional macroscopic fields (internal variables). In liquid crystal theory, the fields of order parameters are called the alignment tensors of different orders. In the isotropic phase, all alignment tensors are zero, whereas in the liquid crystalline phases, at least

230

12 Outlook: Mesoscopic Theory of Complex Materials

some alignment tensors are non-zero. The alignment tensors are internal variables. In equilibrium, they are determined by the equilibrium variables, mass density and temperature.

12.4.5 Differential Equation for the Distribution Function and for the Alignment Tensors The orientation distribution function has been defined as the mass fraction: f (x, n, t) =

ρ(x, n, t) . ρ(x, t)

(12.28)

For the macroscopic mass density, the balance of mass with the assumption of incompressibility (∇ · v = 0) is applied, and for the mesoscopic mass density, we have the mesoscopic balance of mass. We obtain for the distribution function: ∂ f (x, n, t) + v(x, n, t) · ∇ f (x, n, t) + ∇n · (u(x, n, t) f (x, n, t)) = 0. (12.29) ∂t The differential equation (12.29) for the ODF allows derivation of a system of differential equations for the alignment tensors. In these equations, the alignment tensors of all orders are coupled. In general, a closure relation is needed in order to deal with only a limited number of moments (see, for instance, [43]). Together with such a closure relation, these equations are the relaxation equations for the internal variables. In general, we have (in Cartesian components), for the kth-order alignment tensor, by taking the kth moment of (12.29): dan(k) 1 ...n k dt  ∇α

S2

− k (ω × a (k) ) δ vˆ

α

n m 1 . . . n ml

 2k + 1  (2l − 1)!! k!4π l,even n n 1 . . . n n k d2 n am(l)1 ...m l

n 1 ...n k

=

 + ∇α δ vˆ α n m 1 . . . n m l n n 1 . . . n n k d2 n(ln ρ) S2    n m 1 . . . n m l n n 1 . . . n n k ∇n × δ ω ˆ · dn am(l)1 ...m l + 2 S   ˆ × a (l) m 1 ...m l d2 n −l n m 1 . . . n ml n n1 . . . n nk δω 2  S δ vˆ α n n 1 . . . n n k d2 n∇α (ln ρ) + S2

12.4 Example: Mesoscopic Theory of Uniaxial Liquid Crystals

231

 + ∇α δ vˆ α n n 1 . . . n n k d2 n S2    n n 1 . . . n n k ∇n × δ ω ˆ · dn . +

(12.30)

S2

For the second-order alignment tensor, this has the form of a relaxation equation: da(2) − 2 (ω × a(2) ) = m(a(k) , k = 2, . . . ; u), dt

(12.31)

where the production G of the kth-order alignment tensor depends on all alignment tensors of any order and on the orientation change velocity u. A closure relation expresses the higher order alignment tensors a (k) (k = 4, 6, . . . ) in terms of the second-order one.

12.4.6 Summary The mesoscopic background gives a definition of order parameters, which are the internal variables in a macroscopic description. It also allows equations of motion for these order parameters to be derived.

12.4.7 A Constitutive Equation from the Mesoscopic Background: An Example We will discuss briefly, as an example, the derivation of the macroscopic constitutive equation for the viscous stress tensor from a mesoscopic one [44]. For the mesoscopic state space, we use one including macroscopic and mesoscopic quantities: Z = {(x, t), T (x, t), ∇v(x, t) , n, N := n˙ − (∇ × v(x, t)) × n}.

(12.32)

Then, a representation theorem for the stress tensor linear in the velocity gradient reads: t(n, x, t) =

(n, x, t)  tˆ(, T ) + α1 nnnn : (∇v)sym + α2 nN + α3 N n (x, t) +α4 (∇v)sym + α5 nn · (∇v)sym + α6 n · (∇v)sym n  +ξ nn : (∇v)sym δ + ξ nn∇ · v + ξ ∇ · vδ , 1

2

3

(12.33)

232

12 Outlook: Mesoscopic Theory of Complex Materials

where the coefficients αi may depend on the scalar quantities macroscopic density and temperature. If the deviation of the mesoscopic velocity from the macroscopic velocity can be neglected, then the macroscopic stress tensor is the average of the mesoscopic one (see 12.22): 

(n, x, t)  tˆ(, T ) + α1 nnnn : (∇v)sym + α2 nN 2 (x, t) S +α3 N n + α4 (∇v)sym + α5 nn · (∇v)sym + α6 n · (∇v)sym n  +ξ nn : (∇v)sym δ + ξ nn∇ · v + ξ ∇ · vδ d 2 n

t(x, t) =

1

2

3

= tˆ(, T ) + α1 nnnn : (∇v)sym − α2 nn × (∇ × v) +α3 (∇ × v) × nn + α4 (∇v)sym  T +α5 nn · (∇v)sym + α6 nn · (∇v)sym +ξ1 nn : (∇v)sym δ + ξ2 nn∇ · v + ξ3 ∇ · vδ  1 sym ˆ = t (, T ) + α1 nnnn : (∇v) − α2 a + δ × (∇ × v) 3  1 +α3 (∇ × v) × a + δ + α4 (∇v)sym 3   T 1 1 a + δ · (∇v)sym +α5 a + δ · (∇v)sym + α6 3 3   1 1 +ξ1 a + δ : (∇v)sym δ + ξ2 a + δ ∇ · v + ξ3 ∇ · vδ. (12.34) 3 3 The brackets   denote averages with the orientation distribution function:  nn =

S2

f (n, x, t)nnd2 n

(12.35)

and analogously for the fourth-order tensor. The averages of products of the microscopic director have been written in terms of the second- and the fourth-order moment. If the distribution function is rotation symmetric, this can be simplified to expressions in terms of the macroscopic director and the scalar order parameters S (2) and S (4) . The result is a macroscopic constitutive equation for the stress tensor. It is almost in the form of the Leslie expression for the stress tensor [13], except for the director dependence of the terms involving the rotation of the velocity field ∇ × v. Comparing the expression (12.34) with the Leslie stress tensor, there results a theoretical prediction of the order parameter dependence of the viscosity coefficients. This information is not available from a purely macroscopic theory, and the result of the mesoscopic background theory has been compared to measurements [44], using the affine transformation model (assuming ellipsoidal particles) introduced in [45, 46]. The agreement with the experimental results is good.

12.5 Summary of the Mesoscopic Theory

233

12.5 Summary of the Mesoscopic Theory The mesoscopic concept introduces field quantities defined on an enlarged domain, the mesoscopic space, which takes into account the internal structure of the complex material. In addition, a mesoscopic distribution function is introduced as a statistical element. The advantages of such a mesoscopic background theory over a purely macroscopic description are as follows: • From the internal structure, it is clear which balance equations have to be taken into account. For liquid crystals, for example, the balance of angular momentum is relevant. • Based on the mesoscopic distribution, function order parameters are defined that are the internal variables in a purely macroscopic description. From the mesoscopic balance equations, one can derive equations of motion for the internal variables. • The mesoscopic constitutive theory gives the order parameter dependence of macroscopic constitutive quantities. • Different physical systems can be considered within the same or analogous mesoscopic theories. So, for instance, the orientation of a biaxial molecule and the orientation of the crystal axes of a micro-crystallite are described by the same mesoscopic variable. In addition, this mesoscopic variable can be an element of S 3 [21], which makes the theory for biaxial liquid crystals analogous to the theory of uniaxial liquid crystals.

References 1. I. Müller, Grundzüge der Thermodynamik (Springer-Verlag, Berlin, Heidelberg, 1994) 2. M. Doi, S.F. Edwards, The Theory of Polymer Dynamics (Clarendon, Oxford, 1986) 3. M. Doi, Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases. J. Polym. Sci. Polym. Phys. 19(2), 229–243 (1981) 4. M. Kröger, Rheologie und Struktur von Polymerschmelzen. Ph.D. Thesis, Technische Universität Berlin, W und T Verlag, Berlin, 1995 5. P.-G. De Gennes, J. Prost, The Physics of Liquid Crystals, 2nd edn. Monographs on Physics. (Clarendon Press, Oxford, 1995) 6. H. Kelker, R. Hatz, Handbook of Liquid Crystals (Verlag Chemie, Weinheim, Deerfield, 1980) 7. W. Muschik, Internal variables in non-equilibrium thermodynamics. J. Non-Equilib. Thermodyn. 15, 127–137 (1990) 8. G.A. Maugin, W. Muschik, Thermodynamics with internal variables, part I. General concepts. J. Non-Equilib. Thermodyn. 19, 217–249 (1994) 9. G.A. Maugin, W. Muschik, Thermodynamics with internal variables, part II. Applications. J. Non-Equilib. Thermodyn. 19, 250–289 (1994) 10. G.A. Maugin, The Thermomechanics of Plasticity and Fracture (Cambridge University Press, Cambridge, 1992) 11. C. Truesdell, W. Noll, Non-Linear Field Theories of Mechanics, Encyclopedia of Physics, Vol. III/3, Sect. 98. (Springer, Berlin, 1965) 12. J.L. Ericksen, Anisotropic fluids. Arch. Rat. Mech. Anal. 4, 231 (1960)

234

12 Outlook: Mesoscopic Theory of Complex Materials

13. F.M. Leslie, Some constitutive equations for liquid crystals. Arch. Rat. Mech. Anal. 28, 265– 283 (1968) 14. S. Hess, Irreversible thermodynamics of nonequilibrium alignment phenomena in molecular liquids and in liquid crystals. Z. Naturforsch. 30a, 728–733 (1975) 15. S. Hess. Irreversible thermodynamics of nonequilibrium alignment phenomena in molecular liquids and in liquid crystals ii. Z. Naturforsch. 30a, 1224–1232 (1975) 16. S. Hess. Fokker-Planck-Equation Approach to Flow Alignment in Liquid Crystals. Z. Naturforsch. 31a, 1034–1037 (1976) 17. G.A. Maugin, R. Drouot, Thermodynamic modelling of polymers in solution. in Constitutive Laws and Microstructure, Axelrad, W. Muschik (Springer Verlag, Berlin, Wien New York, 1988), pp.137–161 18. C. Papenfuss, chapter Dynamics of Internal Variables from the Mesoscopic Background for the Example of Liquid Crystals and Ferrofluids, in Applied Wave Mathematics - Selected Topics in Solids, Fluids, and Mathematical Methods, (Springer, Berlin Heidelberg, 2009), pp.89–125 19. S. Blenk, H. Ehrentraut, W. Muschik, A continuum theory for liquid crystals describing different degrees of orientational order. Liq. Cryst. 14(4), 1221–1226 (1993) 20. C. Papenfuss, Mesoscopic continuum theory for liquid crystals. AAPP Atti Della Accad. Peloritana Dei Pericolanti 97(S1), A21 (2019) 21. H. Ehrentraut. A unified mesoscopic continuum theory of uniaxial and biaxial liquid crystals. PhD Thesis, Technische Universität Berlin, W und T Verlag, Berlin, 1996 22. S. Blenk, H. Ehrentraut, W. Muschik, Orientation balances for liquid crystals and their representation by alignment tensors. Mol. Cryst. Liqu. Cryst. 204, 133–141 (1991) 23. H. Ehrentraut, W. Muschik, C. Papenfuss, Mesoscopically derived orientation dynamics of liquid crystals. J. Non-Equilib. Thermodyn. 22, 285–298 (1997) 24. C. Papenfuss, Nonlinear dynamics of the alignment tensor in the presence of electric fields. Arch. Mech. 50(3), 529–536 (1998) 25. W. Muschik, C. Papenfuss, H. Ehrentraut, Sketch of the mesoscopic description of nematic liquid crystals. J. Non-Newton. Fluid Mech. 119(1–3), 91–104 (2004) 26. S. Blenk, W. Muschik, Orientational balances for nematic liquid crystals. J. Non-Equilib. Thermodyn. 16, 67–87 (1991) 27. C. Papenfuss, Theory of liquid crystals as an example of mesoscopic continuum mechanics. Comput. Mater. Sci. 19, 45–52 (2000) 28. W. Muschik, C. Papenfuss, H. Ehrentraut, Mesoscopic theory of liquid crystals. J. Non-Equilib. Thermodyn. 75–106 (2004) 29. T. Carlsson, F.M. Leslie, Behaviour of biaxial nematics in the presence of electric and magnetic fields. Liq. Cryst. 10(3), 325–340 (1991) 30. H. Brand, H. Pleiner, Hydrodynamics of biaxial discotics. Phys. Rev. A 24(5), 2777–2779 (1981) 31. S. Blenk, H. Ehrentraut, W. Muschik, Macroscopic constitutive equations for liquid crystals induced by their mesoscopic orientation distribution. Int. J. Eng. Sci. 30(9), 1127–1143 (1992) 32. S. Blenk, H. Ehrentraut, W. Muschik, C. Papenfuss, Mesoscopic orientation balances and macroscopic constitutive equations of liquid crystals, in Proceedings of 7th International Symposium on Continuum Models of Discrete Systems. Materials Science Forum, vol. 123-125, (Paderborn, Germany, June 1992), pp. 59–68 33. S. Blenk, W. Muschik, Orientational balances for nematic liquid crystals describing different degrees of orientational order. ZAMM 72(5), T400–T403 (1992) 34. S. Blenk, W. Muschik, Mesoscopic concepts for constitutive equations of nematic liquid crystals in alignment tensor formulation. ZAMM 73(4–5), T331–T333 (1993) 35. W. Muschik, C. Papenfuss, H. Ehrentraut, Alignment tensor dynamics induced by the mesoscopic balance of the orientation distribution function. Proc. Estonian Acad. Sci. Phys. Math. 46, 94–101 (1997) 36. W. Muschik, H. Ehrentraut, C. Papenfuss, S. Blenk, Mesoscopic theory of liquid crystals, in 25 Years of Non-Equilibrium Statistical Mechanics, Proceedings of the XIII Sitges Conference,vol. 445, ed. by J.J. Brey, J. Marro, J.M. Rubi, M. San Miguel (Sitges, 13–17 June 1994), Lecture Notes in Physics, (Springer, 1995), pp. 303–311

References

235

37. E. Cosserat, F. Cosserat, Sur la mechanique énérale. Acad. Sci. Paris 145, 1139–1142 (1907) 38. A.C. Eringen, Micropolar theory of liquid crystals, in Liquid Crystals and Ordered Fluids, vol. 3, ed. by J.F. Johnson, R.S. Porter (Plenum Press, New York, 1978), pp. 443–474 39. A.C. Eringen, J.D. Lee, Relations of two continuum theories of liquid crystals, in Liquid Crystals and Ordered Fluids, vol. 2, ed. by J.F. Johnson, R.S. Porter (Plenum Press, New York, 1974), pp. 315–330 40. A.C. Eringen, Theory of micropolar fluids. J. Math. Mech. 16, 1–18 (1966) 41. A.C. Eringen, Simple microfluids. Int. J. Eng. Sci. 2, 205–217 (1964) 42. S. Blenk, H. Ehrentraut, W. Muschik, Statistical foundation of macroscopic balances for liquid crystals in alignment tensor formulation. Physica A 174, 119–138 (1991) 43. C. Papenfuss, W. Muschik, Orientational order in free standing liquid crystalline films and derivation of a closure relation for higher order alignment tensors. Mol. Cryst. Liq. Cryst. 330, 541–548 (1999) 44. H. Ehrentraut, S. Hess, Viscosity coefficients of partially aligned nematic and nematic discotic liquid crystals. Phys. Rev. E 51(3), 2203–2212 (1995) 45. D. Baalss, The viscosity coefficients of biaxial-nematic liquid crystals. Phenomenology and affine transformation model. Z. Naturforsch. 45a(1), 7–13 1990 46. D. Baalss, S. Hess, The viscosity coefficients of oriented nematic and nematic discotic liquid crystals; affine transformation model. Z. Naturforsch 43a, 662–670 (1988)

Index

A After effect, 77, 80 Alignment tensor, 61, 229 Anisotropic viscosity, 62 Antisymmetric stress tensor, 89

D Díffusion flux, 53 Deformation gradient, 6 measure, 8 Dufour effect, 53

B Balance of angular momentum, 14, 18 charge, 26 energy, 14, 19 entropy, 21 internal energy, 19 mass, 12, 17 momentum, 13, 17 spin, 18 Body, 7, 17 Burgers’ equation, 184

E Entropy condition, 193 Equilibrium, 29 conditions, 135 state space, 29 Equipresence, 114 Eulerian description, 6 Extended thermodynamics, 43, 161 of irreversible processes (TIP), 164

C Cattaneo, 163 Cauchy problem, 182 Cauchy strain, 9 Characteristics, 179 Class of materials, 126 Colloid suspensions, 71 Configuration mapping, 6 Constitutive equations linear, 47 Couette flow, 74 Couple stresses, 89 Curie principle, 47

F Flexible fibers, 80 Flow birefringence, 70 Flux convective, 17 non-convective, 17 Fourier’s law, 50 of heat conduction, 161

G Gibbs’ equation, 32, 46 Gibbs surface, 200 Gyarmati’s wave approach, 106

H Heat-conducting fluid, 130

© Springer Nature Switzerland AG 2020 C. Papenfuß, Continuum Thermodynamics and Constitutive Theory, https://doi.org/10.1007/978-3-030-43989-7

237

238 Higher derivatives, 131 Hugoniot equation, 188

I Index notation, 7 Infinite speed, 162 of propagation, 162 Internal variable, 42, 59, 137

L Lagrangian description, 6 Landau theory, 62, 68 Large state space, 36 Law of thermodynamics second, 48 Lax criterion, 192 Liquid crystals, 60 Liu relations, 132 Liu technique, 129 Local equilibrium, 45, 133

M Material damage, 141 frame indifference, 115 symmetry, 116 time derivative, 8 Mechanical model systems, 95 Mesoscopic balance equations, 223 theory, 219 Micropolar continuum, 89 Mixture, 22, 50 Momentum, balance of, 17

N Newtonian fluid, 50 Non-Newtonian fluid, 76 Non-reversible direction axiom, 127, 128

O Objectivity, 115 Observer, 5 Onsager relations, 47 Orientation distribution function, 225

Index P Parabolic differential equation, 162 Partial balance equations, 22 Peltier effect, 56 Process direction, 127

R Rational extended thermodynamics, 170 thermodynamics, 113 Relaxation of the alignment tensor, 68 Representation theorem, 120 Residual inequality, 145 Reynolds’ transport theorem, 15

S Schottky systems, 11 Second law, 48 of thermodynamics, 31, 126 Seebeck effect, 55 Shape memory alloy, 40 Small state space, 36 State space, 35 Surface geometry, 201 Suspensions of flexible fibers, 147 Symmetric hyperbolic system, 174

T Thermal diffusion, 53 Thermodynamics of irreversible processes (TIP), 45 Thermoelement, 54

V Viscosity, 71 Viscous fluid, 130 Viscous, heat-conducting fluid, 48

W Wave equation, 181

Z Zeroth law of thermodynamics, 29