Continuum Thermodynamics - Part Ii: Applications And Examples

Part II: Applications and Examples

Bettina Albers, Krzysztof Wilmanski

• Publisher: WSPC
• Year: 2014
• eBook ISBN: 9789814412391
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

This second part of Continuum Thermodynamics is designed to match almost one-to-one the chapters of Part I. This is done so that the reader studying thermodynamics will have a deepened understanding of the subjects covered in Part I. The aims of the book are in particular: the illustration of basic features of some simple thermodynamical models such as ideal and viscous fluids, non-Newtonian fluids, nonlinear solids, interactions with electromagnetic fields, and diffusive porous materials. A further aim is the illustration of the above subjects by examples and simple solutions of initial and boundary problems as well as simple exercises to develop skills in the construction of interdisciplinary macroscopic models.

Contents:

• Introduction
• Geometry of Deformations of Solids
• Kinematics of Continua in Different Descriptions
• Balance Equations
• Some Solutions for Fluids and Solids
• Stability
• Thermodynamical Problems
• Extended Thermodynamics of Jou–Casas-Vázquez–Lebon
• Dislocations
• Acoustic Waves
• Interaction of Ponderable Bodies with Electromagnetic Fields
• Mechanics of Porous Materials
• Thermodynamics of Porous Materials with the Porosity Balance
• Appendices:
  • Basic Notions
  • Integral Transforms
  • Green Functions
  • Bessel Functions and Bessel Equation
  • Basic Physical Units

Readership: Materials scientists and physicists.
Key Features:

• A unique combination of continuum thermodynamics and numerous models of practical bearing
• Confrontation of various approaches to modern continuum thermodynamics
• New and unpublished results on complex models of immiscible mixtures

Chapter 1

Introduction

The intention by writing Part II of the book on continuum thermodynamics was the deepening of some issues covered in Part I as well as a development of certain skills in dealing with practical problems of macroscopic processes. However, the main motivation for this part is the presentation of main facets of thermodynamics which appear when interdisciplinary problems are considered. There are many monographs on the subjects of solid mechanics and thermomechanics, on fluid mechanics and on coupled fields but most of them cover only special problems in great details which are characteristic for the chosen field. It is rather seldom that relations between these fields are discussed. This concerns, for instance, large deformations of the skeleton of porous materials with diffusion (e.g. lungs), couplings of deformable particles with the fluid motion in suspensions, couplings of adsorption processes and chemical reactions in immiscible mixtures with diffusion, various multi-component aspects of the motion, e.g. of avalanches, such as segregation processes, etc.

As mentioned in the preface, Part II has the same structure as Part I. This means that the basic notions, the foundations and some further considerations related to a certain topic which is treated in the present part can be found in the corresponding chapter in Part I. However, we are endeavored to present a summary of the basics also at the beginning of the chapters in Part II. The theoretical expositions in Part I often go beyond the necessary theory used in practical examples.

In some applications we rely on the work of other scientists. In order not to adulterate such theories and examples, we have predominantly kept the original notation. Therefore it may happen that the notation throughout the book is not uniform in all places. However, where possible we tried to conform the notation of Part II to that of Part I.

Contents of the book

In the following three chapters some foundations of continuum mechanics are repeated.

Chapter 2 deals with the geometry of deformations of solids. We introduce reference and current configurations, the function of motion and the corresponding deformation gradient. Different measures of deformation are compared and a generalized measure is presented and illustrated. The polar decomposition of the deformation gradient is addressed. This important property is taken up in Appendix A.1 where mathematical basics are shown on an example of polar decomposition. Furthermore, geometrical aspects of universal solutions are discussed. Such solutions are the foundation of experimental verification of constitutive relations for many materials under static conditions. In the literature classified families are introduced and corresponding deformation tensors are shown. The description comes about in rectangular Cartesian, cylindrical polar or spherical polar coordinate systems. These and some other coordinate systems are the topic of Appendix A.2.

In Chapter 3 the time dependence of motion is also accounted for. The kinematics of continua in both material or Lagrangian and in spatial or Eulerian description is presented. Also the transformation properties of vectors and tensors are shortly addressed. For porous media the Lagrangian description is often used with the skeleton as reference. We show an example clarifying the Lagrangian description of relative motion.

Chapter 4 is concerned with balance equations. Global and local balance equations for regular and singular points are summarized and the relation between Cauchy and Piola-Kirchhoff stresses is pointed out.

In Chapter 5 we focus for the first time on material behavior. The discussion of ideal fluids is preceded by the introduction of the d’Alembert paradox. We discuss its origin within the frame of the general momentum conservation law. Due to its role in the theory of boundary layers and in the linear modeling of porous materials, in regard to ideal fluids a simple example is presented which is connected with the d’Alembert paradox and the added mass effect. Afterward the Navier-Stokes equation for the description of viscous fluids, its thermodynamical properties and the uniqueness of solutions are addressed. We mention two types of viscous flow, namely lamellar and creeping flows. One section is devoted to the boundary layer theory basing on ideas of Ludwig Prandtl. These considerations are followed by the investigation of Maxwell and N-th grade fluids. Several examples for viscometric flows, namely plate-and-plate, cone-and-plate, Couette flow and Poiseuille flow are considered. In the section on nonlinear elastic solids rubber-like materials are studied. As examples of homogeneous deformations, for isochoric extension and simple shear, stresses for compressible and incompressible materials are investigated. We introduce Ericksen’s Theorem and close the section with an example on heterogeneous deformations, namely pure torsion of a circular cylinder. In the following section viscoelastic solids are considered. They belong to the broad class of simple materials in which the set of constitutive variables consists of the deformation gradient, the temperature and the gradient of temperature. Formally, their response in all processes is determined by the response to all homogeneous thermokinetic processes. Examples of simple rheological models indicate that the constitutive relations have the form of evolution equations. We introduce the Kramers-Kronig relation which states that viscoelastic materials are inherently dispersive, i.e. the propagation speed of the mechanical disturbance is frequency dependent. Furthermore, we introduce the correspondence principle. It states that the complex viscoelastic moduli can be replaced by those of the elasticity theory. This reveals the possibility of converting numerous static solutions of elasticity into quasi-static solutions of viscoelasticity. We demonstrate the application of the correspondence principle on a simple example, the axial symmetric problem of a cylinder under given radial loading on both lateral surfaces.

Chapter 6 is devoted to the stability analysis of continua. We show a few characteristic examples, as for instance, the stability properties of some flows of fluids (the torsional Couette flow and the Rayleigh-Bénard problem), large static deformations of solids (stability of a nonlinear elastic strip) and thermodynamic equilibrium states of some continuous systems (second-grade fluids). The concern of the stability analysis of thermomechanical systems is not only with applications and particular engineering problems. It yields as well important information on fundamental properties of thermodynamical models. In particular, it prescribes the ranges of material parameters in which such models are physically acceptable and relevant.

In Chapter 7 some thermodynamical problems are presented and solved. The first section concerns heat conduction problems described in Cartesian, cylindrical and spherical coordinates. While for some problems the solution involves only one space variable and the time, in others two or more space variables are involved. There are different methods to obtain solutions. The simplest case is present if the solutions can be expressed as a product of solutions of one-variable problems. Moreover, multiple Fourier series or their generalizations or Green’s function can be used. Another possibility is the direct application of the Laplace transformation method. All the mentioned methods are treated in the Appendix. Fourier transforms are the topic of Appendix B.1, Laplace transforms of Appendix B.2. Appendix C is devoted to Green functions – both the static and dynamic cases of isotropic elastic materials are considered. Finally, in Appendix D Bessel functions and the Bessel equation are introduced because they are needed in Chapter 7 to describe the solutions of some problems. Of course, the solutions for steady temperature are less complex than those for variable temperature. Examples for both situations are presented. Since they have considerable importance in practice, in the next section anisotropic media are investigated. As an example, the conduction in a thin crystal plate is studied – the general theory of flow, without any assumptions on symmetry is developed. The third section of this chapter is devoted to thermal boundary layers. They appear in many practical applications such as phase transformations (melting, evaporation, solidification, condensation, etc.) but also problems of heat transfer between civil engineering constructions and environment, air conditioning systems, etc. contain field equations whose boundary conditions concern transition regions in which thermal boundary layers appear. Composite beams with embedded shape memory alloy form the last thermodynamical problem demonstrated in this chapter. Shape memory alloys constitute a class of functional, smart materials which have found many technological applications and offer innovative solutions in the design of adaptive structures. They may undergo a temperature- or stress-induced martensitic phase transformation resulting in the shape memory effect and pseudoelastic behavior.

Chapter 8 reveals an introduction to Extended Thermodynamics in the version of Jou-Casas-Vázquez-Lebon. They were primarily motivated by the non-equilibrium statistical mechanics and, in particular, by the so-called Fluctuation-Dissipation Theorem. The ideas of this microscopic theory are presented, and, in conclusion we comment on common points and on main differences between this version of Extended Thermodynamics and that of Müller-Liu-Ruggeri which was the subject of Chapter 8 of Part I. As was done in Part I the example of ideal gases is studied.

It is well known that dislocations are the source of plastic deformation. In Chapter 9 we present some properties of discrete dislocations as well as a continuum model of these defects in crystalline materials. Dislocations are line-defects characterized by the Burgers vector which, in turn, is defined in a crystal by the Burgers contour. However, dislocations do not only play a role in crystal materials but also in the modeling of rupture of tectonic plates yielding earthquakes. This application is briefly presented at the end of Chapter 9. Various other defects may exist which are not that easily to describe. In spite of their practical importance, such defects are not described in this book. However, in some way we are coming back to such a problem in Chapter 12 where in the description of freezing and thawing processes also the creation of microcracks is of importance.

Chapter 10 is on acoustic waves. We start with a general discussion on parabolic and hyperbolic models. Afterward the propagation of acoustic waves in nonlinear materials with memory is accounted for. An approximate solution in the vicinity of the front is constructed. Several dynamical problems of continua are the topic of Section 4. Not only waves in fluids and fluid layers with different boundaries are addressed but also waves in linear solids are considered. Both bulk and surface waves appear in different situations under consideration. Also cylindrical surfaces are mentioned because they appear quite often in geotechnics – for instance, they occur in the analysis of wave in boreholes. A particular class of waves, leaky waves, is mentioned in Section 5. These are such waves whose energy is transferred on some other modes. The last section of this chapter is devoted to bulk and surface waves in viscoelastic solids. The motivation for these investigations is the modeling of soils and rocks by means of viscoelastic materials. Such models are not multi-component but the diffusion process yields naturally a viscous character of the material modeled by a single component continuum.

In Chapter 11 interactions of ponderable bodies with electromagnetic fields are examined. Modern technologies yield discoveries of effects and devices whose description requires more sophisticated models than these considering thermomechanical properties alone. Many questions of the construction of macroscopic models can be answered only with the help of modern continuum thermodynamics. Some of these questions have already been mentioned in Part I. In this chapter we extend the subject and discuss also some issues of plasmas. To this aim at first the Maxwell theory of electromagnetism is summarized. Afterward the coupling of thermomechanical and electromagnetic fields is addressed. Finally, several magnetohydrodynamical models of plasmas are introduced and the stability of plasmas is discussed.

Chapters 12 and 13 deal with porous materials. While in Chapter 12 a few problems of special behavior of multi-component porous materials are shown, Chapter 13 focuses on chosen examples for which the model with the balance equation of porosity introduced by K. Wilmanski is used.

Thus, Chapter 12 begins with a summary of the theory of immiscible mixtures which is the basis of many models used for porous materials, and with an outline of the description of multi-component porous materials in Lagrangian and in Eulerian way. In the next section two-component models with constitutive relations for the porosity are introduced. Both models for incompressible and compressible components are considered and their thermodynamic admissibility is inspected. A further section of this chapter is devoted to double- and multi-porosity models. First, the original field of application of double porosity is given attention to: fissured rocks. Such materials consist of pores and permeable blocks, the blocks separated from each other by a system of fissures. Thus, the coefficient of fissuring of the rock builds one porosity and the porosity of the individual blocks is the second one. We repeat here Barenblatt’s model for such media which, obviously, not only contains two different porosities but also two permeabilities, two pressures etc. A second example of multi-porosity models is presented, namely swelling media which appear mainly in bioengineering. We are concerned with ionized porous structures imbibed with electrolyte solutions in which interfacial phenomena often determine the macroscopic behavior. A further biomechanical example is presented: topics of the next section are soft tissues. They behave anisotropically because their fibers have preferred directions. They undergo large deformations and some of them show viscoelastic behavior. We point out the structure of these materials, introduce a model and show an example.

Chapter 13 starts with a summary of the balance equation of porosity and associated models. In the following sections these models are applied. The first examples are freezing and thawing processes. An iterative procedure for the calculation of the mechanical properties is shown. It incorporates two different stages of the process according to the actual temperature. First, isothermal diffusion in the poroelastic range without freezing is considered while the second range contains the process of freezing. The model for the latter range is based on the Gurson-Tvergaard-Needleman theory for plastic deformations. The measure of damage is described by the extent of the porosity changes caused by freezing. Section 3 is concerned with the linear stability of a 1D flow under transversal disturbance with adsorption. The disturbances satisfy equations of the model for multi-component systems with adsorption. It is considered that a fluid/adsorbate mixture flows through the channels of a skeleton. In this case a kinematic nonlinearity acts against the permeability of the medium. Adsorption processes contribute in a nonlinear way to the field equations and essentially influence the stability properties. In Section 4 we study the wave propagation in porous media with anisotropic permeability. We investigate a model in which the stress-strain relations are isotropic but the tortuosity is not. The anisotropy of tortuosity yields essential changes of the attenuation of the waves depending on the direction of propagation in relation to the principal directions of tortuosity and on the mode of the wave. Also in the last section the wave propagation is discussed. A linear model for three-component media is shown. In such media the speeds and attenuations of the waves depend not only on the frequency but also on the degree of saturation. The capillary pressure between the pore fluids is one of the most important quantities entering the hyperbolic model.

In the end, in Appendix E, the basic physical units are listed which are used throughout the book in the presented examples and applications.

Chapter 2

Geometry of deformations of solids

2.1 Summary: Geometry

For a better understanding, some basics of continuum mechanics concerning the geometry are summari...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Dedication
• Preface
• Contents
• 1 Introduction
• 2 Geometry of deformations of solids
• 3 Kinematics of continua in different descriptions
• 4 Balance equations
• 5 Some solutions for fluids and solids
• 6 Stability
• 7 Thermodynamical problems
• 8 Extended thermodynamics of Jou–Casas-Vázquez–Lebon
• 9 Dislocations
• 10 Acoustic waves
• 11 Interactions of ponderable bodies with electromagnetic fields
• 12 Mechanics of porous materials
• 13 Thermodynamics of porous materials with the porosity balance
• Appendices
• Bibliography
• Index of Researchers
• Subject Index