Mechanics of mixtures

MECHANICS /MIXTURES

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Series on Advances in Mathematics for Applied Sciences - Vol. 35

MECHANICS "/MIXTURES K. R. Rajagopal Department of Mechanical Engineering University of Pittsburgh, USA

L. Tao Viscoustech Inc., Pittsburgh, USA

World Scientific Singapore • New Jersey • London •• Hong Kong

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British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Library of Congress Cataloging-in-Publication Data Rajagopal, K. R. (Kumbakonam Ramamani) Mechanics of mixtures / K.R. Rajagopal, L. Tao. p. cm. — (Series on advances in mathematics for applied sciences ; vol. 35) Includes bibliographical references and index. ISBN 9810215851 1. Fluid dynamics. 2. Continuum mechanics. I. Tao, L. II. Title. III. Series. QA911.R27 1995 532'.05-dc20 95-16400 CIP

Copyright © 1995 by World Scientific Publishing Co. Pte. Ltd. AH rights reserved. This book, or parts thereof may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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This book is dedicated to my parents Hemalata and K. R. Ramamani, with love and affection, K. R. Rajagopal, and to my late uncle Ruantao Zhang,

L. Tao.

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Preface An author usually faces the onerous task of justifying the need for his or her treatise, as there are few subjects left where not enough has been written. Fortunately for us, the continuum theory of mixtures is one of the few areas where it is unnecessary to plead the case for a book. While there are many mathematical treatments on the subject matter of this book, their approach is quite different from that which is presented here. A few review articles on mixture theory have come close to providing a comprehensive overview of the theoretical foundations of the topic; however they do not discuss, in any detail, initial and boundary value problems within the context of the theory. It is to this aspect of the subject that our book is primarily devoted. Our bias towards applications can be felt throughout the book, though the last two chapters are devoted to developing new approaches to modelling a mixture of two fluids and a fluid infused with solid particles with a special class of applications in mind. We have tried to inject new ideas or look at old problems within the context of a new perspective, wherever possible in the book. For instance, the exact boundary conditions that are to be prescribed for boundary value problems involving mixtures in general are far from settled. Boundary conditions based on thermodynamic argu­ ments have been used previously to analyze steady state problems. These boundary conditions are valid when the boundary of the mixture is saturated. However, there are situations when such a condition does not obtain on the boundary. Thus, we have revisited some steady state boundary value problems and solved them within the con­ text of a new set of boundary conditions that are based on a very different philosophy from that adopted in earlier attempts at resolving these problems. Similarly, in the treatment of a mixture of a fluid infused with solid particles in this book, we present an approach that is markedly different from my earlier treatment of such problems with Johnson and Massoudi. We envisage a different class of problems where we feel our present approach is applicable. The presentation here concerning a. mixture of two fluids also sets forth an approach that might be extended to a turbulent motion of the mixture of two fluids. Here, we recognize that the turbulent motion of a single constituent is as yet not well understood. However, given the fact that mixtures do undergo turbulent motion, it is necessary to address this problem even if the attempt is crude at best. This book is not meant to be a text on the theory of mixtures, as it does not VII

Vlll

Preface

develop the basic frame-work for the study of mixtures in the kind of detail and completeness that a beginning graduate student needs. The book presumes that the reader is quite conversant with continuum mechanics. No exercises or examples are provided, and in many instances, results from papers published recently in the literature are used, without recording them in any detail. Issues that we consider to be unresolved are discussed with the hope that it might stimulate the reader to think about them. A great many assumptions, approximations and simplifications, possibly oversimplifications are made in order to put the theory to use, presenting the opportunity for one with more ability to address the same problems in a more rational and rigorous manner. My interest in the continuum theory of mixtures stems from my interactions with Alan Wineman at the University of Michigan at Ann Arbor. The numerous discus­ sions that we had have influenced my thinking about the mechanics of mixtures, and is reflected in all my work on mixtures. I would be remiss if I did not acknowledge the contributions of my teachers and collaborators, too numerous to mention individ­ ually, from whom I have learnt a great deal of continuum mechanics over the years. Of course, it goes without saying, that any errors or misunderstandings of the subject are solely attributable to me.

K. R. Rajagopal Pittsburgh January 1995

Contents 1

INTRODUCTION

1

2

PRELIMINARIES 2.1 Kinematics 2.2 Partial Tractions and Partial Stresses . . . 2.3 Balance of Mass 2.4 Balance of Linear Momentum 2.5 Balance of Angular Momentum 2.6 Conservation of Energy 2.7 Second Law of Thermodynamics 2.8 Volume Additivity Constraint . . . . .

... ... . . . .

. . . .

. . .. .. .. ..

7 7 10 10 11 14 15 16 18

3

D I F F U S I O N O F A F L U I D T H R O U G H A SOLID U N D E R G O I N G LARGE DEFORMATIONS: CONSTITUTIVE RESPONSE FUNC­ TIONS 21

4

S T E A D Y STATE PROBLEMS 4.1 Boundary Conditions 4.2 Diffusion of a Fluid Through a Rubber Slab 4.3 Diffusion of a Fluid Through a Rubber Annulus . .

29 30 37 46

5

DIFFUSING SINGULAR SURFACE 5.1 Analysis of a Time-dependent Diffusing Singular Surface 5.2 Unsteady Diffusion of a Fluid Through a Nonlinearly Elastic Solid Slab 5.3 Unsteady Diffusion of a Fluid Through a Nonhnearly Elastic Solid Cylindrical Annulus . . . . . . .

57 58 75

6

W A V E P R O P A G A T I O N I N SOLIDS I N F U S E D W I T H F L U I D S 6.1 Biot's Equations 6.2 Counterpart of Biot's Equations 6.3 Wave Propagation in Anisotropic Swollen Mixtures ix

87 99 100 105 110

x

7

8

MECHANICS OF

M I X T U R E OF T W O N E W T O N I A N F L U I D S 7.1 Basic Balance Equations 7.2 Constitutive Variables 7.3 Constitutive Modelling 7.4 Two Special Motions M I X T U R E OF A F L U I D A N D SOLID PARTICLES 8.1 Turbulence Modelling in Newtonian Fluids 8.2 Balance Equations for the Mixture 8.3 On Constitutive Modelling 8.4 Specific Model

A S O M E RESULTS F R O M D I F F E R E N T I A L G E O M E T R Y

MIXTURES

. . . .

...

123 124 125 128 131 139 142 149 154 158 163

B STATUS OF D A R C Y ' S LAW W I T H I N T H E C O N T E X T OF M I X ­ TURE THEORY 167 Bibliography

171

Index

191

Chapter 1 INTRODUCTION The Oxford English Dictionary [l] defines a mixture as "a product of mixing, a com­ plex unity or aggregate (material or immaterial) composed of various ingredients or constituent parts mixed together." It goes on to describe a mixture as a "mixed state or condition; co-existence of different ingredients or of different groups or classes of things mutually diffused through each other." Most bodies, astrophysical, geological, biological or otherwise are, in the spirit of the above definition, mixtures in which two or more constituents co-exist; if one of the constituents is preponderant and the other constituents essentially insignificant, the body is usually assumed to be of the pre­ dominant single constituent. However, there are innumerable situations where none of the constituents present in the body can be ignored with respect to the others; plasmas and gaseous mixtures that surround stars, biological tissues and muscles, slurries and suspensions, porous rocks and soil infused with water or oil, and gaseous mixtures that occur naturally are examples that come immediately to mind. Some­ times, the constituents in a mixture undergo chemical reactions, usually resulting in an interconversion between the constituents and the formation of new compounds. However, there is a large class of mixtures whose constituents do not react chemically and it is such mechanical mixtures that will be studied in this book. The importance of the study of mixtures was recognized long ago and mathe­ matical models have been developed to describe the flow of such mixtures. In this context the pioneering works of Fick [2] and Darcy [3] deserve special mention. These early attempts were aimed at describing the diffusion or flow of one constituent of the mixture through another, and they are all the more remarkable in that the predic­ tions of such theories provide useful insights for a large class of problems. Due to the ubiquity of mixtures, there have been innumerable studies that have drawn upon the fundamental works of Fick and Darcy and extended and modified them. There have also been innovative and novel approaches to the study of mixtures that do not follow from the studies of Fick and Darcy. These studies are too numerous to be cited. We shall mention some of them with the caveat that many important and even landmark works on mixtures are not cited here and there is no attempt at completeness with 1

2

MECHANICS OF MIXTURES

respect to the references to previous work. The treatises on the flow of homogeneous fluids through porous media that deserve mention are the early text by Muskat [4] and the more recent ones by Polubarinova-Kochina [5] and Scheidegger [6]. Detailed treatment of various aspects of diffusion can be found in Barrer [7], Jost [8], Bosworth [9], Carman [10], and Crank [11]. A detailed discussion of diffusion involving chemical reaction can be found in the treatise by Frank-Kamenetskii [12]. The early papers on statistical approaches to solutions that take into account intermolecular interactions are due to Kirkwood and Buff [13], Pearson and Rushbrooke [14] and Mazo [15]. A succinct discussion of the statistical approaches to mixtures can be found in the article by Mansoori and Matteoli [16]. Early mathematical studies of diffusion, basically the heat equation or some gen­ eralizations of it, bear the illustrious names of Fourier [17], Stokes [18], Lame [19], Stefan [20], Kirchhoff (cf. Kirchhoff and Hanseman [21]), Rayleigh [22], Sommerfeld [23], Boltzmann [24], Poincare [25], Boussinesq [26], and Volterra [27], to name just a few. Extensive literature relating to the heat equation can be found in Carslaw [28] and Carslaw and Jaeger [29]. The works of Fick and Darcy were extended in several ways, for a variety of mixtures. Graham [30] studied the diffusion of gases in rubber, Wroblewski [31] carried out some of the earliest experiments for such diffusion processes. Later, Hufher [32] also studied the diffusion of gases in rubber. Important early contributions to the diffusion of one solid through another are due to Graetz [33] and Arrhenius [34]. Copious references to studies of interdiffusion of solids can be found in the treatise by Barrer [7]. While the predictions of Fick's law and Darcy's law provide good agreement in many physical situations, large departures from the predictions of these laws are also possible in practice. Various generalizations have been proposed to Fick's law and Darcy's law so that they can provide a better account of the mechanics of diffusion in which inertial effects of the diffusant are significant. Also, generalizations have been proposed, in problems associated with the diffusion of fluids through a porous solid that is undergoing large deformations, which usually involve the diffusivity or perme­ ability to depend on various kinematical quantities. However, such generalizations are always almost ad hoc. Additionally, Fick's law and Darcy's law are quite inadequate in providing some important information on diffusion processes. For instance, they cannot predict the stresses or strains in the solid, or the flow field of the fluid. Such information is critical in many problems where the solid may fracture, rupture or be­ come otherwise damaged due to the stresses or strains that are induced in them and we can easily recognize the importance of these issues in the mechanics of geological and biological materials. We can overcome these inadequacies, by appealing to the theory of mixtures. The theory of mixtures or mixture theory provides a rational framework for study­ ing problems involving mixtures and traces its origins to the seminal work of Fick (cf. Truesdell [35,36]). The basic premise of the theory is that the space occupied by

INTRODUCTION

3

a mixture can be considered as being occupied co-jointly by the various constituents of the mixture, each considered as a continuum in its own right. Thus, at each point in the space occupied by the mixture, there will be a particle belonging to each of the constituents. This presupposes that each constituent is sufficiently dense in the mixture, and not all real mixtures or suspensions qualify to be modeled thus. Here, we shall only consider mixtures wherein the constituents are dense enough that they can be homogenized over the region of the mixture as a continuum. Each of these homogenized continua move relative to the others, due to the application of forces. For each constituent we can define its motion, mass density, partial stress tensor, internal energy, temperature, entropy and other relevant physical quantities, as we do for a single constituent continuum. The theory of mixtures allows for the interconversion of mass between the vari­ ous constituents. Balance laws for mass, linear momentum, angular momentum and energy take into account contributions of any of these quantities with regard to a particular constituent due to the influence of the other constituents. An appropriate thermomechanical setting can also be provided within which to study mixtures that takes into account the basic laws of thermomechanics. It has been shown that ap­ propriate approximations to such a general theory of mixtures lead to the classical theories due to Darcy and Fick. We shall discuss that approximation procedure in some detail later. Great strides have been made in developing a rational theoretical basis for studying the mechanics of mixtures starting with the pioneering work of Truesdell [35,36], fol­ lowed by those of Truesdell and Toupin [37], Truesdell [38], Adkins [39,40,41,42], Kelly [43], Green and Naghdi [44,45,46,47,48,49], Hayday [50,51], Bowen [52,53,54,55,56], Muller [57,58], Oliver [59], Gurtin, Oliver and Williams [60], Williams [61,62], Sampaio [63], Atkin and Craine [64,65] and others. Variational approaches have also been used to formulate the basic equations for the mechanics of mixtures, and references to these can be found in Bedford and Drumheller [66] and Bedford [67]. Much of the theoretical progress in the field up to 1984 is discussed in great detail in the various appendices in the book on Rational Thermodynamics by Truesdell [68] and the review articles by Bowen [56], Atkin and Craine [64], and Bedford and Drumheller [69]. A very succinct account of the early historical development of the theory of mixtures can be found in the article by Bowen [70]. Such progress in developing the theory notwithstanding, there are few solutions to initial-boundary value problems within the context of the theory. This, in some part, is due to the difficulty in the prescrip­ tion of boundary conditions in traction and mixed boundary value problems. This difficulty stems from the fact that while we know the total traction, which is the sum of the partial tractions, that act on the boundary, we do not know the individual par­ tial tractions. We do not encounter such difficulties in displacement boundary value problems or those in which the velocities are prescribed on the boundary. Even so, there are few boundary value problems that have been solved wherein displacements or velocities are prescribed on the boundary. Furthermore, the resolution of initial-

4

MECHANICS OF MIXTURES

boundary value problems within the context of any theory requires knowledge of the properties of the material being used, that can be only gleaned from careful experi­ ments. This is no mean task, even for a single constituent, if the material's response is sufficiently nonlinear. In the case of a mixture, this difficulty is greatly exacerbated as the theory involves material moduli not only for the individual constituents but also for the interactions between the constituents. Several variants of the theory of mixtures have been propounded to study prob­ lems of diffusion, phase change, and the flow of coexistent phases. A particularly interesting formulation is due to Roberts and his co-workers who have studied melt­ ing and solidification of alloys and other mushy mixtures with a modified version of the theory of mixtures. Loper and Roberts [71] model an alloy undergoing melt­ ing/solidification as a mixture of fine particles in a fluid. This was later extended by Hills, Loper and Roberts [72] and Hills and Roberts [73]. Hills and Roberts [74] studied stress-relaxation of a mushy region which responds like a viscoelastic mate­ rial. A review of the theories for modelling two phase mixtures can be found in Hills and Roberts [75]. Also worth mentioning is the analysis of Voller et al. [76] who study the mechanics of a mushy region by introducing terms in the balance of linear momentum that allows the equations to mimic Darcy's law. There is a great deal of other literature that use two phase models to study the problem of phase change but we shall not discuss them here. In Chapter 2 we introduce the basic notions of mixture theory. We briefly discuss the kinematics of mixtures and document the basic balance laws for the mass, linear momentum, angular momentum, and energy, as well as the second law of thermo­ dynamics. A very detailed discussion of the material presented in Chapter 2 can be found in the excellent review articles by Bowen [56] and Atkin and Craine [64]. We record the field equations relevant to the diffusion of a fluid through an elastic solid undergoing large deformations in Chapter 3. We keep our discussions to a minimum as lengthy treatments of the pertinent issues can be found in Shi [77], Shi, Rajagopal and Wineman [78], and Gandhi, Rajagopal and Wineman [79]. An early attempt to study traction boundary value problems of a mixture of a solid infused with a fluid undergoing finite deformations, which is not some ad hoc generalization of Fick's law, was that by Shi, Rajagopal and Wineman [78,80]. They augment the boundary conditions by appealing to a thermodynamic argument: they assume that the portion of the boundary of the mixture that is in contact with a pure liquid is saturated and thereby assume that the Flory-Huggins equation (cf. Treloar [81]) holds on the boundary. Later, Rajagopal, Wineman and Gandhi [82], motivated by the analysis of Treloar [81], derived a boundary condition based on the assumption that the boundary is saturated, which in turn implies that the variation in the Gibbs free energy of dilution is zero. This requires a relation between the stresses acting on the boundary, the various kinematical quantities and the specific volume of the fluid. Using this relationship as an additional boundary condition, various boundary value problems were solved (cf. Gandhi, Rajagopal and Wineman [79], Dai and Rajagopal

INTRODUCTION

5

[83], Dai, Rajagopal and Wineman [84]). As we shall see later, such an approach is not without its difficulties. Recently, Tao and Rajagopal [85] have developed an alternate method to obtain an additional boundary condition without appealing to the constraint of saturation and have solved boundary value problems using this condition. We shall discuss the difference between these approaches later. Suffice to say that both these methods provide a means for overcoming the indeterminacy in prescribing boundary conditions, and depending on the physical situation, one or the other could be employed. A detailed discussion of the method for generating additional boundary conditions and the resolution of several steady state boundary value problems are presented in Chapter 4. Unsteady motions of porous solids infused with fluids have relevance to problems in seismology, geophysics and groundwater motion. In such problems, chemical reac­ tions are unimportant and the problem can be studied within a mechanical setting. Such problems can involve the movement of singular surfaces in a mixture or singular fronts behind which we have a mixture and ahead of which there is a single con­ stituent. Recently, Tao, Rajagopal and Wineman [86] have developed a framework to study unsteady problems involving singular surfaces. The theory is general enough to account for curvature effects of the singular surface, capillarity and surface tension effects. This theory can be used to study several technologically important problems like groundwater motion, the spreading of contaminants, the salination of fresh water, dyeing of textiles, to name just a few examples. We shall discuss the work of Tao, Rajagopal and Wineman [86] in some detail, in Chapter 5. In the fifties, Biot [87,88] proposed a theory to describe the propagation of waves in a porous solid infused with a fluid. His theory is however based on the assumption that the solid undergoes infinitesimal strains, in addition to several others. Rajagopal and Tao [89] have studied the propagation of waves in both isotropic and anisotropic solids infused with fluids. They clearly delineate the numerous simplifying assumptions that need to be introduced to the general theory in order to obtain Biot's theory as a proper approximation. Clearly, Biot's theory cannot be used if the many assumptions that are tacit to his theory do not obtain, as is indeed the case in many interesting problems. We shall discuss the recent advances that have been made in this area, in Chapter 6. Numerous mathematical models have been developed to describe the mechanics of two fluids. In Chapter 7, we present an approach for studying the mixture of two fluids by appealing to some ideas of Soo [90]. Finally, in Chapter 8, we present a new theory for modelling the mechanics of a fluid infused with solid particles. We borrow ideas liberally from turbulence theories to develop our theory, and we propose a specific model that incorporates the effects due to virtual mass and drag. Finally, we wish to reiterate that the special emphasis of this book is the solution of initial-boundary value problems within the context of the theory of mixtures. The ideas developed here apply specifically to a mixture of two constituents, as additional constituents introduce additional difficulties associated with specifying boundary con-

6

MECHANICS OF MIXTURES

ditions for traction and mixed boundary value problems, as well as determining the constitutive functions. Much remains to be done within the context of mixture the­ ory, several interesting issues remain unresolved and we hope that the book serves to pinpoint some of them.

Chapter 2 PRELIMINARIES 2.1

Kinematics

A fundamental assumption in the theory of mixtures is that each point in the space occupied by the mixture is occupied by a particle belonging to each constituent. Such an assumption is clearly physically impossible, but it is appropriate if each of the constituents is dense enough in the sense that at each point in space, each neighborhood of the point (not too small) can be considered in a homogenized sense as a continuum of each constituent. Thus, in essence we have several co-existent continua in the space occupied by the mixture. As the mixture deforms, these co­ existent continua deform with respect to each other. We can now generalize the basic ideas in the mechanics of a single continuum (cf. Truesdell [91]) to that of several co-existent continua. Let us consider a mixture of n constituents. We shall denote by X 1 , i = 1, ..., n, a typical particle belonging to each constituent in the reference state. At time t, let these typical particles occupy the position x in the three dimensional Euclidean space. The motion of each constituent is defined through x = x'(x\t),

i = l,...,n.

(2.1)

The usual summation convention on repeated indices will not be adopted in this chapter. When summation is intended, it will be made explicit. We shall assume for each i, \ l >s one-to-one, continuous and invertible. The velocity of each of these particles, denoted by v l , is given by

v* = - £ , at

i = l,...,n.

(2.2)

We can define a mean velocity associated with the mixture in the following manner. Let pzR denote the mass density of the i-th constituent in its reference configuration, and let pl denote the mass density of the i-th. constituent in the current configuration, in the mixed state. The total mass density of the mixture p is given by 7

8

MECHANICS OF

n

I

= £/>'■ £p pP ==

MIXTURES

(2-3) (2-3)

The mean velocity of the mixture v is given by 1

n

. .

v = -£>V. V=

(2.4) (2.4)

P i=l

Of course there are different ways in which a mean velocity can be defined. Equation 2.4 gives the mass weighted average. The diffusion velocity u 1 of the i-th constituent is defined as u\x; { == v* v*---vv,,

(2.5) (2.5)

which is the velocity of the i-th constituent relative to the mean velocity. The deformation gradient F* associated with the i-th constituent is given by

dx { rp* = = a§ dXiy'

1 = 1 1,. .,n. *'* »-»»-

6 (2-(2.6) )

We shall assume that the deformation is such that the deformation gradient is nonsingular, i.e., detFVO, detF i 7 ^ 0 ,

i = l,...,n. l,. .,n.

(2.7) (2.7)

The left Cauchy-Green stretch tensor B* and the right Cauchy-Green stretch tensor C associated with the i-th constituent are defined through T

i C e i ==( F(F*) ' ) T F F\ ,

i i Ti ! r B Bi i = ¥Fi{F ( F) , ) ,

(2.8) (2.8)

l

the Green-St. Venant strain E is defined as

W-E? ==i i ( C(& ' - l-) , -*). v 2 1

(2.9) (2.9)

and the principal invariants of B are defined through

-tr( f

1

IP = -- - >(f)) 22 -- tr ( BB f ==t rtrB', B ' , IP ' ) 2 ,, ~ 2

r --

1

IIP IIP = = detB detB1 .

(2.10) (2.10)

We shall find it convenient, following Bowen [56], to introduce the following nota­ tions. Let be any quantity (scalar, vector or tensor) defined at the point x in the mixture at time t. Then, we define -■

{l)

D®d> D 4>

d

i ./ .

N N

¥-^ = -D (x'(3C (*,*).*) ,t).') • Dtr ==^|*(x* X'fixed

(2.11) (2-11)

A 1 fixed

That is, <j? or D^<j)/Dt denotes the variation of with time as noted by an observer who moves with the particle X 1 . Thus,

PRELIMINARIES

9 ^ D^4> 9 , , —4>i t f ( xt), t ) ++ [ g|rga dr 0( X( xt ), ]t )V] (x v*(x,t).

d

+ [ [grad(/> g r a d < ( (x ] v /((xx ,t), ^ x ' t )*)]' ' ^'

~~~Dt=di ~Dt ==4>{*,t) > ,*) +

(2 13)

-

and thus or D/Dt is the variation of 4> with respect to time as noted by an observer at x moving with the mean velocity of the mixture. The acceleration a* of the i-th constituent is defined through

a

1

(2-14)

= -£>

a =

92r

at2'

and the velocity gradient L l through \

(2.20)

\

(2-21)

t = ('>/>' — — - + p'divv* = m\

(2.24)

In addition to the balance of mass of each constituent, we also have the balance of mass for the mixture as a whole and thus d

— [ pdv+ [ pv ■ da = 0, at Jp Jdv

(2.25)

whose local form is p + pdivv = 0.

(2.26)

We note that the balance of mass for the mixture as a whole has the same form as that for a single constituent continuum. Also, it follows from summing Eq. 2.24 over all i = 1, ..., n and Eq. 2.26 that n

^m

l

= 0,

(2.27)

i=i

which states that there is no net production of mass. The above representations for the balance of mass are from an Eulerian point of view. However, in many applications involving the diffusion of fluids through a solid, it is convenient to express the balance of mass for the solid constituent in the Lagrangian form. In this situation, the global balance of mass for the z-th constituent takes the form

4- f pldV = [ mldV,

(2.28)

dt Jn Jn here, d/dt denotes the material (total) time derivative. The local form of the above balance equation is p 1 detF i = mMetF*

2.4

Balance of Linear Momentum

The balance of linear momentum for the z-th constituent states that

(2.29)

MECHANICS OF MIXTURES

12

- [ pWdv + I p V v 1 ■ da = f Uf dt Jv Jav Jar v '

da + [ ( p V + m' + m V ) ^.. Jv v '

(2.30)

In the above equation, b* is the specific external body force, and m ' is the momentum supply due to the interaction between the i-th constituent and other constituents (interactive body force). The term m V is the contribution to the momentum due to the mass production m1 of the i-th constituent. If the integrands of the above integrals are continuous and smooth, then it follows from standard arguments that — ( p V ) + div ( p V ® v l ) = div ( 0,

(2.50)

with =

1 -

y_ zz U=i=i Pp V

(251)

pP The entropy inequality 2.47 is quite general and allows for a different temperature field to be associated with each constituent. However, in all the problems considered

MECHANICS

18

OF

MIXTURES

herein, we shall a s s u m e a single c o m m o n t e m p e r a t u r e for all t h e c o n s t i t u e n t s . Read­ ers interested in t r e a t m e n t s w h e r e a different t e m p e r a t u r e field is associated w i t h each c o n s t i t u e n t a r e referred t o t h e works by I n g r a m a n d E r i n g e n [150], Steel [151], Bowen a n d G a r c i a [152,153], C r a i n e , G r e e n a n d N a g h d i [154], G u r t i n a n d P e n h a [155], Bed­ ford a n d S t e r n [156], Bowen a n d R a n k i n [157], Pecker a n d Deresiewicz [158], Bowen a n d C h e n [159], a n d Bowen a n d Reinicke [160].

2.8

Volume Additivity Constraint

Consider two fluids t h a t a r e incompressible in their p u r e s t a t e s a n d s u p p o s e t h a t we m i x t h e m t o g e t h e r . T h e n t h e volume of t h e m i x t u r e would e q u a l t h e volumes of t h e two c o n s t i t u e n t s . 1 T h i s v o l u m e a d d i t i v i t y in t h e m i x t u r e places restrictions on t h e m o t i o n of t h e m i x t u r e . Mills [161] was t h e first t o s t u d y this issue in d e t a i l . As we are p a r t i c u l a r l y interested in a m i x t u r e of a p o r o u s solid c o n s t i t u e n t a n d a fluid c o n s t i t u e n t , w i t h o u t m a s s conversion, we shall discuss t h e consequence of t h e restriction of v o l u m e a d d i t i v i t y in this case. Let us s u p p o s e t h a t t h e solid h a s a n a t ­ u r a l p o r o u s s t a t e w i t h uniform macroscopic porosity f3. T h e n t h e v o l u m e additivity constraint takes t h e form (cf. Shi [77]) P3

Pf

!L(l-0) + !L. = l, PO

(2.52)

PR

where p j = (1 — j3) psR is t h e effective m a s s density of t h e solid in its n a t u r a l p o r o u s s t a t e . Here, we i n t r o d u c e p j since we will consider t h e n a t u r a l p o r o u s s t a t e as t h e reference s t a t e of t h e a s s u m e d solid c o n t i n u u m in which t h e solid c o n t i n u u m is defor­ m a t i o n free. Notice t h a t in t h e case of (3 ^ 0 t h e a s s u m e d solid c o n t i n u u m does not u n d e r g o deformations (ps = pp) if t h e a s s u m e d fluid c o n t i n u u m h a s m a s s density of p? = f3pR, t h a t is, t h e fluid j u s t fills t h e macroscopic voids c o n t a i n e d in t h e p o r o u s solid. It follows from Eq. 2.52 t h a t 1 - /3 dp'

1

dp1

at ' p ^ = 0 '

(2 53)

- / v p * + i - V p / = 0. Po pR

(2.54)

PI

-

and

'Even in the case of a single constituent, incompressibility is an idealization. In the case of a mixture of two incompressible constituents, there is an inherent difficulty at the conceptual level; the two constituents have to occupy the whole volume of the mixture as they have to co-exist at each point in the mixture, which would violate the assumption that each constituent is incompressible. However, within the context of homogenized equivalent constituents such an idea can be given meaning.

PRELIMINARIES

19

Combining Eqs. 2.53, 2.54 and 2.24 and assuming mf = m" = 0 yields divv* + — " — ^ d i v v ' + V ( — ) • ( v s - v / ) = 0.

(2.55)

Po Similarly, for a mixture of two incompressible fluid constituents, we have (cf. Mills

[161]) P1 P2 £- + £- = 1, PR

(2.56)

PR

and ^ d i v v 1 + 4 d i w 2 + V (4] PR

PR

\PRJ

■ (v 1 - v 2 ) = 0. V

;

(2.57)

While we shall use the volume additivity constraint to study several problems involving a mixture of a solid infused with a fluid, a mixture of two fluids and a mixture of a fluid and solid particles, there are other constraints that could be used when considering mixtures. We notice that the volume additivity constraint leads to a Lagrange multiplier into the field equations, and the constraint does no work. Green, Naghdi and Trapp [162] have considered internal constraints which produce no entropy in thermodynamic processes. The interesting study of Liu [163] which shows how to exploit the entropy inequal­ ity with the Lagrange multiplier method warrants special mention. However, we shall not delve into this rather lengthy procedure because our main interest is to obtain the specific constitutive forms consistent with the entropy inequahty. The recent paper by Hutter and Svedsen [164] discusses several issues for constrained mixtures. So far we have summarized the basic field quantities and the basic balance equa­ tions in the theory of mixtures. The theory is not complete because it contains undetermined quantities such as m1, ) > 0.

(3.6)

Introducing a" = a3 - p3 (A3 - A)l,

af = af - p* [A* - A) I,

m=m-X/(p'(A3-A)),

(3.7)

we can rewrite Eqs. 3.3 through 3.6 in the form: P P1

3

-pr=div(^)

D

= div ( y j + af)

T

- m + p 3 b,

(3.8)

+m + pfb,

(3.9)

=a3 + a^

(3.10)

CONSTITUTIVE

MODEL OF A FLUID DIFFUSION THROUGH A SOLID

23

and pA + tr (cr3L3) + tr ( V l / ) + m ■ (v s - vf)

> 0.

(3.11)

We shall assume that the solid and the fluid are incompressible in their pure states. Then, the motion of the mixture has to satisfy the volume additivity constraint 2.52. Therefore, we have

-0) + ^ MlPo «l

(3.12)

= l,

and

-pA + tr (a3L3) + tr ( ^ / L / ) + m ■ (v s - v ' ) 1

+P —trl/ + Pi

p\.Tf PpRT

t,L' +, v niP K -(v-vO 'Wo

>o,

(3.13)

where P is the Lagrange multiplier associated with the constraint 3.12. We now need to postulate the constitutive structures for A3, A*, A, cf", a1 and m. From our experience with both an elastic solid single continuum and a linearly viscous fluid continuum, we might expect the constitutive response functions to depend on p3, F3, pf and I / . We also expect the various interactive mechanisms that were discussed in Chapter 2 to come into play and thus the constitutive relations may also depend on the relative velocity v* — v-^, a properly frame invariant form of the relative acceleration a and possibly other quantities like higher spatial gradients of F 3 and the velocity gradient of the solid I / . For the sake of simplicity, and motivated by earlier studies by Shi [77] and Shi, Rajagopal and Wineman [78], we shall assume that (A3,Af,A)

= (^,A',A)(/,F',v*-v'),

( K - *) +pf P

^ a ^1

i a^a^

K -pfdF^dH P -(v^

dFfj dxk

^ dF^ dxk

(Dlfttf

f0

Defining cf , a

) (

/}

2 dxk a ^ v m zX»>«n 3 +P a

-"M-l**--D^J -pH-{ m 3 s0

(

vl) 2 Pdxk dF% v 1 " -vi) *W (r < » -*V

dAodF

0]

( < -- -^)>o. a^ (7j\ (« a^ # *°' (3-18)

and m° through

- ~vl)V *« = ^ % + ^ | f + \pF*^Ff (< ~vi)(v' " V-) (< " ") <j

=

n

+ps s(v' +p (i* t--vl)H w/) fl^ - v() + *f3, jk(v'(t£ k-vl)+rr°°

„ 1 4 =- T^7 **i = i-p 'Pp

/

(

S 8ii RP R

-

(3.19)

1 f9A0c sdHmn(s PP S Sii Pp V pp PP v r5p7 ) I + 2/z 2 D / - c ( w * - W ' ) ,

(3.23)

°f° = 73 (trD 3 ) I + 2 M 3 D ' +

74

( t r D ' ) I + 2 / J 4 D ' - c ( w > - W')

(3.24)

,

and P" PS I , f) mr=a—-j\v--V). (3.25) PoPR A lengthy but straightforward application of the reduced entropy inequality leads to

Pi > 0,

7J

2 + -Mi > 0,

2 74 + -M4 > 0,

/i 4 > 0, 2

4/Ul/i4 > (/u2 + /i 3 ) ,

' ! -n +T7M1J (73 + ~Pz; 3 7 V 3 c > 0, a > 0.

2 . 72 + 73 + - (Ma + Ms) (3.26)

In later applications, we will often choose cr3° = a^° = 0, that is, we will not incorporate the effects of viscosity in a" and a* This does not mean that the fluid diffusing through the solid is inviscid. The effect of the viscosity enters the theory through the interaction term m ° in Eq. 3.21 and Eq. 3.25 that accounts for the drag. The virtual mass tensor H will be assumed to be given by pH = p3pfU0.

(3.27)

In this equation, H 0 is constant, symmetric and positive semi-definite or zero. The assumption is motivated by the fact that in the hydrodynamics of a solid body moving in a Newtonian fluid, the virtual mass is proportional to the mass density of the fluid and the volume occupied by the solid body.

MECHANICS OF MIXTURES

26

The anisotropy of the solid can be captured by choosing the appropriate form for A0 in terms of the appropriate integrity basis (cf. Spencer [195]). If the solid is orthotropic with xr, x2 and a 3 -axes as its three principal directions, the appropriate form for ^o is (see Dai [196], Dai, Rajagopal and Wineman [84]) A0 = A0 (pf, I, II, EU,E22,

£33, {El2f

, (El3f

, (E23f)

,

(3.28)

where

/ = tr [F3 (F*)T] ,

II = i {[tr (Fs ( F ' ) T ) ] 2 - tr (Fs (F'fY} ,

E = - [(Fa)TF3-l] .

(3.29)

In the derivation of Eq. 3.28, we have appealed to the principle of material frame indifference, and the third principal invariant III = det \F3 (F3) = Jpo/p" does not appear in the expression for A0, as it can be replaced in terms of p? by virtue of Eq. 3.12. If the solid is transversely isotropic with a;3-axis being its principal direction, then A0 takes the form (cf. Dai and Rajagopal [83]), AQ = A0 [pf, I, II, E33, {El3f

+ (E23f)

.

(3.30)

In general, for the diffusion of a fluid through an anisotropic solid, the form for the specific Helmholtz free energy is too complicated and involves too many material constants for whose values we have very little, if any, experimental guidance. In fact, even in the simplest of problems of a transversely isotropic solid considered by Dai and Rajagopal [83], the expression for A0 is A0 = Kt (I - 3) + #2^33 + K3 {II -S) + KA(I+K6 (E213 + E\3) + a A

+ a 2 l?j)

.

3) 2 +

K ^ (3.31)

We see that there are as many as eight material constants. If the solid is isotropic, equation 3.30 reduces to (cf. Shi, Rajagopal and Wineman [78], Gandhi, Rajagopal and Wineman [79]), A0 = A0(p},I,Il)

.

(3.32)

Generally, we have to determine AQ from experiments. However, such experimen­ tal data is not available at the present moment. Shi, Rajagopal and Wineman [78], based on the theoretical work of Wall and Flory [197] on rubber elasticity, suggested the following constitutive expression for A0 for the mixture of an incompressible New­ tonian fluid and an incompressible isotropic rubber-like elastic solid with (3 = 0 and H = 0,

CONSTITUTIVE MODEL OF A FLUID DIFFUSION THROUGH A SOLID

A> K I+ lInn (1 Ao = K J - -33 + f l -- ^ - j

P'JI

,

J

27

(3.33)

where A' = 1ZT/2MC, 1Z is the gas constant, T the absolute temperature of the mixture, Mc the molecular weight between cross-links of the solid. Later, Gandhi, Rajagopal and Wineman [79] proposed another expression, A0 = K(I-3)

+

TIT ri —

-v

b ( l - v ) + X(l-i/)

(3.34)

Here, Vx is the molar volume of the fluid, x ls a mixing parameter which depends on the particular solid-fluid combination and v = p3 j p3R is the solid volume fraction. The models 3.33 and 3.34 are modifications to account for the infusion of a fluid in a neo-Hookean material. Generalizations to the Mooney model (see Mooney [198]) and the model due to Gumbrell, Mullins and Rivlin [199] have been used to study swelling. Of course, there is no dearth of models which could be used to study the problem of swelling or diffusion through nonlinearly elastic solids by appropriately modifying the numerous models available in the literature (cf. Valanis and Landel [200], Ogden [201], Knowles [202]). In order to provide closure to the theory, we also need to know A* or A3. In general, A* and A3 depend on p', F" and v" — v-' (see Eq. 3.14), and there is no reason to suppose A* and A3 have a similar structure. We shall assume special forms for A! and A' in later applications for the sake of simplicity and the purpose of illustration. The above formulation will be applied to study the problems of steady diffusion, unsteady diffusion with the existence of a moving singular surface and wave propa­ gations in the next three chapters. It would be appropriate for us to conclude this chapter by pointing out that the above formulation reduces to the constitutive formulation of an incompressible elastic solid undergoing large deformations in the absence of the fluid (cf. Green and Adkins [204]). In fact, much of the argument presented in this chapter is based on the pioneering papers of Rivlin [205,206,207] and the subsequent work by Smith and Rivlin [208], Smith [209], Pipkin and Wineman [210,211] and Truesdell [212] and generalizes them to account for the existence of a fluid.

Chapter 4 STEADY STATE P R O B L E M S As we have observed earlier, much of the developments in the theory of mixtures, since 1957, have been mainly restricted to the general formulations of the basic equations and constitutive models. Very few studies have been carried out which apply the theory to solve practical problems. One main reason for this lacunae of concrete applications is the difficulty in specifying boundary conditions for problems involving mixtures. In the case of a mixture composed of a solid constituent and a fluid constituent, while there are the balance of linear momentum equations for each constituent, usu­ ally only the total traction on the boundary of the mixture is known and this, in general, is not sufficient to determine the motion of the mixture. Since the individual partial tractions are not known, one needs a method for splitting the total traction on the boundary, and this should be derived on a physically sound basis. One way to achieve this is to assume that the boundary of the mixture under consideration is in a saturated state (cf. Treloar [81], Rajagopal, Wineman and Gandhi [82]). Such a state is characterized by the variation of the Gibbs free energy of dilution being zero, which leads to a condition that relates the stretch ratios of the solid and the applied surface tractions (the details are presented in Section 4.1). Following this treatment, several steady diffusion problems of a fluid through an isotropic (or anisotropic) nonlinearly elastic solid have been analyzed (cf. Shi, Rajagopal and Wineman [78], Rajagopal, Shi and Wineman [80], Gandhi, Rajagopal and Wineman [79], Gandhi, Usman, Wine­ man and Rajagopal [216], Rajagopal and Wineman [203], Dai and Rajagopal [83], Dai, Rajagopal and Wineman [84]). In this chapter, another way of furnishing boundary conditions to study the dif­ fusion of a fluid through an elastic solid is proposed, that is, the partial traction of the fluid constituent on the boundary is assumed to be equal to the product of the surface fraction occupied by the fluid on the boundary and the fluid pressure in the pure fluid region. Furthermore, the surface fraction of the fluid on the boundary is assumed to be equal to the volume fraction of the fluid on the boundary. This method of sphtting the traction may be expected to be appropriate for processes in which the 29

MECHANICS OF MIXTURES

30

diffusion is slow. A detailed treatment of the method of splitting the total traction is presented in Section 4.1. Based on this idea and the constitutive model presented in Chapter 3, the steady diffusion of a fluid through an isotropic rubber slab and the steady diffusion of a fluid through an isotropic rubber cylindrical annulus are studied in Section 4.2 and Section 4.3, respectively. In these applications, the diffusion processes will be assumed to be steady, isothermal and slow, the virtual mass effect and the external body forces will be neglected and the specific Helmholtz free energy function A is assumed to be described by Eq. 3.34. It is hoped that the results presented in this chapter will help to determine, albeit partially, the applicability of different constitutive models proposed and to design experiments to find material parameters that appear in these constitutive models.

4.1

Boundary Conditions

As we have remarked earlier, one of the thorny obstacles when it comes to putting mixture theory to use, is our inability to prescribe boundary conditions for traction boundary value problems. In the following, we shall discuss two different ways of augmenting the boundary conditions. The first method appeals to thermodynamics and derives a relation amongst the boundary tractions, the deformation of the solid constituent of the mixture, the mass densities of the constituents, etc., that holds when the mixture on the boundary is saturated (cf. Rajagopal et al. [82]). While such a boundary condition has been used with some degree of success in studying the large deformations of solids infused with fluids there are situations wherein it is inapplicable; in fact it leads to inconsistencies in the of boundary value problems A second method of augmenting boundary conditions has been nrnnnsed recently based on the sohtting of the traction adduced to above that hi LeW mechanical underpinning (cf Tao and Raiaeopal \S5]) It is also possihle iTJZZ boundary conditions by requiring continu tv in the'chemical n o W ^ l h ! ! S not investigate this possibility here. potential,

Boundary condition based on t h e notion of saturation Before embarking into the method for generating boundary conditions, a few words about the swelling of solids are warranted. As we shall be concerned with the diffusion of solvents through rubber and polymeric materials, our discussions will be pertinent to the phenomenon of swelling of such materials. In studying elastic solids from the perspective of continuum mechanics, we approach it from a phenomenological point of view. Rubber like elastic solids have also been modeled using a statistical approach (cf. Kuhn [217], Wall [218], James and Guth [219,220], Flory [221,222], Wall and Flory [197], Treloar [81,223], Yasuda et al. [224]), and in fact it is one of the few great success stories where phenomenological and statistical modelling speak with one voice. The study of the swelling of rubber due to a solvent, which is in

STEADY

STATE

PROBLEMS

31

equilibrium, was first studied by Flory [225] and Huggins [226]. They determined the configurational entropy due to mixing, using statistical methods, and used this to determine a formula for the Gibbs free energy of dilution. This equation bears the name the Flory-Huggins equation. The Gibbs free energy of dilution is assumed to be the sum of a part due to mixing and the other due to the strain, and setting the variation in the Gibbs free energy of dilution to be zero at equilibrium leads to a relation between the applied traction and the stretch in the solid (cf. Flory and Rehner [227], Gee [228]). The phenomenon of swelling from the above perspective is discussed in great detail in the book by Treloar [81]. Our discussion of boundary conditions here draws liberally from the previous work on swelling, but the problem is studied within the context of mixtures, and the intent is completely different; the aim here is to develop boundary conditions. Let us consider an incompressible isotropic solid in contact with an incompressible fluid in a fluid bath. The experimental work of Southern and Thomas [229] indicates that as the fluid diffuses, the boundary is saturated. By saturation, we mean a state in which a small element of the solid adjacent to the fluid is in a state in which it cannot absorb any more fluid, that is whatever fluid enters the elemental volume along the boundary has to exit through the elemental volume so that there is no accumulation of the fluid. Southern and Thomas [229] observed "When a sheet of rubber vulcanizate is immersed in a solvent, the surface layer takes up its equilibrium amount of the liquid, virtually instantaneously, the liquid subsequently diffuses into the bulk of the rubber.'' We interpret the notion of equilibrium to indicate that the mixture adjacent to the fluid, in its swollen state (extent of deformation) and under the applied stress, is in a saturated state. Thus, an elemental volume of the mixture in such a saturated state cannot be diluted further, i.e., it cannot absorb any additional fluid. This condition can be expressed in the form that the variation of the Gibbs free energy of dilution is zero. Assuming that the whole mixture is in a state of saturation, Treloar [81] derived an equation for the saturated state that is essentially the Flory-Huggins equation. And such an approach leads to the governing equation (field equation) for the mix­ ture as a whole. Our intent here is quite different. The appropriate field equations are the balance equations presented in Chapter 3. However, if the boundary alone can be assumed to be saturated, then a procedure similar to that adopted by Treloar [81] leads to a boundary condition, which can be used to augment the usual trac­ tion boundary condition. The fact that we use the notion of saturation to generate boundary conditions, and not field equations, cannot be overemphasized. Suppose we have a cube of material that is in the same state of deformation as that of a material element on the boundary of the mixture. The cube infused with the fluid in its swollen state is in equilibrium and the deformation of the cube is homogeneous. Let us now carry our thought experiment a little further. We shall suppose that the saturated cube which is in equilibrium is isolated in the sense that no fluid could enter or leave it (we could think of the cube infused with the fluid to

MECHANICS

32

OF

MIXTURES

be covered by an impermeable elastic sheet). Furthermore, suppose that we shghtly deform the cube and this deformation is carried out isothermally. Since the cube is isolated, by the first law of thermodynamics, we have 8W + 8Q = Ss%

(4.1)

where SW is the work done on the swollen cube, SQ is the heat transfered and Se* the change in the internal energy of the swollen cube. Since the Helmholtz free energy A* is defined through A' = e'-Tn',

(4.2)

SW + SQ = 8 A* + TSrf.

(4.3)

we have

In deriving the above equality, we have used the fact that the process is isothermal at absolute temperature T. If we now suppose that the deformation is reversible, and this is a critical assumption, then SQ = T6n\

(4.4)

SW = SA*.

(4.5)

and thus equation 4.3 reduces to

Let us take our thought experiment one step further. To the cube that is in equilibrium in a saturated state, of total mass m and specific Helmholtz free energy of the mixture A, let us suppose that we add a small amount of mass of fluid such that the new mass of the cube is m + 5m and the corresponding specific Helmholtz free energy is A + SA, so that the Helmholtz free energy of the cube is (m + Sm)(A + 8A). Ignoring higher order terms and equating the variation in the work done to the variation in the Helmholtz free energy of the cube, we obtain SW = m (SA) + (Sin) A.

(4.6)

The above relationship of course holds only under the special assumptions that axe supposed to hold in our thought experiment. In a real problem, an element on the boundary of the mixture adjacent to the fluid may or may not meet such restrictions. Yet, we shall assume that the relationship 4.6 holds, which leads to a relation on the boundary that can be used as a boundary condition. It is worthwhile to remark that equation 4.5 is derived on the assumption that we have a closed system, but equation 4.6 is obtained by treating the system as open, as we have added a mass 8m. Whether such an approximation is reasonable can only be borne out by the predictions of the assumption.

33

STEADY STATE PROBLEMS

The above motivation is slightly different from the derivation provided by Treloar [81] where in a saturated mixture, the variation in the Gibbs free energy of dilution being zero is employed. Under appropriate assumptions and approximations this also leads to Eq. 4.6. In the approach used by Treloar, however, we are faced with defining the pressure P of the swollen cube, which is taken to be the pressure of the bath. However, in mixtures of non-linear materials the pressure and the normal stresses are not the same and certain care has t o be exercised. We shall not get into a detailed discussion of these issues here as it can be found in standard treatises devoted to non-linear mechanics. Moreover, Treloar does not have a mixture in our sense and there is no cognizance of partial tractions and their distinction from the total traction to which the fluid pressure in the bath is related. Now, consider a unit cube, which after absorbing the fluid and attaining a state of saturation is of length At, A2 and A3 in the Xi, £ 2 a n d £3 directions, respectively. Suppose the tractions acting along the x% direction on the appropriate faces of the cube are given by t^. Let us suppose that the edge of the cube A; elongates by 5A; due to the addition of a small amount of the fluid. A trivial calculation yields SW — H ' +/ + &dpi ' ■. 81 dll

SA =

"