INTRODUCTION CONTINUUM MECHANICS FOR ENGINEERS (REVISED EDITION)

Transcription

1 INRODUCION O CONINUUM MECHANICS FOR ENGINEERS (REVISED EDIION) 007 By Ray M. Bowen Professor of Mechanical Engineering President Emeritus exas A&M University (First Edition Originally Published By Plenum Press 1989 as Volume 39) (of Mathematical Concepts Methods in Science Engineering) (ISBN ) (Series Editor: Angelo Miele) Copyright Ray M. Bowen Updated Revised 008

2 Preface (First Edition) his textbook is intended to introduce engineering graduate students to the essentials of modern Continuum Mechanics. he objective of an introductory course is to establish certain classical continuum models within a modern framework. Engineering students need a firm understing of classical models such as the linear viscous fluids (Navier-Stokes theory) infinitesimal elasticity. his understing should include an appreciation for the status of the classical theories as special cases of general nonlinear continuum models. he relationship of the classical theories to nonlinear models is essential in light of the increasing reliance, by engineering designers researchers, on prepackaged computer codes. hese codes are based upon models which have a specific limited range of validity. Given the danger associated with the use of these computer codes in circumstances where the model is not valid, engineers have a need for an in depth understing of continuum mechanics the continuum models which can be formulated by use of continuum mechanics techniques. Classical continuum models others involve a utilization of the balance equations of continuum mechanics, the second law of thermodynamics, the principles of material frameindifference material symmetry. In addition, they involve linearizations of various types. In this text, an effort is made to explain carefully how the governing principles, linearizations other approximations combine to yield classical continuum models. A fundamental understing of these models evolve is most helpful when one attempts to study models which account for a wider array of physical phenomena. his book is organized in five chapters two appendices. he first appendix contains virtually all of the mathematical background necessary to underst the text. he second appendix contains specialized results concerning representation theorems. Because many new engineering graduate students experience difficulties with the mathematical level of a modern continuum mechanics course, this text begins with a one dimensional overview. Classroom experience with this material has shown that such an overview is helpful to many students. Of course, more advanced students can proceed directly to the Chapter II. Chapter II is concerned with the kinematics of motion of a general continuum. Chapter III contains a discussion of the governing equations of balance the entropy inequality for a continuum. he main portion of the text is contained in Chapter IV. his long chapter contains the complete formulation of various general continuum models. hese formulations begin with general statements of constitutive equations followed by a systematic examination of these constitutive equations in light of the restrictions implied by the second law of thermodynamics, material frame-indifference material symmetry. Chapter IV ends with an examination of the formal approximations necessary to specialize to the classical models mentioned above. So as to illustrate further applications of continuum mechanics, the final chapter contains an introductory discussion of materials with internal state variables. he book is essentially self contained should be suitable for self study. It contains approximately two hundred eighty exercises one hundred seventy references. he references at the end of each chapter are divided into References General References. he ii

3 iii Preface References are citations which relate directly to the material covered in the proceeding chapter. he General References represent additional reading material which relate in a general way to the material in the chapter. his text book evolved over an extended period of time. For a number of years, early versions of the manuscript were used at Rice University. I am indebted for the assistance my many students gave me as the lecture notes evolved into a draft manuscript. he final manuscript has been utilized at the University of entucky by my colleague, Professor Donald C. Leigh, in an introductory graduate course. I am indebted to him for his many comments suggestions. Preface (004 Revised Edition) Ray M. Bowen Lexington, entucky his electronic textbook is a revision to the textbook, Introduction to Continuum Mechanics which was published by Plenum Press in A small amount of new material has been added in Chapters 1, 3 4. In addition, an effort has been made to correct numerous typographical errors that appeared in the first edition. It is inevitable that other typographical errors creep into the manuscript when it is retyped. I hope there has been a net reduction in these kinds of errors from the first edition to this revised edition I remain indebted to my colleagues that have pointed out errors over the years. A special mention needs to be made to my good friends Dr. C.-C. Wang of Rice University Dr. Donald C. Leigh of the University of entucky. Not only were they kind enough to adopt the first edition as a textbook, they informed me of many corrections improvements that could be made. I am also indebted to my students at exas A&M University that endured my teaching from the revised edition after being out of the classroom for many years. It is my desire intention that this revised textbook be made available for free to anyone that wishes to have a copy. For the immediate future, the access will be provided by posting it on the website of the Mechanical Engineering Department of exas A&M University. I would appreciate being informed of any typographical other errors that remain in the posted version. Preface (007 Revised Edition) he 007 revisions mainly involve the correction of typographical errors that have entered the text as it has been retyped revised over the years. A small amount of new material has been added in Chapter 1. he index able of Contents have been revised to reflect the addition of this material. For those that acquire the text as a pdf file, the search utility within Adobe Acrobat provides an excellent alternative to the index. Ray M. Bowen rbowen@tamu.edu College Station, exas

4 Contents 1. One-Dimensional Continuum Mechanics 1.1. inematics of Motion Strain Balance of Mass Balance of Linear Momentum Balance of Energy General Balance he Entropy Inequality Example Constitutive Equations hermodynamic Restrictions Small Departures from hermodynamic Equilibrium Small Departures from Static Equilibrium Some Features of the Linear Model 38 Bibliography 47. inematics of Motion.1. Bodies Deformations 48.. Velocity, Acceleration Deformation Gradients ransformation of Linear, Surface Volume Elements Strain inematics Infinitesimal Strain inematics 69 References 7 Bibliography 7 3. Equations of Balance 3.1. Balance of Mass Balance of Linear Momentum Balance of Angular Momentum Balance of Energy he Entropy Inequality Jump Equations of Balance - Material Versions 100 References 106 Bibliography Models of Material Behavior 4.1. Examples Isothermal Elasticity - hermodynamic Restrictions Isothermal Elasticity - Material Frame Indifference Isothermal Elasticity - Material Symmetry Incompressible Isothermal Elasticity hermoelastic Material with Heat Conduction Viscous Dissipation-Constitutive Assumptions hermoelastic Material with Heat Conduction Viscous Dissipation-General hermodynamic Restrictions 15 iv

5 v Contents 4.8. hermoelastic Material with Heat Conduction Viscous Dissipation-Equilibrium hermodynamic Restrictions hermoelastic Material with Heat Conduction Viscous Dissipation-Material Frame Indifference hermoelastic Material with Heat Conduction Viscous Dissipation-Material Symmetry Constitutive Equations for a Compressible, Conducting, Viscous Fluid Constitutive Equations for an Isotropic Linear hermoelastic Solid with Heat Conduction 196 References 03 Bibliography Materials with Internal State Variables 5.1. Constitutive Assumptions hermodynamic Results Maxwell-Cattaneo Heat Conductor Maxwellian Materials Closing Remarks-Alternate Forms of the Entropy Inequality 3 References 36 Bibliography 37 Appendix A. Mathematical Preliminaries A.l. Vector Spaces 38 A.. Linear ransformations 43 A.3. Inner Product Spaces 49 A.4. Components of Vectors Linear ransformations 56 A.5. Cross Products, Determinants the Polar Decomposition heorem 58 A.6. Multilinear Functionals ensor Algebra 76 A.7. Euclidean Point Spaces, Coordinate Systems 78 A.8. Vector Analysis 8 References 87 Bibliography 88 Appendix B. Representation heorems 89 Bibliography 94 Index 95

6 1 One-Dimensional Continuum Mechanics It is often not clear to engineering students that there is a common basis for their courses in thermodynamics, fluid mechanics elasticity. he pace of most undergraduate curriculums is such that there is no opportunity to stress the common features of these courses. In addition, many undergraduate engineering students have limited skill with vector analysis Cartesian tensor analysis. hese problems make it awkward to teach a modern introductory course in Continuum Mechanics to first year engineering graduate students. Experience has shown that an elementary preview of the modern course can be a great asset to the student. his chapter contains such a preview. It is a brief survey of the elements of continuum mechanics presented for one dimensional continuous bodies. his survey allows the student to encounter a new notation several new concepts without the problem of learning three dimensional vector tensor analysis. his chapter contains a development of the one dimensional forms of the equations of balance of mass, momentum energy. he entropy inequality is presented, it is utilized to derive the thermodynamic restrictions for a particular material model inematics of Motion Strain We shall denote by U the one dimensional body. he symbol denotes an element or particle of the body U. It is useful at this point not to distinguish between the body U the portion of one dimensional space it occupies. hus, is a real number. It is customary to refer to as the material or Lagrangian coordinate of the particle. he set of material coordinates is a subset of the real numbers called the reference configuration. If t denotes the time e is the set of real numbers, then the deformation function is a function χ(, t): U e which, for each t, maps U into its present configuration. We write x = χ( t, ) (1.1.1) where x is the spatial position or coordinate of the particle at the time t. he spatial coordinates are also called Eulerian coordinates. We shall assume that for each t, χ has an inverse 1 χ such that = χ 1 ( xt, ) (1.1.) heses assumptions insure that x are in one to one correspondence for each t are, in effect, a statement of permanence of matter. he particle cannot break into two particles as a result of the deformation, two particles 1 cannot occupy the same spatial position x at the same instant of time. he velocity of at time t is 1

7 Chapter 1 χ(, t) x = (1.1.3) t he acceleration of at the time t is χ(, t) x = (1.1.4) t he displacement of at the time t is w= χ(, t) (1.1.5) Because of (1.1.), we can regard x, x w to be functions of ( x, t ) or (, t ). he pair (, t ) are called material variables, the pair ( x, t ) are called spatial variables. Clearly, by use of (1.1.1) (1.1.) any function of one set of variables can be converted to a function of the other set. If ψ is a function of (, t ), then its material derivative, written ψ, is defined by If the function ψ of ( x, t ) is defined by (, t) ψ = ψ (1.1.6) t ψ ψ χ 1 ( x, t) = ( ( x, t), t) (1.1.7) an elementary application of the chain rule yields ψ (, t) ψ( x, t) ψ( x, t) ψ = = + x (1.1.8) t t x Equation (1.1.8) gives the material derivative in terms of spatial derivatives. For notational simplicity, we shall write (1.1.8) as ψ ψ ψ = + x (1.1.9) t x where it is understood that ψ / t is computed at fixed x, ψ / x is computed at fixed t. As an illustration of (1.1.9), we can take ψ = x obtain x x x = + x (1.1.10) t x

8 One-Dimensional Continuum Mechanics 3 he deformation gradient is defined by χ(, t) F = (1.1.11) Since χ has an inverse, it is trivially true that 1 χχ ( (,),) x t t = x, thus, χ 1 ( t, ) χ ( xt, ) x = 1 (1.1.1) Equation (1.1.1) shows that F 0 F χ ( x, t) = = F x (1.1.13) he displacement gradient is defined by H wt (, ) = (1.1.14) It follows from (1.1.11), (1.1.14), (1.1.5) that H F are related by H = F 1 (1.1.15) If d denotes a differential element of the body at, then it follows from (1.1.1) (1.1.11) that dx = Fd (1.1.16) Equation (1.1.16) shows that if d is a material element at then Fd = dx is the deformed element at x. herefore, F measures the local deformation of material in the neighborhood of. If x is expressed as a function of ( x, t ), then x L = x (1.1.17) is the velocity gradient. Because χ(, t) x x χ(, t) = = t x (1.1.18)

9 4 Chapter 1 it follows from (1.1.11), (1.1.17) (1.1.6) that F = LF (1.1.19) Exercise Show that 1 1 F F L = (1.1.0) Next we shall state prove an important result known as Reynold s theorem or the transport theorem. Consider a fixed portion of the body in 1. he deformation function, for each t, deforms this portion into a region in e. Without loss of generality, we can take this region to be x1 x x, where x1 = χ( 1, t), x = χ(, t) x = χ( t, ). Reynold s theorem states that if Ψ is a sufficiently smooth function of x t, then i x x Ψ( xt, ) Ψ (,) x tdx = dx+ψ( x,)( txx,) t Ψ( x1,)( txx 1,) t (1.1.1) t x1 x1 he derivation of (1.1.1) is best approached by utilizing a formalism which does not depend upon the use of a deformation function, its associated motion. We first let Φ ( x, t) be the indefinite integral, with respect to x, of a function Ψ ( x, t). In other words, let Φ ( xt, ) = Ψ ( xtdx, ) + constant (1.1.) It follows then that, for two points x 1 x, Φ( xt, ) x =Ψ( x, t) (1.1.3) x Ψ ( x, tdx ) =Φ( x, t) Φ( x1, t) (1.1.4) x1 If we allow the limits in (1.1.4) to be functions of time, i.e., x () t Ψ ( x, tdx ) =Φ( x( t), t) Φ( x1( t), t) (1.1.5) x1 () t x () t he total derivative of Ψ( x, tdx ) is x () t 1

10 One-Dimensional Continuum Mechanics 5 d () x t dφ( x( t), t) dφ( x1( t), t) Ψ ( xtdx, ) = dt x1 () t dt dt Φ( x(),) t t Φ( x(),) t t dx() t = + t x dt Φ( x1(),) t t Φ( x1(),) t t dx1() t t x dt (1.1.6) If (1.1.3) (1.1.4) are used, (1.1.6) can be written d x () t Φ( x(),) t t dx() t Φ( x1(),) t t dx1() t Ψ ( xtdx, ) = +Ψ( x( t), t) Ψ( x1( t), t) x () t dx() t dx1() t = Ψ ( xtdx, ) +Ψ( x( t), t) Ψ( x1( t), t) t x1 () t dt dt dt x1 () t t dt t dt (1.1.7) he next formal step is to interchange the order of integration on x with the partial differentiation with respect to t. he result of this final step is d x () t x () t ( x, t) dx () t dx () t dt t dt dt 1 Ψ ( x, t) dx = ( ( ), ) ( 1( ), ) x1() t dx +Ψ x t t Ψ x t t (1.1.8) x1() t he result (1.1.8) is a mathematical identity that holds for the function Ψ of ( x, t ) functions x 1 x of t that are sufficiently smooth that the various derivatives above exist. In calculus, it sometimes goes by the name Leibnitz s rule. It is one of the stard results one needs in applied mathematics for the differentiation of integrals. We have not been precise about the smoothness assumptions sufficient to give the result (1.1.8). More rigorous derivations of this result can be found in many Calculus textbooks. References 1 contain this derivation. In any case, we are interested in this result in the special case where the functions x 1 x of t are derived from the deformation function χ. In this case, (1.1.8) immediately becomes (1.1.1) Exercise 1.1. Show that (1.1.1) can be derived by a change of variables in the integral first step is to write x = χ (, t) x = χ (, t) 1 1 Ψ( x, tdx ). he x x= χ (, t) Ψ (,) x tdx = Ψ (,) xtdx = Ψ( χ( t,),) tfd (1.1.9) x1 x1= χ ( 1, t) 1 Differentiate (1.1.9) rearrange the result to obtain (1.1.1). he name "transport theorem" is suggested by viewing the last two terms in (1.1.1) as the net transported out of the spatial region x1 < x< x by the motion of the material.

11 6 Chapter 1 Equation (1.1.8) its special case (1.1.1) assume that Φ is differentiable, thus, from (1.1.3), Ψ continuous as a function of x in x1 < x< x. Next we wish to remove the assumption that Ψ is continuous in x1 < x< x derive a generalization of (1.1.1). he derivation simply utilizes the general result (1.1.8) in two different intervals carefully joins the results so as they apply to the interval x1 < x< x. We assume Ψ is continuous except at a point yt () in the interval x1 < x< x. As shown in the following figure, at the point yt () the function Ψ is allowed to undergo a discontinuity as a function of x at a fixed time t. As indicated on the figure Ψ = lim Ψ ( x, t) (1.1.30) x y() t + Ψ = lim Ψ ( x, t) (1.1.31) x y() t We use the notation [ Ψ] to denote the jump of Ψ defined by + [ Ψ ] =Ψ Ψ (1.1.3) Ψ 1 Ψ + Ψ Ψ x1 yt () x Exercise Figure If Ψ Φ undergo jump discontinuities at yt () show that

12 One-Dimensional Continuum Mechanics 7 [ ΨΦ ] = [ Ψ] Φ + Ψ [ Φ ] = Ψ [ Φ ] + [ Ψ] Φ + + = [ Ψ][ Φ ] +Ψ [ Φ ] + [ Ψ] Φ = ( Ψ +Ψ )[ Φ ] + [ Ψ] ( Φ +Φ ) (1.1.33) Exercise If Ψ undergoes a jump discontinuity at y( t ), show that Ψ = Ψ Ψ + Ψ x x dx ( x, t) ( x1, t) [ ] (1.1.34) x1 Given a function Ψ which undergoes a jump discontinuity at y( t ), Reynold s theorem takes the form i x x Ψ Ψ ( x, t) dx = dx +Ψ( x, t) x ( x, t) t Ψ ( x, txx ) (, t) + [ Ψ] y x1 x1 1 1 (1.1.35) where y ( t) is the velocity of the point y( t ). he derivation of (1.1.35) is elementary. Because Ψ is differentiable in x1 < x< y() t in y() t < x< x, (1.1.7) yields d dt Ψ( x, t) t y y Ψ ( x, tdx ) = dx y ( x1, txx ) ( 1, t) x +Ψ Ψ (1.1.36) 1 x1 d dt Ψ( x, t) t x x + Ψ ( x, tdx ) = dx+ψ( x, txx ) (, t) Ψ y y (1.1.37) y he addition of (1.1.36) (1.1.37) yields (1.1.35). If (1.1.34) is used, (1.1.35) can also be written i x x Ψ Ψx Ψ ( x, tdx ) = + dx Ψ( x y ) (1.1.38) t x [ ] x1 x1 1.. Balance of Mass In this section we shall state the one dimensional form of the equation of balance of mass. his equation is the first of four fundamental principles which form the basis of continuum

13 8 Chapter 1 mechanics. he others are balance of momentum, balance of energy the entropy inequality. hese equations of balance will be discussed in subsequent sections. We shall denote by ρ the mass density (mass/length) of U in its deformed configuration. herefore, he corresponding quantity in the reference configuration is ρ = ρ(, t) (1..1) ρ = ρ ( ) (1..) R R Balance of mass is the simple physical statement that the mass of the body any of its parts are unaltered during a deformation. If an arbitrary part of the body is defined by 1 < <, then it is deformed into x1 < x< x by the deformation, where x1 = χ( 1, t) x = χ(, t). Balance of mass is the assertion that x ρ d = ρdx (1..3) R 1 x1 for all parts of the one dimensional body U. Because the left side of (1..3)is independent of t, an alternate form of balance of mass is i x x1 ρ dx = 0 (1..4) Next we shall derive the local statement of balance of mass. he statement is local in the sense that it holds at an arbitrary point at an arbitrary time t rather than for an interval < <. By use of (1.1.16), (1..3) can be written 1 1 ( ρf ρ ) R d = 0 (1..5) If we assume the integr ρf ρr is a continuous function of, the fact that (1..5) must hold for every interval 1 < < forces the following local statement of balance of mass: ρf = ρ (1..6) R For reasons that will become clear later, we shall refer to (1..6) as the material form of the local statement of balance of mass. Other local statements follow by differentiation of (1..6). It follows from (1..6) that i ρf = ρf + ρf = 0 (1..7)

14 One-Dimensional Continuum Mechanics 9 If we now use (1.1.19) the fact that F 0, (1..7) yields Exercise 1..1 Show that (1..8) can be written in the alternate forms ρ + ρl = 0 (1..8) i 1 1 = L ρ ρ (1..9) ρ ρ x + = 0 t x (1..10) Equations (1..8), (1..9) (1..10) are local statements of balance of mass which hold at points where ρ x are differentiable. If there is a point for which ρ x undergo a jump discontinuity, we must proceed from (1..3) more carefully. If ρ x suffer a jump discontinuity at yt (), it follows from (1..4) (1.1.38) that ρx dx ρ( x y ) 0 x + = 1 t x [ ] (1..11) x ρ Equation (1..11) holds for all ( x1, x ). he integr is assumed to be continuous at all points except yt (). If ( x1, x ) is an arbitrary interval which does not contain yt (), then (1..11) implies (1..10). Given (1..10), (1..11) then yields [ ρ( x y )] = 0 (1..1) at x = yt ( ). he physical meaning of (1..1) is quite clear. It simply states that the flux of mass across the point y ( t) is continuous. Exercise 1.. Use (1..1) show that the jump in specific volume, [ 1/ ρ], is given by Exercise ρ ( x + y)1/ [ ρ] = [ x ] (1..13)

15 10 Chapter 1 Use (1..10) show that where Ψ is any function of ( x, t ). Exercise 1..4 Use (1..14) show that ρψ ρψx ρψ= + (1..14) t x i x x ρ Ψ ( x, tdx ) = ρ Ψ dx [ ρ Ψ( x y )] (1..15) x1 x Balance of Linear Momentum In the three dimensional theory, the statement of balance of momentum consists of two parts. he first is the statement concerning the balance of linear momentum, the second is a statement concerning the balance of angular momentum. For a one dimensional theory the concept of angular momentum does not arise. he linear momentum of the part of U in x1 < x< x is x ρxdx x1 he rate of change of linear momentum is required to equal to the resultant force on the part of U. he formal statement is written x x1 i ρ xdx = f (1.3.1) for all parts of the body U, where f is the resultant force acting on the part. We shall assume that f consists of two contributions write 1 x f = ( x, t) ( x, t) + ρbdx (1.3.) x1 he quantity bxt (, ) is called the body force density (body force/mass), the integral of ρ b is the resultant body force acting on the part of U in x1 < x< x. he quantity ( x, t ) is a contact force. It results from the contact of the part of U in x1 < x< x with that not in x1 < x< x. ( x, t ) is the one dimensional counterpart of stress. If ( x, t ) > 0 (< 0) the material point is in tension (compression). If we combine (1.3.1) (1.3.), we obtain

16 One-Dimensional Continuum Mechanics 11 i x x ρxdx = ( x, t) ( x, t) + ρbdx (1.3.3) 1 x1 x1 Next, we wish to deduce from (1.3.3) a local statement of balance of linear momentum. For the sake of generality, we allow, ρ x to suffer jump discontinuities at a point y( t ) in U. It easily follows from (1.1.34) that x ( x, t) ( x1, t) = dx [ ] (1.3.4) x1 x, from (1..15), that i x x ρ xdx = x 1 x1 ρ xdx [ ρ x( x y )] (1.3.5) hese results allow (1.3.3) to be written x ρx ρb dx ρx( x y) 0 x + = 1 x [ ] [ ] (1.3.6) Since (1.3.6) must hold for all parts of U, it follows by the same argument that produced (1..10) (1..1) from (1..11) that for all x yt ( ), at x = yt ( ) Exercise Show that an alternate form of the acceleration x is ρ x = + ρb (1.3.7) x [ ρxx ( y )] [ ] = 0 (1.3.8) ρ x ρ x ρ x = + (1.3.9) t x Exercise 1.3. Show that (1.3.8) can be written

17 1 Chapter 1 [ ρ( x y ) ] = 0 (1.3.10) γ[ x ] [ ] = 0 (1.3.11) + + where γ = ρ ( x y) = ρ ( x y ). Exercise Show that γ = [ ] [ 1/ ρ] (1.3.1) Equation (1.3.1) shows that [ ] [ 1/ ρ] must have the same sign. Equation (1.3.1) is known as the Rankine-Hugoniot relation. Just as (1..6) is the material version of balance of mass, equation (1.3.7) can be manipulated into a material version of balance of linear momentum. his material version is ρ Rx = + ρrb (1.3.13) Equation (1.3.13) follows by multiplication of (1.3.7) by F making use of (1..6) Balance of Energy Balance of energy, or the First Law of hermodynamics, is the statement that the rate of change of total energy equals the rate of work of the applied forces plus the rate of heat addition. he total energy includes the sum of the internal energy the kinetic energy. If ε ( x, t) is the internal energy density (internal energy/mass), x 1 ρε+ x x1 ( ) dx is the total energy of the part of U in x1 < x< x. he rate of work or power of the applied forces is x (, txx ) (, t) x (, txx ) (, t) ρ xbdx x x1

18 One-Dimensional Continuum Mechanics 13 he rate of heat addition arises from heat generated at points within the body U from contact of the part of U in x1 < x< x with that not in x1 < x< x. If rxt (, ) denotes the heat supply density (rate of heat addition/mass), then x x1 ρrdx is the rate of heat addition resulting from heat generated within the body. he rate of heat addition from contact is written qx (, t) qx (, t) 1 where qxt (, ) is the heat flux. he mathematical statement which reflects balance of energy is therefore i 1 x dx x t x x t x t x x t ρxbdx x x ρε ( + ) = (, ) (, ) ( 1, ) ( 1, ) + x 1 x1 1 x + qx (, t) qx (, t) + ρrdx x1 (1.4.1) Exercise Show that (1.4.1) implies i ρ 1 ( ε x q + x ) = + xb r x x ρ + ρ (1.4.) for all x yt ( ) [ ρε+ 1 ( )( ) 0 x x y x + q] = (1.4.3) at x = yt ( ). Exercise 1.4. Show that when [ q ] = 0 (1..1) (1.3.8) can be used to write (1.4.3) in the forms 1 ( ) [ ε + x y ] = 0 (1.4.4) ρ

19 14 Chapter 1 ε 1 1 ( ) 0 + [ ] + [ ] = (1.4.5) ρ Equation (1.4.5) is known as the Hugoniot relation. It is often written in terms of a different thermodynamic quantity than the internal energy density ε. he quantity that is used is the enthalpy density defined by Exercise Given the definition (1.4.6), show that (1.4.5) takes the form Exercise Derive a material version of (1.4.). χ = ε (1.4.6) ρ [ χ] + + = 0 + ρ ρ [ ] (1.4.7) Next we shall use (1.3.7) to derive from (1.4.) a thermodynamic energy equation. If (1.3.7) is multiplied by x, the result can be written 1 ρ x = x + ρ xb (1.4.8) x If this equation is subtracted from (1.4.), the result is q ρε = L + ρr (1.4.9) x where the definition (1.1.17) has been used. he term the contact forces. Since x in (1.4.) arose from the rate of work of x x = x + L (1.4.10) x x

20 One-Dimensional Continuum Mechanics 15 this rate of work decomposes into a part which changes the mechanical energy, x, a part x which changes the internal energy, L. he term L is sometimes called the stress power. Exercise Derive the material version of (1.4.9). Exercise On the assumption that none of the field quantities undergo jump discontinuities, show that i x x x ρεdx = q( x, t) q( x, t) + ρrdx + Ldx 1 x1 x1 x1 (1.4.11) i 1 x dx x t x x t x t x x t ρxbdx x x ρ( ) = (, ) (, ) ( 1, ) ( 1, ) + x 1 x1 x x1 Ldx (1.4.1) Equations (1.4.11) (1.4.1) show how the stress power couples the internal energy the kinetic energy of the part of the body U in x1 < x< x General Balance he reader has probably noticed a formal similarity between the three balance equations discussed thus far in this chapter. Each balance equation is a special case of the following equation of general balance: i x x ρψdx =Γ( x, t) Γ ( x, t) + ρϕdx 1 x1 x1 (1.5.1) In equation (1.5.1) the left side represents the rate of change of the amount of ψ in x1 < x< x. he term Γ( x, t) Γ ( x1, t) represents the net influx of ψ, the last term presents the supply of ψ. he following table shows the choices of ψ, Γ ϕ appropriate to the equations of balance of mass, momentum energy.

21 16 Chapter 1 ψ Γ ϕ Mass Momentum x b Energy 1 ε + x x q r+ xb Exercise Derive the following local statements of the general balance: Γ ρψ = + ρϕ (1.5.) x for x yt ( ), [ ρψ ( x y ) Γ ] = 0 (1.5.3) for x = yt ( ). Equation (1.5.3) is a one dimensional version of a result known as otchine's theorem. Exercise 1.5. Derive a material version of (1.5.) he Entropy Inequality he entropy inequality is the mathematical statement of the Second Law of hermodynamics. In order to state this inequality, we introduce three new quantities. he entropy density (entropy/mass) is denoted by η ( x, t). he entropy flux is denoted by hxt (, ), the entropy supply density is denoted by kxt. (, ) hese three quantities are required to obey the following entropy inequality or Clausius-Duhem inequality: i x x ρηdx h( x, t) h( x, t) + ρkdx 1 x1 x1 (1.6.1) for all parts of the body U. he temperature is introduced by forcing the ratio of entropy flux to heat flux to equal the ratio of entropy supply density to heat supply density. he temperature θ ( x, t) is defined to be the common value of these two ratios, i.e., θ q r ( xt, ) = h = k (1.6.)

22 One-Dimensional Continuum Mechanics 17 We also require θ to be a positive number, thus, qxt (, ) hxt (, ) = θ ( x, t) (1.6.3) rxt (, ) kxt (, ) = θ ( x, t) (1.6.4) Given (1.6.3) (1.6.4), (1.6.1) becomes i x ( 1, ) (, ) x ρηdx qx t qx t ρr + dx θ( x, t) θ( x, t) θ x1 x1 1 (1.6.5) By a now familiar argument, (1.6.5) can be written x q/ θ ρr q ρη dx ρη( x y) 0 x + + x θ [ θ ] (1.6.6) for all parts of U. he following local inequalities are valid: q/ θ ρr ρη + 0 (1.6.7) x θ for x yt (), q [ ρη( x y ) + ] 0 (1.6.8) θ for x = yt ( ) If (1.6.7) is written q θ 1 q ρη + ρr 0 (1.6.9) θ x θ x the term q ρ r x can be eliminated by use of (1.4.9). he result of this elimination is q θ ρθη ( ε) + L 0 (1.6.10) θ x

23 18 Chapter 1 A more convenient version of (1.6.10) results if we introduce the Helmholtz free energy density defined by his definition allows (1.6.10) to be written Exercise If θ 0 is a positive number, show that ψ = ε ηθ (1.6.11) q θ ρψ ( + ηθ ) + L 0 (1.6.1) θ x x x1 i 1 ( ( ) x ) ρψ+ ηθ θ + dx x (, txx ) (, t) x (, txx ) (, t) θ 0 θ 0 + qx ( 1, t) 1 qx (, t) 1 θ( x1, t) θ( x, t) x x θ0 + ρxbdx + 1 ρrdx x 1 x 1 θ (1.6.13) Equation (1.6.13) is a useful representation of the entropy inequality (1.6.5) when one wants to study the stability of certain types of bodies. Exercise 1.6. Use (1.6.1) prove that the Helmholtz free energy density cannot increase in an isothermal constant deformation process. Exercise he enthalpy density χ was defined by equation (1.4.6). he Gibbs function is defined by ς = ψ (1.6.14) ρ Use these definitions show that q θ ρθη ( χ) 0 (1.6.15) θ x

24 One-Dimensional Continuum Mechanics 19 Exercise q θ ρς ( + ηθ ) 0 (1.6.16) θ x Use (1.6.16) prove that the Gibbs function cannot increase in an isothermal constant stress process. Exercise Use the definition (1.6.11) show that (1.4.9) can be written Exercise Show that material versions of (1.6.7), (1.6.1) (1.6.17) are respectively. Exercise q ρθη = ρ( ψ + ηθ ) + L + ρr (1.6.17) x q / θ ρrr ρη R + 0 (1.6.18) θ ( ) q θ ρ 0 R ψ + ηθ + F (1.6.19) θ q ρrθη = ρr( ψ + ηθ ) + F + ρrr (1.6.0) hroughout this chapter, we have developed jump expressions which govern balance of mass, momentum energy across a jump discontinuity. We have also, with (1.6.8), developed a jump inequality which follows from the entropy inequality. It is interesting to develop material versions of these jump equations. As a first step, combine (1..6) (1..1) show that [ F 1 ( x y )] = 0 (1.6.1) he physical quantity 1 ± Y ± = F ( y x ± ) is the velocity in the reference configuration of the image of the spatial discontinuity. he notation ± means that the equation 1 ± Y ± = F ( y x ± ) is really two equations, one evaluated on the + side of the discontinuity one evaluated on the side.

25 0 Chapter 1 In any case, it follows from (1.6.11) that conservation of mass forces [ Y ] = 0. hus, the physical quantity Y is actually continuous across the discontinuity. Show that material versions of equations (1.3.8), (1.4.3) (1.6.8) are Exercise ρ Yx [ ] + [ ] = R 0 (1.6.) ρ Y [ ε + 1 R 0 x ] + [ x ] [ q] = (1.6.3) q ρry [ η] [ ] 0 θ (1.6.4) Use the definition of Y given above show that the jumps [ x ] [ F] are related by Exercise Derive the material version of the Rankine-Hugoniot relation [ x ] = YF [ ] (1.6.5) ρ RY = [ ] (1.6.6) [ F] Equation (1.6.6) is useful in the study of one dimensional shock waves in certain types of materials. It gives the velocity of the shock in the reference configuration in terms of jumps in stress jumps in deformation. It is, in reality, the material version of (1.3.1). Exercise Show that material versions of (1.4.5) (1.4.7) are 1 ( + ρr[ ε] + ) [ F] = 0 (1.6.7) As with (1.4.4) (1.4.5), these results also assume [ q ] = ρr[ χ] + ( F + F )[ ] = 0 (1.6.8)

26 One-Dimensional Continuum Mechanics Example Constitutive Equations he equations of balance are indeterminate in that they involve more variables than there are equations. his indeterminacy is to be expected since the balance equations apply to every continuous body. Experience shows that continuous bodies behave in radically different ways. here must be equations of state or constitutive equations which distinguish various types of materials. An important part of continuum mechanics is the study of constitutive equations. In this section we shall give examples of constitutive equations which define certain well know types of materials. he first example is taken from gasdynamics. he material defined by the constitutive equations to be listed below is a heat conducting compressible gas with constant specific heats. he constitutive equations which define this material are ε = c v θ + ε + (1.7.1) η = c lnθ Rln ρ+ η + (1.7.) v = π = ρrθ (1.7.3) q = θ (1.7.4) x where c v is a positive constant that represents the specific heat at constant volume, R is a positive constant that represents the gas constant, ε + is a constant representing the reference internal energy, η + is a constant representing the reference entropy, π is the one dimensional pressure θ (, ρ ) is the one dimensional thermal conductivity. he one dimensional pressure is the force of compression on the gas. It is a property of the thermal conductivity that ( θ, ρ) 0 (1.7.5) As our notation indicates, can depend upon θ ρ. It follows from (1.7.1), (1.7.) (1.6.11) that It follows from (1.7.6), (1.7.) (1.7.3) that ψ = cθ θc lnθ + θrln ρ+ ε θη (1.7.6) v v ψ η = (1.7.7) θ

27 Chapter 1 π = ρ ψ ρ (1.7.8) herefore, the Helmholtz free energy, as a function of ( θ, ρ ), determines η π. hus, ψ is a thermodynamic potential for the material defined by (1.7.1) through (1.7.4). Given ψ, then η π are determined by (1.7.7) (1.7.8). he internal energy density ε is then determined from (1.6.11). It is reasonable to question why (1.7.7) (1.7.8) happen to hold. Other questions one θ could ask are why is it only the heat flux that depends upon x, why does q vanish when θ x vanishes, why must the thermal conductivity be nonnegative. In the next section we shall show that the entropy inequality places restrictions on the constitutive equations. In particular, the restrictions (1.7.5), (1.7.7) (1.7.8) are consequences of the entropy inequality. How one establishes these results will be explained in the next section. It is important to note that certain of the features of (1.7.1) through (1.7.4) are not a consequence of the entropy inequality. For example, c v is required to be a positive number. his requirement is a consequence of thermodynamic stability considerations. For reference later, we shall give several other example constitutive equations. he example stress constitutive equations are the following: 1. Linear elasticity w = E (1.7.9) where E is a material constant called the one-dimensional modulus of elasticity.. Linear viscoelasticity (Volterra material) wt (, ) wt (, s) = E(0) + E ( s) ds (1.7.10) 0 where Es ( ) is the stress relaxation modulus. 3. Linear Viscous Material (Voight or elvin Material) w w = E + μ (1.7.11) where E μ are material constants. he constant μ is the coefficient of viscosity. 4. Maxwellian Material

28 One-Dimensional Continuum Mechanics 3 where τ E are material constants. 5. Linear hemoelasticity w τ + = E (1.7.1) w = E β ( θ θ0) (1.7.13) where E is the isothermal modulus of elasticity β is a constant that can be related to the coefficient of thermal expansion. Example constitutive equations for the heat flux are the following: 1. Nonconductor. Fourier Heat Conductor 3. Maxwell-Cattaneo Heat Conductor 4. Gurtin-Pipkin Heat Conductor q q = 0 (1.7.14) = θ (1.7.15) x τq + q = θ (1.7.16) x θ ( xt, s) q= a() s ds (1.7.17) 0 x In the formulation of any theory of material behavior there are certain principles which restrict constitutive equations. he first is a requirement of consistency. his requirement is that constitutive assumptions must be consistent with the axioms of balance of mass, momentum energy with the entropy inequality. his requirement will be the one we begin to investigate in the next section. When we consider three dimensional models, the requirements of material frame indifference material symmetry will be used to restrict constitutive equations. As an operating procedure, we shall utilize the concept of equipresence. his concept states that an independent variable present in one constitutive equation should be present in all unless its presence can be shown to be in contradiction with consistency,, for three dimensional models, material frame indifference or material symmetry.

29 4 Chapter hermodynamic Restrictions In this section we shall establish the type of thermodynamic restrictions described in the last section. We shall illustrate our results by examining the thermodynamic restrictions which follow for a particular set of constitutive assumptions. By a thermodynamic process, we mean a set consisting of the following nine functions of (, t ): χ, θψη,,, q,, ρ, r b. he members of this set are required to obey balance of mass, balance of momentum balance of energy. It is convenient to introduce a special symbol for the function whose value is θ write θ = Θ (, t) (1.8.1) he constitutive equations we shall study in this section are characterized by requiring ψ, η, q to be determined by the functions Θ χ. Formally, we shall write ( ψ ( t,), η( t,), t (,), qt (,)) = f( Θ (,), χ(,)) (1.8.) he function f is called the response function. An admissible thermodynamic process is a thermodynamic process which is consistent with (1.8.). If we regard ρ R ( ) as given, then for every choice of Θ χ there exists an admissible thermodynamic process. o prove this assertion, we must construct from Θ χ the seven remaining functions ψ, η, q,, ρ, r b such that balance of mass, momentum energy are satisfied. Given (1.8.), Θ χ determine the four functions ψ, η, q. he function ρ is determined by balance of mass (1..6) written ρ = ρ R / F (1.8.3) he function b is determined by balance of linear momentum (1.3.7) written 1 b= x (1.8.4) ρ x Finally, the function r is determined by balance of energy (1.6.17) written 1 1 q r = θη+ ψ + ηθ L+ (1.8.5) ρ ρ x he impact of the last argument is that when (1.8.) is given, the balance equations are always satisfied no matter how we select Θ χ. is As yet, we have not made use of the entropy inequality. he inequality (1.6.1), rewritten,

30 One-Dimensional Continuum Mechanics 5 where 1 ρψ ( + ηθ ) + L qg 0 (1.8.6) θ g θ = (1.8.7) x If we were to substitute (1.8.) into (1.8.6), the resulting inequality depends, in a complicated way, on Θ χ, the response function f. We can view (1.8.6) as a restriction of the response function f or a restriction on the functions Θ χ. We shall require that (1.8.6) be a restriction on f. As an illustration, consider the case where (1.8.) specializes to ψ = ψθ (, gff,, ) (1.8.8) η = η( θ, gff,, ) (1.8.9) = ( θ, g, F, F ) (1.8.10) q= q( θ, g, F, F ) (1.8.11) hese constitutive assumptions clearly contain (1.7.1) through (1.7.4) as a special case. hey define a nonlinear one dimensional material that is compressible, viscous heat conducting. Our objective is to determine how (1.8.6) restricts the functions ψ, η, q. First we differentiate (1.8.8) to obtain ψ = ψ θ + ψ g + ψ F + ψ F θ g F F (1.8.1) If this result, along with (1.8.9), (1.8.10) (1.8.11), are substituted into (1.8.6), the result is ψθ (, gff,, ) ψθ (, gff,, ) ρ + η( θ, gff,, ) θ ρ g θ g ψθ (, gff,, ) 1 ψθ (, gff,, ) + ( θ, g, F, F) ρf FF ρ F F F 1 gq( θ, g, F, F ) 0 θ (1.8.13)

31 6 Chapter 1 Equation (1.8.13) is required to hold for every choice of the functions Θ χ. By selecting a family of functions Θ χ each having the same θ, gf, F, the quantities θ, g F can be assigned any value. In particular they can be assigned values which violate the inequality (1.8.13) unless ψθ (, gff,, ) + ηθ (, gff,, ) = 0 (1.8.14) θ ψ( θ, gff,, ) = 0 g (1.8.15) ψ( θ, gff,, ) = 0 F (1.8.16) herefore, ψ = ψθ (, F) (1.8.17) ψ ( θ, F) η = η( θ, F) = θ (1.8.18) hus, η is determined by ψ both quantities cannot depend upon g F. Given (1.8.17) (1.8.18), (1.8.13) reduces to ψθ (, F) 1 1 ( θ, gff,, ) ρf FF gq( θ, gff,, ) 0 (1.8.19) F θ Because can depend on g q can depend on F, it is not possible to conclude that the two terms in (1.8.19) are separately positive. It does follow from (1.8.19) that ψθ (, F) 1 ( θ,0, F, F ) ρf FF 0 (1.8.0) F 1 gq( θ, g, F,0) 0 (1.8.1) θ

32 One-Dimensional Continuum Mechanics 7 Equation (1.8.1) shows that when F = 0, the heat flux must be opposite in sign from the ψ ( θ, F) temperature gradient. We shall show below that ρf is the stress in a state of F thermodynamic equilibrium. Equation (1.8.0) shows that, when g = 0, the stress in excess of ρ ψ ( θ, F) F necessarily has a nonnegative stress power. F Next we shall derive the equilibrium restrictions from (1.8.19). As a function of ( θ, gff,, ), the left side of (1.8.19) is a minimum at ( θ,0, F,0) for all θ F. Because of this fact, the material defined by (1.8.8) through (1.8.11) is said to be in thermodynamic equilibrium when g = F = 0. If we define a function Φ of ( θ, gff,, ) by ψθ (, F) 1 1 Φ ( θ, gff,, ) = ( θ, gff,, ) ρf FF gq( θ, gff,, ) 0 (1.8.) F θ then Φ is a minimum at ( θ,0, F,0). herefore, dφ( θλ, a, F, λa d λ λ = 0 = 0 (1.8.3) d Φ( θλ, a, F, λa dλ λ= 0 0 (1.8.4) for all real numbers a A. Since dφ( θλ, a, F, λa Φ( θ,0, F,0) Φ( θ,0, F,0) = a+ A d g F (1.8.5) λ λ = 0 (1.8.3) is equivalent to Φ( θ,0, F,0) Φ( θ,0, F,0) = = 0 g F (1.8.6) Equation (1.8.4) is equivalent to the requirement that the x matrix

33 8 Chapter 1 Φ( θ,0, F,0) Φ( θ,0, F,0) g g F Φ( θ,0, F,0) Φ( θ,0, F,0) g F F is positive semi-definite. It easily follows from (1.8.) (1.8.6) that q( θ,0, F,0) = 0 (1.8.7) (, F) ( θ,0, F,0) = ρf ψ θ F hus, the equilibrium heat flux must vanish the equilibrium stress must equal If we define a function 0 ( θ, F) by 0 (, F) ( θ, F) = ρ F ψ θ F (1.8.8) ρ ψ ( θ, F) F. F (1.8.9) e a function ( θ, g, F, F ) by e 0 ( θ, g, F, F) = ( θ, g, F, F ) ( θ, F) (1.8.30) 0 then is the equilibrium stress e that vanishes in equilibrium. Exercise e is the dissipative or extra stress. he result (1.8.8) shows Calculate the elements of x matrix defined above show that the matrix is positive semidefinite if only if q( θ,0, F,0) 0 g (1.8.31)

34 One-Dimensional Continuum Mechanics 9 e 1 1 q( θ,0, F,0) ( θ,0, F,0) F θ g F e 1 ( θ,0, F,0) 1 1 q( θ,0, F,0) F 4 g θ F (1.8.3) Note, in passing, that (1.8.31) (1.8.3) combine to yield e 1 ( θ,0, F,0) F 0 F (1.8.33) Exercise 1.8. Show that when the constitutive equations (1.8.8) through (1.8.11) are independent of F, that (1.8.17), (1.8.18), (, F) ( θ, F) = ρf ψ θ F (1.8.34) 1 gq( θ, g, F) 0 (1.8.35) θ are the thermodynamic restrictions. Exercise Show that when the constitutive equations (1.8.8) through (1.8.11) are independent of g, that (1.8.17), (1.8.18), q = 0 (1.8.36) (1.8.37) e 1 ( θ, FFFF, ) 0 are the thermodynamic restrictions. Exercise Express the formula (1.8.9) in terms of the variables θ ρ show that

35 30 Chapter 1 where 0 = π (1.8.38) ψ ( θ, ρ) π = ρ ρ (1.8.39) Exercise If (1.8.18) (1.8.9) are used, the derivative of (1.8.17) is ψ ηθ ρ 0 FF 1 = + (1.8.40) Equation (1.8.40) is called the Gibbs relation. Show that ε θη ρ 0 FF 1 = + (1.8.41) Also, on the assumption that (1.8.18) can be solved for θ as a function of ( η, F), show that ε ( η, F) θ = η (1.8.4) 0 (, F) ( η, F) = ρf ε η F (1.8.43) Exercise Use the results of Exercise show that the energy equation (1.6.17) (or (1.4.9)) reduces to ρθη q x e = L + ρr (1.8.44) Exercise Show that the material version of (1.8.44) is e q ρrθη = F + ρrr (1.8.45) Exercise 1.8.8

36 One-Dimensional Continuum Mechanics 31 Derive the thermodynamic restrictions for a material whose constitutive equations are ψ = ψθθ (,, gff,, ) (1.8.46) η = η( θ, θ, gff,, ) (1.8.47) = ( θθ,, g, F, F ) (1.8.48) q= q( θθ,, g, F, F ) (1.8.49) he material model defined by these constitutive equations could possibly yield a hyperbolic partial differential equation for θ. he model defined by (1.8.8) through (1.8.11) yields a parabolic equation which has the undesirable feature that thermal disturbances propagate with infinite speed. Will the above model yield a hyperbolic equation? 1.9. Small Departures from hermodynamic Equilibrium he mathematical model which results from the constitutive equations (1.8.8) through (1.8.11) is quite complicated. A less complicated model results if we assume that the departure from thermodynamic equilibrium is small. In this case we can derive approximate formulas for q e. Departures from the state ( θ,0, F,0) are measured by a positive number defined by = g + F (1.9.1) Given e e = ( θ, g, F, F ) (1.9.) q= q( θ, g, F, F ) (1.9.3) we can write the following series expansions e e e ( θ,0, F,0) ( θ,0, F,0) = F + g+ O F g ( ) (1.9.4)

37 3 Chapter 1 q( θ,0, F,0) q( θ,0, F,0) q= g+ F + O g F ( ) (1.9.5) he leading terms in both expansions vanish because of the equilibrium results (1.8.7) (1.8.8). he coefficients in the expansions (1.9.4) (1.9.5) correspond to material properties of the body. We shall write q( θ,0, F,0) θ (, F) = g λθ (, F) = F e ( θ,0, F,0) F (1.9.6) (1.9.7) q( θ,0, F,0) αθ (, F) = F F (1.9.8) ν ( θ, F) = e ( θ,0, F,0) g (1.9.9) herefore, (1.9.4) (1.9.5)can be rewritten e = λl+ νg + O( ) (1.9.10) q g αl O( ) = + (1.9.11) where (1.1.19)has been used. he quantity is the coefficient of thermal conductivity λ is the bulk coefficient of viscosity. he coefficients ν α are zero in many of the stard applications of our model, thus, are not given names which will be familiar to the reader. Material symmetry considerations, which we have not discussed, show that ν α must vanish for materials with a center of symmetry. his means that the constitutive equations are invariant under an inversion in the reference configuration. In any case, the material coefficients in (1.9.10) (1.9.11)must obey the restrictions (1.8.31) (1.8.3). hese restrictions yield ( θ, F) 0 (1.9.1)

38 One-Dimensional Continuum Mechanics 33 Because of (1.9.1), it follows from (1.9.13)that 1 α θ (, F) λθ (, F) θ ν+ (1.9.13) 4 θ λθ (, F) 0 (1.9.14) Equations (1.9.1) (1.9.14)are the classical results that the thermal conductivity the viscosity cannot be negative. When the remainder terms are omitted from (1.9.10) (1.9.11), the result is a material model with linear dissipation. he field equations which result from utilizing these approximations are still nonlinear. In the next section we shall proceed one additional step assume the departure from a static solution θ = const F 1 is small. he resulting constitutive equations are linear yield a set of linear governing partial differential equations Small Departures from Static Equilibrium If we consider the state of constant temperature constant deformation defined by Θ ( t, ) = θ + (1.10.1) χ (, t) = (1.10.) it follows that, in this state, F = 1, F = 0 g = 0. It immediately follows from (1.8.17), (1.8.18) 0 (1.8.9)that ψ, η are constants in the state defined by (1.10.1) (1.10.). Also, e from (1.8.7), (1.8.8), (1.8.9) (1.8.30) it follows that q vanish. It follows from (1.3.7) that, in the state defined by (1.10.1) (1.10.), Likewise, the energy equation (1.8.44) yields b = 0 (1.10.3) r = 0 (1.10.4) herefore, given (1.10.1)through (1.10.4), the field equations are identically satisfied. Such a solution is appropriately called a static solution. Our objective in this section is to derive the approximate constitutive field equations which are valid near the static solution. Departures from the static solution are measured by a positive number 1 defined by

39 34 Chapter 1 = θ θ + + g + F + F (1.10.5) 1 ( ) ( 1) Note that the static solution is trivially a thermodynamic equilibrium state. In order to obtain 0 expressions for η which are correct up to terms of order O( 1), we must have a representation for ψ ( θ, F) correct up to terms O( ). he necessary expansion of (1.8.17) is ψθ (,1) + ψθ (,1) ψ ( θ, F) = ψ( θ,1) + ( θ θ ) + ( F 1) θ F ψθ (,1) + ψθ (,1) + + ( θ θ ) + ( θ θ )( F 1) θ θ F + 1 ψθ (,1) 3 + ( F 1) + O ( 1 ) F (1.10.6) For the sake of a simplified notation, (1.10.6) shall be written ψ ( θ, F) = ψ η ( θ θ ) + ( F 1) ρ 1 c β F θ v + + ( θ θ ) ( θ θ )( 1) + ρr 1 λ 3 + ( F 1) + O( 1 ) ρ R R (1.10.7) where + + ψ = ψθ (,1) (1.10.8) η = + + = ρr + ψθ (,1) θ + ψθ (,1) F (1.10.9) ( ) c v = θ + + ψθ (,1) θ ( ) + ψθ (,1) β = ρr θ F (1.10.1)

40 One-Dimensional Continuum Mechanics 35 + ψθ (,1) λ = ρr F ( ) From (1.8.18) (1.8.9), η + + are the entropy stress in the equilibrium state. he coefficient c v is the specific heat at constant volume, λ is the isothermal modulus of elasticity β is a constant that is related to the coefficient of thermal expansion. he exact relationship to the coefficient of thermal expansion is not important to us at this point. As follows from (1..6), ρ R is the density in the static equilibrium state. Given (1.10.7), it follows from (1.8.18) (1.8.9)that + cv + β η = η + ( θ θ ) + ( F 1) + O( 1 ) ( ) + θ ρ R = βθ ( θ ) + λ( F 1) + O( ) ( ) Next, we need expressions for e q valid near the static solution. Recalling that both e q must vanish whenever g F are zero, we obtain from (1.9.) (1.9.3) e e e ( θ +,0,1,0) ( θ +,0,1,0) = g+ F + O( 1 ) g F ( ) q( θ +,0,1,0) q( θ +,0,1,0) q = g+ F + O( 1 ) g F ( ) By use of (1.8.7) (1.1.11), g θ θ θ 1 x 1 + ( F 1) 1 = = F = θ = + O( 1 ) ( ) his result, along with (1.9.6) through (1.9.9) allows equations ( ) ( ) to be written e θ = ν + + λ + F + O( 1 ) ( )