# muschik_non-equilibrium thermodynamics with applications to solids

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内容提示： CISM COURSES AND LECTURES Series Editors: The Rectors of CISM Sandor Kaliszk:y - Budapest Horst Lippmann - Munich Mahir Sayir - Zurich The Secretary General ofCISM Giovanni Bianchi - Milan Executive Editor Carlo Tasso- Udine The series presents lecture notes, monographs, edited works and proceedings in the field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series in to make known in the international scientific and technical community results obtained in some of the...

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CISM COURSES AND LECTURES Series Editors: The Rectors of CISM Sandor Kaliszk:y - Budapest Horst Lippmann - Munich Mahir Sayir - Zurich The Secretary General ofCISM Giovanni Bianchi - Milan Executive Editor Carlo Tasso- Udine The series presents lecture notes, monographs, edited works and proceedings in the field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series in to make known in the international scientific and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences.
INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECfURES- No. 336 ~ V NON-EQUILIBRIUM THERMODYNAMICS WITH APPLICATION TO SOLIDS DEDICATED TO TI-IE MEMORY OF PROFESSOR TI-IEODOR LEHMANN EDITED BY W.MUSCHIK TECHNICAL UNIVERSITY, BERLIN Springer-Verlag Wien GmbH
Le spese di stampa di questo volume sono in parte coperte da contributi del Consiglio Nazionale delle Ricerche. This volume contains 33 illustrations. This work is subject to copyright. Ali rights are reserved, whether the whole or part of the material is concemed specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 1993 by Springer-Verlag Wien Original1y pub1ished by CISM, Udine in 1993. In order to make this volume available as economically and as rapidly as possible the authors' typescripts have been rcproduced in their original forms. This method unfortunately has its typographical limitations but it is hoped that they in no way distract the reader. ISBN 978-3-211-82453-5 DOI 10.1007/978-3-7091-4321-6 ISBN 978-3-7091-4321-6 (eBook)
PREFACE This monograph is made up of notes for the Advanced School on "Non-Equilibrium Thermodynamics with Applications to Solids" we gave at the International Centre for Mechanical Sciences in Udine in September 1992. This school was prepared together with our late colleague Theodor Lehmann. Thus we all dedicated this school to the memory of him. Because non-equilibrium thermodynamics is a still growing discipline in vivid development it was the aim of the school to put emphasis on the basic ideas behind the different approaches and on their application, especially to solids. Therefore starting out witlz fundamentals of non-equilibrium thermodynamics a general part follows on thermodynamics of solids, including thermoplasticity. Basic ideas, phenomenological as well as microscopic ones, underlying extended thermodynamics and its application are discussed. Non-equilibrium tlzermodynamics of electromagnetic solids, especially dielectric relaxation and elastic superconductors are treated. The important question of stability in plasticity is investigated with re gard to constitutive equations. Obviously it is imposszble to present in 4-5 days all aspects of non-equilibrium thermodynamics of solids. But as the detailed discussion on a high level between tlze lecturers and tlze participants demonstrates, tlze sclzool gave way for a new understanding andfor further individual studies.
First of all 1 thank warmly my colleagues P. Haupt, G. Lebon, G.A. Maugin, and H. Petrykfor their encouragement in organizing, preparing, and peiforming the schoo/. Then 1 have to thank the participantsfor stimulating questions and vivid discussions, and fast not /east, it is my pleasant duty to thank the authorities of CISM for inviting us to present this school and for the great hospitality of the Centre. In particular 1 want to express my gratitude to Professor Sandor Kaliszky for his idea and sponsoring to deliver such a school. W. Muschik
Dedication to Theodor Lehmann * 10 August, 1920 t 29 August, 1991 This volume is dedicated to Professor Dr.-lng. Dr. h.c. Theodor Lehmann. The subject of Theodor Lehmann was engineering science and natural philosophy. He saw the motivation and the origin of engineering mechanics in practica! applications as well as in theoretical physics. He studied mechanical engineering in Breslau and Hannover, doctorated in the field of fluid mechanics and then went to an industrial manufactory to work on metal forming processes. There he might have felt that the foundations of thermoplasticity are insufficient: He moved back to the University of Hannover and wrote a Habilitation thesis, entitled "Einige Betrachtungen zu den Grundlagen der Umformtechnik" (i.e ,"Some Remarks to the Foundations of Metal Forming"). For a period of 35 years he worked in general mechanics, continuum mechanics, thermodynamics and plasticity theory. Lehmann had a widespread experience in quite different branches of his subject and a deep knowledge of its history and literature. He played an active and creative role in the development of modem continuum mechanics. He was appreciated all over the world as a great scientist and quite often as a very good friend. Theodor Lehmann was open minded and helpful to everyone; he was always willing to share his knowledge and experiences with colleagues and especially with young people. Originally, he promised to participate in this summer school, and we were looking forward to his cooperation and the opportunity to leam from him - not only in the field of science. We are not able to replace him.
CONTENTS Page Preface Dedication to Theodor Lehmann Fundamentals of Nonequilibrium Thermodynarnics by W. Muschik ....................................................................................... ! Thermodynamics of Solids by P. Haupt ......................................................................................... 65 Extended Thermodynamics by G. Lebon ....................................................................................... 139 Nonequilibrium Thermodynarnics of Electromagnetic Solids by G.A. Maugin .................................................................................. 205 Stability and Constitutive Inequalities in Plasticity by H. Petryk ...................................................................................... 259
Abstract Starting out with a survey on different thermodynamical theories concepts of nonclassical thermodynamics such as state space, process, projection, and nonequilibrium contact quantity are discussed. Using the dissipation inequality for discrete systems the existence of a non-negative entropy production is investigated. Interna! variables are defined by introducing concepts concerning their properties. Then field formulation of thermo-dynamics is achieved by use of nonequilibrium contact quantities intro-duced above. Extended thermodynamics is shortly discussed in compari-son with the non-extended one. Material axioms and how to exploit the dissipation inequality are items of the last two chapters. FUNDAMENT ALS OF NONEQUILIBRIUM THERMODYNAMICS W.Muschik Technical University of Berlin, Berlin, Germany
2 W. Muschik 1 Survey Tbermodynamics is concerned witb tbe general structure of Scbottky systems [1]. By definition tbese systems excbange beat, work and material witb tbeir environment. Tbeir states are described by state variables wbicb are elements of a suitable state space. It is useful to distinguisb between considering only one Scbottky system in an equilibrium environment, a so-called discrete system, and between dealing witb a set of sufficiently small Scbottky systems, eacb denoted by its position and time, excbanging beat, work and material witb its adjacent Scbottky systems. In doing so we get a so-called field formulation of tbermodynamics wbicb needs different matbematical tools witb regard to tbe description of discrete systems. But tbe general tbermodynamical fundamentals are tbe same in botb cases, altbougb tbey may appear in a different form. Traditionally tbese fundamentals of tbermodynamics are formulated by "Laws" wbicb are enumerated from zero to tbree. Here tbese laws are discussed in tbe framework of different tbermodynamical tbeories of wbich a survey is given as an introduction. 1.1 Probabilistic and Deterministic Theories Thermodynamical tbeories can be divided into two classes, tbe probabilistic and tbe deterministic tbeories (Fig. 1.1). Tbe probabilistic theories tbemsel ves arc de-composed into stocbastic, statistica!, and transporttbeoretical brancbes wbicb are operating witb totally different concepts. Tbe deterministic tbeories are split up into those which describe discrete systems and others which deal with continuumthcore-tical concepts. Stocbastic tbermodyna.mics [2] [3] is characterized by a measure space [0, A, P] -a Kolmogorov probability algebra- which is defined on the state space of a microdo-main. These microdomains form mesodomains to which the probabilistic description of tbe microdomains is transferred by introducing suita.ble mea.n values. Tbe site of tbe mesodomain is identified witb tbe material coordina.te so getting the connection between stocbastic and deterministic thermodynamics. Statistica! tbeories are marked by a distribution function or by a density operator wbicb may depend on a relevant set of observables (so-called Beobachtungsebene) [4]. We use sucb a description for tbe foundation of nonequilibrium contact quantities, such as contact temperature or nonequilibrium chemical potentials [5]. Transporttheoretical methods also use a distribution function f(p,q,t) which is defined in contrast to tbe distribution functions of tbe statistica! methods on the (2f+1)-dimensional p-space of the single molecule baving f degrees of freedom [6]. In contrast to tbe probabilistic theories the deterministic or pbenomenological theories do not take into consideration tbe molecular structure of materials, whe-reas probabilistic tbeories even embrace this structure by microscopic models using master equations, molecular dynamics or density operators. The basic concept of
Fundamcntals of Nonequilibrium Thermodynamics phenomenological theories is that of the macroscopic variable. These quantities describe the state of the system which can be retraced immediately to measuring quantities of the system. Examples are volume, pressure, temperature, mass density, charge density, magnetization, pressure tensor, interna! energy, etc. 1 Thermodynamics / 1 ~ - ~ r o b a b i l i s t i c deterministic ~ stochastic [Q, lf ,P] stati sti cal p( G 1 , .... Gk) transporttheoretical f([?, g, t) discrete systems (9, .o, U) continuumstheoretical p ( ~ , t ) , E ( ~ , t ) , Y ( ~ , t ) ••• Figure 1.1: Diagram showing distinct classes of thermodynamical theories. As mentioned above deterministic theories are divided into those which des-cribe discrete systems, these are Schottky systems, and those which apply continu-umtheoretical methods. As Fig. 1.2 shows there exist a lot of similar, but different deterministic thermodynamical theories. 3
4 W. Muschik Deterministic Theories l phenomenological Theories l continuumstheoretical + discrer systeml 1 Thermostatics Non-Ciassical Thermodynamics Linear lrreversible Thermodynamics Rational Thermodynamics Non-Ciassical Theories Theories using Evolution Criteria Theories using Variational Principles Extended Ther1nodynamics Figure 1.2: Family of deterministic thermodynamical theories. It should be shortly motivated why we have such a variety of phenomenological nonequilibrium theories: The transition from mechanics to thermostatics is achic-ved by adding thermodynamical quantities to the mechanical ones. Besides othcr quantities especially temperature and entropy are added. Because both these quan-tities are defined by measuring rules in equilibrium, the transition from mcchanics to thermostatics is possible without any problem. Now the question arises how to define temperature and entropy in nonequilibrium? In principle this question can be answered differently, and therefore no natural extension of thermostatics to thermodynamics exists [7]. Either temperature and entropy will redefined for nonequilibrium, or they are taken for primitive concepts, i.e. their mathematical
Fundamentals of Nonequilibrium Thermodynamics existence is presupposed and first of all a physical verification remains open. Here we deal mainly with a non-classical approach to thermodynamics starting out with discrete systems and transferring the results to continuum thermodynamics. Non-classical thermodynamics is characterized by a dynamical nonequilibrium concept of temperature, whereas other theories use the hypothesis of local equilibrium (irre-versible thermodynamics) or introduce temperature as a primitive concept (rational thermodynamics). 1.2 Discrete Systems and Field Formulation A system Q which is separated by a partition 89 from its surroundings 9 is called a discrete system or a Schottky system [1], if the interaction between Q and g can be described by the heat exchange Q, the power exchange W, and the material exchange ne (Fig. 1.3). • Q •e. n -Figure 1.3 : A Schottky system Q exchanges heat, power, and material through ag with its surroundings Q. The description of a thermodynamica.l system by a discrete one is restricted be-cause the system is taken for a black box exchanging quantities with its surroundings. As we will see below ( sect. 1.2.) the state variables of the discrete system belong to the system as a whole. The exchange quantities between the discrete system and its vicinity depend also on the state of the vicinity. Because the concept of a discrete system is so easy, we can use it for general considerations as introducing states, processes, exchange and contact quantities, and for formulating the laws of ther-modynamics. Because no gradients appear in the description of a discrete system, this concept is often used in thermostatics. But also complex machines in enginee-ring sciences are described by discrete systems, if only the exchange quantities are of interest. Thus for fundamental considerations and for a reduced description of complex situations, the concept of a discrete system is a very useful one. 5
6 W. Muschik For getting more information the Schottky system can be divided into discrete subsystems (Fig. 1.4) which exchange heat, power, and material with the adjacent discrete systems. Figure 1.4: A Schottky system divided into discrete subsystems. Therefore the state of the Schottky system is described by the states of the subsystems depending now on position. This gives risc to introduce a local state, if the division into subsystems is sufficiently small. The process taking place in thc Schottky system is characterized by the exchanges betwecn its subsystems. These exchange quantities are in some case nonlocal because they depend on the states of two subsystems. Therefore the exchanges will depend on gradients, if introducting a field formulation, i.e. if we replace the division into subsystems by fields. It is easy to see we will get different theories whether the fields describing the exchanges will be independent variables themselves or will be dependent on the gradients of the independent local state variables. One of the differences between extcnded thermodynamics and the usual one is caused by this fact. As we will see below (sect. 1.6.) the transfer from discrete systems to fi ele! formulation gives some insight in how to introduce the fields of temperature and entropy. Both quantities are strictly defined only in equilibrium, but in field formu-lation they should be applicable to nonequilibrium. How to define them in this case is a question of non-classical theories. But most of the theories in field formulation use temperature and entropy as primitive concepts, as we will see in the next section. 1.3 Rational Thermodynamics Historically Rational Thermodynamics [8] is the thermal extension of Rational Me-chanics. This extension is simply performed by adding the fields of temperature, entropy, and entropy flux density as primitive concepts, i.e. these fields are not defined physically. Although manifold, some representative prototypes of Rational Thermodynamics can be specified [9]. We distinguish between (Fig. 1.5) Clausius-Duhem theories which all use an in time local dissipation inequality as an analytical
Fundamentals of Nonequilibrium Thennodynamics expression describing the Second Law, and between other theories which use dissi-pations inequalities being global in time or which use evolution criteria [10], [11]. Here we will not treat the latter ones and also not those Clausius-Duhem theories which are global in position [12]. Field Formulation ~ , .,, -,r CLAUSIUS • DUHEM Theories (III) local intime Dissipation lnequalities (III) global in time Evolution Criteria (III) + local in position ~ . global in position Figurc 1.5: Different theori<'s in field formulation. Clausius-Duhem theorics presuppose t.hc cxistcncc of thc in time local Clattsius-Duhem dissipalion inequality from which thcy got their namc: p . ~ + V · .P - 1 = u ~ O. (1) [[('re thc cntropy dcnsity s and thc ('ntropy flux density .P=qf8+k (2) are considcr('d a.s constitutive equations [13] (k /:-o) oras being determined by heat flux ovcr tcmperature [14] (k =o) (p is thc mass density,/ is the entropy supply,u is thc cntropy production density). Thc in timc local dissipation inequality ( 1) rcpresents only a sufficient formula-t ion of thc second law bccausc this is an in time global statement (sec sect. 1.4.2). Fidd formulat.ions taking this into account are givcn by [15], [16] (3) 7
8 W. Muschik and f[(l/p)\1· (q/0)- (r/0)] dt ~ O . (4) In both inequalities an undefined nonequilibrium temperature 0 appears, which is taken for a primitive concept. In (3) an equilibrium entropy seq is used, which can be interpreted as the entropy of an accompanying process [17], [18] (see sect. 1.5.3). This concept originates from non-classical thermodynamics which is characterized by a dynamical concept of temperature, with which we will become acquainted, when discussing nonequilibrium contact quantities (sect. 1.3.2). Notice that ( 4) is formulated for a cyclic process and does not include any entropy. The Second Law for discrete systems needs a discussion in more detail and is treated in section 1.5.3. U p to now we can classify phenomenological thermodynamical theories in the following way (Fig. 1.6): The used dissipation inequality may be local or global in time; temperature et!ld entropy may be introduced as primitive concept oras derivecl quantities; the relation (2) between entropy and heat flux density may be universal or material dependent; and finally, the introduced state spaces (sect. 1.2.) may be small or large. Dissipation { local . Inequahty } . t. In 1me global (5) Temperature { primitive concept Entropy derived quantity (6) re atwn +-+ 'P . 1 . ""' { universal matenal dependent q (7) { small l arge State Space (8) Figure 1.6: Categories for classifying phenomenological thermodynamical theo-nes. 1.4 Different Formulations of the Second Law Formulations of the second law for a discrete system are historically the oldest ones [19]. First of all, we have to distinguish between formulations which use work criteria, the so-called Sears-Kestin statement (Fig. 1. 7) [20]. This states that, for adiabatic processes which are cyclic in the work variables, the work performed on the system ist not negative. At first sight it seems that the Sears-Kestin statement - because
Fundamentals of Noncquilibrium Thcnnodynamics 9 it deals especially with adiabatic processcs - is only necessary but not sufficient for dcscribing the physical content of the Second Law [21] which is expressed by the Clausius inequality f[Q(t)/T(t)- s(t) · ne(t)]dt :S O (9) (T thermostatic temperature of the system 's vicinity and its s molar entropy ). But by adding two further axioms relate to reversible processes and to compound systems - general items not affecting or affccted by the Second Law and often presupposed tacitly- Sears-Kestin statement becomes equivalent to Clausius inequality [22]. Here we will not refer to formulations of the Second Law which use work criteria. Although there is a huge flood of papers concerning the second law, two streams can easily be distinguished (Fig. 1. 7): one stems from Clausius and Lord Kelvin [23], the other from Caratheodory [24] anei Born [2.5]. Discrete Systems ~, CARATHEODORY-BORN FALK ·JUNG SERRIN- SILHAVY LANDSBERG CLAUSIUS · KEL VIN thermostatic temperature both s1gns positve + -+ -+ ® nonequilibrium tempera ture both signs positve ----® -® --? SEARS- KESTIN statements Only reversible processes method + inaccessibility + branching chain accumulat10n funCIIOn --eHic1ency -no-go principles 1 1 Figure 1. 7: Classification for formulations of the Second Law for discrete systems. Although both formulations are equivalent in the field of thermodynamics, they are, in general, different [26]. Those in the Caratheodory-Born stream [27] make use of a statement of the adiabatic inaccessibility of states in the vicinity of each state [28]. But this statement is only valid for reversible processes, and yields a foliation of the state space into surfaces of constant entropy which is, of course, the thermostatic entropy. Also the existence of the thermostatic temperature can be proved, but only for positive absolute temperatures.
10 W. Muschik Those in the Clausius-Kelvin stream state that selected irreversible processes operating between two heat reservoirs, which need not have only positive absolute temperatures, are existent or non-existent [29]. Therefore, the Clausius-Kelvin for-mulation, representing a no-go principle for certain processes, is not restricted to reversible processes and positive absolute temperatures. The temperatures which appear in this formulation of the second law are thermostatic temperatures of the heat reservoirs controlling the process. Also in the case of negative absolute tempe-ratures [30] the processes are presupposed to be so fast - and therefore irreversible-that the nonequilibrium systems of negative absolute temperature can be taken for reservmrs. The question arises: Is it possible to extend the Clausius-Kelvin formulation to nonequilibrium analogues of the thermostatic temperature [31]? This can indeed be proved by the use of an additional axiom of the simulation of processes [32]. We get for closed systems f Q(t) dt < f Q(t) dt <o T(t) -0(t) -(10) Here Q(t) is the heat exchange at time t between the system and its environment which is in equilibrium at the thermostatic temperature T(t). The integral of thc reduced heat exchange Q fT along a cyclic process defined in a sui table state space is not positive according to the Clausius-Kelvin statement for closed systems (9). If we use the contact temperature e of the discrete system (sect. 1.3.2), we will get the extended inequality on the r.h. si de of ( 10). We now discuss two other formulations (Fig. 1.7): that of Falk-Jung [33], which belongs to the Caratheodory-Born type, and tha.t of Serrin-Silhavy [34], which is related to the Clausius-Kelvin type. The Falk-Jung formulation of the second law is part of an axiom system of thcr-modynamics represented by algebraic methods. In this approach thermodynamical variables are constructed by two operations: the composition of systems which lcads to extensive variables, and the contacting of systems which defines intensive varia-bles. Processes are described by a transition relation, by means of which chains of transitions can be defined. The adiabatic isolation of a system has the structure of a chain of transitions, with no branching, by means of which the empirica! entropy is defined. This procedure which uses the adiahatical isolation to determine the empi-rica! entropy, reduces the Falk- Jung formulation to reversible processes. Thcrcfore with this technique, only the first part of the second law can be formulatecl. The Serrin-Silhavy formulation uses temperature, work and heat as primitive concepts. It is usual for work and heat to be primitive concepts in thermodynami-cal theories, but - as we discussed above - temperature should not be a primitive concept without any physical interpretation. So Serrin introduces a "thermal mani-fold consisting of the set of hotness levels". Of course, the hotness level rcpresents an empirica! temperature, but nothing is said about what empirica! temperature
Fundamentals of Nonequilibrium Thennodynamics Il represents; it could be that of the discrete nonequilibrium system or that of its environment. The latter is, of course, the correct interpretation. After this, Ser-rin introduces the so-called accumulation function celebrated by Man [35] which in usual terminology is given by Q(P, r) := P f Q ( t ) ~ [ r - T(t)]dt, with the step function ~ ( x ) = { 1, o, x>O X::::; 0. The second law can now be formula.tcd as Q(P,·) ~O ~ Q(P,·) =O, which is only valid for positive absolute tempera.tures [36]. As we saw above, a.ll formulations of the second la.w for discrete systems are global in timc (Cia.usius-Kelvin type) or presuppose topologica.! properties of the state space (Caratbeodory-Born type) which are only given in equilibrium. Thus in thermodynamics of discrete systcms no in time local formula.tion of the second law cxists. Consequently intime local formulations as the dissipa.tion inequality (1) necd additional assumptions beyond the second law. (11) (12) (13) 2 State Spaces, Processes, and Projections For dcscribing the state of a. discrete syst.cm we need a. state spa.ce. The chemical composition of the system is dctermincd by the mole numbers n, mecha.nical and geometrica! propcrties are described by the work variables a. \Ve get a general state by adding thcrmodyna.mica.l varia.bles z Z := (a,n,z) E Z (14) Zis callcd (nonequilibrium)sla/c spacc of a disc1·ete system [17]. The power cxchange of thc discrete systcm g with ils vicinit.y g is homogenC'ous in the time rates of the work varia.bles ~ V ( t ) := F(t). a(t) (15) (F(t) arc the generalizcd forces bclonging to g). Thc mole number of thc mass cxchange is dcfined by (16)
12 W. Muschik (ni are the time rates of the mole numbers due to chemical reactions,·e are the reaction speeds, ,;+ is the transposed matrix of the stoichiometric equations &i · M =o, (17) (M are the mole masses). The heat exchange Q(t) is a quantity which can be measured by calorimetry and which is introduced here as a primitive concept. The thermodynamical variables z include the interna! energy U of Q, the contact temperature E> which is independent of U [17], and if necessary, the interna! variables a. Contact temperature and interna! variables will be treated in more detail in sect.l.3.2. and in 1.5. We presuppose that we know what isolating, adiabatic, power-insulating, and material- insulating partitions are,and how to define them (for more details sec [37], sect. 1.1.1 ). Then the equilibrium conditions for discrete systems are easy to specify: Def: Time-independent states of isolated systems are called states of equilibrium. Especially simple syste::m are those which do not contain adiabatic partitions: Def: A system is called thermal homogeneous, if it does not contain any adiabatic partitions in its interior. Obviously for describing a state of cqnilibrium we do not need as many varia-bles as in nonequilibrium. Thcrefore the qucstion ariscs how many variables span the equilibrium subspace? The answcr is given by the Zeroth Law which bases on experience and which is here formulatcd as an "cmpircm" (axiom whose validity is founded on experience). Empirem: (01h Law): The state space of thermal homogencons system in equili-brium is Z* := (a,n,*;am(a,n,*)). * is exact one additional thermodynamical variablc, and am is one of the k possiblc equilibrium values of the interna! variables a which are determined hy (a, n, * ). As we will see below, this thcrmodynamical variable * rnay be t.he interna! energy U of the system or its thermostatical temperature T. 1f the state of a system changes in time, we say the systern undergocs a process: . Def: A path in state spacc is called a process ( 18) Z(·) :=(a, n, z)(·) (19) especially Z(t) =(a, n, z)(t), (20) Experience shows that equal processes in different systems can induce different values of other variables not included in the state spacc. A vcry simple exarnple are two gaseous systems having equal volume and tempera.ture but different pressure. Of course each of the two systems consists of another gas. Different materials differ
Fundamentals of Nonequilibrium Thermodynamics 13 from each other by different material properties which are described by a map M called the constitutive map. The domain of M depends on the chosen state space: There are state spaces so that _,\.;f is local in time as in thc theory of thermoelastic materials or in theories using interna) variables. There are other state spaces on which M is not local in time as in materials showing after-effects or hysteresis. Therefore we classify [38]: , , , •' , , '2(·) - -e Z(t) ~ MCtJ Figure 1.8: The material property M ( t) is only determined by the state Z ( t). ztc·J t Figure 1.9: The history Z1(·) of the process Z(·) is defined by the path of Z(·) up to time t. Def.: A state space is called large, if material properties M are defined by maps local in time (Fig. 1.8) M : Z(t)--. M(t), for aii t. (21) For defining after-effects we need the concept of the history: Def.: For a fixed time t and real s ~ O Z1(s) := Z(t- s), sE [O,r], (22) is called the history of the process Z( ·) between t -Def.: A state space is called small, if material properties M are defined by maps on process histories (Fig. 1.10) T and t (Fig. 1.9). M: Z1(·)--. M(t), for aii t. (23)
14 W. Muschik fv1 (t) Figure 1.10: In small state spaces the domain of the constitutive map is the history of the process. According to the zeroth law the space of the states of equilibrium has less dimen-sions as the general state space. Therefore projections exist mapping the general state space onto the equilibrium subspace. By these projections to each nonequili-brium process an equilibrium "process" or better a trajectory in equilibrium sub-space is attached (Fig. 1.11 ). Figure 1.11: Equilibrium subspace represented as hypcrsurfacc in state spacc. The projection P* maps Z(t) into Z*(t) so inducing the accompanying proccss :;::• from the real process :F.
Fundamentals of Nonequilibrium Thermodynamics 15 Def.: A trajectory Z*( ·) in the equilibrium subspace induced by a projection P*, (P*)2 = P*, of a process Z(·) P* Z(t) = Z*(t) P*(a,n,z)(·) = (a,n,*iOm)(·) is called an accompanying process [39]. Because in equilibrium material properties do not depend on process histories ( that is a demand for constructing the state space) (2-!) (25) M*(t) = M(Z*(t)) (26) we get by the definition of M* due to (23) (2i) the following property of thc constitutive map M(Z*1(·)) = M(Z*(t)). (28) Additiona.lly thc cquilibrium values llf* of .M have to be compatible wit.h its none-quilibrium valucs M by satisfying the Embcdding A:riom: F {B it(l)dt = 1 \ { ~1 -}A 1 \ 1 . ~ ' 1 = F* {B il*(t)dt 14 A, il E equil. subsp. (29) H wc cspccially apply this embedding axiom !.o a quant.ity which is defined on equi-librium subspacc, anei we look for an cxtcnsion of this equilibrium quantity to no-ncquilibrium, this cxtcnsion has to satisfy thc embcdding axiom, because otherwise it would not bc compa.t.iblc with the carlicr dcfined cquilibrium quant.ity: i.e. a no-ncquilibrium cntropy - in wha.tcvcr wa.y dcfincd - has to obey thc embedding axiom in ordcr to bc in agrecmcnt wit h t.hc well known concept of equilibrium entropy. ny cxpericncc we know that in adiahatic isolated system the work exchange is independent of thc proccss, if thc initial and thc final state are the same in ali proccsscs undcr considcration: If ad: 1 -+ 3 -+ 2, ad: 1 -+ 4 -+ 2 (30) arc two adia.batic proccsscs, thcn the work exchange (31) is independent of thc path in thc state spa.cc betwcen 1 and 2. This can be formulated a.s
16 W. Muschik Empirem: (lst Law): A function of stateU(Z) exists, so that for adiabatic pro-cesses a ad: 1 ~ 2 the work W1'2 in a system at rest is represented by W1c; =Uz- U ~ , Ui := U(Zi), ad: W = U, in a restsystem. If the process constraint of being adiabatical is cancelled, we define: Def.: In closed restsystems the beat exchange is defined by Q := (;- w, closed: Qfz = Uz- U1 - W1c;, in a rcstsystem. Proposition: If E is the total energy E := U + Ekin + Y, Y = potential energy, and L' the power of the fc:-ces without a potential, the first law for closeei systems writes (32) (33) (34) (35) (36) By (32) we defined by mechanics thc interna! energy of al! states Zz which are adiabatically connected with Z...