Modern Developments in the Mechanics of Continua

201 Pages·1966·8.078 MB·English

by  Salamon Eskinazi (Eds.)

Contributors to This Volume
Th. Alziary de Roquefort E. W. Merrill
D. R. Axelrad H. S. Mickley
Jose Barberan F. A. J. A. Morgan
Vaughn C. Behn A. C. Pipkin
Bernard D. Coleman F. W. Smith
Subhendu K. Datta K. A. Smith
A. G. Fabula Gerald A. Strobel
R. L. Fosdick W. D. Taylor
R. Goethals George B. Thurston
Morton E. Gurtin C. Truesdell
Ismael Herrera R. P. S. Virk
W. Jaunzemis C.-C. Wang
J. L. Lumley R. N. Yong
M o d e rn D e v e l o p m e n ts
in t he M e c h a n i cs of C o n t i n ue
Proceedings of an International Conference on Rheology
Held at the Pinebrook Conference Center of
Syracuse University
Pinebrook, New York, August 23-27, 1965
Supported by The New York State Science and Technology Foundation
Edited by SALAMON ESKINAZI
Department of Mechanical and Aerospace Engineering
Syracuse University
Syracuse, New York
ACADEMIC PRESS New York London 1966
COPYRIGHT © 1966, BY ACADEMIC PRESS INC.
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List of Contributors
Numbers in parentheses indicate the pages on which the authors' contributions begin.
Th. Alziary de Roquefort , E.N.S.M.A., University of Poitiers, Poitiers,
France (139)
D. R. Axelrad, Department of Mechanical Engineering, McGill Univer­
sity, Montreal, Canada (183)
Jose Barbera n F., Instituto de Geofisica, Universidad Nacional de
Mexico, Mexico City, Mexico (175)
Vaughn C. Behn, School of Civil Engineering, Cornell University, Ithaca,
New York (193)
Bernar d D. Coleman , Mellon Institute, Pittsburgh, Pennsylvania (165)
Subhendu K. Datta,t Department of Aerospace Engineering Sciences,
University of Colorado, Boulder, Colorado (73)
Fabu4la
A. G. The Pennsylvania State University, University Park,
Pennsylvania (145)
R. L. Fosdick, Department of Mechanics, Illinois Institute of Technology,
Chicago, Illinois (109)
R. Goethals, E.N.S.M.A., University of Poitiers, Poitiers, France (139)
Morton E. Gurtin , Brown University, Providence, Rhode Island (165)
Ismael Herrer a R., Instituto de Geofisica, Universidad Nacional de
Mexico, Mexico City, Mexico (175)
W. Jaunzemis , Department of Engineering Mechanics, The Pennsylvania
State University, University Park, Pennsylvania (65)
J. L. Lumley, Department of Aeronautical Engineering, The Pennsylvania
State University, University Park, Pennsylvania (145)
t Present address: Department of Aeronautical Engineering, Indian Institute of Higher
Technology, Khampur, India.
% Present address: United States Naval Ordnance Test Station, Pasadena, California.
ν
VI List of Contributor s
E. W. Merrill, Department of Chemical Engineering, Massachusetts
Institute of Technology, Cambridge, Massachusetts (37)
H. S. Mickley, Department of Chemical Engineering, Massachusetts
Institute of Technology, Cambridge, Massachusetts (37)
A. J. A. Morgan , Department of Engineering, University of California,
Los Angeles, California (53)
A. C. Pipkin, Division of Applied Mathematics, Brown University,
Providence, Rhode Island (89)
F. W. Smith, Division of Mechanical Engineering, National Research
Council of Canada, Ottawa, Ontario, Canada (13)
K. A. Smith, Department of Chemical Engineering, Massachusetts Insti
tute of Technology, Cambridge, Massachusetts (37)
Gerald A. Strobel ,t School of Civil Engineering, Cornell University,
Ithaca, New York (193)
W. D. Taylor , Department of Biophysics, The Pennsylvania State Univer
sity, University Park, Pennsylvania (145)
George B. Thurston , Physics Department, Oklahoma State University,
Stillwater, Oklahoma (19)
C. Truesdell, Department of Mechanics, The Johns Hopkins University,
Baltimore, Maryland (1)
P. S. Virk, Department of Chemical Engineering, Massachusetts Institute
of Technology, Cambridge, Massachusetts (37)
C.-C. Wang, Department of Mechanics, The Johns Hopkins University,
Baltimore, Maryland (129)
R. N. Yong, Department of Civil Engineering and Applied Mechanics
and Soil Mechanics, McGill University, Montreal, Canada (183)
t Present address: Shellfisheries Management Unit, State of New York Conservation
Department, Oakdale, Long Island, New York.
Preface
The papers collected in this volume were presented at the Pinebrook
Rheology Conference held on August 23-27, 1965 and sponsored by
the Department of Mechanical and Aerospace Engineering at Syracuse
University.
These papers were presented and were part of a larger effort at promot
ing national and international interest in mechanics of continua. During
the period of August 9 to September 24, 1965, Dr. C. Truesdell, Distin
guished Visiting Professor at Syracuse University and Professor in the
Department of Mechanics at The Johns Hopkins University, presented
a series of lectures in the form of a course in mechanics of continua.
The Pinebrook Conference was held at one of Syracuse University's
Adirondacks Conference Centers located on Upper Saranac Lake in
Pinebrook, New York.
S. ESKINAZI
December, 1966
vii
Thermodynamics of Deformation
C. TRUESDELL
Department of Mechanics
The Johns Hopkins University
Baltimore, Maryland
On an occasion such as this, the speaker is expected to disclose some
earth-shaking discovery of his own. Great international conferences are
scheduled at decent intervals, nowadays no less than a week apart, so
as to allow such discoveries to be made from one to the next. The only
way to be snobbish today in circles of science is to refuse to make any
more discoveries and instead devote one's time to learning the state of
the subject, like an old-fashioned teacher from the days before research
had become a racket for hoodwinking the public into allowing profes­
sors something approaching one third of a living wage. With the per­
mission of the experts here assembled, I will explain thermodynamics to
the best of my ability to understand it. While I shall present some
unpublished material, it will be what I have learned from others.
The danger of crossing the mouth of the dismal cavern of thermo­
dynamics is well known. As I wrote once, but was too tactful ever to
publish, "Indeed, there are many who claim to understand thermo­
dynamics, but it is best for them by common consent to avoid the topic
in conversation with one another, since it leads to consequences such
as can be expected from arguments over politics, religion, or the canons
of female beauty. Honesty compels me to confess that in several attempts,
made over decades, I have never been able to understand the subject,
not only in what others have written on it, but also in my own earlier
presentations."
While any physicist selected at random understands thermodynamics
perfectly and has only contempt or pity for those unfortunate and
unphysical persons who express doubt, if we cast the die a second time
and draw a second physicist, we can expect to hear a lively clash as the
words "reversible," "irreversible," "adiabatic," "isolated," "thermal
contact," "reservoir," "cyclic," "quasistatic," "ideal engine," "boiler,"
"piston," "condenser," and "universe" contribute to the entropy of the
acoustic medium. To overcome the difficulties it is best to follow a simple
ι
2 C. Truesdell
rule for study of the history of science: setting aside what an author says,
see what he does.
Since the earliest thermodynamic researches, two kinds of phenomena
have been considered. In the former, different equilibria are compared,
without regard to the means, if any, of passing from one to the other.
In the latter, the process of interconversion of heat and work is described
as it proceeds. As is well known, the former, which really ought to be
called thermostatics, has lent itself to a modest and successful mathe­
matical theory, elaboration of which fills the major parts of most books
on the subject, while the latter, which deserves the name thermodynamics,
has remained mainly verbal and far from complete. We are not to be
deceived by the words "quasi-static process" or "reversible process,"
for these are mere names for parametric families of equilibria—if we
prefer, names to cover absence of natural processes. Carnot, Clausius,
and most of the classic writers on thermodynamics maintain an illusion
of dealing with natural changes because they couch their comparisons
of equilibria in terms of processes obeying, or usually obeying, thermal
and caloric equations of state. The mathematics of thermostatics was
made explicit and precise in 1875 by Gibbs, today often cited but
apparently little read. He divided the subject into two parts: that based
on formal manipulation of equations of state, and that based on true
minimal principles. The true minimal principles are stronger: they imply
not only the formal manipulations, which all amount to reinterpretation
of the fact that the gradient vanishes at a minimum, and that partial
differentiations commute and obey chain rules, but also conditions of
stability and various consequent inequalities.
Planck in 1887 formulated the following inequality connecting dif­
ferentials of energy E, temperature 0, and entropy Η with the work W:
dE - 0dH^ W, (1)
a relation which he regarded as a summary of "all conclusions with
regard to thermodynamic chemical changes, hitherto drawn by dif­
ferent authors in different ways...." Here, in general, no equation of
state holds. Since differentials of independent variables are arbitrary
quantities, the relation as it stands makes no sense at all. Planck claimed
to deal with "any infinitesimal change" of "any system in nature,"
including "any homogeneous or heterogeneous system of bodies at a
common temperature." Thus I believe he referred to time rates in actual
processes. If so, we can find meaning for his postulate (1) if we replace
it by
Ε - ΘΗ ^ P, (2)
Thermodynamic s of Deformation 3
where the dots indicate the time rates and where Ρ is the rate of working
of the mechanical forces.
Perhaps Planck meant by energy what we now call total energy,
Ε + X, where Κ is the kinetic energy. The equation of balance of
energy then is
Κ + Ε = Ρ + Q, (3)
where Q is the rate of increase of energy through nonmechanical effects
such as heat conduction and radiation. If so, we can write Planck's
postulate in the two forms:
ΘΗ^>Ε + Κ- Ρ = ζ), (4)
each of which may be called the Clausius-Planck inequality.
The thermodynamics of deforming materials begins with gas dynamics.
While this discipline goes back to Euler's day or earlier, the thermo­
dynamic aspect of it was first made clear by Hadamard in his great
treatise on waves in 1903. The total internal energy and total entropy
of finite amounts ^ of a gas are additive set functions ("extensive
variables" in Maxwell's terms):
Ε = ε dm, Η = \ η dm, (5)
and the corresponding densities ε and η, the specific internal energy and
specific entropy, are assumed related to the density ρ by a local caloric
equation of state:
Άε,η,ρ) = 0. (6)
Hadamard remarked that there was no basis in experiment for carrying
over the static equations of state to "gases in more or less rapid motion."
Nevertheless, from that day on, gas dynamics has rested upon free use
of state, which enable all the formal apparatus of thermostatics to be
incorporated thoughtlessly. The more recent "thermodynamics of
irreversible processes," as developed by Eckart, Meixner, and others,
rests upon transferring to more complicated phenomena the conceptual
basis already familiar in gas dynamics and hence does not question the
foundations. Hadamard's doubt remains. Why should static equations
of state be valid in large and rapid deformations? More important than
the doubt itself is the problem raised by a negative or conditioned
answer. If a caloric equation of state is not valid, so that the classical
thermostatic apparatus may not be applied to elements of mass in the
deforming material, what can be said? In other words, what are the
4 C. Truesdell
relations between thermal and mechanical quantities in severe and rapid
deformations? More briefly, what are the laws of thermodynamics!
On the basis of the new continuum mechanics Coleman was able, last
year, to formulate a thermodynamics of clarity, scope, and precision
corresponding to Noll's theory of purely mechanical phenomena in
simple materials.
Coleman's brilliant work rests on use of the most complicated and dif­
ficult aspects of functional analysis yet applied to continuum mechanics.
However, the results he discovered are simple and easy to understand.
Bowen and Wang, during their polite and silent attendance at some of
the duller lectures at Bressanone in June, saw that a more general con­
ception makes it possible to obtain the basic theory by use of no more
than ordinary calculus as a tool. With their permission, I shall outline
their procedure.
First we must agree on a starting point. As I have said for many years,
disregarding the scoffs of those endowed with physical intuition, tem­
perature and entropy join mass and place and time as primitive, un­
defined variables, described only by such properties as are laid down for
them. Whenever anyone tries to do anything rational with temperature
and entropy, some physicist rises to ask whether a temperature always
exists, and how to define entropy. The same physicist if asked whether
forces exist will reply that he is not interested in philosophical questions,
and if asked how to define mass and force is likely to reply that mass is
a measure of the quantity of matter in a body, while force represents a
push or a pull in a given direction. Such a person ought to be satisfied
if I answer that temperature is a measure of how hot a body is, while
entropy represents how much heat has gone into a body at a given
temperature. Just as we expect that the student in his elementary courses
in physics has already acquired, regarding mechanics, some truths we
need to reinforce and some errors we need to correct, so also we may
presume that he has had the benefit of the vocabulary and confusion
imparted by an elementary course in thermodynamics.
Thus we are free to speak of the temperature θ and the specific entropy
η and to lay down for them, without further ado, conditions which
embody and generalize the results of more than a century's experience
with simple cases. In the equation of balance of energy (3), we now express
Q in terms of an efflux of energy h, intended to represent conduction of
heat in broad form, and a supply of energy, q, intended to account for
radiation and similar effects:
= I
Q h · η da + \ q dm. (7)