Thoughts on Thermomechanics
Walter Noll
Department of Mathematical Sciences
Carnegie Mellon University, Pittsburgh, PA 15238, USA
Abstract
First, I give a short description of the history of the interpretation of the second law of thermo-
dynamics as a restriction on the kind of material properties that physical systems occurring in nature
can have. Second, I describe the desirability of developing a framework, using the concept of a
state
of a
material element, for developing a general thermomechanical theory of simple materials.Third, I give a
taste of the mathematical infrastructure needed for such a theory. Finally, I give an outline of the paper
[NS] by Brian Seguin and myself which may be a model for the first few chapters of future textbooks on
continuum thermomechanics.
Key Words:
Thermomechanics, Second Law, Simple materials, mathematical infrastructure.
1. History and Outlook
Until about 50 years ago, the overlap between
thermodynamics and mechanics was very limited. It
dealt mostly with the laws relating thermal energy
and mechanical energy. However, the first and sec-
ond laws of thermodynamics were not connected with
continuum mechanics. Courses in thermodynamics,
elasticity, and hydrodynamics had very little connec-
tion.
The very term “thermomechanics” had not
been invented. Even now the term is still not in the
dictionary. Yet there are books with this title, and
typing this term into Google yields 65 000 hits.
Of course, the development of statistical me-
chanics in the 19th century yielded an important con-
nection between particle mechanics and thermody-
namics, but this did not seem to have much relevance
to continuum mechanics.
When working on
The Non-Linear Field Theo-
ries of Mechanics
[NLFT] with Clifford Truesdell in
the early 1960’s, I encountered the problem of justi-
fying the formula relating the stress in elasticity to
the gradient of the internal energy with respect to
the transplacement gradient
1
. First, I naively con-
jectured that this formula can be derived from the
law of conservation of energy. It cannot. What is
needed is an appropriate interpretation of the second
law of thermodynamics.
Here is the interpretation:
Dissipation principle:
For all thermody-
namic processes that are admissible for a given con-
stitutive assumption, the entropy production must be
positive or zero.
The decisive word in this postulate is the quan-
tifier
all
. It makes the postulate a restrictive con-
dition on the constitutive assumptions that can be
imposed on systems of the type under consideration.
Indeed, if constitutive assumptions are laid down at
will and without restriction, the entropy production
can be expected to be positive or zero only for some
but not for all admissible processes. Thus, the sec-
ond law is not a restriction on the kind of processes
that can occur in nature, but a restriction on the
kind of material properties that physical systems oc-
curring in nature can have. This is not made clear in
most explanations of the second law in the textbook
literature on thermodynamics. An exception is the
Lectures on Physics
by R. Feynman. He writes: “So
we see that a substance’s properties must be limited
in a certain way; one cannot make up anything he
wants .... This principle, this limitation, is the only
real rule that comes out of thermodynamics.” (See
pp. 44-6 and 44-7 of reference [F].)
What happened next is described by the fol-
lowing excerpt from Bernard Coleman’s
Memories of
Clifford Truesdell
[C]:
“In the early 1960’s Walter Noll put to me the
idea, as if it should be obvious to everyone, that the
inequality is a restriction on all processes that are ad-
missible in a material of which the body is composed,
and, because one defines each material by giving a
set of constitutive relations, the Clausius-Duhem in-
equality, as it must hold for all processes compatible
with these relations, becomes a
restriction on con-
stitutive relations.
In another act of great generos-
ity, Walter suggested that we develop the idea to-
gether. It took a while to sort the argument out and
to present it in a way that would convince the wary.”
The result of our efforts was the paper
The
1
In 1958, I introduced the unfortunate terms “configuration” and “deformation” for what are now called “placement” and
“transplacement”. I apologize. Since these old terms were used in [NLFT], they were widely accepted, and I am now in the
ironic position to fight against something that I started.
1
Thermodynamics of Elastic Materials with Heat Con-
duction and Viscosity
[CN]. There we gave rigor-
ous mathematical proofs that the second law implies,
among other things, that the heat conduction coeffi-
cients and the viscosities in fluids must be positive or
zero. This paper became a citation classic, and the
procedure used in this paper has been dubbed the
“Coleman-Noll procedure”.
Typing this term into
Google yields about 3500 hits.
In
The Non-Linear Field Theories of Mechanics
,
published in 1965, Clifford Truesdell and I presented
what we called “The general theory of material be-
havior” (Chapter C of [NLFT]). About seven years
later, in about 1971, I realized that this “general the-
ory” cannot be the final word. (See the
Preface to
the Second Edition
, published in 1992.) What was
needed is a general framework for the formulation of
constitutive laws, using the concept of a
state
of a ma-
terial element. I presented such a framework in the
paper
A New Mathematical Theory of Simple Mate-
rials
, 50 pages long, published in 1972 [N1]. This
framework dealt only with purely mechanical phe-
nomena. However, in the end (Sect.21), I proposed
that this framework should be generalized to become
a general thermomechanical theory. It took 35 years
before somebody started to develop such a general-
ization. It was my doctoral student Brian Seguin,
now at the end of his fourth year as a graduate stu-
dent at CMU. His work could be of groundbreaking
significance.
I was disappointed that my 1972 paper [N1]
mentioned above generated very little resonance. The
reason is that it required familiarity with a mathe-
matical infrastructure that was and still is foreign to
most people interested in thermomechanics. Here is
an example: A letter from the editor of the
Reviews of
Modern Physics
, written in 1988, informed me that a
1961 paper by Bernard Coleman and me had become
a citation classic and that they would welcome re-
ceiving other papers from me. In 1995 I submitted a
paper entitled
On Material Frame Indifference
to this
journal. I thought that this paper should be of inter-
est to an audience wider than just those interested
only in the mathematics of continuum physics. How-
ever, my paper was rejected even though the editor
conceded that “the article is clearly written”. Here
are some quotes from the reviewer:
“I enjoyed reading this paper and very much
would like to see it published.
I am afraid, how-
ever, that the Reviews of Modern Physics is not the
appropriate place. I believe that the overwhelming
majority of the readers of the journal will consider
the paper unreadable. Not because the material pre-
sented is intrinsically difficult, but rather because the
author’s individual form of the ‘Bourbakian’ style is
far removed from anything that physicists are will-
ing to digest. .... Professor Noll is highly respected
in the mathematical community and has more than
once proved himself to be ahead of his time. ...”
The paper is now part 2 of my
Five Contribu-
tions to Natural Philosophy
[FC].
In a recent exchange of e-mails, another scien-
tist also told me that he has “reservations for the
Boubakization of mechanics”. Here is my answer to
him:
“My more recent work is almost entirely ex-
pressed in terms of the stage 3 infrastructure of math-
ematics.
This is not just a matter of notation or
style. The language of stage 2 mathematics is sim-
ply insufficient to describe my ideas. I am convinced
that this stage 3 infrastructure will eventually pre-
vail, even among physicists and engineering scientist,
but it may take another 40 years. To call it ‘Bour-
bakization’ is a gross mischaracterization.”
The stage 2 mathematical infrastructure is
based on the concepts of variables, constants, and
parameters.
The stage 3 mathematical infrastruc-
ture is based on the concepts of sets and mappings.
For a detailed explanation, see my essay entitled
The
Conceptual Infrastructure of Mathematics
[N2].
One of the issues that has created much con-
fusion in the literature on thermomechanics is the
Principle of Material Frame Indifference
. It is often
confused with material symmetry. I deal with this
issue in part 2 of the
Five Contributions to Natu-
ral Philosophy
[FC] mentioned above. The best way
to avoid using this Principle and not confuse it with
material symmetry is to formulate constitutive laws
without using a frame of reference. This has been
done in [N1], [FC], [N3], and also in Brian Seguin’s
work to be described in a later presentation.
I am tired of being ahead of my time and hope to
induce some people to become familiar with some of
the stage 3 mathematical infrastructure that will be
needed, for example, to understand the paper
Basic
Concepts of Thermomechanics
[NS1] by Brian Seguin
and me outlined in Section 3 below. A detailed treat-
ment of some of this infrastructure is presented in my
textbook [FDS], now available free of charge on the
internet. The following gives a taste of it.
2. Mathematical infrastructure
2.1. Sets and Mappings
The set of real numbers will be denoted by R
I .
The set of positive numbers (including zero) will be
denoted by P
I
while the set of strictly positive num-
bers (excluding zero) will be denoted by P
I
×
.
2
In order to specify a
mapping
f
:
A −→ B
,
one first has to prescribe two sets,
A
and
B
, and then
a definite procedure, called the
evaluation rule
of
f
, which assigns to each element
a
∈ A
exactly one
element
f
(
a
)
∈ B
. The set
A
is called the
domain
of
f
and the set
B
is called the
codomain
of
f
. We
say that
f
is a mapping from
A
to
B
or maps
A
to
B
.
For every set
A
we have the
identity map-
ping
1
A
:
A −→ A
of
A
, defined by 1
A
(
a
) :=
a
for
all
a
∈ A
.
The
composite
g
◦
f
:
C −→ A
of two mappings
f
:
A −→ B
and
g
:
B −→ C
is defined by
(
g
◦
f
)(
a
) :=
g
(
f
(
a
))
for all
a
∈ A
.
Now let a
mapping
f
:
A −→ B
be given. We
say that
f
is
injective
if, for every
b
∈ B
, there is
at
most one
a
∈ A
such that
f
(
a
) =
b
. We say that
f
is
surjective
if, for every
b
∈ B
, there is
at least one
a
∈ A
such that
f
(
a
) =
b
. The mapping
f
is both
injective and surjective if and only if, for every
b
∈ B
,
there is
exactly one
a
∈ A
such that
f
(
a
) =
b
. In
that case, we say that
f
is
invertible
and we define
the
inverse
f
←
:
B −→ A
by the procedure which
associates with each
b
∈ B
the only
a
:=
f
←
(
b
)
∈ A
which satisfies
f
(
a
) =
b
. We then have
f
◦
f
←
= 1
B
and
f
←
◦
f
= 1
A
.
The set
Rng
f
:=
{
f
(
a
)
∈ B |
a
∈ A}
is called the
range
of
f
. We have Rng
f
=
B
if and
only if
f
is surjective.
The
mapping
f
induces
a
mapping
f
>
: Sub
A −→
Sub
B
, from the set Sub
A
of all
subsets of
A
to the set Sub
B
of all subsets of
B
. It
is defined by
f
>
(
A
) :=
{
f
(
a
)
∈ B |
a
∈ A}
for all
A
∈
Sub
A
and called the
image mapping
of
f
.
Given two sets,
A
1
and
A
2
, one can form the
set-product
A
1
× A
2
of
A
1
and
A
2
. It is defined by
A
1
× A
2
:=
{
(
a
1
, a
2
)
|
a
1
∈ A
1
, a
2
∈ A
2
}
.
Given
h
1
:
D −→ A
1
and
h
2
:
D −→ A
2
one
can construct the
term-wise evaluation mapping
(
h
1
, h
2
) :
D −→ A
1
× A
2
by
(
h
1
, h
2
)(
d
) := (
h
1
(
d
)
, h
2
(
d
))
for all
d
∈ D
.
Conversely, given
h
:
D −→ A
1
× A
2
then there are
mappings
h
1
:
D −→ A
1
and
h
2
:
D −→ A
2
such
that
h
= (
h
1
, h
2
). The mappings
h
1
and
h
2
are called
the
component mappings
of
h
. A similar result
holds when the codomain of
h
is the product of more
the two sets.
2.2. Linear Algebra
Here we deal only with finite-dimensional real
linear spaces. Let
T
1
and
T
2
be such linear spaces.
We use the notation Lin (
T
1
,
T
2
) for the set of all
linear mappings from
T
1
to
T
2
.
This set also has
the structure of a linear space and dim Lin (
T
1
,
T
2
) =
dim
T
1
×
dim
T
2
. Given
L
∈
Lin (
T
1
,
T
2
) and
v
∈ T
1
we denote by
Lv
the element of
T
2
that
L
assigns to
v
.
If
L
1
and
L
2
are both linear mappings such that
the composite
L
1
◦
L
2
is meaningful then we will de-
note this composite simply by
L
1
L
2
. If a linear map-
ping
L
is invertible, we denote its inverse by
L
−
1
. We
denote by Lis (
T
1
,
T
2
) the set of all invertible linear
mappings, i.e.
linear isomorphisms
, from
T
1
to
T
2
.
This set is non-empty if and only if dim
T
1
= dim
T
2
.
We use the abbreviations
Lin
T
:= Lin(
T
,
T
) and Lis
T
:= Lis(
T
,
T
)
.
The second of these sets forms a group with respect
to composition, called the
linear group
of
T
.
The
dual
of a linear space
T
is defined by
T
∗
:= Lin(
T
,
R
I )
.
In accordance with the general rule of denoting the
evaluation of linear mappings, the value of
λ
∈ T
∗
at
v
∈ T
will be be denoted simply by
λ
v
. The dual
T
∗∗
of the dual space
T
∗
will be identified with
T
in
such a way that the value at
λ
∈ T
∗
of the element
of
T
∗∗
identified with
v
∈ T
is
v
λ
:=
λ
v
. We have
dim
T
∗
= dim
T
.
Given
v
∈ T
2
and
λ
∈ T
∗
1
we define the
tensor
product v
⊗
λ
∈
Lin (
T
1
,
T
2
) of
v
and
λ
by
(
v
⊗
λ
)
u
:= (
λ
u
)
v
for all
u
∈ T
1
.
The dual of Lin
T
contains a special element
tr
∈
(Lin
T
)
∗
called the
trace
which is characterized
by the property
tr(
v
⊗
λ
) =
λ
v
for all
v
∈ T
,
λ
∈ T
∗
.
Another mapping of interest is the
determinant
det : Lin
T −→
R
I .
2
We use the notation
Unim
T
:=
{
L
∈
Lis
V | |
det
L
|
= 1
}
for the
unimodular group
, which is a subgroup of
Lis
T
, and the notation
Unim
+
T
:=
{
L
∈
Lis
V |
det
L
= 1
}
2
For a matrix-free definition see Sect.14 of [FDS], Vol.II
3
for the
proper unimodular group
, which is a sub-
group of Unim
T
.
To every
L
∈
Lin (
T
1
,
T
2
) one can associate
exactly one element
L
>
∈
Lin (
T
∗
2
,
T
∗
1
), called the
transpose
of
L
, characterized by the condition that
λ
(
Lv
) = (
L
>
λ
)
v
for all
v
∈ T
1
,
λ
∈ T
∗
2
.
The space Lin (
T
,
T
∗
) will be identified with the
space of all bilinear forms on
T
. The subspace
Sym (
T
,
T
∗
) :=
{
L
∈
Lin (
T
,
T
∗
)
|
L
>
=
L
}
of Lin (
T
,
T
∗
) will be identified with the space of all
symmetric bilinear forms. The subset
Pos
+
(
T
,
T
∗
) :=
{
G
∈
Sym (
T
,
T
∗
)
|
(
Gv
)
v
>
0
for all
v
∈ T
with
v
6
=
0
}
of Sym (
T
,
T
∗
) will be identified with the set of all
strictly positive bilinear forms. It is an open subset
and a
linear cone
in Sym (
T
,
T
∗
), but not a sub-
space. We note that Pos
+
(
T
,
T
∗
)
⊂
Lis (
T
,
T
∗
).
A (genuine)
inner-product space
V
is a linear
space endowed with additional structure by singling
out a specific element ip
∈
Pos
+
(
V
,
V
∗
), called the
inner-product
. The inner-product is used to iden-
tify the linear space
V
with its dual
V
∗
. It is custom-
ary to use the notation
v
·
u
:= (ip
v
)
u
for all
v
,
u
∈ V
.
The
magnitude
|
u
|
of an element
u
∈ V
is defined
by
|
u
|
:=
√
u
·
u
.
If
T
is just a linear space without inner product
then the entire theory of inner-product spaces can
be applied relative to any
G
∈
Pos
+
(
T
,
T
∗
).
3
For
example, for each
G
∈
Pos
+
(
T
,
T
∗
) one can define
Orth
G
:=
{
A
∈
Lis
T |
A
>
GA
=
G
}
,
Orth
+
G
:=
{
A
∈
Orth
G
|
det
A
= 1
}
.
The first of these is the
orthogonal group
of
G
,
a subgroup of Unim
T
, and the second of these is
the
proper orthogonal group
of
G
, a subgroup of
Unim
+
T
. The following two facts about orthogonal
groups are of interest:
(1) For all
G
1
,
G
2
∈
Pos
+
(
T
,
T
∗
) we have
Orth
G
2
= Orth
G
2
⇐⇒
G
1
=
c
G
2
for some
c
∈
P
I
×
.
(2) The groups Orth
G
are all maximal subgroups of
Unim
T
.
4
In the applications to thermomechanics, a three-
dimensional linear space
T
represents an
infinites-
imal element
of a continuous body.
The elements
G
∈
Pos
+
(
T
,
T
∗
) represent the
configurations
of
the element. A
deformation
is simple a change of
configuration, and a
deformation process
is a map-
ping
P
:
I
−→
Pos
+
(
T
,
T
∗
), where
I
is a non-empty
real interval.
3. Basic Thermomechanics
3.1. Description
Here is an outline of the paper entitled
Basic
Concepts of Thermomechanics
[NS1] by Brian Seguin
and me, which is intended to serve as a model for
the first few chapters of future textbooks on contin-
uum mechanics and continuum thermomechanics. It
differs from most existing such textbooks in several
important respects:
1) It uses the stage 3 mathematical infrastruc-
ture mentioned above.
2) It is completely coordinate-free and R
I
n
-free
when dealing with basic concepts.
3) It does not use a fixed
physical space
. Rather,
it employs an infinite variety of
frames of reference
,
each of which is a Euclidean space. The motivation
for avoiding physical space can be found in Part 1,
entitled
On the Illusion of Physical Space
, of [FC].
Here, the basic laws are formulated without the use
of a physical space or any external frame of refer-
ence. The only frames of reference used there are
internal ones, generated by the configurations of the
body systems used.
4) It considers inertia as only one of many ex-
ternal forces and does not confine itself to using only
inertial frames of reference.
Hence kinetic energy,
which is a potential for inertial forces, does not ap-
pear separately in the energy balance equation. In
particle mechanics, inertia plays a fundamental role
and the subject would collapse if it is neglected. Not
so in continuum mechanics, where it is often appro-
priate to neglect inertia, for example when analyzing
the motion of toothpaste when it is extruded slowly
from a tube.
3.2. Physical Systems
In the mid 1950’s I regularly taught courses for
engineering students with the titles
Statics
and
Dy-
namics
. In
Statics
, the students were asked to con-
sider some system (a building, a bridge, or a ma-
chine), draw
free-body diagrams
, and apply to each
of these the balance of forces and torques. This of-
ten gave enough linear equations to determine the
stresses in each of the pieces of the system. In
Dy-
namics
, the students were asked to apply the same
procedure as in Statics, except that inertial forces are
3
This insight, to the best of my knowledge, was first employed in [N1].
4
The proof is given in [N5].
4
taken into account. This often led to linear differen-
tial equations. I then wondered what the underlying
conceptual background of all this was. Here is the
result:
Definition 1.
An ordered set
Ω
with order
≺
is said
to be
materially ordered
if the following axioms
are satisfied:
(MO1)
Ω
has a maximum
ma
and a minimum
mn
.
(MO2) Every doubleton has an infimum.
(MO3) For every
p
∈
Ω
there is exactly one member
of
Ω
denoted by
p
rem
, such that
inf
{
p, p
rem
}
= mn
and
sup
{
p, p
rem
}
= ma
.
(MO4)
(inf
{
p, q
rem
}
= mn)
=
⇒
p
≺
q
for all
p, q
∈
M .
The mapping
rem := (
p
7→
p
rem
) : Ω
−→
Ω
is
called the
remainder mapping
in
Ω
.
Here Ω is considered to consist of the whole sys-
tem and all of its parts. Given
a, b
∈
Ω,
a
≺
b
is read
“
a
is a part of
b
”. The maximum ma is the “material
all”, i.e. the whole system, and the minimum mn is
the “material nothing”. The inf
{
a, b
}
is the overlap
of
a
and
b
, and
a
rem
is the part of the whole system
ma that remains after
a
has been removed. With this
in mind, the two conditions (MO3) and (MO4) above
are very natural.
Theorem 1:
Let
Ω
be a materially ordered set.
Then
Ω
has the structure of a Boolean algebra with
p
∧
q
:= inf
{
p, q
}
and
p
∨
q
:= sup
{
p, q
}
for all
p, q
∈
Ω
,
(1)
The proof is highly non-trivial. The best version is
given in [NS2].
3.3 Additive Mappings a and Interactions
Let Ω be a materially ordered set and
W
a linear
space. We say that the parts
p
and
q
are
separate
if
p
∧
q
= mn. We use the notation
(Ω
2
)
sep
:=
{
(
p, q
)
∈
Ω
2
|
p
∧
q
= mn
}
.
(2)
Definition 2.
A function
H
: Ω
−→ W
is said to be
additive
if
H
(
p
∨
q
) =
H
(
p
) +
H
(
q
)
for all
(
p, q
)
∈
(Ω
2
)
sep
.
(3)
For every
p
∈
Ω we put Ω
p
:=
{
q
∈
Ω
|
q
≺
p
}
,
which again has the structu of a materially ordered
set.
Definition 3.
A function
I
: (Ω
2
)
sep
−→ W
is said
to be an
interaction
if, for all
p
∈
Ω
, both
I
(
·
, p
rem
) : Ω
p
−→ W
and
I
(
p
rem
,
·
) : Ω
p
−→ W
are additive.
The
resultant
Res
I
: Ω
→ W
of a given inter-
action
I
is defined by
Res
I
(
p
) :=
I
(
p, p
rem
)
for all
p
∈
Ω
.
(4)
We say that a given interaction is
skew
if
I
(
q, p
) =
−
I
(
p, q
) for all (
p, q
)
∈
(Ω
2
)
sep
.
(5)
Theorem 2:
An interaction is skew if and only if its
resultant is additive.
The proof of this result is fairly easy but not
entirely trivial.
Remark 1:
The concept of an interaction is an ab-
straction. Its values may have the interpretation of
forces, torques, or heat transfers. In most of the past
literature these cases were treated separately even
though much of the underlying mathematics is the
same for all. Thus, this abstraction, like most others,
is a labor saving device.
3.4. Continuous Bodies
A continuous body system is a set
B
of
mate-
rial points
, endowed with structure by the speci-
fication of a non-empty set Conf
B
, whose elements
are called
configurations
of
B
. These configurations
are Euclidean metrics and the set Conf
B
is subject to
several conditions. In order to state these, one needs
several concepts:
1) A class Fr of
fit regions
, which are sub-
sets Euclidean spaces. These regions are those that
a body system can occupy.
Intuitively, the term
“body” suggest that the regions it can occupy are
connected. We do not assume this but, for simplic-
ity, we will use the term “body” rather than “body
system” from now on.
2) A class Pl
B
of
placements
of the body,
which are invertible mappings from the body
B
to a
fit region in some Euclidean space, called the
frame-
space
of the placement.
3) A class Tr of
transplacements
, which are
invertible mappings from a fit region in some frame-
space to a fit region in the same or another frame-
space.
The precise definitions and conditions that these
concept must satisfy are, to some extent, quite tech-
nical and can be found in [NS1].
Each configuration
δ
∈
Conf
B
can be used to
construct a Euclidean space
E
δ
and and an
imbed-
ding
imb
δ
, which is a special placement of the body
in the frame-space
E
δ
. These special frame-spaces are
internal, because they constructed from configura-
tions; they are not external, i.e., introduced from the
5
outside. In general, placements are in frame-spaces
introduced from the outside.
Every placement
µ
induces a configuration
δ
µ
,
but infinitely many placements all induce the same
configuration.
Given a continuous body
B
, consider the set Ω
B
defined by
Ω
B
:=
{P ∈
Sub
B |
imb
δ>
(
P
)
∈
Fr
for every
δ
∈
Conf
B}
.
(6)
In (3.6) “every” can be replaced by “some” without
change of meaning.
The members of Ω
B
are called
parts
or
sub-
bodies
of
B
.
Ω
B
is
materially ordered
, as defined in Sect.3.1,
by inclusion. It has the structure of a Boolean algebra
with
P ∧ Q
:=
P ∩ Q
,
(7)
P ∨ Q
:= Int Clo(
P ∪ Q
)
,
(8)
P
rem
:= Int(
B\P
)
,
(9)
The proof of this highly non-trivial result can
be found in [NV].
3.5. Densities and Contactors
We assume that a continuous body
B
is given.
For each part
P ∈
Ω
B
and each placement
µ
∈
Pl
B
we use the notation
P
µ
:=
µ
>
(
P
) and, in particu-
lar.
B
µ
:= Rng
µ
. We say that a part
P ∈
Ω
B
is
internal
if, for every placement
µ
∈
Pl
B
, we have
Clo
P
µ
⊂ B
µ
. We denote the set of all internal parts
by Ω
int
B
.
Definition 4.
An additive mapping
H
: Ω
B
−→ W
is said to have
densities
if, for every
µ
∈
Pl
B
, there
is a continuous mapping
h
µ
:
B
µ
−→ W
such that
H
(
P
) =
Z
P
µ
h
µ
for all
P ∈
Ω
int
B
.
(10)
We call
h
µ
the
density
of
H
in the placement
µ
.
Definition 5.
We say that an interaction
I
:
(Ω
B
)
2
sep
−→ W
has
contactors
if, for every place-
ment
µ
, there is a
C
1
mapping
C
µ
:
B
µ
−→
Lin (
V
µ
,
W
)
such that
I
(
P
,
Q
) =
Z
Rct
µ
(
P
,
Q
)
C
µ
n
P
µ
for all (
P
,
Q
)
∈
(Ω
B
)
2
sep
with
P ∈
Ω
int
B
.
(11)
We call
C
µ
the
contactor
of
I
in the placement
µ
.
In both these definitions, “every” can be re-
placed by “some” without change of meaning.
Theorem 3:
Given an interaction
I
: (Ω
B
)
2
sep
−→
W
with contactors and an additive mapping
H
: Ω
B
−→ W
with densities, the following three
conditions are equivalent:
1) We have
Res
I
(
P
) +
H
(
P
) =
0
for all
P ∈
Ω
int
B
.
(12)
2) For every placement
µ
∈
Pl
B
, we have
div
C
µ
+
h
µ
=
0
.
(13)
where
h
µ
is the density of
H
in the placement
µ
, and
C
µ
is the
contactor
of
I
in the placement
µ
.
3) Condition 2) holds with with “every” replaced by
“some”.
3.6. Balance of Forces and Torques
It is often useful to fix a Euclidean space
E
, with
translation space
V
, and confine one’s attention to
placements whose range space is
E
. It is then useful
to consider force systems with values in
V
, indepen-
dent of the choice of a configuration, as follows:
Definition 6.
A
force system
in the space
V
is a
pair
(
F
i
,
F
e
)
, where
F
i
: (Ω
B
)
2
sep
−→ V
is an interac-
tion and
F
e
: Ω
B
−→ V
is additive. The mapping
F
i
is called the
internal force system
in
V
and
F
e
is
called the
external force system
in
V
.
Let a force system (
F
i
,
F
e
) in
V
be a given. The
first fundamental law of mechanics, called the
Bal-
ance of Forces
, says:
Res
F
i
(
P
) +
F
e
(
P
) =
0
for all
P ∈
Ω
B
.
(14)
We say that the system (
F
i
,
F
e
) is
force-
balanced
if (14) holds.
Since
F
e
is additive, the following
Law of Ac-
tion and Reaction
is an immediate consequence of
(14) and Theorem 2:
Every balanced balanced internal force system
is skew, i. e. ,
F
i
(
P
,
Q
) =
−
F
i
(
Q
,
P
)
for all
(
P
,
Q
)
∈
(Ω
B
)
2
sep
.
(15)
Remark 2:
The law of action and reaction is often
referred to as
Newton’s Third Law
. Thus, if one as-
sumes the balance law (3.14), one can prove Newton’s
Third Law instead of assuming it a priori, as New-
ton and many physics textbooks since Newton have
done. The balance of forces has been understood by
engineers, if only implicitly, since antiquity.
We now assume now that
F
i
has contactors and
F
e
has densities..
Let
µ
be a placement of the body in
E
and put
B
µ
:=
µ
>
(
B
). Let
T
µ
:
B
µ
:
−→
Lin
V
denote the
contactor for
F
i
and let
b
µ
:
B
µ
:
−→ V
denote the
6
density of
F
e
in the given the placement
µ
. It follows
from Theorem 3 that the force balance, restricted to
internal parts
P
, is equivalent to
div
T
µ
+
b
µ
=
0
.
(16)
It is usual to assume that the force system is not
only force-balanced but also
torque-balanced
. We
will not give a precise definition here but note that
the condition
Rng
T
µ
⊂
Sym
V
(17)
is equivalent to balance of torques for internal parts.
From here on we will assume that (16) and (17)
are valid. We adjust the codomain of
T
µ
to Sym
V
without change of notation and call
T
µ
:
B
µ
:
−→
Sym
V
the
Cauchy stress
of the force system in the
placement
µ
and the mapping
b
µ
:
B
µ
−→ V
the
external body-force density
in the placement
µ
.
Let a configuration
δ
∈
Conf
B
be given. As in
Section 3.4, we denote the imbedding space for
δ
by
E
δ
and its translation space by
V
δ
, and we use the
results above in the case when
µ
:= imb
δ
and write,
for simplicity,
δ
rather than imb
δ
as a subscript.
Definition 7.
A
force system in the configura-
tion
δ
is a pair
(
F
i
δ
,
F
e
δ
)
which is a force system, both
force-balanced and torque-balanced, in the space
V
δ
in the sense of Definition 6.
Let such a force system (
F
i
δ
,
F
e
δ
) in
V
δ
be given.
We assume that
F
i
δ
has contactors and that
F
e
δ
has
densities. The results (3.16), and (17) remain valid
when the subscript
δ
is used instead of
µ
, when
T
δ
is
interpreted to be the contactor of
F
i
δ
in the placement
imb
δ
, and when
b
δ
is interpreted to be the density of
F
e
δ
in the placement imb
δ
. We may call
T
δ
the
con-
figurational stress
and imb
δ
the
configurational
body force density
for
δ
.
Definition 8.
A
mechanical process
is a time-
family
((¯
δ
t
,
¯
F
i
t
,
¯
F
e
t
)
|
t
∈
I
)
of triples, where
(¯
δ
t
|
t
∈
I
)
is a deformation process and, for every
t
∈
I
,
( ¯
F
i
t
,
¯
F
e
t
)
is a force system, both force-balanced and torque-
balanced, in the configuration
¯
δ
t
, as defined by Defi-
nition 7.
3.7. Energy Balance
Energy and heat are fundamental concepts in
thermodynamics.
Definition 9.
A
heat transfer system
is a pair
(
Q
i
, Q
e
)
, where
Q
i
: (Ω
B
)
2
sep
−→
P
I
×
is an interac-
tion and
Q
e
: Ω
B
−→
P
I
×
is additive. The mapping
Q
i
is called the
internal heat transfer
and
Q
e
is
called the
external heat absorption
.
We assume that
Q
e
has densities and
Q
i
has
contactors. Hence, given any placement
µ
in a frame-
space
E
with translation space
V
, we obtain a den-
sity
r
µ
:
B
µ
−→
R
I
and a contactor
−
q
µ
:
B
µ
−→
Lin (
V
.
R
I )
∼
=
V
such that
Q
e
(
P
) =
Z
P
µ
r
µ
for all
P ∈
Ω
int
B
.
(18)
Q
i
(
P
,
Q
) =
−
Z
Rct
µ
(
P
,
Q
)
q
µ
·
n
P
µ
for all (
P
,
Q
)
∈
(Ω
B
)
2
sep
with
P ∈
Ω
int
B
.
(19)
r
µ
is called the
heat absorption field
and
q
µ
the
heat flux
of the given heat transfer system in
the placement
µ
.
Definition 10
An
energetic process
is a sextuple
((¯
δ
t
,
¯
F
i
t
,
¯
F
e
t
,
¯
Q
i
t
,
¯
Q
e
t
,
¯
E
t
)
|
t
∈
I
)
of time-families such
that
((¯
δ
t
,
¯
F
i
t
,
¯
F
e
t
)
|
t
∈
I
)
is a mechanical process,
(( ¯
Q
i
t
,
¯
Q
e
t
)
|
t
∈
I
)
is a time-family of heat transfer sys-
tems, and
( ¯
E
t
|
t
∈
I
)
is an appropiatly defined dif-
ferentiable time-family of additive mappings, called
the
internal energy
.
We
say
that
a
given
energetic
process
((¯
δ
t
,
¯
F
i
t
,
¯
F
e
t
,
¯
Q
i
t
,
¯
Q
e
t
,
¯
E
t
)
|
t
∈
I
) is
energy-balanced
if
¯
P
t
(
P
) + Res
¯
Q
i
t
(
P
) + ¯
Q
e
t
(
P
) = ¯
E
•
t
(
P
)
for all
P ∈
Ω
B
and
t
∈
I ,
(20)
where ( ¯
P
t
|
t
∈
I
) is the power-family determined by
the mechanical process ((¯
δ
t
,
¯
F
i
t
,
¯
F
e
t
)
|
t
∈
I
).
Under suitable conditions, the energy balance
law can be given a local form
3.8. Temperature and Entropy
Theorem 6.
Let a heat transfer system
(
Q
i
, Q
e
)
, as
defined by Definition 10, and a function
θ
:
B −→
P
I
×
of class
C
1
, called
temperature
, be given, and as-
sume that
Q
i
has contactors and
Q
e
has densities.
Then there is a pair
(
H
i
, H
e
)
, where
H
i
: (Ω
B
)
2
sep
−→
R
I
is an interaction and
H
e
: Ω
B
−→
R
I
is the ad-
ditive function such that, for every placement
µ
we
have
H
i
(
P
,
Q
) =
−
Z
Rct
µ
(
P
,
Q
)
q
µ
θ
µ
·
n
P
µ
for all (
P
,
Q
)
∈
(Ω
B
)
2
sep
with
P ∈
Ω
int
B
(21)
and
H
e
(
P
) =
Z
P
µ
r
µ
θ
µ
for all
P ∈
Ω
int
B
,
(22)
where
q
µ
is the heat flux in the placement
µ
,
r
µ
is the external heat absorption density in
µ
and
θ
µ
:=
θ
◦
µ
←
. The pair
(
H
i
, H
e
)
is called the
entropy
7
transfer system
generated by the heat transfer sys-
tem
(
Q
i
, Q
e
)
and the temperature
θ
.
H
i
is called the
internal entropy transfer
, and
H
e
is called the
external entropy absorption
.
The proof is not very difficult.
Definition 11.
Let by a time-famiy
(( ¯
Q
i
t
,
¯
Q
e
t
)
|
t
∈
I
)
of heat transfer systems and a time-family of tem-
peratures
(¯
θ
t
|
t
∈
I
)
be given, and let
(( ¯
H
i
t
,
¯
H
e
t
)
|
t
∈
I
)
be the resulting entropy transfer system as de-
scribed in Theorem 6. Let
( ¯
N
t
: Ω
B
−→
R
I
|
t
∈
I
)
be a differentiable time-family of additive mappings,
called the
internal entropy
.
We say that
(( ¯
H
i
t
,
¯
H
e
t
,
¯
N
t
)
|
t
∈
I
)
is a
dissipa-
tive entropical process
if
¯
N
•
t
(
P
)
≥
Res
¯
H
i
t
(
P
) + ¯
H
i
t
(
P
)
for all
P ∈
Ω
int
B
and
t
∈
I .
(23)
Under suitable conditions, the energy balance
law can be given a local form.
Using the local form of the energy balance law
and the local form of the dissipation inequality, one
can obtain the following
intrinsic reduced dissi-
pation inequality
:
¯
ρ
m
(¯
θ
¯
η
•
−
¯
•
) +
1
2
tr(¯
S
¯
G
•
)
−
1
¯
θ
¯
γ
¯
h
≥
0
,
(24)
where ¯
γ
(
X, t
) :=
∇
X
¯
θ
(
·
, t
)
∈ T
∗
X
for all
X
∈ B
,
t
∈
I
, and where the
intrinsic stress
¯
S
and the
in-
trinsic heat flux
¯
h
are related to the Cauchy-stress
¯
T
and the heat flux ¯
q
by
¯
M
¯
S
¯
M
>
= ¯
T
m
,
¯
Mh
= ¯
q
m
,
(25)
respectively, with ¯
M
(
X, t
) :=
∇
X
µ
t
∈
Lis (
T
X
,
V
)
for all
X
∈ B
, t
∈
I
.
The inequality (24) involves only the (internal)
thermomechanical process
(¯
δ,
¯
θ,
¯
F
i
,
¯
Q
i
,
¯
N ,
¯
E
)
and not the external forces, nor the external heat
transfer, nor any external frame of reference. It is
(24) that is used to determine restrictions on frame-
free formulated constitutive laws, as will be explained
in a later presentation by Brian Seguin.
References:
[NS1] W. Noll and B. Seguin:
Basic Concepts of
Thermomechanics
, 20 pages (2009). Published as B5
on the website
www.math.cmu.edu/
∼
wn0g/noll
.
[NLFT] C. Truesdell and W. Noll:
The Non-
Linear Field Theories of Mechanics
, Encyclopedia
of Physics, Vol. III/3, 602 pages. Springer-Verlag,
1965. Second Edition, 1992. Translation into Chi-
nese, 2000. Third Edition, 2004.
[F] R. P. Feynmann, R.B. Leighton. and M. Sands,
The Feynmann Lectures on Physics, Vol. I
, Addison-
Wesley, Reading 1963.
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Memories of Clifford Truesdell
,
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70
, 1-13 (2003).
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The Thermody-
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, Archive for Rational Mechanics and
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13
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48
, 1-50 (1972).
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Five Contributions to Natural Phi-
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, 73 pages (2004). Published as B1 on the
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www.math.cmu.edu/
∼
wn0g/noll
.
[N2] W. Noll:
The Conceptual Infrastructure of Math-
ematics
, 5 pages (1995).
Published as A2 on the
website
www.math.cmu.edu/
∼
wn0g/noll
.
[N3]
W.
Noll:
A Frame-Free Formulation of
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.
Journal of Elasticity
83
,
291-307
(2006).
Also published as B4 on the website
www.math.cmu.edu/
∼
wn0g/noll
.
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, Archive for
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18
, 100-102 (1965).
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W.
:
Finite-Dimensional
Spaces:
Algebra,
Geometry,
and
Analysis,
Vol.I
and
Vol.II
. Published as C1 and C2 on the website
www.math.cmu.edu/
∼
wn0g/noll
.
( Vol.I was pub-
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has about 110 pages and is growing.
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Monoids, Boolean Al-
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, International Jour-
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37
, 187-
202 (2007).
Also published as C4 on the website
www.math.cmu.edu/
∼
wn0g/noll
.
[NV] W,Noll and E.Virga:
Fit regions and functions
of bounded variation
, Archive for Rational Mechanics
and Analysis 102, 1-21 (1988),
8