Ferrarese G. (editor) Wave Propagation
Подождите немного. Документ загружается.
254
e1asticity,Int.J.Engng
Sci.
l
(1965)
561-596.
7 -
P.Chadwick-P.K.Currie
: The
propagation
and
growth
of
acce=
1eration
waves
in
heat-conducting
elastic
materials,
Arch.
Rat.Mech.An 49
(1972)
137-158.
8 -
P.J.Chen
Growth and
decay
of
waves
in
solids,Enc.of
Phy=
sics,
VI
a/
3
Springer
Verlag,
Berlin,
1973.
9 -
P.J
.Chen
-
M.E.B:urtin
: On wave
propagation
in
inextensi=
ble
elastic
bodies,
Int.J.Solids
Struct
.1Q
(1974)
275-281.
10 -
P.Chen-J.W.Nunziato
:On wave
propagation
in
perfectly
heat
conducting
inexteneible
elastic
bodies,J.of
El.5(1975)
155-160.
11 -
B.D.Coleman,
M.E.Gurtin,I.Herrera,C.Truesdell
:
Wave
prop~
gation
in
dissipative
materials,A
reprin'
of
five
memoirs,
Springer
Verlag,New
York
1965.
12 -
WgD.Collins
:
One-dimensional
non
linear
wave
propagation
in
incompressible
elastic
materials,Q.J.Mech.appl.Math.19
(1966)
259-328.
13 - J •Dunwoody : One
dimensional
shock
waves
in
heat
conduct!
ng
materials
with
memory :
1-
Thermodynamics-Arch.Rat.Mech.
An.~7
(1972)
117-148;
2-Shock
analysis,
ibidem,
192-204;
3-
Evolutionary
behaviour,
ibidem
50,278-289.
14 - J.Dunwoody:On weak
shock
waves
in
thermoelastic
solids,Q.
J.Mech.appl.Math.30
(1977)
203-208.
15 -
J.L.Eriksen:On
the
propagation
of
waves
in
isotropic
incom=
pressible
perfectly
elastic
materials,J.Rat.Mech.An.
2 ,
(1953),329-338.
16 -
A.E.Green-P.M.Naghdi:A
derivation
of
jump
condition
for
entropy
in
thermomechanics,J.of
Elast.
8
(1978)
179-182.
17 -
E.Inan:Decay
of
weak
shock
waves
in
hyperelastic
solids,
Acta
Mech. 23
(1975)
103-112.
255
18 -
T.Manacorda:On
the
propagation
of
discontinuity
w
ave~
in
thermoelastic
incompressible
solids,Arch.Mech.Stos.(2),
24
(1971)
277-285.
19 -
T.Manacorda:
Zagadnienia
Elastodynamiki,Ossolin
eum,Varsa=
via,1978~
Cfr.anche
A -
15.
20 -
R.W.Ogden:Growt~
and
decay
of
acceleration
waves
in
incom=
pressible
elastic
solids,Q.J.Mech.appl.Math.
27
(1974)
451-464.
21 -
N.Scott:Acceleration
waves
in
constrained
elastic
materials
Arch.
Rat.Mech.An.
58
(1975)
57-15.
22 -
N~Scott:Acceleration
waves
in
incompressible
elastic
so=
lids,Q.J.Mech.appl.Math.
29
(1976)
295-310.
23 - N.Sc
_ott-M.Hayes:Small
vibrations
of
a
fiber-reinforced
composite,Q.J.Mech.appl.Math.
29
(1916)
467-486.
24 -
C.Trimarco:Onde
di
accelerazione
in
materiali
termoelasti=
ci
con
vincolo
di
inestendibilita,in
pubbl.
su
Atti
Acc.
Sci.Modena.
25 -
C.Truesdel1:General
and
exact
theory
of
waves
in
finite
elastic
strain,Arch.Rat.Mech.An.
8
(1961),263-296.
CENTRO
INTERNAZIONALE
MATEMATICO
ESTIVO
(C.I.M.E.)
"ENTROPY
PRINCIPLE"
AND MAIN
FIELD
FOR
A NON
LINEAR
COVARIANT SYSTEM
TOMMASO
RUGGERI
INTERNATIONAL
MATHEMATICAL
SUMMER
CENTER
(C
.I
.M
.E.)
1"
1980
C.1.M.E.
Session:
"Wave
propagation".
Bressanone
8-17
giugno
1980 .
"ENTROPY
PRINCIPLE"
AND
MAIN
FIELD
FOR A
NON
LINEAR
COVARIANT
SYSTEM
by
TONNASO
RUGGERI
Istituto
di
Matemat
ica
Applicata
-
Universita
di
Bologna
Via
Vallescura
2 - 40136 Bologna
(Italy).
I.Introduction
My
lecture
is
complementary
to
the
lectures
given
by
G.Boillat
i n
the
first
part
of
this
course.
In
part~cular
I am
shall
deal
with
some
problems
concer-
ning
quasi-linear
hyperbolic
system
compatible
with
a
supplementary
conservation
law;
relativistic
theories
will
be
cons
idered
with
special
emphas i s .
I
start
with
a
brief
bibliographical
introduction
to
the
subject
I
shall
be
concerned
with.
In
1970-71
1.
MUller,
in
some works [1] on
"rational"
thermomechanics
of
con-
tinuous
media,
proposed
the
"entropy
principle"
as
a
criterion
for
selecting
the
constitutive
equations.
This
author
considers
the
equation
governing
the
evolution
of
a thermomechanic
system
: a)
balance
of
momentum, b)
balance
of
mass and c)
balance
of
energy
equations
. Adding
the
constitutive
equations
to
the
prev
ious
syste
m one
gets
a
system
of
5
equations
in
5 unknowns. Each
so-
lution
of
this
system
is
called
a "thermodynamic
process"
. Then MUller
postu-
-
lates
the
existence
of
an
additive
function
cr
(entropy)
such
that
:
d
+ (PSv. +
~
.)
=
cr
dX
i
~
~
o
lJ t her modynami c
process
.
(l)
Furthermore
he
supposes
that
both
the
entropy
density
S and
the
flux
~i
are
....
constitutive
functions
(p and v
are
respectively
the
mass
density
and
the
ve-
locity).
Hence, from
(1)
further
constrains
arise
for
the
constitutive
rela-
260
tions,
besides
the
usual
ones
which
can
be imposed
according
to
the
principle
of
material
objectivity.
In
particular
the
author
shows
the
existence
of
a
universal
function,
which he
identifies
with
the
absolute
temperature
;
hence,
he
deduces
the
first
principle
of
thermodynamics
.
In
a
different
conte>ct
in
1971 K
.O.Friedrichs
and
P.D.Lax
[2] and
later
the
former
in
1974 [3J examined a
similar
problem.
In
particular
in
[3]
Friedrichs,
in
a
covariant
formalism,
considers
a
conservative
quasi-
(*)
linear
hyperbolic
system
of
r
first
order
equations
of
the
type
o
a
~
(U)
=
.2(U)
a
(2)
In
(2).
N
eqs.
may
be
identified
with
the
field
equations.
while
the
remaining
r - N
are
supplementary
conservation
laws.
Then
comp~
tibility
conditions
are
required
in
order
that
the
system
has
a
so-
lution.
In
particular,
when
r=N+l
(one
supplementary
law),
as
the
system
is
quasi-linear,
compatibility
is
ensured
by
the
existence
of
an
r-vector
z(U).
such
that:
a
I·aa~
=
1-.2
I,l U,
Introducing
the
operator
V = a/au we
have
I,l a u.
a
(condition
I)
= 0
(3)
Moreover
Friedrichs
supposes
another
condition
holds:
it
exist
at
least
a
time-like
covector
{~
},
independent
of
the
field,
such
that
the
quadratic
a
form
(condi
tion
II)
is
positive
definite
.
(4)
Here
oU
is
an
arbitrary
variation
of
the
field
and
o2~t
= su -
VV
ga su .
Using
condition
I and
II
Friedrichs
shows
that
the
system
of
the
field
equat
ions
is
a
hyperbolic
symmetric
system
.
(*)
To avo
id
misunderstanding
the
vectorErn
r
are
underlined
.
261
Later
several
authors
[4J.
[5], [6J
provided
further
contribut
ions
on
th
is
s
ubject.
especially
concerning
shock
waves
in
non-covariant
formalism.
Now
we
shall
obtain
the
above
mentioned
results
in
an
explicitly
covariant
formalism.
dealing
with
the
physically
relevant
case
of
one
supplementary
law.
The
covariant
formulation
allows
to
apply
the
results
to
explicitly
covariant
theories
and.
moreover.
to
emphasize
some
conceptual
aspects
that,
in
our
opi-
nion.
have
not
yet
been
pointed
out.
2.
Main
field
and
Covariant
convex
density.
Let
V~
be a
C·,
4-dimensional
mani fol d and x a
point
of
v~.
x
a
being
local
coordinates
of
x. The
manifold
is
supposed
to
endowed
with
a
pseudo-Riemannian
a a8
metric
.
In
the
local
coordinates
x • g
represents
the
components
of
the
metric
tensor
of
signature
(+ - -
-)
.
On
V~
we
consider
a
quasi-linear
conservative
system
of
N
first
order
partial
differential
equations
for
the
unknown N-ve
ctor
U(xa)E~N
(5)
the
components
of
Fa and U
are
contravariant
tensors
and d
a
is
intended
as
a
covariant
derivative
operator.
We
suppose
that
the
system
(5)
is
hyperbolic.
Le.
:
.:J
a
time-like
covector
{Cal.
such
that
the
following
two
statements
hold
:
i)
(6)
ii)
V
covector{~a}
of
space-like
the
eigenvalue
problem
(~
- llC ) A
a
d = 0
(7)
a . a -
has
only
real
proper
solutions
ll(k) and a
set
of
linearly
independent
eigen-
(k)
vectors
d
(k=I.2
•••••
N).
The
covectors{~a
-
llC
a}
• where II
is
solution
of
(7)
are
called
"characteristic".
while
the{c
a}
fUlfilling
L},
ti)
are
said
"subcharacteristic
".
262
When
a
differentiability
conditions
holds,
let
us
suppose
that
(5)
is
compatible
with
a
supplementary
conservation
law
a
aa
h
(U)
=
g(U),
(8)
h
a
being
a
contravariant
vector
and g a
covariant
scalar.
In
this
case
we may
write
the
conditions
I and
II
of
Friedrichs
in
a more
convenient
form.
We
have:
ga - (J
E -
(:
)
Since
by
(3)
I
is
defined
up
to
a
scalar
factor,
we may
write
I
='
then
Friedrichs
conditions
lock
like
U'
• VF
a
= V ha,
(9)
uv-r =
g,
(10)
We
remark
that
(9),
multiplied
by 6 U,
can
be
written
equivalently:
V s u.
(12}
The
identity
(12)
show
the
first
important
result
:
,
U' i s
invariant
with
respect
to
field
transformatio~:in
fact
U'
depends
only
<in
i" a and
-it
and
then
it
does
not
depend
on
the
choice
of
the
field
variables.
By
applying
the
operator
0
to
(12)
and
replacing
into
(ll}
we
get
Q = 6U" OF
a(
>0
\l
6U
(13)
a
Hence
(13)
too
is
independent
of
the
choice
of
the
field;
we may
then
choose
the
field
in
the
convenient
form
(14)
263
We
put
also
a
h = h I;a '
and
contracting
(12)
with
I;awe
have
U' -OU
=oh
+~ U'
=Vh
(15)
(16)
For
the
particular
choice
(14)
of
the
field
variables,
U'
is
espressed
by
the
gradient
of
a
covariant
scalar
function
h
only.
In
the
case
of
continuum
mechanics,
the
expression
(16)
is
equivalent
to
the
first
principle
of
thermodynamics.
We
point
out
that
the
components
of
U'
play
the
sa
me
role
of
the
Lagrange
multipliers
introduced
by
I-Shih
Liu
[7J
in
the
context
of
entropy
principle
of
MUller.
Condition
(13)
is
now
equivalent
to
Q =oU' - tSu =
tS
2
h
>
0,
(17)
i.e
.
to
the
convexity
of
the
covariant
scalar
density
h=hal;
with
respect
to
a
a
the
field
U = F
I;a'
If
for
a
system
(5)
there
exists
a
vector
U' and
at
leaet: a eo
oeetor
(I;a}suah
that
(12)
and
(17)
ho
ld,
we
say
that
the
system
is
a aonvex
aovariant
d
ensity
system
.
Conditions
(16)
and
(17)
ensure
also
that
the
mapping
U'+~U
is
globally
invertible
, becauseVU' =VVh and
th
is
gradient
matrix
is
symmetric
and
positive
definite;
then
for
a
theorem
about
globally
univalence
([8J)
U'
+~
U
is
glo-
bally
univalent
in
every
convex open domain D
~
R
N
Therefore
it
is
possible
to
choose
the
vector
U·
itself
as
field
variables
(t)
and
prove
that
in
this
case
system
(5)
has
the
form
L
{U'}
f(U'
)
(18)
where
the
operator
~
is
given
by
A' a a
~
a
(19)
(t)
For
the
proof
of
the
statements
proposed
in
this
lecture
one may
see
[9J •
264
and
(20)
case
we
have
='J'
'J'h',
a
system
A,a a U' = f
is
sim-
- a
is
positive
definite,
and
in
our
but
from
(20)
h
,a
~
=
h'
=
a
hyperbolic
:
in
fact
aT a
A' and A'
~
- - a
System
(18)
is
symmetric
a
metric
hyperbolic
if
~'
a
A'
~
~
a
= U'·U - h
is
the
Legendre
conjugate
function
of
h and
then
it
is
a convex
function
of
U' •
We
remark
also
that
the
differential
operator
in
(19)
depends
only
on
one-
four-vector
h
ia
and
this
justifies
our
definition
of
"four
veator
generating funa-
tion"
for
the
symmetric
system
.
We
have
seen
that
any convex
covariant
density
system
is
endowed
with
a
vec-
tor
U'
that
may be
expressed
as
a
function
of
the
field
variable
and
is
in-
variant
with
respect
to
transformations
of
field.
In
fact
it
is
determined
completely
only
by
the
conservative
system
(5)
and
the
supplementary
law
(8).
Moreover we
pointed
out
that,
when U'
is
chosen
as
field,
the
system
assumes
a
symmetric
form,
so
that
the
local
Cauchy
prob
lem
is
well
posed.
Such
remarkable
properties
suggest
us
to
call
U'
the
"main
field"
of
the
system.
We
remark
that
not
only
on
the
mathematical
point
of
view
U'
and
h,a
possess
a
special
role
with
respect
to
other
quanti
ties,
but
also
from
the
physical
point
of
view,
they
are
privileged,
since
they
are
related
to
the
"observables"
of
the
physical
system,
as
we
shall
see
later.
System
(5),
compatible
with
(8),
is
riducible
to
the
form
(18)
for
a
sui
table
choice
of
the
field
variables
and
viceversa;
in
fact
it
is
eas
y r:o
prove
that
the
system
(18)
provides
always
a
supplementary
law
(8)
:
let
h
a
= U' •
'J'h'Cl
_
h,a,
we
have
a h
Cl
a
'J'h
Cl
• a U'
c
U' • f { U' }
U'·f
=g.
Finally
we
have
shown
that
265
A ne
cessary
and s
ufficient
condi t i ons f or
the
sys
t em ( 5 )
to
be compat i
ble
with
a suppleme
ntary
conservati
on law (8 )
with
h ( U) convex fun c
ti
on wi
th
r es-
pect
to
U = F
a
(
(con
vex
covariant
dens
ity
systems
) ,
is
t hat there e
xists
a
a
choice
of
the
f i e
ld
U ' (
invariant
r
espect
field
transf
orma
ti
ons and
indep
en-
dent
of
t he congruence
defined
by
the
time
-like
covector
{(
a
})'
so t hat t he
syst
em
(5
) assum
es
the
sy
rmzetric f orm (
18)
with
h'=h,
a ( conv
ex
fu nct i on of U' •
a
Th
is
p
ropos
ition
is
a f
ir
s t c
ontr
ibut
ion
of
the
que s t l on
pro
pose
d by
LM
Ulle
r
("a
c
hallenge
to
ma
themati
cia
ns"
[lJ
) .
At
least
we
point
out
t
hat
if
in
(8
) we impo
se
the
c on
di
t i on g >
0,
t hen , by
(1
0)
all
solutions
of
(18)
sa
tisfy
3 .
Shock
Waves
Theory
for
Convex
Covariant
Density
Systems.
i)
Entropy
growth
across
a
shock
wave.
Let
Il a
connected
open
i n t o
two
open
subset
Il
l.
Il
z
.
se
t
of
v"
and
r
a m
L
et
~(
x
) = 0 ,
~
E C
a h
ype
rs
u
rface
cutting
Il
(m>
2)
,
be
the
equ
ation
of
r : we
shall
in
d
en
t
if
y r
wit
h a
sh
ock
h
yp
ersur
f a
ce
for
the
fie
l d U.
It
is
kno
wn
th
at
the
Rank
ine-
Hugon
io
t c
ondit
ions
mus t ho
ld
[ F
a]
~
=
0,
on r ,
(2
1)
a
where
brac
kets
denot
e t
he
jum p a c
ro
s s r
and
~
a = aa
~
.
Formally
t
he
R
ank
i n
e-
Hug on
io
t
equation
s a
re
ob t a i ne d
from
the
field
e qs .
(5)
thro
ug
h
the
correspo
nd e n
ce
rule
a
...
~
[ J
(22)
a a
However
this
r ul e
do
es
n
ot
w
ork
wh
en
a p
plied
to
the
s up
plem
en
tary
equation
(8);
in
fact
n =
[h
a ]
~
, on r , (
23)
a
does
not,
in
general
, v
an
i
sh
. Fu
rt
he
rmo r e 1 t i s
possib
le
to
sh
ow
th
a t
n i s
non
negative.
Th i s r
esult
wa s pr o
ven
i n a
non
c ovar i an t f
ormal
i sm
by P
.D.Lax
[4J
introducing
an
ar
tifi
c
al
vi
scosity
in
the
f i
el
d e qu
ations;
a
diff
erent
proof
was
given
in
[51.
It
is
know
tha
t
th
e pos i t i ve s i gna t ur e
of
n
fo
r t
he
non r e l a
tiv
i
st
ic
pe
rfect
fluid
implies
the
gr
owt
h
of
t
he
rmo -
d
yn
a m
ic
e
ntr
opy
a
cr
os
s t
he
s hoc
k.
That
i s
the
reason
why
the
c
ond
it
i on
n
> 0 i s of t
en
c
alle
d
in
the
literatur
e "
ent
r opy gr o
wth
c
on
di
tion"