Ferrarese G. (editor) Wave Propagation
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288
Expand
ing
the
first
two
terms
on
the
right
gives
(3.10)
[:X
(:~)]
a
2y
=
'ilFX~F
[F
X
X]
(3.11
)
(::2
(:;X
)]
[ f a
2y
a
3y}
a
3y
[Fh]
=
F
--10
2 - -
10--
XX]
afaF
aF
2aF
aF3
X X X
+ a
2y
[F
]
aF2
XXX
X
These
expressions
are
inserted
into
(3.9)
and
together
with
(3.5)
and
the
relation
(3.12)
If
the
region
0,
V
is
[11]
•
+
F
XX
=
2M~+
cSt
(3.13)
we may
derive
the
following
ordinary
differential
equation
for
the
ampli
tude,
aCt)
=
[F
X
] ;
10
{
M
cSV
2 a
3y
F
XX
a
3y}
a
---1o-F
---+2---
V
cSt
V X aF
2aF
V aF3
X X
1 a
3y
2
+
--
a =O.
V2
aF3
X
Again,
th
is
equation
may be so
lve
d
as
in
Chen
ahead
of
the
wave
is
in
constant
strain,
F;
=
constant
and
then
aCt)
=
a(O)/
(1
+ t
~
a
3y).
mv
2
aF3
X
a
3
y
Obviously,
under
the
right
conditions
on
a(D),
M, V and
--
there
aF3
X
will
be breakdown
of
aCt).
In
both
th
is
and
the
preceding
it
is
evid
ent t
hat
hi
gher
order
waves ,
i.
e.
of
one
order
high
er
than
what i s
usually
def i ned
as
an
accelerat
ion
wa
ve,
suffer
a
type
of
breakdown ana
lo
gous
to
that
o
bserved
in
acceleration
waves
in
classical
theori
e
s,
usually
thought
to
be a
ssociated
with
formatio
n
of
a wave of lower
or
der.
If
the
behav
iour
encount
er
ed h
er
e i s not
to
be acc ep
ted,
t hen one
289
may
argue:
(a)
the
dipolar
theories
should
not
be
used
as
exact
theories
for
wave
propagation
problems.
(b)
the
classical
case
is
the
first
order
approx-
imation
and
dipolar
terms
should
be
included
in
an expansion
procedure.
If
(a)
is
thought
to
be
the
case
then
perhaps
one
has
to
look
at
a
viscoelastic
type
theory
in
which
gradients
of
all
orders
are
present.
Certainly,
it
would
appear
necpssary
to
invest-
igate
higher
order
theories.
If
we
try
to
study
shock waves
or
acceleration
waves
in
the
theories
considered
here,
we
find
that
the
basic
gov~rning
equations
are
not
in
a
suita
~le
form. For example,
equation
(3.1)
contains
terms
of
second,
third
and
fourth
orders
and
it
is
not
obvious
how
to
proceed
in
that
case.
It
is
perhaps
likely
that
the
higher
order
nature
of
the
equation
prevents
the
formation
of
waves
of
lower
order.
Certainly
the
phen-
omenon
of
focussing
(see
ego
Koops
and Wilkes
[l~J)
is
not
likely
to
occur.
To
see
this
and complete
this
work we
consider
a
linear
dipolar
elastic
material
and show
that,
unlike
classical
linear
elasticity,
stability
in
the
CO
norm
is
possih1e.
(This
possibility
was
pointed
out
by
Koops
and Wilkes [14J and
Koops
and
Straughan
[lSJ).
4.
Uniform
stabil;ty
of a
linear
d;polar
elast;c
mater;al.
The
appropriate
equations
for
an
anisotropic
l
inear
dipolar
elastic
material,
which may be
derived
from
the
cited
papers
of
Green and Naghdi and Green and
Riv1in,
are
(see
[l SJ )
pU. -
pf
•••
- T
•••
+
l:(k.).
1 • =- pF
.••
+
pr·,
1-
J1..J
J1..J
::; 1..,<J 1
-J.J
1.
where u
i
are
the
components
of
displacement
about
a
reference
configuration
and
the
other
terms
are
named
in
§1. Here,
the
inertia
coefficient
satisfies
The
constitutive
equations
for
Tij
and
L(kj)i
290
(4.2)
r
j i
= m
j k
(1$) u
i
•
k
'
where m
i j
is
a symmetric
tensor
which
defines
a
non-negative
bilinear
form.
are
(4.4)
E(kj)i
= h
pqi j k
Up~q
+
CpqmiJk
Up~qm'
where
the
"elasticities"
satisfy
the
following
symmetries
w
ith
U.
1.
C
i j k
hmn
=chmnijk
We
suppose
(4.1)
are
defined
and
aU
i
prescribed
on
the
au
Ui
=
qi
(~.t)
au.
__
'l. =
hi
(~.t)
an
on a bounded
region
0
of
R3
boundary
of
o. r. Say
on
r
)(
[0
...
).
(Green and
Rivlin
[1]
suggest
that
(4.6)2
should
be
replaced
by
conditions
on
the
dipolar
traction.
However. we have chosen
these
conditions
so
that
we may
apply
the
appropriate
Sobolev
inequality
for
illustration
of
the
new
stability
effect.)
Let
Vi
and
V~
be two
solutions
to
(4.1)
in
0
with
the
same
body
forces
and same boundary
data
(4
.6).
Define
u. =v* -
V.
'l. i 'l.
and
then
a
routine
calculation
enables
us
to
show
the
total
energy
E(t)
is
constant
in
time.
i.e.
def
(4.7)
E(t)
=
~
f (pll
.ll.
+ pm'
k
U.
.U.
k
p 'l. 'l. J 'l.. J 'l.• •
+ a
kmij
Up..mUi.j + Cmnqijh
Um.nqU
i•
j k
+ 2
b1'8tij
U
tU'
. )dx
1'.8
'l..J
=
E(O).
'V
t
~
0.
If
now,
the
potential
energy
is
positive
definite
in
the
following
sense
(4.8)
291
J
(a'_
'J' Uk
u,
,+2b
t"
U t
U
,
,+c '.'k U .
.».
'k)dx
1<1111-
,m
1-
, J
1'8
1-J
1',8
1-,J
T11I1lj1-J
171,,",!
1-,J
Q
(A >
0)
.,
we deduce 'from
(4.7)
that
(4.9)
E(O)
~
A J
u,
'k
u
,
'k
liz.
1-.J 1-,J
fl
However,
for
Q C R3,
~,2
(Q)
C CO(Q),
see
Gilbarg
~nd
Trud~nger
[16],
corollary
7.11,
and so
there
is
a cons
rerrt
u depending on
Q, A such
that
(4.10)
,
[E(O»)'
~
\l sup IY(lS,t)l.
fl
Thus. by making
F(O)
arbitrarily
small
lui
is
likewise
a~bi~rarily
small
and so
the
solution
is
stable
in
the
CO
norm.
Acknowledgments
I
should
like
to
express
my
s
incere
thanks
to
Professor
D.
Graffi
for
his
kind
invitation
to
study
in
Italy
in
June -
July
19BO
and
to
both
Professor
Graffi
and
Professor
G.
Ferrarese
for
my
appreciation
to
Professors
D.
Graffi
and
M.
Dr. F.
Franchi
for
many
stim
ulating
discu
ssions.
the
opportunity
wish
to
express
Fabrizio
and
to
to
hold
a
seminar
in
Bressanone.
finall
y,
I
292
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Arch. Rational
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Anal.
16
(196~),
325-353.
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W.
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•
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~.
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E
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