Civil Engineering Reference
In-Depth Information
v
n
1
≤
c
1
for
all
n
and some suitable
c
1
>
0. Since
H
1
()
is compact in
H
0
()
, there is a
subsequence of
(v
n
)
which converges w.r.t. the
Because of the hypothesis on
0
, Friedrichs' inequality implies
·
0
-norm. With the constant
c
2
1
2
0
2
0
from Theorem 3.1, we have
c
v
n
−
v
m
≤
ε(v
n
−
v
m
)
+
v
n
−
v
m
≤
2
2
2
0
2
2
0
2
2
0
.
The
L
2
-convergent subsequence is thus a Cauchy sequence in
H
1
()
, and so
converges in the sense of
H
1
to some
u
0
. Hence,
ε(v
n
)
+
2
ε(v
m
)
+
v
n
−
v
m
≤
n
+
m
+
v
n
−
v
m
0,
and
|
u
0
|
1
=
lim
n
→∞
|
v
n
|
1
=
1. By Remark 3.2, we deduce from
ε(u
0
)
=
0 that
u
0
has the form (3.17). In view of the zero boundary condition on
0
, it follows
that
u
0
ε(u
0
)
=
lim
n
→∞
ε(v
n
)
=
=
0. This is a contradiction to
|
u
0
|
=
1.
1
Korn's inequality asserts that the variational problem (3.8) is elliptic. Thus,
the general theory immediately leads to
3
be a domain with piecewise smooth bound-
ary, and suppose
0
has positive two-dimensional measure. Then the variational
problem (3.8) of linear elasticity theory has exactly one solution.
3.4 Existence Theorem.
Let
⊂ R
3.5 Remark.
In the special case where Dirichlet boundary conditions are pre-
scribed (i.e.,
0
=
and
H
1
=
H
0
), the proof of Korn's first inequality is
simpler. In this case
|
v
|
1
,
≤
√
2
for all
v
∈
H
0
()
3
.
ε(v)
0
,
(
3
.
19
)
It suffices to show the formula for smooth vector fields. In this case we have
(s)
v
:
(s)
v
(
div
v)
2
.
2
∇
∇
−∇
v
:
∇
v
=
div[
(v
∇
)v
−
(
div
v)v
]
+
(
3
.
20
)
Here
(v
∇
)
is to be interpreted as
i
v
i
∂
∂x
i
. The formula (3.20) can be verified,
for example, by solving for all terms in the double sum. Since
v
=
0on
∂
,it
follows from the Gauss integral theorem that
div[
(v
∇
)v
−
(
div
v)v
]
dx
=
[
(v
∇
)v
−
(
div
v)v
]
nds
=
0
.
∂
Integrating (3.20) over
,wehave
(s)
v
2
0
2
1
(
div
v)
2
dx
≥
∇
−|
v
|
=
2
0
,
and (3.19) is proved.
Note that the constant in (3.19) is independent of the domain. If we are given
Neumann boundary conditions on a part of the boundary, the constant can easily
depend on
. We will see the consequences in connection with the locking effect
for the cantilever beam shown in Fig. 58 below. — On the other hand, for the pure
traction problem, i.e., for
0
=∅
, there is again a compatibility condition; see
Problem 3.17.
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