Civil Engineering Reference
In-Depth Information
For a proof, see Duvaut and Lions [1976], Nitsche [1981], or the end of this §.
Its structure is similar to the proof that the divergence satisfies an inf-sup condition
as a mapping of
H
1
()
d
into
L
2
()
; cf. Ch. III, §6. A special case in which the
inequality can be easily verified is dealt with in Remark 3.5 below.
3.2 Remark.
If the strain tensor
E
of a deformation is trivial, then by Remark 1.1
the deformation is an affine distance-preserving transformation. An analogous as-
sertion holds for the linearized strain tensor
ε
:
Let
⊂ R
3
be open and connected.
Then for v
∈
H
1
(),
ε(v)
=
0
,
if and only if
3
.
v(x)
=
a
∧
x
+
b
with a, b
∈ R
(
3
.
16
)
For the proof, we note that
∂
2
∂x
i
∂x
j
v
k
=
∂
∂x
i
ε
jk
+
∂
∂x
j
ε
ik
−
∂
∂x
k
ε
ij
=
0
(
3
.
17
)
in
H
−
1
()
if
ε(v)
=
0. From this we conclude that every component
v
k
must be
a linear function. But then a simple computation shows that a displacement of the
form
v(x)
=
Ax
+
b
can only be compatible with
ε(v)
=
0if
A
is skew-symmetric.
This leads to (3.16).
On the other hand, it is easy to verify that the linear strains for the displace-
ments of the form (3.16) vanish.
Korn's inequality is simplified for functions which satisfy a zero boundary
condition. In the sense of Remark II.1.6, it is only necessary that
v
vanishes on
a part
0
of the boundary, and that
0
possesses a positive
(n
−
1
)
-dimensional
measure.
3
be an open bounded
set with piecewise smooth boundary. In addition, suppose
0
⊂
∂ has positive
two-dimensional measure. Then there exists a positive number c
=
c
(,
0
) such
that
3.3 Korn's Inequality
(Korn's second inequality). Let
⊂ R
ε(v)
:
ε(v)dx
≥
c
v
2
1
for all v
∈
H
1
().
(
3
.
18
)
Here H
1
() is the closure of
{
v
∈
C
∞
()
3
;
v(x)
=
0
for x
∈
0
}
w.r.t. the
·
1
-norm.
Proof.
Suppose that the inequality is false. Then there exists a sequence
(v
n
)
∈
H
1
()
with
ε(v
n
)
:
ε(v
n
)dx
≤
1
n
2
ε(v
n
)
0
:
=
and
|
v
n
|
1
=
1
.
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