Civil Engineering Reference
In-Depth Information
As deformations we admit functions
ψ
which satisfy Dirichlet boundary conditions
on
0
along with the local injectivity condition det
(
∇
ψ(x)) >
0. - We introduce
appropriate function spaces later.
The expression (2.4) refers to the variational formulation for the displace-
ments. We note that frequently the stresses are also included as variables in the
variational problem. Because of the coupling of the kinematics with the constitu-
tive equations, we get a saddle point problem, and thus mixed methods need to be
applied.
2.2 Remark.
The properties of the material laws discussed in §1 may be rediscov-
ered in analogous properties of the energy functionals. To save space, we present
them without proof.
For an objective material,
W(x,
·
)
is a function of only
C
=
F
T
F
:
W(x,F)
=
W(x,F
T
F)
and
2
∂ W(x,C)
∂C
˜
3
(x,C)
=
for all
C
∈ S
>
.
The dependence of
C
can be made more precise.
W
depends only on the principal
invariants of
C
, i.e.,
W(x,C)
=
W(x,ı
C
)
for
C
∈ S
3
>
. Analogously, for isotropic
materials, we have
W(x,F)
=
W(x,FQ)
3
+
3
+
for all
F
∈ M
,Q
∈ O
.
In particular, for small deformations,
λ
W(x,C)
=
2
(
trace
E)
2
+
µE
:
E
+
o(E
2
)
(
2
.
5
)
with
C
=
I
+
2
E
. Here, as usual,
trace
(A
T
B),
A
:
B
:
=
A
ij
B
ij
=
ij
for any two matrices
A
and
B
.
2.3 Examples.
(1) For St. Venant-Kirchhoff materials,
λ
2
(
trace
F
W(x,F)
3
)
2
=
−
+
µF
:
F
(
2
.
6
)
λ
2
(
trace
E)
2
=
+
µ
trace
C.
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