Biomedical Engineering Reference
In-Depth Information
with
∇
0
u
T
,
F
=
I
+
(13.2)
reflecting history-dependent material behaviour. In addition the local balance of
momentum has to be satisfied (see Chapter
11
), as well as the local mass balance:
F
−
T
· ∇
0
=
ρ
¨
·
σ
+
ρ
q
u
,
(13.3)
with
ρ
0
det(
F
)
.
ρ
=
(13.4)
The equations given above form a set of non-linear, coupled partial differential
equations (derivatives with respect to the three material coordinates in
x
0
and the
time
t
are dealt with). Consequently, for a unique solution of the displacement field
u
(
x
0
,
t
) boundary conditions and initial conditions are
required. With respect to the boundary conditions, for all
t
at every point of the
outer surface of
V
0
three (scalar) relations have to be specified: either completely
formulated in stresses (dynamic or natural boundary conditions), completely for-
mulated in displacements (kinematic or essential boundary conditions) or in mixed
formulations. With respect to the initial conditions, at the initial time point (
t
x
0
,
t
) and the stress field
σ
(
0),
for all the points in
V
0
, the displacement and velocity have to be specified. If the
initial state is used as the reference configuration,
u
(
x
0
,
t
=
0)
=
0 for all
x
0
in
V
0
.
=
13.2.2
Geometrical linearity
Provided that displacements, strains and rotations are small (so
F
≈
I
and conse-
quently det(
F
)
1) the general set for solid continuum problems as presented in
the previous section can be written as
≈
σ
(
x
0
,
t
)
=
G
{
ε
(
x
0
,
τ
);
τ
≤
t
}
,
(13.5)
where
F
{}
, as used in Eq. (
13.1
), has been replaced by
G
{}
due to adaptations in
the argument, with
u
T
,
u
1
2
∇
0
∇
0
ε
=
+
(13.6)
and
∇
0
·
σ
+
ρ
0
q
=
ρ
0
¨
u
.
(13.7)
Again these equations have to be satisfied for all
x
0
in
V
0
and for all times
t
.
The first equation (a formal functional expression) indicates that the current local
