Biomedical Engineering Reference
In-Depth Information
Thus, the directional change (rotation) of the considered line segment, of the
current state with respect to the reference state, is described by the difference
of the unit vectors
e
and
e
0
. For the relation between the components of
d
x
at
the (fixed) current time
t
and the components of the accompanying
d
x
0
it can be
written, using the chain rule for differentiation for (total) differentials:
x
∂
x
0
dx
0
+
∂
x
∂
y
0
dy
0
+
∂
x
∂
z
0
dz
0
∂
dx
=
y
∂
x
0
dx
0
+
∂
y
∂
y
0
dy
0
+
∂
y
∂
z
0
dz
0
∂
dy
=
∂
z
∂
x
0
dx
0
+
∂
z
∂
y
0
dy
0
+
∂
z
∂
z
0
dz
0
,
dz
=
(10.5)
for which a Lagrangian description has been taken as the point of departure
according to:
x
=
x
(
x
0
,
t
). In a more compact form it can be formulated as
⎡
⎣
⎤
⎦
=
∂
x
∂
x
0
∂
x
∂
y
0
∂
x
∂
z
0
T
T
∂
y
∂
x
0
∂
y
∂
y
0
∂
y
∂
z
0
d
∼
=
F
d
∼
0
with
F
=
∼
0
∼
(10.6)
∂
z
∂
x
0
∂
z
∂
y
0
∂
z
∂
z
0
and in tensor notation:
∇
0
x
T
dx
=
F
·
dx
0
with
F
=
.
(10.7)
The tensor
F
, the deformation tensor (or deformation gradient tensor), with
matrix representation
F
, was already introduced in Section
9.6
. This tensor com-
pletely describes the (local) geometry change (deformation and rotation). After
all, when
F
is known, it is possible for every line segment (and therefore also for a
three-dimensional element) in the reference configuration, to calculate the accom-
panying line segment (or three-dimensional element) in the current configuration.
The tensor
F
describes for every material line segment the length and orientation
change:
F
determines the transition from
d
0
to
d
e
.
Fig.
10.2
visualizes a uniaxially loaded bar. It is assumed, that the deformation
is homogeneous: for every material point of the bar the same deformation tensor
F
is applicable. It can simply be verified, that the depicted transition from the
reference configuration to the current configuration is defined by
and the transition from
e
0
to
F
=
λ
x
e
x
e
x
+
λ
y
e
y
e
y
+
λ
z
e
z
e
z
,
(10.8)
with the stretch ratios:
A
A
0
.
0
λ
x
=
and
λ
y
=
λ
z
=
(10.9)
