Biomedical Engineering Reference
In-Depth Information
2
4
3
5
h
x
1x3
1x3
2
3
½
½
ou
ox
ov
ox
o
w
ox
000 000
000
o
u
oy
h
y
1x3
1x3
4
5
½
½
o
v
oy
o
w
oy
000
000 000
o
u
oz
h
z
1x3
1x3
ov
oz
o
w
oz
½
½
A ¼
¼
ð
2
:
79
Þ
h
y
h
x
ou
oy
ov
oy
o
w
oy
o
u
ox
o
v
ox
o
w
ox
000
1x3
½
000
o
u
oz
o
v
oz
o
w
oz
ou
oy
ov
oy
o
w
oy
h
z
h
y
1x3
ou
oz
ov
oz
o
w
oz
000
o
u
ox
ov
ox
o
w
ox
½
h
z
h
x
1x3
½
The deformation matrix B, dependent of u, can be defined as,
B ¼ B
0
þ
B
NL
ð
u
Þ
ð
2
:
80
Þ
since it varies with the deformation of the solid. The linear part of the deformation
matrix is represented by B
0
and the nonlinear contribution by B
NL
. For the three-
dimensional case,
2
4
3
5
o
u
ox
00
o
u
oy
o
u
oz
0
B
0
¼
o
u
oy
o
u
ox
o
u
oz
0
0
0
ð
2
:
81
Þ
00
o
u
oz
o
u
oy
o
u
ox
0
and
B
NL
¼ AG
ð
2
:
82
Þ
The nonlinear deformation is actualized through matrix A, which contains the
displacement current configuration.
2.3.2 Stiffness Matrix
The tangential stiffness matrix K
T
can be determined considering the variation of
the virtual work of Eq. (
2.71
), in order to the generalized displacements du,
2
4
3
5
dL
1
¼ d
Z
X
B
T
rdX
ð
2
:
83
Þ
