Biomedical Engineering Reference
In-Depth Information
2
3
s
ðÞ
:
¼
Y
ðÞþ
Z
t
4
5
I
j t
ðÞ
Yt
t
ð Þ
dt
0
t
0
¼
0
8
<
9
=
dt
0
Z
t
þ
2
3
c t
ðÞ
^
U
1
t
t
ð Þ
tr B
ðÞ
1
3
W t
t
ð Þ
C
1
t
t
ð Þ
I
:
;
t
0
¼
0
2
4
3
5
B
ðÞ
ð
3
:
345
Þ
^
U
1
ðÞþ
Z
t
c t
ðÞ
^
U
1
t
t
ð Þ
dt
0
þ
2
t
0
¼
0
Z
t
2
3
c t
ðÞ
^
U
1
t
t
ð Þ
W t
t
ð Þ
C
1
t
t
ð Þ
dt
0
t
0
¼
0
with the definition
h
i
W t
t
ð Þ
tr C
1
t
t
ð Þ
B
ðÞ
ð
3
:
346
Þ
where B :
¼
F
F
T
J
2
=
3
B is the left deviatoric C
AUCHY
strain tensor and B
I
:
¼
tr B is its first invariant.
3.3 Finite Element Method
3.3.1 Introduction
From the vast amount of literature on finite element methods, overviews can be
found in Bathe (1996), Zienkiewicz and Taylor (2000a, b), Parisch (2003) and
Wriggers (2008).
The finite element method (FEM) is a numerical approach and discretization
technique to establish an approximate solution of the governing equations where a
closed form solution is not feasible. In mechanical applications, these are usually
partial differential equations including local balance of linear momentum, kine-
matical relations and the constitutive equation (boundary value problem); in multi-
physics problems other equations apply. The complexity of the majority of these
problems involving complex geometries and/or boundary conditions in two or
three dimensions requires the use of numerical methods, since analytical solutions
do not exist. The fundamental concept of the finite element method is that the
underlying differential equations describing a physical problem are transformed to
a numerically solvable discrete formulation. Possible applications of FEM are to
predict stress fields and potential deformation within solid structures subjected to
external forces. Fields of application may involve heat transfer, electromagnetic
fields and fluid flow, among others.
