Biomedical Engineering Reference
In-Depth Information
2 Quadratic Corotated Finite Elements
2.1 Elasticity Model
In elasticity theory, material laws relate the deformation of an object to the stress
applied on it. In order for a deformation measure to be rotation invariant, it has to be
nonlinear. Consequently, linear elastic models cannot be used if an object is
subjected to large deformations, regardless of the material properties (geometric
nonlinearity). However, if a fully nonlinear formulation is discretized using the
FEM and an implicit time integration scheme, a nonlinear system of equations has
to be solved for each time step. The corotational FE formulation offers an attractive
alternative to this computationally expensive approach.
The semidiscrete nonlinear equation of motion can be written as
f
int
f
ext
M
x
€
þ
¼
(1)
with the mass matrix
M
, the internal nodal forces
f
int
, the external nodal forces
f
ext
and the nodal spatial coordinates
x
In classic linear elasticity, the Cauchy strain
tensor is used to obtain the tangent stiffness matrix
f
int
iI
K
iIjJ
¼
@
x
jJ
;
(2)
@
which is subsequently used to derive the internal nodal forces
f
int
iI
x
jJ
¼
K
iIjJ
x
jJ
:
(3)
The basic concept of the co-rotated FE method is to linearize the equation of
motion by extracting the rotational component
R
of the deformation gradient
r' ¼
RS
(4)
while using the remaining stretch matrix
S
as the deformation measure in order to
derive the internal nodal forces. This approach can be described as rotating the
deformation field into the initial configuration, calculating the nodal forces using
the linear Cauchy strain tensor and finally rotating the forces back to the deformed
configuration. The elemental internal nodal forces are thus given by
;
f
int
e
R
e
K
e
R
e
x
x
0
¼
(5)
where
R
e
is the elemental rotation matrix which is detailed in Sect.
2.2
. Inserting (
5
)
into (
1
) and adding the damping term
C
e
_
x
gives rise to the linearized elemental
equation of motion
