Biomedical Engineering Reference
In-Depth Information
We assume an exponential behaviour for the damage function
f
d
(ʺ)
=
1
−
d
=
exp
(ʷ
d
[
ʺ
d
−
ʺ
]
),
(25)
∂
d
ʺ
=[
ʷ
d
f
d
(ʺ)
]
−
1
with
ʷ
d
>
0 so that
>
0 and introduce an initial damage thresh-
old
ʺ
d
, which must be exceeded in order to activate damage evolution. Furthermore,
we include a saturation parameter
ʷ
d
. It becomes apparent that larger values of
ʷ
d
accelerate the damage process, whereas larger values of
ʺ
d
lead to a delay of the dam-
age initiation. Note, that for the limiting case
0, damage is initiated from the
very beginning of the loading process, whereas damage does not evolve for
ʺ
d
=
ʺ
d
ₒ∞
.
4 Finite Element Discretisation
This section deals with the spatial finite element discretisation of the underlying
coupled system of non-linear equations. This includes a combination of tri-quadratic
serendipity interpolation functions with respect to the displacement field, and tri-
linear interpolation functions with respect to the non-local damage field variable
where we outline an efficient and compact FE-implementation using a common
Voigt-notation-based vector-matrix-format.
4.1 Discretisation
=
n
el
e
h
0
We discretise the domain
B
0
by
n
el
finite elements, so that
B
0
≈
B
1
B
0
e
,
=
B
0
e
is characterised by
n
en
placement-nodes and
n
en
non-
local-damage-nodes. According to the isoparametric concept, we interpolate the field
variables
where every finite element
as well as the geometry
X
by the same shape functions
N
˕
˕
X
h
N
˕
·
h
N
˕
·
˕
e
,
X
≈
=
X
e
,
˕
≈
˕
=
(26)
ʾ
:=
and transform them to a hexahedral reference element with natural coordinates
{
ʾ,ʷ,ʶ
}∈
B
B
:= {
ʾ
∈ R
3
denotes the
reference domain. In the present context, the number of displacement-nodes and non-
local-damage-nodes per element—and consequently the related shape functions—do
not necessarily have to coincide, i.e.
n
en
=
, where
|−
1
≤
ˇ
≤+
1
;
ˇ
=
ʾ,ʷ,ʶ
}
n
en
and
N
˕
=
N
ˆ
. We approximate the
associated field variables, i.e. the placement
˕
and the non-local damage variable
ˆ
by means of the product of shape functions
N
ʱ
(
ʾ
)
ʱ
=
˕, ˆ
,
and discrete element
˕
e
and
ˆ
e
, i.e.
degrees of freedom
N
˕
·
˕
e
,
N
ˆ
·
ˆ
e
,
h
h
˕
=
ˆ
=
(27)
