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manual-cohesive_elements.tex
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\subsection
{
Cohesive Elements
}
The cohesive elements that have been implemented in
\akantu
are based
on the work of Ortiz and Pandolfi~
\cite
{
ortiz1999
}
. Their main
properties are reported in Table~
\ref
{
tab:coh:cohesive
_
elements
}
.
\begin
{
table
}
[!htb]
\begin
{
center
}
\begin
{
tabular
}{
l|llcc
}
\toprule
Element type
&
Facet type
&
Order
&
\#
nodes
&
\#
quad. points
\\
\midrule
\texttt
{
\_
cohesive
\_
1d
\_
2
}
&
\texttt
{
\_
point
\_
1
}
&
linear
&
2
&
1
\\
\hline
\texttt
{
\_
cohesive
\_
2d
\_
4
}
&
\texttt
{
\_
segment
\_
2
}
&
linear
&
4
&
1
\\
\texttt
{
\_
cohesive
\_
2d
\_
6
}
&
\texttt
{
\_
segment
\_
3
}
&
quadratic
&
6
&
2
\\
\hline
\texttt
{
\_
cohesive
\_
3d
\_
6
}
&
\texttt
{
\_
triangle
\_
3
}
&
linear
&
6
&
1
\\
\texttt
{
\_
cohesive
\_
3d
\_
12
}
&
\texttt
{
\_
triangle
\_
6
}
&
quadratic
&
12
&
3
\\
\bottomrule
\end
{
tabular
}
\end
{
center
}
\caption
{
Some basic properties of the cohesive elements in
\akantu
.
}
\label
{
tab:coh:cohesive
_
elements
}
\end
{
table
}
\begin
{
figure
}
\centering
\includegraphics
[width=.6\textwidth]
{
figures/cohesive2d
}
\caption
{
Cohesive element in 2D for quadratic triangular elements
T6.
}
\label
{
fig:smm:coh:cohesive2d
}
\end
{
figure
}
Cohesive element insertion can be either realized at the beginning of
the simulation or it can be carried out dynamically during the
simulation. The first approach is called
\emph
{
intrinsic
}
, the second
one
\emph
{
extrinsic
}
. When an element is present from the beginning, a
bilinear or exponential cohesive law should be used instead of a
linear one. A bilinear law works exactly like a linear one except for
an additional parameter
$
\delta
_
0
$
separating an initial linear
elastic part from the linear irreversible one. For additional details
concerning cohesive laws see Section~
\ref
{
sec:cohesive-laws
}
.
\begin
{
figure
}
\centering
\includegraphics
[width=.75\textwidth]
{
figures/insertion
}
\caption
{
Insertion of a cohesive element.
}
\label
{
fig:smm:coh:insertion
}
\end
{
figure
}
Extrinsic cohesive elements are dynamically inserted between two
standard elements when
\begin
{
equation
}
\sigma
_
\mathrm
{
eff
}
>
\sigma
_
\mathrm
{
c
}
\quad\text
{
with
}
\quad
\sigma
_
\mathrm
{
eff
}
=
\sqrt
{
\sigma
_
\mathrm
{
n
}^
2 +
\frac
{
\tau
^
2
}{
\beta
^
2
}}
\end
{
equation
}
in which
$
\sigma
_
\mathrm
{n}
$
is the tensile normal traction and
$
\tau
$
the resulting tangential one (Figure~
\ref
{
fig:smm:coh:insertion
}
).
For the static analysis of the structures containing cohesive
elements, the stiffness of the cohesive elements should also be added
to the total stiffness of the structure. Considering a 2D quadratic
cohesive element as that in Figure~
\ref
{
fig:smm:coh:cohesive2d
}
, the
opening displacement along the mid-surface can be written as:
\begin
{
equation
}
\label
{
eq:opening
}
\vec
{
\Delta
}
(s) =
\llbracket
\mat
{
u
}
\rrbracket
\,\mat
{
N
}
(s) =
\begin
{
bmatrix
}
u
_
3-u
_
0
&
u
_
4-u
_
1
&
u
_
5-u
_
2
\\
v
_
3-v
_
0
&
v
_
4-v
_
1
&
v
_
5-v
_
2
\end
{
bmatrix
}
\begin
{
bmatrix
}
N
_
0(s)
\\
N
_
1(s)
\\
N
_
2(s)
\end
{
bmatrix
}
=
\mat
{
N
}^
\mathrm
{
k
}
\mat
{
A U
}
=
\mat
{
PU
}
\end
{
equation
}
The
\mat
{
U
}
,
\mat
{
A
}
and
$
\mat
{N}^
\mathrm
{k}
$
are as following:
\begin
{
align
}
\mat
{
U
}
&
=
\left
[
\begin
{
array
}{
c c c c c c c c c c c c
}
u
_
0
&
v
_
0
&
u
_
1
&
v
_
1
&
u
_
2
&
v
_
2
&
u
_
3
&
v
_
3
&
u
_
4
&
v
_
4
&
u
_
5
&
v
_
5
\end
{
array
}
\right
]
\\
[1ex]
\mat
{
A
}
&
=
\left
[
\begin
{
array
}{
c c c c c c c c c c c c
}
1
&
0
&
0
&
0
&
0
&
0
&
-1
&
0
&
0
&
0
&
0
&
0
\\
0
&
1
&
0
&
0
&
0
&
0
&
0
&
-1
&
0
&
0
&
0
&
0
\\
0
&
0
&
1
&
0
&
0
&
0
&
0
&
0
&
-1
&
0
&
0
&
0
\\
0
&
0
&
0
&
1
&
0
&
0
&
0
&
0
&
0
&
-1
&
0
&
0
\\
0
&
0
&
0
&
0
&
1
&
0
&
0
&
0
&
0
&
0
&
-1
&
0
\\
0
&
0
&
0
&
0
&
0
&
1
&
0
&
0
&
0
&
0
&
0
&
-1
\end
{
array
}
\right
]
\\
[1ex]
\mat
{
N
}^
\mathrm
{
k
}
&
=
\begin
{
bmatrix
}
N
_
0(s)
&
0
&
N
_
1(s)
&
0
&
N
_
2(s)
&
0
\\
0
&
N
_
0(s)
&
0
&
N
_
1(s)
&
0
&
N
_
2(s)
\end
{
bmatrix
}
\end
{
align
}
The consistent stiffness matrix for the element is obtained as
\begin
{
equation
}
\label
{
eq:cohesive
_
stiffness
}
\mat
{
K
}
=
\int
_{
S
_
0
}
\mat
{
P
}^
\mathrm
{
T
}
\,
\frac
{
\partial
{
\vec
{
T
}}}
{
\partial
{
\delta
}}
\mat
{
P
}
\,\mathrm
{
d
}
S
_
0
\end
{
equation
}
where
$
\vec
{T}
$
is the cohesive traction and
$
\delta
$
the opening
displacement (for more details check
Section~
\ref
{
tab:coh:cohesive
_
elements
}
).
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "manual"
%%% End:
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