Biomedical Engineering Reference
In-Depth Information
and Voigt-notation is denoted by the superscript v. Applying the fundamental lemma
of calculus of variations, this results in the residual form of the present coupled
problem
f
e
int
−
f
e
ext
=
r
e
=
0
,
(37)
f
e
int
−
f
e
ext
=
r
e
=
0
.
(38)
4.2 Linearisation
The governing system of Eqs. (
37
) and (
38
) is coupled with respect to the motion
˕
and the non-local damage field
ˆ
, which itself is linked to the local damage variable
ʺ
by means of the penalty term. To solve this highly non-linear system of equations,
we use an incremental-iterative Newton-Raphson scheme where we subsequently
omit the subscript index
n
1 associated with time
t
n
+
1
for the sake of readability.
A Taylor series expansion around the solution at the current iteration step
k
up to the
linear term gives
+
d
r
˕
d
d
r
˕
d
r
ek
+
1
=
r
ek
+
·
ʔ
˕
+
·
ʔ
ˆ
=
,
0
(39)
˕
ˆ
d
r
ˆ
d
d
r
ˆ
d
r
ek
+
1
=
r
ek
+
·
ʔ
˕
+
·
ʔ
ˆ
=
0
.
(40)
˕
ˆ
Herein, the increments
ʔ
ˆ
=
ˆ
k
+
1
−
ˆ
k
represent the differ-
ence of the discrete nodal degrees of freedom at iteration-step
k
ʔ
˕
=
˕
k
+
1
−
˕
k
and
1 and
k
. Assuming
'dead loads', we deduce the element-specific sub-matrices of the Jacobian as
+
B
˕
t
I
d
d
r
˕
d
K
˕˕
e
v
B
˕
+[
G
˕
·
˃
·
G
˕
t
=
=
·
·
]
v,
(41)
e
˕
t
B
d
v
d
r
˕
d
˃
B
˕
t
K
˕ˆ
N
ˆ
d
=
=
·
·
v,
(42)
e
ˆ
d
ˆ
t
B
2
d
y
d
g
v
d
r
ˆ
d
t
K
ˆ˕
e
N
ˆ
B
˕
d
=
=
·
·
v,
(43)
˕
t
B
N
ˆ
t
d
y
d
v
d
y
d
v
B
ˆ
d
d
r
ˆ
d
t
K
ˆˆ
e
N
ˆ
+
B
ˆ
=
=
·
·
·
v,
(44)
ˆ
ˆ
∇
x
ˆ
t
B
allowing us to express the geometrical contribution of
K
˕˕
conveniently by means of
the Kronecker product
. Furthermore,
e
,d
˃
/
d
ˆ
,2d
y
/
d
g
,d
y
/
d
ˆ
and d
y
/
d
∇
x
ˆ
