Biomedical Engineering Reference
In-Depth Information
Likewise, the weighting function
w
is written as
w
=
w
x
(
x
,
y
)
e
x
+
w
y
(
x
,
y
)
e
y
.
(18.33)
w
are
In the plane strain case, the matrices associated with the tensors
ε
and
ε
given by
⎡
⎤
⎦
⎡
⎤
⎦
xx
xy
ε
xx
ε
xy
0
ε
xy
ε
yy
0
000
ε
ε
0
⎣
⎣
w
yy
0
000
xy
ε
=
,
ε
=
ε
ε
.
(18.34)
w
:
σ
equals
Consequently, the inner product
ε
w
:
w
w
w
ε
σ
=
ε
xx
σ
xx
+
ε
yy
σ
yy
+
2
ε
xy
σ
xy
.
(18.35)
Notice the factor 2 in front of the last product on the right-hand side due to the
symmetry of both
w
and
ε
σ
. It is convenient to gather the relevant components of
w
,
ε
ε
(for future purposes) and
σ
in a column:
xy
w
)
T
xx
yy
(
∼
=
ε
ε
2
ε
ε
xy
,
T
∼
=
ε
xx
ε
yy
2
(18.36)
xy
and
ε
xy
) and
(notice the 2 in front of
ε
σ
xy
.
T
∼
=
σ
xx
σ
yy
(18.37)
w
:
This allows the inner product
ε
σ
to be written as
w
:
σ
=
(
∼
w
)
T
ε
∼
.
(18.38)
Step 3
The constitutive equation according to Eq. (
18.4
) may be recast in the
form
∼
=
H
∼
.
(18.39)
Dealing with the isotropic Hooke's law and plane strain conditions and after
introduction of Eqs. (
18.12
) and (
18.13
) into Eq. (
18.4
), the matrix
H
can be
written as
⎡
⎣
⎤
⎦
+
⎡
⎣
⎤
⎦
110
110
000
4
−
20
G
3
H
=
K
−
240
003
.
(18.40)
Consequently
w
:
w
)
T
H
ε
σ
=
(
∼
∼
.
(18.41)
