Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

arising from integrating by parts the viscous contribution. Since the residual on the
boundaries can be neglected, other boundary contributions from the stabilizing terms
are negligible. Note from Eq. (2.91) that the whole residual appears in the stabilizing
term. However, we have omitted higher-order terms in the above equation for clarity.
As mentioned in Chapter 1, it is convenient to use matrix notation when the ®nite
element formulation is carried out. We start here from Eq. (1.7) of Chapter 1 and we
repeat the deviatoric stress and strain relations below
_" kk
ij ˆ 2 _" ij ÿ ij
…3:33†
3
where the quantity in brackets is the deviatoric strain. In the above
and
_" ij ˆ 1
2
@ui ‡
@xj @uj @xi …3:34†
_" ii ˆ @ui …3:35†
@xi We now de®ne the strain in three dimensions by a six-component vector (or in two
dimensions by a three-component vector) as given below (dropping the dot for
simplicity)
e ˆ ‰ " 11 " 22 " 33 2" 12 2" 23 2" 31 Š T ˆ " x " y " z 2" xy 2" xz 2" zx
T
…3:36†
with a matrix m de®ned as
m ˆ ‰ 1 1 1 0 0 0Š
T
We ®nd that the volumetric strain is
…3:37†
" v ˆ " 11 ‡ " 22 ‡ " 33 ˆ " x ‡ " y ‡ " z ˆ m T e …3:38†
The deviatoric strain can now be written simply as (see Eq. 3.33)
e d ˆ e ÿ 1
3 m" v ˆ I ÿ 1
3 mmT
ÿ
e ˆ Ide …3:39†
where
Id ˆ I ÿ 1
3 mmT
ÿ
…3:40†
and thus
Id ˆ 1
2
2 ÿ1 ÿ1 0 0
3
0
6 ÿ1
6 ÿ1
6
3 6 0
6
4 0
2
ÿ1
0
0
ÿ1
2
0
0
0
0
3
0
0
0
0
3
0 7
0 7
0 7
05
…3:41†
0 0 0 0 0 3
If stresses are similarly written in vectorial form as
r ˆ ‰ 11 22 33 12 23 31 Š T
Characteristic-based split (CBS) algorithm 71
…3:42†
where of course 11 is identically equal to x and is also equal to 11 ÿ p with similar
expressions for y and z, while 12 is identical to 12, etc.