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

72 A general algorithm for compressible and incompressible ¯ows
Immediately we can assume that the deviatoric stresses are proportional to the
deviatoric strains and write directly from Eq. (3.33)
r d ˆ Idr ˆ I0e d ˆ I0 ÿ 2
3 mmT
ÿ
_e …3:43†
where the diagonal matrix I0 is
2
2
3
6
I0 ˆ 6
4
2
2
1
1
7
5
…3:44†
1
To complete the vector derivation the velocities and strains have to be appropriately
related and the reader can verify that using the tensorial strain de®nitions we
can write
_e ˆ Su …3:45†
where
T
u ˆ u1 u2 u3 …3:46†
and S is an appropriate strain matrix (operator) de®ned below
2
3
@
6 0 0
@x 7
6 1
7
6
@
7
6
7
6 0 0 7
6 @x2 7
6
7
6
@ 7
6 0 0 7
6
@x 7
6
3 7
S ˆ 6
@ @
7
6
7
6
0 7
6 @x2 @x1 7
6
7
6
@ @ 7
6 0
7
6 @x3 @x 7
2 7
6
4 @ @
7
5
0
@x 3
@x 1
…3:47†
where the subscripts 1, 2 and 3 correspond to the x, y and z directions, respectively.
Finally the reader will note that the direct link between the strains and velocities will
involve a matrix B de®ned simply by
B ˆ SNu …3:48†
Now from Eqs. (3.30), (3.32) and (3.43), the solution for Ui in matrix form is:
Step 1
~U ˆÿM ÿ1
u t …C u ~U ‡ K ~u ÿ f †ÿ t…K u ~U ‡ f s† n
…3:49†
where the quantities with a ~ indicate nodal values and all the discretization matrices
are similar to those de®ned in Chapter 2 for convection±di€usion equations (Eqs. 2.94