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

When such vector forms are used we can write the strain rates in the form
_e ˆ Su …1:5†
where S is known as the stain operator and u is the velocity given in Eq. (1.1).
The stress±strain relations for a linear (newtonian) isotropic ¯uid require the
de®nition of two constants.
The ®rst of these links the deviatoric stresses ij to the deviatoric strain rates:
ij ij ÿ ij
kk
3 ˆ 2 _" _" kk
ij ÿ ij
3
…1:6†
In the above equation the quantity in brackets is known as the deviatoric strain, ij is
the Kronecker delta, and a repeated index means summation; thus
ii 11 ‡ 22 ‡ 33 and _" ii _" 11 ‡ _" 22 ‡ _" 33 …1:7†
The coe cient is known as the dynamic (shear) viscosity or simply viscosity and is
analogous to the shear modulus G in linear elasticity.
The second relation is that between the mean stress changes and the volumetric
strain rates. This de®nes the pressure as
p ˆ ii
3 ˆÿ _" ii ‡ p 0
…1:8†
where is a volumetric viscosity coe cient analogous to the bulk modulus K in linear
elasticity and p 0 is the initial hydrostatic pressure independent of the strain rate (note
that p and p 0 are invariably de®ned as positive when compressive).
We can immediately write the `constitutive' relation for ¯uids from Eqs (1.6) and
(1.8) as
or
ij ˆ 2 _" ij ÿ ij _" kk
3
‡ ij _" kk ÿ ijp 0
ˆ ij ÿ ijp …1:9a†
ij ˆ 2 _" ij ‡ ij… ÿ 2
3 † _" ii ‡ ijp 0 …1:9b†
Traditionally the Lame notation is often used, putting
ÿ 2
3
…1:10†
but this has little to recommend it and the relation (1.9a) is basic. There is little
evidence about the existence of volumetric viscosity and we shall take
_" ii 0 …1:11†
in what follows, giving the essential constitutive relation as (now dropping the su x
on p 0)
ij ˆ 2 _" ij ÿ ij _" kk
3
without necessarily implying incompressibility _" ii ˆ 0.
The governing equations of ¯uid dynamics 5
ÿ ijp ij ÿ ijp …1:12a†