Environmental Engineering Reference
In-Depth Information
2 Mathematical Models
2.1 Perzyna's Model
Based on themathematical relation of Naghdi andMurch ( 1963 ) between viscoplastic
strain rate and the derivative of a potential function respect to the deviatoric stress
tensor, Perzyna ( 1966 ) proposed a model in which the strain rate tensor
vp
ij depends
ʵ
on the yield stress k 1 as
1
2 μ
k 1
J 2 .
vp
ij
ʵ
=
S ij
1
(1)
stands for Macauley brackets; m is the material shear modulus; S ij is the
stress deviator; and J 2 is the second invariant. Based on the expressions for effective
stress and effective strain rate (Bathe 1996 ), Perzyna ( 1966 ) obtained a relationship
for the viscosity μ in terms of the effective stress
Here
˃ and effective strain rate ʵ :
3 ʵ .
μ =
(2)
2.2 Viscosity Power Law Model
One of the simplest and most widely used equations for modelling viscosity in a
non-Newtonian fluid flow with shear thickening and dilatant behaviour was devised
by Ostwald ( 1975 ):
n
1
μ =
m
ʳ
,
(3)
m 2
where m
and n are the consistency and index coefficients, respectively.
The m parameter depends on temperature as follows:
(
Nsn
/
)
m 0 exp Δ
1
T
E
R
1
T 0
m
=
,
(4)
where m 0 is the viscosity at T 0 ,
E is the activation energy for the process, R is
the universal constant of gases, T 0 is a reference temperature and T is the process
temperature. By making some arrangements the following expression is obtained:
Δ
m
=
m 0 exp [
a
(
T
T 0 )
]
.
(5)
2.3 Heat Generation Schmidt Model
Schmidt and Hattel ( 2008 ) proposed that heat generation during FSW is a function
of tool geometry, tool plunge force and material yield stress. Viscous dissipation was
computed in his work as ˄ :∇ V . So, heat generation by friction q f [W m 2 ]inthe
tool/workpiece interface is:
 
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