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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