3 Modeling of Friction Stir Welding Processes
117
where K wt and h t are the conductance and convective heat transfer coefficient, T t is
the tool temperature, and T is the workpiece temperature. The heat dissipation from
the top surface to the environment consists of both convective and radiation losses
expressed as.
Q wa = σ b ε b
T
4
− T
4
a
+ h b (T − T a ),
(3.90)
where Q wa is heat loss through top surface to surrounding, σ b is the Stefan–Boltzmann constant, h b is the heat transfer coefficient at the top surface, T a is the ambient
temperature, and ε b is the emissivity. The heat loss at the workpiece bottom surface
and the backing plate Q wb is modeled by considering a suitable convective heat
transfer coefficient with equivalent conductive heat transfer during experiment. The
heat dissipation to backing plate is expressed as
Q wb = K wb
∂ T
∂z
= h b (T − T b ),
(3.91)
where K wb and h b are the conductance and convective heat transfer coefficient for the
heat interaction of workpiece and the backing plate, and T b is the temperature of the
backing plate.
3.5.3 Material Flow Modeling
In order to obtain high structural efficiency of the welds and for optimal tool design,
it is necessary to understand the material flow. Computational fluid dynamics (CFD)
using commercial software Fluent was employed by Colegrove and Shercliff [21] to
model the material flow in FSW. They developed a ‘slip’ model where the local shear
stresses govern the interface conditions. It was assumed that the shear stress experienced by the workpiece is different for different tool materials. The surface shear
stress was assigned maximum limiting value. Two different boundary conditions were
used: (a) stick condition, where the shear stress was lower than the limiting shear
stress, and (b) slip condition, where for stick condition to exist, the shear stress necessary was more than the limiting shear stress,however, the applied shear stress was
limited to the maximum shear stress value, thus resulting in slipping at the interface.
A three-dimensional thermo-mechanical FE model was developed by Jain et al. [44]
to predict the material flow and forces in FSW. The model used Lagrangian formulation with a built-in feature of DEFORM-3D called point tracking tool to analyze
the velocity and material flow as shown in Fig. 3.8.
Sellars-Tegart law is also used to model steady-state flow stress. Sellars-Tegart
law [74] considers the material to be incompressible viscous non-Newtonian fluid.
It puts forward a relation between temperature T and rate of deformation ˙
ε by using
the Zener-Hollomon parameter as
117
where K wt and h t are the conductance and convective heat transfer coefficient, T t is
the tool temperature, and T is the workpiece temperature. The heat dissipation from
the top surface to the environment consists of both convective and radiation losses
expressed as.
Q wa = σ b ε b
T
4
− T
4
a
+ h b (T − T a ),
(3.90)
where Q wa is heat loss through top surface to surrounding, σ b is the Stefan–Boltzmann constant, h b is the heat transfer coefficient at the top surface, T a is the ambient
temperature, and ε b is the emissivity. The heat loss at the workpiece bottom surface
and the backing plate Q wb is modeled by considering a suitable convective heat
transfer coefficient with equivalent conductive heat transfer during experiment. The
heat dissipation to backing plate is expressed as
Q wb = K wb
∂ T
∂z
= h b (T − T b ),
(3.91)
where K wb and h b are the conductance and convective heat transfer coefficient for the
heat interaction of workpiece and the backing plate, and T b is the temperature of the
backing plate.
3.5.3 Material Flow Modeling
In order to obtain high structural efficiency of the welds and for optimal tool design,
it is necessary to understand the material flow. Computational fluid dynamics (CFD)
using commercial software Fluent was employed by Colegrove and Shercliff [21] to
model the material flow in FSW. They developed a ‘slip’ model where the local shear
stresses govern the interface conditions. It was assumed that the shear stress experienced by the workpiece is different for different tool materials. The surface shear
stress was assigned maximum limiting value. Two different boundary conditions were
used: (a) stick condition, where the shear stress was lower than the limiting shear
stress, and (b) slip condition, where for stick condition to exist, the shear stress necessary was more than the limiting shear stress,however, the applied shear stress was
limited to the maximum shear stress value, thus resulting in slipping at the interface.
A three-dimensional thermo-mechanical FE model was developed by Jain et al. [44]
to predict the material flow and forces in FSW. The model used Lagrangian formulation with a built-in feature of DEFORM-3D called point tracking tool to analyze
the velocity and material flow as shown in Fig. 3.8.
Sellars-Tegart law is also used to model steady-state flow stress. Sellars-Tegart
law [74] considers the material to be incompressible viscous non-Newtonian fluid.
It puts forward a relation between temperature T and rate of deformation ˙
ε by using
the Zener-Hollomon parameter as