120
N. Bhardwaj et al.
for precipitate dissolution, and the gas constant is denoted by R. The thermal history
of the process is discretized into N total small steps where equivalent time for each
step is calculated followed by summation of all the time steps to arrive at the total
equivalent time. The fraction of hardening precipitates f
f 0 is expressed as.
f
f 0
= 1 − X d = 1 − t
n
eq ,
(3.100)
where the value of n is usually considered as 0.5 [39]. Linear interpolation between
original and fully dissolved state is used to predict the final hardness distribution as
H V = (H V max − H V min )
f
f 0
+ H V min ,
(3.101)
where HV max and HV min are hardness of the material at fully hardened and softened (original) conditions, respectively. The modeling of mechanical response and
hardness accompanied by the thermal modeling was used for prediction of residual
stresses by Feng et al. [30]. The prediction of residual stresses improved significantly
with inclusion of metallurgical model under hot welding conditions [2].
Apart from the thermal and elasto-plastic mechanical models, a model to analyzing
the microstructure of the welded material is also important. Coupling of metallurgical
model improves the efficiency of the model at predicting the evolution of mechanical
properties. The next section discusses modeling of microstructure during FSW.
3.5.5 Cellular Automata Modeling of FSW
A cellular automata (CA) model uses an algorithm to represent discrete spatial and/or
temporal evolution of a complex system by application of deterministic/probabilistic
transformation rules to the lattice locations [62]. Cellular automata (CA) models
are generally used to relate the initial microstructure to evolving microstructure
during materials processing techniques such as rolling, extrusion, welding and casting
solidification. It is a collection of ‘colored’ cells on a grid of specified shape that
evolves through a number of discrete time steps according to a set of rules based
on the states of neighboring cells. The rules are applied iteratively for as many time
steps as required. Figure 3.9 shows a schematic of the working of CA with a set
of simple rules as an example. Often, CA is combined with finite element method
(FEM) to provide robust simulation methodology called the cellular automata finite
element (CAFE) method. In this method, CA cells containing properties of microfeatures (like dislocation density, initial grain size) are connected to the integration
point of elements that are once again related to macro-, micro-outputs (like stress,
strain, etc.).
For several years, the applications of CAFE models include modeling mixed
microstructures [23], evolution of microstructure during rolling [90], effect of
N. Bhardwaj et al.
for precipitate dissolution, and the gas constant is denoted by R. The thermal history
of the process is discretized into N total small steps where equivalent time for each
step is calculated followed by summation of all the time steps to arrive at the total
equivalent time. The fraction of hardening precipitates f
f 0 is expressed as.
f
f 0
= 1 − X d = 1 − t
n
eq ,
(3.100)
where the value of n is usually considered as 0.5 [39]. Linear interpolation between
original and fully dissolved state is used to predict the final hardness distribution as
H V = (H V max − H V min )
f
f 0
+ H V min ,
(3.101)
where HV max and HV min are hardness of the material at fully hardened and softened (original) conditions, respectively. The modeling of mechanical response and
hardness accompanied by the thermal modeling was used for prediction of residual
stresses by Feng et al. [30]. The prediction of residual stresses improved significantly
with inclusion of metallurgical model under hot welding conditions [2].
Apart from the thermal and elasto-plastic mechanical models, a model to analyzing
the microstructure of the welded material is also important. Coupling of metallurgical
model improves the efficiency of the model at predicting the evolution of mechanical
properties. The next section discusses modeling of microstructure during FSW.
3.5.5 Cellular Automata Modeling of FSW
A cellular automata (CA) model uses an algorithm to represent discrete spatial and/or
temporal evolution of a complex system by application of deterministic/probabilistic
transformation rules to the lattice locations [62]. Cellular automata (CA) models
are generally used to relate the initial microstructure to evolving microstructure
during materials processing techniques such as rolling, extrusion, welding and casting
solidification. It is a collection of ‘colored’ cells on a grid of specified shape that
evolves through a number of discrete time steps according to a set of rules based
on the states of neighboring cells. The rules are applied iteratively for as many time
steps as required. Figure 3.9 shows a schematic of the working of CA with a set
of simple rules as an example. Often, CA is combined with finite element method
(FEM) to provide robust simulation methodology called the cellular automata finite
element (CAFE) method. In this method, CA cells containing properties of microfeatures (like dislocation density, initial grain size) are connected to the integration
point of elements that are once again related to macro-, micro-outputs (like stress,
strain, etc.).
For several years, the applications of CAFE models include modeling mixed
microstructures [23], evolution of microstructure during rolling [90], effect of