3 Modeling of Friction Stir Welding Processes
95
3.3 Basic Governing Equations of the Process
As discussed earlier, modeling is an important aspect of analysis of a process in order
to gain understanding as well as for optimization of the process. Process modeling and
optimization help reduce expensive and time-consuming experiments. Two aspects
should be modeled in FSW—plastic deformation and heat transfer. This section
discusses both the aspects.
3.3.1 Modeling of Plastic Deformation
In order to formulate plastic deformation problems, the governing equations used
are discussed in this section. When a body undergoes plastic deformation, it retains
the deformed configuration after removal of load and cannot go back to its original form. Unlike the elastic region, the stress–strain relationship does not follow
a linear relation in plastic region. The material behavior is also different in case of
loading and unloading. It is generally observed that for small deformation, a material behaves elastically at the initial stage, and with further deformation, the material
behaves plastically. A yield criterion is set to determine the end of elastic behavior
and onset of plastic behavior. For continued further plastic deformation after (initial)
yield point, additional stress is required suggesting that the yield condition keeps on
changing with plastic deformation. This is known as strain hardening. A criterion for
subsequent yielding is required for modeling the hardening behavior. An accurate
definition of material behavior is required for mathematical modeling of plastic analysis. Mathematically, yield criterion, in the form of a scalar function, is represented
as
f (σ i j ) = 0,
(3.1)
where f is a yield function, and σ ij are Cauchy stress tensor components. Two
commonly used yield criteria for representing material behavior in the context of
FSW are discussed, viz. von Mises and Tresca yield criteria [26]. Along with yield
criteria, hardening associated with plastic deformation is also discussed.
3.3.1.1 The Maximum Distortion Energy Yield Criterion
The maximum distortion energy (or von Mises) yield criterion is represented as
f (σ i j ) ≡ 3J 2 − σ
2
Y = 0,
(3.2)
where σ Y is the yield stress, and J 2 is an invariant of the Cauchy stress tensor’s
deviatoric component [27]. J 2 is defined as
95
3.3 Basic Governing Equations of the Process
As discussed earlier, modeling is an important aspect of analysis of a process in order
to gain understanding as well as for optimization of the process. Process modeling and
optimization help reduce expensive and time-consuming experiments. Two aspects
should be modeled in FSW—plastic deformation and heat transfer. This section
discusses both the aspects.
3.3.1 Modeling of Plastic Deformation
In order to formulate plastic deformation problems, the governing equations used
are discussed in this section. When a body undergoes plastic deformation, it retains
the deformed configuration after removal of load and cannot go back to its original form. Unlike the elastic region, the stress–strain relationship does not follow
a linear relation in plastic region. The material behavior is also different in case of
loading and unloading. It is generally observed that for small deformation, a material behaves elastically at the initial stage, and with further deformation, the material
behaves plastically. A yield criterion is set to determine the end of elastic behavior
and onset of plastic behavior. For continued further plastic deformation after (initial)
yield point, additional stress is required suggesting that the yield condition keeps on
changing with plastic deformation. This is known as strain hardening. A criterion for
subsequent yielding is required for modeling the hardening behavior. An accurate
definition of material behavior is required for mathematical modeling of plastic analysis. Mathematically, yield criterion, in the form of a scalar function, is represented
as
f (σ i j ) = 0,
(3.1)
where f is a yield function, and σ ij are Cauchy stress tensor components. Two
commonly used yield criteria for representing material behavior in the context of
FSW are discussed, viz. von Mises and Tresca yield criteria [26]. Along with yield
criteria, hardening associated with plastic deformation is also discussed.
3.3.1.1 The Maximum Distortion Energy Yield Criterion
The maximum distortion energy (or von Mises) yield criterion is represented as
f (σ i j ) ≡ 3J 2 − σ
2
Y = 0,
(3.2)
where σ Y is the yield stress, and J 2 is an invariant of the Cauchy stress tensor’s
deviatoric component [27]. J 2 is defined as