98
N. Bhardwaj et al.
curve of the material is used to determine these constants by fitting the function to
the curve.
In case of kinematic hardening, with increase in plastic strain, the yield locus
translates in the direction of incremental plastic strain. The kinematic hardening
model also includes the Bauschinger effect. For subsequent yielding, yield criterion
is represented as
f (σ i j − α i j ) = 0,
(3.14)
where α i j is called the back stress. It denotes the yield locus’s incremental translation
while retaining its shape Fig. 3.2b. After implementing kinematic hardening, the
yield criterion for further yielding according to the maximum distortion energy yield
criterion is expressed as
σ
i j − dα i j
σ
i j − dα i j
−
2
3
σ
2
Y = 0,
(3.15)
where dα i j represents the incremental back stress. According to Prager’s hardening
law, translation of yield locus occurs in the direction of plastic strain. However,
Ziegler refined the hardening model by stating that the yield locus translates along
the line joining the stress tensor and the center of the yield locus. Ziegler’s hardening
law in linear form is given as
dα i j =
H
σ Y
σ i j − α i j
dε
p
eq ,
(3.16)
where H
is the derivative of stress with respect to equivalent plastic strain,
σ i j − α i j
is a vector along the incremental translation, and incremental equivalent plastic strain
dε
p
eq is expressed as
dε
p
eq =
2
3
dε
p
ij dε
p
ij .
(3.17)
Figure 3.2c shows a schematic of combination of isotropic and kinematic
hardening, often called as combined hardening.
Two different approaches are taken when formulating the governing equations of
a mathematical model for evaluating stresses, strains, displacement or other parameters. One is the Lagrangian approach, in which the region of analysis keeps changing
continuously; the other is the Eulerian approach where analysis is performed in
a fixed region of space. Eulerian formulation assumes a fixed space known as the
control volume, where the entire analysis takes place. It is assumed that the deforming
material flows through the control volume behaving like a non-Newtonian fluid. This
approach is also referred to as flow formulation to imply the flowing nature of the
deformed material. It is essential to compute the material boundary at each time increment, since material and element boundaries in Eulerian approach do not correspond
N. Bhardwaj et al.
curve of the material is used to determine these constants by fitting the function to
the curve.
In case of kinematic hardening, with increase in plastic strain, the yield locus
translates in the direction of incremental plastic strain. The kinematic hardening
model also includes the Bauschinger effect. For subsequent yielding, yield criterion
is represented as
f (σ i j − α i j ) = 0,
(3.14)
where α i j is called the back stress. It denotes the yield locus’s incremental translation
while retaining its shape Fig. 3.2b. After implementing kinematic hardening, the
yield criterion for further yielding according to the maximum distortion energy yield
criterion is expressed as
σ
i j − dα i j
σ
i j − dα i j
−
2
3
σ
2
Y = 0,
(3.15)
where dα i j represents the incremental back stress. According to Prager’s hardening
law, translation of yield locus occurs in the direction of plastic strain. However,
Ziegler refined the hardening model by stating that the yield locus translates along
the line joining the stress tensor and the center of the yield locus. Ziegler’s hardening
law in linear form is given as
dα i j =
H
σ Y
σ i j − α i j
dε
p
eq ,
(3.16)
where H
is the derivative of stress with respect to equivalent plastic strain,
σ i j − α i j
is a vector along the incremental translation, and incremental equivalent plastic strain
dε
p
eq is expressed as
dε
p
eq =
2
3
dε
p
ij dε
p
ij .
(3.17)
Figure 3.2c shows a schematic of combination of isotropic and kinematic
hardening, often called as combined hardening.
Two different approaches are taken when formulating the governing equations of
a mathematical model for evaluating stresses, strains, displacement or other parameters. One is the Lagrangian approach, in which the region of analysis keeps changing
continuously; the other is the Eulerian approach where analysis is performed in
a fixed region of space. Eulerian formulation assumes a fixed space known as the
control volume, where the entire analysis takes place. It is assumed that the deforming
material flows through the control volume behaving like a non-Newtonian fluid. This
approach is also referred to as flow formulation to imply the flowing nature of the
deformed material. It is essential to compute the material boundary at each time increment, since material and element boundaries in Eulerian approach do not correspond