Thixoforming : Semi-solid Metal Processing

174j 6 Modelling the Flow Behaviour of Semi-solid Metal Alloys
Dk
¼ cð _
Dt g Þ keð ½ g_ Þ kðtÞŠ
ð6:5Þ
where c is the rate constant of approaching the equilibrium value. The rate constant
appears from the experiments to be lower for high shear rates and is modelled as
b _
c ¼ ae g
ð6:6Þ
6.1.3
Approach from Solid-state Mechanics
The essential properties of a material that behaves as a plastic are the existence of a
yield point and the occurrence of permanent deformation when the stresses have
reached the material s yield point. The simplest description of the plastic behaviour is
expressed by the elastic–ideal plastic model (Prandtl and Reuss) and rigid–ideal
plastic (Levy and von Mises) material model. Based on these constitutive equations in
the metal forming the elastic-plastic also the viscoplastic constitutive equations are
used, which are primarily based on the plasticity theory of von Mises. By modifying
these models, the processes with work hardening or softening effects can also be
calculated.
The ideal plastic constitutive equation of Levy and von Mises describes the
correlation between stress and deformation by a linear relationship between the
strain rate _eij and the deviatory stress sij:
_eij ¼ _ ffiffiffiffiffiffiffiffiffiffiffiffi
l sij with l_ 1 3
¼
2 _eij
r
_eij
t0
ð6:7Þ
The location-dependent proportionality factor _ l is a function of all components of
the strain rate tensor _e and the yield stress t0. The yield stress is described by the
so-called flow curve and depends on the current state of deformation, strain rate
and temperature.
Since the yield stress is defined for the uniaxial stress condition, but in most real
processes a triaxial stress condition is present, an equivalent stress sV is calculated
from the stress tensor. This equivalent stress can be compared with the yield stress. If
the equivalent stress reaches the value of the yield stress (sV ¼ t0) at a given
equivalent strain eV and equivalent strain rate _eV, plastic deformation occurs.
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sV ¼ ð Þ 2 þ ðs22 s33Þ
2 þ ðs33 s11Þ
2
h i
þ 3 t 2
r
12 þ t 2
23 þ t 2
31
1
2 s11 s22
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
_e V ¼
2
3
_e 2
11 þ _e 2
22 þ _e 2 1
33 þ
2 _
s
g 2
12 þ g_ 2
23 þ g_ 2
31
eV ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
e 2
11 þ e 2
22 þ e 2 1
33 þ g 2
12 þ g 2
23 þ g 2
31
3 2
ð6:8Þ
ð6:9Þ
ð6:10Þ