a coarser mesh away from
these zones (Figure 2.a).
The level of refinement
shown in Figure
321
Modelling of Heat Transfer
and Phase Transformations
in the Rapid Manufacturing
of Titanium Components
8 Will-be-set-by-IN-TECH
2.a is necessary if certain
aspe
the 20th layer, where a
beam power of 50 W was
reached and kept constant
for the rest of the
process. An average
absorptivity of 15 % was
considered in the
calculations, according to
the
results of Hu et al. (Hu &
Baker, 1999) regarding the
laser depositi
allowing for a high stability
and dimensional accuracy
in the manufacture of the
parts. The
deposition was conducted
using a Ti-6Al-4V powder
with a particle size in the
range 25-75
m fed through a capillary
at a mass flow rate of 0.14
g/min.
(a) (b)
Fig.
(a) (b)
Fig. 8. (a) Cooling rates
experienced by the
material deposited in the
different layers. (b)
Temperature evolution of
the material deposited in
the first layer for the first
120 s of the
fabrication process.
Tempering of the
martensite takes place
component of arbitrary
geometry can be found by
implementing Equation 16
as a computer
code.
2.3 Representation of the
physical domain
A finite element model
should ideally describe the
geometry of the substrate
and the tracks
as closely as possible.
Freq
12.a). The phase results
primarily
330 Convection and
Conduction Heat Transfer
Modelling of Heat Transfer
and Phase Transformations
in the Rapid Manufacturing
of Titanium Components 17
from the tempering of
martensite, which is a slow
process when compare
stainless steel or CoCr
alloys, Ti-6Al-4V allows the
production of much
stronger, lighter and
less stiff implants and with
improved biomechanical
behaviour.
Ti-6Al-4V is an / titanium
alloy that contains 6% of
the -phase stabilising
element Al,
and 4% of
hardness.
The variation of the
volume fraction of along
the height of the wall is
small but compares
well with the values
predicted by the model.
The calculated Youngs
modulus and
hardness show an overall
good correlation with the
experimental values and
(T0) (19)
where k0 and n0 are the
reaction rate constant and
Avrami exponent at the
temperature T0,
respectively. In the next
interval, [t1, t2[, the
transformation is assumed
to take place at the
temperature T1, but one
must take into
consideration the
proportion of (Figure 3).
Thus, the martensite
volume fraction is given
by:
f
_ (T) = f
_ (T0) + ( f(T0) fr) [1
exp (Ms T)] , (26)
with f
_ (T0) the volume fraction
of
_
phase present in the alloy
prior to quenching.
Similar phase
transformations will o
solution for the heat
transfer problem.
In the model proposed in
this chapter, Equation 16 is
solved iteratively for each
element in
the step by step approach
described in the previous
section. Addition of
material is taken
into account by activating
at e
mm/s). For this scanning
speed the cooling rates are
much higher than the
critical cooling
rate for the martensitic
transformation (410 C/s)
and asymptotically
approach a limit value
between 1500 C/s (for t
= 2 s) and 1900 C/s (for t
> 30 s) as the number
are 117, 82 and 114 GPa
respectively and the
Vickers hardnesses are
320, 140 and
350 HV.
326 Convection and
Conduction Heat Transfer
Modelling of Heat Transfer
and Phase Transformations
in the Rapid Manufacturing
of Titanium Components 13
4. Results
4.1 E
and Phase Transformations
in the Rapid Manufacturing
of Titanium Components
18 Will-be-set-by-IN-TECH
direction of the substrate
and on its mid plane so
that a symmetry plane
exists and only half
of the geometry needs to
be considered for
calculation purp
(Equation 18). The values
of k and
n in Equation 18 for this
reaction were determined
by Mur et al. (Mur et al.,
1996). If the
decomposition is
incomplete, tempering
results in a three-phase
microstructure consisting
of
_
+ + .
3.3 Phase transformations
d
elements for the proper
representation of the 3-D
features of the tracks, as
shown in Figure
2.a.
1
2
3
(a) (b)
Fig. 2. (a) Finite element
mesh of substrate and
tracks. (b) Step-wise
approach to simulate the
addition of material. New
elements are activate
of 130 W which can be
focused to a spot of 0.3
mm in diameter by means
of a ZnSe lens with
a focal length of 63.5 mm.
The system employs a
closed loop online control
system whereby
329
Modelling of Heat Transfer
and Phase Transformations
in the Rapid Manu
of reducing considerably
the number of elements in
the mesh, and as a
consequence the
number of calculations and
the computational time
necessary to resolve the
problem. Several
authors have developed
finite element models
which use simple cubic
elements
Fig. 20. (a) Contour plot of
the cooling rate as a
function of v and t for
Tsub = 20C. The
cooling rate is mostly
dependent on the scanning
speed, as evidenced by the
constant cooling
rate lines being almost
vertical. (b) Contour plot
showing the dependen
and decomposes by a
martensitic
transformation. The
proportion of
transformed
into martensite (
_
) depends essentially on
the undercooling below the
martensite start
temperature (Ms) and is
given by (Koistinen &
Marburger, 1959):
f
_ (T) = 1 exp [(Ms
T
a microstructure composed
of 0.92 + 0.08 in this
region (Figure 18.b). As a
result, the final
part presents a nonuniformdistribution of
hardness, 350 HV in the
bottom layers and 305 HV
333
Modelling of Heat Transfer
and Phase Transformations
in the Rapid
interaction zone, a lower
temperature gradient
in the build direction slows
down the heat flow,
causing a reduction of the
cooling rate which
is approximately given by:
T
t
=
k
cp
2T
x2 , (27)
where xx_ is the build-up
(vertical) direction. Figure
18.a sh
3.2 Phase transformations
during re-heating
When new layers are
added to the part, the
previously deposited
material undergoes
heating/cooling cycles that
may induce microstructural
and properties changes. If
the
microstructure formed in
first thermal cyc
the 15th layer, as depicted
in the
plot of Figure 9.a.
This facilitates tempering
because, as heat
accumulates in the part,
the material residence
time in the tempering
temperatures range
increases from less than 1
s in the first cycles to
approximately 4
deposition of consecutive
layers
(t) and substrate
temperature (Tsub) to the
microstructure, hardness
and Youngs modulus
distributions in parts
produced by laser powder
deposition. A summary of
these results
has been published
elsewhere (Crespo & Vilar,
2
titanium alloys worldwide,
a market estimated at
more than $2,000 million
(Leyens & Peters,
2003). This predominance
is mainly due to Ti-6Al-4V
having the best all-around
mechanical
characteristics for
numerous applications.
This alloy is extensively
used
the final microstructure
consists of and
because the
transformation does not
reach
completion. In isothermal
condition the kinetics of
this transformation is
described by the
Johnson-Mehl-Avrami (JMA)
equation:
f (t) = 1 exp (jtn) , (18)
where f(t), k an
is called the -transus
temperature (Polmear,
1989). The proportion of
phase in equilibrium
depends on the
temperature, varying from
approximately 0.08 at
room temperature to 1.00
at
the -transus, and is given
by (R. Castro, 1966):
f eq
(T) =
_
0.925
0.
334 Convection and
Conduction Heat Transfer
Modelling of Heat Transfer
and Phase Transformations
in the Rapid Manufacturing
of Titanium Components 21
ab
Fig. 18. (a) Cooling rate
experienced by the
material deposited in the
last layer as a function
of the
asymptotically approaches
a value below the
martensite critical cooling
rate (410 C/s).
Therefore, the deposition
of additional layers would
likely lead to the
suppression of the
martensitic transformation
in the top layers of the
part.
The thermal cycles