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Abstract and Figures

In this paper, a multi-physics numerical model for multi-track-multi-layer laser powder bed fusion (L-PBF) process is developed and used for analysing the formation and evolution of porosities caused by lack of fusion and improper melting. The simulations are divided into two categories: first and foremost, a multi-physics thermo-fluid model in meso-scale, and second, a mechanical model based on the concept of a unit cell. The thermo-fluid model is used to track and observe the formation of the porosities, and considers phenomena such as multi-phase flow, melting/solidification, radiation heat transfer, capillary and thermo-capillary (Marangoni effect) forces, recoil pressure, geometry dependant absorptivity, and finally evaporation and evaporative cooling. The results for the investigated process parameters indicate that the porosities are mainly formed due to improper fusion of the particles. The probability of presence of pores is also observed to be higher in the first layers. Moreover, the lack of fusion zones are seen to become smaller in the subsequent layers, largely due to better fluid flow and higher temperatures in those layers. Based on the porosity levels determined from the thermo-fluid model, a unit cell mechanical model with an equivalent amount of porosity has been made and subsequently subjected to loading for analysing the part's mechanical behaviour. The unit cell results show that an increase in the porosity can highly affect and deteriorate the part's elastic modulus and its yield strength, as well. The combination of the thermo-fluid and the mechanical unit cell model establishes a direct link between process parameters and mechanical properties for L-PBF. INTRODUCION Laser powder bed fusion (L-PBF) method, is categorized as a metal additive manufacturing process [1], where the metallic parts are produced in a layer-wise manner. A simple view of a typical L-PBF machine is shown in Fig. 1. In this process, first a layer of fine spherical metallic particles, whose diameters typically span from approximately 10 µm to 60 µm [2], is distributed on a base plate (which in turn is attached to a build platform) by means of a controllable coating mechanism. Then, the laser starts scanning predefined locations based on the data provided in the CAD files [3]. While the laser scans these zones, the fine metallic particles will get coalesced together, either by being sintered or fully-molten, depending on
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MULTI-SCALE MULTIPHYSICS SIMULATION
OF METAL L-PBF AM PROCESS AND
SUBSEQUENT MECHANICAL ANALYSIS
M. BAYAT*, D. DE BAERE*, S. MOHANTY* AND J. H. HATTEL*
* Department of Mechanical Engineering, Technical University of Denmark, building 425, room 225, Lyngby, Denmark
ABSTRACT
In this paper, a multi-physics numerical model for multi-track-multi-layer laser powder bed fusion (L-PBF)
process is developed and used for analysing the formation and evolution of porosities caused by lack of fusion
and improper melting. The simulations are divided into two categories: first and foremost, a multi-physics
thermo-fluid model in meso-scale, and second, a mechanical model based on the concept of a unit cell. The
thermo-fluid model is used to track and observe the formation of the porosities, and considers phenomena
such as multi-phase flow, melting/solidification, radiation heat transfer, capillary and thermo-capillary
(Marangoni effect) forces, recoil pressure, geometry dependant absorptivity, and finally evaporation and
evaporative cooling. The results for the investigated process parameters indicate that the porosities are mainly
formed due to improper fusion of the particles. The probability of presence of pores is also observed to be
higher in the first layers. Moreover, the lack of fusion zones are seen to become smaller in the subsequent
layers, largely due to better fluid flow and higher temperatures in those layers. Based on the porosity levels
determined from the thermo-fluid model, a unit cell mechanical model with an equivalent amount of porosity
has been made and subsequently subjected to loading for analysing the part’s mechanical behaviour. The unit
cell results show that an increase in the porosity can highly affect and deteriorate the part’s elastic modulus
and its yield strength, as well. The combination of the thermo-fluid and the mechanical unit cell model
establishes a direct link between process parameters and mechanical properties for L-PBF.
Keywords: process-property correlation, multi-physics model, thermo-fluid model, multi-layer L-PBF,
porosity formation, unit cell approach.
INTRODUCION
Laser powder bed fusion (L-PBF) method, is categorized as a metal additive manufacturing
process [1], where the metallic parts are produced in a layer-wise manner. A simple view of a
typical L-PBF machine is shown in Fig. 1. In this process, first a layer of fine spherical metallic
particles, whose diameters typically span from approximately 10 µm to 60 µm [2], is
distributed on a base plate (which in turn is attached to a build platform) by means of a
controllable coating mechanism. Then, the laser starts scanning predefined locations based on
the data provided in the CAD files [3]. While the laser scans these zones, the fine metallic
particles will get coalesced together, either by being sintered or fully-molten, depending on
the intensity of the laser power. After the first layer is shaped, the build platform moves down
and another layer of powder is laid down. Subsequently, the powder feeding and laser
scanning continues until the whole part is manufactured [4].
L-PBF has many advantages over the conventional production methods (e.g. Casting, milling,
etc) and outweighs them in certain areas such as design freedom, material waste,
manufacturing time, etc [5]. However, some of the parts that are manufactured by this process,
still suffer from major defects, such as, denudation zones [6], keyhole porosities [7], surface
porosities [8] and voids [9].
Fig. 1 L-PBF machine along with its components; (left) cross section and (right) 3-D views.
Depending on the range of the processing parameters (e.g. scanning speed, laser power, hatch
distance, etc), the shape and dimensions of the defects might differ considerably. For example,
low linear energy density (LED) can result in surface defects as well as lack of fusion zones
with unsintered particles [8]. Adoption of high LED levels cause overheated sites that
ultimately lead to internal metallurgical or keyhole voids [10].
In the recent years, advanced online monitoring systems have been developed for analysing
the thermal maps [11], the solidification time as well as the keyhole evolution [12] during the
L-PBF process. However, these research methods, depending on their illumination source and
degree of complexity, can either be highly expensive or constitute high safety risks [13].
In the current scenario, developing a multi-physics numerical model for L-PBF process that
can predict the evolution of the internal pores can improve the predictability of the process to
a large extent. From a numerical point of view, L-PBF contains many physical phenomena,
such as: evaporation, melting/solidification, thermal radiation, laser absorption, liquid metal
flow, recoil pressure, etc. Numerical models have been developed to address these
phenomena, and depending on the complexity and governing physics, they can be categorised
into thermal models [14–16], thermo-fluid models [17–19], thermo-microstructural models
[20,21] and thermo-mechanical models [22]. In particular, thermo-fluid models are best suited
for observing the formation and evolution of porosity during the laser powder bed fusion
process. Such information regarding the distribution of voids/pores can further be utilized to
predict the mechanical response of the corresponding parts through the representative volume
elements and unit cells methods . Accordingly, by means of these approaches, one might be
able to investigate the effect of different patterns of voids or added elements on the overall
mechanical response of the parts [23,24].
In this paper, a multi-physics numerical model for the L-PBF process has been developed
which covers multi-phase flow, solidification/melting, evaporation, recoil pressure, capillary
forces, Marangoni effect, buoyancy effect, ray tracing method, radiation and flow in porous
medium. The formation of lack-of-fusion defects during the multi-track and multi-layer L-
PBF process has been investigated using said model. Moreover, to simulate the actual
stochastic distribution of powder layers, a discrete element method (DEM) has been
implemented that tracks the position of the powder particles during the powder-feeding step.
To consider the effects of surface elevation of the previous sintered/fused layers on the current
powder layer’s distribution, the results of thermo-fluid simulation of each layer have been
exported as CAD data for usage in the DEM simulations. Eventually, the calculated porosity
level for all layers has been used to make an equivalent unit cell to study the effect of the
porosity on the mechanical behaviour of the samples. An image analysis has been performed
on the fluid fraction data to calculate the porosity levels needed for unit cell model.
NUMERICAL MODEL
During the L-PBF process, usually a relatively high amount of power (50-1000 W) [2] is
concentrated on a small surface area, which can easily melt down and even evaporate the
powder layer. Physical phenomena such as recoil pressure, capillary and thermo-capillary
forces prevail in those harsh conditions and highly affect the material’s flow behavior. To
determine the temperature field and observe the material’s flow inside the melt regions, it
becomes necessary to solve the coupled partial differential equations of balance of mass, linear
momentum and energy together. This has been implemented in the thermo-fluid model
developed using the commercial software Flow-3D, which in-turn is based on the finite
volume method (FVM). Moreover, to better simulate the thermal interaction between laser
rays and the fluid (powder and base metal), a ray-tracing method along with a geometry-
dependant absorptivity has been implemented. In the following section of the paper, the
governing equations for the thermo-fluid model are first described and a detailed description
about the laser-material model is also given. Subsequently, the mechanical model for
analysing the material behaviour with the unit cell approach and its governing equations are
described. The mechanical model has been implemented on the commercial software
ABAQUS version 6.16 which works based on the finite element method.
GOVERNING PHYSICS: THERMO-FLUID MODEL
Fluid flow motion
The computational domain is divided into two immiscible phases i.e. the ambient air and
the metal regions (labelled as void and fluid respectively). Several methods are available in
literature to track the free surface of each phase, such as level-set [25], phase field and volume
of fluid (VOF) [26] in a fixed mesh (Eulerian framework) approach and Stephan’s problem in
a moving mesh (Lagrangian framework) approach. In this work, the VOF method has been
used for tracking the phase interface. According to the VOF method, the mass balance
equation for two-phase flow can be expressed as [26]:
()+
.
=
(1)
the subscript ( ) stands for the fluid phase, (
) is the density and  ( 
.) is the
volumetric rate of mass loss by evaporation.  shows the volume fraction of this fluid phase.
To solve equation (1), it is necessary to fulfil the condition that the summation of volume
fractions of all present phases equals unity:
+= 1
(2)
where the subscript ( ) demonstrates the void phase. This condition also holds an important
physical meaning that the cells must not become empty of a medium.
To determine the pressure-velocity field, the system of equations of balance of linear
momentum should be solved along with (1) and (2):

+
.
=
+++
(3)
is the velocity vector and the two vectors
and
stand for the volumetric forces caused
by the spatial variation in fluid stress tensor and solidification drag forces based on Darcy’s
law for porous medium [27]. The vector of the volumetric forces caused by the spatial
variation in internal stresses is defined as:
=
. []
(4)
where [] is the internal stress tensor and is defined as:
= 21
2,+,1
3,
(5)
 is the delta Dirac function. The last term in equation (5) will diminish due to the
incompressibility of the present fluid. Since the flow is two-phase, the average values for
viscosity and density are used, defined by means of a simple mixing rule:
=+
(6)
=+
(7)
To model the moving solidification front, the volumetric forces are defined to be highly
dependent on the liquid fraction of the fluid. These forces, also called the solidification drag
forces, are supposed to free and freeze the fluid flow depending upon the fluid liquid fraction
[28]:
=(1)
+.
(8)

. and [] are the Karman-Cozeny constants, which, depending on the application,
are in the range of 1010and 1010 respectively. Interestingly, the solidification
drag forces, according to equation (8), are in the opposite direction of the velocity vectors.
Thus, a decrease in liquid volume fraction dramatically increases these forces to an extent that
is sufficiently high to freeze the fluid motion.
The last term in equation (3) is the volumetric force which is being imposed on the fluid due
to the buoyancy effect. In this work, since the flow is assumed to be incompressible, the
Boussinesq approximation is used for calculating the buoyancy force:
=()
(9)
where (
) and are the expansion coefficient and the fluid’s liquidus temperature,
respectively, and (
) is the gravitational acceleration.
Heat transfer and solidification
The energy balance equation is applied to the computational domain and subsequently
coupled to the velocity field, to find the temperature field and the fluid’s fusion condition
(liquid or solid state):
()
 +
.
=
. (
)
(10)
 (
) is the volumetric heat sink due to the evaporative cooling of fluid and (
) is
called the sensible enthalpy that can be expressed as:
=+ 
 +
(11)
where  ( 
) and  (
) stand for the reference specific enthalpy of the fluid and the
latent heat of fusion, respectively. The enthalpy-porosity method, which has been widely used
for melting/solidification modelling [29], is also used in this study. It is assumed that, for the
current metal, liquid volume fraction is a linear function of temperature during the
solidification interval:
Since the problem is assumed to be a multi-phase flow, the average sensible enthalpy is used
in the heat balance equation, which can be determined by a mixing rule as:
=0
1
,<
,<<
,<.
(12)
=+
+
(13)
Recoil, capillary and thermo-capillary forces
During the L-PBF and welding processes, since high-density heat fluxes are imposed on
the fluid’s surface, elevated temperatures are attained which cause the material to boil and
evaporate. The vaporization of the metal will impart a negative force against the fluid surface,
largely due to momentum balance with the escaping vaporized metal, called as the recoil
pressure which in extreme cases can even lead to formation of deep keyholes. On the other
hand, localized elevated temperature fields will cause large thermal gradients which will
subsequently result in thermally-induced shear stresses on the exposed surfaces of the fluid
(Marangoni effect). Furthermore, due to the large deflections of the fluid surfaces, capillary
forces are also dominant and can affect the fluid motion, especially on the places where big
curvatures are formed.
In principal, the saturation pressure of a fluid can be determined by means of thermodynamic
relations and equations of state. However, empirical correlations can also be used to find the
saturation pressure with a good accuracy, such as the Clausius-Clapeyron equation:
 and in equation (14) are the atmospheric pressure and the fluid’s saturation
temperature (boiling temperature) at this pressure, respectively.  (
) is the latent heat of
vaporization and ( 
.) is the specific heat capacity of vaporized fluid at constant volume.
There have been a number of investigations on the recoil pressure, and in this study the recoil
pressure is assumed to be a function of saturation pressure [30]:
The recoil pressure will thus increase exponentially with increase in temperature beyond the
boiling point. According to the statistical thermodynamics, the evaporative mass rate can be
approximated as [18]:
where  and ( 
.) are the accommodation coefficient and the universal gas
constant respectively. The thermally-induced shear stresses can be expressed as:
In equation (17),
(
) shows the planar (tangent to the interface) temperature gradient of
the fluid and 
 (
.) represents the temperature dependency of the surface tension.
Furthermore, as mentioned before, since the fluid surface might not be smooth, the capillary
forces are also present:
()=exp 
(1) 1
1
,
(14)
()= 0.54(),
(15)
=.
2.(),
(16)
=
.
.
(17)
=..
(18)
In equation (18),
and
are the surface tension between the two phases and the
curvature of the free surface respectively. From a mathematical point of view, the mentioned
triple forces can be expressed in a single vector form as:
According to equation (19), the recoil and capillary effects act as an jump in the pressure on
the void/fluid interface and are always normal to the free surface [31]. On the other hand, the
thermally-induced shear stresses, as can be seen on the right side of equation (19), are tangent
to the free surface. Also, depending on the sign of temperature dependency of the surface
tension, the flow might be driven radially outward or inward.
is the void/fluid interface
normal vector and is calculated as:
is a scalar function which shows the elevation of the void/fluid interface. The curvature
mentioned in equation (20) can be determined by:
The mentioned forces are all imposed on the boundary, shown in Fig. 2, along with the
imposed thermal boundary conditions.
Fig. 2 Schematic view of the recoil pressure, capillary and thermo-capillary forces along with the
thermal boundary conditions on the Ω boundary.
Fig. 2 also shows an overall view of the present boundary conditions, where the top boundary
is assumed to be a pressure outlet boundary condition, and set to ambient atmospheric
pressure allowing both phases to escape the control volume if necessary.
/ ()=[+.]

(
.
)

(19)
=


(20)
=.
|
|=1
|
|
.
|
||
|(
.
)
(21)
Furthermore, the laser also irradiates the interface, so the thermal boundary condition on this
layer can be expressed as:
and (
.) are the fluid’s surface emissivity and the Stephan-Boltzmann constants
respectively. (
.) stands for the convective heat transfer coefficient of the void. 
in equation (22) is the heat flux of the laser ray “” after” number of recorded collisions.
Laser-material interaction
Depending on the application, there are several ways to model the effect of the laser on the
thermal behaviour of the material, such as moving point source, moving heat flux, moving
volumetric heat source, equivalent volumetric heat source based on optical penetration depth
(OPD) and ray tracing methods. In this work, to better simulate the actual laser-material
interaction, the ray-tracing method, along with a geometry-dependent absorptivity, are used.
The laser heat flux, which is being emitted from the focusing lens shown in Fig. 2, is assumed
to have a planar Gaussian distribution with the following mathematical function:
where is the beam radius and , show the position of the moving laser beam.
Depending on the resolution of the computational domain, the analytical expression of the
heat flux in equation (23) will be divided into a number of smaller parts while retaining the
total beam energy. For each incident ray”, the new direction of the reflected rays , for
the “” th collision is determined based on the ray-tracing method as follows:
The absorptivity of the laser ray at each collision is then assumed to be a geometry-dependant
cosine function of the incident angle in this work..
GOVERNING PHYSICS: MECHANICAL MODEL
The equilibrium equation is the principal governing partial differential equation which defines
the relationship between the applied forces and the material’s mechanical response,
in which  is the stress tensor in equation (25). The total strain tensor 
 can be decomposed
into elastic and plastic parts:
where ( ) , ( ) are subscripts standing for elastic and plastic behaviour respectively. The
elastic strain is calculated based on the general Hooke’s law:


+(
)+()+,= 0
(22)
(,)=2
exp 2(()+())
,
(23)
,=,2(,.
,)
,
(24)
.= 0,
(25)

=
+
,
(26)
(Pa) and are elastic modulus and Poisson’s ratio, respectively. The J2 flow theory defines
the mechanical behaviour of the material in the plastic zone. The corresponding plastic strain
increment is calculated as:
(Pa) is the tangent modulus of the material,  is the deviatoric stress tensor and is the
equivalent Mises stress.
METHODOLOGY
As already discussed in the introduction, the L-PBF process consists of two primary steps,
namely powder-laying and fusion. In this work, to determine the position of the solid particles
during the powder-laying step, a discrete element method (DEM) is used, as shown in Fig. 3.
After the DEM simulation, the CAD file containing the laid powder layer is extracted by an
in-house code and imported as a fluid region into the thermo-fluid model (fusion model). After
the thermo-fluid calculations, the CAD file of the fluid region, along with its stored thermal
data, are extracted and the former is fed to the next powder-laying calculation model where it
serves as a solid wall with a prescribed restitution coefficient. The latter is used as the initial
thermal and fluid boundary condition for the next thermo-fluid (fusion) model. This procedure
is repeated for three layers.

=
1 + 1
2++
12.
(27)

=9
41
1

.
(28)
=1
3,
(29)
=3
2.
(30)
Fig. 3 The methodology used for the powder-laying and fusion calculations.
To assess the influence of the lack-of-fusion induced porosities on the mechanical behaviour
of the manufactured samples, a unit cell approach is implemented in this work. This unit cell
method is used to study the effect of porosities on both the yield stress and the elastic modulus
of the samples. First, several cross-sections are made from the 3D CAD geometry extracted
from the thermo-fluid model, and the section contours (showing the value of fluid fraction)
are transformed into monochromic images. Subsequently, the images are processed by a
numerical code developed to determine the amount of porosity in these cross-sections.
Next, a simplified unit cell geometry, i.e. a square with 500 μm edge length, is constructed
with a hole( equivalent to the determined porosity) in the middle of the sample. This unit cell
is then subjected to the mechanical boundary conditions and loads shown in Fig. 4. The left
and bottom boundaries are x-symmetry and y-symmetry respectively, while a uniform x-
displacement is imposed on the right boundary in an incremental manner. While the cell is
being loaded, the average force on the right boundary is determined. At the same time, the
average strain is also calculated, which is defined as the ratio of the x-displacement to the
characteristic length of the unit cell.
Fig. 4. Mechanical boundary conditions for the unit cell.
RESULTS AND DISCUSSIONS
THERMAL FIELDS AND FLUID FLOW
The process parameters and the laser specifications used in this study are all gathered in Table
1. The temperature field during the scanning of first layer along with the melt region are both
shown in Fig. 5 for different times.
Table 1. Process parameters for the multi-layer simulation
Parameter
Value
Parameter
Value
Hatch spacing
100 µm
Scanning speed
1600 mm/s
Beam size
60 µm
Layer thickness
40 µm
Beam power
150 W
Scan length
1000 µm
According to Fig. 5(a), the maximum temperature zone is formed very close to the centre of
the moving laser beam. At 300 μs, most of the powder and base-plate are still at their initial
temperature of 300 K. Due to the relatively high scanning velocity of the laser and the presence
of air between the powders, the flow (propagation) of the heat in the transverse direction is
much slower. On the other hand, the thermal resistance of the material (at t > 300 μs) on the
right side of the scanning beam, is much lower than that of the unmelted powders due to the
dense morphology of the metal.
The melt region contour during the scanning of the first layer is also shown in Fig. 5(b). At t
= 860 μs, it can be clearly observed that the melt region of the two neighbouring tracks do not
reach each other and accordingly, lack of fusion sites and unsintered particles are formed
between the tracks.
Fig. 5 Thermal field and melt region during the scanning of the first layer of the powder.
Corresponding to Fig. 5(c) at t = 1400 μs, a hole is formed close to the position of the centre
of the laser beam. When the temperature in the region goes above the saturation temperature
of the metal, the recoil pressure becomes dominant, pushes the deformable liquid down and
makes a keyhole. The size of the keyhole increases with further increase in temperature due
to greater recoil pressure.
The melting of powder particles during L-PBF is caused either via direct contact with the laser
rays or via contact with a hot return flow formed due to the negative pressure gradient present
on the fusion front. Starting with the former pathway, when the laser irradiates the powder for
the first time, the powder begins to melt, as observed in Fig. 6 (a) and (b), and gains a high
flowability. Due to the presence of recoil pressure, the Marangoni effect and the
incompressibility of the flow, a large pressure is formed on this front (i.e. the melting front),
as shown in the pressure contours in Fig. 7 in the x-z plane.
Fig. 6. Fluid velocity magnitude and melt region for the first layer at (a) 1420 μs and (b) 1430 μs.
This high-pressure front formed ahead of the laser beam drives the hot fluid (metal) backwards
to the already molten zones (see Fig. 7 (a)), thus widening the melt track. As shown in Fig.
7, the coexistence of a high pressure zone in front of the melt pool and a low-pressure one at
its rear, forms a local pressure gradient which serves as a driver of hot metal flow back to the
relatively colder melted regions.
Fig. 7. Pressure front formed ahead of the laser beam for x-z plane.
TRACKING POROSITIES
Fig. 8. Fluid region, temperature contour and melt pool dynamics during the scanning of the second
layer at x=200 μm (transverse cross-section) for (a): 320 µs, (b) 540 µs, (c) 890 µs, (d) 1160 µs, (e)
1440 µs, (f) 1540 µs, (g) 1960 µs, (h) 2050 µs, (i) 2110 µs.
To track how the lack-of-fusion porosities are formed and how they evolve, multiple cross-
sections are made in different planes from the 3D domain. Fig. 8 shows the fluid region along
with its thermal field during the scanning of the second layer at different times on one of the
transverse cross-sections (with respect to direction of laser movement). At t = 320 μs, the
centre of the laser beam reaches the position of the cross-section, and correspondingly the
highest temperatures are observed.
At t = 540 μs, when the laser is farther away from this plane, the melt pool starts to cool down
and solidify at the same time. At t = 890 μs, the laser again reaches the plane while scanning
the second track. The keyhole formed due to presence of recoil pressure becomes sufficiently
deep to capture some of the pores from the previous layer.
At t = 1160 μs, subsequent to the collapse of the keyhole walls and solidification of the melt
region, another pore is observed to have been formed at the top right side of the track.
Simultaneously, some of the pores initially formed at t = 890 μs are now observed to have
shrunk in size. At t = 1960 μs and 2110 μs, a number of pores are also formed due to the air
being trapped below the melt pool. Although these pores have a strong potential to escape the
fluid region (because of large density-driven buoyancy force), they cannot move upwards as
they are entrapped in either the mushy or the solid zone.
On the other hand, there are a number of occasions where the pores formed during the scanning
of the previous layers coalesce or escape the fluid surface, as shown in Fig. 9.
Fig. 9. Fluid region, temperature contour and melt pool dynamics during the scanning of the second
layer at y=200 μm (longitudinal cross-section) at: (a) 730 µs, (b) 790 µs, (c) 830 µs and (d) 1010 µs.
Fig. 9 also shows the temperature contour, melt and fluid region in the x-z longitudinal cross-
section at y=200 μm. Although the particles are molten by the laser scanning, the keyhole is
not formed at this position since the cross-section is almost 50 μm away from the hotspots
formed closer to the center of the beam. At t = 730 μs, there are still a number of remnant
pores from the previous layer where the entrapped air could not escape the fluid surface a
phenomena which is again observed at t = 790 μs since the melt region could not penetrate
deep enough to capture the void between particles. At t = 830 μs, the pore which had been
shaped at t =790 μs is observed to escape the fluid surface. By t = 1010 μs, when the laser has
finished scanning the second track, some of the pores trapped from the first layer have escaped
the melt region.
To demonstrate the entire scanned block’s porosity profile, the fluid region along with its
temperature contour are depicted on a number of cross-sections in Fig. 10 at the end of the
process for all three layers. A shown in Fig. 10, the pores mostly have irregular shapes, with
round edges and sizes spanning between 5 μm to 40 μm. Moreover, the first layer has more
lack of fusion zones compared to the second and third layers.
Fig. 10. Fluid region, temperature contour and melt pool dynamics for the whole three layers, at the
end of the fusion process for different cross-sections at x: (a) -90 µm, (b) -70 µm, (c) -50 µm, (d) -
44 µm, (e) -20 µm and (f) +94 µm.
Furthermore, as observed in Fig. 10 (b), (e) and (f), the pores are mostly created in the area
between the melting tracks and stack above each other in a pattern, especially on the lower
levels. This stacking pattern of pores is a well-known outcome observed in samples made
without rotation or shifting of scanning strategies between successive layers. Based on the
image analysis performed on different cross-sections, the porosity due to the lack of fusion is
found to be in the range of 0.43 % to 2.54 %.
MECHANICAL ANALYSIS: UNIT CELL
The mechanical response of the unit cell for the upper and lower bounds of porosities are
depicted in Fig. 11 and indicate that the porosities have a big influence on the mechanical
behaviour of the samples. As seen in Fig. 11, the increase in porosity can reduce the stiffness
of the material from a value of 456.7 GPa to 403.2 GPa. The yield limit of the part is affected
by the imposed porosities as well. declining from 913 MPa for the unit cell with 0.43 %
porosity to 725.8 MPa at 2.54 % porosity. Consequently, these results show that both stiffness
and yield limits of the additive manufactured parts can be highly deteriorated because of the
presence of even a small amount of porosity.
UNIT CELL AND MECHANICAL ANALYSIS
Fig. 11. Calculated average stress-strain curves for the unit cell for different porosities.
CONCLUSION
In this work, a multi-physics numerical model for the L-PBF process is developed which
accounts for multi-phase flow, fluid flow in porous medium, capillary and thermo-capillary
forces, solidification and melting, particle-particle interaction, geometry-dependant
absorptivity (via ray-tracing method), radiation, , recoil pressure, evaporation and evaporative
cooling. The model also implements surface tracking algorithms to capture the evolution of
the voids and porosities, which are mainly caused by the lack of fusion zones. The results
show that a steep pressure gradient prevails on the melt pool front which pumps the fluid back
into the molten regions. Furthermore, the results suggest that the lack of fusion defects are
mostly found in the lower layers, where the thermal energy of the fluid is lower, compared to
higher levels. A detailed discussion about the formation of the pores is also given, which
reveals that the pores might form, shrink, coalesce and disappear during the course of the
process. Moreover, it is found that most of the pores are formed in the lower levels, where
more lack of fusion defects are present. The paper also presents a unit cell model developed
to capture the effect of the porosities on the mechanical properties, where it is found that an
increase in porosity, decreases both the elastic modulus and the yield strength of the material
at the same time. Through the combination of the thermo-fluid model and the mechanical unit
cell model, a direct link between process parameters of L-PBF and the emergent mechanical
properties of the component has been established. Further developments of the meso-scale
thermo-fluid model by inclusion of the plasma plume formation, and by tracking the
movements of particles and metal droplets dragged by the vapour/gas flow, would lead to
better prediction of the surface of the part being manufactured by L-PBF as well as allow a
more accurate capturing of the keyhole voids formation/evolution. The usage of representative
volume elements to model the mechanical response of the L-PBF parts with porosity will also
potentially lead to better results. Further, adding a metallurgical model able to predict phase
fractions of the metallic alloy, would provide greater inputs to the mechanical analysis and
complete the chain of process-materials-property.
APPENDICES AND ACKNOWLEDGEMENTS
This work has received funding from the European Union Horizon 2020 Marie
Skłodowska_Curie ITN PAM^2 project under grant agreement number 721383.
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