Fundamental consolidation mechanisms during selective beam

Modelling and Simulation in Materials Science and Engineering

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Fundamental consolidation mechanisms duringselective beam melting of powdersTo cite this article Carolin Koumlrner et al 2013 Modelling Simul Mater Sci Eng 21 085011

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IOP PUBLISHING MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING

Modelling Simul Mater Sci Eng 21 (2013) 085011 (18pp) doi1010880965-0393218085011

Fundamental consolidation mechanisms duringselective beam melting of powders

Carolin Korner Andreas Bauereiszlig and Elham Attar

University of Erlangen Materials Science Department Martensstr 5 91058 Erlangen Germany

E-mail carolinkoernerwwuni-erlangende

Received 18 February 2013 in final form 30 August 2013Published 8 November 2013Online at stacksioporgMSMSE21085011

AbstractDuring powder based additive manufacturing processes a component is realizedlayer upon layer by the selective melting of powder layers with a laser or anelectron beam The density of the consolidated material the minimal spatialresolution as well as the surface roughness of the resulting components arecomplex functions of the material and process parameters So far the interplaybetween these parameters is only partially understood

In this paper the successive assembling in layers is investigated with arecently described 2D-lattice Boltzmann model which considers individualpowder particles This numerical approach makes several physical phenomenaaccessible which cannot be described in a standard continuum picture eg theinterplay between capillary effects wetting conditions and the local stochasticpowder configuration In addition the model takes into account the influenceof the surface topology of the previous consolidated layer on the subsequentpowder layer

The influence of the beam power beam velocity and layer thickness onthe formation and quality of simple walls is investigated The simulationresults are compared with experimental findings during selective electron beammelting The comparison shows that our model although 2D is able to predictthe main characteristics of the experimental observations In addition thenumerical simulation elucidates the fundamental mechanisms responsible forthe phenomena that are observed during selective beam melting

(Some figures may appear in colour only in the online journal)

Content from this work may be used under the terms of the Creative Commons Attribution30 licence Any further distribution of this work must maintain attribution to the author(s) andthe title of the work journal citation and DOI

0965-039313085011+18$3300 copy 2013 IOP Publishing Ltd Printed in the UK amp the USA 1

Modelling Simul Mater Sci Eng 21 (2013) 085011 C Korner et al

1 Introduction

Beam and powder based additive manufacturing methods are relatively novel technologiesthat can build parts in layers from powder material by local melting with an electron or laserbeam [1] Examples of commercialized selective beam melting (SBM) processes are selectivelaser beam melting (SLM) and selective electron beam melting (SEBM) During SLM orSEBM the surface of a powder bed is selectively scanned with a beam energy source to heatand eventually melt the powder Thin molten tracks develop and are combined to form a 2Dlayer of the final part After completion of one layer the whole powder bed is lowered about 20to 100 microm and a fresh powder layer is spread out on the building zone The selective meltingprocess is repeated until the component is completed

Generally for different materials different powder consolidation mechanisms are operative[2 3] For metal powders melting and re-solidification are the underlying mechanisms toconsolidate the powder particles to form a functional part Typical material defects associatedwith SLMSEBM are residual porosity not molten powder or not fully connected layers Inaddition SLMSEBM suffer from a high surface roughness which is much larger than expectedfrom the mean powder particle diameter Finding state-of-the-art processing parameters for anew material is still a trial-and-error practice since a deep understanding of the consolidationprocess is yet incomplete

SLMSEBM is rather complex and involves many different physical phenomena [4]absorption of the beam in the powder bed melting and solidification the dynamics of the meltpool wetting of solid powder particles by the melt diffusive and radiative heat conductionwithin the powder bed diffusive and convective heat conduction in the melt pool capillaryeffects gravity etc The melt pool generated by the beam is highly dynamic The movementis driven by the high surface tension in combination with the low viscosity of liquid metalsAs a result the consolidated surface shows a stochastic nature ie it is irregular and lookscorrugated After finishing one layer a new powder layer is applied on the corrugated surfaceleading to a new powder layer with strongly varying thickness which might result in the typicalmaterial defects mentioned above

Finite element or finite difference methods based on a homogenized picture have beenused in order to develop a better understanding of the underlying consolidation process [5ndash12]These approaches are suitable for modeling the global temperature field and energy dissipationbut are unsuitable to reproduce stochastic powder effects like wetting variations of the localdensity of the powder bed dynamic shrinkage etc The homogenized approaches alwayspredict well defined melt pool geometries without the stochastic behavior that is experimentallyobserved [9] since they are not considering individual powder particles That is the existingnumerical approaches are of little help to understand binding errors or rough surfaces and theinfluence of the process and material parameters on the consolidation process

The aim of this paper is to gain a much better understanding of the fabrication processin layers during SLMSEBM In contrast to other numerical approaches in the literature ourapproach is based on a lattice Boltzmann model [13ndash16] where the effect of individual powderparticles is considered [17] The beam is absorbed by the powder particles which are heatedand eventually become molten After solidification a new stochastic powder layer is appliedon the rough surface consisting of powder and locally consolidated regions

The paper is organized as follows after a short introduction into the physical and numericalmodel the algorithm for the generation of stochastic powder layers on a stochastic layer isexplained in detail in section 2 Section 3 describes the experimental approach by SEBM andthe physical and numerical parameters used Section 4 is devoted to the influence of the processparameters (layer thickness beam energy and beam velocity) on the appearance of walls The

2


Figure 1 Additive manufacturing by SBM of powder

numerical results are compared with analogous experiments from SEBM Section 5 is devotedto discussing the underlying physical phenomena with the help of dimensionless numbers andthe characteristic time scales of the process

2 Physical model

SBM of powders involves many different physical processes [4] see figure 1 The beam isabsorbed in the powder bed and heats the powder particles which eventually start to melt andcoalesce The melt pool is highly dynamic driven by capillary forces Due to the hydrodynamicmovement the shape of the melt pool constantly changes during the process Dependent onthe parameters and environment this reshaping can range from little deviations from a quasi-stationary melt pool shape to significant changes in geometry Sometimes the disintegrationof the melt pool into spherical droplets called balling and commonly denoted as Rayleighinstability [18] is observed In addition the shape of the re-solidified melt pool is stronglydependent on the wetting characteristics of the melt with the powder particles [4] Afterfinishing one layer a new powder layer is applied on the irregular and corrugated surfaceConsequently the thickness of the new powder layer is strongly varying which might result intypical process defects like binding faults [3 4] see figure 1 To ensure that two consecutivelayers form a sound bonding sufficient remelting of the previous layer has to take place Inthe following the underlying physical model is described The detailed numerical approachto solve this model which is based on a lattice Boltzmann approach is described in [17]

21 Geometry and beam definition

Since our numerical approach is 2D the problem has to be reduced to 2D Figure 2 shows thegeometric situation The buildup direction is denoted as z and the building plane is definedby x and y The simulation plane is either spanned by the x and z coordinate or y and z

coordinate In the first case (figure 2(b)) the beam is not moving and only the beam power isacting in a time span given by the y-velocity of the beam In the second case (figure 2(c)) thebeam is moving with the velocity of the beam In this paper we are only considering buildingsituations as depicted in figure 2(b)

The electron beam can be described by a two dimensional Gaussian power distribution

I (x y t) = P

2πσ 2middot exp

(minus (x minus vx t)2 + (y minus vy t)2

2 σ 2

) (1)

3


Figure 2 Geometric situation during SBM and reduction to 2D for numerical simulation

where vx and vy are the x- and y-components of the beam velocity v σ is a sixth of the meanwidth of the beam and P is the beam power In order to generate a 2D formulation we firstdefine a beam with an arbitrary power distribution along the y axis

I (x y t) = Pradic2πσ

h(y t) middot exp

(minus (x minus vx t)2

2 σ 2

) (2)

After the beam has traversed through the 2D simulation plane this power distribution hasthe same energy input into the domain as the 2D Gaussian power distribution Thus it shallobey the following conditionint infin

minusinfinI (x y t) dy =

int infin

minusinfinI (x y t) dy (3)

Hence the function h(y t) is chosen so that I (x y t) represents an averaged beamconsisting of a 1D Gaussian power distribution along the x axis and a step function along they axis With w as the width of the beam equaling 6 σ we choose h(y t) to be

h(y t) =

1

w 0 vy t minus y w

0 else(4)

Inserting this and choosing vx = 0 and the simulation plane to be at y = 0 we arrive atthe 2D formulation

I (x t)

Pradic2πσw

middot exp

(minus x2

2 σ 2

) 0 vy t w

0 else

(5)

The line energy EL is given by

EL = P

v (6)

The beam penetrates the powder bed In the case of an electron beam the energy is nearlycompletely absorbed at the position where it has first contact with the powder The absorptionprocess for laser radiation is more complex due to multi-reflection processes causing theradiation transport to have shadowed powder particles [19] Our model does not take reflectionprocesses into account but is able to handle the transient nature of the absorbing surface due tomelting see figure 1 Thus this paper focuses on the simulation and validation of the electron

4


beam melting process where reflection processes do not have to be taken into account Whenthe beam touches a powder particle the energy absorption is modeled by the exponentialabsorption law

dI

dz= λabsI (7)

where λabs denotes the absorption coefficient of the radiation

22 Energy transfer and conservation equations

The beam energy is absorbed in the powder bed the powder temperature increases andthe thermal energy spreads by heat diffusion When the temperature exceeds the solidustemperature of the metal the solidndashfluid phase transformation starts thereby consuming latentheat L When the local liquid phase fraction exceeds a given threshold value the solid startsto behave as a liquid The liquid material is governed by the NavierndashStokes equations Heattransport in the liquid is either by diffusion or by convection Radiation and convection of heatfrom the liquid surface are neglected so that the excess heat of the liquid must be dissipatedby heat conduction into the powder bed in order to re-solidify the melt pool The neglectof convection is justified since the EBM process is under a vacuum Radiation vaporizationand marangoni convection can have an essential effect and will be taken into account in afurther work

The underlying continuum equations of convectionndashdiffusion transport are founded on anenthalpy based methodology The single-phase continuum conservation equations to simulatethermo-fluid incompressible transport comprising melting and solidification are given by

nabla middot u = 0 (8)

partu

partt+ (u middot nabla) u = minus 1

ρnablap + νnabla2u + g (9)

part E

partt+ nabla middot (u E) = nabla middot ( knabla E) + (10)

where nabla is the gradient operator t the time u the local velocity of the melt p the pressure andν the kinematic viscosity Gravity acceleration is denoted by g Surface tension effects aretaken into account via the boundary conditions at the free surface Wetting effects between themelt and the solid phase are also taken into account Details are described in [20] The thermaldiffusivity is designated by k = k(E) The energy source describes the energy depositedin the material by the beam Viscous heat dissipation and compression work are neglected inthis model The thermal energy density E is given by

E =int T

0ρ cp dT + ρ H (11)

where cp is the specific heat at constant pressure T is the temperature and H is the latententhalpy of a computational cell undergoing phase change For a multi component metal alloyH is a complex function of the temperature In a simple approximation it can be expressedas follows

H(T ) =

L T Tf

T minus Ti

Tf minus Timiddot L Ti T lt Tf

0 T lt Ti

(12)

5


with Ti and Tf representing the beginning and the end of the phase transformation respectivelyL is the latent heat of the phase change Denoting ξ as the liquid fraction in a cell

ξ(T ) = H(T )

L (13)

The latent enthalpy is taken up into an effective specific heat cp

E =int T

0ρ cp dT + ρ H =

int T

0ρ cp dT (14)

with

cp =

cp T Tf

cp +L

Tf minus Ti Ti T lt Tf

cp T lt Ti

(15)

The thermal diffusivity k is related to the heat conductivity λ by

k(E) = λ(E)

ρ cp(E) (16)

23 Boundary conditions and interface treatment

The surface between liquid and atmosphere is accounted for with the volume of fluids methodDependent on the fluid motion the fluid fraction of a volume element increases or decreasesWhen a cell is entirely filled or emptied the surface moves accordingly allowing for a freelymoving surface Thermally the liquid-atmosphere surface is perfectly insulating The effectof the surface tension is treated as a local modification of the gas pressure pG acting at theinterface ie the gas pressure is replaced by

pprimeG = pG minus κ middot γ (17)

where κ and γ denote the curvature and the surface tension respectivelyThe dependence of the surface tension on the temperature and the high temperature

gradients in the melt pool induce a hydrodynamic flow perpendicular to the surface Thisphenomenon commonly denoted as Marangoni convection has not been taken into accountfor the simulations presented in this paper This flow would lead to an increase of the transportof heat away from the center of the beam increasing effective heat conduction and resultingin a different melt pool shape Unlike in welding the main effect leading to the growth of themelt pool in a powder bed is the wetting of neighboring powder particles Hence Marangoniconvection can be regarded as a secondary effect increasing the melt pool life span and thereforeits size Due to the neglect of this phenomenon the melt pool size in the simulation might beunderestimated

Thermally the solid-atmosphere surface is perfectly insulating while the solid phase isconductive Hence the thermal conduction between two powder particles is determined bythe contact area between the individual particles unlike in homogenized approaches

The solidndashliquid interface is treated as a hydrodynamic no-slip boundary conditionHowever the solid phase is assumed to be immobile Accordingly the force resulting fromthe no-slip boundary condition is only applied to the liquid phase not to the solid phase Forthe SEBM process that this paper focuses on solid movement can be neglected as the powderis pre-sintered in the process resulting in an immobile powder bed For simulating SLMa mobile solid phase might be necessary Furthermore wetting is included in the numericalmodel Details of the algorithm and its validation are given in [20] The wetting angle betweenfluid and solid is set to 0 for the simulations assuming perfect homologous wetting

6


Figure 3 Powder bed generation (a) Rain model packing algorithm (b) powder layer with aGaussian size distribution (c) powder layer with adapted relative density by removing a certainpercentage of the particles (d) local consolidation by melting (e) generation of virtual particles(black dots) on the solidified melt and the surface particles (f ) generation of a new powder layerand (g) adaption of the relative density see (c)

24 Layer upon layer random powder bed generation

The rain model packing algorithm [21] is used to generate the random powder bed Inthis model particles follow definable trajectories to find a resting place in the powder bedParticles from a given size distribution (eg Gaussian distribution or a bimodal distribution)are placed one by one in randomly selected positions above the packing space see figure 3(a))The newly introduced particle falls downward until it comes into touch with a stationaryparticle Subsequently it attempts to minimize its vertical coordinate by rolling around thecircumference of the stationary particle and any other particle that it comes into contact withMovement ceases when no further downward movement is possible and the particle reachesthe nearest local minimum When no contacted particle is found the particle is deposited onthe basal line

We use a 2D formulation of the rain packing algorithm which results in a powder packingdensity much higher than in the experiments In order to adapt the density of the new powder

7


layer (figure 3(b)) to the experimental density a certain percentage of the particles is removed(figure 3(c)) Local melting of the particles leads to stochastic geometries of the solidifiedmaterial (figure 3(d)) Virtual particles are generated on the surface of the consolidated materialand the surface particles (figure 3(e)) and a new powder layer is applied (figure 3(f )) Againthe density is adapted (figure 3(e))

3 Experimental approach and simulation parameters

In this section a general description of the experimental procedure and the simulationparameters which are identical for all simulations and experiments are given

31 Experiments

All experiments described in this work are carried out with the Arcam A2 EBM machine whichis based on SEBM The SEBM process used for rapid component prototyping is operationallysimilar to the scanning of an electron beam in a scanning electron microscope and it can beconsidered as a variant of SLM Similar to the SLM process metal powders are selectivelymolten in paths traced by the electron beam gun In all cases the width of the Gaussian beamis 350 microm

The SEBM machine consists of an evacuated building tank with an adjustable processplatform two powder dispensing hoppers and a rake system for spreading the powders Theelectron beam is generated by heating a tungsten filament The acceleration voltage of theelectrons is 60 kV The electrons are focused and deflected by electromagnetic lenses andrelease their kinetic energy to the powder particles which causes them to heat

The base material for the process is gas atomized pre-alloyed Tindash6Alndash4V powder with aGaussian particle size distribution between 45 and 115 microm Vertical walls are deposited ona 10 mm thick stainless steel plate The platform is first heated with the defocused electronbeam to a temperature of 760 C Subsequently a layer of Tindash6Alndash4V powder is spread overthe platform Again the entire powder bed on the platform is preheated by scanning withthe defocused electron beam During this procedure the powder is sintered [17] increasingthermal and electrical conductivity and immobilizing the powder

Following this preheating step the beam scans the powder bed in order to melt the powderat predefined positions line after line Here perpendicular single line walls are produced Aftercompletion of the layer the platform is lowered by one layer thickness and the next layer ofpowder is applied This process is repeated until the walls have reached their desired height

A series of single line walls is manufactured with line energies of 05 10 and 20 J mmminus1

using different combinations of beam power and scan speed see table 1 Two different layerthicknesses 70 and 100 microm are considered The samples are cross-sectioned mounted andpolished in order to compare the experimental results with the simulation

32 Simulation parameters

The numerical simulation approach is based on the lattice Boltzmann method (LBM) which isan explicit finite difference method of second order accuracy in time and space [14] The LBMis characterized by dividing the simulation space in cells which all have the same size Allthese cells have the same properties but may assume different kinds of states eg solid liquid orgas The size of the cells defines the length scale In addition a mass scale m a temperaturescale T and a time scale t have to be defined x = 50 times 10minus6 m m = 50 times 10minus13 kgT = 154 times 103 K and t = 22 times 10minus7 s The scales have to be defined since the lattice

8


Table 1 Process settings for the multi-layer experiments and the equivalent simulation parameters

Experiment Simulation

Beam Scan Beam Line Beam ScanProcess current speed power energy energy speedsettings (mA) (mm sminus1) (W) (J mmminus1) 1t xt

F2 2 240 120 06 001F3 5 600 300 05 15 00264F4 10 1200 600 30 00528

F6 2 120 120 06 000528F7 5 300 300 10 15 00132F8 10 600 600 30 00264

F10 2 60 120 06 000264F11 5 150 300 20 15 00066F12 10 300 600 30 00132

Table 2 Physical parameters of Tindash6Alndash4V [22] and corresponding dimensionless parameters forthe LBM simulation

Physical properties Experiment LBM

Density (liquid) 4000 kg mminus3 10Viscosity 0005 Pa s 0011Surface tension 165 N mminus1 015Gravitational acceleration 981 m sminus2 10minus7

Thermal diffusivity of solid 783 times 10minus6 m2 sminus1 0068Thermal diffusivity of liquid 993 times 10minus6 m2 sminus1 0087Solidus temperature 1878 K 122Liquidus temperature 1928 K 125Preheat temperature 1023 K 0664Latent heat 037 times 106 J kgminus1 03λabs 04 micromminus1 20Specific heat 700 J kgminus1 Kminus1 10

Boltzmann simulation uses dimensionless parameters Thus all material parameters have to beexpressed in dimensionless form Table 2 lists the physical parameters and their correspondingvalues used in the simulation

The dimensionless quantities (marked with lowast) follow by multiplying the materialparameters with the relevant scales such as

ρlowast = ρx3

m νlowast = ν

t

x2 σ lowast = σ

t2

m glowast = g

t2

x

klowast = kt

x2 etc (18)

The absorption depth for 60 kV electrons is about 10 microm Since the size of a cell is 5 micrommost of the energy is absorbed within two cells

4 Results

Figure 4 shows the layer by layer building process of a vertical wall The first layer is generateddirectly on the building plate with the preheating temperature T0 = 760 C The first powder

9


Figure 4 Formation of a single wall for different time steps from left to right during the formationof several layers Melting of the powder takes place parallel to the exposure of the beam (200times400cells = 1 mm times 2 mm beam width 70 cells (350 microm) parameters F6 (table 1) layer thickness20 cells = 100 microm)

layer is rather scarce since the total layer thickness is only 100 microm The beam does notmove and the resulting melt pool geometry has a stochastic nature After solidification thesubsequent powder layer is applied on the previous solidified melt pool In each layer theappearance of the melt pool is completely different Thus the subsequent powder layer isapplied upon a stochastic geometry which might be either concave or convex As a resulteach new layer looks different This stochastic behavior is essential for the resolution of thebeam building process

In the following the influence of layer thickness line energy and beam velocity on theappearance of vertical walls is investigated

41 Influence of the layer thickness

The layer thickness is one of the main process parameters for layer based additivemanufacturing processes For SEBM the layer thickness is typically varied between 50 and150 microm for SLM the layer thickness is generally much smaller between 20 and 50 micromsometimes even less than 20 microm In order to investigate the effect of the layer thickness onthe resolution different layer thicknesses between 35 and 150 microm are considered see figure 5The quality of the walls decreases dramatically with increasing layer thickness For layerthicknesses smaller than 70 microm the surface quality is not further improved but the wall widthincreases due to the increasing total energy input with the increasing number of layers Thiseffect becomes clear comparing 70 microm with 35 microm walls

In figure 6 the line energy is adapted to the layer thickness in such a way that the totalenergy input is the same for all layer thicknesses That is the line energy multiplied with thenumber of layers is constant For 50 microm layers the number of lines is doubled compared to100 microm layers but the line energy is halved As a consequence the wall thickness and surfaceroughness is strongly increased for 100 microm layers On the other hand the comparison betweenthe 70 and 50 microm layers reveals that the improvement of the surface roughness is eventuallylimited by the stochastic powder bed

10


Figure 5 Influence of the layer thickness on the wall quality (200 times 400 cells (1 mm times 2 mm)beam width 72 cells (350 microm ) parameters F8 (table 1))

Figure 6 Influence of the layer thickness on the wall quality for constant total energy input(400 times 400 cells (2 mm times 2 mm) beam width 72 cells (350 microm) parameters F8 for 70 microm Theline energy for the others is modified in such a way that the number of layers times the line energyis constant)

42 Influence of the beam line energy and velocity

Figure 7 shows the influence of the beam line energy and beam velocity for two layer thicknesseson the resulting walls As expected the results for the 70 microm layers are better than for 100 micromThis observation holds for the binding faults as well as the surface roughness The meanthickness of the walls increases with an increasing line energy At constant line energy theappearance of the walls is also dependent on the velocity With an increasing beam velocityie an increasing beam power the wall thickness as well as the surface roughness increasesThat is the numerical results predict a dependence on the beam velocity The slower the beamthe better are the results

Figure 8 shows the buildup of the F7100 microm1 J mmminus1 wall in 18 layers In contrast tofigure 4 beam velocity and beam power are increased by a factor of 25 Thus the interactiontime between beam and powder is reduced but the total energy input is the same The firstlayers are still relatively defined but after 10 layers the building process gets more and moreundetermined especially between the 14th and 18th layer Due to the strong geometry changesof the solidified melt pool the new applied powder layer is sometimes much thicker than200 microm which we expect on average for a relative powder packing density of 50

43 Experiment versus numerical simulation

In order to verify the numerical results experiments analogous to the numerical simulations offigure 7 were performed see figure 9 The wall quality strongly increases by decreasing

11


Figure 7 Wall formation as a function of beam velocity line energy and layer thickness (200times400cells = 1 mm times 2 mm beam width 72 cells (350 microm) layer thickness 20 cells = 100 microm)

Figure 8 Layer by layer formation of a single wall (200 times 400 cells = 1 mm times 2 mm beamwidth 72 cells (350 microm) parameters F7 layer thickness 20 cells = 100 microm)

the layer thickness In addition the wall quality decreases with an increasing beamvelocity

In order to have a quantitative measure available to compare simulation and experimentas a function of the process parameters the mean wall thickness is determined The wallthickness is measured geometrically for more than 60 points perpendicular to the wall center

12


Figure 9 Wall formation as a function of beam velocity line energy and layer thicknessExperimental results for Ti6AlV4

Figure 10 Experimental and simulated wall thickness as a function of the scan speed for constantline energies Powder layer thickness 70 microm Error bars indicate the standard deviation of theresults

line and the mean value is calculated Figure 10 shows the mean value of the wall thicknessfor the 70 microm layer walls The simulation results are in good agreement with the experiments

5 Discussion

The experimental findings of figure 9 demonstrate in an impressive way the sensitivity of theresults to the parameters In addition it becomes clear that the resulting surface roughness

13


is always much higher than expected from the mean powder particle diameter An importantquestion is what the origin of this high surface roughness is At first glance position accuracyand the quality of the beam are good candidates to cause the roughness During wall buildingthe beam has to come back to its initial position in each layer If this accuracy is bador if the beam quality changes with time the layers will be inaccurate resulting in a highroughness Numerically positioning accuracy as well as beam quality are perfect Comparingthe experimental results with the numerical ones it becomes clear that there must be anothermechanism that is responsible for the rough surface

In the following the fundamental mechanisms governing the consolidation process duringSBM are discussed with the help of the relevant time scales Material consolidation duringSBM takes place on the micro-scale where surface forces dominate over volume forces Thatis the forces resulting from the surface tension are much larger than gravity or viscous forcesIn order to get a more quantitative measure it is instructive to consider dimensionless numbersA measure to compare gravity and surface tension effects is the Bond number Bo

Bo = ρ middot g middot L2σ (19)

where ρ g L σ denote the density gravitational acceleration the relevant length scale andthe surface tension respectively

The Laplace number La is used to characterize free surface fluid dynamics and representsthe ratio of inertia and capillary effects to viscous forces

La = σ middot ρ middot L

η2 (20)

where η is the viscosity For titanium Bo asymp 10minus4 and La asymp 105 That is during powdermelting the system is dominated by surface tension effects gravity or viscous effect are small

In order to gain a better understanding of the powder particle melting process and theevolution of the melt pool it is instructive to introduce relevant time scales that allow one tocompare different processes The interaction time tint

tint = D

v (21)

is a measure for the contact time of the powder layer with the beam with diameter D andvelocity v

The diffusion time tdiff

tdiff = d2l

k (22)

is the characteristic time necessary to transport heat over a distance dl the layer thicknesswith the thermal diffusivity k The capillary time tcap

tcap = η middot L

σ (23)

is the time necessary for an interface to regain its equilibrium shape after a perturbation TheRayleigh time tray

tray =radic

ρ middot L3

σ (24)

is the time scale which characterizes the relaxation of an interface perturbation under the actionof inertia and surface tension forces [23ndash25]

14


Figure 11 Temperature field during powder layer consolidation Gray denotes the beam andthe black line shows the solidndashliquid interface The time scale for melt pool reshaping is smallcompared to the velocity of the melting front (from numerical experiment F2)

51 Thermal diffusion time versus interaction time

During tint the beam heats the powder and eventually melts it The characteristic time necessaryfor the deposited energy to reach and partly remelt the preceding layer is tdiff If tint tdiff energy distribution is diffusion limited The surface gets strongly superheated and eventuallyevaporation starts On the other hand if tint tdiff energy is lost by thermal diffusion and themelting process becomes ineffective Thus tint asymp tdiff is expected to be a reasonable choicefor the interaction time

v = D middot k

d2l

(25)

For dl = 100 microm the suggested beam velocity and the interaction time would be about200 mm sminus1 and 6 ms respectively It is essential to realize that the optimal velocity is afunction of 1d2

l For dl = 20 microm a beam velocity of 5000 m sminus1 is recommended Thusmuch higher velocities without the danger of overheating and evaporation can be realized withthinner layers

52 Melting versus reshaping

An essential question is whether coalescence is governed by thermal diffusion and melting orreshaping due to the surface tension The time scale for two molten powder particles to mergeis determined by tray if inertia or tcap if viscous forces dominate

For a length scale of the order of 100 microm the capillary time is about 10minus7 s which is rathersmall in comparison with the Rayleigh time of about 10minus4 s Thus reshaping is governed byinertia effects rather than viscous ones

Nevertheless tray is more than one order of magnitude smaller than the thermal diffusiontime for 100 microm layers That is coalescence of the powder particles follows the meltingprocess nearly instantaneously Thus the consolidation is limited by thermal diffusion andnot by hydrodynamics which becomes visible in figure 11 In the time span between 680and 1000 micros the solidndashliquid interface moves only a little while the melt pool shape changesdrastically due to coalescence and the movement induced by surface tension

53 Surface beads and extrusions

The strong influence of the process parameters on the surface quality was demonstrated insection 43 Here the fundamental mechanisms leading to the experimental observations arediscussed

The underlying principle of additive manufacturing is that the component is built layerby layer Thereby it is assumed that layer consolidation is in the vertical direction During

15


Figure 12 Formation of beads and extrusion and intrusions (from numerical experiment F8)

consolidation the powder particles are molten coalesce and the whole layer consolidatesdriven by the surface tension This is a very vigorous process and does not necessarily lead tomere vertical movement it might also result in a horizontal movement of the melt

Figure 12 elucidates the evolution of beads and extrusionsIn both cases larger droplets form due to the surface tension and are accelerated to the left

where they stick to powder particles as a result of the wetting conditions Due to overheatingof the droplets powder particles far away from the range of the beam are also molten ForF7 the droplet is not able to regain contact to the rest of the melt pool and the bead sticksat the surface The situation for F8 is different Here further melting eventually leads to thecoalescence of the droplet with the lower melt pool Nevertheless although the surface tensionpulls the melt downward the dip further sticks to the powder particle and forms a pronouncedextrusion

Figure 12 reveals as a cause for the development of the surface defects a horizontalmovement of the melt which is triggered by the stochastic powder configuration in combinationwith the progression of the melt pool In both cases the powder configuration is such thatthe contact in the horizontal direction is better than in the vertical direction The danger forhorizontal movement increases with increasing the powder layer thickness decreasing therelative density of the powder and increasing the power density of the beam The thicker thepowder layer is the larger is the probability that there is no continuous contact of the powderparticles in the vertical direction This observation explains the strong influence of the layerthickness dl on the surface quality of the walls The number of superimposed powder particlesfor dl = 100 microm is about 3ndash4 for dl = 70 microm it is about 2 The decrease of the relative densityhas a similar effect as an increase of the powder layer thickness

The dependence on the power density at constant line energy is more complex In thiscase the temporal evolution of the melt pool changes with increasing the power density Forlow densities the melt pool starts at the center of the beam and expands horizontally andvertically For high power densities melting starts all over the surface and the danger of ahorizontal movement of the melt pool increases In addition overheating of the melt increaseswith increasing the power density Thus melt sucked into the surrounding powder bed willmelt the neighboring particles and thus increase the horizontal deviation

16


If the powder layer is reduced to a mono-layer of particles the danger of horizontalmovement is strongly reduced since the direct contact to the consolidated bottom drives themolten particles downward Thus we recommend choosing the mean powder diameter andthe layer thickness in such a way that the applied powder layer is nearly a mono-layer Thismeans that for a mean powder diameter of 70 microm the layer thickness should be about 35 microm

6 Summary

This paper describes a numerical approach to simulate beam and powder bed additivemanufacturing processes at the scale of individual powder particles A numerical tool onthe basis of a lattice Boltzmann method (LBM) was developed which allows one to simulatelayer by layer fabrication in 2D The influence of the process parametersmdashbeam power beamvelocity and layer thicknessmdashon the resulting quality for vertical walls is investigated andcompared with experimental results from selective electron beam melting (SEBM)

The numerical results demonstrate the nature of the buildup process which is stronglydetermined by the stochastic nature of the powder bed in each layer In addition the numericalsimulation reveals the underlying physical phenomena that govern the process The highsurface forces in combination with wetting of a stochastic loosely packed powder bed areidentified as the origin of the high roughness and the occurrence of binding faults From thesimulation results we conclude that higher velocities at constant line energy are unfavorablesince the overheated surfaces layer behaves in an unpredictable way In order to increasebuilding accuracy and velocity we recommend adapting the powder size in such a way thatmono-layers of particles are applied during building

The powder-level simulation has turned out to be an essential means to reveal themechanisms and parameters influencing consolidation during selective beam melting (SBM)of powders

Acknowledgments

The authors gratefully acknowledge funding of the German Research Council (DFG) withinthe Collaborative Research Center 814 lsquoAdditive Manufacturingrsquo project B4 and within theframework of its lsquoExcellence Initiativersquo supports the Cluster of Excellence lsquoEngineering ofAdvanced Materialsrsquo at the University of Erlangen-Nuremberg

References

[1] Levy G N Schindel R and Kruth J P 2003 Rapid manufacturing and rapid tooling with layer manufacturing(LM) technologies state of the art and future perspectives CIRP AnnmdashManuf Technol 52 589ndash609

[2] Kruth J-P Mercelis P Van Vaerenbergh J Froyen L and Rombouts M 2005 Binding mechanisms in selectivelaser sintering and selective laser melting Rapid Prototyping J 11 26ndash36

[3] Kruth J-P Levy G Klocke F and Childs T H C 2007 Consolidation phenomena in laser and powder-bed basedlayered manufacturing CIRP AnnmdashManuf Technol 56 730ndash59

[4] Das S 2003 Physical aspects of process control in selective laser sintering of metals Adv Eng Mater 5 701ndash11[5] Williams J D and Deckard C R 1998 Advances in modeling the effects of selected parameters on the SLS process

Rapid Prototyping J 4 90ndash100[6] Bugeda G Cervera M and Lombera G 1999 Numerical prediction of temperature and density distributions in

selective laser sintering processes Rapid Prototyping J 5 21ndash6[7] Zhang Y and Faghri A 1999 Melting of a subcooled mixed powder bed with constant heat flux heating Int J

Heat Mass Transfer 42 775ndash88[8] Shiomi M Yoshidome A Osakada K and Abe F 1999 Finite element analysis of melting and solidifying processes

in laser rapid prototyping of metallic powders Int J Mach Tools Manuf 39 237ndash52

17


[9] Tolochko N K Arshinov M K Gusarov A V Titov V I Laoui T and Froyen L 2003 Mechanisms of selectivelaser sintering and heat transfer in Ti powder Rapid Prototyping J 9 314

[10] Matsumoto M Shiomi M Osakada K and Abe F 2002 Finite element analysis of single layer forming on metallicpowder bed in rapid prototyping by selective laser processing Int J Mach Tools Manuf 42 61

[11] Dai K and Shaw L 2004 Thermal and mechanical finite element modeling of laser forming from metal andceramic powders Acta Mater 52 69ndash80

[12] Kolossov S Boillat E Glardon R Fischer P and Locher M 2004 3D FE simulation for temperature evolution inthe selective laser sintering process Int J Mach Tools Manuf 44 117ndash23

[13] Benzi R Succi S and Vergassola M 1992 The lattice Boltzmann equation theory and applications Phys Rep222 145ndash97

[14] Chen S and Doolen G D 1998 Lattice Boltzmann method for fluid flows Ann Rev Fluid Mech 30 329ndash64[15] He X and Luo L-S 1997 A priori derivation of the lattice Boltzmann equation Phys Rev E 55 R6333ndash6[16] Korner C Thies M Hofmann T Thurey N and Rude U 2005 Lattice Boltzmann model for free surface flow for

modeling foaming J Stat Phys 121 179ndash96[17] Korner C and Attar E 2011 Mesoscopic simulation of selective beam melting processes J Mater Process

Technol 211 978ndash87[18] Gusarov A V Yadroitsev I Bertrand Ph and Smurov I 2007 Heat transfer modelling and stability analysis of

selective laser melting Appl Surf Sci 254 975ndash9[19] Zhou J Zhang Y and Chen J K 2009 Simulation of laser irradiation to a randomly packed bimodal powder bed

Int J Heat Mass Transfer 52 3137ndash46[20] Attar E and Korner C 2009 Lattice Boltzmann method for dynamic wetting problems J Colloid Interfaces Sci

335 84ndash93[21] Meakin P and Juillien R 1987 Restructuring effects in the rain model for random deposition J Physique

48 1651ndash62[22] Iida T and Guthrie R I L 1988 The Physical Properties of Liquid Metals (Oxford Clarendon)[23] Spiegelberg S H Ables D C and McKinley G H 1996 The role of end effects on measurements of extensional

viscosity in filament stretching rheometers J Non-Newtonian Fluid Mech 64 229ndash67[24] Eggers J 1997 Nonlinear dynamics and breakup of free-surface flows Rev Mod Phys 69 865ndash929[25] Steinhaus B Shen A Q and Sureshkumar R 2007 Dynamics of viscoelastic fluid filaments in microfluidic devices

Phys Fluids 19 073103

18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References

IOP PUBLISHING MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING

Modelling Simul Mater Sci Eng 21 (2013) 085011 (18pp) doi1010880965-0393218085011

Fundamental consolidation mechanisms duringselective beam melting of powders

Carolin Korner Andreas Bauereiszlig and Elham Attar

University of Erlangen Materials Science Department Martensstr 5 91058 Erlangen Germany

E-mail carolinkoernerwwuni-erlangende

Received 18 February 2013 in final form 30 August 2013Published 8 November 2013Online at stacksioporgMSMSE21085011

AbstractDuring powder based additive manufacturing processes a component is realizedlayer upon layer by the selective melting of powder layers with a laser or anelectron beam The density of the consolidated material the minimal spatialresolution as well as the surface roughness of the resulting components arecomplex functions of the material and process parameters So far the interplaybetween these parameters is only partially understood

In this paper the successive assembling in layers is investigated with arecently described 2D-lattice Boltzmann model which considers individualpowder particles This numerical approach makes several physical phenomenaaccessible which cannot be described in a standard continuum picture eg theinterplay between capillary effects wetting conditions and the local stochasticpowder configuration In addition the model takes into account the influenceof the surface topology of the previous consolidated layer on the subsequentpowder layer

The influence of the beam power beam velocity and layer thickness onthe formation and quality of simple walls is investigated The simulationresults are compared with experimental findings during selective electron beammelting The comparison shows that our model although 2D is able to predictthe main characteristics of the experimental observations In addition thenumerical simulation elucidates the fundamental mechanisms responsible forthe phenomena that are observed during selective beam melting

(Some figures may appear in colour only in the online journal)

Content from this work may be used under the terms of the Creative Commons Attribution30 licence Any further distribution of this work must maintain attribution to the author(s) andthe title of the work journal citation and DOI

0965-039313085011+18$3300 copy 2013 IOP Publishing Ltd Printed in the UK amp the USA 1


1 Introduction







2




2 Physical model






I (x y t) = P

2πσ 2middot exp


2 σ 2

) (1)

3





h(y t) middot exp


2 σ 2

) (2)



int infin



h(y t) =

1

w 0 vy t minus y w

0 else(4)


I (x t)

Pradic2πσw

middot exp

(minus x2

2 σ 2

) 0 vy t w

0 else

(5)


EL = P

v (6)


4



dI

dz= λabsI (7)






partu



part E



E =int T

0ρ cp dT + ρ H (11)


H(T ) =

L T Tf

T minus Ti


0 T lt Ti

(12)

5



ξ(T ) = H(T )

L (13)


E =int T

0ρ cp dT + ρ H =

int T

0ρ cp dT (14)

with

cp =

cp T Tf

cp +L


cp T lt Ti

(15)


k(E) = λ(E)

ρ cp(E) (16)








6






7





31 Experiments









8





F2 2 240 120 06 001F3 5 600 300 05 15 00264F4 10 1200 600 30 00528

F6 2 120 120 06 000528F7 5 300 300 10 15 00132F8 10 600 600 30 00264

F10 2 60 120 06 000264F11 5 150 300 20 15 00066F12 10 300 600 30 00132







ρlowast = ρx3

m νlowast = ν

t

x2 σ lowast = σ

t2

m glowast = g

t2

x

klowast = kt

x2 etc (18)


4 Results


9








10









11






12





5 Discussion


13








η2 (20)



tint = D

v (21)



tdiff = d2l

k (22)


tcap = η middot L

σ (23)


tray =radic

ρ middot L3

σ (24)


14





v = D middot k

d2l

(25)










15








16



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References


1 Introduction







2




2 Physical model






I (x y t) = P

2πσ 2middot exp


2 σ 2

) (1)

3





h(y t) middot exp


2 σ 2

) (2)



int infin



h(y t) =

1

w 0 vy t minus y w

0 else(4)


I (x t)

Pradic2πσw

middot exp

(minus x2

2 σ 2

) 0 vy t w

0 else

(5)


EL = P

v (6)


4



dI

dz= λabsI (7)






partu



part E



E =int T

0ρ cp dT + ρ H (11)


H(T ) =

L T Tf

T minus Ti


0 T lt Ti

(12)

5



ξ(T ) = H(T )

L (13)


E =int T

0ρ cp dT + ρ H =

int T

0ρ cp dT (14)

with

cp =

cp T Tf

cp +L


cp T lt Ti

(15)


k(E) = λ(E)

ρ cp(E) (16)








6






7





31 Experiments









8





F2 2 240 120 06 001F3 5 600 300 05 15 00264F4 10 1200 600 30 00528

F6 2 120 120 06 000528F7 5 300 300 10 15 00132F8 10 600 600 30 00264

F10 2 60 120 06 000264F11 5 150 300 20 15 00066F12 10 300 600 30 00132







ρlowast = ρx3

m νlowast = ν

t

x2 σ lowast = σ

t2

m glowast = g

t2

x

klowast = kt

x2 etc (18)


4 Results


9








10









11






12





5 Discussion


13








η2 (20)



tint = D

v (21)



tdiff = d2l

k (22)


tcap = η middot L

σ (23)


tray =radic

ρ middot L3

σ (24)


14





v = D middot k

d2l

(25)










15








16



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References




2 Physical model






I (x y t) = P

2πσ 2middot exp


2 σ 2

) (1)

3





h(y t) middot exp


2 σ 2

) (2)



int infin



h(y t) =

1

w 0 vy t minus y w

0 else(4)


I (x t)

Pradic2πσw

middot exp

(minus x2

2 σ 2

) 0 vy t w

0 else

(5)


EL = P

v (6)


4



dI

dz= λabsI (7)






partu



part E



E =int T

0ρ cp dT + ρ H (11)


H(T ) =

L T Tf

T minus Ti


0 T lt Ti

(12)

5



ξ(T ) = H(T )

L (13)


E =int T

0ρ cp dT + ρ H =

int T

0ρ cp dT (14)

with

cp =

cp T Tf

cp +L


cp T lt Ti

(15)


k(E) = λ(E)

ρ cp(E) (16)








6






7





31 Experiments









8





F2 2 240 120 06 001F3 5 600 300 05 15 00264F4 10 1200 600 30 00528

F6 2 120 120 06 000528F7 5 300 300 10 15 00132F8 10 600 600 30 00264

F10 2 60 120 06 000264F11 5 150 300 20 15 00066F12 10 300 600 30 00132







ρlowast = ρx3

m νlowast = ν

t

x2 σ lowast = σ

t2

m glowast = g

t2

x

klowast = kt

x2 etc (18)


4 Results


9








10









11






12





5 Discussion


13








η2 (20)



tint = D

v (21)



tdiff = d2l

k (22)


tcap = η middot L

σ (23)


tray =radic

ρ middot L3

σ (24)


14





v = D middot k

d2l

(25)










15








16



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References





h(y t) middot exp


2 σ 2

) (2)



int infin



h(y t) =

1

w 0 vy t minus y w

0 else(4)


I (x t)

Pradic2πσw

middot exp

(minus x2

2 σ 2

) 0 vy t w

0 else

(5)


EL = P

v (6)


4



dI

dz= λabsI (7)






partu



part E



E =int T

0ρ cp dT + ρ H (11)


H(T ) =

L T Tf

T minus Ti


0 T lt Ti

(12)

5



ξ(T ) = H(T )

L (13)


E =int T

0ρ cp dT + ρ H =

int T

0ρ cp dT (14)

with

cp =

cp T Tf

cp +L


cp T lt Ti

(15)


k(E) = λ(E)

ρ cp(E) (16)








6






7





31 Experiments









8





F2 2 240 120 06 001F3 5 600 300 05 15 00264F4 10 1200 600 30 00528

F6 2 120 120 06 000528F7 5 300 300 10 15 00132F8 10 600 600 30 00264

F10 2 60 120 06 000264F11 5 150 300 20 15 00066F12 10 300 600 30 00132







ρlowast = ρx3

m νlowast = ν

t

x2 σ lowast = σ

t2

m glowast = g

t2

x

klowast = kt

x2 etc (18)


4 Results


9








10









11






12





5 Discussion


13








η2 (20)



tint = D

v (21)



tdiff = d2l

k (22)


tcap = η middot L

σ (23)


tray =radic

ρ middot L3

σ (24)


14





v = D middot k

d2l

(25)










15








16



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References



dI

dz= λabsI (7)






partu



part E



E =int T

0ρ cp dT + ρ H (11)


H(T ) =

L T Tf

T minus Ti


0 T lt Ti

(12)

5



ξ(T ) = H(T )

L (13)


E =int T

0ρ cp dT + ρ H =

int T

0ρ cp dT (14)

with

cp =

cp T Tf

cp +L


cp T lt Ti

(15)


k(E) = λ(E)

ρ cp(E) (16)








6






7





31 Experiments









8





F2 2 240 120 06 001F3 5 600 300 05 15 00264F4 10 1200 600 30 00528

F6 2 120 120 06 000528F7 5 300 300 10 15 00132F8 10 600 600 30 00264

F10 2 60 120 06 000264F11 5 150 300 20 15 00066F12 10 300 600 30 00132







ρlowast = ρx3

m νlowast = ν

t

x2 σ lowast = σ

t2

m glowast = g

t2

x

klowast = kt

x2 etc (18)


4 Results


9








10









11






12





5 Discussion


13








η2 (20)



tint = D

v (21)



tdiff = d2l

k (22)


tcap = η middot L

σ (23)


tray =radic

ρ middot L3

σ (24)


14





v = D middot k

d2l

(25)










15








16



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References



ξ(T ) = H(T )

L (13)


E =int T

0ρ cp dT + ρ H =

int T

0ρ cp dT (14)

with

cp =

cp T Tf

cp +L


cp T lt Ti

(15)


k(E) = λ(E)

ρ cp(E) (16)








6






7





31 Experiments









8





F2 2 240 120 06 001F3 5 600 300 05 15 00264F4 10 1200 600 30 00528

F6 2 120 120 06 000528F7 5 300 300 10 15 00132F8 10 600 600 30 00264

F10 2 60 120 06 000264F11 5 150 300 20 15 00066F12 10 300 600 30 00132







ρlowast = ρx3

m νlowast = ν

t

x2 σ lowast = σ

t2

m glowast = g

t2

x

klowast = kt

x2 etc (18)


4 Results


9








10









11






12





5 Discussion


13








η2 (20)



tint = D

v (21)



tdiff = d2l

k (22)


tcap = η middot L

σ (23)


tray =radic

ρ middot L3

σ (24)


14





v = D middot k

d2l

(25)










15








16



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References






7





31 Experiments









8





F2 2 240 120 06 001F3 5 600 300 05 15 00264F4 10 1200 600 30 00528

F6 2 120 120 06 000528F7 5 300 300 10 15 00132F8 10 600 600 30 00264

F10 2 60 120 06 000264F11 5 150 300 20 15 00066F12 10 300 600 30 00132







ρlowast = ρx3

m νlowast = ν

t

x2 σ lowast = σ

t2

m glowast = g

t2

x

klowast = kt

x2 etc (18)


4 Results


9








10









11






12





5 Discussion


13








η2 (20)



tint = D

v (21)



tdiff = d2l

k (22)


tcap = η middot L

σ (23)


tray =radic

ρ middot L3

σ (24)


14





v = D middot k

d2l

(25)










15








16



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References





31 Experiments









8





F2 2 240 120 06 001F3 5 600 300 05 15 00264F4 10 1200 600 30 00528

F6 2 120 120 06 000528F7 5 300 300 10 15 00132F8 10 600 600 30 00264

F10 2 60 120 06 000264F11 5 150 300 20 15 00066F12 10 300 600 30 00132







ρlowast = ρx3

m νlowast = ν

t

x2 σ lowast = σ

t2

m glowast = g

t2

x

klowast = kt

x2 etc (18)


4 Results


9








10









11






12





5 Discussion


13








η2 (20)



tint = D

v (21)



tdiff = d2l

k (22)


tcap = η middot L

σ (23)


tray =radic

ρ middot L3

σ (24)


14





v = D middot k

d2l

(25)










15








16



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References





F2 2 240 120 06 001F3 5 600 300 05 15 00264F4 10 1200 600 30 00528

F6 2 120 120 06 000528F7 5 300 300 10 15 00132F8 10 600 600 30 00264

F10 2 60 120 06 000264F11 5 150 300 20 15 00066F12 10 300 600 30 00132







ρlowast = ρx3

m νlowast = ν

t

x2 σ lowast = σ

t2

m glowast = g

t2

x

klowast = kt

x2 etc (18)


4 Results


9








10









11






12





5 Discussion


13








η2 (20)



tint = D

v (21)



tdiff = d2l

k (22)


tcap = η middot L

σ (23)


tray =radic

ρ middot L3

σ (24)


14





v = D middot k

d2l

(25)










15








16



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References








10









11






12





5 Discussion


13








η2 (20)



tint = D

v (21)



tdiff = d2l

k (22)


tcap = η middot L

σ (23)


tray =radic

ρ middot L3

σ (24)


14





v = D middot k

d2l

(25)










15








16



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References









11






12





5 Discussion


13








η2 (20)



tint = D

v (21)



tdiff = d2l

k (22)


tcap = η middot L

σ (23)


tray =radic

ρ middot L3

σ (24)


14





v = D middot k

d2l

(25)










15








16



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References






12





5 Discussion


13








η2 (20)



tint = D

v (21)



tdiff = d2l

k (22)


tcap = η middot L

σ (23)


tray =radic

ρ middot L3

σ (24)


14





v = D middot k

d2l

(25)










15








16



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References





5 Discussion


13








η2 (20)



tint = D

v (21)



tdiff = d2l

k (22)


tcap = η middot L

σ (23)


tray =radic

ρ middot L3

σ (24)


14





v = D middot k

d2l

(25)










15








16



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References








η2 (20)



tint = D

v (21)



tdiff = d2l

k (22)


tcap = η middot L

σ (23)


tray =radic

ρ middot L3

σ (24)


14





v = D middot k

d2l

(25)










15








16



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References





v = D middot k

d2l

(25)










15








16



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References








16



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References



6 Summary




Acknowledgments


References









17
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References
















18

1 Introduction

2 Physical model






31 Experiments


4 Results




5 Discussion




6 Summary

Acknowledgments

References

Fundamental consolidation mechanisms during selective beam

Documents

Transcript of Fundamental consolidation mechanisms during selective beam