Superalloys 2016: Proceedings of the 13th International Symposium on Superalloys. Edited by: Mark Hardy, Eric Huron, Uwe Glatzel, Brian Griffin, Beth Lewis,.
Superalloys 2016: Proceedings of the 13th International Symposium on Superalloys Edited by: Mark Hardy, Eric Huron, Uwe Glatzel, Brian Griffin, Beth Lewis, Cathie Rae, Venkat Seetharaman, and Sammy Tin TMS (The Minerals, Metals & Materials Society), 2016
A MULTI-SCALE MULTI-PHYSICS APPROACH TO MODELLING OF ADDITIVE MANUFACTURING IN NICKEL-BASED SUPERALLOYS C. Panwisawas1,*, Y. Sovani1, M.J. Anderson1, R. Turner1, N.M. Palumbo2, B.C. Saunders2, I. Choquet3, J.W. Brooks1 and H.C. Basoalto1 1
School of Metallurgy and Materials, University of Birmingham, Birmingham B15 2TT, UK 2 Rolls-Royce plc, PO Box 31, Derby DE24 8BJ, UK 3 Department of Engineering Science, University West, 461 86 Trollhättan, Sweden
Keywords: Multi-scale modelling, Additive manufacturing, Thermal fluid dynamics, IN718, Aerospace component. most recent methodologies used to unlock benefits through costeffective, efficient materials and process design. With a requirement to consider the modelling of small-scale interactions between heat and powder particles [5-7] as well as the mechanical simulation of distortion, a well-established framework is rarely reported in literature. In this work, a multi-scale, multi-physics modelling approach (considering both small-scale interactions and large-scale component analyses) has been employed for the simulation of AM in nickel-based superalloys.
Abstract A multi-scale, multi-physics modelling framework of selective laser melting (SLM) in the nickel-based superalloy IN718 is presented. Representative powder-bed particle distribution is simulated using the measured size distribution from experiment. Thermal fluid dynamics calculations are then used to predict melting behaviour, sub-surface morphology, and porosity development during a single pass scanning of the SLM process. The results suggest that the pores and uneven surface structure are exacerbated by increasing powder layer thicknesses. Predicted porosity volume fraction is up to 12% of the single track when 5 statistical powder distributions are simulated for each powder layer thickness. Processing-induced microstructure is predicted by linking cellular automatons – finite element calculations indicate further that the cooling rate is about 4400 oC/s and grain growth strongly follows the thermal gradient giving rise to a columnar grain morphology if homogeneous nucleation is assumed. Random texture is likely for as-fabricated SLM single pass with approximately 8 Pm and 6 Pm grain size for 20 Pm and 100 Pm powder layer thickness fabrication. Use has been made of the cooling history to predict more detailed microstructure using a γ" precipitation model. With the short time scale of solidification and rapid cooling, it becomes less likely that γ" precipitation will be observed in the condition investigated unless a prolonged hold at temperature is carried out. Future work on extension of the proposed multiscale modelling approach on microstructure predictions in SLM to mechanical properties will be discussed.
Selective laser melting (SLM) [8-10] and direct laser deposition (DLD) [11-12] are both versatile methodologies for the fabrication of components with arbitrary and complex threedimensional geometry. Both methods use consecutive layer-bylayer deposition and consolidation using a prescribed laser scanning path, with the assistance of a computer aided design (CAD) model. The interaction of the laser heat source with the superalloy powder particles (and of powder particles with each other), is a complex fundamental mechanism upon which the SLM and DLD processes are reliant. However, this complex phenomenon can be simulated via thermal fluid flow modelling, in order for the evolution of the metallic/gaseous interface to be predicted and better understood. To better understand the melt flow mechanics of the powder particles within SLM, one first needs to envisage a representative statistical distribution of the powder particles using a randomly packed powder bed using a 2D lattice Boltzmann [13] or a “particle-raining” model presented here, whereby the powder particles are dropped in to place onto a packed configuration. It is understood that (i) the size distribution of powder particles, (ii) packing density [14] and (iii) deposited layer thickness all provoke distinct powder packing configurations and hence statistical characteristics. From these statistical descriptions it is possible to determine process induced pore distributions. Experimentally measured size distributions of powder particles of the IN718 polycrystal superalloy have been used as a sample space for picking up a randomly selected particle, and dropping them one-by-one into the deposition layer configuration, until this built layer is filled to the desired thickness. After pre-spreading of powder feedstock, the laser heat source adopted from refs. [5-6] is scanned across the layer configuration, and the subsequent melt flow behavior is predicted using computational fluid dynamics to qualify and quantify the densification as a function of laser scanning speed and laser power. Microstructure variations induced by the solid-liquid interaction during the SLM process can be predicted using the temperature history and melt pool information from the thermal fluid flow calculation.
Introduction Additive manufacturing (AM) is a novel processing route of increasing interest to a wide number of industries with a considerable number of potential applications, including those for safety-critical aerospace and gas turbine components. Metallic components, especially superalloy parts, can be both fabricated and repaired using the AM technique; however this requires stringent quality control measures to assure part integrity, as the process can be susceptible to defects such as sub-surface voids, poor surface finish, and is also likely to produce some level of residual stress and distortion. Computational modelling therefore can provide an important tool to better understand the physical phenomena associated with the AM process, thus acting as a precursor to tailored experimental procedures [1-2]. In recent years, modelling of additive manufacturing processes has received considerable interest from both the industrial and academic communities. Driven by the rapid developments and increasing use of additive manufacturing techniques, there is a need for the development of modelling capabilities that complement experimental characterisation [3-4]. Integrated computational materials science/engineering (ICME) framework is one of the
The component level response will depend on the emerging microstructure [15]. Spatial distributions of process induced
1021
Figure 1: Proposed multi-scale framework to modelling of selective laser melting in nickel-based superalloy pores, second phase particles and grain size will result in variations in the flow stress of the material and consequently impact on the evolution of residual stresses and component level distortions. A state variable approach has been developed for the plastic behavior of the material that explicitly links the local alloy microstructure to the constitutive response of the material.
encountered. Several potential obstacles and corresponding trajectories of a falling particle are illustrated in Figures 2(c). The dropping sequence of powder particles was determined using a face-centred cubic (FCC) lattice, which ensures that the particles were dropped layer-by-layer. The exact location of the x-y drop coordinate was offset by a random amount in a random direction. The distance was a maximum of 20% of the lattice parameter used to generate the lattice.
The aim of this paper is to present a multiscale material modelling approach to additive manufacturing for IN718 nickel-based superalloy. A summarised framework is presented as illustrated in Figure 1.
The algorithm detected any obstacle encountered by the falling particle along its trajectory. A new trajectory was determined based upon the number of obstacles encountered, as illustrated in Figures 2(c). A particle can only come to rest if it encounters three stable obstacles or falls on the floor. The particles were dropped parallel to the z axis with periodic boundary conditions being applied along the x and y axis. The powder dropping model provides a calculation domain through which the desired layer thicknesses can be achieved and the thermal fluid flow calculation can be performed. Examples of 50 Pm and 100 Pm powder layer thickness model are shown in Figures 2(d) and 2(e), respectively.
Method Powder size distribution model To investigate the melting behaviour of powder particles during SLM it necessary to numerically generate similar powder particle arrangements for simulation. The generated and measured particle powder distribution is anticipated as seen in Figure 2(a). The methodology is successfully implemented in Panwisawas et al. [16] and it is here estimated that the as-received IN718 powder particle size falls into the range of 10 – 40 μm in diameter based on scanning electron micrographs [8]. A probability density function (PDF) based on a Gaussian distribution was used to represent the probability of a finding a particle with radius lying between R and R + dR. Integration of the PDF over particle sizes gives the cumulative distribution function of the measured powder particle size distribution. The measured and fitted cumulative distribution functions are compared in Figure 2(b).
Melt flow To predict the surface morphology of the build, porosity and microstructure development, a computational fluid dynamics (CFD) approach using the C++ open source CFD toolbox socalled Open Field Operation and Manipulation (OpenFOAM ®) has been developed to model the interaction between the laser heat source and the randomly distributed IN718 powder materials, which is illustrated in Figure 3(b). In the model, all interfacial phenomena, including surface tension (capillary force), Marangoni’s flow (thermo-capillary force), recoil pressure, drag force due to solid/liquid transition via Darcy’s term, and buoyancy force, present within the SLM process have been included in the calculations. The energy dissipation in the mushy zone during
Simulation of the spatial arrangements of particles resulting from deposition was approximated by tracking particles dropped onto a substrate [17]. This approach ignores the motion of obstacles encountered by the falling particles and the trajectory of a particle is determined based on the number of obstacles being
1022
Figure 2: (a) comparison of powder particle distribution between experiment [8] and modelling, (b) cumulative distribution function of IN718 particle, (c) schematic diagram of particle arrangement, (d) 50 Pm powder layer thickness model, and (e) 100 Pm powder layer thickness model. melting, and heat loss due to evaporation, conduction, convection and radiation have also been taken into account in this work. Accounting for energy reflections of the laser beam is not treated in the present paper.
݉ሶ ൌ ට
ଶగಳ ்
and the recoil pressure is phenomenologically described by: οு ଵ ଵ ሺܶሻ ൌ ቄ ೇ ቀ െ ቁቅ ோ
ഥ࢛ డఘ డ௧
்
સ ή ሺߩҧ ࢛۪࢛ሻ ൌ െસ સ ή ܶധ ߩҧ ݃ࢋොࢠ ߚሺܶ െ ܶ୰ୣ ሻ െ
ܭ ቀ
ሺଵିಽ ሻమ ಽయ ା಼
ቁ ࢛ ቂߪߢ ෝ
ෝ ሺ ॴ ή
ௗఙ
൫સܶ െ ෝ ሺ ෝ ή સܶሻ൯
ௗ் ഥ ଶఘ ෝ ሻቃ ȁસߙଵ ȁ ሺఘ భ ାఘమ ሻ
(3)
Equation (3) implies that the rate of change of fluid momentum on the left hand side (LHS) is driven by all interfacial forces on the RHS. Divergence of stresses is comprised of hydrostatic pressure, p, and the viscous deviatoric stress tensor, ܶധ, expressed as: ଵ ଵ ଵ ܶധ ൌ ʹߤҧ ቂቀ સ࢛ ሺસ࢛ሻ் ቁ െ ሺસ ή ࢛ሻॴቃ
For this CFD calculation, it is assumed that the alloy is incompressible, so that the flow velocity ࢛ satisfies the continuity condition: સ ή ࢛ ൌ Ͳ (1) The volume-of-fluid equation used to predict the evolution of the liquid/gas interface: డఈభ ሶ સ ή ሺߙଵ ࢛ሻ ൌ െ ೇ (2) డ௧
்ೇ
where, οܪ and ܴ are atmospheric pressure, enthalpy change due to evaporation and universal gas constant, respectively [1922]. As reported previously [5-6], at the start of the interaction between the heat source and the materials during SLM, the model predicts the kinetics of the melt pool. All interfacial forces present during the SLM process have been inserted into the conservation of momentum or Navier-Stokes equation, which is in the form:
The computation domain is divided into a solid/liquid α1 and vapour α2 regions. The solid and liquid constitutive behaviours are defined within α1 by introducing appropriate phase transformations depending on the temperature being above or below the solvus temperature. Above the vaporisation temperature the transformation α1 o α2 is allowed to take place corresponding to molten metal transforming to metal vapour. The latter phase is donated by α2. Modelling the dynamics of the phases during the process is achieved by tracking the evolution of the α1 and α2 phases by a continuity condition with appropriate source/sink terms corresponding to phase transitions. By definition, the summation of metallic ߙଵ and gaseous phases ߙଶ is always unity, i.e. ߙଵ ߙଶ ൌ ͳǡ in every fluid element. In this study, the gaseous phase is generally air atmosphere, unless specified otherwise. Additionally, a weight function of any parameter ݔis used to smear out the effect of metallic and gaseous phases, defined as, ݔҧ ൌ ݔଵ ߙଵ ݔଶ ߙଶ .
ଶ
ଶ
ଷ
Here, ߩҧ is the density, ߤҧ is viscosity and ॴ is the identity matrix. The third term on RHS of Equation (3), namely ߩҧ ݃ࢋොࢠ ߚ൫ܶ െ ܶ ൯, is used to describe the buoyancy force caused by density differences due to thermal expansion, where ݃and ݁Ƹ௭ are the magnitude and unit normal of gravitational force, ߚ is thermal expansion coefficient, ܶ is temperature field and ܶ୰ୣ is the
ఘమ
where ݐis time, and the sink term in the right hand side (RHS) describes the loss of metallic phase due to evaporation when the evaporation temperature ܶ is reached. In this work, ߩଶ refers to the density of metal vapour, which is no different from atmospheric gas phase as chemical species are not distinguished here. The mass evaporation rate ݉ሶ is defined as:
reference temperature. Darcy’s term, െܭ ቀ
ሺଵିಽ ሻమ ಽయ ା಼
ቁ ࢛ , which is
energy dissipation or sink (damping) terms in the mushy zone is modelled in this work by making use of the Carman-Kozeny equation [23], where ܭ ǡ ݂ and ܥ are the permeability coefficient, fraction of liquid metal and a constant, respectively. There are three surface force terms considered here:
1023
(i) ߪߢ ෝ is the surface tension term (capillary force) [24-25], with surface tension ߪ, acting on surface curve ߢ at the unit normal ෝ, ௗఙ ൫સܶ െ ෝ ሺ ෝ ή સܶሻ൯ is the Marangoni force (thermo(ii) ௗ் capillary force) [26-28], taking into account the effect of temperature gradient on the liquid/gas interface, ෝ ሻ is the recoil pressure when evaporation occurs. (iii) ෝ ሺ ॴ ή
where ݂௦ is the volume fraction of solid already formed. With the aid of the KGT model by [33], growth kinetics of a dendritic tip, described by the dendritic tip velocity, ݒሺοܶ ሻ, related to its undercooling can be expressed in Equations (5)-(8) of [34-35], which can be approximated to be: ݒሺοܶ ሻ ൌ ܽଶ οܶ ଶ ܽଷ οܶ ଷ
All surface forces have been applied only at the interface indicated by the interface term ȁસߙଵ ȁ and the sharp surface force ഥ ଶఘ term ሺఘ is used to smear out between hard (metallic) and soft ሻ
The parameters a2 and a3 have been calculated using an IN718 thermodynamics database, as described in [15]. In a CAFE model, the finite element (FE) calculation of the thermal field has been coupled with the CA grain growth model. The CA dendritic growth algorithm was dependent upon the thermal gradient and isotherm velocity which in turn gave the undercooling temperature and had been solved in the thermal FE scheme. The thermal field from the tetrahedral FE mesh could then be interpolated into the grain growth hexahedral CA mesh to obtain dendritic growth, provided that a homogeneous nucleation condition was assumed. Grain morphology and texture can be predicted thereafter.
భ ାఘమ
(gaseous) phases. The conservation of total energy is then written as: ഥ ҧ ் డఘ డ௧
ഥ οு డఘ
െ સ ή ൫ߩҧ ࢛οܪ ൯ સ ή ൫݇തસܶ൯ ସ ൯ ܳ ൯ȁસߙଵ ȁ െ െൣ൫݄ ൫ܶ െ ܶ ൯ ߪ௦ ߳൫ܶ ସ െ ܶ
સ ή ൫ߩҧ ࢛ܥҧ ܶ൯ ൌ െ
்ܳ ൧
ഥ ଶҧ ఘ
డ௧
(4)
ҧ ఘభ ାమ ҧ ఘమ ൯ ൫భ
The thermal energy is balanced between heat input due to the heat source term ்ܳ [29] and heat losses, namely heat loss due to; (i) conduction, સ ή ൫݇തસܶ൯, (ii) convection, ݄ ൫ܶ െ ܶ ൯, (iii) ସ ൯ and (iv) evaporation effects, radiation, ߪ௦ ߳൫ܶ ସ െ ܶ ܳ ̱݉ሶ οܪ [19-22].
Note that the FE thermal model is calculated in ABAQUS to obtain the temperature profile for particle precipitation and mechanical models. γ" precipitation model
Here,ܥҧ is specific heat for the mixture, ݇ത is thermal conductivity of the mixture, οܪ is the enthalpy change due to fusion, ݄ is heat transfer coefficient, ߪ௦ is Stefan-Boltzmann constant, and ߳ is emissivity. By solving the set of Equations (1), (2), (3) and (4), the evolution of melt kinetics and liquid/gas interface change can be analysed and rationalised. A detailed model description is to be published elsewhere [5-6,18] and model parameters were adopted from [30-32]. The CFD model has been applied to the generated powder particle distribution with a calculation domain of 250 μm × 1000 μm × 250 μm, containing 4 million elements with the constant hexahedral mesh size of 2.5 μm, and a laser heat source of 400 W with a scanning speed of 2400 mm/s to simulate the processing condition. The modelling results are compared with the relevant experimentation.
IN718 superalloy is a γ" precipitate strengthened alloy. The nucleation and growth of γ" has been modelled to investigate the particle dispersions that form within the solidified component. Prediction of the particle dispersion is important in determining location specific properties such as the yield behaviour, which in turn can affect process-induced residual stress and component distortions. A multi-component mean field model of the particle size distribution has been implemented and applied to describe the formation of γ" within solidified material during additive manufacture. The expression for the particle growth rate used in the mean field model is based on that proposed by SvobodaFischer-Fratzl-Kozeschnik (SFFK) [37]. Shape factors have been used to describe the γ" precipitates as cylindrical discs [38]. The aspect ratio of the diameter and height of cylindrical particles as a function of size have been obtained from Fisk et al. [39].
Evolution of grain microstructure
The evolution of the particle size distribution is determined by solving: ߲ܨሺܴǡ ݐሻ ߲ሾܨሺܴǡ ݐሻܸሺܴǡ ݐሻሿ ൌ ܫሺܴǡ ݐሻሺͺሻ ߲ܴ ߲ݐ where ܨሺܴǡ ݐሻ refers to the particle number density within a representative volume, ܴ is the radius of a sphere of equivalent volume of the particle, ܸ is the particle growth rate, ܫis the nucleation rate and ݐis time. The general form for the particle growth rate is given by: ଵ ଵ ܸ ൌ ቀ െ ቁݖ (9)
During the solidification of an AM-built structure, grain nucleation and growth phenomena occur. This is simulated using a three-dimensional cellular automaton – finite element (CAFE) code [33-36]. The molten single track and temperature history along the scanning path were exported from the OpenFOAM CFD model outlined in the previous section. This was used to define the solidification domain and the dynamic temperature boundary conditions required to simulate the moving laser heat source and temperature profile. In the CAFE model, the cellular automaton grid was re-meshed with a hexagonal mesh size of 2 μm along the molten scanning track. A continuous nucleation distribution, ௗ , is used to describe the increase of grain density, ݀݊,
ோ ோ
induced by the increase in the undercooling, ݀ሺοܶ ሻ. The total density of grains at a specific ሺοܶ ሻ , is defined by: ο்
and
ௗ ௗሺο் ሻ
ൌ
ௗ ௗሺο் ᇲ ሻ
ౣ౮ ο்బ ξଶగ
ሾͳ െ ݂௦ ሺܶ ᇱ ሻሿ݀ሺοܶ ᇱ ሻ ቀെ
തതതതത ଶ ο் ିο் ξଶο்బ
ቁ
ோ
where the term ܣcontains the diffusivity of particle forming species, ܴ is the critical radius for growth, and ݖis the correction factor for describing non-dilute particle dispersions. The terms for ܣand ܴ including the shape factors within the SFFK model are given in Equations (10) and (11),
ௗሺο் ሻ
݊ሺοܶ ሻ ൌ
(7)
ଶఊೖ ௌೖ
(5)
ܣൌ
(6)
ܴ ൌ
1024
ைೖ ோ ்
ቂσୀଵ ଶఊೖ ௌೖ
ሺೖ ିబ ሻమ ିଵ బ బ
ି σ సభ ೖ ሺఓೖ ିఓబ ሻ
ቃ
(10) (11)
where ߛ is the interfacial energy, ܴ is the gas constant, ܶ is the absolute temperature . If there are ݊ many alloying elements, then the ݅ ୲୦ component has a concentration within the matrix and particle phase given by ܥ and ܥ respectively. The chemical potential of the particle and matrix phases are given by ߤ and ߤ . The terms ܵ and ܱ are shape parameters which are a function of the aspect ratio of the cylinder. The Marqusee and Ross [40] ݖcorrection factor was used, and is given below: ݖൌ ͳ ܴඥͶߨܰ௩ ܴത (12) where the mean particle radius and concentration is given by ܴത and ܰ௩ , respectively. Nucleated particles are treated as spherical, using classical nucleation theory; ିఛ ܫൌ ܼܰߚ ݔ݁ כቀ ቁ (13) ௧ where ܼ refers to the Zeldovitch parameter, ܰ is the number density concentration of nuclei, ߚ כis the atomic attachment rate and ߬ is the incubation time. Thermocalc [41] with the TTNi8 database was used to calculate chemical potentials and particle compositions. The output was calibrated with an energy contribution of 277.7 J/mol to the Gibbs free energy of the γ" phase, obtaining a γǯǯ solvus temperature of 940 °C, similar to that
reported by de Jaeger et al. [42]. A pseudo-binary form of Equation (13) was used, with an effective diffusivity calibrated to capture the particle nucleation, growth and coarsening data collated by Fisk et al., [43]. Material data for Inconel 718 can be found from [44-48]. Results This section focuses on the implementation of the proposed multiscale materials framework of the SLM process of nickel-based superalloys. To begin with the model considers the interaction between the heat source and superalloy powder particles. Figure 3 shows the morphological development after the single pass scanning. It is clear that the sub-surface porosity can be observed in models, and these are potentially exacerbated when the powder layer thickness increases. The results suggest further that surface roughness depends upon the powder particle distribution, packing density and hence the layer thickness, as simulated by CFD model for 5 layers of each considered layer thickness. Variation in porosity volume fraction is observed up to 12%, see Figure 4. Even with the energy density kept constant for all thicknesses, the thicker layers give the higher porosity fraction.
Figure 3: Predicted sub-surface morphology after single pass scanning for (a) 20 Pm, (b) 40 Pm, (c) 50 Pm, (d) 60 Pm, (e) 80 Pm, and (f) 100 Pm powder layer thicknesses. The molten zone of a single-pass track can be extracted from the CFD results and written in a geometry type format. It can be observed that porosity becomes trapped inside the track and the amount of porosity varies with the powder layer thickness. The cooling history associated with the molten track was assigned on the surface boundary, as seen in Figure 5. Some temperature variations have been observed when different powder distribution was used and compared between the molten tracks of 50 Pm (Figure 5(a)) and 100 Pm (Figure 5(b)) layer thicknesses. Prediction from the cellular automata results shows the grain morphology and texture information during solidification. The general features of a section through a portion of the single pass of 50 Pm and 100 Pm layer thicknesses are illustrated in Figures 6(a) and 6(b), respectively. The grain growth follows the thermal gradient and solidification front, resulting in columnar grains along the build and some equiaxed grains around the pores.
Figure 4: Prediction of porosity volume fraction along the single track as a function of powder layer thickness using different statistical powder distribution.
1025
Figure 5: Three-dimensional view of molten zone after single track scanning for (a) 50 Pm and (b) 100 Pm layer thicknesses. The liquidus temperature is 1609 K (1336 oC).
Figure 6: CA results show the predicted grain structure: (a) 50 Pm single track, (b) 100 Pm single track, (c) fraction solid and misorientation prediction of solidification behavior of 50Pm single track, (d) fraction solid and misorientation prediction of solidification behavior of 100 Pm single track. The solidification model reveals further that the texture of the single-pass is random in general, backed up by the pole figure For the single-pass simulation with 50 Pm layer thickness and 250 plots of 50 Pm layer thickness track and 100 Pm layer thickness Pm length, the model predicts a solidification time of about 17 track (Figures 7(a) and 7(b)). The mean angle of misorientation is ms. Whereas for the 100 Pm layer thickness and 125 Pm length about 31º and 29º for 50 Pm and 100 Pm thickness track track the predicted time to solidify also is approximately 17 ms. respectively, see the misorientation maps of Figures 7(c) and 7(d). Therefore, the model predicts solidification time to be doubled for In terms of grain size, the model predicts as-fabricated structure double the layer thickness. This is equivalent to the solidification corresponds to prior beta grains, which are about 6.5-8.3 Pm in speed of 68 m/s and 136 m/s and cooling rate of 4368 ºC/s and size – see Figure 7(e) and 7(f). 4444 ºC/s for 50 Pm and 100 Pm layer thicknesses, respectively.
1026
where ȫ ା ൫ߪǡ ܶǡ ݀ ǡ ߣ ൯ is a generation rate for the mobile dislocation density,ߩ௦ the number of sources, and ݀ ǡ is the mean grain size and ߣ is a mean free path associated with annihilation of dislocations (i.e., formation of dipoles). Values of ݀ are obtained from the CAFE model predictions (see Figure 7). A more detailed description of the modelling approach and the results is published elsewhere [15].
influence the structural integrity. In this work, one scanning speed, one laser power and six powder layer thicknesses have been studied. The present paper has focused on simulating the microstructures arising from the additive manufacture. To determine the mechanical response of the component, this information must be coupled to a constitutive description of the elastic and plastic behaviour of the alloy. A state variable formulation can be used to achieve this aim. Such an approach has been successfully applied to a range of precipitate strengthened nickel-based superalloys to model their viscoplastic behaviour [49,50]. The constitutive description of the hardening behaviour is associated with a kinematic stress, ߪ , arising from stress partitioning between the J and particle phases (Jc and Jcc) and dislocation interactions. A number of strain softening mechanisms are considered and include the mobile dislocation density, ߩ , as well as the increasing plastic rates due to presence of voids. A proposed form for the plastic rate,ߝሶ , is [51], ߝሶ ൌ
ሺఙିఙೖ ሻఒ మ ସగఘ ೡ ܨ൫߶ ൯ ൬ ൰ ഥ ഥ ் ெ ൫ଵିథೝ ൯ெ
െ ͳ൨
Summary and conclusion A multi-scale, multi-physics approach to modelling of selective laser melting has been presented. Specific conclusions can be drawn from this work. 1. A powder particle model has been developed to study the statistical distribution of the SLM process. Variations in size and position give rise to different powder arrangement. 2. Thermal fluid flow calculation indicates that sub-surface porosity and surface structure of the single track are exacerbated by thick powder layer thickness. Cave-like pores and discontinuous track can be observed at thick powder layer. 3. Porosity volume fraction is predicted to vary from 2% up to 12% at the range of 20-100 Pm layer thicknesses. 4. The 50 Pm and 100 Pm thick molten tracks solidify with the velocity of solidification front of 68 m/s and 136 m/s, and cooling rates of 4368 ºC/s and 4444 ºC/s, respectively. 5. The predicted grain structure is equiaxed close to the substrate. Columnar grains grow along the build direction. Texture is predicted to be random with the mean size of 8.3 Pm and 6.5 Pm for 50 Pm and 100 Pm powder layer thicknesses, respectively. 6. The γ" precipitation model predicts little variation in mean particle size and negligible γ" particle volume fraction as the cooling rate is high in this process. 7. The linkage between micro-scale predictions of porosity, grain size and particle precipitation and the simulation of macro-scale response was discussed and a potential constitutive framework identified.
(14)
Here, the volume diffusivity is ܦ௩ , particle volume fraction is ߶ , the inter-particle spacing is ߣ , Young’s modulus (assumed equal, ഥ , ܴ is the gas matrix/particle) isܧ, the mean Taylor factor is ܯ constant, ݇ is Boltzmann’s constant, ܾ the magnitude of burger vector and ߶ the pore volume fraction. The ߶ can be calculated using the relation, ߶ ൌ
ସగ ଷ
ஶ
݂ሺݎሻ ݎଷ ݎ
(15)
where ݂ሺݎሻ is a probability density function for the pore volume fractions derived from the numerical outputs of the CFD model (see Figure 4) outlined in Method section. As already mentioned, the kinematic back stress evolution has two contributions associated with particle/matrix stress transfer and dislocation interactions. We can write the following differential equation for ߪ , ߪሶ ൌ ߶ ܧቀͳ െ
ఙೖ ு כఙ
ቁ
ఈீ ଶඥఘ
ߩሶ
Acknowledgments
(16)
The authors would like to acknowledge Professor Håkan Nilsson of Chalmers University of Technology for useful discussion and Professor Hrvoje Jasak of University of Zagreb for fruitful discussion on OpenFOAM coding. C.P, M.J.A, Y.S, R.P.T, H.C.B, J.W.B thank the support by the Centre for Advanced Simulation and Modelling collaborative project between RollsRoyce plc, Manufacturing Technology Centre and the University of Birmingham and the support by the European Regional Development Fund.
with ܧൌ ܧ ൫ͳ െ ߶ ൯
(17)
The maximum stress transfer is given by ߪ כ ܪ, where כ ܪis a function of the particle volume fraction [50], כܪൌ
ଶథ
(18)
ଵାଶథ
References
The particle dispersion parameters such as ߶ and ߣ are obtained from the mean field modelling framework (see Figures 8 and 9) described earlier in the Method section. The dislocation density evolution is modelled in terms of a continuity relationship for the generation and annihilation/trapping of dislocation, i.e., ߩሶ ൌ ߩ௦ ȫ ା ൫ߪǡ ܶǡ ݀ ǡ ߣ ൯ െ ߩ
௩ത ఒ
[1] C.D. Boley et al., “Modelling of powder absorption in additive manufacturing,” Optical Society of Merica CLEO, (2014). [2] S.A. Khairallah, and A. Anderson, “Mesoscopic simulation model of selective laser melting of stainless steel powder,” Journal of Materials Processing Technology, 214, (2014), 2627-2636.
(19)
1028
[3] X. Tan et al., “Graded microstructures and mechanical properties of additive manufactured Ti-6Al-4V via electron beam melting,” Acta Materialia, 97, (2015), 1-16. [4] X. Zhou et al., “3D-imaging of selective laser melting defects in a CoCr-Mo alloy by synchrotron radiation micro-CT,” Acta Materialia, 98, (2015), 1-16.
sheets,” Production Engineering: Research and Development, 3, (2009), 127-136. [22] M. Courtois et al., “A new approach to compute multi-reflection of laser beam in a keyhole for heat transfer and fluid flow modelling in laser welding,” Journal of Physics D: Applied Physics, 46, (2013), 505305(14pp).
[5] C. Panwisawas et al., “On the role of thermal fluid dynamics into the evolution of porosity during selective laser melting,” Scripta Materialia, 105, (2015), 14-17.
[23] F. Rösler, and D. Brüggemann, “Shell-and-tube type latent heat thermal energy storage: numerical analysis and comparison with experiments,” Heat Mass Transfer, 47, (2011), 1027-1033.
[6] C.L. Qiu et al., “On the role of melt flow into the surface structure and porosity development during selective laser melting,” Acta Materialia, 96, (2015), 72-79.
[24] Y. Sun, and C. Beckermann, “Diffuse interface modelling of twophase flows based on averaging mass and momentum equations,” Physica D, 198, (2004), 281-308.
[7] F. Verhaeghe et al., “A pragmatic model for selective laser melting with evaporation,” Acta Materialia, 57, (2009), 6006-6012 .
[25] J.U. Brackbill et al., “A continuum method for modelling surface tension,” Journal of Computational Physics, 100, (1992), 335-354.
[8] K.N. Amato et al., “Microstructures and mechanical behaviour of Inconel 718 fabricated by selective laser melting,” Acta Materialia, 60, (2012), 2229-2239.
[26] L.E. Scriven, and C.V. Sternling, “The Marangoni effects,” Nature, 187, (1960), 186-188.
[9] Q. Jia, and D. Gu, “Selective laser melting additive manufacturing of Inconel 718 superalloy parts: Densification, microstructure and properties,” Journal of Alloys and Compounds, 585, (2014), 713-721.
[27] T. Ytrehus, and S. Østomo, “Kinetic theory approach to interphase processes,” International Journal of Multiphase Flow, 22, (1996), 133155.
[10] L.N. Carter et al., “The influence of the laser scan strategy on grain structure and cracking behaviour in SLM powder-bed fabricated nickel superalloy,” Journal of Alloys and Compounds, 615, (2014), 338-347.
[28] H. Ki et al., “Modelling of high-density laser-material interaction using fast level set method,” Journal of Physics D: Applied Physics, 34, (2001), 364-372.
[11] L.L. Parimi et al., “Direct laser fabrication of Inconel-718: Effects on microstructure and distortion,” Superalloy2012, ed. by E.S. Huron et al. (Warrendale, PA : TMS, 2012), 511-519.
[29] G.X. Xu et al., “Adaptive volumetric heat source models for laser beam and laser + pulsed GMAW hybrid welding process,” The International Journal of Advanced Manufacturing Technology, 57, (2011), 245-255.
[12] L.L. Parimi et al., “Microstructural and texture development in direct laser fabricated IN718,” Materials Characterization, 89, (2014), 102-111.
[30] J.Z. Li et al., “Thermal expansion of liquid Ti-6Al-4V measured by electrostatic levitation,” Applied Physics Letter, 89, (2006), 111913(2pp).
[13] C. Körner et al., “Mesoscopic simulation of selective beam melting processes,” Journal of Materials Processing Technology, 211, (2011), 978-987.
[31] R. Rai et al., “Heat transfer and fluid flow during keyhole mode laser welding of tantalum, Ti-6Al-4V, 304L stainless steel and vanadium,” Journal of Physics D: Applied Physics, 40, (2007), 5753-5766.
[14] J. Zhou et al., “Numerical simulation of random packing of spherical particles for powder-based additive manufacturing,” Journal of Manufacturing Science and Engineering, 131, (2009) 031004.
[32] R. Rai et al., “Heat transfer and fluid flow during electron beam welding of 21Cr-6Ni-9Mn steel and Ti-6Al-4V alloy,” Journal of Physics D: Applied Physics, 42, (2009), 025503(12pp).
[15] Y. Sovani et al., “Property switching element approach to modelling of additive layer manufacturing,” in preparation, (2016).
[33] W. Kurz et al., “Theory of microstructural development during rapid solidification,” Acta Metallurgica, 34, (1986), 823-830.
[16] C. Panwisawas et al., “An integrated computational materials engineering approach to modelling of selective laser melting in nickelbased superalloys,” in preparation, (2016).
[34] M. Rappaz, and Ch.-A. Gandin, “Probabilistic modelling of microstructure formation in solidification processes,” Acta Metallurgica et Materialia, 41, (1993), 345-360.
[17] J. Zhou et al., “Numerical simulation of random packing of spherical particles for powder-based additive manufacturing,” Journal of Manufacturing Science and Engineering, 131, (2009) 031004.
[35] Ch.-A. Gandin, and M. Rappaz, “A coupled finite element-cellular automaton model for the prediction of dendritic grain structures in solidification processes,” Acta Metallurgica et Materialia, 42, (1994), 2233-2246.
[18] C. Panwisawas et al., “Physics-based keyhole modelling of laser fusion welding in titanium alloys: Keyhole dynamics, thermal fluid flow, and porosity formation,” in preparation, (2016).
[36] Ch.-A. Gandin et al., “A three-dimensional cellular automaton-finite element model for the prediction of solidification grain structures,” Metallurgical and Materials Transactions A, 30, (1999), 3153-3165.
[19] X. Jin et al., “A study on Fresnel absorption and reflections in the keyhole in deep penetration laser welding,” Journal of Physics D: Applied Physics, 35, (2002), 2304-2310.
[37] J. Svoboda et al. “Modelling of kinetics in multi-component multiphase systems with spherical precipitates: I: Theory,” Materials Science and Engineering A, 385(1–2), (2004), 166-174.
[20] X. Jin et al., “Multiple reflections and Fresnel absorption in an actual 3D keyhole during deep penetration laser welding,” Journal of Physics D: Applied Physics, 39, (2006), 4703-4712.
[38] E. Kozeschnik et al., “Shape factors in modeling of precipitation,” Materials Science and Engineering A, 441(1-2), (2006), 68-72.
[21] M. Geiger et al., “A 3D transient model of keyhole and melt pool dynamics in laser beam welding applied to the joining of zinc coated
1029
[39] M. Fisk et al., “Precipitate evolution in the early stages of ageing in Inconel 718 investigated using small-angle x-ray scattering,” Materials Science and Engineering A, 612, (2014), 202-207. [40] J.A. Marqusee, and J. Ross, “Theory of Ostwald ripening – competitive growth and its dependence on volume fraction,” Journal of Chemical Physics, 80(1), (1984), 536-543. [41] J. O. Andersson et al., “THERMO-CALC & DICTRA, computational tools for materials science.” Calphad-Computer Coupling of Phase Diagrams and Thermochemistry, 26(2), (2002), 273-312. [42] J. de Jaeger et al., “Inconel 718 single and multipass modelling of hot forging,” Superalloy2012, ed. by E.S. Huron et al. (Warrendale, PA : TMS, 2012), 663-672. [43] M. Fisk et al., “Flow stress model for IN718 accounting for evolution of strengthening precipitates during thermal treatment,” Computational Materials Science, 82, (2014), 531-539. [44] L. Camus, and G. Engberg, “Elevated temperature deformation of cast Inconel 718,” (1986), 1411-1422. [45] C. Slama, and M. Abdellaoui, “Structural characterization of the aged Inconel 718,” Journal of Alloys and Compounds, 306(1-2), (2000), 277284. [46] Y. F. Han et al., “Coarsening behavior of gamma’’ particles and gamma’ particles in Inconel alloy -718,” Metal Science, 16(12), (1982), 555-561. [47] C. Slama et al., “Aging of the Inconel 718 alloy between 500 and 750 degrees C,” Journal of Materials Research, 12(9), (1997), 2298-2316. [48] M. Sundararaman et al., “Precipitation of the δ-Ni3Nb phase in two nickel base superalloys,” Metallurgical and Materials Transactions A, 19(3), (1988), 453-465. [49] H. Basoalto et al., “A generic microstructure-explicit model of creep in nickel-base superalloys,” Superalloys 2004, ed. by K. Green et al. (Warrendale, PA : TMS, 2012), 897-906. [50] B.F. Dyson, “Microstructure based creep constitutive model for precipitation strengthened alloys: Theory and application,” Materials Science and Technology, 25, (2009) 213-220. [51] H. Basoalto, “A micromechanics description of high temperature deformation of 2-phase alloys,” Internal Report PR-2015-002, PRISM2, University of Birmingham, 2015.
1030
