Modeling of Powder Bed Fusion Additive Manufacturing

Modeling of Powder Bed Fusion

Additive Manufacturing

Doctoral Thesis in Industrial Engineering

University of Pisa

Department of Civil and Industrial Engineering Doctoral Program in Industrial Engineering

Mechanical Engineering Curriculum

Modeling of Powder Bed Fusion

Additive Manufacturing

A thesis submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Industrial Engineering

by

Mattia Moda

Prof. L. Bertini

Prof. B. D. Monelli

Mattia Moda

Residual stresses and distortions are key issues in Powder Bed Fusion (PBF) processes, affecting both the manufacturability and the mechanical strength of 3D printed parts. Their prediction via physical modeling could help reduce the number of failed iterations often needed before achieving a successful build. The first part of this thesis concerns a multi-scale simulation strategy developed for assessing residual stresses and distortions on scanning lengths ranging from one to over one billion times the beam diameter. A meso-scale thermo-structural model evaluates the process-induced temperature, stress, and strain fields on limited scanning lengths. At the other end of the spectrum, a less detailed and purely structural macro-scale model computes stresses and distortions on significant build volumes. The Pointwise Strain Superposition Method defines the incompatible strain and the initial state of the macro-scale model based on the meso-scale results and the given scanning strategy. This simulation method was validated by comparing its results with the stresses and distortions measured on several specimens made of selective laser melted Inconel 718. Stresses were measured through the blind hole drilling method on a cylindrical specimen printed with two different scanning strategies, while distortions were measured on a hollow cylinder and on a cantilever-shaped specimen after removing its supports. In both cases, the simulation showed first-or higher-first-order accuracy despite the significant uncertainties regarding the input parameters and material properties. This robustness, coupled with its computational efficiency, suggests that the presented simulation strategy may be a valid alternative to the calibration-based Inherent Strain Method for enhancing the process optimization and better understanding the underlying physical phenomena along with their effects on the manufactured components. The final part of the thesis concerns the Rosenthal solution for a moving point heat source in steady state on a semi-infinite solid. It is aimed at introducing a theoretical framework for the analysis and optimization of melting processes that use focused moving heat sources, including welding and PBF. Firstly, we analyze the feasibility of the thermal problem while constraining the melt pool size and aspect ratio. We then evaluate the maximum allowable velocity and the corresponding power as elementary functions of the constraints and material properties. Finally, we examine a wide range of melting processes within a dimensionless framework derived from the above solution. After completing the thermal analysis, we proceed to define the misfit strain, i.e., the inelastic strain induced by the gradient of thermal expansion at solidification. This enables us to compute the residual stress field associated with the Rosenthal solution through a quasi-analytical method that exploits its symmetry under the linearity assumption. Ultimately, the proposed approach is not intended to replace other numerical and experimental procedures, but rather to complement their capabilities and encourage more efficient use of available resources. In addition, reframing seemingly different problems within a common perspective can generally improve understanding, reveal new levels of similarity, and sometimes even allow for global solutions.

This thesis condenses three years of research conducted at the University of Pisa and Baker Hughes, whose TPS Innovation Department funded the development of the multi-scale simulation method described in Chapter 2 and thus holds the patent on the scaling strategy designed and implemented by the author (Section 2.3). Chapter 1 delineates the state of the art in process simulation for Selective Laser Melting, focusing specifically on residual stresses and part distortions. Lastly, Chapter 3 represents the author’s contribution towards defining a theoretical framework for the analysis and optimization of melting processes that use focused moving heat sources. It combines many ideas conceived by the author over the course of his doctoral studies with an extensive review of the relevant scientific literature carried out by Andrea Chiocca and Giuseppe Macoretta under the supervision of Bernardo D. Monelli. Ultimately, this document is not intended to be a report on the author’s research activities but rather a compendium of his most significant findings, some of which have also been published separately.

1 State of the art 1

1.1 Introduction . . . 1

1.1.1 Process parameters . . . 2

1.1.2 Laser-matter interaction . . . 4

1.1.3 Heat transfer mechanisms . . . 4

1.1.4 Workpiece integrity and issues . . . 5

1.2 Simulation strategies . . . 6 1.2.1 Methods . . . 6 1.2.2 Scalability . . . 6 1.2.3 Coupling . . . 7 1.3 Meso-scale . . . 7 1.3.1 Thermal problem . . . 8 1.3.2 Structural problem . . . 18 1.4 Macro-scale . . . 21

1.4.1 Dynamic mesh coarsening . . . 21

1.4.2 Physics-based approaches . . . 22

1.4.3 Calibration-based approaches . . . 23

1.4.4 Validation . . . 23

1.5 Conclusions . . . 23

2 Multi-scale simulation strategy 31 2.1 Introduction . . . 31

2.2 Meso-scale . . . 31

2.2.1 Model . . . 32

2.2.2 Heat transfer mechanisms . . . 32

2.2.3 Thermophysical properties . . . 33

2.2.4 Thermal simulation . . . 33

2.2.5 Heat source calibration . . . 33

2.2.6 Mechanical properties . . . 35

2.2.7 Structural simulation . . . 35

2.2.8 Results . . . 35

2.3 Scaling strategy . . . 37

2.3.1 Inherent versus incompatible strain . . . 38

2.3.2 Interpolation functions . . . 39

2.3.3 Sample points . . . 39

2.3.4 Scan lines . . . 40

2.3.5 Pointwise strain superposition . . . 40

2.3.6 Limitations . . . 42 vii

2.4 Macro-scale . . . 43

2.4.1 Model and mesh . . . 43

2.4.2 Material properties . . . 43 2.4.3 Simulation . . . 43 2.4.4 Convergence . . . 44 2.5 Validation . . . 45 2.5.1 Surface stress . . . 45 2.5.2 Cantilever deflection . . . 47

2.5.3 Hollow cylinder distortion . . . 47

2.6 Conclusions . . . 49

3 Linear steady state analysis 51 3.1 Introduction . . . 51

3.2 Thermal analysis . . . 51

3.2.1 Melt isotherm . . . 52

3.2.2 Power-velocity optimization . . . 54

3.2.3 Exploratory data analysis . . . 57

3.3 Structural analysis . . . 60

3.3.1 Misfit strain . . . 60

3.3.2 Formalization of the linear problem . . . 65

3.3.3 Axisymmetric sub-problem . . . 66

3.3.4 Boundary sub-problem . . . 70

3.3.5 Superposition . . . 74

3.3.6 Results and discussion . . . 74

3.4 Conclusions . . . 79

A Structural solution procedure 81 A.1 Annular solution for the axisymmetric sub-problem . . . 81

A.2 Core solution for the axisymmetric sub-problem . . . 83

A.3 Annular solution for the boundary sub-problem . . . 84

A.4 Core solution for the boundary sub-problem . . . 86

B Software implementation 89 B.1 Meso-scale simulation . . . 89

1.1 Spot center and beam diameter. . . 3

1.2 Schematic cross-section of a single track. . . 3

1.3 Thermo-structural simulation scheme – the dashed line represents the growing domain. 8 1.4 Top view of the uniform heat source proposed by Matsumoto [1] and Contuzzi [2]. . 9

1.5 Reference frame moving along with the heat source at the scanning speed v. . . . 9

1.6 Truncated conical heat source [3]. . . 10

1.7 Surface-volumetric heat source [4]. . . 10

1.8 Goldak heat source – isosurface with q000_{= q}000 maxe−3. . . 11

1.9 Heat transfer mechanisms modeled in the reviewed literature. . . 11

1.10 Example of domain and mesh refinement strategy for a meso-scale FE model. The modeled domain can be halved for symmetric simulations involving only a single scan line. . . 16

1.11 Schematic representation of a meso-scale simulation procedure. . . 17

1.12 Current applicability of meso- and macro-scale models in terms of total scanning volume. . . 21

1.13 Scaling strategies emerged from the reviewed literature. . . 22

2.1 MZ transverse cross-section with boundary sampling. . . 34

2.2 Geometric parameters of the objective function. . . 34

2.3 Meso-scale simulation of a single scan line along the x-axis performed on Inconel 718 with the following parameters: sm= 40 µm, P = 285 W, v = 0.96 m s−1, a = 0.9, r_f = 77 µm, r_r = 197 µm, ry = 57 µm, rz = 189 µm, Tp = 353 K. The results are represented in the half-domain y ≥ 0 since they are symmetric about y = 0. . . 36

2.4 Data-flow diagram of the PSS scaling strategy. . . 37

2.5 Schematic representation of the slice activation procedure. . . 44

2.6 Validation specimens – the blue arrow in (b) indicates the wire cut performed on the cantilever supports after the building process (the other two specimens remain fully attached to the base plate). . . 46

2.7 Computed σ(c) _{and measured σ}(m) _{stress components near the axis of the cylindrical} specimen. The z-axis is aligned with the build direction and originates at the interface between the part and the base plate. The results shown at the top of the figure were obtained with a hatch angle θ = 67°, while the others refer to a uniform scanning strategy (θ = 0°) oriented in the x-direction. . . 46

2.8 Measured δ(m)z and computed δ(c)z end deflection varying the number of slices with a fixed mesh for two scanning strategies of hatch angle θ = 0° and 67°. . . 48

2.9 Computed δr(c) and measured δr(m) distortions of the hollow cylinder, beside the

approximate build time per layer, as functions of the height above the base plate z. The distortion δr is defined as the difference between the as-built and nominal outer

radii. . . 49 3.1 Eulerian reference frame employed to express the thermal field produced by a point

heat source located at its origin O and moving along its x-axis with constant velocity of magnitude U relative to the semi-infinite solid z ≤ 0. . . 52 3.2 Schematic representation of the melt isotherm T = Tm produced by a moving point

heat source in steady state on a semi-infinite solid, where L and R are its maximum length and half-width, respectively. . . 52 3.3 Exemplified representation of the feasible region F and optimal operating condition.

The contour map shows the hyperbolic isolines of Ar. . . 56

3.4 Dimensionless power and velocity corresponding to the optimal operating condition varying the maximum allowable aspect ratio. . . 56 3.5 Dimensionless nominal operating conditions retrieved from the reviewed literature

(Table 3.1). The error bars represent the estimated power losses, i.e., the power fractions not contributing to the heating process. The solid line R = ¯R is the

feasibility boundary defined by the dimensionless constraint (3.22), while the contour map shows the isolines of aspect ratio. . . 58 3.6 Longitudinal cross-section of a steady state melt pool produced by a point heat

source located at O and moving along the x-axis. Melting and solidification occur, respectively, when crossing the leading (red) and trailing (blue) portions of the melt isotherm. The detail view shows the cross-section of an infinitesimal material element undergoing solidification. . . 60 3.7 Nonzero misfit strain components varying the dimensionless radius ˜r = r

R for

different values of Ry. Based on Eq. (3.12), Ry = {1, 10, 100} corresponds to

A_r≈ {1.2, 2.1, 5.9}, while Ry → 0 is equivalent to A_r →1. . . 64 3.8 Decomposition of the linear problem in the axisymmetric and boundary sub-problems.

The first concerns the compatibility of the axisymmetric inherent strain ε∗ _{on an}

infinite domain. The second restores the free surface of the semi-infinite domain by canceling the hoop stresses σθθ(a) through the pressure load p. The superposition

principle guarantees that σ = σ(a)_{+ σ}(b) _{if expressed in the same reference frame. . 66}

3.9 Radial (u(a)r ) and axial (u(a)x ) displacements resulting from the solution of the

ax-isymmetric sub-problem for different values of Ry. The displacement components are represented in dimensionless form varying the dimensionless radius ˜r = r

R. . . 67

3.10 Decomposition of the system (3.78–3.81) in the core and annular sub-sub-problems. Each of them has a nonzero inherent strain only inside the k-th subdomain Ωk for

k= 0, . . . , n_a, and ε∗_iiis a local approximation of ε(mi)_ii for i = r, θ, x. . . 68

3.11 Decomposition of the symmetric infinitesimal boundary sub-problem in its right (b+₎

and left (b−_{) components. . . 70}

3.12 Nonzero residual stress components for Ry → 0 (i.e., Ar → 1) and ν = 0.3. The

results are represented in dimensionless form on the fourth quadrant of the ˜y˜z-plane, where ˜y = y

R and ˜z = z

R. Under the adopted assumptions, σ is independent

of x and symmetric about y = 0, i.e., σii(−˜y, ˜z) = σii(˜y, ˜z) for i = x, y, z and

3.13 Nonzero residual stress components for Ry = 100 (i.e., Ar ≈ 5.9) and ν = 0.3.

The results are represented in dimensionless form on the fourth quadrant of the ˜y˜z-plane, where ˜y = y

R and ˜z = z

R. Under the adopted assumptions, σ is independent

of x and symmetric about y = 0, i.e., σii(−˜y, ˜z) = σii(˜y, ˜z) for i = x, y, z and

σyz(−˜y, ˜z) = −σyz(˜y, ˜z). . . 76

3.14 Vertical (y = 0) and surface (z = 0) profiles of the longitudinal (σxx) and transverse

(σyy) residual stress components for different values of Ry with ν = 0.3. Both the

results and coordinates are expressed in dimensionless form; in particular, ˜y = y R and

˜z = z

R. The dotted lines symbolize the loss of validity in proximity to the core Ω0

(3.82), whereas the cross marks indicate the theoretical stress values for r → 0 as defined by Eqs. (3.137, 3.138). . . 77 3.15 Dimensionless value and z-coordinate of the hydrostatic stress peak varying Ry

for three different Poisson’s ratios. The hydrostatic stress peak is located on the symmetry plane y = 0 at z = ˜zhR. For reference, the top abscissa shows the aspect

1.1 Main SLM process parameters based on the reviewed literature. . . 2

1.2 Constitutive laws for non-advective heat transfer mechanisms. . . 13

1.3 Properties needed for the thermal simulation that are not directly related to the heat source. . . 13

1.4 Partial summary of the reviewed literature. . . 25

1.4 Partial summary of the reviewed literature (continued). . . 26

1.4 Partial summary of the reviewed literature (continued). . . 27

1.4 Partial summary of the reviewed literature (continued). . . 28

1.4 Partial summary of the reviewed literature (continued). . . 29

2.1 Main process parameters for the SLM of Inconel 718 [5]. . . 45 3.1 Summary of the reviewed literature. Manufacturing processes are classified into four

categories: Arc Welding (AW), Beam Welding (BW), Laser Metal Deposition (LMD), and Selective Laser Melting (SLM); materials are classified based on the main alloy elements into six macro-categories and, where appropriate, additional sub-categories. 59

AM Additive Manufacturing. 1, 11, 51, 54, 57, 60, 75, 79 AW Arc Welding. xiii, 57, 59

BC Boundary Condition. 8

BCM Beaded Cylinder Morphology. 54, 59 BHD Blind Hole Drilling. 21, 45, 47, 49 BW Beam Welding. xiii, 57, 59

CAD Computer Aided Design. 90 CCM Crack Compliance Method. 21

CFD Computational Fluid Dynamics. 26–28

CMM Coordinate Measuring Machine. 23, 26, 47, 48 DIC Digital Image Correlation. 21, 28

DOE Design of Experiments. 79 DoF Degree of Freedom. 7, 20

DSC Differential Scanning Calorimetry. 15

DVRT Differential Variable Reluctance Transducer. 29 EBFF Electron Beam Freeform Fabrication. 27

EBM Electron Beam Melting. 10, 26–28, 31, 57 EBW Electron Beam Welding. 57

EZ Elastic Zone. 5

FDM Finite Difference Method. 6

FE Finite Element. ix, 8, 16, 18, 24, 31, 33, 39, 40, 43, 45, 47, 49, 89 FEM Finite Element Method. 6, 26, 28

GMAW Gas Metal Arc Welding. 54 HAZ Heat Affected Zone. 5

IC Initial Condition. 8

ISM Inherent Strain Method. 23, 31, 38, 43, 50 LADMD Laser-Aided Direct Metal Deposition. 25 LBM Lattice Boltzmann Method. 6, 16

LBW Laser Beam Welding. 57 LDS Laser Displacement Sensor. 23 LMD Laser Metal Deposition. xiii, 57, 59 MMLD Multi-Material Laser Densification. 25 MZ Melted Zone. ix, 5, 8, 18, 20, 34, 89

ND Neutron Diffraction. 21, 25, 26, 28 OM Optical Microscopy. 25–29

PBF Powder Bed Fusion. 1, 2, 24, 31, 38, 40, 49, 50 PDE Partial Differential Equation. 6, 8

PSS Pointwise Strain Superposition. ix, 31, 37, 39, 40, 44, 45, 49, 89 PZ Plasticized Zone. 5

SEM Scanning Electron Microscopy. 25–27

SLM Selective Laser Melting. xiii, 1–3, 6, 23, 25–29, 31, 45, 57, 59 SLS Selective Laser Sintering. 14, 25, 26

TEM Transverse Electric and Magnetic mode. 2, 4 XRD X-Ray Diffraction. 21, 25

General symbols

¯ Constrained, average, or equivalent quantity ˜ Dimensionless quantity

(a) Axisymmetric sub-problem (b) Boundary sub-problem

b+
Right boundary sub-problem

(b−_{) Left boundary sub-problem}

I Identity matrix i,..., l Iterators

n_a Number of annular subdomains n_l Number of scan lines

δij Kronecker delta

τ Precision of a normal distribution

Operators

: Inner product of second-order tensors

∼ Asymptotic or, in the absence of the limit, order-of-magnitude equivalence ≈ Approximate equality

∧ Logical conjunction ∨ Logical disjunction ∆ Difference operator

∇ Vector differential operator m−1

k·k Euclidean norm tr(·) Trace operator

W₀(·) Principal branch of the Lambert W function

Mathematical and physical constants e Euler’s number 2.718281828... σ Stefan–Boltzmann constant 5.670374419... × 10−8W m−2K−4 Dimensionless groups Bi Biot number Fo Fourier number Gr Grashof number Ma Marangoni number Nu Nusselt number Ry Rykalin number

Geometric and physical quantities

a Energy absorptivity A_r Melt pool aspect ratio

C Compliance elastic tensor Pa−1

c_p Specific heat at constant pressure J kg−1K−1

D Average diameter of powder particles m

d Generalized beam diameter m

dO Distance from the point heat source m

d_t Distance between adjacent scan lines (hatch distance) m

E Young’s modulus Pa

EDV Volume energy density J m

−₃

g Standard acceleration due to gravity m s−2

H Enthalpy per unit volume J m−3

h Heat transfer coefficient W m−2K−1

K Stiffness elastic tensor Pa

L Melt pool length m

` Characteristic length m

ˆ

n Surface normal unit vector

P Power W

P₀ Line load magnitude N m−1

P_opt Optimal power W

Q Coordinate system transformation matrix

q00 Heat flux density W m−2

q000 Heat generation rate per unit volume W m−3

q_I00 Irradiance W m−2

qk Ratio of Rk+1 to Rk

R Melt pool half-width m

r, θ, x Cylindrical coordinates along ˆr, ˆθ, ˆx m

r Position vector m

R_e End curvature radius of the melt isotherm m

Rk k-th knot radius m

S Surface m2

s Dimensionless x-coordinate

¯s Nominal layer thickness m

S_n+ Surface entity with outward-pointing normal ˆn S_n− Surface entity with inward-pointing normal ˆn

T Temperature K

t Time s

t Cauchy traction vector Pa

T₀ Initial temperature K

T_boil Boiling point K

TH Reference temperature for calculating enthalpy K

T_liq Liquidus K

T_m Melting point K

T_p Preheating temperature K

T_ref Reference temperature for calculating thermal strains K

T_sol Solidus K

U Velocity magnitude m s−1

¨u Acceleration m s−2

U_opt Optimal velocity magnitude m s−1

V Volume m3

v Scanning speed m s−1

X x-coordinate of melt pool half-width R m

x, y, z Cartesian coordinates along ˆx, ˆy, ˆz m

X+ Maximum x-coordinate of the melt isotherm m

X− Minimum x-coordinate of the melt isotherm m

z_h z-coordinate of the hydrostatic stress peak m

α Coefficient of thermal expansion (α_sec for secant) K−1

α Thermal expansion tensor (α_sec for secant) K−1

β Blending coefficient for R

γ Surface tension N m−1

∆Hf Enthalpy of fusion per unit volume J m−3

∆Hv Enthalpy of vaporization per unit volume J m−3

δ_p Optical penetration depth m

δR_e Necking of the melt isotherm m

δT_e End temperature perturbation K

∆Tm Temperature difference Tm− T0 K

ε Emissivity of the grey body

ε Total strain ε∗ Inherent strain

ε(∗∗) Strain component of initials ∗∗

ε(pl) Equivalent plastic strain

η Energy reduction factor or blending coefficient for A_r

κ Thermal diffusivity m2s−1

λ Thermal conductivity W m−1K−1

µ Dynamic viscosity Pa s

ν Poisson’s ratio

σ Cauchy stress tensor Pa

σ_h Hydrostatic stress Pa

φ, θ, n Local coordinates along ˆφ, ˆθ, ˆn m

ϕ Porosity

Ωk k-th subdomain

Ω∞ Subdomain r > R

ω Elastic energy density J m−3

Finite Element symbols

[CT] Thermal specific heat matrix J K−1

[CT u] Thermoelastic damping matrix N

[Cu] Structural damping matrix kg s−1

{F_g} Thermal gradient force vector W

{F_q} Thermal body force vector W

{F_T} Thermal load vector W

{F_u} Structural load vector N

[KT] Thermal conductivity matrix W K−1

[Ku] Structural stiffness matrix N m−1

[KuT] Thermoelastic stiffness matrix N K−1

{N } Element shape functions

{T } Nodal temperature vector K

State of the art

1.1

Introduction

In recent years Additive Manufacturing (AM) technologies have experienced rapid growth in terms of development, availability, and spread. Owing to the following strengths:

• limited constraints1 _{on the part geometry}

• complete control of local mechanical properties2

• manufacturing cost substantially independent of the geometric complexity • extreme flexibility to project modifications

• reduction (possibly removal) of assembly operations • short lead-time between design and finished product • limited material waste

they are becoming a valid alternative to traditional processes in many areas, including aerospace, biomedical, and robotics industries. Among these technologies, Powder Bed Fusion (PBF) is of particular interest for metallic and ceramic applications.

This review focuses on Selective Laser Melting (SLM), a PBF process in which a laser beam melts thin powder layers over specific regions of the workpiece section. The powder deposition and scanning phases are repeated alternately, and the part is built layer by layer according to the given scanning path. Finally, the workpiece is removed from the base plate and, possibly, subjected to post-processing operations. A detailed description of the process is provided by Sames et al. [9].

Currently, the following critical issues severely reduce the convenience of the SLM technology, sometimes to the point of making it practically unviable:

• low productivity

• high feedstock and machine cost

• complex control and optimization of process parameters • limited build volume

1_{Defined by the design rules for AM [6–8].}

2_{Evolution of the composite material concept.}

• poor as-built3 _{mechanical properties (mainly concerning fatigue strength)}

• poor dimensional accuracy and high roughness values • limited choice of materials.

Moreover, due to the novelty of the technology, the number of process parameters, and the complexity of the underlying physics, there can be noticeable differences even between parts manufactured with different units of the same machine. These problems may compromise both reliability and flexibility, despite the latter being considered the main advantage of additive-centered manufacturing paradigms. As a result, a successful build can currently require several iterations that weigh on its final cost. Reducing (ideally replacing) these trial and error procedures through process simulation would greatly benefit SLM and PBF technologies in general, especially in light of their high-mix low-volume nature.

Due to the length and time scale differences between the main physical phenomena and the entire building process, predicting all of the possible manufacturing issues with a single model is inefficient and even practically infeasible in some cases. Hence, various tailored models have been proposed to assess the following categories of issues separately:

1. Porosity and surface roughness 2. Residual stresses and distortions.

This review focuses on simulation strategies aimed at predicting issues of the second category. Firstly, we present an overview of the main process parameters and physical phenomena. Particular attention is paid to the heat transfer mechanisms and the estimation of their relative significance. Then follows a qualitative description intended to provide the reader with an intuitive understanding of the physics behind the genesis of residual stresses. Finally, the available simulation strategies are reviewed and classified based on the length scales of the modeled phenomena.

1.1.1 Process parameters

The process parameters related to the laser beam are scanning speed, power, generalized beam diameter, wavelength, TEM4_{, and scanning strategy. These, together with layer thickness, powder}

granulometry, build chamber atmosphere, and preheating temperature, constitute the main control variables (see Table 1.1 for some of their typical values).

Table 1.1 Main SLM process parameters based on the reviewed literature.

Parameter Typical range

Nominal layer thickness 10–50 µm Generalized beam diameter 50–200 µm

Power 100–300 W

Scanning speed 0.3–3.0 m s−1

Preheating temperature 80–200◦_C

3_{Properly designed heat treatments may guarantee static mechanical properties comparable to those of conventional}

forged or cast materials (usually maintaining higher tensile strengths but lower fracture strains) [10–13].

Figure 1.1 Spot center and beam diameter. Figure 1.2Schematic cross-section of a single track.

To describe the SLM process, we need to track the position of the beam spot in an absolute reference frame. The spot center C (Fig. 1.1) corresponds to the centroid of the irradiance5

distribution q00

I(r) measured on a plane S perpendicular to the beam direction:

rC = 1

P

Z Z

S

rq00_I(r) dS (1.1)

while the generalized beam diameter d (hereinafter beam diameter) is defined as 2√2 times the standard deviation of the same distribution [15]:

d= 2√2 s 1 P Z Z S kr − r_Ck2q_I00(r) dS (1.2)

In the case of a normal irradiance distribution, d corresponds to the diameter of the circular area intercepting the 1 − e−2 _{fraction (about 86%) of the total beam power.}

An elementary procedure for estimating the main process parameters can be outlined as follows (see Section 3.2.2 for a rigorous analytical approach6_):

1. Choose the beam diameter and layer thickness as a trade-off between the required resolution and productivity

2. Approximate the scanning speed by equating the characteristic time of laser-matter interaction

t_int= d_v with that of thermal diffusion through one layer t_diff = ¯s_κ2 [16]: v ≈ κd

¯s2 (1.3)

3. Determine the power needed to produce a volume energy density EDV sufficient to melt the

powder layer and ensure its adhesion to the already consolidated material:

P ≈ vd¯s

a EDV (1.4)

It should be noted that many other ED formulations (volumetric, areal, and linear [17]) are available in the literature, with the most common being the macroscopic volume-based energy density [18–20], defined as the laser power divided by the product of the scanning speed, hatch distance (i.e., the distance between adjacent scan lines), and layer thickness [21].

5_{Density of radiation incident on a given surface (W m}−2_{) [14].}

6_{Which helps understand the underlying physics and formulate educated guesses, although parameter fine-tuning}

Referring to the second step of the above procedure, if tint  tdiff, the energy distribution is

diffusion-limited, which may result in significant evaporative heat loss depending on the build chamber atmosphere [16, 22]. In addition, the melt pool aspect ratio increases with both the laser power and the scanning speed (Section 3.2.2), eventually leading to fluid dynamic instabilities and ensuing humping defects [23]. On the other hand, if tint  tdiff, the heat can diffuse across larger

volumes, resulting in limited temperature peaks and possibly poor adhesion.

Regarding the other process parameters, the laser TEM and wavelength mainly affect the energy absorption, while the scanning strategy has a significant effect on both the thermal history and the residual stress field [24, 25]. Selecting the powder granulometry requires a trade-off between quality and cost-effectiveness, as small particles are pricier but allow for thinner layers, which improve the surface finish at the expense of productivity [9]. The build chamber is usually filled with inert gas or kept under vacuum to prevent high-temperature oxidation. In the first case, the shielding gas flows above the scanning surface to clear spatters and metal fumes that could affect the process stability and undermine the quality of the final product [26]. Lastly, preheating at substantial fractions of the melting point reduces the thermal gradients and consequently residual stresses (Section 3.3.6), thereby improving the microstructure homogeneity and mechanical properties [27].

1.1.2 Laser-matter interaction

The laser power incident on a surface can either be reflected, absorbed, or transmitted. Reflection depends on the material (mainly on its chemical composition and microstructure), surface conditions, temperature, and laser wavelength. Transmission through the solid material (substrate and powder particles) is often negligible, as the metal skin depth7 _{[28] is comparable to the wavelength of the}

incident electromagnetic radiation [29] and typically many orders of magnitude smaller than the length scale of the analyzed phenomena. It follows that significant optical penetration below the powder bed surface is possible only inside the pores [30] or within the melt pool depression.

1.1.3 Heat transfer mechanisms

Among the heat transfer mechanisms (i.e., advection, conduction, convection, and radiation), conduction is generally dominant due to the high thermal gradients ensuing from the small length scale of the heating process. Hence, we will take it as a reference for estimating the relative significance of the other heat transfer mechanisms.

The Biot number, which can be interpreted as the ratio between the conductive and convective thermal resistances, is defined as:

Bi = Rcond R_conv =

h`

λ (1.5)

Considering a characteristic length ` ∼ 100 µm (about the typical beam diameter), we obtain

Bi ∼10−5–10−3, suggesting that convection is negligible at the laser spot scale. This also implies

that the gas flow induced above the scanning surface does not significantly affect the cooling process, and its main purpose is to prevent surface oxidation while clearing spatters and metal fumes [26].

At high temperatures, radiation is more effective than convection but less than conduction, as can be proven by estimating the ratio between the conductive and radiative thermal resistances:

R_cond R_rad ≈ εσ λ (T1+ T2)  T₁2+ T₂2` (1.6)

where the two subscripts identify the high-temperature region surrounding the laser spot and the build chamber walls. Equation (1.6) derives from the radiative heat transfer between two gray, diffuse, and opaque surfaces forming an enclosure with view factor F12= 1 [31], assuming that the

first surface area is negligible compared to the second one. Considering T1∼1000–3000 K as the

typical temperature range of the melted zone, λ ∼ 10–100 W m−_1K−_{1, and ε ∼ 0.1–1, the resistance}

ratio (1.6) has an order of magnitude comprised between 10−_{5 and 10}−_2.

Advection can be significant inside the melt pool and near the laser spot, where evaporation occurs. Besides the latter, which is generally the most effective in limiting the peak surface temperature [32], it comprises density- and surface-tension-driven advection (also known as thermo-capillary convection or Marangoni effect). The following dimensionless number, obtained by dividing the Marangoni number by the Grashof number, expresses the ratio between surface tension and buoyancy forces: Ma Gr = − ∂γ ∂T µ gκαρ2 1 `2 (1.7)

thus showing that the Marangoni effect can become dominant over buoyancy at small length scales.

1.1.4 Workpiece integrity and issues

Discontinuity issues (e.g., porosity, cracks, and delamination) should be theoretically avoidable by selecting the correct set of process parameters, while field issues (e.g., residual stresses and distortions) are intrinsic to the process and thus can only be reduced. Porosity and delamination are consequences of incomplete powder melting, gas entrapment, and spatter ejection [9]. Cracking can occur at the microstructural level, due to the fast thermal cycles, or at the macroscopic scale, due to porosity combined with positive hydrostatic stress. Residual stresses are a side effect of thermal expansion and, as such, should be theoretically avoidable only by reducing the process scale down to the atomic level, which is still infeasible for macro-scale parts.

Before delving into the simulation methods, it may be worthwhile to outline the physics behind the genesis of residual stresses. In this regard, Fig. 1.2 shows the schematic cross-section of a single track perpendicular to the cutting plane, in which we can distinguish three regions:

• the Melted Zone (MZ), where temperature exceeds the solidus, and we find a combination of thermal, plastic, and elastic residual strains

• the Plasticized Zone (PZ), where the material undergoes significant plastic deformations that persist even after cooling

• the Elastic Zone (EZ), where the inelastic residual strains are negligible.

After the phase transition, the solidified MZ material experiences thermal contraction while cooling down to room temperature. Such contraction is constrained by the surrounding domain, which gives rise to elastoplastic strains for the sake of geometric compatibility. In addition, the PZ material expands during the heating process, thus moving towards the free surface. Although the associated thermal strains disappear upon cooling, the plastic ones may result in a net unrecovered upward displacement and ensuing strain incompatibilities. Combined, these two phenomena produce a self-balanced residual stress field, whose hydrostatic component is positive around the MZ and negative in the subsurface region. Neighboring scan tracks alter the local residual stress field in nontrivial ways, depending on the process parameters, scanning path, and material properties (which in turn may depend on the metallurgical transformations occurring inside the HAZ8_).

1.2

Simulation strategies

SLM process simulation has three primary sources of complexity. First, as inferable from the previous section, the underlying physical phenomena are numerous and difficult to model. Second, and maybe more important, the length and time scales of the above phenomena are several orders of magnitude smaller than those of the entire process. Last, the experimental validation (as well as material characterization) can be expensive and time-consuming.

1.2.1 Methods

We can distinguish two types of simulation methods:

1. Analytical: closed-form solutions of PDEs9 _{that generally require deep simplifications (often}

resulting in very limiting assumptions) in exchange for efficiency and usability

2. Numerical: more accurate and flexible discrete/iterative solutions, whose main limitations are machine precision, input data approximation, and computational cost.

Analytical solutions inherited from welding [33] are mainly used for estimating the thermal field, while FEM, FDM, and LBM (i.e., Finite Element, Finite Difference, and Lattice Boltzmann Methods) can model virtually any aspect of the manufacturing process, although with significant scalability issues.

1.2.2 Scalability

The problem size pj varies throughout the building process, as it depends on the number of deposited

layers j: pj ≈ Vj `3 ≈ j X i=1 ¯sSi `3 (1.8)

where Vj is the part volume deposited before and including the j-th layer, while Si is the scanning

area of the i-th layer. Moreover, the number of load-steps lj required for simulating the j-th scanning

phase can be estimated as a function of the time scale ∆t:

lj ≈

Sj

d_tv∆t (1.9)

Multiplying the problem size by the number of load-steps, we obtain the approximate memory requirements (strongly related to the computational cost):

M_R ≈ n X j=1 ljpj ≈ 1 `3∆t ¯s d_tv n X j=1 j X i=1 SiSj (1.10)

which are inversely proportional to the time scale and the cubic power of the length scale. For thermal problems dominated by conduction, the Fourier number correlates the two scales:

Fo= κ∆t

`2 (1.11)

and substituting ∆t = Fo `2

κ in Eq. (1.10) results in MR ∝ `

−5_{. Hence, given a fixed build volume}

and process parameters, the length scale heavily affects the computational cost.

In view of this, we classify the simulation methods into the following categories based on their length scales (as proposed by Sames et al. [9]):

• micro-scale (10−9_–10−6_{m): simulation of the microstructure evolution}

• particle-scale (10−_6–10−_{3m): simulation of the beam-powder interaction, phase transitions,}

and melt pool dynamics

• meso-scale (10−_5–10−_{3m): simulation of one or more layers using continuous models (powder}

is considered a uniform solid with homogenized properties)

• macro-scale (10−_{3–1 m): simulation of the entire process with continuous models that}

pos-sibly stack up several layers, thus waiving resolution on thermal gradients and micro-scale phenomena.

Of these, micro- and particle-scale models [16, 32, 34–36] are the most suitable for assessing microstructure and porosity (together with surface roughness and the overall stability of the melting process), respectively, as they can substantially prescind from the macro-scale geometry. However, high computational costs and significant scalability issues (owing mainly to their multiphase nature) currently limit their applicability in favor of experimental procedures [37–42]. On the other hand, owing to their greater scalability, meso- and macro-scale models are emerging as the de facto standard for investigating the genesis and effects of residual stresses.

1.2.3 Coupling

Thermo-structural coupling is strong only inside the melt pool (where advection occurs), as distortion and heat generation by plastic deformation are both negligible elsewhere. This allows for any of the following coupling methods:

• strong coupling: the thermal and structural problems are solved together simultaneously • weak coupling: the thermal and structural problems are solved together, but the mutual

effects are applied as loads and thus delayed by one load-step

• one-way coupling: the thermal analysis is performed first, and the resulting thermal history is then used to define the structural loads and boundary conditions.

The third method is the most flexible in terms of space and time discretizations, which can be useful since a full 3D structural simulation has three times the DoFs10 _{of its thermal counterpart, meaning}

that using different meshes can improve the overall model efficiency. Furthermore, excessive mesh distortion may produce convergence issues, whose solution is generally easier in case of uncoupled discretizations.

1.3

Meso-scale

Meso-scale models (Fig. 1.3) simulate the scanning process with enough detail to capture the thermal gradients at the melt pool level. Powder is typically considered a continuum with homogenized properties to improve scalability and limit the computational cost. However, this simplification prevents the direct simulation of the beam-powder interaction, depression (or even keyhole) formation, denudation, spatter ejection, and dynamic evolution of the melt pool.

The following is an in-depth review of the meso-scale models available in today’s literature framed within a broader overview of the underlying thermo-structural problem. Please note that the thermal and structural analyses are presented separately for clarity purposes, regardless of the specific coupling method employed in the reviewed works.

Figure 1.3 Thermo-structural simulation scheme – the dashed line represents the growing domain.

1.3.1 Thermal problem

The thermal problem is described by the heat equation along with the associated initial and boundary conditions: PDE: ρcp ∂T ∂t = ∇(λ∇T ) + q 000 _(1.12) IC: T (r, 0) = T0(r), r ∈ Ω ∪ ∂Ω (1.13) BC: T (r, t) = T (r, t), r ∈ ΓT (1.14) λ∇T + q00(T, r, t) = 0, r ∈ Γq (1.15)

where Ω is the domain of boundary ∂Ω, while ΓT and Γq are the boundary portions subjected to

essential and natural boundary conditions, respectively [43]. The discretized FE equations have the following form [44]:

[CT]{ ˙T } + [KT]{T } = {Fg}+ {Fq} (1.16)

where the gradient force vector {Fg}is composed of either surface heat sources, unknown heat flux

values (e.g., for constant temperature boundary conditions), or convection terms that need to be equilibrated with the flux at a boundary node (for evaporation, radiation, and convection). The thermal body force vector {Fq}is the expanded sum of the element heat generation loads {Fq}e,

which in turn depend on the heat generation rate per unit volume q000_:

{F_q}e= Z Z Z

Ve

q000{N }edV (1.17)

where Ve is the volume of the e-th element, and {N}e are its shape functions.

The FE system is generally solved via time discretization by managing nonlinearities through iterative methods (e.g., Newton-Raphson).

Heat source

The heat source reproduces the thermal effects of laser-matter interaction and advective phenomena occurring inside the melt pool, as their detailed modeling would often require expensive character-izations and constitutive parameters that might not be available in practice [30, 32, 34]. Not to mention that the homogeneous powder assumption would still limit the achievable accuracy.

Heat is typically applied with a predefined (surface or volumetric) space distribution, which needs to be converted into an equivalent nodal thermal flux in FE analyses. Calibrating such distribution by comparing the MZ measurable through optical microscopy with that resulting from the thermal simulation can significantly reduce the model uncertainties [2–4, 45–48] (Section 2.2.5).

Figure 1.4 Top view of the uniform heat source

pro-posed by Matsumoto [1] and Contuzzi [2].

Figure 1.5 Reference frame moving along with

the heat source at the scanning speed v.

There follows a detailed description of the available heat source types, whose formulations are scaled to make sure they all correspond to the same absorbed power aP .

Uniform heat sources The volumetric heat source proposed by Foroozmehr et al. [46] has the

following form: q000=    4aP πd2ηδp if x 2_{+ y}2 _≤ d2 4 ∧ −ηδp< z ≤0 0 otherwise (1.18)

where the correction factor η takes into account the uniform heat generation rate, while the optical penetration depth δp is defined as the depth at which the radiation power decreased to the e−1

fraction of its surface value (i.e., half the skin depth according to the Beer-Lambert law). However, the optical properties used for this model differ significantly from those of the solid material, as the laser penetration depth inside the powder bed is comparable to the layer thickness due to multiple reflections in an open pore system [49].

Matsumoto [1] and Contuzzi [2] applied a uniform heat source over five elements, as represented in Fig. 1.4. This solution is relatively easy to implement and calibrate at the expense of flexibility.

Gaussian heat source The most common heat source type [48, 50–67] has a Gaussian distribution

over the scanning surface z = 0:

q00= −8aP πd2 exp −8x2+ y2 d2 ! ˆ k (1.19)

where x and y are expressed in the reference frame of Fig. 1.5 originating at the spot center C, while ˆk is the z-axis unit vector of the same reference frame. Assuming that the beam power and diameter are known with sufficient accuracy, the only parameter to be estimated (or characterized experimentally) is the energy absorptivity, which depends on a myriad of factors (Section 1.1.2) but should at least remain constant under steady state conditions.

The main limitations of this heat source type are the enforced axisymmetry, as the energy absorptivity can vary significantly within the heated region, and the underestimated heat penetration. In fact, no surface heat source can reproduce the optical penetration inside the powder layer or melt pool depression (unless complemented by a detailed evaporation model).

Figure 1.6 Truncated conical heat source [3]. Figure 1.7 Surface-volumetric heat source [4].

Modified Gaussian heat sources To overcome the above limitations, Papadakis et al. [3] used

a truncated conical heat source (Fig. 1.6) formulated as follows:

q000=    q₀exp−x2+y2 r2 0  if zi< z ≤0 0 otherwise (1.20) where: r₀= r_e+ z z_i(ri− re) (1.21)

and by enforcing a total heat generation rate of aP :

q₀= − 3aP

πz_i(r2_e+ r_er_i+ r_i2) (1.22)

The additional adjustable parameters re, ri, and zi make the heat distribution (1.20) adaptable to

deep penetration processing conditions.

Mukherjee et al. [68] implemented a simplified version of the above solution with ri= re and

z_i= −¯s but made the beam intensity distribution factor τ explicit:

q000=    4τaP πd2¯s exp  −4τx2_d+y2 2  if − ¯s < z ≤ 0 0 otherwise (1.23)

Furthermore, they expressed the total absorptivity as a function of the energy fraction absorbed by the powder layer ap and the absorptivity of the substrate as:

a= a_p+ a_s(1 − a_p) (1.24)

Cheng et al. [24] used a Gaussian volumetric heat source with τ = 0.5 inherited from EBM11

simulation [69]: q000= Iz(z) 2aP πd2δ_p exp −2 x2+ y2 d2 ! (1.25)

Figure 1.8 Goldak heat source – isosurface with q000= q000maxe−3.

Figure 1.9 Heat transfer mechanisms modeled in

the reviewed literature.

whose dependence on the z-coordinate is expressed by the Iz(z) function:

Iz(z) =    −3z_δ22 p −2 z δp + 1 if − δp< z ≤0 0 otherwise (1.26)

Masoomi et al. [70] opted for a similar solution with an exponential Beer-Lambert attenuation along z, while Weirather [4] combined two different heat sources (Fig. 1.7) to manage surface heating and heat penetration separately. More specifically, the beam power intercepted by a circular area of diameter d

2 around C (i.e., the 1 − e− 1

2 fraction, or about 39%, for a normal irradiance

distribution) is applied as a uniform cylindrical heat distribution of depth zi:

q000=     e−1 2 −1  16aP πd2zi if x 2_{+ y}2 _≤ d2 16∧ zi< z ≤0 0 otherwise (1.27)

and the remaining fraction as a truncated Gaussian heat source of inner diameter d

2: q00=    −8aP πd2 exp  −8x2_d+y2 2  ˆ k if x2+ y2> d₁₆2 ∧ z= 0 0 otherwise (1.28)

Modified Gaussian heat sources provide higher flexibility than the simple version and are thus better suitable for modeling high penetration and keyhole mode heat transfer. Also, their lower thermal gradients generally improve the convergence behavior. On the flip side, the increased number of parameters may lead to more complex characterization procedures.

Goldak heat source All the solutions presented so far share axisymmetry as a common

de-nominator. However, welding simulation suggests this may be less realistic the wider the range of phenomena reproduced by the heat source (which, besides advection and laser-matter interaction, may also include energy absorption in both vapor and liquid phases). To break the symmetry about the yz-plane, Goldak [71, 72] proposed a double ellipsoidal power density distribution, which has become quite popular for AM simulation [45, 73–76] despite being originally intended for welding.

If expressed in the reference frame of Fig. 1.5, the heat generation rate per unit volume has the following form: q000=            6√3aP ψf π √ πr_fryrz exp " −3 x2 r2_f + y2 r_y2 + z2 r_z2 !# if x ≥ 0 6√3aP ψr π √ πr_rryrz exp " −3 x2 r2_r + y2 r_y2 + z2 r_z2 !# otherwise (1.29)

where, referring to Fig. 1.8:

• 0 < ψf <2 is a multiplier modulating the power applied to the half-space x ≥ 0

• ψr= 2 − ψf is a multiplier modulating the power applied to the half-space x < 0

• rf, ry, rz are the semi-axes of the frontal ellipsoidal isosurface with q000= q_max000 e−3

• rr, ry, rz are the semi-axes of the rear ellipsoidal isosurface with q000= q_max000 e−3.

The following condition guarantees the continuity of q000_{across the plane x = 0:}

ψ_f r_f =

ψ_r

r_r (1.30)

and by substituting ψr= 2 − ψf, it results:

ψ_f = 2rf

r_f+ r_r (1.31)

ψ_r= 2rr

r_f+ r_r (1.32)

The dimensional parameters rf, rr, ry, rz, and the energy absorptivity a are process- and

material-dependent quantities that can be estimated through the calibration procedure of Section 2.2.5.

Gusarov heat source Following an in-depth analysis of the interaction between laser radiation

and powder beds [30, 49, 77, 78], Gusarov et al. formulated an equivalent heat source taking into account many additional parameters, including:

• the hemispherical reflectivity of the powder material in dense form • the average diameter of powder particles

• the extinction coefficient

• the optical thickness of the powder layer.

Uniform temperature input Lastly, some authors used uniform temperature constraints [79, 80]

or initial conditions [81] instead of a proper heat source. These solutions are the easiest to implement but come with significant limitations. In particular, temperature constraints require calibration to ensure the correct power input, and, for the same reason, uniform temperature initial conditions should satisfy the following equivalence:

m _H 1 ρ₁ − H₀ ρ₀  = aP ∆t (1.33)

Table 1.2 Constitutive laws for non-advective heat

transfer mechanisms.

Mechanism Constitutive law q00

Conduction Fourier −λ∇T

Convection Newton h∆T ˆn

Radiation Stefan-Boltzmann εσT4nˆ

Table 1.3 Properties needed for the thermal

simula-tion that are not directly related to the heat source.

Property T-dependence Sensitivity

ρ Low High λ High High cp High High ∆Hf - Low ∆Hv - High T_sol - High T_liq - High T_boil - High ε High Low h Low Low

where m is the mass of material deposited in the time interval ∆t, and the two subscripts identify its properties at preheating (0) and deposition (1). When combining a uniform temperature initial condition with other heat sources (whether surface q00_{or volumetric q}000_{), the above equivalence}

becomes: m _H 1 ρ₁ − H₀ ρ₀  +Z ∆t Z Z S q00nˆdS + Z Z Z V q000dV  dt = aP ∆t (1.34)

Heat transfer models

Figure 1.9 shows a schematic overview of the heat transfer mechanisms modeled in the reviewed literature, highlighting that radiation and convection were implemented in 51% and 64% of cases, respectively. Their formulations were generally derived by discretizing the constitutive laws listed in Table 1.2. Advection was either neglected or integrated into the heat source, while heat dissipation through powder was typically modeled as conduction under the homogeneous powder assumption.

Thermophysical properties

Table 1.3 lists the main properties needed for the thermal simulation while providing a qualitative estimate of their temperature dependence and associated model sensitivity [56]. As luck would have it, h and ε are very hard to measure but not particularly significant, as convection and radiation were sometimes even neglected in the reviewed meso-scale models.

In the following paragraphs, we examine the above properties from a modeling perspective.

Density The variability of density with temperature has much greater structural than thermal

effects. In fact, thermal expansion is the root cause of residual stresses and part distortions but hardly affects the thermal field of the solid phase. This allowed many authors to use density values independent of temperature [29, 52, 54, 55, 79, 80, 82].

As over 99% densities are achievable in most cases, approximating the as-built density with that of the powder material is a common practice [9, 83, 84].

and porosity ϕ, defined as the void12 _{volumetric fraction:}

ϕ= Vv V_b = 1 −

ρ_b

ρ_p (1.35)

Bugeda et al. [85] considered the dynamic change in porosity during Selective Laser Sintering (SLS) through the Scherer [86–89] and Mackenzie-Shuttleworth [90] models. However, the vast majority of the reviewed models switched from a constant porosity value to ϕ = 0 at the melting point (in accordance with the full density assumption).

Thermal conductivity Its strong temperature dependence and substantial thermal effects make

thermal conductivity one of the most critical properties, meaning that inaccurate input values will likely compromise the simulation results. Despite this, some authors [1, 2, 77] used constant thermal conductivities, mainly due to the lack of reliable experimental data.

The as-built material is generally considered thermally isotropic with the same conductivity as the powder material. This implies neglecting the residual porosity and assuming that conduction is mainly due to the free flow of electrons [31], which is substantially independent of the molecular arrangement (affecting the lattice vibrational component).

The powder conductivity depends mainly on porosity, pore geometry [91], and filling gas [92]. In the absence of experimental measurements, many authors relied on theoretical and empirical formulations of the bulk conductivity λb, as its low value compared to that of the consolidated

material reduces the associated model sensitivity. This allows for constant or even null powder conductivities [81, 93] with minor simulation errors.

Thümmler and Oberacker [91] estimated λb as:

λ_b≈ λ_p(1 − ϕ) (1.36)

where λp is the particle thermal conductivity. Despite being used by some authors [51, 52, 67],

this model overestimates the powder conductivity, as λb should generally be about one order of

magnitude lower than λp[45, 65]. Badrossamay and Childs [94] improved the above solution through

an empirical model:

λ_b≈ λp(1 − ϕ)

1 + cϕ0.78 (1.37)

whose parameter c should be adjusted to fit the experimental data.

Several authors [62–64, 69, 70, 76, 80, 95] implemented a theoretical model derived from the Zehner-Schlünder-Damköhler (ZSD) equation [96] assuming spherical particles:

λ_b λ_g ≈  1 −p 1 − ϕ 1 + ϕλr λ_g ! +p 1 − ϕ   2 1 −λg λp   1 1 − λg λp log λp λ_g ! −1  + λ_r λ_g   (1.38)

where λg is the thermal conductivity of the filling gas, and λr ≈4F σT3Dis that due to radiation

among particles (with view factor F ≈ 1 3).

Gusarov [77] and Roberts [50] stated that the powder thermal conductivity depends mainly on the filling gas, porosity, size and morphology of particles, but not significantly on their material. Lastly, Shapiro et al. [92] and Hsu et al. [97] considered the finite area contacts between powder particles. Nevertheless, the common practice remains to switch from λb to λp instantly at melting.

Specific heat and phase transition properties The specific heat capacity (at constant

pres-sure) cp affects the thermal diffusivity. Being a continuum property, cp depends on the molecular

nature [98], and thus, in the absence of experimental measurements, it can be estimated from the

c_pi of the constituents and their mass fractions f_i through the Kopp-Neumann law [99]: c_p≈

n X

i=1

c_pifi (1.39)

The vast majority of the reviewed publications used temperature-dependent specific heat capacities measured through Differential Scanning Calorimetry (DSC). However, as cp rises sharply

at phase transitions (becoming virtually infinite for pure substances), and stiff material properties can undermine the simulation accuracy or even cause significant convergence issues, enthalpy is highly preferable for modeling the energy storage.

If the powder bed is considered homogeneous, the Kopp-Neumann law is again applicable for estimating its apparent specific heat capacity [64, 100]:

cp_b≈ cp_pfp+ cp_gfg ≈ cp_p(1 − fg) + cp_gfg ≈ cp_p (1.40)

where the two subscripts identify the powder particles (p) and filling gas (g), and fg is assumed to

be negligible. This explains why all the reviewed works with homogeneous powder properties took the specific heat capacity of the particle material as cp_b.

Emissivity Radiation can be significant inside the melt pool due to its dependence on T4. In

this region, however, the thermal field is mainly affected by the heat source and evaporative heat loss [32]. Hence, it may be convenient to integrate radiation into the calibrated heat source without characterizing ε directly.

Regardless of the above, the emissivity of the powder bed is greater than that of the corresponding solid [80] and follows the expression [101]:

ε_b = ϕSεh+ (1 − ϕS)εp (1.41)

where ϕS is the surface fraction occupied by pores, while εp and εh are the emissivities of holes and

particles, respectively. For randomly packed single-sized gray spheres [101]:

ϕS ≈ 0.908ϕ 2 1.908ϕ2−2ϕ + 1 (1.42) ε_h≈ ε_p  2 + 3.082_1−ϕ ϕ 2 ε_p  1 + 3.082_1−ϕ ϕ 2 + 1 (1.43)

which constitute the theoretical model used by Day and Shaw [80] and Denlinger et al. [76].

Heat transfer coefficient Despite being substantially negligible at the meso-scale level,

convec-tion was featured in 64% of the reviewed works. Its heat transfer coefficient can be expressed as a function of the Nusselt number (where λg is the thermal conductivity of the inert gas):

h= Nu λg

` (1.44)

which is then related to the Reynolds, Prandtl, and Grashof numbers through the convective heat transfer correlations [102].

Figure 1.10Example of domain and mesh refinement strategy for a meso-scale FE model. The modeled

domain can be halved for symmetric simulations involving only a single scan line. Model and mesh

A first categorization of the available models can be made based on the number of dimensions. Körner et al. [16] developed a 2D LBM particle-scale model to simulate a multi-layer building process considering capillary effects, wetting conditions, and a stochastic particle configuration. Criales et al. [56] used a 2D in-house FE model to investigate the temperature profile on the powder surface, while Matsumoto et al. [1] performed a 2D thermo-structural simulation under the plane stress assumption for the consolidated layer. Apart from these, all the other reviewed meso-scale models were three-dimensional.

The domain usually comprises the scanning region and a surrounding volume large enough to allow for a boundary condition of constant temperature (or adiabaticity [56]) over those boundaries that do not face towards the build chamber. This condition implies that the semi-infinite domain assumption is applicable. Hence, the domain size should be subjected to a sensitivity analysis ensuring its effects on the thermal solution are negligible.

As thermal gradients decrease moving away from the scanning region, thus allowing a progressive mesh coarsening towards the domain boundaries, the vast majority of the reviewed models used one of the following meshing strategies:

• compatible mesh technique: every node is a vertex of its adjacent elements, which results in gradual mesh size transitions (Fig. 1.10)

• boundary condition technique: a bonded contact condition (or constraint equation) is applied at the interface between two meshes that may not share the interface nodes, which allows for abrupt mesh size transitions.

The mesh refinement factor, defined as the ratio between the beam diameter and the element characteristic length, ranged from 1 to over 10 in the reviewed models. Coarse meshes were limited

Figure 1.11 Schematic representation of a meso-scale simulation procedure.

to hatch-scale models [1, 2, 73], while highly refined meshes were mostly used for single scan lines [57] but sometimes pushed up to larger scales through dynamic mesh coarsening or refinement strategies (i.e., merging the elements of previous layers at a given distance from the powder bed [75, 76] or moving a refined subdomain along with the heat source [60, 65]). In this regard, Zeng et al. [60] analyzed the sensitivity of the heat flux density associated with a discretized Gaussian heat source to variations of the mesh refinement factor.

Simulation

Considering a single scan line under steady state conditions, the problem (1.16) becomes pseudo-static (i.e., its state variables are time-independent for an observer moving along with the heat source) [85], which allows for a single load-step solution, although with some limitations. In particular, the steady state assumption does not apply to the scan line endpoints nor to neighboring scan lines in case of a limited delay, leading the vast majority of the authors to reject it.

Figure 1.11 shows a standard simulation procedure based on the “element birth and death” technique. After generating the entire mesh, all the powder elements are deactivated; namely, their thermal conductivity is multiplied by a severe reduction factor, specific heat capacity set to zero, and thermal nodal loads (both direct and equivalent) removed. At deposition, all the elements belonging to the powder layer are activated by restoring their properties and nodal loads. Once a powder element is crossed by the heat source, or its temperature exceeds Tsol, porosity is set to

zero, and its material properties become as-built. Where possible, neglecting the powder thermal conductivity [81] allows to skip the powder deposition phase, remove all powder elements outside of the scanning region, and activate the melted ones instead of changing their material properties.

Due to the fast dynamics of the thermal field, a fine time discretization is needed during and immediately after the scanning phase. Then, the cooling rate decreases, and longer time steps are allowed without significant loss of detail. It should also be noted that time discretization causes the heat source to advance by v∆t at each load-step, thus resulting in non-uniform heat distribution over the scanning trajectory. This produces temperature oscillations of amplitude proportional to