A Review on Modelling and Simulation of Laser Additive Manufacturing: Heat Transfer, Microstructure Evolutions and Mechanical Properties

Abstract

Modelling and simulation are very important for revealing the relationship between process parameters and internal variables like grain morphology in solidification, precipitate evolution, and solid-state phase transformation in laser additive manufacturing. The impact of the microstructural changes on mechanical behaviors is also a hot topic in laser additive manufacturing. Here we reviewed key developments in thermal modelling, microstructural simulations, and the predictions of mechanical properties in laser additive manufacturing. A volumetric heat source model, including the Gaussian and double ellipsoid heat sources, is introduced. The main methods used in the simulation of microstructures, including Monte Carlo method, cellular automaton, and phase field method, are mainly described. The impacts of the microstructures on mechanical properties are revealed by the physics-based models including a precipitate evolution based model and dislocation evolution based model and by the crystal plasticity model. The key issues in the modelling and simulation of laser additive manufacturing are addressed.

1. Introduction

Additive manufacturing (3D printing) is different to the traditional subtractive manufacturing in which the usable part is obtained by cutting off from a large block of material. The object can be directly built layer by layer with direct combinations of digital models formed in computer-aided design [1,2,3,4,5] in additive manufacturing. Laser additive manufacturing takes the dominant role for fast development and industrial applications of this technology [6,7,8,9,10]. According to the usage modes of powders and lasers in additive manufacturing, this technology can be divided into three main categories: selective laser sintering; selective laser melting; and laser deposition melting where the laser is considered to be the main heat source. Based on ISO/ASTM52900–15, laser deposition melting is formally named as directed energy deposition, in which focused thermal energy is used to fuse materials as being deposited. Selective laser melting/sintering is formally named as powder bed fusion, in which focused thermal energy selectively fuses regions of the powder bed.

Directed energy deposition can be used to produce high property components with relatively large volume or fix components for re-uses with improvement on mechanical properties. Powder bed fusion is fit for producing components with complex geometrical shapes with high efficiency. It is widely used in aerospace [11,12], automotive [13,14,15], aeronautics [16,17], bioengineering industries [18,19,20] and metamaterials [21,22,23,24,25,26,27].

Different to the consolidation processes in the melting pool in welding and in casting, the cooling rate in the consolidation process in laser additive manufacturing is very high. The high values of cooling rates from 10²~10⁶ K/s lead to new mechanisms on the formations of columnar grains and small equiaxed grains in the consolidation process [28,29,30] and on the phase transformations in stage of the solid stage [31,32,33]. The formed microstructures can determine the final mechanical properties of the additive manufactured components. Numerical methods can provide insight into the additive manufacturing process and become an efficient tool revealing the mechanism on the connections between the microstructures and the mechanical properties. Then, the process can be optimally controlled for the improvement of end-part uses of the additively manufactured components.

Due to the high demands on the investigations on the formations of microstructures and the changes of mechanical properties caused by the microstructural changes, it is necessary to combine the existing experimental and theoretical knowledge on the metallurgy for the explanations of the new generated phenomena in additive manufacturing. So, the main progress on the numerical modelling and simulation of additive manufacturing is reviewed. The main focus of this review relies on the relationship between the micro-structures and mechanical properties for better understanding of the mechanisms on the better microstructural controlling in additive manufacturing.

2. Thermal Modelling

The temperature variation in laser additive manufacturing is the key factor determining the microstructural changes and the final mechanical properties [34,35,36,37,38]. The unexpected defects and deteriorated properties can be caused by inappropriate selection of process parameters, which can be connected with the heat and mass transfer and the non-equilibrium solidification by the heating and cooling rates and temperature gradient in laser additive manufacturing [39]. This leads to the challenges in revealing how the temperature changes in the additive manufacturing process and further influences on the microstructures and properties. The in-situ monitoring provides direct observation of the temperature distributions [40]. However, the numerical modelling provides all the data necessary for investigations on the further influences [41]. This causes the motivation for the numerous thermal modellings of laser additive manufacturing to reveal the temperatures in the time and space domain in laser additive manufacturing. The main existing heat source models of the laser based additive manufacturing include volumetric heat source model and heat source model considering laser-particle interactions.

2.1. Volumetric Heat Source Model

2.1.1. Gaussian Heat Source Model

The volumetric heat source model mainly includes the Gaussian and double ellipsoid types. The heat flux in a Gaussian heat source model follows the Gaussian distribution [42],

$$I\left(x,y,z\right)=q_0\times \exp \left(-2\frac{x^2+y^2}{r_0^2}\right)$$

(1)

$$r_0=r_e+\frac{z}{H}\left(r_e-r_i\right)$$

(2)

where I is the heat flux intensity and q₀ is the maximum value. r_e and r_i represent the radii of the top and bottom profiles of the heat source. H indicates height dimension in the heat source. x, y, z are coordinates.

The further consideration of the heat source moving in the heat source model can lead to slight change of the equation form [43],

$$q(x,y,z)=\frac{2AP}{\pi r^2\eta }\exp \left[-\frac{2\left({\left(x-vt\right)}^2+y^2\right)}{r^2}\right]\exp \left(-\frac{\left|z\right|}{\eta }\right)$$

(3)

where A is the laser energy absorptivity, P is the power of laser, r is the radius of laser, v is the laser scanning speed, η is the laser penetration depth of powder, t is time.

In the polar coordinate system, the Gaussian distribution of heat flux can be written as [44],

$$q=\frac{2a{P}_{\mathrm{laser}}}{\pi w_b^2}\exp \left(-\frac{2r^2}{w_b^2}\right)$$

(4)

where P_laser is the laser power. w_b is the laser beam radius. r is the distance from the power surface to the laser central location. a is the metal powder absorption.

The heat losses can be found on the boundaries by convective and radiative heat transfers,

$$q_{\mathrm{c}}=h\left(T-T_{\mathrm{a}}\right)$$

(5)

$$q_{\mathrm{r}}=\sigma \epsilon \left(T^4-T_{\mathrm{a}}^4\right)$$

(6)

where h is the convective heat transfer coefficient. ε is the emissivity of the material being dependent on material, temperature, and environment [45,46]. q_c is the boundary of convection. q_r is the boundary of radiation. T is temperature and T_a is ambient temperature. Due to the difficulties in direct measurement of emissivity, the value is usually estimated by both experiment and numerical methods [47,48]. σ is the Stefan–Boltzmann constant.

Li et al. [43] used Gaussian heat source model for the simulations of the temperature variations and the post heat treatment in selective laser melting additive manufacturing of aluminum alloys, as shown in Figure 1. Both the temperature distributions and the temperature histories can be obtained by the simulation and then the obtained data can be available to be integrated with the microstructural evolutions for the prediction of the final mechanical properties.

2.1.2. Double Ellipsoid Heat Source Model

By consideration of the melt pool geometry, double ellipsoid heat source model was originally proposed by Goldak et al. [49] and then widely used in welding simulations [50,51,52,53]. The double ellipsoid heat source model in laser additive manufacturing can be written as [54,55,56],

$$q_{\mathrm{f}}=\frac{6\sqrt{3}{\eta }_{\mathrm{s}}{\eta }_{\mathrm{d}}{Q}_{\mathrm{t}}{f}_{\mathrm{f}}}{a_{\mathrm{f}}bc\pi \sqrt{\pi }}\exp \left\{-3\left[{\left(\frac{x+vt}{a_{\mathrm{f}}}\right)}^2+{\left(\frac{y}{b}\right)}^2+{\left(\frac{z}{c}\right)}^2\right]\right\}$$

(7)

$$q_{\mathrm{r}}=\frac{6\sqrt{3}{\eta }_{\mathrm{s}}{\eta }_{\mathrm{d}}{Q}_{\mathrm{t}}{f}_{\mathrm{r}}}{a_{\mathrm{r}}bc\pi \sqrt{\pi }}\exp \left\{-3\left[{\left(\frac{x+vt}{a_{\mathrm{r}}}\right)}^2+{\left(\frac{y}{b}\right)}^2+{\left(\frac{z}{c}\right)}^2\right]\right\}$$

(8)

where a_f and a_r are the lengths in x direction. b and c are the lengths along y and z directions; f_f and f_r are the heat-input parameters for the semi-ellipsoids. Here, b is the laser beam radius; v is transverse velocity of heat; and c is melt depth. a_f can be selected to be c, and a_r can be times of c [57]. f_f and f_r are calculated based on a_f and a_r, respectively [58].

2.2. Heat Source Model Considering Laser-Particle Interactions

All the above-mentioned heat source models are widely and successfully used in traditional welding simulations. However, additive manufacturing is basically different to the welding process due to the existing of powders. The flow of powders is one of the main topics for the improvement of the better controlling of additive manufacturing and has been widely studied by both experimental and numerical methods [59,60,61,62,63,64]. To reveal the effects of powders in the modelling and simulation of laser additive manufacturing, two methods were proposed. One method is to consider the powder effects in the physical properties and the other one is to consider the powder effects in the heat transfer.

When the powders stay in solid state, it can be treated as a porous material. The porosity can be determined by [44]

$$\phi =\frac{{\rho }_{\mathrm{bulk}}-{\rho }_{\mathrm{powder}}}{{\rho }_{\mathrm{bulk}}}$$

(9)

$${\rho }_{\mathrm{powder}}=\left(1-\phi \right){\rho }_{\mathrm{bulk}}, T<T_{\mathrm{m}}$$

(10)

$$k_{\mathrm{powder}}=\left(1-\phi \right)k_{\mathrm{bulk}}, T<T_{\mathrm{m}}$$

(11)

where ρ is density and k is thermal conductivity. T_m is the melting point of the material. When the temperature is further increased to be higher than T_m but smaller than T_L in the mushy zone around the melt pool, the alloy started to be melted but has not been totally transformed from solid to liquid. Here T_L is the temperature where the alloy is totally changed from solid state to liquid. In this stage, the density and the thermal conductivity can be defined as [44],

$${\rho }_{\mathrm{eff}}=\frac{{\rho }_{\mathrm{bulk}}\left(T_{\mathrm{L}}\right)-{\rho }_{\mathrm{powder}}\left(T_{\mathrm{m}}\right)}{T_{\mathrm{L}}-T_{\mathrm{m}}}\left(T-T_{\mathrm{m}}\right)+{\rho }_{\mathrm{powder}}\left(T_{\mathrm{m}}\right), T_{\mathrm{m}} < T < T_{\mathrm{L}}$$

(12)

$$k_{\mathrm{eff}}=\frac{k_{\mathrm{bulk}}\left(T_{\mathrm{L}}\right)-k_{\mathrm{powder}}\left(T_{\mathrm{m}}\right)}{T_{\mathrm{L}}-T_{\mathrm{m}}}\left(T-T_{\mathrm{m}}\right)+k_{\mathrm{powder}}\left(T_{\mathrm{m}}\right), T_{\mathrm{m}} < T < T_{\mathrm{L}}$$

(13)

where T_m is the temperature at the start of the melting and T_L is the temperature at the end of the melting. k_bulk is density of bulk and k_powder is density of powder. k_bulk is density of bulk and k_powder is density of powder.

The other method considering the powder effects comes from the laser-particle interaction-based heat source model. Taking directed energy deposition as example, the flight of powders in the laser beam forms the sheltering effects to the irradiation of laser on the surface of the melt pool. The temperature of the elements being activated in the simulation is not ambient temperature. The preheating from the laser irradiation in the flight stage of powder particles should be considered. An electromagnetic model was established in ref. [65] to consider the phenomena of laser-particle interactions in directed energy deposition. When the laser is treated as an electromagnetic wave, the heating of the flight particles inside the laser beam can be calculated,

$$q_{\mathrm{rh},\mathrm{powder}}=\frac{1}{2}\mathrm{Re}(J_{\mathrm{powder}}\times E_{\mathrm{powder}})$$

(14)

$$q_{\mathrm{ml},\mathrm{powder}}=\frac{1}{2}\mathrm{Re}(i\omega B_{\mathrm{powder}}\times H_{\mathrm{powder}})$$

(15)

where E is the electrical intensity. B is the magnetic intensity. H is the magnetic field intensity. J is the conduction current density vector in media.

The whole power density in one powder particle can be divided into two parts

$$q_{\mathrm{total},\mathrm{powder}}=q_{\mathrm{rh},\mathrm{powder}}+q_{\mathrm{ml},\mathrm{powder}}$$

(16)

Then, the integral of the total powder can lead to the heating power and the reduction part can be also obtained

$$Q_{\mathrm{powder}}={\displaystyle \underset{V_{\mathrm{p}}}{\int }{q}_{\mathrm{total},\mathrm{powder}}}d{V}_{\mathrm{p}}$$

(17)

$$Q_{\mathrm{reductioon}}={\displaystyle \sum _{i=1}^n{\displaystyle \int q_{\mathrm{total},\mathrm{powder},i}}}d{V}_{\mathrm{p},i}$$

(18)

where V_p is volumetric part of particle and n is particle number.

The division of reduction part and the total part of powder can lead to the ratio written as

$$\xi =Q_{\mathrm{reduction}}/Q_{\mathrm{total}}$$

(19)

This parameter can be then adopted in the Gaussian/double ellipsoid model to consider the powder effects in the simulation of the temperature variations.

The temperature rises of the powder particles interacted with laser can lead to temperature rises of elements being activated. The temperature rise of elements can be calculated as [66],

$$\Delta T=\frac{{\eta }_{\mathrm{m}}\times {\eta }_{\mathrm{s}}\times \frac{P}{\pi r_{\mathrm{b}}^2}\times \left(2\pi r_{\mathrm{p}}^2\right)\tau }{\left(\frac{4}{3}\pi r_{\mathrm{p}}^3\right)\times C_{\mathrm{p}}\times {\rho }_{\mathrm{p}}}$$

(20)

where ΔT is the temperature rise in powder flight in laser. η_m represents interference coefficient. η_s represents energy absorption fraction by particles. τ is the flight time in the laser beam. P means power. r_b and r_p represent radii of laser and particles respectively. C_p means the specific heat. ρ_p means density.

Figure 2 reveals the integration of the powder flow behaviors, the electromagnetic wave model and the double ellipsoid heat source model for the prediction of the temperatures in directed energy deposition. In the printing plane, the numbers of the particles can be counted. Its effect can be considered in the electromagnetic model for the simulation of the temperature rises. Then, the average temperature rises are assigned to the elements being activated. The energy loss during this process and absorbed energy are also calculated. The laser irradiated on the surface of the melt pool is then updated with consideration of the obtained data. The updated energy is applied to the melt pool surface to form the double ellipsoid heat source model [67]. Finally, the temperatures on the melt pool surface of the deposited layer can be obtained.

The direct coupling of powder particles with the melting region provides significant progress on the evaluations of powder effects on the final quality of additively manufactured components. Recently, Yan et al. [68,69] proposed coupling model directly revealing the coupling effect between powder and melting pool under laser. The model of the thermally fluid flow in powder scale was utilized to obtain the physically data of temperature in powder bed fusion. The powder spreading process was simulated by discrete element method. The key physical factors include thermal conduction, latent heat of melting, viscosity, Marangoni effect, surface tension, evaporation and recoil pressure are considered. This work provides significant insight into the powder bed fusion.

The scanning strategy and the laser parameters can be very important determining the temperature variations and distributions in thermal modelling. Detailed info can be found in review of thermal analysis of laser additive manufacturing [70,71,72,73,74]. The obtained temperature variations in time and the temperature distributions in space consist of the main input for the further modelling and simulations of microstructural changes and mechanical properties in laser AM.

The applications of different heat source models in the simulations of laser AM are summarized in Table 1.

Table 1. Summaries of heat source models.

Heat Source Model	AM Type	AM Material	AM Parameters	Temperature	Reference
Gaussian heat source	PBF	Ti6Al4V	P = 3 W d = 0.1 mm v = 1 mm/s dp = 30 μm	2425–2450 K	[75]
Gaussian heat source	DED	316L	P = 600 W d = 1.2 mm v = 6 mm/s dp = 80–120 μm	2000 °C	[76]
Surface heat source	DED	Ti-22Al-25Nb	P = 1000 W d = 4 mm v = 3 mm/s dp = 38–160 μm	~2300–2800 °C	[77]
Surface heat source	DED	Ti6Al4V	P = 400–600 W d = 1.74 mm v = 0.2–0.4 m/min	2139–2390 K	[78]
Double ellipsoid heat source	DED	Ti6Al4V	P = 1000 W d = 3 mm v = 5 mm/s dp = 45–150 μm	~2950 °C	[79]
Gaussian heat source	SLS	AlSi10Mg	P = 100 W d = 0.2 mm v = 100 mm/s	1761 °C	[80]
Semi-spherical power distribution model	SLS	Ti6Al4V	P = 270 W d = 0.2 mm v = 1 m/s	4000–7000 K	[81]
Gaussian heat source	PBF	Inconel 625	P = 2 W d = 0.025 mm v = 1 mm/s dp = 30 μm	~2000 °C	[82]
Gaussian heat source	PBF	Ti6Al4V	P = 3 W d = 0.1 mm v = 1 mm/s dp = 30 μm	2500–3000 K	[83]
Gaussian heat source	PBF	Hastelloy X	P = 150 W d = 0.1 mm v = 1–1.6 mm/s	~2700–3300 K	[84]
Gaussian heat source	PBF	Mg2Si with nanoparticles (Si)	P = 6.5–25 W d = 0.6 mm v = 4.23 mm/s	1200–2500 K	[85]
Surface heat source	SLS	Ti6Al4V	P = 170 W d = 0.1 mm v = 1.25 m/s	1650 °C	[86]
Gaussian heat source	SLS	AlSi10Mg	P = 700–1900 J/m d = 0.1 mm v = 0.1 m/s	731–2672 °C	[87]
Gaussian heat source	-	Ti6Al4V SS316L Al7075	P = 100 W d = 0.2 mm v = 4 m/s	2369 °C 1790 °C 969 °C	[88]
Different heat source models	PBF	SS17-4PH	P = 170–220 W d = 0.1 mm v = 0.6–1.3 m/s dp = 16–64 μm	~4500–6500 K	[89]
Gaussian heat source	PBF	Ti-6Al-4V	P = 100 W d = 0.2 mm v = 0.2 m/s	2311–2474 °C (depending on layer thickness)	[90]

P-laser power, d = laser diameter, v = scanning speed, d_p = particle size.

Table 1. Summaries of heat source models.

Heat Source Model	AM Type	AM Material	AM Parameters	Temperature	Reference
Gaussian heat source	PBF	Ti6Al4V	P = 3 W d = 0.1 mm v = 1 mm/s dp = 30 μm	2425–2450 K	[75]
Gaussian heat source	DED	316L	P = 600 W d = 1.2 mm v = 6 mm/s dp = 80–120 μm	2000 °C	[76]
Surface heat source	DED	Ti-22Al-25Nb	P = 1000 W d = 4 mm v = 3 mm/s dp = 38–160 μm	~2300–2800 °C	[77]
Surface heat source	DED	Ti6Al4V	P = 400–600 W d = 1.74 mm v = 0.2–0.4 m/min	2139–2390 K	[78]
Double ellipsoid heat source	DED	Ti6Al4V	P = 1000 W d = 3 mm v = 5 mm/s dp = 45–150 μm	~2950 °C	[79]
Gaussian heat source	SLS	AlSi10Mg	P = 100 W d = 0.2 mm v = 100 mm/s	1761 °C	[80]
Semi-spherical power distribution model	SLS	Ti6Al4V	P = 270 W d = 0.2 mm v = 1 m/s	4000–7000 K	[81]
Gaussian heat source	PBF	Inconel 625	P = 2 W d = 0.025 mm v = 1 mm/s dp = 30 μm	~2000 °C	[82]
Gaussian heat source	PBF	Ti6Al4V	P = 3 W d = 0.1 mm v = 1 mm/s dp = 30 μm	2500–3000 K	[83]
Gaussian heat source	PBF	Hastelloy X	P = 150 W d = 0.1 mm v = 1–1.6 mm/s	~2700–3300 K	[84]
Gaussian heat source	PBF	Mg2Si with nanoparticles (Si)	P = 6.5–25 W d = 0.6 mm v = 4.23 mm/s	1200–2500 K	[85]
Surface heat source	SLS	Ti6Al4V	P = 170 W d = 0.1 mm v = 1.25 m/s	1650 °C	[86]
Gaussian heat source	SLS	AlSi10Mg	P = 700–1900 J/m d = 0.1 mm v = 0.1 m/s	731–2672 °C	[87]
Gaussian heat source	-	Ti6Al4V SS316L Al7075	P = 100 W d = 0.2 mm v = 4 m/s	2369 °C 1790 °C 969 °C	[88]
Different heat source models	PBF	SS17-4PH	P = 170–220 W d = 0.1 mm v = 0.6–1.3 m/s dp = 16–64 μm	~4500–6500 K	[89]
Gaussian heat source	PBF	Ti-6Al-4V	P = 100 W d = 0.2 mm v = 0.2 m/s	2311–2474 °C (depending on layer thickness)	[90]

P-laser power, d = laser diameter, v = scanning speed, d_p = particle size.

Although significant process has been achieved for the modelling and simulation of laser AM, the direct correlation between powders and final quality are still on the way. It is obvious that the thermal model considering powder particle effects can be exactly used to reveal the physical nature of the laser additive manufacturing. However, the powder particles interact with laser source and melt pool. It is the main factor in determining that additive manufacturing is different to the traditional welding process, in which it is complex to reveal all the effects of powders in the modelling and simulation. How to consider the powder effects in the modelling and simulation of laser AM reflecting the physics of nature in laser AM is still challenging. The further combination of the powder-based heat source model and the powder-based simulation microstructures and mechanical properties is strongly necessary in order to realize this objective.

3. Microstructure Evolutions

3.1. Monte Carlo Model

Monte Carlo model is established following probabilistic strategy on selections of grain coarsening orientations, which leads to high efficiency for the problems related to multi length and time scales. In the Monte Carlo model, the computational domain can be discretized into lattices and random numbers from (1–q) are assigned to the lattice representing the random grain orientations [91,92,93], and 0 is usually specially selected to represent the state of liquid.

In the solidification in the modelling, the new grain sites are determined by the defined probability [65,94],

$$p_{\mathrm{s}}=\frac{\delta N_S}{N_{\mathrm{MC}}}=\delta n_{\mathrm{s}}{V}_{\mathrm{MC}}$$

(21)

where δN_s is nucleation rate per time, N_MC is the lattice number, V_MC is lattice volume. δn_s is grain density increase. The lattice point represents a real lattice in the discretized domain and expressed as a number in the matrix for computations. The nucleation rate can be then obtained by the grain density increase and the volume of each lattice point. The increment of grain density can be calculated by the Gaussian distribution on the mold surface in the melting region according to Ref. [90].

In Monte Carlo method, the lattice points are used. The grain boundary energy (GBE) on lattice sites is,

$$E=-J{\displaystyle \sum _{j=1}^m({\delta }_{S_{\mathrm{i}}{S}_{\mathrm{j}}}-1)}$$

(22)

where J represents constant. m is lattice site number around the current selected point. δ is the Kronecker symbol. S_i and S_j are orientations of current lattice site and its neighbour.

When the orientation of the current site is changed, the total energy can be also changed with this re-orientation. When the energy change becomes smaller, it is definite to accept the re-orientation. When the energy change becomes increased, the re-orientation is accepted with Boltzmann probability,

$$p=\{\begin{array}{l}1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Delta E\le 0\\ e^{-\frac{\Delta E}{k_{\mathrm{B}}T}}=e^{-\frac{\left(m_2-m_1\right)J}{k_{\mathrm{B}}T}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Delta E>0\end{array}$$

(23)

where ΔE is GBE change in re-orientation. k_B is the Boltzmann constant. m₁ is the orientation number before re-orientation and m₂ is the different orientation number after re-orientation. T is temperature.

MC method has been adopted to simulate solidification as well as solid-state phase transformation in laser AM [33,95,96,97]. In Ref. [33], a cross sectional MC model was developed to reveal the solidification phenomenon on the cross section, on which temperature variated largely. The fact that the different thermal histories on different sizes lead to different MCS in different selected sites. So, the discrete form of MCS can be expressed as

$$\left(MCS\right)=\left\{\begin{array}{cccc}{\mathrm{MCS}}_{1,1}& {\mathrm{MCS}}_{1,2}& \cdots & {\mathrm{MCS}}_{1,\tilde{m}}\\ {\mathrm{MCS}}_{2,1}& {\mathrm{MCS}}_{2,2}& \cdots & {\mathrm{MCS}}_{2,\tilde{m}}\\ ⋮& ⋮& ⋮& ⋮\\ {\mathrm{MCS}}_{\tilde{n},1}& {\mathrm{MCS}}_{\tilde{n},2}& \cdots & {\mathrm{MCS}}_{\tilde{n},\tilde{m}}\end{array}\right\}$$

(24)

By normalization of MCS data on the cross section, the uniform format for MCS can be re-generated by the introduction of $MC{S}_{\tilde{n},\tilde{m}}/MC{S}_{\max }=p_{\tilde{n},\tilde{m}}$

$$M\left(\tilde{n},\tilde{m}\right)=\left(MCS\right)=\left\{\begin{array}{cccccc}{p}_{1,1}& p_{1,2}& \cdots & p_{1,m^{\ast }}& \cdots & p_{1,\tilde{m}}\\ p_{2,1}& p_{2,2}& \cdots & p_{2,m^{\ast }}& \cdots & p_{2,\tilde{m}}\\ ⋮& ⋮& \cdots & ⋮& \cdots & ⋮\\ p_{n^{\ast },1}& p_{n^{\ast },1}& \cdots & 1& \cdots & p_{n^{\ast },\tilde{m}}\\ ⋮& ⋮& \cdots & ⋮& \cdots & ⋮\\ p_{\tilde{n},1}& p_{\tilde{n},1}& \cdots & p_{\tilde{n},m^{\ast }}& \cdots & p_{\tilde{n},\tilde{m}}\end{array}\right\}.$$

(25)

The MCS can be directly correlated with time

$${\left(\mathrm{MCS}\right)}_{{\Omega }_1}^{{}^{\left(n^{\prime }+1\right)n_1}}={\left(\frac{d_0}{K_1\lambda }\right)}^{n^{\prime }+1}+\frac{\left(n^{\prime }+1\right){\alpha }_{\mathrm{mcs}}{C}_1^{n^{\prime }}}{{\left(K_1\lambda \right)}^{n^{\prime }+1}}{\displaystyle \sum _{i=1}^{n_t}\left[{\exp }^n\left(-\frac{Q}{R{T}_{{\Omega }_1,i}}\right)\Delta t_{{\Omega }_1,i}\right]}$$

(26)

where d₀ is initial size of grain. K₁ is constant and n₁ is constant. They can be obtained by the introduction of regression algorithm. Q is the activate energy. R is the gas constant. and α_mcs and n’ are scaling factors. λ is discrete grid spacing in utilized grid model. The definition of C₁ can be found in Ref. [33].

Figure 3 shows the solidification simulated by Monte Carlo model and observed in experiment. The comparison shows good agreement between numerical results by Monte Carlo model and experiments.

3.2. Cellular Automaton Model

The basic theory on the simulation of the grain coarsening comes from the decrease of the free energy of total computational domain. The change of the grain diameter (D) follows,

$$D_t^2-D_0^2=k{t}^n$$

(27)

where k and n are constants related to material.

When the atoms jump from one side of the grain boundary to another, energy barrier (ΔG_A) should be overcome. Considering the probability for the atom jumps across grain boundary, the probability functions can be found to be fit for both MC and CA [98],

$$P_1=\exp \left(-\Delta G_{\mathrm{A}}/RT\right)$$

(28)

$$P_1=\exp \left(-\left(\Delta G_{\mathrm{A}}+\Delta G\right)/RT\right)$$

(29)

where ΔG is the free energy difference between grains.

CA method includes basic elements like cell, space, neighbor, and rules, as shown in Figure 4. Taking a two-dimensional case as an example, the discretization of two-dimensional space can lead to different grids with positive triangle, square and regular hexagon elements. As the form of expression is easier to implement, here square grid is adopted for the explanation of the simulation. The selection of the neighborhood types includes Neumann type, Moore type and alternating Moore type.

The CA method is discussed in a square grid. For the Moore neighbors type, the transformation rules are selected as follows [30],

• If the state value of a cells e is the same as that of the eight adjacent neighbors, then the state value of the next step is constant;
• (a) If any 3 of the cell b, d, f, and h is A States, the state of the cell e is converted to A at the next CAS;
  (b) If any 3 of the cell a, c, g, and j are A States, the state of the cell e is converted to A at the next CAS;
• If the above conditions are not satisfied, each cell overcomes the energy barriers to change randomly into 8 adjacent cells, calculate the grain boundary energy change ΔE, the GBE for each site can be calculated by the Hamiltonian,

  $$E=J{\displaystyle \sum _j^m\left(1-{\delta }_{s_i{s}_j}\right)}.$$

  (30)

The conversion probability is obtained by the following formula:

$$p=\{\begin{array}{c}1\hspace{1em}\Delta E\le 0\\ \exp \left(-\frac{\Delta E}{kT}\right)\Delta E>0\end{array}.$$

(31)

For the Neumann neighbor type, the rules are as follows:

• If the state value of a cells e is the same as that of the four adjacent neighbors, then the state value of the next step is constant;
• If any 3 of the cell b, d, f, and h is A States, the state of the cell e is converted to A at the next CAS. It can be expressed as (1);
• If the above conditions are not satisfied, each cell overcomes the energy barriers to change randomly into 4 adjacent cells, the rules of calculation are the same as formulas (3) and (4);
• If the above conditions are not satisfied, randomly changes to the value of a neighbor cell.

Cellular automaton has been applied in laser AM for the simulations of dendrite formation in the solidification process [99,100,101,102].

3.3. Phase Field Model

Phase field (PF) method can be applied to the simulation of field dynamics simulation like impurity concentration, crystalline order, orientation and re- orientation, formation of dendrite. The spinodal decomposition and order-disorder kinetics can be also realized by the utilization of phase filed method [103]. The basic theory of phase field method is from the dissipative minimization of free energy. The microstructures are mathematically described as order parameters in the PF method. The order parameter can be treated as a continuous function. In a solidification case, its value can be variated from −1 representing liquid to +1 denoting completely solid state. It is unnecessary to track the movement of the boundaries of the phase changes. In MC and CA, the steps in simulation can only afterwards be related to the real time. However, in PF, the microstructure evolutions can be directly simulated towards the thermodynamic equilibrium with a real time solution. This is the advantage of PF but this point can lead to the obvious increase of computational time. As a computationally more demanding method, the domain for the simulation is generally small. So, the requirement of equilibrium on the computational domain and the full consideration of features on temperature changes should be designed in the extension of the thermodynamic energy for homogeneous system on interfaces. Free energy (FE), as the key for PF, can be expressed as [104,105],

$$F(t)={\displaystyle \int \left[f_0({\eta }_1(r,t),{\eta }_2(r,t),\dots ,{\eta }_{\mathrm{Q}}(r,t))+{\displaystyle \sum _{q=1}^Q\frac{k_q}{2}}{(\nabla {\eta }_{\mathrm{q}}(r,t))}^2\right]}dr$$

(32)

where f₀ is the local FE density. k_q is coefficient of gradient energy related to GBE. η_q is continuous order parameter representing the crystallographic orientation of each grain. The local FE density can be written as

$$f_0(\{{\eta }_q(r,t)\})=-\frac{a}{2}{\displaystyle \sum _{q=1}^Q{\eta }_q^2(r,t)}+\frac{b}{4}{({\displaystyle \sum _{q=1}^Q{\eta }_q^2(r,t)})}^2+(c-\frac{b}{2}){\displaystyle \sum _{q=1}^Q{\displaystyle \sum _{s>q}^Q{\eta }_q^2(r,t){\eta }_s^2(r,t)}}$$

(33)

where a = b and c > a/2 being constants. When η_q = ±1 and η_s = 0 for all s ≠ q, f₀ reaches 2Q minima.

It is generally to use Allen-Cahn equation for the solution of the order parameters η_q,

$$\frac{\partial {\eta }_q\left(r,t\right)}{\partial t}=-L_q(T)\frac{\delta F(t)}{\delta {\eta }_q\left(r,t\right)}\hspace{0.17em}\hspace{0.17em}(q=1,2,\dots ,Q)$$

(34)

where L_q denotes GB mobility.

Then, the modified Arrhenius type equation can be utilized for the description on the temperature and thermal gradient contributions to the GE mobility,

$$L_q(T)=L_0{(\frac{T}{T_a})}^m\exp (-\frac{\Delta Q}{R_{\mathrm{g}}T})$$

(35)

where L₀ and m are constants. ΔQ is the activate energy. R_g = 8.314 J/(mol·K) is gas constant.

Figure 5 reveals the dendrite grain structures coming from the PF model. In comparison with the experimental observations, the phase field model can be efficient to reveal the formation of dendrites in solidification in laser AM.

PF method is widely used in the simulation of microstructural evolutions in laser AM [107,108,109,110,111,112]. The integration of the phase field model into the crystal plasticity model [28,113,114,115] provides an efficient to correlate the microstructural changes with the mechanical properties and remains a hot topic for modelling and simulations in laser additive manufacturing.

3.4. Precipitate Evolution Model

The precipitation evolution model [116,117,118] can be used with combination of the LAM process model to calculate the evolutions of the second phase. Al-Mg-Si alloy is considered as an example of the material for LAM. The nucleation and the following coarsening of precipitates are the key process for the final formation of precipitates in precipitate hardening alloy like Al-Mg-Si. In the phase transformation process, the energy barriers include the interfacial energy and the elastic coherency strain energy. If the coherency strains are neglected on the nucleated sites of particles, the nucleation equation can be written as,

$$j=j_0\exp (-\frac{\Delta G_{\mathrm{het}}^{\ast }}{RT})\exp (-\frac{Q_{\mathrm{d}}}{RT})$$

(36)

where j₀(#/(m³·s)) is initial nucleation site number. Q_d (J mol⁻¹) is the activate energy of element diffusion. According to the classical nucleation theory, an appropriate expression for $\Delta G_{\mathrm{het}}^{\ast }$ is used for the critical energy barrier against heterogeneous nucleation,

$$\Delta G_{\mathrm{het}}^{\ast }=\frac{{\left(A_0\right)}^3}{{\left(RT\right)}^2{\left[\ln \left(\overline{C}/C_{\mathrm{e}}\right)\right]}^2}$$

(37)

where $\overline{C}$(wt.%) is the mean concentration of element in matrix in current stage. A₀ (J mol⁻¹) is the nucleation energy barrier. C_e (wt.%) is the solute concentration at the precipitate/matrix interface in equilibrium state,

$$C_{\mathrm{e}}=C_{\mathrm{s}}\exp (-\frac{Q_{\mathrm{s}}}{RT})$$

(38)

where C_s is pre-exponential term of Mg equilibrium concentration. Q_s is the enthalpy of the solvus boundary.

The particle can be coarsening in a speed that can be calculated as,

$$v=\frac{dr}{dt}=\frac{\overline{C}-C_i}{C_{\mathrm{p}}-C_i}\frac{D}{r}$$

(39)

where C_i (wt.%) is the solute concentrations on the interface between particles and matrix. C_p represents the solute concentration of element inside the particles. Generally, Mg₅Si₃ can be considered to be the only stoichiometric composition of particles in simulation. D (m²·s⁻¹) is a coefficient for diffusion of elements. r is the particle radius.

The solute at interface can be re-written with consideration of the Gibbs-Thomson equation,

$$C_i=C_{\mathrm{e}}\exp (\frac{2\gamma V_{\mathrm{m}}}{rRT})$$

(40)

where γ (J·m⁻²) is interface energy between particle and matrix, and V_m (m³·mol⁻¹) represents precipitation molar volume.

The precipitate stops to continuously grow in the case of equilibrium of $\overline{C}=C_e$. In this case, the critical radius in equilibrium state can be written as,

$$r^{\ast }=\frac{2\gamma V_{\mathrm{m}}}{RT}{\left(\ln \left(\frac{\overline{C}}{C_e}\right)\right)}^{-1}$$

(41)

In the following stage, the new nucleation is added in each time step as j∆t. The initial size of the newly generated particle should be larger than the critical value and it is recommended to be defined as 1.05 r*. The solute can exist in either the matrix or precipitates. The mass balance can be calculated,

$$f{C}_p+\left(1-f\right)\overline{C}=C_0$$

(42)

where f can then be written as,

$$f=\left(C_0-\overline{C}\right)/\left(C_p-\overline{C}\right)$$

(43)

Figure 6 reveals the discretization of the precipitates in the dimension of the radius. The precpitate can be assumed to be sphere-shaped and the volume can be calculated as,

$$f={\displaystyle \sum _i\frac{4}{3}\pi r_i^3{N}_i}$$

(44)

where N_i is particle number and r_i is the corresponding radius.

From the above models, it can be seen that different models reveal different advantages and disadvantages in modelling and simulation of the microstructural evolutions in laser AM. MC and CA reveal high computational efficiency and PF shows its high ability revealing the physical nature of the process. For MC and CA, how to incorporate the physical nature in solidification and solid-state phase transformation is challenging. However, the problem related to robustness of the algorithm and the establishment of its model is still challenging for PF method, which can be the focus for the modelling and simulations of laser additive manufacturing.

4. Mechanical Properties

4.1. Physics Based Model

Physics-based models for the prediction of mechanical properties are established on the strengthening mechanism in materials science. This means that the model can be greatly different for different materials with different strengthening mechanisms in materials science. For precipitate strengthening materials like aluminum alloys, the physics-based model should mainly consider precipitate evolutions and correlate the precipitate evolutions with mechanical properties. For materials with grain and phase strengthening like titanium alloys, the mechanical response is established on the basis of dislocation density and its interactions with grain structures. Here we introduced the two models with the mentioned two strengthening effects.

4.1.1. Precipitate Evolution Based Model of Mechanical Property

The yield strength of material can be divided into three parts,

$${\sigma }_s={\sigma }_0+{\sigma }_{ss}+{\sigma }_{preci}$$

(45)

where σ_ss represents the contribution of solid solution and σ_p represents the contribution from the precipitates. σ₀ represents the contribution from the pure grains. The contribution from the strenthening item from the dislocation density, σ_dd, should be included in this item, as well as the contribution from the frictional stress, σ_f.

$${\sigma }_{dd}=M\alpha Gb\sqrt{\rho }$$

(46)

where M is the Taylor factor. α ranges from 0.2 to 1.5 [119]. G is the shear modulus. b is the Burgers vector and ρ is the dislocation density. The contribution from the dislocation density is close to 1 MPa and can be negligible. ${\sigma }_f$ is about 10 MPa for pure aluminum crystals.

The contribution from the solid solution [117] can be computed by,

$${\sigma }_{ss}={\displaystyle \sum _{i=1}^3{k}_i{C}_i^{2/3}}$$

(47)

where k_i represents the contributions from the different alloying element i. C_i is the corresponding solute concentration of element i. In current case, i = 1, 2 and 3 representing the contributions from the main elements like Mg, Si and Cu. The precipitate hardening is given by [117],

$${\sigma }_{\mathrm{p}}=\frac{M\overline{F}}{b\overline{L}}$$

(48)

where $\overline{L}$ is mean distance between particles in the direction of the dislocation line. $\overline{F}$ represents the strength value to overcome obstacles,

$$\overline{F}=\frac{{\displaystyle \sum _i{N}_i\left(r\right)F_i\left(r\right)}}{{\displaystyle \sum _j{N}_j\left(r\right)}}$$

(49)

where F_i(r) is the discretized obstacle strength of i-th class. Depending on the particle sizes, the movement of the dislocation on the particles can be classified into two modes, shearing or bypassing particles. For precipitates with small sizes, the dislocation can cut off the precipitates by shearing. For precipitates with big sizes, the dislocation can bypass the precipitates to form dislocation loop. Both the cases can lead to the multiplication of dislocations and can be the basic strengthening mechanism for this kind of alloy. This also means that the critical value of r_c should be defined to distinguish the two ways for the material strengthening. For the former case,

$$F_i=2\beta G{b}^2\left(\raisebox{1ex}{$r_i$}\!\left/ \!\raisebox{-1ex}{$r_c$}\right.\right)$$

(50)

For the latter case,

$$F_i=2\beta G{b}^2$$

(51)

where β is the dislocation tension [120,121].

$${\sigma }_p=\frac{M}{b\overline{r}}{\left(2\beta G{b}^2\right)}^{-1/2}{\left(\frac{3f}{2\pi }\right)}^{1/2}{\overline{F}}^{3/2}$$

(52)

When nano particles exist as strengthening items, the strengthening effect can be written as [122],

$${\sigma }_n=\frac{M}{b{\overline{r}}_n}{\left(2\beta G{b}^2\right)}^{-1/2}{\left(\frac{3f_n}{2\pi }\right)}^{1/2}{\left(\frac{{\displaystyle \sum _i{N}_i{F}_i}}{{\displaystyle \sum _i{N}_i}}\right)}^{3/2}.$$

(53)

where ${\overline{r}}_n=\xi r_p$. $\xi$ is defined to be the parameter related to the distribution of nanoparticles. r_p is the radius of the nanoparticles. f_n is the volume fraction of nanoparticles.

4.1.2. Dislocation Density Evolution Based Model of Mechanical Property

The flow stress is determined mainly by the interactions of immobile dislocation in long range σ_L and the dislocates interaction with short obstacles σ_s [123,124],

$${\sigma }_{\mathrm{y}}={\sigma }_{\mathrm{L}}+{\sigma }_{\mathrm{S}}$$

(54)

$${\sigma }_{\mathrm{S}}={\sigma }_{\mathrm{ath}}{\left\{1-{\left[\frac{k_{\mathrm{b}}T}{\Delta f_{0}G{b}^3}\ln \left(\frac{{\dot{\epsilon }}^{\mathrm{ref}}}{{\dot{\overline{\epsilon }}}^{\mathrm{p}}}\right)\right]}^{\frac{1}{q}}\right\}}^{\frac{1}{p}}$$

(55)

$${\sigma }_L=m\alpha Gb\sqrt{{\rho }_i}$$

(56)

where ${\sigma }_{\mathrm{ath}}$ is the shear strength being dependent on temperature. ${\dot{\epsilon }}^{\mathrm{ref}}$ is the reference strain rate [125]. ΔF = Δf₀Gb is the activation energy overcoming the obstacles.

In the deformation of tension, the dislocation multiplication and the dislocation annihilation coexist, as below,

$${\dot{\rho }}_i={\dot{\rho }}_i^{+}-{\dot{\rho }}_i^{-(\mathrm{glide})}-{\dot{\rho }}_i^{-(\mathrm{climb})}-{\dot{\rho }}_i^{-(\mathrm{globularization})}$$

(57)

where the dislocation annihilation is caused by glide, climbing and globularization.

The mobile dislocation density can be directly related to the plastic strain rate,

$${\dot{\rho }}_i{}^{+}=\frac{m}{b}\frac{1}{\Lambda }{\dot{\overline{\epsilon }}}^{\mathrm{p}}$$

(58)

where Λ is the average moving distance of dislocation before it is annihilated or is transformed into immobile dislocation. It can be determined by [126],

$$\frac{1}{\Lambda }=\frac{1}{g}+\frac{1}{s}$$

(59)

where g and s are the sizes of grains and sub-grains.

The evolution of sub-grains can be described by [127],

$$\frac{1}{s}=\frac{\sqrt{{\rho }_i}}{K_{\mathrm{c}}}$$

(60)

where K_c represents the parameter related to temperature.

The grain size is dependent on the time for heating,

$$g^n-g_0^n=Kt$$

(61)

where g₀ is the initial size of grain and K is the material parameter.

The dislocation annihilation is the main mechanism for dynamic recovery in deformation [128],

$${\dot{\rho }}_i^{-(\mathrm{glide})}=\Omega {\rho }_i{\dot{\overline{\epsilon }}}^{\mathrm{p}}$$

(62)

$${\dot{\rho }}_i{}^{-\left(\mathrm{climb}\right)}=2c_{\mathrm{r}}{D}_{\mathrm{app}}\frac{G{b}^3}{kT}\left({\rho }_i^2-{\rho }_{\mathrm{eq}}^2\right)$$

(63)

where Ω is a constant. c_r is material parameter. D_app is the diffusion rate.

Grain globularization observed in experiment in high temperature [129] can lead to the dislocation annihilation [130,131],

$${\dot{\rho }}_i{}^{-\left(\mathrm{globularization}\right)}=\psi {\dot{X}}_{\mathrm{g}}\left({\rho }_i-{\rho }_{\mathrm{eq}}\right)\hspace{1em}\mathrm{until}\hspace{1em}{\rho }_i\le {\rho }_{\mathrm{eq}}$$

(64)

$${\dot{\rho }}_i{}^{-\left(\mathrm{globularization}\right)}=0, \mathrm{when} {\rho }_i<{\rho }_{\mathrm{cr}}$$

(65)

where ${\dot{X}}_g$ is the globularization rate and ψ is the correction coefficient. And A the globularization rate can be divided into two parts [132],

$$X_{\mathrm{g}}=X_{\mathrm{d}}+\left(1-X_d\right)X_s$$

(66)

where

$${\dot{X}}_s=M\frac{\dot{g}}{g}$$

(67)

$${\dot{X}}_d=\frac{B{k}_{\mathrm{p}}{\dot{\overline{\epsilon }}}^{\mathrm{p}}}{{\left({\overline{\epsilon }}^{\mathrm{p}}\right)}^{1-k_{\mathrm{p}}}{e}^{B{\left({\overline{\epsilon }}^{\mathrm{p}}\right)}^{{\mathrm{k}}_{\mathrm{p}}}}}$$

(68)

where M, B and k_p are material constants.

4.2. Crystal Plasticity Model

A simple decomposition of deformation gradient in large deformation can be found [133,134],

$$F=F^{\mathrm{e}}{F}^{\mathrm{p}}$$

(69)

$$\dot{F}=LF$$

(70)

Assuming that the plastic state without consideration of the lattice rotation is the intermediate configuration (~ is added for the expressions in intermediate configuration), then [135,136],

$${\tilde{L}}^{\mathrm{p}}={\tilde{D}}^{\mathrm{p}}+{\tilde{W}}^{\mathrm{p}}={\displaystyle \sum _{\alpha }{\dot{\gamma }}_{\alpha }({\tilde{n}}_{\alpha }\otimes {\tilde{m}}_{\alpha })}$$

(71)

where $n$ is the normal of the slipping plane and $m$ is the slipping direction.

$${\dot{\tilde{F}}}^{\mathrm{p}}={\tilde{L}}^{\mathrm{p}}{\tilde{F}}^{\mathrm{p}}$$

(72)

In the current configuration,

$${\tilde{m}}_{\alpha }=F^{\mathrm{e}}{\tilde{m}}_{\alpha }^0$$

(73)

To keep the orthogonal of the slipping direction and slipping plate normal, i.e., $n\cdot m=0$, we have

$${\tilde{n}}_{\alpha }={\tilde{n}}_{\alpha }^0{F}^{\mathrm{e}}{}^{-1}$$

(74)

The slip rate on the αth slipping system can be represented as the power law function of the resolved shear stress and the slip resistance,

$${\dot{\gamma }}^{\alpha }={\dot{\gamma }}_0{}^{\alpha }{\left(\frac{{\tau }^{\alpha }}{g^{\alpha }}\right)}^{1/m}$$

(75)

where ${\dot{\gamma }}_0{}^{\alpha }$ is the referenced shearing rate. m represents the sensitivity factor. The slipping resistance is divided into two items,

$$g^{\alpha }=\sqrt{{\left({\tau }^{\ast }\right)}^2+{\left({\tau }^{\mathrm{obs}}\right)}^2}$$

(76)

where [137]

$${\tau }^{\ast }={\tau }^{\mathrm{cr}}{\left(1-{\left(\frac{k_{B}T}{H_0}\ln \left(\frac{{\dot{\epsilon }}_0}{\dot{\gamma }}\right)\right)}^{1/q}\right)}^{1/p}$$

(77)

$${\tau }^{\mathrm{obs}}=A\mu b\sqrt{{\displaystyle \sum _{\beta =1}^{NS}{\rho }^{\beta }}}$$

(78)

where H₀ is the activation enthalpy. τ^cr, ε₀, p, q and A are constants, μ is the shear modulus. NS is the slipping system number, and ρ^β is the dislocation density on β-th slipping system.

The use of finite elements in crystal plasticity model provides the possibility of the direct combination with the results of Monte Carlo model, phase field model, and cellular automaton. In Ref. [138], the lattices used in Monte Carlo simulation are directly projected to the finite elements in the crystal plasticity model for the studies on the influences of microstructures on mechanical behaviors in deformation of tension, as shown in Figure 7 and Figure 8. The crystal plasticity model can be extended into a three-dimensional case. Zhang et al. [133,134] proposed a three-dimensional case for the crystal plasticity model. The in-situ synchrotron X-ray diffraction was adopted to explain the mechanism for micromechanical behavior of AlSi3.5Mg2.5 (wt.%) alloy produced by PBF in two different states, as-built state, and direct-aged state.

The combination of crystal plasticity with the microstructural evolution model is one of the key problems connecting the microstructure and mechanical behavior in laser additive manufacturing, which can be important for the evaluation of additive manufacturing quality. This is the reason that this aspect has been becoming a hot topic in the simulation of the laser additive manufacturing in recent years. Among them, the combination of phase field model and the crystal plasticity model is very popular [28,114,139,140] due to the wide use of PF method in the simulation of the microstructural evolutions. Although key processes have been achieved, the integration of the microstructure and the mechanical response with consideration of the more details like defect formation, anisotropy and non-uniformity are strongly necessary. Moreover, the considerations of the formation of grain boundaries and the barrier energy and their relationship with dislocations in such a simulation is also a potential topic for further studies.

The physics-based model reveals the ability connecting the physical nature occurring in laser additive manufacturing including the changes of precipitates and the changes of the dislocations. However, the crystal plasticity model directly shows its ability connecting the grain morphologies with the mechanical responses of the grain clusters. In fact, the plasticity behavior in deformation is basically determined by the dislocation evolution and its interactions with the precipitate, grains, and defects. The bridging of the two models in modelling and simulation of mechanical properties is still challenging.

The changes of mechanical properties in laser additive manufacturing can be important factor determining the final dimensional accuracy incorporating the residual distortions [141] and can affect the further processing properties [142]. Residual stresses and distortions can be usually predicted by inherent strain method [143,144,145,146,147,148,149,150,151]. Although the method is valid and successful in many cases in current stage, the consideration of the mechanical property changes in the model using inherent strain method is still a challenge for the further development of models of residual states, which can be meaningful for the accurate controls of dimensional accuracy in laser additive manufacturing.

The temperature obtained from heat transfer analyses can be adopted for the further simulations on microstructures. The results of microstructural simulations can further affect the prediction of mechanical properties. How to integrate them together is still challenging. The process-structure-property relationship was well reviewed and summarized in Refs. [152,153,154,155,156,157]. The further combination of the reviewed methods to focus on the topic of process-structure-property relationship in laser additive manufacturing is hot topic and needs to be well studied in future.

5. Summaries and Perspectives

Process-structure-property relationship is the main topic for the modelling and simulation of laser additive manufacturing. The investigation on the process-structure-property relationship relies on the development of modelling and simulations in heat transfer, microstructural evolution, and the mechanical properties. The volumetric heat source model includes the Gaussian heat source model and the double ellipsoid heat source model. The combination of heat source models with powder features provides the possibility connecting the powders with the temperatures in laser additive manufacturing. The powder particles can be interacted with laser source and melt pool. It is the main factor in determining that additive manufacturing is different to the traditional welding process, in which it is complex to reveal all the effects of powders in the modelling and simulation. How to consider the powder effects in the modelling and simulation of additive manufacturing reflecting the physics of nature in additive manufacturing is still challenging. The microstructure evolutions can be simulated by Monte Carlo models, cellular automaton, and phase field models. The consideration of the powder and laser features can enhance the reliability of such models in comparison with experimental processes. The mechanical property is strongly affected by the material types. For MC and CA, how to incorporate the physical nature in solidification and solid-state phase transformation is challenging. However, the problem related to robustness of the algorithm and the establishment of its model is still challenging for PF method, which can be the focus for the modelling and simulations of laser additive manufacturing. The precipitate strengthening material can be predicted by the precipitate evolution-based model and the material related grains and phases can be predicted by the dislocation density based model. The further combination of the physics underlying the materials for strengthening effects with mechanical property should be developed in consideration of the different materials in laser additive manufacturing. The bridging of the two models in modelling and simulation of mechanical properties is still challenging.