Additive particle deposition and selective laser processing-a computational manufacturing framework

Abstract

Many additive manufacturing technologies involve the deposition of particles onto a surface followed by selective, targeted, laser heating. This paper develops a modular computational framework which combines the various steps within this overall process. Specifically, the framework synthesizes the following:

• particle dynamics, which primarily entails: (a) the movement of the particles induced by contact with the surface, (b) particle-to-particle contact forces and (c) near-field interaction and external electromagnetic fields.

• laser-input, which primarily entails: (a) absorption of laser energy input and (b) beam interference (attenuation) from particles and

• particle thermodynamics, which primarily entails: (a) heat transfer between particles in contact by conduction and (b) subsequent thermal softening of the particles.

Numerical examples are provided and extensions are also addressed for two advanced processing scenarios involving solid-liquid-gas phase transformations.

Appendices

Appendix 1: Contact area parameter and alternative models

1.1 Contact area parameter

Following Zohdi [116], and referring to Fig. 12, one can solve for an approximation of the common contact radius $a_{ij}$ (and the contact area, $A^c_{ij}=\pi a_{ij}^2$) by solving the following three equations,

$$\begin{aligned} a_{ij}^2+L_i^2=R^2_i, \end{aligned}$$

(8.1)

and

$$\begin{aligned} a_{ij}^2+L_j^2=R^2_j, \end{aligned}$$

(8.2)

and

$$\begin{aligned} L_i+L_j=||{\varvec{r}}_i-{\varvec{r}}_j||, \end{aligned}$$

(8.3)

where $R_i$ is the radius of particle $i$, $R_j$ is the radius of particle $j$, $L_i$ is the distance from the center of particle $i$ and the common contact interpenetration line and $L_j$ is the distance from the center of particle $j$ and the common contact interpenetration line, where the extent of interpenetration is

$$\begin{aligned} \delta _{ij}=R_i+R_j-||{\varvec{r}}_i-{\varvec{r}}_j||. \end{aligned}$$

(8.4)

The above equations yield an expression $a_{ij}$, which yields an expression for the contact area parameter

$$\begin{aligned} A^c_{ij}=\pi a_{ij}^2=\pi (R^2_i-L^2_i), \end{aligned}$$

(8.5)

where

$$\begin{aligned} L_i=\frac{1}{2}\left( ||{\varvec{r}}_i-{\varvec{r}}_j||-\frac{R^2_j-R^2_i}{||{\varvec{r}}_i-{\varvec{r}}_j||}\right) . \end{aligned}$$

(8.6)

Fig. 12

An approximation of the contact area parameter for two particles in contact (Zohdi [116])

1.2 Aternative models

One could easily construct more elaborate relations connecting the relative proximity of the particles and other metrics to the contact force, $\varvec{\Psi }^{con,n}_{ij}\propto \mathcal{F}({\varvec{r}}_i, {\varvec{r}}_j, {\varvec{n}}_{ij},R_i,R_j, \ldots )$, building on, for example, Hertzian contact models. This poses no difficulty in the direct numerical method developed. For the remainder of the analysis, we shall use the deformation metric in Eq. 2.8. For detailed treatments, see Wellman et al. [91–95] and Avci and Wriggers [6]. We note that with the appropriate definition of parameters, one can recover Hertz, Bradley, Johnson–Kendel–Roberts, Derjaguin–Muller–Toporov contact models. For example, Hertzian contact is widely used, with the assumptions being

• frictionless, continuous, surfaces,

• each of contacting bodies are elastic half-spaces, whereby the contact area dimensions are smaller radii of the bodies and,

• the bodies remain elastic (infinitesimal strains),

results in the following contact force:

$$\begin{aligned} \varvec{\Psi }^{con,n}_{ij}=\frac{4}{3}(R^*)^{1/2}E^*\delta ^{3/2}_{ij}, \end{aligned}$$

(8.7)

which has the general form of $\varvec{\Psi }^{con,n}=K^*_{ij}\delta ^p_{ij}$, where

• $R^*=\left( \frac{1}{R_i}+\frac{1}{R_j}\right) ^{-1}$ and

• $E^*=\left( \frac{1-\nu _i^2}{E_i}+\frac{1-\nu _j^2}{E_j}\right) ^{-1}$,

where $E$ is the Young’s modulus and $\nu$ is the Poisson ratio, The contact area with such a model has already been incorporated in the relation above, and is equal to $A^c_{ij}=\pi a^2$ where $a=\sqrt{R^*\delta _{ij}}$. For more details, we refer the reader to Johnson [48]. Furthermore, we remark that the normal contact between a particle and a wall, with a Hertzian model is given by

$$\begin{aligned} \varvec{\Psi }^{wall,n}_i=\frac{4}{3}(R^*)^{1/2}E^*\delta ^{3/2}_{iw}=K^*_{iw}\delta ^p_{iw}, \end{aligned}$$

(8.8)

where $R_j=R_w=\infty$ (see Eq. 8.7)

• $R^*=R_i$ and

• $E^*=\left( \frac{1-\nu _i^2}{E_i}+\frac{1-\nu _j^2}{E_j}\right) ^{-1}$.

It is obvious that for a deeper understanding of the deformation within a particle, it must be treated as a deformable continuum, which would require a highly-resolved spatial discretization, for example using the finite element method for the contacting bodies. This requires a large computational effort. For the state of the art in finite element methods and Contact Mechanics, see the books of Wriggers [97, 98]. For work specifically focusing on the continuum mechanics of particles, see Zohdi and Wriggers [111].

Appendix 2: Phase transformations

To include phase transformations, we consider seven cases, which are implemented in a predictor-corrector manner by first solving

$$\begin{aligned} m_iC_i\dot{\theta }_i=\mathcal{Q}_i+\mathcal{H}_i \end{aligned}$$

(9.1)

to obtain predicted temperature, and then checking the following:

• Solid $\rightarrow$ solid-no melting with $C_i=C_S$: If $\theta (t)<\theta _m$ and $\theta (t+\Delta t)<\theta _m$ then retain Eq. 9.1 with $C(\theta )=C_S$,

• Solid $\rightarrow$ liquid-melting with $C_i=C_S$: If $\theta (t)<\theta _m$ and $\theta (t+\Delta t)\ge \theta _m$ then re-solve Eq. 9.1 with $C(\theta )=C_S+\frac{\delta \mathcal{P}^{S\rightarrow L}}{\delta \theta }$,

• Liquid $\rightarrow$ liquid-melted with $C_i=C_L$: If $\theta (t)\ge \theta _m$ and $\theta (t+\Delta t)\ge \theta _m$ then retain Eq. 9.1 with $C(\theta )=C_L$,

• Liquid $\rightarrow$ solid-solidification with $C_i=C_L$: If $\theta (t)\ge \theta _m$ and $\theta (t+\Delta t)<\theta _m$ then re-solve Eq. 9.1 with $C(\theta )=C_L+\frac{\delta \mathcal{P}^{L\rightarrow S}}{\delta \theta }$,

• Liquid $\rightarrow$ vapor-vaporization with $C_i=C_L$: If $\theta (t)<\theta _v$ and $\theta (t+\Delta t)\ge \theta _v$ then re-solve Eq. 9.1 with $C(\theta )=C_L+\frac{\delta \mathcal{P}^{L\rightarrow V}}{\delta \theta }$,

• Vapor $\rightarrow$ vapor-remains a vapor with $C_i=C_V$: If $\theta (t)\ge \theta _v$ and $\theta (t+\Delta t)\ge \theta _v$ then retain Eq. 9.1 with $C(\theta )=C_V$,

• Vapor $\rightarrow$ liquid-condensation with $C_i=C_v$: If $\theta (t)\ge \theta _v$ and $\theta (t+\Delta t)<\theta _v$ then re-solve Eq. 9.1 with $C(\theta )=C_V+\frac{\delta \mathcal{P}^{V\rightarrow L}}{\delta \theta }$,

where $C_S$ is the heat capacity of the solid, $C_L$ is the heat capacity of the liquid and $C_V$ is the heat capacity of the vapor and

• $0<\delta \mathcal{P}^{S\rightarrow L}$ is the latent heat of melting,

• $0<\delta \mathcal{P}^{L\rightarrow S}$ is the latent heat of solidification,

• $0<\delta \mathcal{P}^{L\rightarrow V}$ is the latent heat of vaporization,

• $0<\delta \mathcal{P}^{V\rightarrow L}$ is the latent heat of condensation and

• $0<\delta \theta$ is small and can be thought of as a “bandwidth” for a phase transformation. For more details on melting processes, see Davis [15].

We note that latent heats have a tendency to resist the phase transformations, achieved by adding the positive terms in the denominator, thus enforcing reduced temperature (during the phase transformation).⁶ This approach is relatively straightforward to include within the staggering framework. As a consequence, the number of particles in the system and their heat capacities will also change in the algorithm, and Eq. 4.10 becomes

$$\begin{aligned} \theta _i(t+\Delta t)&= \theta _i(t)+\frac{1}{m_iC_i}\left( \int _{t}^{t+\Delta t} \mathcal{Q}_i\, dt+ \int _{t}^{t+\Delta t} \mathcal{H}_i\, dt\right) \nonumber \\&\approx \theta _i(t)+ \frac{\Delta t\left( \phi (\mathcal{Q}^K_i(t+\Delta t)+\mathcal{H}^K_i(t+\Delta t))\right) }{m_iC^K_i(t+\Delta t)} \nonumber \\&+ \frac{\Delta t\left( \phi (\mathcal{Q}_i(t)+\mathcal{H}_i(t))\right) }{m_iC_i(t)}. \end{aligned}$$

(9.2)

The subsequent convergence of the thermal calculation is rather quick, since the time steps are extremely small. For more details on convergence on iterative time-stepping schemes, see Zohdi [107–116].