Thermal Analysis of Metal Additive Manufacturing
Theory: the temperature field of a moving heat source
Overview
Parts from our metal 3D printer warp and crack. What can thermal analysis tell us?
In laser powder bed fusion (LPBF) a melt pool about 0.1 mm across moves at around 1 m/s, melting and freezing millions of times. The temperature changes are so abrupt that solidified material tries to shrink, creating residual stress that becomes warping or cracking. Thermal analysis answers three questions: how big the melt pool is (do layers fuse properly), how fast it cools (microstructure and stress), and how heat accumulates in the whole part.
Moving point-source solution (Rosenthal)
$P$ is laser power, $A$ absorptivity, $k$ thermal conductivity, $\alpha$ thermal diffusivity, $v$ scan speed, $\xi$ the position along the scan direction relative to the source (negative behind) and $R$ the distance from the source. It is the steady temperature field, seen from coordinates moving with the source, of a point source travelling at constant speed over a thick body. It omits latent heat, flow in the pool and keyhole melting, but it is good enough to estimate pool size and cooling rate.
Cooling rate and energy density
Cooling rate is proportional to scan speed and inversely proportional to source strength. $E_v$ is the volumetric energy density, with hatch spacing $h$ and layer thickness $t$. It is widely used to compare process settings, but the same $E_v$ can give different pool shapes depending on the power–speed combination.
A welding formula revived by 3D printing
The formula was derived in the 1930s–40s to estimate temperatures in welds. Before computers it let engineers predict the width of the heat-affected zone and the cooling rate of arc welds by hand, and it has stayed in welding textbooks ever since. Metal additive manufacturing is, in effect, millions of fine laser-weld beads laid side by side to build a part, so a welding formula more than half a century old is used directly for first estimates when searching for process settings.
Worked examples
Example 1: melt pool by material and scan speed
Melt pool size and cooling rate at solidification from the Rosenthal solution, at 200 W with the plate at 25 °C (absorptivity assumed 0.4 for Ti-6Al-4V and 316L, 0.3 for AlSi10Mg).
| Material | Scan speed | Width | Length | Depth | Cooling rate |
|---|---|---|---|---|---|
| Ti-6Al-4V | 0.5 m/s | 165 μm | 348 μm | 82 μm | 2.6×10⁶ K/s |
| Ti-6Al-4V | 1.0 m/s | 120 μm | 334 μm | 60 μm | 5.2×10⁶ K/s |
| Ti-6Al-4V | 1.5 m/s | 99 μm | 328 μm | 49 μm | 7.8×10⁶ K/s |
| 316L | 1.0 m/s | 105 μm | 386 μm | 53 μm | 3.7×10⁶ K/s |
| AlSi10Mg | 1.0 m/s | 156 μm | 200 μm | 78 μm | 3.8×10⁶ K/s |
Tripling the speed barely changes the length.
At higher speed heat has less time to spread sideways and downward, so width and depth shrink. The length, though, is set roughly by power ÷ (2π × conductivity × temperature rise to melting) and depends little on speed — for Ti-6Al-4V, 80 W ÷ (2π × 25 × 1630) ≈ 310 μm. You want a depth of about twice the layer thickness (30 μm here), so 49 μm at 1.5 m/s leaves little margin. Without good fusion to the layer below, lack-of-fusion voids form.
Example 2: effect of preheating
| Plate temperature | Pool width | Cooling rate |
|---|---|---|
| 25 ℃ | 120 μm | 5.2×10⁶ K/s |
| 200 ℃ | 127 μm | 4.2×10⁶ K/s |
| 500 ℃ | 143 μm | 2.6×10⁶ K/s |
Ti-6Al-4V at 200 W and 1 m/s. Cooling rate scales with the square of the temperature difference to melting, so preheating to 500 °C roughly halves it. The smaller temperature difference also reduces thermal contraction, helping to suppress residual stress and cracking.
Example 3: energy density
With 100 μm hatch spacing and 30 μm layers, 200 W at 1 m/s gives 67 J/mm³, 200 W at 0.5 m/s gives 133 J/mm³ and 300 W at 1 m/s gives 100 J/mm³. Too low causes lack of fusion; too high causes keyhole melting that traps gas pores. The right window depends on material and machine, so confirm it with test coupons.
How to approach the analysis
Three scales
| Scale | Target | Main method |
|---|---|---|
| Melt pool (0.1 mm) | Penetration, porosity, microstructure | Moving-source conduction, CFD with flow |
| A few layers (1 mm) | Reheating between layers, heat build-up | Thermal analysis following the scan path |
| Part (100 mm) | Distortion, residual stress, supports | Lumped-layer thermo-mechanical, inherent strain |
A 100 mm tall part in 30 μm layers has about 3,300 layers, each with metres of scan lines. Tracking the laser directly is feasible only for a few layers. At part scale, tens of layers are lumped into one analysis layer and heated at once. Checking that results don't change with the number of lumped layers plays the role of a mesh-convergence check.
Cautions
- Powder conductivity: powder conducts about 1/100 as well as bulk metal. Overhangs above powder trap heat and overheat or distort.
- Temperature-dependent properties: conductivity and specific heat change greatly around melting; include latent heat over the solidification range.
- Absorptivity: differs between powder and molten metal and rises sharply in keyhole mode. Calibrate against melt-pool cross-sections.
“Smaller parts overheat more”
Even with identical settings, thin pillars and small parts sometimes come out with rough, discoloured surfaces. Each layer scans so quickly that the next layer arrives before the previous one has cooled, and heat builds up. Building them alongside larger parts helps: the laser spends time on other parts, giving them time to cool. Inter-layer dwell does not appear in melt-pool analysis, but it is obvious in a part-scale thermal model.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Choosing settings by energy density alone | Same value, different pool shape | Check power and speed separately |
| Treating powder as bulk | Overhang overheating missed | Use low powder conductivity |
| Assuming constant absorptivity | Wrong penetration depth | Calibrate with cross-sections |
| Not checking lumped-layer count | Distortion prediction off | Vary lumping to check convergence |
| Ignoring inter-layer dwell | Heat build-up in small parts missed | Model the actual build sequence |
I'd like to compute melt pool size myself.
Try the melt pool simulator and Rosenthal moving heat source tool for pool and temperature field, the weld cooling rate calculator for cooling rate, and the AM thermal distortion tool for distortion. Stress and distortion are covered in AM residual stress, and the welding background in welding thermal analysis.
Related Topics
Experience the theory firsthand with the interactive simulator for this field
All Simulators