Catching Heat for Metal Additive Manufacturing

Engineers, scientists, and manufacturing experts around the world are in a frustrating spot when it comes to additive manufacturing (AM). We have an incredibly powerful technology in our hands, unmatched in its ability to create complex geometries. It can build parts directly from a computer drawing, so the gap between what’s in our minds and what’s in our hands has never been smaller. Think of the cost savings, the labor reduction, the potential empowerment of rural communities. The rewards are tantalizing. But alas, the technology does not yet work quite as it should.

Additive manufacturing of metals has nearly achieved mainstream status in industry. Although the technology is still hard to predict, a well-equipped manufacturing facility is now expected to have some AM capability, which is then promptly relegated to niche applications. This unpredictability is primarily due to repeated melting and solidification of metal, which is how metal is fused together in most AM processes. This repeated heating and cooling of metal introduces a host of unpredictable behavior; the part could become distorted or even fracture due to its frequent expansion and contraction. Even when a part is successfully built to the right shape and size, it could have different mechanical properties than one made using conventional subtractive manufacturing; it could be more brittle than expected, or more elastic. In many applications, this is unacceptable.

So, how do manufacturers overcome these complications? Well, the easiest way to get around AM’s unpredictability is to just try and see! The trial-and-error approach, however, can take days and cost tens of thousands of dollars for a single part. In the cut-throat world of industrial manufacturing, this seductively intuitive approach is simply too costly. 

When trial-and-error won’t do, engineers often resort to numerical models. In brief, a numerical model is a set of mathematical equations that predicts how a system will behave by solving the relevant physical equations. Have you ever calculated how long it will take you to get someplace, knowing how far your destination is and how fast you expect to drive? That’s basically a tiny numerical model you’re solving in your head. Engineers use numerical modeling for a plethora of applications, from predicting the outcome of a vehicle collision to optimizing the interior design of an aircraft. Numerical modeling can be applied to AM to predict trouble before it happens. 

AM presents unique challenges to numerical models, however. Unlike a head-on collision, AM consists of multiple highly sophisticated physical phenomena that occur nearly simultaneously, across several orders of magnitude. At the moment a laser comes in contact with metal powder, the powder surface immediately begins to melt, causing its thermal and physical properties to change rapidly. The molten metal behaves like a thick liquid and can flow, on a microscopic scale, in response to changes in temperature and surface tension within (this is known as the Marangoni effect). As the laser travels across the powder, the melt pool follows along while parts of it begin to cool, eventually fusing together to form a solid. Later, when another layer of powder has been deposited on top, this fused solid will be reheated as the overhead powder is melted. In some cases, the fused solid is re-melted itself. 

We simply do not have the computational power to model all of these phenomena at once. In other words, our computers are not powerful enough to simulate a process that spans hours by analyzing every single microsecond. Even supercomputers shudder at the thought. To overcome these computational limitations, engineers use empirical relations. Empirical relations are correlations that have been observed through experimentation but are not necessarily backed by theory. Rather than attempting to model every single micro-event, engineers instead prefer to lump their effect into empirical constants. One great example of this is the use of a “laser penetration” term. 

Technically, a laser is just a concentrated beam of light—so when it comes in contact with an object, only the object’s surface is affected. However, when the laser is as powerful as those used in AM, the laser is often observed to “dig through” the surface of the material that it travels across. On the microscopic scale, what is happening here is that the laser is indeed interacting with only the surface of the material. The surface is quickly melted, however, resulting in near-immediate reduction in its density and allowing the laser’s heat to travel deeper beneath the surface. In some cases, metal is even vaporized entirely, although this is highly undesirable in AM. This is why, in the span of half a second, the laser may very well be penetrating beyond the surface of the material—and engineers who are modeling the AM process on a macroscopic scale are inclined to ignore the messy microscopic details. But the question is: how far does the laser penetrate? That is an empirical relation. Engineers must test their specific laser, at the specific operating conditions for its intended use, and experimentally observe how far the laser appears to penetrate. Only then can they elegantly avoid the messy microscopic details. That is an example of one of many empirical constants used by macroscopic AM models.

The problem with empirical relations is that they are highly situational. Running the laser a little hotter, using a different build material—or any other seemingly minor change—can have a significant effect on the laser penetration constant. There are countless researchers across the world today developing AM models, with tens of these models currently in the literature. My concern is that each research group may be using empirical relations that are only suited to their specific application, and that some of the nuance of empiricism may be lost as researchers communicate indirectly through scientific literature. Our best work, as a collective global community of researchers, is dependent on our ability to consolidate our individual efforts.

In my recent paper, I attempt to facilitate some of this consolidation. I focus specifically on one component of AM models known as the heat source model. A heat source model is basically a mathematical equation that describes how the laser’s heat is spread out into the build material. I analyze the heat source models in the literature mathematically, by comparing the amount of heat in the laser beam as it is flying toward the build material with the amount of heat that subsequently gets transmitted into the material. Basically, the total heat coming out of the laser must be equal to the total heat going into the build material. Otherwise, the heat source model fails the law of conservation of energy, and its past success is likely due to empiricism. 

First, I present all the heat source models I could find in AM literature. I classify them into two categories: two-dimensional and three-dimensional. Two-dimensional models are those that distribute the laser’s heat solely on the build surface, neglecting any “penetration”. These models work well for lower-powered lasers with high travel velocities; the “digging” effect of the laser is minimal under those circumstances. Most researchers agree that for AM applications, a Gaussian distribution is an accurate two-dimensional representation of the laser’s heat—that is, a distribution where the laser’s power is maximal at the center of the laser spot, exponentially decreasing radially until it is near zero at the edges of the laser spot. Two additional two-dimensional heat source models have been used in AM literature, and my investigation shows that they appear to be highly situational.

Next, I investigate three-dimensional models, where laser penetration is taken into account. The precise depth of penetration is not of interest, of course, because we know it is a situational empirical constant. What I’m more concerned about is the profile of laser power in the depth dimension, as portrayed by each heat source model. Does it decrease linearly? Quadratically? And does it abide by the law of conservation of energy? These are the questions that determine whether or not a heat source model is robust enough to be used widely in any AM modeling scenario. 

The Gaussian ellipsoid and Gaussian cone models are two primary contenders. Both of these models assume a Gaussian distribution of heat on the build surface, but they differ in how they model the laser profile in the depth axis. The former assumes that laser power decreases exponentially as it penetrates into build material, and the latter assumes a linear decrease. Does that sound pedantic? It sort of is! These two models achieve similar results and have been used extensively in the AM literature. The accuracy of an AM model should remain largely unaffected between these two heat source models. Other three-dimensional models are also discussed, but their success is limited. 

You can read the entire investigation, in all its mathematical glory, here. AM is one of the fastest growing technologies in the world today, and I hope that my work has helped streamline its continued development.