## Abstract

Additive manufacturing (AM) method has attracted huge interest in the past decade due to its ability in building complicated geometries with a much lower cost than conventionally produced parts. In AM, the final mechanical properties can be controlled by the AM process parameters. In other words, the AM process parameters control the amount of energy that is transferred into the powder and consequently the resulting microstructure. In this study, the correlation between melt pool geometry and mechanical properties of selective laser melted (SLM) Ti–6Al–4V samples is investigated.

## 1 Introduction

Selective laser melting (SLM) is a well-known method in additive manufacturing (AM) for producing near fully dense parts [14]. AM process parameters control the solidification process and the mechanical properties [3,512]. The mechanical properties of materials can be simulated using their microstructures [1320]. However, as in AM, the microstructure is a function of process parameters, and it would be beneficial to find a direct relationship between the AM process parameters and their mechanical properties.

Many studies worked on some analytical- or physics-based frameworks to predict the mechanical properties for any sets of AM process parameters [2123]. However, any changes in the process parameters result in a new microstructure that needs a new simulation for mechanical properties prediction. These models generally use a physics-based model to predict the temperature profile, and then, the microstructure. Accordingly, the mechanical properties can be predicted using the resulting microstructure. Therefore, these models are computationally expensive.

In order to have a time-efficient model for predicting a mechanical property, such as yield strength, a model is needed that outputs the material property as a function of AM process parameters. In this regard, it is needed to track the effectiveness of different AM process parameters on the mechanical properties. However, this model would need enough data to build an efficient model. On the other hand, the AM process parameters particularly define the amount of energy that is transferred into the material during the manufacturing process. In this study, a physics-based model was used to predict the resulting thermal gradient and, then, found a relationship between the AM process parameters and the yield strength.

In this study, ten different SLM Ti–6Al–4V samples were used. Depending on the AM process parameters, SLM Ti–6Al–4V parts can have high yield strength of 1300 MPa [24], their high strength-to-weight ratio makes them a good fit for a wide range of applications in the industry [2527]. The morphology of phases, grain sizes, and crystallographic orientation of grains, which are the main factors in the mechanical properties of materials, is directly related to the AM process parameters.

In this paper, the yield strength of SLM Ti–6Al–4V parts for different sets of process parameters was used to define a model that predicts the yield strength of SLM parts for any sets of process parameters. This model uses the AM process parameters to predict the temperature profile of the materials under the manufacturing process. Then, the thermal gradient and melt pool geometry were used to predict the yield strength.

## 2 Methodology and Materials

### 2.1 Temperature Profile and Melt Pool Geometry.

In SLM, for thermal simulation, a moving point heat source solution is used. Carslaw and Jaeger [28] by assuming a three-dimensional semi-finite body developed this linear solution for the moving point heat source:
$T(x,y,z)=Pη4πKx2+y2+z2exp(−V(x2+y2+z2+x)2κ)+T0$
(1)
where x, y, z are the distances from the moving point heat source, P is the laser power, η is the laser absorption, K is the thermal conductivity, V is the scanning speed, T0 is the initial temperature of solid, and κ is the thermal diffusivity, which is expressed as
$κ=KρC$
(2)
where ρ is the density and C the is heat capacity. Equation (1) calculates the three-dimensional temperature profile during the additive manufacturing process in which the melt pool can be determined by comparing it to the material melting temperature. Figure 1 illustrates a typical melt pool geometry which indicates melt pool length (L), melt pool width (W), and melt pool depth (D).

Table 1 shows the required properties of Ti–6Al–4V for Eq. (1) which were captured from the literature.

### 2.2 Materials.

In this paper, the tensile test results of 10 different near fully dense SLM Ti–6Al–4V (grade 23) samples were studied. The tensile tests were performed in the building direction to be consistent because of the directional dependency of additively manufactured parts. Table 2 illustrates the AM process parameters for the SLM Ti–6Al–4V samples in this study.

Figure 2 represents the tensile test results of the Ti–6Al–4V samples mentioned in Table 1. In Fig. 2, the stress at zero plastic strain represents the yield strength of that sample. The onset of plastic strain is right after the yield strain in which the deformation strategy is no longer elastic.

## 3 Results: Relationship Between the Melt Pool, Temperature Profile, and Yield Strength

In AM, process parameters control the amount of energy is transferred into the powder. Therefore, there should be a correlation between the amount of energy transferred into the powder and the mechanical properties of the final part. In this regard, to measure the transferred energy, Eq. (1) was used to calculate the temperature profile for each sample. Then, any location with a temperature above the melting temperature would be in the melt pool as depicted in Fig. 1. Any sets of AM process parameters would affect the melt pool geometry (L, D, and W). In this study, the process parameters which were used from the literature (Table 2) were implemented in Eq. (1) to calculate the temperature profile and see the melt pool changes in different circumstances. Figure 3 represents the top view of the calculated melt pools for all ten samples of Table 2.

As seen in Fig. 3, different AM process parameters have a different melt pool geometry. This melt pool geometry may affect the mechanical properties of the final part. To study this idea, different combinations of the melt pool dimensions, as expressed in Table 3, were used in different regression models to predict the yield strength of each sample. In the model building, the cross-validation technique was used in a way that each time, nine samples were used in training to predict the yield strength for the 10th sample. However, the melt pool geometry itself was not sufficient for building a linkage model.

In the next step, it was tried to use more thermal characteristic values that each set of AM process parameters can affect. As it is expected, temperatures in the melt pool vary for different sets of AM process parameters; however, it is probably impossible to use all temperature information in the melt pool for building a regression model. Moreover, many studies have shown that the thermal gradient in the melt pool directly defines the final grain structure during solicitation at the border of the melt pool [37,38]. Also, microstructure and grain size/structure have a direct relationship with the mechanical properties of materials [3942]. Therefore, it is expected that the thermal gradient is directly related to the final mechanical properties. In this regard, the thermal gradient in the laser beam direction was also considered as an important parameter. In order to find one specific thermal gradient parameter for each melt pool, Eq. (3) was used:
$G=∑0L(∑0DLGzDL)L$
(3)
where L is the melt pool length, DL is the melt pool depth at each location on the middle plane of melt pool width, and Gz is the thermal gradient in the laser beam direction. Figure 4 represents a schematic of the plane that the average thermal gradient (Eq. (3)) was calculated. As shown, the plane is in the middle of the melt pool width which passes the laser beam.

Table 4 shows the results of G calculations for all ten samples. Then, different combinations of input parameters among L, D, W, L/D, and G were used for different regression models. Given the relatively small dataset in this study, standard (least-squares) regression techniques using a polynomial model form were used. In each try of model building, the cross-validation technique was used to avoid overfitting in the data. The results showed that a polynomial model with the order of three gives the lowest error for the prediction of the yield strength using L/D and G as the input parameters. Table 4 illustrates the cross-validation results using a third-order polynomial model for yield strength data.

According to Table 4, the highest error is 13.35% which is for sample 4; however, the error for the other nine samples is less than 6% which shows the reliability of this model. Moreover, it should be noted that in this study as we only have 10 samples, nine of them were used in the model building at each step. This means that this model would be much stronger if more data were added to this dataset. Then, to build the final model, all 10 samples used in a third-order polynomial model to report the relationship between yield strength and temperature profile information expressed as follows:
$σy=7.35×(LD)2+21.05×G+0.32×G(LD)2−5.38×G(LD)+638.90$
(4)

Equation (4) gives an average error of 4.5% over the ten samples used in this study. This study opens a new avenue in additive manufacturing in order to correlate the temperature profile to the yield strength. Apparently, similar to all other regression models, this model always can be strengthened by adding more data points. It also should be mentioned that the current methodology is not considering the other local effects such as thermal cycling; however, as it is a regression model and the thermal profile and thermal gradient were used as inputs, the effect of thermal cycling was indirectly considered in the model building. For example, when there is a bigger melt pool with a higher thermal gradient, it causes a more profound thermal effect on the beneath layers, and finally, it affects the yield strength of the final part. But in the regression model, the model itself sees that the higher thermal gradient is correlated to a higher yield strength. So, indirectly, the model considers that part as well.

## 4 Conclusion

This paper investigated a new idea for predicting yield strength of near fully dense SLM Ti–6Al–4V alloy in the building direction. In the proposed model, thermal profile information was used to predict the yield strength. For thermal profile prediction, a physics-based moving heat source solution was used. Then, melt pool geometry, as well as thermal gradient, was used as input parameters. The cross-validation technique was used to avoid overfitting in the model building process. The final model that was reported showed an average error of 4.5% over the ten samples used in this study. In this study, only ten samples were used to validate the idea, which the results were satisfying. This model always can be modified by adding more data points.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper. No data, models, or codes were generated or used for this paper.

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