Predicting the Strength of EBAM 3D Printed Ti-6Al-4V from Processing Conditions

Abstract

In this study, a process-to-property linear regression model was developed to predict the yield and ultimate tensile strengths of as printed Ti-6Al-4V from electron beam additive manufacturing (EBAM). A total of 8 printing conditions such as bead width, wire feed rate, deposition speed were utilized to predict the material properties in three different notional parts produced over a period of several months. It was found that as the precision and variety of processing conditions collected during print improved between prints, so did the predictive ability of the model. In the final print, the model predicted the yield and ultimate strengths of 72 specimens with an R² correlation of 0.8 and 0.6 for the horizontal and vertical test specimens, respectively. Although the current model indirectly accounted for thermal fluctuations, further improvements to the model’s ability to predict material strength are expected with the addition of thermal data captured in subsequent notional parts.

1. Introduction

In recent years, Additive Manufacturing (AM) processes have become increasingly commonplace due to the advantages they offer over traditional manufacturing methods. Arguably one of the most promising AM processes is that of wire-fed Electron Beam Additive Manufacturing (EBAM). This process is similar to electron beam welding and utilizes high-energy electron beams to fuse metal wires together. Similar to other forms of additive manufacturing, wire-fed EBAM allows for the printing of full-density, functional as-printed parts requiring minimal post-processing treatments [1,2].

Wire-fed EBAM processes have the advantage of high production speeds in addition to the ease of feedstock storage, availability and handling compared to powder-based AM. Importantly, wire-fed AM also originates from a feedstock less susceptible to contamination [2,3,4,5,6,7,8]. Finally, the shape and length scale of the feedstock—mm sized diameter wire as opposed to μm sized spherical metal particles [1,3,9,10]—help to improve the density of EBAM prints and diminishes the risk for porosity and inclusions. However, as with all AM processes, there are issues of residual stresses, occasional internal voids as well as significant material property anisotropy. These are inherent to the printing processes and the thermo-mechanical interactions between layers [3,9,11], however they remain some of the most pressing research frontiers for AM to become more widespread.

Generally, AM improves the “buy-to-fly” ratio—or, the mass ratio between the original stock material used to produce the part and that of the final machined part. The buy-to-fly of traditionally manufactured titanium aircraft components ranges from 12:1 to 25:1 [12,13]. This translates to approximately 90% of the initial material being discarded for the final print. AM not only has the potential to reduce a part’s overall life-cycle impact, it also offers engineers and designers improved design freedom to create products that are unique, manufacturable at low volumes, and economical [2]. Williams, Martina, et al. [14] analyzed wire-arc additive manufacturing of Ti-6Al-4V and steel printed parts, determining that depending upon the deposition rate, ranging from 1 kg/h to 4 kg/h, they could achieve a buy-to-fly ratio of less than than 1.5. At higher deposition rates the fidelity of the part is compromised and requires significant machining, deposition of larger amounts of material thus increasing the buy-to-fly ratio to be more in line with traditionally manufactured parts.

The thermo-mechanical cycles and processing parameter variability that is intrinsic to AM printing, subsequently influence all aspects of the final part’s material properties [9,15,16]. Variations in thermal experiences and processing conditions result in microstructural gradients, making the science of predicting the material properties and resulting residual stresses a complex multiphysics simulation problem. One method to experimentally determine the material properties involves the use of destructive testing procedures in order to observe the microstructures associated with specific print geometries and processing parameters [7,17,18,19]. However, destructive testing is costly in both time and resources and negates the improved life-cycle benefits of AM. Ideally, a predictive model could be used instead to predict the material anisotropy based on processing conditions alone, relinquishing the need to destructively investigate each part. Such a parameter-to-property model would by-pass the need for multiscale multiphysics modeling, which remains to this day a time consuming and computationally intensive endeavor.

Processing parameters heavily influence the material properties of the final build. Specifically, heat input, energy distribution, and wire feed have all been shown to play a significant role [5,20]. The proper tuning of energy input and wire feed is essential to the formation and control of a stable melt pool which in turn affects the mechanical properties. As such, efforts were put forth by [11,21,22] to model the effect of thermo-mechanical cycles on the evolution of the material’s localized microstructure. Chekir, Tian, et al. investigated laser wire deposited Ti-6Al-4V of heights between 40 and 65 mm by developing a finite element thermal model in order to predict thermal distributions caused by the deposition process [11]. Their study revealed that processing conditions promoted the coarsening of microstructures due to multiple melting-solidification cycles. The group also investigated the effect of post-processing treatments upon the material grains, concluding that annealing or a hot isostatic pressing followed by aging preserves prior $\beta$ grain morphology with coarsening of the microstructure. Similarly, Sikan, Wanjara, et al. [21] developed a 3D transient, fully coupled thermo-mechanical finite element model that accurately predicted the cooling rates, grain morphology and microstructure of EBAM printed Ti-6Al-4V. The model, when validated against experimental results, had thermal predictions with an average error of 3.7% [21]. Bonifaz and Watanabe examined the micro residual stresses and micro plastic strains in SAE-AISI-1524 gas tungsten arc welded joints. The model uses a transient, non-linear multiscale finite element approach [22] that incorporates anisotropy through the use of the maximum and minimum Young’s modulus in the stiffest and least stiff directions, respectively. These studies demonstrate that by using high resolution, high powered computational modeling it is possible to analyze the melt pool, microstructure and stresses for both as printed and post-processed parts. While these time intensive models are useful, a real-time estimate of the material properties during print would provide the possibility of on-the-fly adjustments to maintain bounds of the material properties.

Given the complexity in predicting the material properties resulting from layer-by-layer printing processes, there has been a shift towards the utilization of Neural Networks (NN), genetic algorithms and other Machine learning approaches to modeling at large [23]. These methods are informed by the physics of the process, but use more direct models that, for specific geometries and builds, can be trained to predict material properties from microstructure [24] or even from the build’s processing parameters alone. Statistical models [24,25] or machine learning methods that make use of long short-term memory (LSTM) networks [26] or other machine learning architectures [23,27] in order to improve prediction models. NNs are useful as they can provide accurate predictions of the printed metal tensile strength by utilizing LSTM architecture, and in situ data or process parameters—such as thermal data, printing speed, and layer height [26]. It is important to note, however, that NNs and hybrid NNs do require a degree of caution due to their tendency to overfit solutions. Additionally, they remain restricted to specific print geometries related to the training data supplied.

Predictive models based on processing parameters alone, such as this one and those by Zhang, Wang, and Gao [26], Weglowski et al. [7] and others that have been previously discussed, bypass the need for destructive testing, saving money and time. However, key processing parameters can vary between the different types of AM and must be incorporated into the models. Yao, Zhou, and Huang [28] analyzed the double-pulsed gas metal arc welding (DP-GMAW) and provided a framework to determine influential process parameters for double-pulsed gas metal arc welding (DP-GMAW). Utilizing their framework, they were able to determine the optimal processing parameters needed to generate optimal weld beads for prints. Their work showed that the most influential factors for welding quality were the welding speed, followed by twin pulse frequency and twin pulse current. Carefully controlling these processing conditions, can lead to more efficient printing, consistent builds and minimizing the need for destructive testing.

The main aim of the current study is to provide the methodology needed to determine the yield and ultimate tensile strengths of Ti-6Al-4V printed parts as a function of the process variables at any given location within the build. The paper is structured as follows: Section 2 presents the materials, printing process, and linear regression model methods used to predict yield and ultimate tensile strengths. Section 3 provides an overview and discussion of the predicted results. Finally, a summary of the work is given along with a discussion of the future direction of the project.

2. Materials and Methods

Three geometrically identical notional parts (NPX) were produced in Ti-6Al-4V using EBAM (Sciaky, Inc., Chicago, IL, USA) to observe printed material properties in a variety of typical geometries. The geometry chosen for the notional parts was meant to simulate a wide range of possible printed geometries expected in industry. These large scale parts measured 40 cm by 72 cm by 18 cm and were printed on both sides of a Ti-6Al-4V build plate using a 4th rotational axis. The notional parts were designated chronologically as NP1, NP2, and NP3. The geometry of NP1, NP2, and NP3 can be seen in Figure 1. The replicated parts enabled process refinement over the course of the three separate builds, with changes including build plate thickness, type and rate of data collection during processing. The evolution of the print quality provided grounds for the development of the process-to-property model developed here.

2.1. Printing Materials and Heat Treatment

A predictive model of the material properties of EBAM printed parts was developed based on the processing conditions monitored during the printing of three large Ti-6Al-4V testing parts NP1, NP2, and NP3. Although the geometry of the part remained consistent between each build, several processing conditions evolved over the refinement of the printed parts between each NP. Metal was deposited on both the top and bottom of a 1.27 cm build plate. A thicker build plate for NP2 to combat plate distortion from residual stresses, which can be seen in Figure 2 for NP1 of 2.54 cm, resulting in an undeflected final build plate.

Printed parts were sectioned down to sub-component geometries for post-processing for three different heat treatment conditions. Heat treatment 1 (HT1) was performed as a low temperature, stress relief anneal, conducted according to the AMS 4999 specification (without the aging step). Heat treatment 2 (HT2) involved the application of a hot isostatic pressing (HIP) in order to hold the samples high in the $\alpha +\beta$ two-phase field per AMS 4999. The third heat treatment, HT3, is a $\beta$ anneal per AMS 4905 in order to minimize porosity effects upon the development of the microstructure [24,25,29].

2.2. Material Testing

A combination of non-destructive evaluation (NDE) and tensile/compression testing was performed across all three printed parts, with specimen testing locations shown in Figure 3. These test locations correlated with locations of geometrical anomalies and critical material hotspots within the figure, such as along the phalanges on the back portion of the part, at corners, and other locations where there is a sharp orientation changes. The test specimen locations were chosen to determine the nature of the relationship between process, parameters, and material properties. The material testing provided results against which the predictive model can be compared for accuracy. Specimens were tested for ultimate tensile strength ($UTS$) and yield strength ($YS$).

2.3. Processing Parameters

Careful consideration was given to processing parameters from the build reports for NP1, NP2, and NP3. A total of 27 variables were recorded during deposition of NP1-3. Of those, 13 variables are used by the software (proprietary software from Sciaky, Inc., Chicago, IL, USA) algorithms to control the machine processing conditions in a real-time feedback loop. Automatic monitoring in this way helped to achieve the desired print geometry. A few of the print conditions monitored in situ included the printed bead width ($BW$), accelerating voltage ($AV$), beam current ($BC$), beam focus ($BF$), and chamber vacuum level ($Vac$). The averages and standard deviations for these print conditions are listed in

Table 1. Measured values and standard deviations of the print conditions.

Measured Parameter	Average	Standard Deviation
$AV$	39.95 kV	0.19 kV
$BF$	3.43 A	0.002 A
$BW$	459.9 pixels	54.52 pixels
$Vac$	42.79 μTorr	16.15 μTorr
$BC$	218.8 mA	15.65 mA

.

Since there was no thermal monitoring during the processing of NP1-3, the $BW$ variable became the most critical for the development of the model. The $BW$ measurement was provided by automated graphical analysis of video images taken in real-time during the deposition. Images were analyzed by an in-house software to determine the melted pool width just behind (trailing by about 2.54 cm) the actively depositing nozzle. However, due to geometry and variations in image contrast, optical measurements used to record the $BW$ variable were at times limited. In the event of turning a corner, for example, optical tracking of the bead would be unable to track the deposited bead and $BW$ measurements would become erroneous. The setpoint value, another recorded variable, reported the intended beadwidth at specific build locations. Setpoint values are a function of the deposition geometry measured in pixels, where 100 pixels = 2.54 mm. When the setpoint is equal to −1, the printer is not actively printing and processing parameters are not being recorded. When the setpoint is equal to a value other than −1 the build control algorithm is operative and the deposition data are considered in the linear regression analysis.

The importance of setpoint values is in their relationship to the measured $BW$ values; EBAM process contains pre-assigned variations in deposition power to keep $BW$ values within an expected range. Figure 4 shows the continuous setpoints for NP3.

Table 2. Total number of data points available for NP1 through NP3. The setpoints 400 through 470 are continuous on the top and bottom of the print, while the points marked with a −1 are when the processing parameters were not used in the predictive model.

Setpoint	NP1	NP2	NP3
400	3446	29,044	6414
410	−	65,750	−
415	−	19,972	−
420	−	9798	17,514
435	153,865	14,178	10,230
450	−	−	2323
470	284,333	258,148	171,818
−1	182,959	134,437	102,053
Total	624,603	531,327	368,079

shows the setpoint values variations from NP1, NP2, NP3.

2.4. Linear Regression Analysis

Destructive tensile testing was performed on NP1, NP2, and NP3. Each of the tested specimens provided a data set that related the $UTS$ and $YS$ to the process variables that were measured in situ during deposition, such as $BW$ (pixels), $AV$ (kV), and $BC$ (mA), within voxels of data identified to each specimen. The process variable data were then averaged for each specimen in order to create representative values of the test specimen. The experimental $UTS$ and $YS$ data, collected along with the aforementioned representative averaged processing variables, were used to create a linear combination of the variables to predict $UTS$ and $YS$. The proposed linear regression was developed by combining relevant variables and suitable functions of other fundamental variables as shown below in Equations (1)–(5).

$$F=a_0+a_1{x}_0+a_2{x}_2+\cdots +a_n{x}_n$$

(1)

where $a_n$ are the parameters, and $x_n$ are the processing variables of interest. A total of 9 variables of interest were selected to model the material properties of both the $UTS$ and $YS$ of 49 specimens for NP1, 58 specimens for NP2, and 72 specimens for NP3. The proposed models for the $UTS$ and $YS$ are as follows in Equations (2) and (3):

$$\begin{array}{c}\hfill UTS=a_0+a_{1}BW+a_{2}W{F}_{ref}+a_{3}Speed+a_{4}ZaxisPos+a_{5}Vac+a_{6}ED+a_{7}CR+a_{8}Vol\end{array}$$

(2)

$$\begin{array}{c}\hfill YS=a_0+a_{1}BW+a_{2}W{F}_{ref}+a_{3}Speed+a_{4}ZaxisPos+a_{5}Vac+a_{6}ED+a_{7}CR+a_{8}Vol\end{array}$$

(3)

The $UTS$ and $YS$ were obtained from both horizontal and vertical test specimens relative to the build plate. For the predictive model, however, it was necessary to separate these specimen orientations into two distinct models due to varying influence of processing conditions on their respective microstructure evolutions. Although the same variables were used to predict the strength of horizontal and vertical specimens (Equations (2) and (3)), the weights (and therefore relative influence) of each term varied (at times greatly) between each model. Most of the variables included in the linear regression analysis are ones that come directly from in situ monitoring of the process, or are reference variables for the printing software. As noted before, $BW$ (pixels) is the width of the bead measured by image analysis of optical monitoring. The variable $W{F}_{ref}$ (in/min) is the wire feed reference speed, $Speed$ (in/min) is the deposition nozzle travel speed, $ZaxisPos$ (in) is the vertical height above the deposition plate, and $Vac$ (μTorr) is the reference vacuum of the printing chamber.

The final three variable in the regression are derived from the other available processing conditions, and meant to stand in for temperature readings in situ. These three variables are $ED$, the energy density (see Equation (4)), $CR$, the approximated cooling rate, and $Vol$, the volume of material below the current print location, representing the heat sink volume below each newly deposited bead.

The Energy Density ($ED$) of deposition processes is dependent upon the power of the beam (found here from the accelerating voltage ($AV$) and beam current ($BC$)) and the area over which this beam is applied (derived from the speed of the nozzle ($Speed$), and beadwidth ($BW$)) as shown in Equation (4) [30]. This equation divides the power of the electron beam—as represented voltage ($AV$) times beam current ($BC$)—by the output speed of the nozzle ($Speed$) and the area of the beadwidth ($BW$).

$$ED=\frac{AV\times BC}{Speed\times B{W}^2}$$

(4)

The variable for $CR$ was added to the linear regression in lieu of real-time cooling rate measurements which are known to affect the tensile strength of deposited Ti-6Al-4V [22,31]. As CR decreases with height away from the quenching base plate, tensile strength is known to decrease, as a result of changes in the microstructure evolution. $CR$ was approximated to be inversely proportional to the amount of time elapsed since onset of deposition ($t-t_0$)—where t is the time at deposition for the specimen and $t_0$ is the time at initiation of the bead. Also inversely proportional to the height away from the base plate (Z). As deposition occurs further above the base plate, the quenching ability of the plate diminishes along with a build up of heat in the upper layers of the deposition. Furthermore, the amount of time elapsed since onset of deposition is a good indication of the interpass temperature of the material on which the deposition occurs. It is known that depositing onto hot material will decrease the rate of cooling, as heat has nowhere to go. As a result, the $CR$ was established using the following equation:

$$CR=\frac{b_0}{\overline{Z}(t-t_0)}$$

(5)

where $b_0$ is a constant determined by linear fit and $\overline{Z}$ is the average vertical height for the specimen. In this equation, the $CR$ variable decays as a function of the Z axis position and time since onset of printing ($t-t_0$), which both influence the interpass temperature, and hence also the cooling rate of the material.

The amount of material surrounding the tensile specimen affects the cooling rate of the metal and is another key element to predicting the values of $UTS$ and $YS$. To approximate this effect, a crude rectangular prism was developed to represent the volume of material available around the current point of deposition. This volume effect on cooling rate is described by $Vol$, which provides further insight into the thermo-mechanical cooling processes during the build process. $Vol$ (Equation (6)) is defined as the number of data points which exist in a specified volume 5.08 cm around and 0.254 cm below the specimen’s current print location, denoted by $x_s$, $y_s$, and $z_s$ in Equation (7).

$$\begin{array}{c}\hfill Vol=n\left(V\right)\end{array}$$

(6)

$$\begin{array}{c}\hfill V=\left\{x\right|x_s-5.08\le x\le x_s+5.08;y|y_s-5.08\le y\le y_s+5.08;z|z_s-0.254\le z\le z_s\}\end{array}$$

(7)

where x (cm), y (cm), and z (cm) are the locations in the X, Y, and Z directions about the current point of specimen analysis.

3. Results

3.1. NP1 Predictive Model Results

NP1 was the first notional part that was printed. It provided foundational knowledge to help further improve the printing process. Following NP1 it was necessary to increase the thickness of the build plate, as previously discussed, in order to prevent plate distortion caused by heat dissipation, residual stresses, and weight of the build part itself; see Figure 2.

A total of 49 test specimens were collected from NP1. Of these, 28 specimens were horizontal and 21 were vertical orientations, with the orientation of the specimen being dependent upon if the cut was taken parallel or perpendicular to the bead deposition location. In order to optimize the correlation coefficient, $R^2$, a variable value range investigation was performed focusing on the volumetric term of the linear regression model. The range of values for the $Vol$ modifier can be seen in

Table 3. $R^2$ values for predicted $UTS$ and $YS$ for the horizontal and vertical components utilizing a range of values for the volume in the z, or vertical, direction in order to optimize the accuracy of the predicted results for all components of the stresses.

$\mathit{Vol}$ Thickness	$\mathit{UTS}$ Horizontal	$\mathit{YS}$ Horizontal	$\mathit{UTS}$ Vertical	$\mathit{YS}$ Vertical	${\mathit{R}}^2$ Sum
0.254	0.48297	0.48039	0.66939	0.59814	2.231
0.508	0.46945	0.46914	0.67837	0.60905	2.226
0.762	0.44363	0.43299	0.66684	0.59553	2.139
1.016	0.45156	0.44167	0.65951	0.58557	2.138
1.27	0.45388	0.44123	0.65836	0.58341	2.137
1.524	0.44743	0.43892	0.65806	0.58263	2.127
1.778	0.43625	0.42538	0.65813	0.58228	2.102
2.032	0.42746	0.41469	0.65846	0.58222	2.083
2.286	0.42276	0.41107	0.65841	0.58222	2.074
2.54	0.41860	0.40741	0.65844	0.58222	2.067
2.794	0.41481	0.40429	0.65857	0.58222	2.059
3.302	0.40580	0.39344	0.65913	0.58234	2.041
3.81	0.39244	0.37627	0.66003	0.58269	2.011
4.572	0.37177	0.34735	0.65927	0.58232	1.961
5.08	0.35905	0.32865	0.65859	0.58222	1.926

, where the optimal $R^2$ value was found for a volume thickness of 0.254 cm. This resulted in an $R^2$ value of 0.48 for the horizontal specimens, 0.669 for the vertical specimen’s $UTS$, and 0.59 for the vertical specimen’s $YS$ as shown in

Table 4. The number of points in the data set, n, for the linear regression model and their associated correlation coefficient, $R^2$, for NP1.

Direction	Stress	n	${\mathit{R}}^2$
horizontal	$UTS$	28	0.48297
horizontal	$YS$	28	0.48039
vertical	$UTS$	21	0.6694
vertical	$YS$	21	0.59814

. The coefficients, $a_0$ through $a_8$, for the linear regression model are listed in

Table 5. Calculated coefficients for the linear regression models of horizontal and vertical components of $UTS$ and $YS$ in NP1.

Component	${\mathit{a}}_0$	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{a}}_5$	${\mathit{a}}_6$	${\mathit{a}}_7$	${\mathit{a}}_8$
$UT{S}_{horiz}$	144.99	195.9	−3.08	8.24	3.04	0.009	298.83	48.67	3.06 $\times {10}^{-3}$
$Y{S}_{horiz}$	961.01	169.1	−10.79	13.58	3.94	−0.008	283.29	70.03	4.89 $\times {10}^{-3}$
$UT{S}_{vert}$	11,855	377.2	0.27	−1.14	−20.33	−0.534	420.48	−28.31	−7543.24
$Y{S}_{vert}$	19,365	586.8	−0.37	2.39	−28.56	−0.728	625.16	−34.92	−12,576.41

and the equations are shown in Equations (8) through (11).

Figure 5 shows the $UTS$ and $YS$ for the horizontally and vertically oriented specimens. The computed $UTS$ and $YS$ using the linear regression model were compared to experimentally determined $UTS$ and $YS$. Each graph also shows the regression line, where by an identical estimated and experimental values would have given an $R^2$ of unity.

$$\begin{array}{c}\hfill UT{S}_{horiz}=144.99+195.9BW-3.08W{F}_{ref}+8.24Speed+3.04ZaxisPos+0.0091Vac\\ \hfill +298.83ED+48.67CR+3.06\times {10}^{-3}Vol\end{array}$$

(8)

$$\begin{array}{c}\hfill Y{S}_{horiz}=961.01+169.1BW-10.79W{F}_{ref}+13.58Speed+3.94ZaxisPos-0.008Vac\\ \hfill +283.29ED+70.03CR+4.89\times {10}^{-3}Vol\end{array}$$

(9)

$$\begin{array}{c}\hfill UT{S}_{vert}=11855+377.2BW+0.27W{F}_{ref}-1.14Speed-20.33ZaxisPos-0.534Vac\\ \hfill +420.48ED-28.31CR-7543.24Vol\end{array}$$

(10)

$$\begin{array}{c}\hfill Y{S}_{vert}=19365+586.8BW-0.37W{F}_{ref}+2.39Speed-28.56ZaxisPos-0.728Vac\\ \hfill +625.16ED-34.92CR-12576.41Vol\end{array}$$

(11)

3.2. NP2 Predictive Model Results

The analysis for NP2 builds upon the work that was done by NP1, including range of value analysis for the volumetric ($Vol$) term in order to optimize the predictive ability of the model. Specifically, the heat sink volume term of the predictive algorithm was varied between 0.254 and 5.08 cm as shown in

Table 6. $R^2$ values for predicted $UTS$ and $YS$ for the horizontal and vertical components utilizing a range of values for the volume in the z, or vertical, direction in order to optimize the accuracy of the predicted results for all components of the stresses.

$\mathit{Vol}$ Thickness	$\mathit{UTS}$ Horizontal	$\mathit{YS}$ Horizontal	$\mathit{UTS}$ Vertical	$\mathit{YS}$ Vertical	${\mathit{R}}^2$ Sum
0.254	0.8204	0.88106	0.57844	0.670265	2.9502
0.508	0.8251	0.8813	0.5776	0.657984	2.94203
0.762	0.8221	0.8809	0.57172	0.65632	2.93115
1.016	0.82104	0.87877	0.5717	0.65632	2.9288
1.27	0.81717	0.87742	0.56872	0.65052	2.9138
1.524	0.8177	0.875617	0.57077	0.65371	2.9178
1.778	0.82217	0.875827	0.56821	0.64999	2.9162
2.032	0.82309	0.87478	0.5679	0.64961	2.9154
2.286	0.82777	0.874	0.56776	0.64963	2.9191
2.54	0.83414	0.87328	0.56583	0.64648	2.9197
2.794	0.84046	0.87343	0.56432	0.64392	2.92215
3.302	0.85283	0.873002	0.56238	0.64025	2.9284
3.81	0.86276	0.87265	0.56184	0.63991	2.9371
4.572	0.869285	0.87195	0.56125	0.639405	2.9419
5.08	0.86991	0.871075	0.56121	0.639317	2.9415

. This analysis led to a volume thickness modifier of 0.254 being selected to favor the vertical prediction due to how it is the more complex of the two directions being examined. The complexity in analyzing the vertically oriented specimens primarily lies in the crossing of several beads.

Figure 6 shows the measured strength when compared to the computed strength $UTS$ and $YS$ for the horizontally and vertically oriented test specimens, respectively, with a line of linear best fit plotted to show the idealized $R^2=1$ test case.

Table 7. Calculated coefficients for the linear regression models of horizontal and vertical components of $UTS$ and $YS$ for NP2.

Component	${\mathit{a}}_0$	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{a}}_5$	${\mathit{a}}_6$	${\mathit{a}}_7$	${\mathit{a}}_8$
$UT{S}_{horiz}$	0	42.5	0.8	0.04	0.6	0.05	80.9	10.3	2.26 $\times {10}^{-3}$
$Y{S}_{horiz}$	0	212.4	−4.06	16.1	1.9	0.2	338.3	18.01	1.51 $\times {10}^{-3}$
$UT{S}_{vert}$	0	254.9	−34.7	28.3	7.5	−0.07	639.4	−4.8	2302.2
$Y{S}_{vert}$	0	259.7	−89.8	10.5	7.9	−0.27	772.8	−2.3	7036.4

and

Table 8. The number of points in the NP2 data set, n, for the linear regression model and their associated correlation coefficient, $R^2$.

Direction	Stress	n	${\mathit{R}}^2$
horizontal	$UTS$	24	0.8204
horizontal	$YS$	24	0.88107
vertical	$UTS$	34	0.57844
vertical	$YS$	34	0.67027

show the calculated coefficients and n and $R^2$ values for NP2 using the chosen Z-axis volume variation of 0.1. Equations (12) through (15) show the final linear regression model equations. As seen, the model’s predictive ability is better for specimens that are oriented in the horizontal direction with an $R^2$ value of 0.8204 for the $UTS$, and 0.88107 for the $YS$. The vertical orientation $UTS$ has an $R^2$ value of 0.57844 and $YS$ has a value of 0.67027. The significant variation in $R^2$ values might be explained by the cooling rates varying significantly in the vertical direction. The representation of the cooling rate needs to be more thoroughly investigated or fully recorded during the deposition process. However, there is a marked improvement in the prediction when compared with the results obtained for NP1 primarily due to the improved processing variable capture in NP2. By better controlling the printing process, the formation of the beadwidth and speed were more consistent throughout the printing process leading to a more consistent value for $ED$.

$$\begin{array}{c}\hfill UT{S}_{horiz}=0+42.5BW+0.8W{F}_{ref}+0.04Speed+0.6ZaxisPos+0.05Vac+80.9ED\\ \hfill +10.3CR+2.26\times {10}^{-3}Vol\end{array}$$

(12)

$$\begin{array}{c}\hfill Y{S}_{horiz}=0+212.4BW-4.06W{F}_{ref}+16.1Speed+1.9ZaxisPos+0.2Vac+338.3ED\\ \hfill +18.01CR+1.51\times {10}^{-3}Vol\end{array}$$

(13)

$$\begin{array}{c}\hfill UT{S}_{vert}=0+254.9BW-34.7W{F}_{ref}+28.3Speed+7.5ZaxisPos-0.07Vac+639.4ED\\ \hfill -4.8CR+2302.2Vol\end{array}$$

(14)

$$\begin{array}{c}\hfill Y{S}_{vert}=0+259.7BW-89.8W{F}_{ref}+10.5Speed+7.9ZaxisPos-0.27Vac+772.8ED\\ \hfill -2.3CR+7036.4Vol\end{array}$$

(15)

3.3. NP3 Predictive Model Results

The analysis of NP3 was done utilizing the same framework that was used for NP1 and NP2. An analysis of a range of values for volume was performed review the heat sink volume thickness to optimize the prediction of the linear regression model in both the horizontal and vertical directions. The results of that analysis can be seen in

Table 9. $R^2$ values for predicted $UTS$ and $YS$ for the horizontal and vertical components utilizing a range of values for the volume in the z, or vertical, direction in order to optimize the accuracy of the predicted results for all components of the stresses.

$\mathit{Vol}$ Thickness	$\mathit{UTS}$ Horizontal	$\mathit{YS}$ Horizontal	$\mathit{UTS}$ Vertical	$\mathit{YS}$ Vertical	${\mathit{R}}^2$ Sum
0.254	0.795	0.769122	0.60208	0.54664	2.71286
0.508	0.79141	0.762864	0.6024	0.54781	2.7045
0.762	0.78721	0.756946	0.59989	0.54614	2.69019
1.016	0.78471	0.752357	0.59515	0.54179	2.67401
1.27	0.78308	0.74943	0.58938	0.53638	2.6582
1.524	0.78201	0.74723	0.58491	0.5325	2.64665
1.778	0.781015	0.745832	0.58086	0.52923	2.6369
2.032	0.78026	0.744923	0.57887	0.52775	2.6318
2.286	0.77975	0.744795	0.57693	0.52642	2.6279
2.54	0.77979	0.745767	0.57483	0.525145	2.6255
2.794	0.78067	0.748375	0.57369	0.52462	2.6274
3.302	0.78539	0.757894	0.57218	0.52444	2.6399
3.81	0.792701	0.7708	0.57164	0.52517	2.6603
4.572	0.80319	0.786155	0.57178	0.52663	2.6878
5.08	0.80905	0.793216	0.572009	0.527404	2.7017

, with optimal volume thickness modifier found to be 0.1, which is consistent with the results obtained for the other notional parts. This value provides the most accurate prediction of stresses in the vertical direction, with other modifiers providing minor improvements to the prediction accuracy in the horizontal direction.

NP3 had a total of 72 specimens tested, with a breakdown of 43 horizontally and 29 vertically oriented test pieces. Figure 7 shows the computed strength of the printed parts, as predicted by the linear regression model, versus the experimentally measured strengths along with a linear fit line to demonstrate the $R^2=1$ ideal test case.

Table 10. Calculated coefficients for the linear regression models of horizontal and vertical components of $UTS$ and $YS$.

Component	${\mathit{a}}_0$	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{a}}_5$	${\mathit{a}}_6$	${\mathit{a}}_7$	${\mathit{a}}_8$
$UT{S}_{horiz}$	97.3	96.33	−0.69	3.2	−1.7	0.061	−40.7	−16.2	−2.17 $\times {10}^{-3}$
$Y{S}_{horiz}$	76.8	119.2	−0.96	4.3	−1.9	0.064	−23	−21.2	−3.01 $\times {10}^{-3}$
$UT{S}_{vert}$	−7960	−166.2	−2.1	5.2	−2.8	0.42	−233.9	−2.31	5288.6
$Y{S}_{vert}$	−8217.7	−160.5	−2.7	6.7	−1.7	0.55	−237.4	7.16	5455.6

shows the calculated coefficients for the linear regression model. Equations (16) through (19) show the final equation forms of the $UTS$ and $YS$ prediction equations. The horizontal specimens were found to have an $R^2$ of 0.79501 and 0.76912, for the $UTS$ and $YS$, respectively while the vertical test specimens had $R^2$ values of 0.60209 and 0.54665, for the $UTS$ and $YS$ respectively.

Table 11. Comparison of the correlation coefficient, $R^2$, between successive prints, NP1, NP2, and NP3.

Direction	Stress	${\mathit{R}}^2$-NP1	${\mathit{R}}^2$-NP2	${\mathit{R}}^2$-NP3
horizontal	$UTS$	0.48297	0.8204	0.79501
horizontal	$YS$	0.48039	0.88107	0.76912
vertical	$UTS$	0.6694	0.57844	0.60209
vertical	$YS$	0.59814	0.67027	0.54665

shows the improved predictive ability of the model for NP3 compared to the original printed part, NP1.

$$\begin{array}{c}\hfill UT{S}_{horiz}=97.3+96.33BW-0.69W{F}_{ref}+3.2Speed-1.7ZaxisPos+0.061Vac-40.7ED\\ \hfill -16.2CR-2.17\times {10}^{-3}Vol\end{array}$$

(16)

$$\begin{array}{c}\hfill Y{S}_{horiz}=76.8+119.2BW-0.96W{F}_{ref}+4.3Speed-1.9ZaxisPos+0.064Vac-23ED\\ \hfill -21.2CR-3.01\times {10}^{-3}Vol\end{array}$$

(17)

$$\begin{array}{c}\hfill UT{S}_{vert}=-7960-166.2BW-2.1W{F}_{ref}+5.1Speed-2.8ZaxisPos+0.42Vac-233.9ED\\ \hfill -2.31CR+5288.6Vol\end{array}$$

(18)

$$\begin{array}{c}\hfill Y{S}_{vert}=-8217.7-160.5BW-2.7W{F}_{ref}+6.7Speed-1.7ZaxisPos+0.55Vac-237.4ED\\ \hfill +7.16CR+5455.6Vol\end{array}$$

(19)

4. Discussion

The linear predictive model is able to predict the $UTS$ and $YS$ of as printed EBAM parts. The model was processing condition dependent, i.e., changed with each NP’s processing conditions.

It was shown that by including the cooling rate term in the linear regression model showed an increased accuracy of the model’s ability to predict the vertical $UTS$ and $YS$ by $5.36\%$ and $2.39\%$, respectively, while the horizontal values were improved by $21.8\%$ and $27.1\%$, respectively. Further, by including the volume variable to represent the surrounding heat sink in the linear regression model improved the accuracy of the predicted vertical $UTS$ and $YS$ by $2.55\%$ and $15.05\%$, respectively, while the horizontal values were improved by $14.69\%$ and $2.20\%$, respectively.

The coefficients of the linear regression model also confirm the interconnection of the processing variables to one another. For instance, if $Speed$ is modulated there will be a subsequent change in the interpass temperature which is represented by the cooling rate. Therefore, there will be a variation in the coefficients of the model between successive printings unless the bulk majority of the processing variables remain constant between prints. The model’s coefficients also provides insight into the relevancy of the processing variables to predicting the material strength; variables with small coefficients, i.e., with magnitudes of less than 10, are nearly insignificant for predicting material strength when compared with other processing variables that relate directly to thermal history.

As mentioned previously, the printed geometry was chosen to help simulate a wide range of possible print geometries expected to be printed in industry. The medley of geometries within the print should allow the linear regression model to be suitable for other print geometries and sizes than the ones presented in this study.

The linear regression model improved as build conditions improved, creating a more reliable—or consistent—EBAM print. Some dependencies and tendency to overfitting due to the number of test samples that are included. NP1, predictably, had the worst predictive results due to the distortion of the build, of the test build while NP2 and NP3 produced linear models that more accurately predicted the $UTS$ and $YS$ for both horizontally and vertically oriented test pieces. However, for both NP2 and NP3 the linear regression model had a significantly higher degree of accuracy when predicting the strengths of horizontally oriented test pieces when compared to their vertically oriented counterparts.

The discrepancy in prediction ability could be explained from the need to improve the model’s ability to predict the cooling rate, or ideally monitor the cooling rate in situ. Presently, the model approximates the cooling rate at a point in the build based upon the amount of time the machine has been running and then takes the average of it. This approximation might oversimplify the complex thermal interactions in the vertical build direction and, subsequently, not accurately represent the cooling rate’s influence upon the final material properties. Similarly, when comparing the weights of each of the terms of the linear model, the volume term has significantly more influence upon the predicted strength in the vertical orientation than in the horizontal. Thus, an examination of the volume might be necessary to further improve the model.

NP3 has the largest number of samples tested for the horizontal direction, while NP2 has the largest number of tested samples in the vertical direction with 43 and 34 samples, respectively. Generally, with the exception of the vertical $YS$ prediction for NP3, the $R^2$ value increases with a decreased number of samples. Therefore, it may be necessary to determine if the linear regression model is overfitting specific data points and put forth further analysis into outliers, or other inconsistencies, in the data that might otherwise skew the results.

Furthermore, it was also shown that the beadwidth and energy density generally stayed within the same order of magnitude between prints. The values also remained in the same order of magnitude regardless of the orientation of the specimen. The general consistency across all of the test runs confirms the importance that beadwidth and energy density have in creating a reliable, consistent print. Similarly, it confirms the necessity of including $BW$ and $ED$ processing variables for accurate material property prediction.

5. Conclusions

The linear regression model shows the relative importance of the processing variables upon the material properties upon the printed part. Based upon the order of magnitudes of the coefficients for the variables, it is seen that comparatively build parameters such as $W{F}_{ref}$ and $Vac$ are less important than the derived variables, such as $ED$, $Vol$, and $CR$. Of the directly monitored process variables, beadwidth ($BW$) is the most significant for predicting material properties. Therefore, being able to more accurately quantify those values—either through in situ measurements or improved model representation—should improve the linear regression model predictions.

It is well established that thermo-mechanical cycles have a large impact upon the development of material microstructures and resulting material properties. The linear regression model confirms this, by the importance of the cooling rate and volume, an indirect measure of the temperature effects, in the vertically oriented specimen’s models. It is expected that in situ infrared temperature recordings will enable us to better predict of failure regions from resulting local material strength by modifying the cooling rate term from in situ real-time, temperature sensing at the melt pool should further improve the predictive capability of the model.

Similarly, by working to incorporate other in situ process monitors, such as novel acoustic sensing, it is anticipated that this model could be significantly improved. The acoustic signatures during print may hold the key to identifying potential material failure points within prints in real-time [32,33,34,35]. The linear regression model will also be applied to smaller-scale geometries and single wall prints in a different material than the Ti-6Al-4V discussed. Three single wall materials have been printed using a gas-metal arc welding (GMAW) printer with addition in situ process monitoring.

Ultimately, this work on wire-fed AM processing variable analysis is expected to provide a holistic view of printing conditions and their effect on material properties. Leading to increased 3D printed material reliability and reduced manufacturing costs.