Thermal history based prediction of interlayer bond strength in parts manufactured by material extrusion additive manufacturing

Material extrusion additive manufacturing, also known as fused filament fabrication (FFF), is currently one of the most widely used technologies. Although promising, the technology is prone to several defects including poor surface quality, low dimensional accuracy, and inadequate mechanical performance caused by weak bonds between successively deposited layers. Studies have shown that bonding between filaments forms above the material’s glass transition temperature which makes it essential to study the thermal history of the printing process. Since interlayer bonding is thermally driven, this study has focused on the development of a regression model to predict the average interlayer bonding strength of a part using the thermal history of the printed layers and the process parameter settings. The process parameters studied are deposition temperature, print speed, and layer thickness. This study relies on using finite element analysis (FEA) to obtain the part’s thermal history and scanning electron microscopy (SEM) to evaluate the bond quality by performing microstructure analysis. The average interlayer bond strength was assessed by measuring the interlayer bond widths and average weld times of all layers in a printed part. The weld time is the time that the temperature of an extruded filament stays above the glass transition temperature when reheated by an adjacent layer. This study includes experimental validation of the developed predictive models to estimate the average weld time and average bond strength of thin wall samples. Results show that the average bond strength is most significantly influenced by two key variables—the average weld time and layer thickness.


Introduction
Over the past few years, there has been a surge in demand for additive manufacturing (AM) due to the several advantages it offers over traditional subtractive methods [1]. AM processes reduce production costs and allow for more design freedom and on-demand fabrication. AM also offers consumers the ability of producing conventionally nonaccessible and high-complexity designs with minimal wastage and fast production [2,3]. Among the various available processes, material extrusion, also known as fused filament fabrication (FFF), is the most commonly used AM technology due to its accessibility, affordability, and ease in fabricating parts/prototypes with a wide range of low-priced feedstock materials [4,5].
FFF is a layer-by-layer additive manufacturing process that involves the extrusion of a thin road of the melted thermoplastic polymer through a computer-controlled heated nozzle onto a heated build plate. The extruded layers quickly cool and solidify, forming the final part [6]. With the expanding use of FFF in many engineering applications, it is essential that parts are manufactured with acceptable dimensional accuracy, surface quality, and mechanical performance. Numerous research studies have been conducted to explore the impact of adjusting process parameters on the overall printing part quality [7][8][9] while also fine-tuning them to meet specific application requirements [10]. FFF is a thermally driven process, and with all the temperature fluctuations developed during the fabrication process, ensuring satisfactory part quality still remains a challenge. During fabrication, the recurrent heating and rapid cooling cycles result in nonuniform thermal gradients, which lead to the accumulation of thermal stresses in the part. These stresses affect the part's dimensional accuracy and cause permanent defects to the part such as warpage, delamination, and weak bonding [11].
It is also important to consider the nature of the thermoplastic materials used in FFF, such as acrylonitrile butadiene styrene (ABS) or polylactic acid (PLA), which are amorphous and semi-crystalline polymers, relatively. During the printing process, a hot filament is extruded onto filaments that were previously extruded and are now cooling down. This new extrusion causes them to be reheated, especially if they were directly in contact with this newly deposited filament. For adequate bonding to take place, the goal is to have them reheated to a temperature higher than the glass transition temperature, T g or the crystallization temperature, and T c for amorphous and semicrystalline materials, respectively [12,13]. Each filament should remain hot enough for a sufficient time during deposition and still cool and solidify as quickly as possible to avoid deformation due to gravity and the weight of subsequent layers [14]. The quality of bonding during extrusion is directly impacted by the process parameters, which heavily influence the thermal history of the filaments. For instance, Fountas et al. discovered that using smaller layer heights, resulting in more layers, leads to greater thermal energy absorption by the printing material and, consequently, better bonding. However, using a relatively higher printing temperature can generate excess thermal energy that may harm the previously deposited strands and weaken the already bonded material [15].
Given the above, it is important to analyze and monitor the temperature evolution of extruded filaments during deposition and cooling and how it is influenced by the printing process parameters. In the literature, many experimental and theoretical efforts have been done to study the temperature profile of a printed part. Bellehumeur et al. [16] developed a lumped capacity model of a single extruded filament road to study the bond formation among extruded polymers through thermal and dynamic analysis. It was shown to have good agreement with experimental studies. However, the model did not consider thermal contacts and hence cannot be used for a sequence of raster lines. Later, Costa et al. [17] expanded the above work and developed an analytical solution for the transient heat transfer during the printing process while accounting for physical contacts between any filament and its neighbors or with the printer bed. The same authors then proposed a coupled analytical and computational method to predict bond quality between adjacent filaments in an FFF part [14] but lacked experimental validation. Thomas and Rodríguez [18] performed a twodimensional (2D) transient heat transfer analysis of the interlayer temperature histories of a single raster but assumed that all previous layers cooled down to ambient temperature before a new layer was deposited. They concluded, however, that lower cooling rates promote stronger interlayer bonding. Compton et al. [19] examined the temperature evolution of a large-scale thermoplastic composite wall by applying a 1D finite difference heat transfer model and validating their results using infrared (IR) thermography. They drew important conclusions by using the T g as a pass/fail cutoff for the likelihood of significant warping or cracking in a part.
The use of IR cameras to monitor and study the temperature profile of extruded filaments during and after printing has become a promising technique used by many researchers in this field. Seppala and Migler [20] used IR thermography under different printing conditions to measure spatial and temporal profiles. They were able to measure the weld time in which each sublayer is reheated to a temperature higher than the material's glass transition. In a later study, they expanded their work by developing an experimental framework to take into consideration the weld formation during the material extrusion in FFF [21]. Malekipour et al. [22] monitored the layer-by-layer fabrication process while varying the process parameters, but their work was not validated by any other method. Using an IR camera, Ferraris et al. [23] recorded the spatial and temperature profiles of a vertical wall and validated their results with a finite difference method (FDM) that ignored the effects of convection and radiation from the surroundings. Although their results were not completely accurate, they presented valuable correlations between the measured interlayer and intralayer times with the bond lines in part by conducting a microscopic analysis. Similarly, Kuznetsov et al. [24] investigated the effects of process parameters on temperature evolutions at the interface of adjacent layers. The average temperature distribution of a part along with the varied process parameters was correlated to the mechanical strength of the fabricated part.
In order to examine more complex parts and actual geometries, researchers began using finite element analysis (FEA) simulations as an alternative method to provide insights on the temperature evolution with respect to time and space. For instance, despite the high computational cost, Cattenone et al. [25] managed to study the effect of various process parameters on the mechanical strength of FFF parts. To simulate the actual printing conditions accurately, they used a meshing technique that matched or was a fraction of the height of the extruded filament. Through a comparison of the simulation results and experimental findings, they could confirm their outcomes and demonstrate the trustworthiness of their simulations. Barocio et al. [26] relied on the use of the FEA tool Abaqus to predict the influence of printing conditions (processing parameters and temperature history) on the interlayer bond strength. Although similar to the study presented in this work, their method of measuring the bonding degree was based on the predicted critical energy release rate. They confirmed their predictions by conducting tests on double cantilever beam (DCB) specimens rather than examining the bonding width.
In this work, the interlayer bonding strength of 3D-printed ABS samples was investigated by conducting a microstructure analysis of the bond width between all layers using scanning electron microscopy (SEM). This work offers a unique insight on the influence of various process parameters along with the average weld time of all extruded layers of a part on the average and minimum interlayer bonding strength of a part. A prediction model of the average interlayer bonding strength dependent on the average weld time and process parameters will be presented. The weld time is the time that the temperature of an extruded filament stays above the material's glass transition temperature when reheated by an adjacent layer, as illustrated in Fig. 1.

Manufacturing
In this study, eighteen thin-walled parts of varying deposition temperature, print speed, and layer thickness were printed using an Ender 3 Pro 3D printer with a nozzle diameter of 1 mm. To ensure consistency and ease of validation, this nozzle diameter was chosen because it matched the maximum layer thickness variation tested and was also the most practical diameter to observe using the infrared camera. The tested variations of the selected printing parameters are summarized in Table 1; all the tested levels are included in the models presented in this paper. The parts were initially designed in SolidWorks, exported as high-resolution stereolithography (STL) files, and converted to G-code using Ultimaker Cura 4.0 software. A commercially available HATCHBOX white acrylonitrile butadiene styrene (ABS) filament with a diameter of 1.75 mm and a melting point between 220 and 230 °C was used to print the parts. The printed parts were of the following dimensions: 50 mm long (x-axis), 16 mm tall (z-axis), and 1 mm wide (y-axis) which is equivalent to the nozzle's diameter. This width was intentionally picked to achieve a single, constant uni-directional tool path for all layers (moving from left to right on the x-axis), as shown in Fig. 1 below. The number of layers in each printed sample varied due to the change in the layer thickness in this study. A layer thickness of 0.4 mm, 0.8 mm, and 1 mm results in a part with 40, 20, and 16 layers respectively. Regarding the controlled printing parameters in this study, (i) the bed temperature was held constant at the maximum possible temperature of 110 °C to promote the best bonding results. (ii) The fan was turned off by setting the speed to 0% to make it similar to the simulation conditions, for accuracy.
A full-factorial design of experiment (DOE) approach was conducted to study the influence of the selected process parameters on the average weld time and average and minimum bond strength of all layers in a part. A full factorial study with a total of 18 parts was conducted, as shown in Table 2.

FFF simulation
FFF is purely a thermal process that is explained by the heat transfer governing partial differential equation (PDE) [11,27,28] given by Eq. (1): where T, ρ, c p , k, and ̇q are temperature, density, specific heat, thermal conductivity, and the internal heat generated, respectively. The initial temperature of the extruded material was set as the deposition temperature. In terms of boundary conditions, the bed temperature was set as a constant temperature boundary condition at the bottom surface of the part's first layer. Throughout the building process, all outer surfaces of the part also experience convection and radiation boundary conditions with the surrounding environment. The initial and boundary condition are described as follows: where q conv , q rad , h, T ∞ , ε, and σ are heat flux due to convection, heat flux due to radiation, the heat convection  coefficient, the ambient temperature of the surrounding air, the material's emissivity, and the Stefan-Boltzmann constant, respectively. A value of 34 W/m 2 K was calculated for the natural convection heat transfer coefficient using the method described in the Appendix.
To perform this heat transfer analysis of the FFF process, the authors took advantage of the additive manufacturing (AM) modeler plug-in offered by the commercial software Abaqus. The FFF process was simulated by element activation and deactivation, in which all elements were initially deactivated. The extrusion of material was then simulated by following the part's generated tool path and activating the elements one by one. The material properties used in setting up these simulations are summarized in Table 3.
In Abaqus, each of the eighteen parts was first imported as a graphic file into Abaqus. The part was then set to have a mesh consisting of 8-noded linear heat transfer brick elements DC3D8, with a seed size equivalent to the layer thickness. The deposition temperature T deposition was applied as a pre-defined temperature field to the elements. In addition, to make the simulation reflect real-life conditions experienced through the FFF process, several boundary conditions (BCs) were defined. Those included (i) applying a constant temperature BC to the bottom surface of the first layer in contact with the print bed to reflect the print bed temperature of 110 °C and (ii) applying convection and radiation effects on all the elements, as they were all exposed to the environment during the building process.
Along with the imported part and defined initial and boundary conditions, the AM modeler requires three more data inputs to successfully simulate the FFF process [29]. Those are (i) event series. The event series path is a table formed by translating the imported G-code into the time (t) and position (x, y, and z) coordinates of the printing path; (ii) property table. The property table is used to define dependent parameters, which can depend on temperature, field, and solution-dependent state variables (SDV). Mainly, the enclosure ambient temperature condition was defined for this simulation; (iii) parameter table. This is used to define the process-specific parameters that are independent of time, space, or material state. For this FFF study, the material deposition direction and bead size equivalent to the 1 mm nozzle diameter were also defined.
Instead of manually defining the inputs and simulating the FFF process of each part, a python script was written to facilitate and speed up the simulations, given that it reads the location and name of the part and G-code file. The FE analysis was performed on an Intel® Xeon® Silver 4116 CPU @ 2.10 GHz, 80 GB RAM with 12 CPUs.
In order to achieve a high level of temperature resolution, a transient thermal analysis was conducted using a time step of 0.01 s. While this may result in a longer processing time and larger storage requirements for the output data, it is critical for accurately capturing the peaks and valleys of the extruded layers' temperatures. This temporal resolution ensures a more accurate measurement of the weld time for the purposes of this study.
As a result of running this thermal analysis, an output database (ODB) file showing the temperature evolution of all the nodes vs time was produced. The temperature evolution curve of all nodes of interest was generated and plotted as shown in Fig. 2. As mentioned before, the points of interest were the nodes at the top and center of each extruded layer (equivalent to the point of interest at which the bond width of each layer was measured). The thermal model required between 30 mins and 4 h to complete the simulation, depending on the process parameter values, mesh size, and the time step size.

Validation
This work relies on obtaining the thermal history of the filaments by using the software Abaqus and calculating the time it takes the temperature of an extruded filament to stay above the glass transition temperature before a new layer is extruded. Only the simulation data of two samples were compared to those obtained using a calibrated infrared (IR) thermography, for validation, as illustrated in section 3.1. The samples were printed near the front edge of the printer's platform, to be near the camera mounted on the printer (exactly 11 cm from the print location), as shown in Fig. 3. As soon as the printing process starts, a video was captured at a rate of 60 frames/s using an A35 infrared camera (FLIR, Wilsonville, OR). From this video, temperature curves of each layer were extracted using the ResearchIR (FLIR) software by placing 1 × 1 pixel cursors on the top center of each printed layer to be compared to the Abaqus nodal temperature results. The IR camera operates with an accuracy of ± 5 °C, a spatial resolution of 0.680 mrad, and a set emissivity calibration of 0.92 [30], equivalent to that of ABS polymer. The IR camera was calibrated by following the  [8] 105°C same method used in the literature [31,32], in which a thin ABS part was positioned on a hot plate. After 20 min of reaching equilibrium, the temperature of the part's top surface was measured using infrared thermography and correlated with the T-type thermocouple measurement at different plate temperatures.

Interlayer bond strength analysis
During the FFF process, the polymer extruded from the nozzle solidifies and bonds with neighboring materials without the assistance of any external energy other than gravity [33,34]. The mechanical strength of an FFF printed part significantly depends on the quality of bonding taking place between neighboring filaments extruded to form the part. Several studies [2,6,[35][36][37] have investigated the interlayer and intralayer bond quality, but a few [38,39] were able to relate the bond formation to experimental data (i.e., different variations of printing parameters, geometry, material properties, and temperature history). The bond formation between polymer filaments in an FFF process consists of three subsequent steps, as shown in Fig. 4: (1) surface contact, (2)  neck growth driven by surface tension, and (3) molecular diffusion at the interface [16,40]. Once surface contact occurs, the interfacial bond strength continues to grow as the polymer chains start to diffuse across the interface until the temperature T < T g [40]. This criterion can be used to predict the interlayer bonding strength by monitoring the temperature evolution of layers during a print. In this work, the interlayer bonding strength of FFF-produced parts is predicted using a simple model proposed in the literature by Coogan and Kazmer [6], which is a modified form of the healing model [41,42]: where W bond is the measured bond width between layers; W is the nominal road width, which is the processing set point (equivalent to the nozzle diameter of 1 mm in this study); σ ∞ is the strength of the fully healed ABS polymer, which was found to be 38 MPa; χ is the interlayer penetration distance of diffusion between layers; and χ ∞ is the diffusion distance at which full strength is achieved and the healing process is complete [6]. Though the study presented here solely measures the bond widths between layers, using the above experimentally validated model provides a good insight and estimate of the part's strength. Given that the model in Eq. (5) is a proportional model using the experimentally measured bond widths, similar trends and relationships found in this study can be derived for both bond widths and bond strengths.

Microstructure analysis
To measure the bond width between layers, W bond , the crosssections of FFF printed parts were observed using a dualbeam scanning electron microscopy, SEM/FIB (Carl Zeiss Auriga Cross Beam FIB-SEM Workstation, Germany). The SEM samples were first prepared by cutting the printed parts at the center to only include the points of interest, using a miter trim cutter. They were then sputtered with silver before imaging. To demonstrate, Fig. 5 shows part #6 after printing, and the lateral view of the cut and sputtered sample on the SEM stub. The red dashed lines mark the width of the SEM sample used.
Some example images illustrated in Fig. 6 show the crosssection of the three printed samples of varying layer thickness: (a) 0.4 mm, (b) 0.8 mm, and (c) 1 mm. The W bond is the narrowest width between each of two adjacent layers of filament, as shown in Fig. 6c. After the bond width between all layers is measured, the average and minimum bond width along a part are derived. The number of layers inspected in each sample varied due to the change in its layer thickness.

Experimental data validation
This study and the presented regression models fully depend on the temperature results obtained from the Abaqus simulations. However, to validate the simulation data, a comparison was made with the temperature data; the data was compared to those measured by a calibrated infrared camera, as mentioned in Section 2.3. Figure 7 illustrates the comparison of both experimental and simulation temperature evolution of the first five extruded layers of (a) run 3 and (b) run 6. The process parameters of the parts printed in Fig. 7 are as follows: (a) run 3 (DT: 220 °C, PS: 5 mm/s, and LT: 1 mm) and (b) run 6 (DT: 220 °C, PS: 10 mm/s, and LT: 1 mm).
Only two runs with a deposition temperature of 220°C and a 1 mm layer thickness but varying print speeds were used to validate the simulation data. Showing validated plots of smaller layer thicknesses, namely, 0.4 mm and 0.8 mm, was a bit challenging due to the infrared camera's focus and manually placing the 1 × 1 pixel cursors on the top center of each printed layer, as previously mentioned in Section 2.3. However, even by varying just the print speed, Fig. 7 shows a noticeable overlap between the temperature changes in the first five extruded layers as captured by the simulation and experiment.
The peaks and troughs in the Abaqus simulation appear to be smoother and less bumpy, suggesting a more idealized situation. This could be attributed to various factors, such as optimal operating conditions (such as ambient and bed temperatures) and the limitations of the model in replicating the material's crystallization behavior. It is evident that the simulation results cooled more quickly, with the temperature curve dropping rapidly after each successive layer was printed, and reaching room temperature faster. This could be used to further improve the convective heat transfer coefficient used in the simulations. However, the fact that the simulation data reaches ambient temperature faster does not have an effect on the weld time measurements between experimental and simulated data. Table 4 and Fig. 7 depict the variance in weld time between experimental and simulation data for runs 3 and 5. The results indicate that there is a slight difference between the two, with discrepancies mostly under 2 min. The difference could be attributed to human error during the experimental measurement process or the idealized cooling conditions assumed in the simulation. However, a general trend of a decreasing average weld time, as the number of printed layers increases from layer 1   Table 5 presents the minimum and average bond strengths, which were determined using experimentally measured bond widths along a printed part, as well as the average weld time obtained using Abaqus, which is defined as the time that the layers remained above the material's glass transition temperature. The standard deviation of both the average weld time and bond strength is also provided in Table 5. The table is reflecting the design of experiment combinations that the L18 mixed factorial design suggested, summarized in Table 2.

Response table with measured experimental and simulation results
The results from the DOE approach presented in Table 5 were evaluated by ANOVA and used to create mathematical regression models to identify significant factors and interactions between the different process parameters on the average weld time, average interlayer bond strength, and minimum interlayer bond strength between the layers of a printed part. The ANOVA results obtained using Minitab® 21 presented for each of the output responses are within a 95% confidence interval. Additional metrics such as R 2 , adjusted R 2 , and projected R 2 , also presented in the ANOVA tables, are indicative of whether the regression models found can be used to provide a good fit for the existing data, along with whether they can make good predictions using other combinations of data that weren't tested.

Effects of process parameters on weld time
The main effects of factorial plots for the average weld time, shown in Fig. 8, demonstrate how the average weld time varies with input parameters. The deposition temperature, print speed, and layer thickness all have a positive correlation with weld time. An increase in any of these input parameters results in an increase in the average weld time of layers in a part. That is because increasing deposition temperature, print speed, and layer thickness will help keep the filaments at a temperature a lot higher than the glass transition upon reheating.
Though the main effect plot is useful in understanding the magnitude of the effects of each individual process parameter on the average weld time, ANOVA tests are important in evaluating the statistical significance of each parameter. Results from the ANOVA and model summary for the average weld time with regard to the three process parameters, deposition temperature, print speed, and layer thickness, are shown in Table 6. In order to identify the most significant variables, an ANOVA test was conducted with a 95% confidence level. The significance of each factor was determined by its respective F-value, while the level of significance was determined by the p-value or probability value. Among the factors tested, deposition temperature, print speed, and layer thickness had high F-values of 22.9, 158.46, and 140.78, respectively. And all the process parameters tested were found to be equally significant in their effect on average weld time, as indicated by a p-value less than 0.05. The resulting regression model proposed in Eq. (6) for finding the average weld time of a printed part achieved a 95.84% R and a 92.53% R 2 (pred), which makes it effective in predicting untested variations of the chosen input parameters.
The average weld time can be predicted using the following equation: Contour plots are an additional set of results that provide us with useful insights on how to choose the printing parameters to achieve a certain desired output response. Figure 9 illustrates the generated contour plot graphs that demonstrate the relationship between a part's average weld time and process variables. Figure 9a shows that the average weld time is highest when both the print speed and deposition temperature increase. Figure 9b shows that the average weld time increases when both the layer thickness and deposition temperature increase, and finally, Fig. 9c shows that an increase in both the layer thickness and print speed will increase the weld time between layers. A part with a significantly smaller layer thickness and printed at a low-speed setting cools faster than other combinations of settings and achieves a relatively lower average weld time (< 2.5 s), as shown in Fig. 9c. By the time the adjacent new layer is deposited, the previous layer has almost completely cooled down and solidified.    Although the effect of an increase in deposition temperature is not as noticeable, a higher deposition temperature slows down the cooling rate of the filaments and ultimately promotes stronger bonds. Similarly, the faster the print, the faster the reheating cycles take place, developing stronger interlayer bonds. However, due to the compact nature of layers in a print with smaller layer thicknesses (i.e., 0.4 mm) as shown in Fig. 6, the bond width is geometrically larger than that in a print with a larger layer thickness (i.e., 1 mm). This resulted in a negative correlation between layer thickness and average interlayer bond strength.

Average interlayer bonding
Results for ANOVA and the model summary for the average interlayer bond strength are shown in Table 7. A 95% confidence level was set for the ANOVA test to identify the most significant variables. Both layer thickness and average weld time had a p-value less than 0.05, indicating that they have the most influence on the average interlayer bond strength of a fabricated part. The significance of layer thickness is evident with a p-value of 0.000, while the average weld time follows closely with a p-value of 0.03, highlighting the necessity of investigating the thermal history of filaments to anticipate their bonding ability. The resulting regression model proposed in Eq. (7) for finding the average bond strength of a printed part achieved a 98.83% R 2 and a 97.70% R 2 (pred) which makes it highly effective in predicting untested variations of the chosen input parameters.
The average interlayed bond strength can be predicted using the following equation: Figure 11 illustrates the contour plot graphs that demonstrate the relationship between a part's average interlayer bond strength and process variables, in addition to the average weld time. The plots in Fig. 11 show that the average weld time plays a huge role in affecting the interlayer bond strength, which iterates the findings shown in the literature. In the plots, the axes of the process parameters are equivalent to the fully tested levels shown in Table 1. Figure 11a shows that an increase in both print speed and deposition temperature promotes stronger interlayer bonds. That is due to the slower cooling and heat accumulation taking place between layers.  Figure 11b, c shows that layer thickness dominates the effect on average bond strength regardless of what the deposition temperature or print speed is. The smaller the layer thickness for any of these combinations, the higher the average interlayer bond strength of a part is. Figure 11d-f provides good insight on the target weld time required to achieve adequate interlayer bonding strength for any deposition temperature, print speed, or layer thickness. This will potentially be useful when trying to manipulate the process parameters to achieve a certain weld time between the layers and ultimately improve the bonding between layers of a part. Though there is not a steady relationship that can be concluded from the contour plots in Fig. 11d, e due to the low significance of deposition temperature and print speed. From Fig. 11d, it can be concluded that although a higher average weld time is often desired to promote stronger bonding between layers, but this is not always the case. With a lower cooling rate and a relatively high average weld time, an extruded layer could remain soft and mellow for a longer period and start deforming in shape, causing a displacement from the intended position. Figure 11e shows that at relatively lower print speeds, the user should target faster cooling and quicker solidification of the extruded layer. Figure 11f shows that printed parts are strongest when printed at a smaller layer thickness and a slower cooling rate or higher average weld time. Contrarily, parts printed with larger layer thicknesses are relatively weaker. This result reiterates the importance of the layer thickness set on the part's strength. The prediction model presented in Eq. (7) was a second-order polynomial estimation due to nonlinear results. Irregularities shown in Fig. 11d-f could be due to a variety of reasons. The common factor in these figures is the average weld time, which in itself is dependent on various factors including deposition temperature, print speed, and layer thickness, resulting in nonlinearity in the contour plots. This nonlinear behavior could also be attributed to the underlying physical process of polymer bonding and its dependency on the filament's thermal history. Figure 12 shows the main effect of factorial plots for the minimum interlayer bond strength. Similar to the average interlayer bond strength of a part, an increase in the deposition temperature and print speed results in an increase, whereas an increase in layer thickness has a negative effect on the minimum interlayer bond strength.

Minimum interlayer bonding strength
A 95% confidence level was set for the ANOVA test to identify the most significant variables that affect the minimum bond strength of layers in a part. From the results obtained, the layer thickness is found to be the most influencing parameter on the minimum bond strength, with a p-value equal to 0.001. The resulting regression model found for minimum bond strength achieved a low R 2 value which makes it a relatively inefficient prediction model. Hence, only the models developed for average weld time and average bond strength were validated in this study, as presented in the following section.

Validation tests
To validate the regression models established in this study, a set of seven samples were used, each containing new and distinct combinations of the process parameters under investigation: a deposition temperature within the initial range of 220-240 °C; a print speed from 5 to 20 mm/s; and a layer thickness from 0.4 to 1 mm. The weld time, T weld was measured using Abaqus and compared against the predicted range determined by Eq. (6). In a similar manner, the average bond strength, σ bond of the validation test parts, was calculated based on the average bond widths of all layers in each part and compared to the predicted range established by Eq. (7). This predicted range, which represents a 95% confidence interval, was tabulated in Table 8 and includes both minimum and maximum values.
The validation results in Table 8 show a comparison of the measured and predicted responses, and they reveal that there was no significant difference between the two. As shown in Fig. 13, the average weld time and average bond strength measured are within the predicted 95% confidence interval, except for one data point from each response. Therefore, the regression models can overall be considered adequate to predict and explain the average weld time and bond strength of the tested part. In analyzing the relationship between average weld time and average interlayer bond  Fig. 13 a, b The comparison between the measured and predicted 95% confidence interval range of average weld time and average interlayer bond strength respectively strength, the contour plots in Fig. 11d-f suggest that the correlation between the two is not purely linear, which helps explain the variability shown in Fig. 13.

Conclusion
The present work provides a unique approach in which the thermal history of an FFF fabricated part was used along with the printing process parameters to predict the average interlayer bond strength. Simulations performed to analyze the temperature evolution of the print were used to obtain the weld time of each layer. In this study, the weld time is defined as the time that the temperature of an extruded filament stays above the glass transition temperature when reheated by an adjacent layer. According to the full factorial DOE adapted, eighteen thin wall parts were printed and observed under the SEM to measure the bond widths and ultimately calculate the interlayer bond strengths of all its layers. The levels of the tested process parameters were DT of 220 and 240 °C, PS of 5, 10, and 20 mm/s, and LT of 0.4, 0.8, and 1 mm. Simultaneously, eighteen heat transfer simulations were performed on Abaqus to obtain the weld time of each layer. Because this study is fully dependent on simulation data, it was important to compare the simulated data to the experimental one, for validation purposes. The average weld time measured from the experimental and simulated data for runs 3 and 5 showed a maximum discrepancy of under 2 min between the two. The difference could be attributed to human error during the experimental measurement process or the idealized cooling conditions assumed in the simulation. This comparison could be used to help improve the convective heat transfer coefficient used in the simulation. But even with such insignificant discrepancy, the validity and reliance of the simulated data have been proven.
The results were used to run an ANOVA study and find regression models that can predict the average weld time, average interlayer bond strength, and minimum bond strength of a part and explain their relation to the set process parameters. All process parameters (deposition temperature, print speed, and layer thickness) had a significant impact on the average weld time of a part with a p-value significantly less than 0.05. Additionally, while a higher deposition temperature and print speed were proven to be more desirable for adequate bonding to take place, but only the layer thickness and average weld time of a part showed the most significant in affecting the average interlayer bond strength, with p-values less than 0.05. The prediction of average weld time was modeled using a linear approach based on the tested process parameters, while the prediction of average interlayer bond strength was modeled nonlinearly, considering the average weld time (thermal history) and process parameters. The nonlinearity can be attributed to the physical process of polymer bonding, which in itself is dependent on the filament's thermal history and affected by various factors such as deposition temperature, print speed, and layer thickness, all of which influence the independent variable of average weld time. This relationship is illustrated in the contour plots, including average weld time as an independent variable.
According to the R 2 (pred) values obtained for each model, it was found that the prediction model suggested for the minimum bond strength was not as accurate or reliable in predicting the desired response. But the prediction models of the average weld time and average interlayer bond strength were deemed reliable by having a 92.53% and 97.70% R 2 (pred), respectively. To validate the models, seven new parts with untested variations of the printing parameters were used. As presented, there is a good agreement between the predicted and measured results, which validates the suggested prediction models. The presented contour plots reflect all the tested levels in this study and hence did not require normalization.
This study can be expanded further to include other materials and more process parameters, covering a larger range of data. In addition, this thermal study can be expanded to include data points in 2D or 3D, giving a more extensive analysis of the part's thermal history and ultimately mechanical performance or bonding. The data found and presented can be valuable in determining the nature of bonding taking place in an FFF part, given a set of known process parameters. Incorporating the thermal history of the filaments in predicting the bonding strength can help us identify the points at which inadequate bonding is expected to take place by knowing the filament's thermal history and finding the average and location of the minimum interlayer bond strength.
Appendix. Estimation of the natural convection coefficient By taking the geometry of each single deposited layer as a horizontal cylinder, the average natural convection coefficient, h can be calculated using the Churchill and Chu correlating equation [43]: where Nu d is the Nusselt number, k f is the thermal conductivity of air, and L c is the characteristic length. Given that the assumed geometry is a horizontal cylinder, the characteristic length becomes the outer diameter of the layer, L c = D = 0.001 m. The corresponding Nusselt number, Nu d , for a horizontal cylinder can be expressed as: where Rayleigh number, Ra D , which also depends on the Prandtl number, Pr, is given by: Equation (9) is solved by finding the properties of air at the film temperature, T f = (T s + T ∞ )/2= (220 + 23)/2= 121.5 ° C, and 1 atm, which are arranged in Table 9. Pr.