A new method for assessing anisotropy in fused deposition modeled parts using computed tomography data
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Abstract
Voids in fused deposition modeled (FDM) parts are assumed to be a key driver for their anisotropic behavior. However, these assumptions are based on investigations of voids using only 2D data (microscopy images). This paper presents a new method to measure such voids by analyzing 3D-data of from X-ray computed tomography (CT), and application of this data for assessment of mechanical parameters. The article is divided into three parts, where the first part elaborates on a proposed method to assess and characterize the void geometry throughout uniaxial printed FDM parts using CT-data. The second part presents an investigation of the void configurations in samples manufactured using different process parameters, aiming to understand how variation in extrusion rate and compensation for non-linear dynamic extrusion behavior affects the void sizes. The third part displays how the information regarding void sizes could be further related to global mechanical properties, using a multiscale finite element approach. The present method of CT-data analysis gives a clear overview of the spatial variation of the void geometry, and findings from the investigated samples suggest that the size of voids have a large non-random spatial variation, highly dependent on the turning points of the toolpaths, and also significantly affected by accumulation of excess material. Printing at a low extrusion rate increases the void sizes considerably, while implementation of an extrusion dynamics compensation algorithm was found to have low impact on the void sizes. The multiscale finite element approach predicts anisotropic elastic behavior, significantly more compliant in the vertical and transversal direction, relative to the printing direction of the infill. It also predicts a non-isotropic strain energy density throughout the specimen, where the location and magnitude of the most energy dense locations vary significantly for different directions of loading, which implicates an anisotropic behavior in terms of failure, in accordance with literature.
Keywords
Fused deposition modeling Multiscale modelling Finite element analysis1 Introduction
A method to assess void geometries throughout the specimen;
A demonstration of how this method can be used to assess influence of process parameters on the voids, and investigate the effect of altering flow rate and pressure advance parameters;
A demonstration of how this method can be used to assess the voids influence on mechanical parameters as stiffness and failure parameters.
The method presented analyzes volumetric data from X-ray computed tomography (CT-scan), and measures the void sizes throughout a small cubic volume of near-dense FDM parts. The data is analyzed using the OpenCV image processing library through Python programming. Volumetric data from parts made by FDM has been researched earlier with a focus on dimensional accuracy and porosity [9, 10]. However, to our knowledge, this has not been done with the focus on voids as a driver for anisotropy. In response to this gap in prior research, this article reports on the first attempt in relating 3D void geometries based on experimental volumetric information, to anisotropy properties.
To demonstrate how this analysis can be used for obtaining mechanical parameters of the resulting cellular material, a multiscale simulation method is developed. This method is capable of capturing the global elastic behavior and local strain energy density distribution, which can further be used to explore mechanical aspects of failure in FDM manufactured specimens. This is achieved through a first-order homogenization method using finite element analysis, using the geometrical configuration of the voids found through the CT-data analysis.
The reminder of the article is structured as follows: Section 2 presents previous literature on void formation in FDM. Section 3 describes the invented method used for analysis. Section 4 gives a brief overview of the process parameters investigated. The multiscale modeling approach is presented in Section 5. Section 6 displays the attributes and procedure for manufacturing of the samples. Experimental results and discussion from void analysis are given in Section 7, while experimental results and discussion from multiscale analysis are presented in Section 8. Finally, summary and key takeaways are found in Section 9.
It is important to note that there is a large variation in the design and performance of FDM systems, so that numerical results would most possibly be different from machine to machines. But we expect the trends to be similar, especially for the most popular Open Source systems, building on the Marlin Firmware, and even more so if also using a direct extruder design with both hot-end and extruder co-located on the print head (as opposed to a Bowden design).
2 Voids in fused deposition modeling
Reduction of cross section due to voids;
Void-induced stress concentrations;
Incomplete interdiffusion of polymer chains.
3 Method of CT data analysis
- 1.
Known in number.
- 2.
Periodically distributed (as opposed to randomly distributed).
- 3.
Possible to isolate (each void is contained by four adjacent lines of material).
- 1.
Obtain volumetric data from specimen.
- 2.
Threshold the data/images, separating air and solid material.
- 3.
Set up a cell-grid, where each cell covers an individual intersection between four lines of filament
- 4.
Characterize the height and width of each void.
- 5.
Identify the largest void in each cell.
- 6.
Analyze the through-thickness properties.
This approach, finding the void geometries rather than the conventional method of analyzing the neck geometry [6, 12, 13] (distance between the voids), allows for more computationally efficient analysis, as one do not need to find the nearest neighboring voids.
When written as a one or two index average value, e.g., \( {\overline{C}}^{ij} \) denotes the average over the missing indexes, in this case, the row index k. As some cells have more than one void (in the case of much noise or irregular void shapes), all voids in each cell are analyzed, and only the maximum values for these coefficients are used.
4 Parameter selected for investigation
Two parameters are selected for a brief investigation of their influence on the void size and spatial distribution, namely flow rate and pressure advance.
Flow rate, also called extrusion multiplier, refers to relative increased or decreased feeding velocity of raw material/filament. Although the rotational velocity of the filament feeder, and its radius will give an approximate relationship between the rotational velocity of the feeder and volumetric extrusion rate of material, experimental results from Bellini et al. [14] show that this deviation could be large. Tuning of this parameter is therefore crucial to achieve the correct amount of the extruded material [15]. This parameter is highly connected with tensile capacity [1, 11], and is therefore assumed to have a significant influence on the void sizes.
Pressure advance is a compensation algorithm for reducing unwanted extrusion defects found in areas of high acceleration or deacceleration and is therefore assumed to decrease eventual variation in void size near the edges. The specific algorithm used in this research was linear advance v1.0 as implemented in the Marlin v1.1.8 firmware. The algorithms’ influence on the mesostructure or mechanical performance of FDM parts has not been investigated earlier.
5 Multiscale simulation approach
- 1.
Use the previously described method to obtain the shape and area of the inherent voids.
- 2.
Find the distribution of void sizes and average void shape.
- 3.
Create unit cell simulation models for cells with a range of void sizes, but same shape as found in step 2, and obtain effective homogenized material constants.
- 4.
Map these onto a finite element model using the distribution data from step 1.
- 5.
Simulate the uniaxial loading response of the model.
- 6.
Find the global stress/strain response and energy storage in each element.
The finite element analysis was performed using the commercial finite element code ABAQUS, release 6.14. Models were created using the Python scripting interface supported in ABAQUS/CAE meshed with 8-node elements of type C3D8R. The resulting stress vectors represent the rows of the stiffness matrix, and the resulting effective elastic properties were extracted from the compliance matrix by inverting the stiffness matrix. Poisson’s ratio in the simulation was set to 0.35 [17]. While irrelevant due to normalization of results, the Youngs modulus was set to 3000 MPa. The effect of element size was investigated in a few convergence studies for selected void sizes, and was found to have insignificant effect on the displacement results on the boundaries. Mapping of the cellular material properties from the cell-level simulations onto the global finite element model was done with reduced resolution in y-direction, so that the model could be simulated with 26 × 100 × 39 elements rather than 26 × 1100 × 39, which was the resolution from the CT scan data analysis. The C_{a} used for each element was therefore the average over neighboring ± 5 data points taken in the y-direction, which was done to reduce the computational load.
6 Manufacturing and CT scanning of samples
Print variations for the three samples
Sample no. | F | K [s] |
---|---|---|
1 | 1 | 0 |
2 | 0.9 | 0 |
3 | 1 | 0.06 |
7 Void analysis results and discussion
The first observation is that the void sizes are not randomly distributed, as they show clear signs of spatial dependency through non-random patterns both in transverse, longitudinal, and vertical directions. The non-random size distribution also implies non-uniform stress distribution during uniaxial loading—especially for the force flow for transverse loading. The results also show that the voids are smaller close to the print bed than further up, especially for the 3–5 first layers, which is in accordance with literature [6] (also illustrated in Fig. 24 as \( {\overline{C}}_w^{\kern0.5em k} \)). As seen from Fig. 19, the void size at the sides where the toolpath changes direction is significantly higher than for the rest of the structure, and creates a relatively stiffer bond on those sides. As the boundary of the analysis is 1 mm from the edge of the specimen, the actual solid turning point is not a part of the analysis. It is therefore believed that the temperature of the end of the previous line is sufficiently high, so that it sinters together with the new line to a higher degree than the more distant areas. In Fig. 19, the void size at the sides where the toolpath changes direction is significantly higher than for the rest of the structure, creating a relatively stiffer bond on those sides. As the boundary of the analysis is 1 mm from the edge of the specimen, the actual solid turning point is not a part of the analysis. It is therefore assumed that the temperature of the end of the previous line is sufficiently high, so that it sinters together with the new line to a higher degree than in the more distant areas.
As seen in Figs. 19, 20, and 21, the voids are increasing in size from the turning point until approximately the mid-plane of the specimen where it reaches a plateau. The oscillation in void size is also significantly higher near the middle of the sample. It is suspected that this is due to the fact that the velocity of the printer is higher in these areas. There is also a high void size at the end-of-print for all specimens, which is assumed to be a print defect due to effects while stopping the material extrusion and removing the nozzle.
For the two specimens with default extruder multiplier, results from the x-z plane show that the void sizes decrease considerably throughout each layer. The most plausible explanation for this effect is accumulation of excess material due to inaccuracies. There are indications of machine or extrusion dynamics playing a role, as there is some oscillating variation along each void, where \( {\overline{\mathrm{C}}}_{\mathrm{a}}^{ij} \) for the two densest samples show oscillating values of 2–3%, as seen in Fig. 23.
Maximum reduction in cross section along each axis. Maximum values for transverse (\( {\overline{C}}_h^{\kern0.5em i} \)), longitudinal (\( {\overline{C}}_a^{\kern0.5em k} \)), and vertical (\( {\overline{C}}_w^{\kern0.5em j} \)) void measures
Sample | Max \( {\overline{\mathrm{C}}}_{\mathrm{h}}^{\kern0.5em \mathrm{i}} \) | Max \( {\overline{\mathrm{C}}}_{\mathrm{a}}^{\kern0.5em \mathrm{k}} \) | Max \( {\overline{\mathrm{C}}}_{\mathrm{w}}^{\kern0.5em \mathrm{j}} \) |
---|---|---|---|
1 | 33% | 4.9% | 29% |
2 | 54% | 16% | 55% |
3 | 35% | 5.1% | 29% |
Total average coefficients
Sample | \( {\overline{C}}_h \) | \( {\overline{C}}_a \) | \( {\overline{C}}_w \) |
---|---|---|---|
1 | 27% | 3.8% | 25% |
2 | 48% | 12% | 49% |
3 | 28% | 4.1% | 26% |
Comparing samples 1 and 3 reveals that incorporating a linear pressure advance of 0.06 s does not impact the void sizes to a large extent, but might increase consistency along the longitudinal axis. Figure 23 shows that the samples incorporating this are slightly less dense in the turning points of the toolpaths (slightly less blue on the right-facing planes). The difference is however marginal, so no firm conclusions are made. Comparing samples 1 and 2, however, reveals that under-extrusion of 10% impacts the void sizes considerably. It triples the void’s cross section, while increasing the height and width approximately 70%, compared with sample 1. Tuning the flow rate correctly is therefore a crucial task for achieving the smallest voids possible, which has been emphasized in literature [11, 21, 22]. Total averages are seen in Table 3, where C_{a} would be equal to the porosity.
8 Results and discussion of multiscale approach
Unit cell simulation models were made for 100 different void sizes, with C_{a} ranging from 0 to 0.16 (equivalent to void cross section areas from 0 to 0.0192 mm^{2}) where the void shape was a scaled version of the geometry seen in Fig. 25, uniformly scaled in x and z-direction. These were then mapped onto the global structure.
Stiffness reduction in each direction compared with bulk material
Direction | Stiffness reduction |
---|---|
X | 10.7% |
Y | 3.8% |
Z | 13.5% |
From the results, it is very clear that one introduces more energy into the system for loading in transverse or vertical direction compared with loading in longitudinal, using the same force. While the most energy dense elements for y-directional loading has approximately 10% more energy than for loading an isotropic cube of bulk material, x- and z-directional loading have 37% and 43%, respectively. This could explain much of the cellular material anisotropy reported from FDM specimens.
The regions of high strain energy density also differ from4 load case to load case. For transverse direction loading, the highest strain energy density levels are found on the compliant sides as discussed in Fig. 19. The strain energy density is also highest for the first columns due to the large void sizes, and on the very last, because of the large defects during print. Conversely, for loading in vertical direction, the high strain energy density levels are found on the stiff sides. For loading in longitudinal direction, the high strain energy density elements are somewhat more randomly distributed, but tend to be higher in the stiff region in the end of each layer (right hand side for each image in Fig. 30).
An important aspect is that many of the high strain energy density areas are located at or near the boundaries of the geometry, especially for loading in the transverse and vertical directions. This observation indicates that edge effects might play a considerable role for eventual crack initiation and growth.
9 Conclusions and further work
This article has presented a novel method for automatically capturing void sizes from CT scan data. In accordance with literature, the method is capable of identifying effects of decreasing flow rate on the void sizes, which increases the void sizes significantly, from approximately 27 and 28% of the cell height and width, to 47 and 49% with a 10% decrease in flow rate. The results also indicate that incorporating pressure control using an advance algorithm increases the void sizes marginally, but not enough to firmly conclude on any relationship. The key takeaway from the results, regarding void sizes, is that they exhibit a clear spatial dependency of void sizes; hence, prior research must be approached with care as most often single microscopy images are used for geometry assessments, and through-thickness variations are thereby omitted.
Further, the research has shown how geometry data of the voids can be used in investigating cellular material properties, linear elastic properties and strain energy density distribution through a novel multiscale approach based on first-order homogenization. This approach is applied on the default material sample under no pressure advance compensation or altering of flow rate. The results show that the introduction of voids make the structure significantly more compliant for loading in the vertical and transverse direction, resulting in 13.5% and 10.7% lower stiffness, respectively, than fully dense cubes with the same material. For longitudinal direction loading, the structure is 3.8% less stiff, which is the same magnitude as the porosity for the sample. The results from the multiscale model also show that the strain energy densities are much higher for loading in the vertical and transverse directions compared with loading in the longitudinal direction. The highest cell-average strain energy densities for the sample investigated is 37% (transverse loading), 11% (longitudinal loading), and 41% (vertical loading) higher than it would have been for a non-porous sample with the same loading. This indicates anisotropic failure properties, but a more complete failure assessment model has yet to be developed.
Further work on the method of capturing the geometry of voids should try to optimize process parameters for minimizing void sizes. Here, it is of interest to investigate whether this has any influence on the strength of FDM specimens by making comparison with experimental data. Another important extension of this work on void analysis would be to further develop the method to be able to analyze CT data from geometries with an alternating layup (e.g., [0°, 90°]) or more complex global geometries.
As failure in unidirectionally printed FDM, specimens would propagate in the interface between layers or columns of filament lines for transverse or vertical loading, further work on the multiscale simulation method should try to establish methods for identifying the weakest layer or column intersection. The approach could also be enhanced to include more geometric variation of the voids in the unit cell response simulation, rather than relying on a characteristic shape. However, as most of the high-strain energy density areas are on the boundaries of the geometry, a complete failure assessment should aim to include edge effects, as they could be essential for crack initiation When such a method is in place, each cross section can be analyzed individually, and be used to map the strain field from the global simulation onto the identified weakest cross section, before further investigating the link between FDM mesostructure and failure.
Notes
Funding information
This research is supported by The Research Council of Norway through BIA project 235410/O30, and is done in collaboration with AquaFence AS. We greatly acknowledge their support.
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