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Verification of stress transformation in anisotropic material additively manufactured by fused deposition modeling (FDM)

The widespread use of additive manufacturing (AM) has been extensively progressed in the past decade due to the convenience provided by AM in rapid and reliable part production. Fused deposition modeling (FDM) has witnessed even faster growth of application as its equipment is environmentally friendly and easily adaptable. This increased use of FDM to manufacture prototypes and finished parts is accompanied by concerns that 3D printed parts do not perform the same as relatively homogeneous parts produced by molding or machining. As the interface between two faces of bonded material may be modeled by stress elements, in theory, by modeling 3D printed layers subjected to tension at varying angles as transformed stress elements, the stress required to break the layer bonds can be determined. To evaluate such a relationship, in this study, the stresses calculated from stress transformation were compared with the behavior of 3D printed specimens subjected to tensile loads. The maximum principal stress was found to be constant relative to the layer angle, regardless of whether the specimen experienced failure at the layer interface or within the layer material, although the specimens with layers 75° relative to the load were notable exceptions to this finding. This failure at much lower stresses for the samples used in the 75° tests may be attributed to a possible environmental factor, such as temperature or humidity change, degrading the samples’ structural integrity.


Introduction
Additive manufacturing (AM) has improved the traditional method of part production through the elimination of several inner-process manufacturing steps, and post-processing like heat treatment [1][2][3][4][5]. The layer-by-layer production fashion of AM can fabricate parts with complicated geometries in a single-step process [6][7][8][9][10][11]. Researchers have extensively investigated the AM from different viewpoints in the past decade. Innovative methods for controlling the process temperature were experimented, which could take the material characteristics into account [12,13]. These methods were taken advantage of when producing electric components through ink writing [14]. Since the surface finish of printed parts was not favorable, using different material thickness for various layers helped with creating high-quality parts with low surface roughness [15]. Also, deployment of tool path planning through addition of extruders to the system improved the quality of part, which caused a considerable difference in printing spherical geometries [16,17]. Fabrication of parts with such complex shapes results in material waste and long production time, which can be avoided by utilizing modular supports [18][19][20][21].
The ease of manufacture of components produced by fused deposition modeling (FDM) out of polymers makes them useful for rapid prototyping and scale testing of mechanisms and structures before mass-production or large-scale installation [22,23]. In addition, considering the process impacts on both user-safety and the environment, the filament format of feedstock material to FDM process is less hazardous than the other formats of feedstock, such as powder in powder-based AM processes [24][25][26][27][28][29][30][31][32]. Before the advent of affordable FDM technology, prototypes may have been manufactured out of polymers using injection molding, which is both time-consuming and expensive on smaller scales [33,34]. FDM allows parts with complex geometries to be rapidly manufactured cheaply, allowing designers to demonstrate and test the functionality of certain arrangements of parts [35].
A review on different aspects of the FDM process is given in [36], where the significant factors determining the quality of parts are discussed, out of which layer thickness was the most influential. Several researchers have also evaluated the tensile loads to investigate the effect of build orientation, raster angle, type of material, and impact speed on the mechanical properties and dimensional accuracy of parts fabricated by FDM [37]. By testing both single-and multi-layer specimens, the raster orientation was found an influential factor while the impact speed had limited effect [38]. In addition, the impact of build orientation on the parts' formation behavior was investigated in [39] by performing the tensile testing experiments on fabricated parts. These behaviors were also numerically studied with finite element analysis, results of which matched with those of experiments [40].
There is debate, however, as to the suitability of objects made using FDM as accurate representations of their production counterparts, in the realm of either AM or traditional manufacturing processes [41][42][43][44][45]. As FDM necessitates forming polymer filament into layers, the mechanical behavior under loading of the object at the layer boundaries may differ from the bulk behavior of the polymer that the part is composed of, making the part's loading behavior anisotropic [46,47]. While many manufacturers provide mechanical properties in the material specifications for their filament materials, the extensiveness of the documentation varies greatly in the tests performed and the print settings specified. For instance, while some filament manufacturers list filament's mechanical properties without mentioning the test standards or print settings, other manufacturers provide extensive documentation on test methods and print settings. Due to the planar nature of FDM, these documentations are still of limited use, as the relationship between the mechanical properties and the layer orientation is unclear. A mathematical model is required for determining such a relationship; the simplest representation can be the transformation of a stress element.
Experimentation has already been performed in the literature to determine the stress-strain behavior and tensile strength of parts manufactured by FDM under certain loading cases. However, only the extreme layer orientation angles were investigated. In this study, through examining the mechanical properties of parts fabricated with FDM at different layer orientations, the stress transformations were used to derive a stress-angle curve.

Stress transformations
An element of the 3D printed material at the layer interface may be represented by a 3D stress element, and as the stress on the surface of the part σ z may be neglected [48], it may instead be represented as a plane stress element as shown in Fig. 1a. When subjected to loads at varying layer orientations, the stress element at the layer interface may be represented by a transformed stress element as shown in Fig. 1b, with the forces acting on the stress element represented by the freebody diagram in Fig. 1c.
As the test samples are considered to be in pure tension, σ x and τ xy may also be neglected, leaving the stresses at the layer interface dependent on only the tensile stress σ y and layer orientation θ in Eqs. (1) through (3). (1) x � y � = y 2 sin 2 The stress transformation approach may be validated by comparing the maximum tensile stress developed at θ = 90° (tension normal to layers) to the transformed stress σ x1 developed at the layer interface for all other layer orientations. The validity of stress transformation will be verified if σ y at θ = 90° is equal to the transformed stress σ x1 normal to the layer for all layer orientations tested.

Experimental procedure
Tensile tests were performed on type I specimens modeled and tested according to the specifications from ASTM D638 [49,50]. Three test specimens per orientation were fabricated to evaluate the results' repeatability. All test specimens were produced from 1.75-mm diameter PETG filament using an Anet A8 3D printer with the following printing parameters: 240 °C printing temperature, 60 °C heated bed temperature, 40-mm/s print speed, 1.2-mm layer thickness, 100% Fig. 2 Stress-strain curves were obtained from samples with layers a perpendicular to the load, b 75°, 60°, 45°, 30°, 15°, and parallel to the load infill density, and concentric infill pattern. The test specimens were produced with printed material layers oriented from 0° to 90° in 15° increments, measured from normal to the direction of tension. With respect to technology-design principles, it is worthwhile mentioning that the designers are limited by the 45° angle when it is necessary to print supports since the orientation and position of the model may not always be appropriate when 3D printing is used for part fabrication. Tensile tests were performed with an Instron 3365 tensile testing apparatus. The specimens were affixed between the grips of the tensile testing machine with a distance of 115 mm between the grips as specified in the ASTM D638, and they were tested under a motion rate of 3 mm/s.

Results and discussion
Unlike most polymers, which usually experience a large amount of yield and elongation before breaking, most samples tested under the specified parameters of this study displayed a brittle fracture. Figure 2 shows the various stress-strain profiles of all of the test samples, grouped by the angle between the layer plane and the load. For angle 75°, while loads of each sample's failure were approximately equal, the extension of trial 1 reached just above 2.25 mm before failure, while the other two trials endured longer to extend to around 3.5 mm and 3.75 mm.
Trial 2's specimen at 60° failed at approximately half of the load and extension necessary to rupture the other two trials. This load was therefore considered an outlier and was not considered in calculations. The behavior of the specimen in trial 3 at 0° in Fig. 3 was particularly noteworthy, because instead of undergoing brittle fracture upon failure, it experienced ductile yielding with an elongation of over 28 mm, and still did not experience complete separation. While its failure behavior was inconsistent with those of other specimens, the failure load was consistent with the other specimens of the set, and thus the strength of this particular sample was considered during calculation.
After obtaining the ultimate strengths of the samples at each layer angle, each set of ultimate strengths were averaged and plotted against the layer angle relative to the load in Fig. 4.
The angle reference was switched from "relative to load" to "relative to the horizontal" axis for easier referencing later in the transformation equations. To find the related tensile stresses, these loads were utilized and were plotted in Fig. 5.  (1) to find the transformed stress σ x' normal to the layer interface and plotting against the layer interface angle in Fig. 6 revealed that the stress was varying.
The remaining stresses σ y' and τ x'y' were calculated and plotted with σ x' in Fig. 7.
Realizing that rather than experiencing single stress, the layers were experiencing combined loading, and the maximum principal stress for each layer angle was calculated using Eq. (4) and plotted in Fig. 8.
This revealed that the maximum principal stress was the constant stress between all test specimens at around 38 MPa, except for the samples tested with layers printed at 75°to the load, which consistently failed at much lower stresses. (4) sin 2

Conclusion
In this study, fused deposition modeling (FDM) additive manufacturing (AM) was used to fabricate parts with different layer orientations. Through examining the mechanical properties of fabricated parts, the stress transformations were used to generate a stress-angle curve. The stresses calculated from stress transformation were compared with the behavior of 3D printed specimens subjected to tensile loads. Based on the results, the initial hypothesis, that the transformation of a stress element could be validated if the calculated transformed stress σ x' normal to the layer interface resulting from tensile tests was constant, was disproven. However, this was the result of failure to consider the other stresses acting at the layer interface rather than a fault in the principles of stress transformation, as the maximum principal stress resulting from the combined calculated transformed stresses relative to the layer angle was constant, regardless of whether the specimen experienced failure at the layer interface or within the layer material. The notable exceptions to this were the specimens with layers 75° relative to the load.
Noting the more pronounced striations along the samples used in the 75° tests, it is possible that an environmental factor during printing, such as temperature or humidity change, may have been responsible for the degradation of the samples' structural integrity, causing them to fail at much lower stresses.
Availability of data and material The raw/processed data required to reproduce these findings will be made available on request.
Code availability Not applicable.

Declarations
Ethics approval The manuscript contains original ideas which have never been published before in other journals.

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This study is not a human transplantation study.
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The authors declare their consent for publication.

Competing interests
The authors declare no competing interests.