The effects of voids on structural properties of fused deposition modelled parts: a probabilistic approach

The effects of voids on structural properties of fused deposition modelled parts: a probabilistic... In the search to understand the functional capabilities and limitations of fused deposition modelling (FDM) manufactured components, control over their structural behaviour is crucial. For example, voids introduced during the production phase are a large contributor to anisotropy, yet the magnitude of this contribution remains unquantified. As a baseline model for quantifying strength reduction due to process-induced voids, a statistical method for evaluation of the minimum residual (net) cross section is proposed and tested. Our new method serves to predict the reduction in ultimate tensile strength of transversely printed specimens relative to solid or longitudinally printed specimens, based on void sizes identified from microscopy images of the centre plane of a tensile specimen. ImageJ is used to identify void sizes from the microscopy images, and residual cross sections are determined using a bit counting MATLAB script. From the distribution of residual cross sections, the weakest link for a given sample size is estimated. The accuracy of the proposed method is determined through comparison with experimental test data for samples of polylactic acid (PLA). The results reveal a close yet slightly under-predicted strength estimate, which for the case considered predicted approximately 5 MPa (12%) lower strength than observed in the experiments. Based on our findings, we have established evidence that the anisotropic behaviour of FDM specimens in PLA can to a large extent be explained by the reduction in residual cross section. This implies that other effects such as fracture mechanics and atomic diffusion of polymer chains play a secondary role for the phenomena observed. . . . . . . Keywords FDM Fused deposition modelling AM Additive manufacturing Voids PLA Polylactic acid 1 Introduction multiple dimensions, and for physical products, these dimen- sions could be: Creating and testing prototypes, as most other experimenta- tion in product development, are mainly an endeavour to re- & Appearance duce uncertainty. How likely is it that the product works as & Dimensions expected? To draw valid conclusions from prototype testing, & Stiffness one would like the performance to be as close to the intended & Weight design as possible. This could require compatibility in & Strength If there are ways in which the prototype performs different Electronic supplementary material The online version of this article than the expected production model, one should at least be (https://doi.org/10.1007/s00170-018-2148-x) contains supplementary aware of the difference and able to estimate the potential de- material, which is available to authorized users. viation [1]. A major topic for prototyping processes over the past sev- * Sigmund A. Tronvoll eral years, probably boosted by the maker-movement, has sigmund.tronvoll@ntnu.no been additive manufacturing (AM). Moreover, due to technol- ogy advances and patent expirations, AM has now become Department of Mechanical and Industrial Engineering, NTNU - affordable for many hardware designers and engineers. For Norwegian University of Science and Technology, Richard many cases, this technology has reduced the need of going Birkelands vei 2B, 7491 Trondheim, Norway 3608 Int J Adv Manuf Technol (2018) 97:3607–3618 through production drawings and highly skilled labour to pro- duce and hence test complex parts. Especially for production components such as injection moulded plastics, it is now pos- sible to generate close-to-final quality-products by “hitting a button” and letting time do the work. The industry surveying Wohlers Report shows that the volume of AM machines is largely driven by sales of consumer-directed machines, sold not only to consumers, but also to industrial customers. In 2015, an estimate of almost 280,000 desktop printers (sub 5000$) were sold worldwide, compared to approximately 13,000 units in the industrial price range [2]. Currently, the consumer seg- ment is dominated by a single process type—namely the fused deposition modelling—in which lines of heated ther- moplastic (called filament) are deposited, fused together Fig. 2 3D printer axis and components and stacked in layers [3]. However, this filament fusing layer depositing method does create several compatibility issues. For the dimensions parameters and strategies to reduce this anisotropy—or gen- and appearance, one is restricted by the filament widths and erally increase the mechanical strength—has therefore been a layer heights, giving a minimum shell thickness and a clear major topic among researchers [8–15]. “layered” look. While for the mechanical performance, one Our research started off likewise, aiming to reduce the an- must tune the build strategy, process parameters and material isotropy through annealing. This has proven to be effective for to achieve the desired behaviour. Therefore, significant effort inter filament bonding in an earlier scientific study [8], but is put into investigating how these factors affect the mechan- also been debated in different forums of the 3D printing com- ical performance. munity. The basic concept is that, when trying to melt together The most apparent topic for investigation of mechanical two lines of filament, one gets a reduction in strength com- strength of FDM parts is the change of tensile capacity for pared to the bulk material due to incomplete diffusion of poly- different build strategies, pioneered by the work of Ahn, mer chains, reduced cross section (introducing voids) and Montero, Odell, Roudy and Wright [4, 5], as well as the in- fracture mechanics type stress concentrations, as seen in vestigation of the mesostructure by Rodríguez, Thomas and Fig. 1. Annealing was therefore introduced to increase atomic Renaud [6, 7]. The anisotropy arises from the fact that the diffusion. However, initial tests indicated no effect on the load-bearing capacity of a filament along its axis of deposition tensile specimens in polylactic acid (PLA). As a result, the differs from the capacity transversely of two filaments melted following question was raised: What is the baseline reduction together (inter filament bonding). Optimization of process in strength due to each mechanism? There are numbers of papers seeking to improve the FDM process [8, 12–14, 16], yet, very few quantify the potential performance increase due to their proposed process enhancements. To better understand the performance of 3D-printed parts, unlike process optimization where one seeks to find the opti- mal process parameters, we would therefore try to answer the Fig. 1 The three main inter filament bonding strength reduction Fig. 3 Close-up of print paths with no perimeter. Colour for contrast only. mechanisms. F denotes load direction Only two last layers shown, for convenience Int J Adv Manuf Technol (2018) 97:3607–3618 3609 Fig. 4 Different area/volume domains of 3D-printed parts following question: Just by visually inspecting the 3D-printed Usually, this is done by using thermoplastics, which are heated specimen, what can we expect of strength reduction due to the up to above-melting temperature and extruded through a noz- reduction in cross section, resulting from the characteristics of zle onto a table or the workpiece as seen in Fig. 2 and Fig. 3. the process? The base material is either supplied as continuous filament As a starting point, we propose a simple engineering meth- through a rolling wheel feeder or as pellets using a hopper od to estimate the nominal reduction in tensile strength due to and a reciprocating screw. The material is deposited layer by voids. The method is meant to predict failure stress of trans- layer in the z-direction, using a 2.5 axis CNC system. versely infilled tensile specimens, based on the statistical dis- As the material is deposited as lines—rather than melting tributions of residual (remaining) cross sections. This will be or curing of volumetric pixels—the material characteristics are achieved through the use of microscopy images processed highly dependent on the strategy for producing these seg- through ImageJ for void identification, combined with a ments. In general, the resulting parts’ structural integrity is MATLAB script for size estimations to give statistical values governed by five characteristics: for cross section reduction. Based on the identified size of voids, their statistical distribution and the sample size, an ex- & Strategy—How are the filament paths placed? pected failure load distribution is created based on the size of & Material—What are the characteristics of the extruded the weakest link. The predicted distribution will then be com- base material? pared with experimental tensile test data for parts in PLA to & Geometry—How are these lines shaped? estimate the accuracy of the proposed method. & Accumulated strain—What strains have been introduced to the part throughout the process? & Inter filament bonding characteristics—How well do these lines stick to other lines? 2 Theory and background of fused deposition modelling The production strategy and material are preset control pa- rameters, while the geometry of the lines of filament, their The basic concept of FDM is manufacturing through deposi- accumulated strains and their bonding are variables, resulting tion of materials in the form of small strips of filament. Fig. 5 45° (diagonal), 0° (longitudinal) and 90° (transverse) directed infill. 0° directed infill is shown printed with four outlines to reduce stress concentrations along the edges on the specimen exterior 3610 Int J Adv Manuf Technol (2018) 97:3607–3618 Fig. 6 Tensile strength vs. Young’s modulus for already in- market FDM materials. Data from the software CES EduPack from Granta Design Limited from the process parameters as layer height, nozzle tempera- with build plate), the overhang or bridges (facing downwards ture, bed temperature, extruder multiplier, overlap, material, into the air, or onto support structure), the top (facing out of z- etc. One would often need to choose a strategy both for cre- plane upwards), as seen in Fig. 4. ating exterior or interior (infill) of a part and what mechanical To create a smooth outer surface, the outline is very often and aesthetic properties these domains should have. The ex- comprised of semi-continuous lines (lines that bite their tail), terior is divided into four sub categories: the outline (the in- while the inner 2D domains are filled to their specified density. plane outward facing domain), the bottom (domain in contact This can be achieved using different geometric patterns, e.g., Fig. 7 Fracture toughness vs. elongation to failure for already in-market FDM materials. Data from the software CES EduPack from Granta Design Limited Int J Adv Manuf Technol (2018) 97:3607–3618 3611 Fig. 10 Geometric measures of voids lowest performance, with a reported degradation of tensile strength from 22 to 90% [4, 6] compared to the bulk material. Some work using PLA reports an 8–16% reduction of strength of transversal specimens compared with longitudinal ones [18, 19]. However, this work seems to suffer from print quality Fig. 8 Void formation between filaments issues and specimen printing orientations requiring support structures, which might have influenced the results. Specimens that are printed out of x-y plane are often omitted, linear raster, honeycomb, Hilbert curve or concentric raster, to possibly due to the non-symmetric manufacturing conditions. create a near solid, or a partially filled structure to reduce When creating on-bed standing tensile specimens, the temper- density/material and cost/build time. ature history, the vibrations and thereby the specimen charac- The prior research on the subject of material mechanics is teristics would vary along its length. Especially voids tend to mainly done using linear raster infill [4–6, 10, 11, 17], where be smaller close to the heat bed than further away [20]. efforts have been made to find the optimal infill types and Many different materials are available on the market; a orientations, or use the results for classical laminate theory. selection of them, alongside some of their mechanical proper- The reasons for not using more complex infill could be that ties, can be seen in Fig. 6 and Fig. 7. These could be provided it would involve more complex analysis, or the fact that this as pure, copolymer or filled (carbon/glass/wood/silica), where was the standard method of filling before honeycomb and the most used materials are unfilled PLA and unfilled ABS. cubic infill became mainstream. The common findings are, Here, the dominant one is PLA due to its relatively low melt- however, that compressive strength is not severely affected ing point and low shrinkage from solidification to room tem- by infill direction, unlike the tensile strength which is highly perature, which make it easy to use for FDM. Compared with dependent. The most used tensile test specimens are 0° ABS, PLA has very good strength, stiffness and fracture (longitudinal) infill, ± 45° and 90° angled (transverse) infill toughness, but low elongation properties make it less suited compared to the axis of loading, shown in Fig. 5.Research on ABS shows that specimens with transverse infill have the Fig. 11 Size and position of the maximum vertical measure of a void, Fig. 9 Near triangular voids in zigzag pattern which will be used later in the paper 3612 Int J Adv Manuf Technol (2018) 97:3607–3618 Table 1 Process Layer height 0.3 mm characteristics for production of specimens Extrusion multiplier 1.0 Nozzle temperature 210 °C Heat bed temperature 55 °C Print speed 60 mm/s Nozzle size 0.4 mm for components that utilise the material for springs and spring- like components (e.g., snap fits). Fig. 13 Ultimate tensile engineering stress for the samples using cross section area based on its exterior dimensions 3 Anisotropy and voids strength increases with decreased void sizes. Moreover, these voids are not rhombic but tend to extend more upwards than Extruded filament lines have a cross section spanning from downwards, forming a kite/diamond shape. Some researchers oval to a near flattened appearance, where the main drivers for report contradicting findings to this, however, suggesting that the geometry are: the voids extend less upward than downward [21, 22], attrib- uted to, e.g., gravitational forces. However, our experience is & Flow rate in accordance with Rodriguez et al. [7], i.e., the observed & Path placement asymmetry increases with increased flow rate or overlap of & Fluid/solid mechanics of the material paths. High flow rate or overlap results in near triangular & Layer height voids, alternating raster directions spread into a zigzag pattern as illustrated in Fig. 9. The origin of the shape can partly be explained from fluid We have defined the following geometric values, as mea- mechanics, and the circular shape of the nozzle as Hagen– sured from the layer boundary or filament boundaries, also Poiseuille flow through the nozzle should be expected, using shown in Fig. 10 and Fig. 11: viscous materials such as molten plastics. This implies that the d maximum upwards extension of void velocity of the material through the nozzle is highest at the d maximum downwards extension of void centre and declining toward the nozzle wall. This, along with d maximum horizontal measure of void the circular shape of the nozzle, results in less extruded mate- d distance from left contact point to position of d D A rial away from the centreline of the extrusion path (or said d maximum vertical measure of void max otherwise, it would be difficult to extrude a perfectly rectan- θ misalignment of filament intersections gular line of molten material using a circular nozzle). In addi- θ misalignment of maximum upwards and downwards tion, the filament is commonly extruded into a corner made up measure by the previous layer and the previous line of filament, Other geometric measures that have a significant effect on constraining the flow of material and hence flattening its fracture behaviour would be the corner radii. boundaries. As these cross sections do not form sharp corners, placing many filaments alongside each other creates an almost uniform pattern of voids, as illustrated in Fig. 8. How these voids form, depending on process characteris- tics, and their effect on mechanical behaviour has been inves- tigated by Rodriguez et al. [6, 7]. Their findings show that the Fig. 14 Results from the transversal specimens compared with the mean Fig. 12 Tensile specimen geometry of the longitudinal ones Int J Adv Manuf Technol (2018) 97:3607–3618 3613 sections along the specimen. We further assume that the resid- ual strength of the specimens compared with the ultimate ten- sile strength of the material is proportional to the estimated residual cross section compared with the net cross section. It is worth noting that the researchers mentioned above have main- ly used ABS for their investigations, whereas we will use PLA in this study. There are other theoretical models for describing fracture due to inherent voids, where the most widespread one is prob- ably the Gurson model [26]. The essence of this model is that it describes the role of hydrostatic pressure in nucleation and growth of voids, hence explaining the pressure dependency of some materials. However, this model is mainly applicable for materials with ductile behaviour. This could exclude PLA, Fig. 15 Approach for analysis of residual cross section which is reported as brittle [27–29], typically worsened by How these voids form, or more correctly, how the bonds ageing and exposure to moisture [30]. Also, because the voids betweenfilaments form, havebeeninvestigatedbymanyre- are not randomly distributed, but regularly structured holes searchers as this is a major factor to the strength of FDM parts. running across the whole cross section, the Gurson model Li et al. [21] used geometric considerations to calculate the would need extensive modification to work for FDM void density and bond geometry. Bellehumeur et al. [20] specimens. modelled the bond formation between two filaments, depend- Another approach for predicting the strength of FDM- ing on temperature, while Sun et al. [13] investigated the tem- printed specimens could be through linear elastic fracture me- perature profile for some printing processes and its effect on chanics (LEFM), as the voids mentioned could be seen as void formation. Coogan and Kazmer [23, 24]modelled the subcases of periodic notches/holes [31]. Notably, methods strength of single filament-to-filament bonds, including the for estimating the stress intensity factors for closely placed contribution of the reduced cross section, and effects of diffu- rhombic holes with sharp edges, based on numerical calcula- sion of polymer chains. tions, are developed, e.g., the work of Savruk and Kazberuk These efforts mainly sought to increase the understanding [32]. Research has also been done on fracture toughness of of the phenomenon of void/bond formation. When expanded FDM parts [8, 19]. However, using this as a predictive ap- to handle more complicated parts than single filament-to- proach—i.e., investigating the development of cracks between each single void—LEFM would need sufficient control over filament bond, they could be of high value for predicting part strength. However, the approaches lack the stochastic perspec- the critical stress intensity factors in each domain of the tensile tive that would need to be incorporated for investigating real- specimen. This would be difficult due to highly non-consistent world applications and performances. As noted by Gurralla thermal history and hence crystallinity and other material pa- and Regalla, the void sizes are not consistent [25], and a de- rameters [33]. terministic approach would therefore be insufficient. It is worth noting that our method is not intended to de- To fill this gap, we would explore the statistical effect of scribe the fundamental material mechanics around the voids, void size distribution on ultimate tensile strength of trans- but rather to work as an engineering assessment of what to versely printed FDM parts. Our hypothesis is that it is possible expect from FDM-printed parts due to reduction in residual to predict with reasonable accuracy the performance of a cross section. Understanding the impact of this factor would transversely printed specimen, compared to a longitudinally be crucial for further investigating the influence of other phe- printed one, from the distribution of the maximum vertical nomena such as diffusion of polymer chains, fracture mechan- measure of voids, and hence the distribution of residual cross ics and residual strain. Fig. 16 Microscopy picture of dimensions 2570 × 724 compiled of three individual pictures 5 mm 3614 Int J Adv Manuf Technol (2018) 97:3607–3618 Fig. 17 Different threshold methods tried out, from upper left corner—Sauvola, Phansalkar, Otsu, Niblack, Midgrey, Median, Mean, Contrast and Bernsen. Auto local threshold method, with local radius of 200 pixels 4 Printing, tensile testing and microscopy Fig. 5, also noted by Ahn et al. [4]), these could have failed preparation of samples prematurely. First, a total of eight transversely printed and eight longitudi- nally printed samples were made simultaneously in an unmod- ified Prusa i3 MK2 printer, with the process specifications 5 Method and analysis given in Table 1. To keep the research as scientifically controlled as possible The proposed method will aim to find the residual cross sec- (introducing few polymer additives), while maintaining it rel- tion through microscopy images of a segment of the tensile evant for most practitioners, uncoloured PLA filament with a specimens. The strategy employed is summarised in Fig. 15. 1.75-mm diameter was chosen for the experiment. The PLA One segment of a single, transversely printed specimen was was stored in vacuum until printing and tested 2 days after cut along its centre axis, sanded and polished for inspection. printing, where stored in an air tight container. Before inspection, the specimen was treated with dye pene- A dog bone geometry based on ISO 527-2-1B was trant for contrast enhancing, however avoiding dye penetrant employed (as seen in Fig. 12), but its clamp section was made extractor (white fluid used for extracting dye penetrant, and 16.4 mm wide (compared with the standard 20 mm clamp hence improve the visibility of cracks and defects) as this section) to make it fit into the clamps of the tensile test bench. tends to give a misleading geometry/size of voids. The lay-up was equal to the 0° and 90° specimens in Fig. 5. Three individual microscopy images of the sample were All samples were tested under quasi-static conditions with taken and combined, giving a total sample length of 29 mm a displacement rate of 0.2 mm/min. An assortment of the and a resolution of 2570 × 724 pixels. Due to global colour results is seen in Fig. 13 and Fig. 14. The average ultimate gradients (colour differences not due to voids but tensile stresses (UTS) were 38.5 and 60.4 MPa for the trans- miscolouring), a simple global greyscale threshold for identify- versal and longitudinal specimens, respectively, with standard ing the voids would lead to misinterpreting the sizes of voids. deviations of 1.2 and 2.4 MPa. Therefore, the image was processed through ImageJ using the The failure load of the transversal specimens falls in be- Auto Local Threshold algorithm, which estimates the suitable tween 61 and 66% compared with the mean UTS of the lon- threshold of each pixel based on the colour of the pixels within gitudinally printed specimens. As the latter failed in the round- a radius of 100 pixels. There are different methods for deciding ed fillet, possibly affected by stress concentrations inherent to the threshold level, where the Contrast method captured the the production method (from the discrete stepping seen in voids more accurately, i.e., giving the largest voids without Fig. 18 Detail of same area with, from left to right—Sauvola, Phansalkar, Otsu, Niblack, Midgrey, Median, Mean, Contrast and Bernsen Int J Adv Manuf Technol (2018) 97:3607–3618 3615 exhibiting unnatural artefacts, and was therefore used in the rest of the study (see Figs. 16, 17,and 18). The image was then divided into cells, containing one “fil- ament intersection” each, as shown in Fig. 19.Eachcell was scanned to identify the vertical pixel column in the cell with ij the highest number of black dots (finding d ,where ij de- max notes the row/column index of the cell), assuming this value to be constant throughout the cross section. The residual cross ij section factor (r ) of each cell was then taken as: Fig. 20 The magnitude and position of each cell residual cross section ij ij d fraction (r ) shown as red crosses, and average over whole column of ij max r ¼ 1− ð1Þ j cells (R ) shown as blue line cell where d is the height of the cell. These values were then cell The next step was then to estimate the weakest link averaged over the column of cells, creating an average resid- W, representing the minimum residual cross section ual cross section factor for each column (R ), as shown in from a given sample size: Fig. 20. Using equal cell heights, R is calculated as: j W ¼ min R ð5Þ j¼1→m ij where m is the total number of columns in a given sample. ð2Þ R ¼ ∑ When estimating the weakest link in a sample size, and not the i¼1 specimen, it is necessary to incorporate a statistical perspec- where n is equal to the number of rows in the speci- tive. The general procedure is assuming a Gaussian distribu- men. This is the factor assumed to be proportional to tion of the residual cross-sectional factors and finding the the ultimate strength of a single cross section, based on distribution of the expected weakest link within a sample. the gross cross section of the specimen divided by the An approximative Gaussian distribution, using mean and stan- strength of the bulk material. dard deviation from the distribution of the residual cross sec- Referring to the measures from Fig. 19, the gross cross section tions, from a microscopy picture, is shown in Fig. 21. of the specimen (A ) and the residual cross section (A )for gross res Denoting the probability density function (PDF) for the each column j read: residual cross section factors and the cumulative distribution function f and F , respectively, these provide the following A ¼ W⋅H ð3Þ gross R R relationship: A ¼ A ⋅R ð4Þ x res gross ð6Þ F ¼ ∫ f dx The approach allowed for the maxima of voids in each cell column to be horizontally misaligned to some The cumulative distribution function (CDF) for the proba- extent without affecting the results. This implies that bility of the weakest link in a sample of size m would then be triaxiality was neglected since this is difficult to incor- the following: porate without considering more complex analysis, such F ¼ 1−ðÞ 1−F ð7Þ as finite element analysis (FEA). Fig. 19 Detail of the cell division of the black and white picture, together Fig. 21 Real distribution of residual cross section minimums, alongside with the assumed geometry of the voids through the cross section. Cells the probability density function of standard Gaussian distribution scaled are shown sliced at d to the same numbers of samples max 3616 Int J Adv Manuf Technol (2018) 97:3607–3618 which is a Weibull distribution, often seen in weakest link problems. Moreover, its associated probability density func- tion is: m m ð8Þ f ¼ F W W dx This gives the probabilities for a sample of arbitrary size m and print quality equal to the printed specimens, shown in Fig. 22. It is observed that as the sample size grows large, alongside its decreasing value of the weakest link, the variance decreases, as showninFig. 23. This should make the failure load estima- tions more correct for larger samples. Due to the weakest link effect, the distribution shows a steeper decline than incline, Fig. 23 Variance as function of sample size indicating very low probability of a high strength outcome. As a “rule of thumb”, for this print quality and size in the range of specimens run along the axis of loading, the volume fraction 100 lines of filament, it would be unlikely for a specimen to would be a sufficient scaling factor. Estimating the volume achieve a strength of more than 70% of the strength of a void fraction from the microscopy picture, yields v = 0.9525. free sample. Also, the print strategy used on the transverse specimens re- sults in a wavy surface on its edges as shown in Fig. 24 (where the nozzle changes direction), which collocates with the cross- 6 Comparison with experimental tensile test section minimums. An average over 15 “valleys” result in a data reduction in cross-sectional area of 5.8% (denoted ε ), and edges variation in this measure is neglected. The above probability values are all compared to a solid cross The test samples had a straight section of 60 mm, which section. Hence, to compare it to longitudinally printed ones, results in m = 150 filament lines, when using a line width of this must be scaled accordingly as these also exhibit cross- 0.4 mm. Moreover, it is assumed that tensile failure would sectional reduction. As voids for longitudinally printed arise when the axial stress in the weakest link reaches a critical level and that this level is the same as for the longitudinal printed specimens. Formally, this can be stated as: F F transverse longitudinal ð9Þ A ⋅v A ⋅W⋅ 1−ε gross edges gross f Under the above assumptions, this yields the probability density function and cumulative distribution function for the ultimate tensile strength shown in Fig. 25 and Fig. 26. Fig. 22 a Cumulative distribution function. b Probability density function for the weakest link in a sample of size n,inpercentage of a Fig. 24 Curvy edges on the sides of the transversely printed specimens, solid cross section narrowing the specimen at the locations of maximum void concentrations Int J Adv Manuf Technol (2018) 97:3607–3618 3617 estimation method using a series of microscopy pictures could also be the origin of this under-prediction of strength. However, according to our proposed model, there is reason to believe that for specimens of PLA, much of the anisotropic behaviour could be explained directly by the reduction in re- sidual cross section. Due to this effect, the probability of achieving a relatively high strength sample diminishes fast with increased specimen length. The proposed method is only suitable for the prediction of failure loads for transversely printed specimens. Also, our method is only assessed for PLA, which is the dominant ma- terial in practitioner’s usage,althoughlessusedinprevious research. Fig. 25 Scaled probability density function (continuous line) for model The method should be further verified by design of exper- alongside experimental data (bars) for ultimate tensile stress using gross iment techniques, such as Taguchi methods or factorial design, cross section (calculated from exterior dimensions of the specimen) of the transversely printed specimens to investigate the influence of different printing conditions on the void sizes and position, and whether the resulting voids can explain the changes in tensile capacity. This approach For our data, the model gives a close yet slightly conserva- could also be validated for different printing orientations, or tive estimate (about 5 MPa discrepancy) and a matching shape for finding the influence of void sizes on other capacity mea- of the continuous distribution. sures such as ultimate compressive strength and ultimate shear strength. Another important aspect that should be investigated is the 7 Discussion, limitations and further work through-thickness properties of the voids. The element of stress triaxiality and fracture could also be investigated, using, The method proposed herein gives a close estimate on the e.g., FEA and experimental fracture toughness tests. The men- expected distribution of failure loads of transversely printed tioned aspects would all be important for future application of specimens based on the failure load of longitudinally printed this model, whose ultimate aim is to link the bulk material specimens. However, in this case, the method predicts a lower properties and the structural properties of a printed part. outcome than the physical experiments. As the approach ne- glects incomplete atomic diffusion and fracture mechanics, it Acknowledgements This research is supported by The Research Council would lead one to assume that the estimate would predict a of Norway through projects 235410 and 267768. We greatly acknowledge their support. higher strength than the physical experiments. The discrepan- cy could be explained by possible premature failure of the longitudinally printed specimens due to stress concentrations Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// in the fillets of the tensile samples. The residual cross section creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Publisher’sNote Springer Nature remains neutral with regard to juris- dictional claims in published maps and institutional affiliations. References 1. Tronvoll SA, Elverum CW, Welo T (2017) Prototype experiments: strategies and trade-offs. Procedia CIRP 60:554–559 2. Wohlers T. (2016) Wohlers report 2016. Wohlers Associates, Inc. 3. Chen L, He Y, Yang Y, Niu S, Ren H (2017) The research status and development trend of additive manufacturing technology. Int J Adv Manuf Technol 89:3651–3660 Fig. 26 Scaled cumulative distribution function (continuous line) for 4. 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Park S-D, Todo M, Arakawa K (2005) Effects of isothermal crys- of fused deposition modelling (FDM) process parameters using tallization on fracture toughness and crack growth behavior of poly bacterial foraging technique. Intell Inf Manag 01:89–97 (lactic acid). J Mater Sci 40:1055–1058 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The International Journal of Advanced Manufacturing Technology Springer Journals

The effects of voids on structural properties of fused deposition modelled parts: a probabilistic approach

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Engineering; Industrial and Production Engineering; Media Management; Mechanical Engineering; Computer-Aided Engineering (CAD, CAE) and Design
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Abstract

In the search to understand the functional capabilities and limitations of fused deposition modelling (FDM) manufactured components, control over their structural behaviour is crucial. For example, voids introduced during the production phase are a large contributor to anisotropy, yet the magnitude of this contribution remains unquantified. As a baseline model for quantifying strength reduction due to process-induced voids, a statistical method for evaluation of the minimum residual (net) cross section is proposed and tested. Our new method serves to predict the reduction in ultimate tensile strength of transversely printed specimens relative to solid or longitudinally printed specimens, based on void sizes identified from microscopy images of the centre plane of a tensile specimen. ImageJ is used to identify void sizes from the microscopy images, and residual cross sections are determined using a bit counting MATLAB script. From the distribution of residual cross sections, the weakest link for a given sample size is estimated. The accuracy of the proposed method is determined through comparison with experimental test data for samples of polylactic acid (PLA). The results reveal a close yet slightly under-predicted strength estimate, which for the case considered predicted approximately 5 MPa (12%) lower strength than observed in the experiments. Based on our findings, we have established evidence that the anisotropic behaviour of FDM specimens in PLA can to a large extent be explained by the reduction in residual cross section. This implies that other effects such as fracture mechanics and atomic diffusion of polymer chains play a secondary role for the phenomena observed. . . . . . . Keywords FDM Fused deposition modelling AM Additive manufacturing Voids PLA Polylactic acid 1 Introduction multiple dimensions, and for physical products, these dimen- sions could be: Creating and testing prototypes, as most other experimenta- tion in product development, are mainly an endeavour to re- & Appearance duce uncertainty. How likely is it that the product works as & Dimensions expected? To draw valid conclusions from prototype testing, & Stiffness one would like the performance to be as close to the intended & Weight design as possible. This could require compatibility in & Strength If there are ways in which the prototype performs different Electronic supplementary material The online version of this article than the expected production model, one should at least be (https://doi.org/10.1007/s00170-018-2148-x) contains supplementary aware of the difference and able to estimate the potential de- material, which is available to authorized users. viation [1]. A major topic for prototyping processes over the past sev- * Sigmund A. Tronvoll eral years, probably boosted by the maker-movement, has sigmund.tronvoll@ntnu.no been additive manufacturing (AM). Moreover, due to technol- ogy advances and patent expirations, AM has now become Department of Mechanical and Industrial Engineering, NTNU - affordable for many hardware designers and engineers. For Norwegian University of Science and Technology, Richard many cases, this technology has reduced the need of going Birkelands vei 2B, 7491 Trondheim, Norway 3608 Int J Adv Manuf Technol (2018) 97:3607–3618 through production drawings and highly skilled labour to pro- duce and hence test complex parts. Especially for production components such as injection moulded plastics, it is now pos- sible to generate close-to-final quality-products by “hitting a button” and letting time do the work. The industry surveying Wohlers Report shows that the volume of AM machines is largely driven by sales of consumer-directed machines, sold not only to consumers, but also to industrial customers. In 2015, an estimate of almost 280,000 desktop printers (sub 5000$) were sold worldwide, compared to approximately 13,000 units in the industrial price range [2]. Currently, the consumer seg- ment is dominated by a single process type—namely the fused deposition modelling—in which lines of heated ther- moplastic (called filament) are deposited, fused together Fig. 2 3D printer axis and components and stacked in layers [3]. However, this filament fusing layer depositing method does create several compatibility issues. For the dimensions parameters and strategies to reduce this anisotropy—or gen- and appearance, one is restricted by the filament widths and erally increase the mechanical strength—has therefore been a layer heights, giving a minimum shell thickness and a clear major topic among researchers [8–15]. “layered” look. While for the mechanical performance, one Our research started off likewise, aiming to reduce the an- must tune the build strategy, process parameters and material isotropy through annealing. This has proven to be effective for to achieve the desired behaviour. Therefore, significant effort inter filament bonding in an earlier scientific study [8], but is put into investigating how these factors affect the mechan- also been debated in different forums of the 3D printing com- ical performance. munity. The basic concept is that, when trying to melt together The most apparent topic for investigation of mechanical two lines of filament, one gets a reduction in strength com- strength of FDM parts is the change of tensile capacity for pared to the bulk material due to incomplete diffusion of poly- different build strategies, pioneered by the work of Ahn, mer chains, reduced cross section (introducing voids) and Montero, Odell, Roudy and Wright [4, 5], as well as the in- fracture mechanics type stress concentrations, as seen in vestigation of the mesostructure by Rodríguez, Thomas and Fig. 1. Annealing was therefore introduced to increase atomic Renaud [6, 7]. The anisotropy arises from the fact that the diffusion. However, initial tests indicated no effect on the load-bearing capacity of a filament along its axis of deposition tensile specimens in polylactic acid (PLA). As a result, the differs from the capacity transversely of two filaments melted following question was raised: What is the baseline reduction together (inter filament bonding). Optimization of process in strength due to each mechanism? There are numbers of papers seeking to improve the FDM process [8, 12–14, 16], yet, very few quantify the potential performance increase due to their proposed process enhancements. To better understand the performance of 3D-printed parts, unlike process optimization where one seeks to find the opti- mal process parameters, we would therefore try to answer the Fig. 1 The three main inter filament bonding strength reduction Fig. 3 Close-up of print paths with no perimeter. Colour for contrast only. mechanisms. F denotes load direction Only two last layers shown, for convenience Int J Adv Manuf Technol (2018) 97:3607–3618 3609 Fig. 4 Different area/volume domains of 3D-printed parts following question: Just by visually inspecting the 3D-printed Usually, this is done by using thermoplastics, which are heated specimen, what can we expect of strength reduction due to the up to above-melting temperature and extruded through a noz- reduction in cross section, resulting from the characteristics of zle onto a table or the workpiece as seen in Fig. 2 and Fig. 3. the process? The base material is either supplied as continuous filament As a starting point, we propose a simple engineering meth- through a rolling wheel feeder or as pellets using a hopper od to estimate the nominal reduction in tensile strength due to and a reciprocating screw. The material is deposited layer by voids. The method is meant to predict failure stress of trans- layer in the z-direction, using a 2.5 axis CNC system. versely infilled tensile specimens, based on the statistical dis- As the material is deposited as lines—rather than melting tributions of residual (remaining) cross sections. This will be or curing of volumetric pixels—the material characteristics are achieved through the use of microscopy images processed highly dependent on the strategy for producing these seg- through ImageJ for void identification, combined with a ments. In general, the resulting parts’ structural integrity is MATLAB script for size estimations to give statistical values governed by five characteristics: for cross section reduction. Based on the identified size of voids, their statistical distribution and the sample size, an ex- & Strategy—How are the filament paths placed? pected failure load distribution is created based on the size of & Material—What are the characteristics of the extruded the weakest link. The predicted distribution will then be com- base material? pared with experimental tensile test data for parts in PLA to & Geometry—How are these lines shaped? estimate the accuracy of the proposed method. & Accumulated strain—What strains have been introduced to the part throughout the process? & Inter filament bonding characteristics—How well do these lines stick to other lines? 2 Theory and background of fused deposition modelling The production strategy and material are preset control pa- rameters, while the geometry of the lines of filament, their The basic concept of FDM is manufacturing through deposi- accumulated strains and their bonding are variables, resulting tion of materials in the form of small strips of filament. Fig. 5 45° (diagonal), 0° (longitudinal) and 90° (transverse) directed infill. 0° directed infill is shown printed with four outlines to reduce stress concentrations along the edges on the specimen exterior 3610 Int J Adv Manuf Technol (2018) 97:3607–3618 Fig. 6 Tensile strength vs. Young’s modulus for already in- market FDM materials. Data from the software CES EduPack from Granta Design Limited from the process parameters as layer height, nozzle tempera- with build plate), the overhang or bridges (facing downwards ture, bed temperature, extruder multiplier, overlap, material, into the air, or onto support structure), the top (facing out of z- etc. One would often need to choose a strategy both for cre- plane upwards), as seen in Fig. 4. ating exterior or interior (infill) of a part and what mechanical To create a smooth outer surface, the outline is very often and aesthetic properties these domains should have. The ex- comprised of semi-continuous lines (lines that bite their tail), terior is divided into four sub categories: the outline (the in- while the inner 2D domains are filled to their specified density. plane outward facing domain), the bottom (domain in contact This can be achieved using different geometric patterns, e.g., Fig. 7 Fracture toughness vs. elongation to failure for already in-market FDM materials. Data from the software CES EduPack from Granta Design Limited Int J Adv Manuf Technol (2018) 97:3607–3618 3611 Fig. 10 Geometric measures of voids lowest performance, with a reported degradation of tensile strength from 22 to 90% [4, 6] compared to the bulk material. Some work using PLA reports an 8–16% reduction of strength of transversal specimens compared with longitudinal ones [18, 19]. However, this work seems to suffer from print quality Fig. 8 Void formation between filaments issues and specimen printing orientations requiring support structures, which might have influenced the results. Specimens that are printed out of x-y plane are often omitted, linear raster, honeycomb, Hilbert curve or concentric raster, to possibly due to the non-symmetric manufacturing conditions. create a near solid, or a partially filled structure to reduce When creating on-bed standing tensile specimens, the temper- density/material and cost/build time. ature history, the vibrations and thereby the specimen charac- The prior research on the subject of material mechanics is teristics would vary along its length. Especially voids tend to mainly done using linear raster infill [4–6, 10, 11, 17], where be smaller close to the heat bed than further away [20]. efforts have been made to find the optimal infill types and Many different materials are available on the market; a orientations, or use the results for classical laminate theory. selection of them, alongside some of their mechanical proper- The reasons for not using more complex infill could be that ties, can be seen in Fig. 6 and Fig. 7. These could be provided it would involve more complex analysis, or the fact that this as pure, copolymer or filled (carbon/glass/wood/silica), where was the standard method of filling before honeycomb and the most used materials are unfilled PLA and unfilled ABS. cubic infill became mainstream. The common findings are, Here, the dominant one is PLA due to its relatively low melt- however, that compressive strength is not severely affected ing point and low shrinkage from solidification to room tem- by infill direction, unlike the tensile strength which is highly perature, which make it easy to use for FDM. Compared with dependent. The most used tensile test specimens are 0° ABS, PLA has very good strength, stiffness and fracture (longitudinal) infill, ± 45° and 90° angled (transverse) infill toughness, but low elongation properties make it less suited compared to the axis of loading, shown in Fig. 5.Research on ABS shows that specimens with transverse infill have the Fig. 11 Size and position of the maximum vertical measure of a void, Fig. 9 Near triangular voids in zigzag pattern which will be used later in the paper 3612 Int J Adv Manuf Technol (2018) 97:3607–3618 Table 1 Process Layer height 0.3 mm characteristics for production of specimens Extrusion multiplier 1.0 Nozzle temperature 210 °C Heat bed temperature 55 °C Print speed 60 mm/s Nozzle size 0.4 mm for components that utilise the material for springs and spring- like components (e.g., snap fits). Fig. 13 Ultimate tensile engineering stress for the samples using cross section area based on its exterior dimensions 3 Anisotropy and voids strength increases with decreased void sizes. Moreover, these voids are not rhombic but tend to extend more upwards than Extruded filament lines have a cross section spanning from downwards, forming a kite/diamond shape. Some researchers oval to a near flattened appearance, where the main drivers for report contradicting findings to this, however, suggesting that the geometry are: the voids extend less upward than downward [21, 22], attrib- uted to, e.g., gravitational forces. However, our experience is & Flow rate in accordance with Rodriguez et al. [7], i.e., the observed & Path placement asymmetry increases with increased flow rate or overlap of & Fluid/solid mechanics of the material paths. High flow rate or overlap results in near triangular & Layer height voids, alternating raster directions spread into a zigzag pattern as illustrated in Fig. 9. The origin of the shape can partly be explained from fluid We have defined the following geometric values, as mea- mechanics, and the circular shape of the nozzle as Hagen– sured from the layer boundary or filament boundaries, also Poiseuille flow through the nozzle should be expected, using shown in Fig. 10 and Fig. 11: viscous materials such as molten plastics. This implies that the d maximum upwards extension of void velocity of the material through the nozzle is highest at the d maximum downwards extension of void centre and declining toward the nozzle wall. This, along with d maximum horizontal measure of void the circular shape of the nozzle, results in less extruded mate- d distance from left contact point to position of d D A rial away from the centreline of the extrusion path (or said d maximum vertical measure of void max otherwise, it would be difficult to extrude a perfectly rectan- θ misalignment of filament intersections gular line of molten material using a circular nozzle). In addi- θ misalignment of maximum upwards and downwards tion, the filament is commonly extruded into a corner made up measure by the previous layer and the previous line of filament, Other geometric measures that have a significant effect on constraining the flow of material and hence flattening its fracture behaviour would be the corner radii. boundaries. As these cross sections do not form sharp corners, placing many filaments alongside each other creates an almost uniform pattern of voids, as illustrated in Fig. 8. How these voids form, depending on process characteris- tics, and their effect on mechanical behaviour has been inves- tigated by Rodriguez et al. [6, 7]. Their findings show that the Fig. 14 Results from the transversal specimens compared with the mean Fig. 12 Tensile specimen geometry of the longitudinal ones Int J Adv Manuf Technol (2018) 97:3607–3618 3613 sections along the specimen. We further assume that the resid- ual strength of the specimens compared with the ultimate ten- sile strength of the material is proportional to the estimated residual cross section compared with the net cross section. It is worth noting that the researchers mentioned above have main- ly used ABS for their investigations, whereas we will use PLA in this study. There are other theoretical models for describing fracture due to inherent voids, where the most widespread one is prob- ably the Gurson model [26]. The essence of this model is that it describes the role of hydrostatic pressure in nucleation and growth of voids, hence explaining the pressure dependency of some materials. However, this model is mainly applicable for materials with ductile behaviour. This could exclude PLA, Fig. 15 Approach for analysis of residual cross section which is reported as brittle [27–29], typically worsened by How these voids form, or more correctly, how the bonds ageing and exposure to moisture [30]. Also, because the voids betweenfilaments form, havebeeninvestigatedbymanyre- are not randomly distributed, but regularly structured holes searchers as this is a major factor to the strength of FDM parts. running across the whole cross section, the Gurson model Li et al. [21] used geometric considerations to calculate the would need extensive modification to work for FDM void density and bond geometry. Bellehumeur et al. [20] specimens. modelled the bond formation between two filaments, depend- Another approach for predicting the strength of FDM- ing on temperature, while Sun et al. [13] investigated the tem- printed specimens could be through linear elastic fracture me- perature profile for some printing processes and its effect on chanics (LEFM), as the voids mentioned could be seen as void formation. Coogan and Kazmer [23, 24]modelled the subcases of periodic notches/holes [31]. Notably, methods strength of single filament-to-filament bonds, including the for estimating the stress intensity factors for closely placed contribution of the reduced cross section, and effects of diffu- rhombic holes with sharp edges, based on numerical calcula- sion of polymer chains. tions, are developed, e.g., the work of Savruk and Kazberuk These efforts mainly sought to increase the understanding [32]. Research has also been done on fracture toughness of of the phenomenon of void/bond formation. When expanded FDM parts [8, 19]. However, using this as a predictive ap- to handle more complicated parts than single filament-to- proach—i.e., investigating the development of cracks between each single void—LEFM would need sufficient control over filament bond, they could be of high value for predicting part strength. However, the approaches lack the stochastic perspec- the critical stress intensity factors in each domain of the tensile tive that would need to be incorporated for investigating real- specimen. This would be difficult due to highly non-consistent world applications and performances. As noted by Gurralla thermal history and hence crystallinity and other material pa- and Regalla, the void sizes are not consistent [25], and a de- rameters [33]. terministic approach would therefore be insufficient. It is worth noting that our method is not intended to de- To fill this gap, we would explore the statistical effect of scribe the fundamental material mechanics around the voids, void size distribution on ultimate tensile strength of trans- but rather to work as an engineering assessment of what to versely printed FDM parts. Our hypothesis is that it is possible expect from FDM-printed parts due to reduction in residual to predict with reasonable accuracy the performance of a cross section. Understanding the impact of this factor would transversely printed specimen, compared to a longitudinally be crucial for further investigating the influence of other phe- printed one, from the distribution of the maximum vertical nomena such as diffusion of polymer chains, fracture mechan- measure of voids, and hence the distribution of residual cross ics and residual strain. Fig. 16 Microscopy picture of dimensions 2570 × 724 compiled of three individual pictures 5 mm 3614 Int J Adv Manuf Technol (2018) 97:3607–3618 Fig. 17 Different threshold methods tried out, from upper left corner—Sauvola, Phansalkar, Otsu, Niblack, Midgrey, Median, Mean, Contrast and Bernsen. Auto local threshold method, with local radius of 200 pixels 4 Printing, tensile testing and microscopy Fig. 5, also noted by Ahn et al. [4]), these could have failed preparation of samples prematurely. First, a total of eight transversely printed and eight longitudi- nally printed samples were made simultaneously in an unmod- ified Prusa i3 MK2 printer, with the process specifications 5 Method and analysis given in Table 1. To keep the research as scientifically controlled as possible The proposed method will aim to find the residual cross sec- (introducing few polymer additives), while maintaining it rel- tion through microscopy images of a segment of the tensile evant for most practitioners, uncoloured PLA filament with a specimens. The strategy employed is summarised in Fig. 15. 1.75-mm diameter was chosen for the experiment. The PLA One segment of a single, transversely printed specimen was was stored in vacuum until printing and tested 2 days after cut along its centre axis, sanded and polished for inspection. printing, where stored in an air tight container. Before inspection, the specimen was treated with dye pene- A dog bone geometry based on ISO 527-2-1B was trant for contrast enhancing, however avoiding dye penetrant employed (as seen in Fig. 12), but its clamp section was made extractor (white fluid used for extracting dye penetrant, and 16.4 mm wide (compared with the standard 20 mm clamp hence improve the visibility of cracks and defects) as this section) to make it fit into the clamps of the tensile test bench. tends to give a misleading geometry/size of voids. The lay-up was equal to the 0° and 90° specimens in Fig. 5. Three individual microscopy images of the sample were All samples were tested under quasi-static conditions with taken and combined, giving a total sample length of 29 mm a displacement rate of 0.2 mm/min. An assortment of the and a resolution of 2570 × 724 pixels. Due to global colour results is seen in Fig. 13 and Fig. 14. The average ultimate gradients (colour differences not due to voids but tensile stresses (UTS) were 38.5 and 60.4 MPa for the trans- miscolouring), a simple global greyscale threshold for identify- versal and longitudinal specimens, respectively, with standard ing the voids would lead to misinterpreting the sizes of voids. deviations of 1.2 and 2.4 MPa. Therefore, the image was processed through ImageJ using the The failure load of the transversal specimens falls in be- Auto Local Threshold algorithm, which estimates the suitable tween 61 and 66% compared with the mean UTS of the lon- threshold of each pixel based on the colour of the pixels within gitudinally printed specimens. As the latter failed in the round- a radius of 100 pixels. There are different methods for deciding ed fillet, possibly affected by stress concentrations inherent to the threshold level, where the Contrast method captured the the production method (from the discrete stepping seen in voids more accurately, i.e., giving the largest voids without Fig. 18 Detail of same area with, from left to right—Sauvola, Phansalkar, Otsu, Niblack, Midgrey, Median, Mean, Contrast and Bernsen Int J Adv Manuf Technol (2018) 97:3607–3618 3615 exhibiting unnatural artefacts, and was therefore used in the rest of the study (see Figs. 16, 17,and 18). The image was then divided into cells, containing one “fil- ament intersection” each, as shown in Fig. 19.Eachcell was scanned to identify the vertical pixel column in the cell with ij the highest number of black dots (finding d ,where ij de- max notes the row/column index of the cell), assuming this value to be constant throughout the cross section. The residual cross ij section factor (r ) of each cell was then taken as: Fig. 20 The magnitude and position of each cell residual cross section ij ij d fraction (r ) shown as red crosses, and average over whole column of ij max r ¼ 1− ð1Þ j cells (R ) shown as blue line cell where d is the height of the cell. These values were then cell The next step was then to estimate the weakest link averaged over the column of cells, creating an average resid- W, representing the minimum residual cross section ual cross section factor for each column (R ), as shown in from a given sample size: Fig. 20. Using equal cell heights, R is calculated as: j W ¼ min R ð5Þ j¼1→m ij where m is the total number of columns in a given sample. ð2Þ R ¼ ∑ When estimating the weakest link in a sample size, and not the i¼1 specimen, it is necessary to incorporate a statistical perspec- where n is equal to the number of rows in the speci- tive. The general procedure is assuming a Gaussian distribu- men. This is the factor assumed to be proportional to tion of the residual cross-sectional factors and finding the the ultimate strength of a single cross section, based on distribution of the expected weakest link within a sample. the gross cross section of the specimen divided by the An approximative Gaussian distribution, using mean and stan- strength of the bulk material. dard deviation from the distribution of the residual cross sec- Referring to the measures from Fig. 19, the gross cross section tions, from a microscopy picture, is shown in Fig. 21. of the specimen (A ) and the residual cross section (A )for gross res Denoting the probability density function (PDF) for the each column j read: residual cross section factors and the cumulative distribution function f and F , respectively, these provide the following A ¼ W⋅H ð3Þ gross R R relationship: A ¼ A ⋅R ð4Þ x res gross ð6Þ F ¼ ∫ f dx The approach allowed for the maxima of voids in each cell column to be horizontally misaligned to some The cumulative distribution function (CDF) for the proba- extent without affecting the results. This implies that bility of the weakest link in a sample of size m would then be triaxiality was neglected since this is difficult to incor- the following: porate without considering more complex analysis, such F ¼ 1−ðÞ 1−F ð7Þ as finite element analysis (FEA). Fig. 19 Detail of the cell division of the black and white picture, together Fig. 21 Real distribution of residual cross section minimums, alongside with the assumed geometry of the voids through the cross section. Cells the probability density function of standard Gaussian distribution scaled are shown sliced at d to the same numbers of samples max 3616 Int J Adv Manuf Technol (2018) 97:3607–3618 which is a Weibull distribution, often seen in weakest link problems. Moreover, its associated probability density func- tion is: m m ð8Þ f ¼ F W W dx This gives the probabilities for a sample of arbitrary size m and print quality equal to the printed specimens, shown in Fig. 22. It is observed that as the sample size grows large, alongside its decreasing value of the weakest link, the variance decreases, as showninFig. 23. This should make the failure load estima- tions more correct for larger samples. Due to the weakest link effect, the distribution shows a steeper decline than incline, Fig. 23 Variance as function of sample size indicating very low probability of a high strength outcome. As a “rule of thumb”, for this print quality and size in the range of specimens run along the axis of loading, the volume fraction 100 lines of filament, it would be unlikely for a specimen to would be a sufficient scaling factor. Estimating the volume achieve a strength of more than 70% of the strength of a void fraction from the microscopy picture, yields v = 0.9525. free sample. Also, the print strategy used on the transverse specimens re- sults in a wavy surface on its edges as shown in Fig. 24 (where the nozzle changes direction), which collocates with the cross- 6 Comparison with experimental tensile test section minimums. An average over 15 “valleys” result in a data reduction in cross-sectional area of 5.8% (denoted ε ), and edges variation in this measure is neglected. The above probability values are all compared to a solid cross The test samples had a straight section of 60 mm, which section. Hence, to compare it to longitudinally printed ones, results in m = 150 filament lines, when using a line width of this must be scaled accordingly as these also exhibit cross- 0.4 mm. Moreover, it is assumed that tensile failure would sectional reduction. As voids for longitudinally printed arise when the axial stress in the weakest link reaches a critical level and that this level is the same as for the longitudinal printed specimens. Formally, this can be stated as: F F transverse longitudinal ð9Þ A ⋅v A ⋅W⋅ 1−ε gross edges gross f Under the above assumptions, this yields the probability density function and cumulative distribution function for the ultimate tensile strength shown in Fig. 25 and Fig. 26. Fig. 22 a Cumulative distribution function. b Probability density function for the weakest link in a sample of size n,inpercentage of a Fig. 24 Curvy edges on the sides of the transversely printed specimens, solid cross section narrowing the specimen at the locations of maximum void concentrations Int J Adv Manuf Technol (2018) 97:3607–3618 3617 estimation method using a series of microscopy pictures could also be the origin of this under-prediction of strength. However, according to our proposed model, there is reason to believe that for specimens of PLA, much of the anisotropic behaviour could be explained directly by the reduction in re- sidual cross section. Due to this effect, the probability of achieving a relatively high strength sample diminishes fast with increased specimen length. The proposed method is only suitable for the prediction of failure loads for transversely printed specimens. Also, our method is only assessed for PLA, which is the dominant ma- terial in practitioner’s usage,althoughlessusedinprevious research. Fig. 25 Scaled probability density function (continuous line) for model The method should be further verified by design of exper- alongside experimental data (bars) for ultimate tensile stress using gross iment techniques, such as Taguchi methods or factorial design, cross section (calculated from exterior dimensions of the specimen) of the transversely printed specimens to investigate the influence of different printing conditions on the void sizes and position, and whether the resulting voids can explain the changes in tensile capacity. This approach For our data, the model gives a close yet slightly conserva- could also be validated for different printing orientations, or tive estimate (about 5 MPa discrepancy) and a matching shape for finding the influence of void sizes on other capacity mea- of the continuous distribution. sures such as ultimate compressive strength and ultimate shear strength. Another important aspect that should be investigated is the 7 Discussion, limitations and further work through-thickness properties of the voids. The element of stress triaxiality and fracture could also be investigated, using, The method proposed herein gives a close estimate on the e.g., FEA and experimental fracture toughness tests. The men- expected distribution of failure loads of transversely printed tioned aspects would all be important for future application of specimens based on the failure load of longitudinally printed this model, whose ultimate aim is to link the bulk material specimens. However, in this case, the method predicts a lower properties and the structural properties of a printed part. outcome than the physical experiments. As the approach ne- glects incomplete atomic diffusion and fracture mechanics, it Acknowledgements This research is supported by The Research Council would lead one to assume that the estimate would predict a of Norway through projects 235410 and 267768. We greatly acknowledge their support. higher strength than the physical experiments. The discrepan- cy could be explained by possible premature failure of the longitudinally printed specimens due to stress concentrations Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// in the fillets of the tensile samples. The residual cross section creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Publisher’sNote Springer Nature remains neutral with regard to juris- dictional claims in published maps and institutional affiliations. References 1. Tronvoll SA, Elverum CW, Welo T (2017) Prototype experiments: strategies and trade-offs. Procedia CIRP 60:554–559 2. Wohlers T. (2016) Wohlers report 2016. Wohlers Associates, Inc. 3. Chen L, He Y, Yang Y, Niu S, Ren H (2017) The research status and development trend of additive manufacturing technology. Int J Adv Manuf Technol 89:3651–3660 Fig. 26 Scaled cumulative distribution function (continuous line) for 4. 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Journal

The International Journal of Advanced Manufacturing TechnologySpringer Journals

Published: May 28, 2018

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