Accepted Manuscript Title: Interference fit of material extrusion parts Authors: L. Bottini, A. Boschetto PII: DOI: Reference: S2214-8604(18)30630-4 https://doi.org/10.1016/j.addma.2018.11.025 ADDMA 596 To appear in: Received date: Revised date: Accepted date: 21 August 2018 31 October 2018 20 November 2018 Please cite this article as: Bottini L, Boschetto A, Interference fit of material extrusion parts, Additive Manufacturing (2018), https://doi.org/10.1016/j.addma.2018.11.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Interference fit of material extrusion parts Bottini L.*, Boschetto A. Sapienza University of Rome, Department of mechanical and aerospace engineering, Via Eudossiana 18, 00184 Rome (Italy) PT luana.bottini@uniroma1.it, alberto.boschetto@uniroma1.it, RI *Corresponding author: luana.bottini@uniroma1.it PT ED M A N U SC Abstract Material extrusion is an Additive Manufacturing process able to fabricate a physical object directly from a virtual model using layer by layer deposition of a thermoplastic filament extruded by a nozzle. The fabrication of functional components implies the need for the assembly with other parts with different properties in terms of material and surface quality. One of the most used assembly method involving plastic materials is the interference fit. It consists of fastening elements in which the two parts are pushed together, by means of a fit force, and no other fastener is necessary. It requires the accurate design of the interference, typically carried out by the designers through diagrams and theoretical formulations supplied by the material manufacturers. At present no theory has been provided for material extrusion parts due to the anisotropic behavior: the mesostructure, the surface roughness and the dimensional deviations mainly depend upon the build orientation. In this work the effects of the surface morphology and the interference grade on the assembly and disassembly forces in an interference fit joint are investigated. For the purpose, a design of experiment with a factorial plan has been carried out. The coupling behavior and the maximum forces are discussed. A new variable namely the real interference has been introduced and a relationship between this variable and the assembly force has been found. Through this model it is possible to know in advance the force necessary to assemble a material extrusion part with an assigned interference grade. CC E Keywords: Material extrusion, Fused Deposition Modeling, Interference fit, Coupling, Fit force. Nomenclature α deposition angle A 𝑖 interference grade 𝑑̂ stratification direction 𝑛̂ normal to the surface λc roughness sampling length λs short wavelength cut-off R2 determination coefficient adj R2 adjusted determination coefficient DF degrees of freedom SS sum of squares SStotal total sum of squares PT SSfactor sum of squares of the generic factor SSfactorA x factorB sum of squares of the interaction RI SSerror sum of squares of the error SC MS mean squares U S within sample standard deviation N F Fisher test statistics P probability value A ∆ℎ(𝛼) deviation from the nominal value as a function of α β0 regression coefficient ε random error CC E 𝜎 2 generic variance PT β1 regression coefficient ED F𝑎 assembly force M 𝑖𝑟 real interference grade ANoVA analysis of variance predicted R2 predicted determination coefficient A Adj SS adjusted sum of squares Adj MS adjusted mean squares Coef regression model coefficients SE Coef standard error of the coefficients Obs unusual observations Fit fitted value by the regression model Resid residual value Std Resid standard residual value R large residual MAE Mean Absolute Error MAPE Mean Absolute Percentage Error A CC E PT ED M A N U SC RI PT 1 INTRODUCTION The Additive Manufacturing (AM) technologies in the last years have been characterized by evolutionary developments in materials and processes with reduction of costs opening up to a wide range of applications. Primarily aimed at producing prototyping models, they are recognized in manufacturing industry including tooling and batch production [1]. This revolution has changed every stage of the product/process development from the design to the assembly [2]. In fact these technologies permit to realize integrated assembly and embedded components modifying the perspective of joining processes: assembly drawings are greatly simplified by the parts count reduction since features can be combined in a single piece and by the ability to produce complex geometry comprising coupling elements [3]. Assembly limitations are lessened by AM, but the need for joining these parts with ones fabricated by other technologies remains. This task must be undertaken considering Design for Assembly methodologies which aim to reduce the time, the cost and the difficulties by minimizing the number of parts and eliminating fasteners [4]. Among the joining processes the interference fit, also referred to as press fit, shows several advantages according to these indications. Screws, metal inserts, adhesives and other means are avoided. It is widely used when plastic materials are involved since it is fast, economical and it requires simple tooling. In an interference fit, the mating hole or the inner part are shaped so that they slightly deviate in size from the nominal dimension. A coupling force is needed to fasten the joint which results in the deformation of both the parts [5]. The interference grade must be accurately designed in order to have a desirable joint stress, ensuring that parts do not crack for excessive stress and loosen the coupling for stress relaxation [6]. For the purpose some specific models are needed to adequately choose the interference. Traditional interference fit design methods are based on the two-dimensional stress analysis for elastic loading which is solved by Lame’s equations [7]. Currently the design of the interference fit between metal parts refers to the so-called tolerance capability charts which are particular diagrams typically supplied by the material manufacturer: these charts allow determining the theoretical interference limits within the coupling that can be considered functional. For plastic materials the manufacturer’s charts are presented in a similar manner even if they differ for some aspects which could be carefully considered: surface defects, direction of fiber orientation, fabrication parameters of some traditional technologies such as the injection molding [8]. In this case the major objective is to minimize the assembly operations by consolidating fasteners directly into the parts themselves: in fact this process is capable of producing extremely complicated part geometries [9]. For this reason the interference fit is one of the most employed methods but its design needs the understanding of joint safety factors, loads, designing algorithms, material properties, stress relaxation over time, creep and moisture conditions [10]. Recently great effort has been paid to this method since it improves the fatigue life of double shear lap and bolted joints; typically the approach consisted in a finite element analysis and experimental confirmations [11-14]. Also through an analytical modeling, some singularities in the radial strains and displacement at joint interface confirmed a coupling behavior very difficult to investigate [15]. This behavior is also influenced by the surface roughness which is generally able to enhance friction effect in the unbounded interference joints [16]. The influence of the A CC E PT ED M A N U SC RI PT roughness in presence of low form defects is presented in [17]: for low tightening, although the asperities quickly become plastics, the influence on the average pressure is relatively little; conversely for higher tightening, large deformation can occur and the principal parameters are the average amplitude, the average width and the shape of the roughness profile. The mean height of the asperities is the major factor contributing to the fit strength making advisable the use of traditional machining as opposite to polishing and grinding [18]. Moreover assemblies characterized by nubs exhibit higher load bearing ability than assemblies with plain surfaces [19]. The scenario markedly complicates when composite materials are involved: the stress distribution is nonuniform and depends upon the laminate properties and the ply orientation [20]; during the installation the material can delaminate for oversized percentage interference fit [21] and a microscopic damage mechanism takes place [22]; the interference fit changes the load transfer in a way which is strongly dependent upon the hole deformation [23]. These recent achievements highlight how materials characterized by mesostructures are very difficult to understand as regards the interference fit. AM parts are particular mesostructured materials due to their layer by layer fabrication and toolpath strategies. Material extrusion process is one of the most widely diffused AM technologies. It was patented in the ‘90s by Stratasys Ltd. with the name of Fused Deposition Modeling (FDM). It forms the parts by means of an extrusion of melted polymer in form of filament. Important mesostructure features are a function of the processing parameters [24] and the strain-stress response of the material is greatly affected by the type of mesostructure [25]. Bellini et al. [26] strived to determine the stiffness matrix of the orthotropic material suggesting some road-paths to reduce the anisotropies. The anisotropic behavior was evidenced in [27] providing some build rules to design FDM parts and Ziemian et al. [28] assessed that the ultimate strength benefits from the alignment of the polymer molecules along the stress axis. The bond strength between filaments and the mesostructure are affected by the fabrication strategy and creep formation [29]. Croccolo et al.in [30] found how the mechanical strength and the stiffness markedly depend upon the number of the contours deposited around the component edge. The deposition angle, i.e. the angle between the stratification direction and the normal to the considered surface, is responsible for the major variation of the part quality characteristics [31]. The surface roughness has been modeled and predicted by several authors [32-34] highlighting that the average roughness varies in the range 12 µm – 70 µm. The studies of the filament morphology in [35] pointed out that the prototype surface is characterized by different profile shapes with different tribology properties: these aspects should be considered when a secondary operation is involved and the machinability assessment is necessary [3637]. In [38] the dimensional deviations of FDM part have been investigated: a relationship between the deposition angle and the layer thickness has been found. In the same study widespread irregularities such as planarity misfits, surface uplifts and local defects have been evidenced. These observations highlight that the behavior of a FDM prototype surface involved in an interference fit must be deeply investigated. There is a lack of technical and scientific knowledge about methodologies predicting the behavior of a FDM parts assembly. The aim of this work is to investigate how the surface morphology, the interference grade and the dimensional deviation from the nominal value affect the assembly and disassembly forces in an interference fit joint. 2 EXPERIMENTATION A design of experiment with a factorial plan was carried out considering two factors i.e. the deposition angle α and the interference grade 𝑖. With reference to Figure 1a the employed specimens were parallelepipeds 10x20x20 mm3. They have been oriented onto the built platform in order to have six levels of the deposition angle α, i.e. the angle between the stratification direction 𝑑̂ and the normal 𝑛̂ to the coupling surface. The levels are 3°, 9°, 15°, 30°, 60°, 90° chosen in accordance with a geometric progression A CC E PT ED M A N U SC RI PT since small angles are characterized by higher dimensional deviations than big ones [38].The distance between the coupling surfaces has been varied by adding the interference grade 𝑖 to the nominal value: the levels (0 mm, 0.1 mm, 0.2 mm) were limited to 0.2 mm which is close to the layer thickness. Each test was replicated two times, for a total of 36 tests. 3. Each specimen had a 6 mm diameter hole to permit the fixing on the experimental equipment: for the purpose an interference fit between the hole and the piston has been provided by designing the fit as ∅6p6/M13 according to the standard nomenclature [39], to the FDM part standard tolerance grade [40] and FDM fundamental deviation derived from [38]. The specimens were fabricated by a Stratasys Dimension bst 768 employing ABS-P400 material. The fabrication parameters were: 0.254 mm layer thickness; solid interior model; break away support generation strategy. The fabricated specimens are reported in Figure 1b. The experimentation has been carried out on this industrial FDM machine with this material since in [34, 38] it has been demonstrated that changing materials (ABS P400, ABSPlus, ULTEM 9085, polycarbonate) and machines (Stratasys Dimension bst 768, Fortus, 360, Fortus 400, Fortus 900) does not significantly modify the surface morphologies. Conversely the deposition angle is the main process parameter that affects the roughness and the dimensional accuracy. In order to investigate the effects of manufacturing issues, a dimensional and a surface characterizations were provided. The former measures the deviation of a workpiece from the intended shape, the latter allows to detect the irregularities at small wavelength resulting from the manufacturing processes [41]. The dimensional characterization was undertaken on the four corners and on the center of the coupling faces before the assembly operation by a standard outside micrometer. Each measurement is performed by placing the anvil and the spindle on the two opposite zones of the coupling faces: e.g. the first corner measurement 1 is taken on the top left and bottom left zones (Figure 2a). The micrometer is characterized by 0.6 μm maximum deviation from flatness and 2 μm maximum deviation from parallelism on anvil and spindle and an accuracy of ±2 μm. A ratchet stop supplying a constant force of 5N allowed to neglect the workpiece deformation since the investigated ABS P400 processed is characterized by an Young modulus ranging within 1,000–1,800 MPa as reported in [42, 43]. The surface characterization was provided by profilometer measurements and 3D maps: for the purpose a Taylor Hobson Talysurf Talyprofile Plus, with a vertical range and a vertical resolution of 2200 µm and 16 nm respectively, was used. According to [44] the roughness sampling length, namely the cutoff λc, and the evaluation length were set to 2.5 mm and 12.5 mm respectively. A 2.5 µm stylus, sampling every 1.5 µm, was employed with a measuring force of 4 mN. The acquired data were processed by a spline profile filter [45] with a short wavelength cut-off λs pair to 8 µm. The 3D maps 10x13mm2 were acquired through the same profilometer and no filter was applied. A flatbed Epson Perfection V850 Pro scanner was used to take optical 9600 dpi macrographies of the surface specimens. The assembly and disassembly forces were provided by an industrial-like hydraulic press instrumented with a position transducer and a piezoelectric charge PCB 201B05 load cell connected to a PCB 442A101 charge amplifier. The load cell could measure compression forces ranging from 45 N to 22 kN and its sensitivity was 224.8 mV/kN; a 10 kN preload was provided in order to measure both compression and tensile forces. The displacement and the force signals were acquired by a National Instruments PCI 6221 analog/digital system controlled in Labview 8.1 environment. The displacement speed and the sampling frequency were chosen at 1 mm/s and 1000 Hz respectively. The specimens were coupled with an AISI 4140 chrome molybdenum high tensile steel mold with coupling surfaces characterized by 1 µm average roughness: this way the mold deformation was negligible and the assembly force was mainly due to the coupling surfaces of the FDM parts since their roughness was much greater than the mold one [35]. A schematization of the fitting and the experimental equipment is reported in Figure 2b and c. The mold dimension has been tolerated in hole-basis fit system [39] assigning and verifying 20H5 which has been coupled with (20+i)m13 for the specimens [38, 40]. This way the mold limit deviations are negligible with respect to the FDM part ones. A CC E PT ED M A N U SC RI PT 3 RESULTS 3.1 Analysis of the assembly and disassembly forces The results of the dimensional characterization are shown in Table 1. It is well evident that the repetitions have very good repeatability with a Root Mean Square Error, calculated zone by zone, less than 0.03 mm in accordance with the FDM accuracy. Also the interference grade shows small errors and, as expected, it does not modify the measurements. Conversely the deposition angle deeply affects the deviations. These assessments are supported by an Analysis of Variance (ANoVA). The ANoVA is a statistical technique that tests if the means of two or more populations are equal: it evaluates the importance of one or more factors by comparing the response variable means at the different factor levels [46]. The null hypothesis states that all population means (factor level means) are equal while the alternative hypothesis states that at least one is different. The main output is typically arranged in a table that lists the sources of variation, their degrees of freedom (DF), the sum of squares (SS), and the mean squares (MS). The total sum of squares (SStotal) describes the total variability in the data which is partitioned into components attributable to the factors (SSfactor), their interaction (SSfactorA x factorB ) and the error (SSError). The effect of a factor on the response variable is called main effect calculated averaging across the levels of any other factors. An interaction occurs when the effect of a factor on the response variable changes depending upon the level of another factor. In order to determine whether factors or interactions are significant their variances are singularly compared with the error one. The variance is estimated by the mean squares (MS) calculated by dividing the sum of squares by the degrees of freedom which are the number of independent values free to vary. This ratio is a F-statistics which is 1 when null hypothesis is true and greater than 1 when alternative is true. The probability of observing a F-statistics as large or larger than the observed one is called p-value [47]. In this work by fixing a significance level of 0.05 a smaller value of p-value are evidence that null hypothesis is incorrect. In Table 2 the ANoVA table for the dimensional measurements is reported: the P-value for α is markedly less than 0.05 thus giving evidence that this factor is strongly significant; the P-values of the factor 𝑖 and the interaction are greater than reasonable choices of level of significance, indicating they do not modify the outcomes. The R2 and the adj R2 are very close and indicate that the predictors explain more than 97% of the variance in dimensional deviations. It is noteworthy that ANoVA test assumes that although different samples can come from populations with different means, they have the same variance [47]. In order to verify this hypothesis a test for equal variances has been undertaken. In Figure 3 the Bonferroni simultaneous confidence intervals for the standard deviations of each specimen measure have been reported. The intervals overlap for deposition angles in the range 3°-60°, conversely 90° population is notably different from the others: this is confirmed by the multiple comparisons and the Levene’s tests with P-values less than 0.05. By splitting data the tests for equal variances are satisfied and ANoVA results confirm previous conclusions. Investigating the face zones, the data deviation is less than 0.05 mm except for 90° which deviates more than 0.1 mm. Moreover a geometrical trend can be observed: the values uniformly increase from the left to the right evidencing that the face is inclined along this direction. This is a shape deviation which can affect the assembly behavior and must be taken into account. According to these findings the deposition angle is the most important geometrical factor since the accuracy is deeply affected by the building orientation. In Figure 4a the 3° deposition angle specimen is investigated by means of the surface topography. Since the profilometric measurements are related to the mean, in the shown graph the profile is moved along the z- A CC E PT ED M A N U SC RI PT axis in order to have the highest peaks at the position achieved by the dimensional characterization, i.e. at about 0.38 mm. In fact, since the micrometer anvil and spindle are bigger than the topographic features, only the highest peaks are detected by the contact. During the coupling such features play an important role as described in the following. In the Figure 4a the staircase effect is well evident: the horizontal flat zone is 4.83 mm in length and the step height is 0.254 mm in accordance with the processing parameters and the geometrical slope. Along the coupling direction (black arrow) a variable force is expected: at the beginning the mold encounters the first part end characterized by a peak height equal to 0.13 mm which gradually increases to the maximum deviation of about 0.38 mm. This morphology can explain the acquired force trend. In the case of 0.0 mm interference grade (Figure 4b), the trend initially shows a quick increase probably due to the entry of the specimen border edge in the mold; then it starts to have a linear growth for about 5 mm of the piston displacement in accordance with the abovementioned surface morphology. Subsequently the step is pushed in the mold and the assembly force markedly increases up to 600 N, then the force trend returns to the previous linear growth reaching a value below 400 N. In the case of 0.1 mm interference grade, the surface morphology is the same but the range of the dimensional deviation is greater than the previous case. The force behavior (Figure 4c) is also in this case characterized by two peaks spaced by about 5 mm: the former is quadruple in value (about 800 N); the latter is about the same in height and wider with respect to 0.0 mm case, probably due to the plastic deformation. When 0.2 mm interference grade is set, the force trend becomes chaotic (Figure 4d): since the real interference is pronounced a chipping of the filaments occurs. In Figure 5 the macrographs of the surfaces before and after the assembly have been reported. The specimen before assembly (Figure 5a) is very close to the one assembled with 0.0 mm interference grade (Figure 5b) with the exception of the central filament width (indicated by the red arrows) and its neighboring filaments: the assembly operation crushed this highest zone leading to an observable widening. Conversely for 0.1 mm interference grade (Figure 5c) some marked damages can be observed: some filaments (blue arrows) moved by some millimeters affecting the external contour as indicated by the red arrows. In the case 0.2 mm interference grade (Figure 5d) a number of defects occur. The internal damages are extended all over the surface (blue arrows); the central filament has been chipped away (black arrows); the entire layer slid by 1 mm uncovering the underlying one characterized by a rotated filling strategy as indicated by the yellow arrows. A lot of chipped filaments have been moved in the assembly direction heaping material near the border (red arrows). In the cases of 9° deposition angle, the coupling surfaces are characterized by the morphology shown in Figure 6a. No planarity loss is present in the measured area and a homogeneous texture is observed. The behavior is periodical with a double-peak period, as well evident from the profile reported in Figure 6b: the highest peak is related to the contour and the lowest one to the filling. This morphology affects the coupling force behavior reported in Figure 6c. A growing trend with a sawtooth pattern is observed: the period is 1.65 mm very close to the measured profile spacing of 1.67 mm, so it can be assessed that the coupling with each peak causes this waviness. The observed pattern amplitude is 250 N mainly due to the deformation of the highest profile peak that is characterized by a width of 855 µm in the bending direction and a height of 400 µm. The slight growing of the mean trend can be explained by the increasing of the contact surface as the coupling proceeds. When the interference grade is set to higher values (Figure 6d for 0.1 mm and Figure 6e for 0.2 mm) an expected increase of the maximum force is observed: notwithstanding a more evident scattering in shape, similar results in term of spacing are obtained. An interesting confirmation of the microgeometry effect on the assembly dynamics is the behavior measured in the range of coupling distance between -4 mm to 0 mm shown in Figure 6e: the observed morphology is even more in accordance with the profile shape. A CC E PT ED M A N U SC RI PT In Figure 7 the specimen characterized by a deposition angle of 15° is considered. This case shows similar features with respect to the previous one: the periodical double-peak morphology with a profile height of 0.4 mm is present (Figure 7a). The measured spacing is 972 µm in accordance with the theoretical one (981 µm) and the highest peak width is 536 µm thus a smaller amplitude force waviness is expected notwithstanding the average roughness is close to the previous case. The Figure 7b shows the trend of the assembly force for 0 mm interference grade: the amplitude is 170 N and the spacing is 964 µm strengthening the previous considerations. The mean trend is slowly growing while for 0.1 and 0.2 mm interference grade a steeper increase is observed at the beginning (Figure 7c and 7d). The maximum assembly force increases by about 100 N for each 0.1 mm grade. For the surface related to 30° deposition angle, the profile is no longer characterized by the double-peak period: this slope is enough to allow the covering of the filling filaments. Here a spacing and peak to valley height lower than the previous analyzed cases lead to a force trend characterized by a little amplitude sawtooth pattern (Figure 8a). The maximum force increases with the interference grade: little increments are observed for 0.0 mm and 0.1 mm interference grades (Figure 8b and Figure 8c) and a marked increase for 0.2 mm one (Figure 8d). The case of 60° deposition angle shows results in accordance with the previous considerations. The profile in Figure 9a is characterized by a spacing of 290 µm and a height less than 10 µm; as evidenced by the force trends (Figure 9b, 9c and 9d) the sawtooth pattern is no longer visible since the macrogeometry is reduced. The maximum assembly forces are 280 N, 380 N and 490 N at the investigated interference grades: they exhibit a linear trend starting from an initial value presumably due to the dimensional deviation from the nominal value pair to 0.1 mm. The results for 90° deposition angle do not follow the considerations made for the previous cases. In particular in Figure 10a, 10b and 10c the force behavior is very close to the 60° case and the maximum force values for the different interference grades (275N, 370N and 485N) underline this likeness. The 3D map undertaken over the 90° deposition angle surface (Figure 10d) shows a regular texture with filaments that lie on a plane inclined with respect to the opposite face of the specimen. If a section is considered (Figure 10e), a peak height variation of 0.1 mm is measured. This value is very close to the dimensional deviation of the 60° specimen thus explaining the force similarity. The spacing of this profile is exactly equal to the layer thickness as expected for vertical wall fabrication, too small to provoke a force waviness. For the disassembly forces the effects of the deposition angle and the dimensional deviation are not meaningful. This is due to the plasticization and the chipping as highlighted before. The 3° deposition angle case (Figure 11a) is characterized by decreasing trends: the curves for the different interference grades are very close and intersecting; as a result the maximum forces are marginally different. The same behavior can be observed for 9° specimens as shown in Figure 11b: the sawtooth pattern observed at the correspondent assembly stage is not present. Conversely the 15° and 30° disassembly cases present a behavior according to the assembly ones; in fact the measured force spacings are 967 µm and 498 µm respectively (Figure 11c and 11d), very close to the geometrical profiles and assembly force spacings. As the deposition angle increases the interference grade begins to affect the disassembly forces, as it is well evident in Figure 11e and 11f. These surfaces are characterized by small dimensional deviations thus only a little plasticization is expected. In fact the maximum disassembly forces of 0.0 mm interference grade are close to the assembly ones confirming this hypothesis; as the interference grade increases, the plasticization reduces the maximum disassembly forces. It is noteworthy to consider that the measured assembly and disassembly forces are smaller than the corresponding forces expected for an identical joint made by injection molding. According to the press fit theory [9, 10] the expected assembly forces in a smooth dry contact coupling are more than twice the FDM case. This is explained by FDM mesostructure and surface morphology. Since FDM is a filament based technology the presence of voids and fiber layout modify the load response of the material [24]. Depending upon the toolpath strategies so called aperture errors can be observed on the single slice [48]. Finally the surface morphology is characterized by peak to valleys on the same order of the interference grade thus deeply reducing the bearing ratio of the profile [31]. The disassembly forces in injection molded part are expected to be on the same order of the assembly one [10]. In the case of FDM fit this is true until the chipping and plastic deformation occur. A CC E PT ED M A N U SC RI PT 3.2 Model development The maximum assembly and disassembly forces are interesting coupling aspects to predict: the former allows knowing the force needed to provide the interference, the latter can be related to the joint resistance, giving important information for the interference fit design. The maximum force values are shown in Figure 12. The assembly outcomes are markedly influenced by the interference grade and the deposition angle according to the previous observations. The maximum forces are detected at low deposition angles because the dimensional deviation is high at these slopes. A nonlinear decreasing is observed as the angle increases. Conversely the interference grade seems to affect the force in a linear way. No trend is evident in the disassembly force with exception of the 60° and 90° deposition angle specimens as previously indicated. As a first approximation it can be assessed that the disassembly forces are almost constant: at low and medium deposition angles most of the assembly force is lost in plasticization and chipping returning about 350 N; at high deposition angles a linear dependency upon the interference grade is observed. In order to measure how the investigated factors affect the maximum forces an ANoVA has been carried out. In Table 3 the results are displayed. There is significant evidence for judging that the assembly force is affected by both the factors since very low P-values (< 0.05) are obtained. A slightly significant effect is present for the interaction. The R2 and the adj R2 claim that more than 96% of the assembly forces are explained by the factors. The ANoVA carried out on the disassembly forces leads to the results shown in Table 4. The F-test gives Pvalues more than 0.05 thus indicating that no factor is significant. The low values of the R2 and the adj R2 confirm that the factors do not acceptably explain the output. The low total variability (68457 with respect to the previous one of 1614955) and the test results are due to the plasticization and the chipping. The interference grade P-value is near to 0.05 thus indicating that some levels could be significantly different: carrying out the analysis in pairs, only 60° and 90° deposition angles are evidently different by means of the interference grade but not different in terms of deposition angle confirming the previous hypothesis. Since the assembly forces are dependent upon the investigated factors, a model can be developed in order to know in advance the force needed to provide the coupling. The FDM dimensional deviation can be predicted by the following model [38]: ∆ℎ(𝛼) = 0.38 𝑐𝑜𝑠 2 (𝛼) (1) where α is the deposition angle. A new variable namely the real interference 𝑖𝑟 can be defined as the sum of this deviation and the assigned interference grade 𝑖 at design stage: (2) 𝑖𝑟 = 0.38 𝑐𝑜𝑠 2 (𝛼) + 𝑖 From a theoretical point of view a linear relationship between the assembly force and the interference can be supposed [49]. Hence a simple linear regression as a function of the real interference is proposed: 𝐹𝑎 = 𝛽0 + 𝛽1 𝑖𝑟 + 𝜀 = 𝛽0 + 𝛽1 (0.38 𝑐𝑜𝑠 2 (𝛼) + 𝑖) + 𝜀 (3) where 𝛽0 , 𝛽1 are the regression coefficients and ε is the random error term with mean zero and variance 𝜎 2 [50]. In Figure 13 the maximum forces are represented by means of the average and standard deviation values as a function of 𝑖𝑟 . The disassembly forces show slight increase for low real interference grades which are ED M A N U SC RI PT related to 60° and 90° deposition angle specimens. Conversely for high values of 𝑖𝑟 , a confusing trend can be observed. Performing the ANoVA test versus this variable there is no significant evidence that the output changes over 𝑖𝑟 (probability value = 0.136). Thus the disassembly forces can be considered constant: a mean value of 308N and a standard deviation of 44N can be assessed. In the assembly case at 60° and 90° deposition angles two distinct trends can be observed. As the interference grade increases, the trends become intricate similarly to the disassembly case. Nevertheless the overall trend appears to be linearly increasing. In order to investigate this behavior a linear regression analysis has been provided according to the model of Eq.3. In Table 5 the output of this analysis has been reported. The obtained equation is: 𝐹𝑎 = 186.37 + 1196.7𝑖𝑟 (4) The ANoVA carried on the regression reveals that the relationship between the assembly force and the 𝑖𝑟 is statistically significant. The chosen predictor is meaningful for the model and the R2 and the adj R2 claim that more than the 89% of the variation in assembly force can be explained by the regression model. The predicted R2 indicates that the model is highly capable of providing valid predictions for new observations: as this value is calculated by systematically removing each observation from the data set and determining how well the newly regressed model predicts the removed observation, the obtained value very close to the regular R2 proves that the data were not overfitted. Two data points are unusual observations: they are related to 90° deposition angle at 0.1 mm interference grade and 3° deposition angle at 0 mm interference grade which are characterized by quasi elastic coupling and plasticization respectively. This last observation is characterized by the greatest deviation (146.4 N) which corresponds to a maximum percentage error of 29.7%. The mean absolute error (MAE) and the mean absolute percentage error (MAPE) are 57.9N and 11.2% respectively highlighting a good quality of the regression model in view of the fact the predicted forces are affected by a number of phenomena arising from microgeometry and mesostructure. Since R2 does not indicate the goodness-of-fit a residual analysis has been carried out. In Figure 14a the data, the regression model and the 95% prediction bands are reported. The equations of upper bound and lower bound are: 𝑢𝑏 = 𝐹𝑎 + 1.8708√5627.7 − 3347.4𝑖𝑟 + 4818.3𝑖𝑟 2 (6) PT 𝑙𝑏 = 𝐹𝑎 − 1.8708√5627.7 − 3347.4𝑖𝑟 + 4818.3𝑖𝑟 2 (5) A CC E The graph highlights that the most of the points lie within 95% confidence bands. Three points are out or very close to these limits: the first two are the ones indicated by the unusual observations; the third is the result at 3° and 0.2 mm interference grade which corresponds to the maximum investigated real interference and a large scattering can be expected for this extreme value. By graphing the quantilequantile plot of residuals (Figure 14b) the distribution of the points is close to the normal line (in red) thus indicating a non-deterministic behavior. This is confirmed by the Anderson-Darling normality test that gives a probability value much more than 0.05 supporting the assessment that the error term is normally distributed and the model has a good fit of the data. Conclusions In this paper an investigation about the interference fit joint of FDM parts is reported. The experimentation highlighted that the coupling behavior is deeply affected by the surface morphology showing equal spacing and correspondent shape between profiles and force trends. The maximum assembly forces are significantly dependent upon the deposition angle and the interference grade as confirmed by the statistical analysis. This information allowed the introduction of a new variable namely the real interference (that considers together the designed interference grade and the deviation introduced by the technology) PT permitting to find a linear relationship between assembly forces and this parameter: this model can help the joint design giving predictions for the contributions of the assigned interference grade and the deviation caused by the FDM fabrication which is affected by the part orientation. The assembly operation is observed to damage the coupling surface in the most of the cases: chipping and plastic deformation occur especially at low deposition angles. As a consequence the disassembly forces are lower than the assembly ones and they do not depend upon the deposition angle because the surface morphology is deeply modified during the assembly step. This gives information about the joint resistance and can be used for the design of the interference fit. Further investigations will regard the extension of this methodology to other joint shape, coupling length, materials and machines. A CC E PT ED M A N U SC RI Acknowledgements The authors wish to thank Prof. F. Veniali for his precious guide and Eng. Daniele Viennese for the help in the experimental campaign. A CC E PT ED M A N U SC RI PT References [1] I. Campbell, D. Bourell, I. Gibson, Additive manufacturing: rapid prototyping comes of age, Rapid Prototyping Journal 18: 4 (2012) 255-258. [2] C.K. Chua, K.F. Leong, C.S. Lim, Rapid Prototyping: Principles and Applications, World Scientific, River Edge, NJ, 2010. [3] Y. Tang, , S. Yang, Y.F. 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Bonferroni simultaneous confidence intervals for the standard deviations with 95% confidence level. PT RI SC A CC E PT ED M A N U Figure 4. 3D map related to 3° deposition angle (a); force trends for specimens characterized by 3° deposition angle and 0.0 mm (b), 0.1 mm (c), 0.2 mm (d) interference grade Figure 5. Macrographs of the surface related to 3° deposition angle before (a) and after the assembly operation (b, c, d) PT RI SC U A CC E PT ED M A N Figure 6. 3D map and profile related to 9° deposition angle (a, b); force trends for specimens characterized by 9° deposition angle and 0.0 mm (c), 0.1 mm (d), 0.2 mm (e) interference grade Figure 7. Surface profile for 15° deposition angle (a); force trends for specimens characterized by 15° deposition angle and 0.0 mm (b), 0.1 mm (c), 0.2 mm (d) interference grade PT RI SC CC E PT ED M A N U Figure 8. Surface profile for 30° deposition angle (a); force trends for specimens characterized by 30° deposition angle and 0.0 mm (b), 0.1 mm (c), 0.2 mm (d) interference grade A Figure 9. Surface profile for 60° deposition angle (a); force trends for specimens characterized by 60° deposition angle and 0.0 mm (b), 0.1 mm (c), 0.2 mm (d) interference grade PT RI SC A CC E PT ED M A N U Figure 10. Force trends for specimens characterized by 90° deposition angle and 0.0 mm (a), 0.1 mm (b), 0.2 mm (c) interference grade; 3D map (d) and profile (e) related to 90° surface Figure 11. Disassembly trends force for different deposition angles PT RI SC PT ED M A N U Figure 12. Assembly and disassembly maximum forces for investigated deposition angles and interference grade A CC E Figure 13. Maximum assembly and disassembly forces as a function of the real interference. PT RI SC A CC E PT ED M A N U Figure 14. Scatter diagram and linear regression of the assembly force as function of the real interference grade (a); quantile-quantile plot of residuals (b) Table 1. Dimensional deviations (mm) from the nominal value taken on the four corners and on the center of the coupling faces for the investigated interference grades and deposition angles. Deposition angle [°] 3° 9° 0.38 0.4 0.36 0.37 0.33 0.11 0.31 0.08 -0.06 0.09 0.03 -0.02 0.37 0.34 0.37 0.36 0.29 0.32 0.08 0.07 -0.07 0.04 0.37 0.39 0.37 0.38 0.36 0.34 0.33 0.35 0.08 0.09 -0.04 0.04 0.36 0.34 0.39 0.35 0.35 0.34 0.35 0.35 0.35 0.38 0.36 0.39 0.35 0.36 0.32 0.32 0.34 0.29 0.35 0.32 0.36 0.34 0.32 0.37 0.11 0.3 0.07 0.33 0.07 0.31 0.32 0.32 0.35 0.34 0.34 0.31 0.01 0.12 0.11 0.1 0.33 0.08 0.34 0.09 -0.05 PT 0.35 0.32 -0.03 0.05 -0.01 0.03 RI 0.35 0.36 0.07 -0.07 0.13 -0.04 0.12 SC 0.35 0.37 0.04 -0.03 -0.01 0.39 0.36 0.33 0.37 0.37 0.33 0.3 0.33 0.08 0.11 -0.08 0.02 0.35 0.37 0.36 0.37 0.34 0.34 0.34 0.32 0.1 0.11 -0.05 0 0.36 0.34 0.35 0.35 0.32 0.29 0.09 -0.08 0.37 0.35 0.38 0.35 0.1 -0.04 0.35 0.37 0.4 0.36 0.38 0.34 0.36 0.36 0.37 0.33 0.32 0.33 CC E PT ED M 0.39 0.35 U 0.38 N 0.38 A 0.3 0.35 90° 0.36 0.37 Interference grade [mm] 0.33 60° 0.37 0.39 0.2 0.34 0.36 0.39 0.1 0.35 30° 0.29 A 0.0 15° 0.3 0.12 0.33 0.08 0.34 0.09 0.33 -0.02 0.09 0.32 0.12 0.03 0.03 0.01 0.09 -0.06 -0.01 Table 2. ANoVA analysis for dimensional deviations versus deposition angle, interference grade and replication. Factor α Table 3. ANoVA for assembly force A CC E PT ED M A N U PT SC Factor Type Levels Values α fixed 6 3; 9; 15; 30; 60; 90 i fixed 3 0.0; 0.1; 0.2 Source DF SS MS F α 5 958890 191778 105.13 i 2 578344 289172 158.52 α*i 10 44885 4489 2.46 Error 18 32836 1824 Total 35 1614955 S = 42.7108 R-Sq = 97.97% R-Sq(adj) = 96.05% P 0.000 0.899 0.970 RI Type Levels Values fixed 6 3; 9; 15; 30; 60; 90 fixed 3 0.0; 0.1; 0.2 𝑖 Source DF SS MS F α 5 4.03285 0.80657 1602.26 2 0.00011 0.00005 0.11 𝑖 10 0.00170 0.00017 0.34 α* 𝑖 Error 162 0.08155 0.00050 Total 179 4.11621 S = 0.0224365 R-Sq = 98.02% R-Sq(adj) = 97.81% P 0.000 0.000 0.046 Table 4. ANoVA for disassembly force P 0.199 0.065 0.244 A CC E PT ED M A N U SC RI PT Factor Type Levels Values α fixed 6 3; 9; 15; 30; 60; 90 i fixed 3 0.0; 0.1; 0.2 Source DF SS MS F α 5 12010 2402 1.65 i 2 9315 4658 3.19 α*i 10 20877 2088 1.43 Error 18 26255 1459 Total 35 68457 S = 38.1919 R-Sq = 61.65% R-Sq(adj) = 25.43% Table 5. Regression report for assembly force as a function of the real interference grade F 297.23 297.23 P 0.000 0.000 RI Std Resid -2.12 R 2.02 R PT P 0.000 0.000 Pred R-Sq=88.45% A CC E PT ED M A N U SC Analysis of Variance Source DF Adj SS Adj MS Regression 1 1459381 1459381 Ir 1 1459381 1459381 Error 34 166938 4910 Total 45 1626319 Model Summary Coefficients Term Coef SE Coef T Constant 186.4 26.8 6.96 Ir 1196.7 69.4 17.24 S = 70.071 R-Sq = 89.74% R-Sq(adj) = 89.43% Fits and Diagnostics for Unusual Observations Obs Fa Fit Resid 2 493.5 639.9 -146.4 34 441.5 306.0 135.4 R Large residual