Analytical and Simulation-Based Prediction of Surface Roughness for Micromilling Hardened HSS

Journal of Manufacturing and Materials Processing

Article Analytical and Simulation-Based Prediction of Surface Roughness for Micromilling Hardened HSS

Alexander Meijer 1,*, Jim A. Bergmann 2 , Eugen Krebs 1, Dirk Biermann 2 and Petra Wiederkehr 1,2

1 Institute of Machining Technology, TU Dortmund University, 44227 Dortmund, Germany 2 Virtual Machining, Chair for Software Engineering, TU Dortmund University, 44227 Dortmund, Germany * Correspondence: [email protected]; Tel.: +49-231-755-5820

Received: 28 June 2019; Accepted: 8 August 2019; Published: 12 August 2019

Abstract: The high quality demand for machined functional surfaces of forming tools, entail extensive investigations for the adjustment of the manufacturing process. Since the surface quality depends on a multitude of inﬂuencing factors in face micromilling, a complex optimization problem arises. Through analytical and simulative approaches, the scope of the experimental investigation to meet the requirements for surface roughness can be signiﬁcantly reduced. In this contribution, both analytical and simulation-based approaches are presented in the context of predicting the roughness of a machined surface. The consideration of actual tool geometry and shape deviations are used in a simulation system to achieve the agreement with experimental results.

Keywords: surface roughness prediction; geometrical simulation; micromilling

1. Introduction Surface roughness is an important quality criterion of functional components [1]. To achieve low roughness values in manufacturing process chains, finishing operations can be used. Thereby, additional manufacturing steps, like polishing, are often costly and still need to be carried out manually. To design an optimized manufacturing line, the resulting roughness values of each process have to be known. However, due to the dependence on a high number of influencing factors in each machining step, the prediction of these values is challenging. Cutting parameters, choice of material, process dynamics and cutting edge geometry are of decisive importance. This is valid for both the geometrically undefined as well as the geometrically defined cutting edges [2]. First approaches for predicting roughness values were developed for turning processes, by considering the engagement situation and the tool geometry in an analytical model. One approach to calculate the theoretical peak-to-valley roughness Rth as r 2 2 f Rth = rε r (1) − ε − 4 was presented by Bauer [3], which takes the corner radius rε and the feed rate f into account. In literature, however, a more frequently applied equation

f 2 Rth = (2) 8 rε ·

J. Manuf. Mater. Process. 2019, 3, 70; doi:10.3390/jmmp3030070 www.mdpi.com/journal/jmmp J. Manuf. Mater. Process. 2019, 3, 70 2 of 17 by Shaw [4] can be found. Martelotti [5] extended this mathematical relationship, by considering the process specific feed per tooth f z, as well as properties of the tool, such as corner radius rε and number of teeth z, as 2 fz Rth = , (3) fz z 8 r · · − π to calculate Rth for milling processes. Although the occurrence of ploughing effects, tool deviations or geometric properties of the cutting edge are not considered, these equations already provide an approximation of the theoretical roughness depth. For the calculation of the theoretical roughness depth in a face circumferential milling processes, the secondary cutting edges need to be taken into account. Since the resulting machining grooves are influenced by the condition and orientation of the secondary cutting edge, it should be considered for a well-founded prediction of the theoretical roughness depth. In order to gain a more detailed understanding of the generation of a workpiece surface, predictions of topographies for complete parts are preferable. A prediction for a ball-end milling process was presented in the approach of Layegh and Lazoglu [6]. Their model considers the trochoidal motion path of the cutting edge in dependency of the feed per tooth f z, the engagement conditions ap and ae, the tool orientation and the runout, which enables the calculation of a three-dimensional topography. In a face milling process using an end-milling cutter with corner radius, the geometric dependencies are much more complex. The surface topography is not defined by a single cut, but results from a superposition of a large number of cutting motions. Hadad and Ramezani [7] showed that different surface topographies could be generated by changed tool inclinations in face milling processes. Freiburg and Biermann [8] presented an analytical model for high feed milling that considers this relationship. Due to the special geometry of the tools, the transferability to conventional processes is limited [9]. Further approaches can be found in the field of geometric simulations. Kundrák and Felho used a CAD system to determine the surface roughness when using face-milling tools with special cutting inserts [10]. However, this approach does not take the microgeometric properties of the tool, i.e., the actual condition and deviation of the ideal shape of the cutting edge, or trochoidal motion paths of the cutting edges into account [11]. Denkena and Böß were pursuing a similar approach applying a process simulation system, in which CAD data of the tool shape can be respected. When mapping the tool shape to the surface topography, only the tool shell is considered [12]. Lavernhe et al. illustrated the influence of microscopic defects at the cutting edge on the generation of surface topographies [13]. The validation was performed for a ball-end milling tool. Denkena et al. combined a material removal simulation with an empirical model to incorporate stochastic variations regarding the machined workpiece material and, thus, achieved a detailed prediction of the surface quality [14]. In summary, promising approaches have been developed for predicting the surface roughness in face milling processes. This was achieved primarily through geometric simulation approaches. However, promising analytical approaches could be identified for special cases. In addition, the studies showed, that the level of detail is essential for a highly valid prediction. In particular, deviations from ideal cutting edges regarding its own macro- and microgeometric properties must be considered. This aspect is gaining importance to ensure predictions for functional surfaces with low roughness values. In this paper, both an analytical and a geometric simulation-based method are presented for the prediction of the surface roughness resulting from face micromilling processes. The analytical model allows a fast but idealized determination of the roughness profile considering special aspects of the face milling process. The geometric simulation approach enables a realistic prediction of a three-dimensional surface topography considering complex engagement conditions and influencing variables like a detailed tool model considering microgeometric properties of the cutting edge, runout, axial offset and specific variations of the process parameter values. In addition, the simulation is used to investigate and visualize the observed effects resulting from the face milling process. J. Manuf. Mater. Process. 2019, 3, 70 3 of 17 J. Manuf. Mater. Process. 2019, 3, x FOR PEER REVIEW 3 of 17

2.2. Experimental Experimental Setup Setup InIn the the following, following, the the experimental experimental setup setup and and boundary boundary conditions conditions of of the the investigation investigation are are introduced.introduced. In In addition addition to a to presentation a presentation of the of machine the machine tool, the tool, cutting the tools,cutting and tools, measuring and measuring systems, thesystems, sample the material sample is brieﬂymaterial characterized. is briefly characterized.

2.1.2.1. Face Face Micromilling Micromilling Process Process TheThe milling milling experiments experiments werewere carriedcarried outout on a machining machining center center Kern Kern HSPC HSPC 2522. 2522. Due Due to tothe themineral mineral cast cast construction construction of ofthe the machine machine stand stand and and the the installation installation in an in anair-conditioned air-conditioned room room on a onpolymer a polymer concrete concrete foundation, foundation, a high-precision, a high-precision, vibration-reduced vibration-reduced process process could could be ensured. be ensured. The Themachining machining process process and and the the basic basic process process pa parametersrameters are are presented presented in inFigure Figure 1.1 .

FigureFigure 1. 1.Schematic Schematic visualization visualization of of a facea face micromilling micromilling process. process.

TheThe cutting cutting tests tests were were conducted conducted using using a microa micro end-mill end-mill with with a diametera diameter of ofd =d 1= mm1 mm and and two two solidsolid CBN CBN (Cubic (Cubic Boron Boron Nitride)Nitride) cutting edges, edges, a acorner corner radius radius of ofrε =r ε0.02= 0.02mm,mm, and an and average an average cutting cutting edge rounding of S = 0.97 0.16 µm. Compared to solid carbide tools, CBN tools have a edge rounding of 𝑆 = 0.97 ± 0.16± µm. Compared to solid carbide tools, CBN tools have a higher higherresistance resistance to wear, to wear, which which limits limits the time the time variance variance on the on determined the determined surface surface characteristics characteristics [15]. [15 The]. Thetool tool was was selected selected taking taking the the wear-related wear-related shape shape chan changege and and its itseffects eﬀects on onthe the surface surface topography topography into intoaccount. account. Thus, Thus, the the shape shape of of the the tool tool could could be be assu assumedmed to bebe constantconstant forfor each each milling milling experiment experiment withinwithin the the scope scope of of this this paper. paper. A toolA tool with with sharp shar cuttingp cutting edges edges was was chosen chosen to reduce to reduce the occurrencethe occurrence of plastic deformation eﬀects in front of the cutting edge [11]. Cutting speeds were set to v = 120 m/min of plastic deformation effects in front of the cutting edge [11]. Cutting speeds werec set to vc = 120 at a width of cut of a 0.1 mm and a cutting depth of a 0.01 mm. m/min at a width ofe =cut of ae = 0.1 mm and a cutting depthp = of ap = 0.01 mm.

2.2.2.2. Workpiece Workpiece Material—1.3344 Material—1.3344/AISI/AISI M3:2 M3:2/S/S 6-5-3 6-5-3 TheThe tool tool steel steel 1.3344 1.3344 (AISI (AISI M3:2) M3:2 was) was used used as as sample sample material material for for the the presented presented investigations. investigations. ThisThis steel steel is a cobalt-free,is a cobalt-free, powder-metallurgically powder-metallurgically produced produced high-speed high-speed steel (HSS), steel which (HSS), was which hardened was to 63 1 HRC. Due to its high strength, and in particular its transverse strength, this high-performance hardened± to 63 ± 1 HRC. Due to its high strength, and in particular its transverse strength, this high- high-speedperformance steel high-speed is preferably steel used is preferably for the manufacturing used for the of manufacturing highly stress loaded of highly dies stress andmolds loaded [16 dies]. Therefore,and molds the [16]. material Therefore, is ideally the suitedmaterial for is the ideally production suited offor cold the workproduction tools such of cold as puncheswork tools and such dies, as toolspunches for cold and extrusion, dies, tools deep for drawing, cold extrusion, sintering deep tools, dr asawing, well assintering conventional tools, cuttingas well toolsas conventional with high demandscutting tools on tool with life. high Due demands to the high on tool strength life. Due and to hardness the high of strength the material, and hardness a lower proportionof the material, of plastica lower deformation proportion occurs of plastic during deformation the machining occurs process during compared the machining to non-hardened process compared steels [17 to] and, non- therefore,hardened a highsteels degree [17] and, of shape therefore, accuracy a high can degr be assumedee of shape regarding accuracy the can geometric be assumed engagement regarding of the the toolgeometric [18]. engagement of the tool [18]. 2.3. Measuring Systems 2.3. Measuring Systems In order to characterize the condition of micro end-mills, the scanning electron microscope Mira3 In order to characterize the condition of micro end-mills, the scanning electron microscope Mira3 from Tescan and the strip projection microscope MicroCAD plus from LMI was used. The latter from Tescan and the strip projection microscope MicroCAD plus from LMI was used. The latter enabled detailed recordings of the cutting edge microgeometry. The determined characteristic enabled detailed recordings of the cutting edge microgeometry. The determined characteristic values values of the microgeometry were then utilized to model the tools in the simulation system used. of the microgeometry were then utilized to model the tools in the simulation system used. The machined surface topography was measured using the confocal white-light microscope µsurf from

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The machined surface topography was measured using the confocal white-light microscope µsurf fromJ. Nanofocus. Manuf. Mater. Process. Analyses 2019, 3, ofx FOR measured PEER REVIEW surface topographies were conducted using the metrology 4 of 17 software MountainsMap Premium 7.4 from Digital Surf. A cut-oﬀ wavelength of λ = 0.25 was selected Nanofocus. Analyses of measured surface topographies were conducted usingc the metrology to determine the surface characteristics of the measured and simulated surfaces. Surface proﬁles were software MountainsMap Premium 7.4 from Digital Surf. A cut-off wavelength of λc = 0.25 was extractedselected from to measurementsdetermine the surface in feed characteristics direction. of the measured and simulated surfaces. Surface profiles were extracted from measurements in feed direction. 3. A New Analytical Approach for Surface Roughness Determination In3. A the New following Analytical section, Approach an analytical for Surface model Roughness for describing Determination the theoretical roughness depth Rth in a circumferentialIn the following face milling section, process an analytical is presented. model for As describing described the in theoretical Equations roughness (1)–(3), thedepth theoretical Rth roughnessJ.in Manuf. a circumferential depth Mater. Process. can be2019 calculatedface, 3, x FORmilling PEER based processREVIEW on is a two-dimensionalpresented. As described cross-section in Equations of the (1)–(3), geometric 4 ofthe 17 tool theoretical roughness depth can be calculated based on a two-dimensional cross-section of the engagementNanofocus. and, Analyses thus, indicatesof measured the highestsurface topographies expected roughness were conducted depth alongusing thethe feedmetrology direction. In additiongeometric to thetool processengagement parameter and, thus, values indicates used th ine highest Equation expected (1), the roughness angle ψ depthof the along face-sided the feed minor softwaredirection. MountainsMapIn addition to the Premium process parame7.4 fromter Digital values Surf.used Ain Equationcut-off wavelength (1), the angle of λ𝜓c of= 0.25the face-was cutting edge (see Figure2c) is taken into account, which plays a decisive role for the resulting surface selectedsided minor to determine cutting edge the (seesurface Figure characteristics 2c) is taken of into the account, measured which and playssimulated a decisive surfaces. role Surfacefor the topographyprofilesresulting inwere surface a circumferential extracted topography from measurements in face a circumferential milling process.in feed face direction. milling process.

3. A New Analytical Approach for Surface Roughness Determination In the following section, an analytical model for describing the theoretical roughness depth Rth in a circumferential face milling process is presented. As described in Equations (1)–(3), the theoretical roughness depth can be calculated based on a two-dimensional cross-section of the geometric tool engagement and, thus, indicates the highest expected roughness depth along the feed direction. In addition to the process parameter values used in Equation (1), the angle 𝜓 of the face- sided minor cutting edge (see Figure 2c) is taken into account, which plays a decisive role for the resulting surface topography in a circumferential face milling process. FigureFigure 2. Schematic 2. Schematic view view on on (a )(a a) a two two dimensional dimensional en engagementgagement situation situation during during milling, milling, (b) two (b) two consecutive tooth feeds, (c) single milling marks with influencing parameters. consecutive tooth feeds, (c) single milling marks with inﬂuencing parameters.

The macroscopicThe macroscopic shape shape of of the the tool tool and and thethe chosenchosen process process parameters parameters determine determine the geometric the geometric cutter-workpiece-engagement and, thus, induce the resulting milling marks, as illustrated in Figure cutter-workpiece-engagement and, thus, induce the resulting milling marks, as illustrated in Figure2. 2. The depth of these marks depends on the feed per tooth fz, the corner radius rε and the angle of the f r The depthminor ofcutting these edge marks ψ. depends While the on feed the feedper pertooth tooth definesz, thethe cornerwidth radiusof the millingε and themarks, angle the of the minorinterrelation cutting edge of theψ. Whilecorner radius the feed and per the tooth angle deﬁnes of the minor the width cutting of edge the determine milling marks, the vertical the interrelation depth. of the corner radius and the angle of the minor cutting edge determine the vertical depth. 𝑦= 𝑚⋅x+b where 𝑚=tan(ψ and 𝑏=0 (4) Figure 2. Schematic view on (a) a two dimensional engagement situation during milling, (b) two consecutive tooth feeds,y (c=) singlem x milling+ b wheremarks withm = influencingtan(ψ) and parameters.b = 0 (4) 𝑟 = (𝑥−𝑥· + (𝑦−𝑦 where 𝑥 = 𝑓 and 𝑟=𝑦 =𝑟 (5) 2 2 2 The macroscopicr = shape(x x mof) the+ tool(y andym the) choswhereen processxm = f zparametersand r = y determinem = rε the geometric (5) − − cutter-workpiece-engagement and, thus, induce the resulting milling marks, as illustrated in Figure Based on the cross-section of the machining mark shown in Figure2, an approach for calculating 2. The depth of these marks depends on the feed per tooth fz, the corner radius rε and the angle of the the maximumminor cutting depth edge of theψ. markWhile was the derived,feed per usingtooth thedefines combination the width of of a linearthe milling and circular marks, equationthe (see Figureinterrelation3). By of inserting the corner Equation radius and (4) the into angle Equation of the minor (6), parameters cutting edge candetermine be substituted the vertical as depth.

𝑦= 𝑚⋅x+b where! 2𝑚=tan(ψ and 𝑏=0 (4) 2 y 2 rε = fz + (y rε) . (6) ( ) − − 𝑟 = (𝑥−𝑥 + (𝑦−𝑦tanψ where 𝑥 = 𝑓 and 𝑟=𝑦 =𝑟 (5)

Figure 3. Illustration of the basic geometric relationship of a machining mark in context of tool shape.

Based on the cross-section of the machining mark shown in Figure 2, an approach for calculating the maximum depth of the mark was derived, using the combination of a linear and circular equation (see Figure 3). By inserting Equation (4) into Equation (6), parameters can be substituted as

𝑟 = − 𝑓 + (𝑦−𝑟. (6) (

Figure Figure 3. Illustration 3. Illustration of the of the basic basic geometric geometric relationship relationship of of a amachining machining mark mark in context in context of tool of shape. tool shape.

𝑟 = − 𝑓 + (𝑦−𝑟. (6) (

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Since the maximum depth of the machining mark is reached at the point of intersection of both equations, y is equivalent to the theoretical roughness depth Rth. By solving for Rth, the theoretical roughness can be calculated as r 2 fz fz 1 2 2 rε + 2 rε + 4 1 + fz · tan(ψ) ± − · tan(ψ) − · tan2(ψ) · Rth = . (7) 2 1 + 1 · tan2(ψ) This simplified consideration of the process can be used to analyze the effects of the macroscopic shape for end milling tools without extensive experimental investigations (see Section5). In addition to conventional characterization parameters of the tool such as the diameter d, the corner radius rε and the number of cutting edges zn, the macroscopic shape also includes shape errors of the tool. These consist of the runout error and the axial offset of both minor cutting edges. While the former can either be caused by the clamping of the tool, the radial runout of the minor cutting edge on end-mills is attributable to production deviations. The microgeometric edge comprises of the edge rounding, which can be described by the average edge rounding S and further deviations, such as the roughness of the cutting edge. This also includes wear-related changes from the original tool shape, such as microscopic defects or larger breakouts. The geometric shape of cutter-workpiece-engagements is also influenced by vibrations during the milling process. Tool deflections have a direct effect on the engagement of the individual cutting edges, which influence the quality of the surface. The material behavior can have a substantial influence on the surface roughness and the surface integrity during cutting, due to the occurrence of smearing or the formation of burrs during cutting [18]. Additional to the overall deformation behavior, the aforementioned cutting edge rounding can lead to the ploughing effect [19], which describes the plastic deformation of the material to be cut in front of the cutting edge (Section 5.4). During machining processes without additional tool inclination and very low static tool deflections, a reduced influence of the back cut can be observed. Depending on the process parameter values used, the back cut of the tool influences the surface topography during traversal of the workpiece. Especially the lateral overlapping of the machining track increases this effect due to a large number of tool engagements, particularly for low values of the width of cut. This implies the need to take a large number of tooth engagements into account when investigating resulting surface topographies. The overall number of consecutive cuts of adjacent cutting paths leads to a significant decrease in surface roughness (see Figure1). Due to the complexity of influencing parameters, a high number of experiments is needed to establish an understanding of the prognosis of the surface quality for a defined machining process. To cope with these limitations, geometric simulations can be used to reduce the number of experiments. In this paper, a model for the prediction of surface topographies is presented, which can be used to reduce the experimental effort.

4. A Geometric Simulation Approach for the Prediction of Surface Topographies To optimize the process parameter values for the manufacturing of high-quality surfaces without the use of preliminary milling experiments and subsequent evaluation of the resulting surfaces, a geometric simulation approach will be presented. Previous approaches of simulation systems used approximations of tool geometries in order to ensure feasible runtimes for the prediction of cutting forces or vibrations during milling [20]. In the presented approach, digitized cutting faces were used to achieve a more realistic prediction of ﬁne-grained details during milling, such as the microgeometry of the used cutting edges.

4.1. Time Step Based Discrete Simulation System For the prediction of surface topographies, a geometric physically-based simulation system [20] was extended to use a time-step-based discretization of NC milling paths. Using a discretization of J. Manuf. Mater. Process. 2019, 3, 70 6 of 17 J. Manuf. Mater. Process. 2019, 3, x FOR PEER REVIEW 6 of 17 J. Manuf. Mater. Process. 2019, 3, x FOR PEER REVIEW 6 of 17 acould ﬁxed be∆t considered.in correspondence As seen with in Figure the rotational 4, the resu speed,lting discretesurface topography substeps for is engagement visualized throughout situations couldthe whole be considered. process during As seen simulation in Figure . After 44,, thethe a resulting resufinishedlting surfacesimulationsurface topographytopography run, the surface isis visualizedvisualized topography throughoutthroughout can be theexported whole processin various during data simulation.simulationformats. By. After using a finished ﬁnishedthe NanonFocus simulation Surface run, the NMS-data surface topography format, simulated can be exportedtopographies in various can be datadataanalyzed formats.formats. utilizing By using MountainsMap, thethe NanonFocusNanonFocus thus, simulated Surface NMS-dataNMS-data and measured format, topographies simulatedsimulated topographiescan be compared. can be analyzed utilizing MountainsMap, thus, simulated and measured topographies can be compared.

Figure 4. Illustration of the applied geometric physically-based simulation system showing two Figure 4. Figureadjacent 4. toolIllustrationIllustration paths with of of the the applied discretized geometric workpiece physically-basedphysic surfaceally-based model simulation (a) Representation system showing of a tool two in adjacent tool paths with the discretized workpiece surface model (a) Representation of a tool in adjacentengagement tool situations;paths with ( bthe) Triangular discretized mesh workpiece representation surface and model surface (a) Representationtopography resulting of a tool from in a engagement situations; (b) Triangular mesh representation and surface topography resulting from a engagementcertain process situations; configuration. (b) Triangular mesh representation and surface topography resulting from a certain process configuration.configuration. 4.2.4.2. DetailedDetailed ToolTool ModelModel Representation Representation Including Including the the Cutting Cutting Edge Edge Microgeometry Microgeometry 4.2. Detailed Tool Model Representation Including the Cutting Edge Microgeometry In order to model the shape of the tool, the number of cutting edges zn, the corner radius of the In order to model the shape of the tool, the number of cutting edges zn, the corner radius of the n tool Inrε, orderas well to asmodel the rake the shapeand flank of the angle tool, were the number considered. of cutting A determined edges z , theradial corner runout radius of 0.2of theµm tool rε, as well as the rake and flank angle were considered. A determined radial runout of 0.2 µm was tool rε, as well as the rake and flank angle were considered. A determined radial runout of 0.2 µm alsowas taken also taken into account into account in the model.in the model. Since the Since shape the of shape the tool of hasthe atool decisive has a influencedecisive influence on the resulting on the was also taken into account in the model. Since the shape of the tool has a decisive influence on the surfaceresulting quality, surface it hasquality, to be it modeled has to be with modeled a suffi cientwith a level sufficient of detail. level This of meansdetail. that,This means in addition that, toin resulting surface quality, it has to be modeled with a sufficient level of detail. This means that, in theaddition macrogeometry, to the macrogeometry, the characteristics the ofcharacteristic the cuttings edges of the had cutting to be considerededges had toto providebe considered a realistic to addition to the macrogeometry, the characteristics of the cutting edges had to be considered to predictionprovide a ofrealistic the surface prediction quality. of This the wassurface achieved quality. by constructingThis was achieved a triangle by mesh, constructing which adequately a triangle provide a realistic prediction of the surface quality. This was achieved by constructing a triangle representedmesh, which the adequately microgeometry represented of the the cutting microgeometry edges (see Figure of the5 cutting). edges (see Figure 5). mesh, which adequately represented the microgeometry of the cutting edges (see Figure 5).

FigureFigure 5.5.Simulation Simulation modelmodel ofof thethe analyzedanalyzed cuttercutter takingtaking thethe microscopicmicroscopicgeometry geometry ofofa acutting cuttingedge edge intoFigureinto account. account. 5. Simulation model of the analyzed cutter taking the microscopic geometry of a cutting edge into account. ForFor preciseprecise modellingmodelling ofof thethe microgeometrymicrogeometry ofof thethe tool,tool, thethe cuttingcutting edgesedges werewere measuredmeasured usingusing thethe stripstripFor precise projectionprojection modelling microscopemicroscope of the LMILMI microgeometry MikroCADMikroCAD plus.ofplus the. TheThetool, valuevalue the cutting determineddetermined edges forwerefor thethe measured edgeedge radiusradius using ofof rtherββ = strip 2.26 projection± 0.45 µmµm microscope was then usedused LMI toto MikroCAD modelmodel thethe plus virtualvirtual. The cuttingcutting value edges. edges.determined InIn addition,addition, for the the theedge determineddetermined radius of ± roughnessrroughnessβ = 2.26 ± 0.45ofof thethe µm cuttingcutting was then edgeedge used ofofR R stos= = model 1.51.5 µµmm the waswas vi appliedrtualapplied cutting toto thethe edges. modelledmodelled In addition, cuttingcutting edgeedgethe determined inin orderorder toto obtainroughnessobtain aa realisticrealistic of the modelcuttingmodel ofedgeof thethe of tool.tool. Rs = TheThe 1.5 modelling modellinµm was gapplied ofof thethe roughnesstoroughness the modelled waswas achieved achievedcutting edge byby projecting projectingin order to aa stochasticobtainstochastic a realistic noisenoise signalsignal model to to of the the the cutting cutting tool. edge Theedge modellin topography. topography.g of the roughness was achieved by projecting a stochastic noise signal to the cutting edge topography. 4.3. Calculation of Intersections with Workpiece Model 4.3. Calculation of Intersections with Workpiece Model For the calculation of the resulting surface topographies, the measured mesh was imported as a triangleFor themesh calculation and represented of the resulting using a surfacehalfedge topo datagraphies, structure the [21]. measured The workpiece mesh was was imported modeled as by a triangle mesh and represented using a halfedge data structure [21]. The workpiece was modeled by

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4.3. Calculation of Intersections with Workpiece Model J. Manuf. Mater. Process. 2019, 3, x FOR PEER REVIEW 7 of 17 J. Manuf.For Mater. the calculationProcess. 2019, 3 of, x theFOR resulting PEER REVIEW surface topographies, the measured mesh was imported 7 of as 17 a trianglea height mesh field. and To representedcalculate the using intersection a halfedge between data structurethe workpiece [21]. Theand workpiecethis tool model, was modeled ray-triangle by a intersectionheight field. ﬁeld. tests To To calculate were used. the Typical intersection implementations between the use workpiece an approach and this to identify tool model, the intersection ray-triangle for intersectionintersectiona plane representing tests were used.used. the Typicaltriangle. implementationsimplementations Afterwards, th usee use coordinates an an approach approach of to tothe identify identify intersection the the intersection intersection are checked, for for a aplane whetherplane representing representing they lie thewithin triangle.the thetriangle. Afterwards,given Afterwards, triangle, the coordinatesfor th inestance coordinates of thevia intersectionbarycentric of the intersection arecoordinates checked, are whether [22]. checked, Due they to whetherlienumerical within they the inaccuracies, given lie within triangle, theintersections for given instance triangle, viaalong barycentric for an inedstancege coordinates between via barycentric two [22]. adjacent Due coordinates to numerical triangles [22]. inaccuracies, can Due lead to to numericalintersectionsduplicate inaccuracies,detections along an or edge noneintersections between at all (see two along Figure adjacent an 6). ed To trianglesge compensate between can leadtwo for toadjacentnumerical duplicate triangles instabilities, detections can each orlead none pairto duplicateatof all adjacent (see detections Figure triangles6). To or compensatewithinnone at this all (seedata for Figure numerical structure 6). To instabilities,had compensate a “dominant each for pairnumerical halfedge”, of adjacent instabilities, which triangles is represented each within pair ofthisusing adjacent data Plücker structure triangles coordinates had within a “dominant [23]. this Thesedata halfedge”, structuredominant whichhad halfedges a is“dominant represented could be halfedge”, used using to Plücker determine which coordinates is whetherrepresented [a23 ray]. usingThesepotentially Plücker dominant intersects coordinates halfedges a triangle. [23]. could These be Since used dominant each to determine dominant halfedges whether edge could was a be ray usedassigned potentially to determine to one intersects distinct whether a triangle.triangle, a ray potentiallySinceduplicate each dominantintersectionsintersects edge a triangle.were was prevented, assigned Since toeach thus, one dominant distinct leading triangle, toedge higher was duplicate numeric assigned intersections stability to one (see distinct were Figure prevented, triangle, 6). duplicatethus, leading intersections to higher were numeric prevented, stability thus, (see Figureleading6). to higher numeric stability (see Figure 6).

Figure 6. Schematic depiction of a problematic scenario during triangle ray intersections (a) where Figure 6. Schematic depiction of a problematic scenario during triangle ray intersections (a) where Figureray-intersections 6. Schematic could depiction slip through of a problematic adjacent triangles scenario F i duringand Fj due triangle to numerical ray intersections inaccuracies (a) (whereb) with ray-intersections could slip through adjacent triangles Fi and Fj due to numerical inaccuracies (b) with ray-intersectionsthe unambiguous could determination slip through of adjacent the cut triangletriangles by Fi usingand F j Plücdue kerto numerical representations inaccuracies of edges. (b) with the unambiguous determination of the cut triangle by using Plücker representations of edges. the unambiguous determination of the cut triangle by using Plücker representations of edges. InIn order order to to determine determine the the intersection intersection of of a a given given ray ray with with a a triangle, triangle, the the approach approach presented presented by by AmanatidesAmanatidesIn order et to et al. determineal. [24 [24]] was was used.the used. intersection By By calculating calculating of a give the the Plückern Plücker ray with inner inner a triangle, product product th of ofe an approachan edge edge with with presented a a potential potential by Amanatidesray,ray, the the sign sign et of ofal. the the [24] resulting resulting was used. product product By calculating determines determines the the Plückerthe side side ofinner of the the twoproduct two given given of raysan rays edge [25 [25].]. with By By calculatinga calculating potential ray,thisthis innerthe inner sign product product of the for resultingfor each each edge edge product of of a a triangle, triangle,determines the the equality thequalitye side of of coe coefficienttheﬃ twocient given signs signs rays hints hints [25]. towards towards By calculating a a possible possible thisintersectionintersection inner product (see (see Figure forFigure each7). 7). edge Thus, Thus of the,a thetriangle, actual actual calculationthe calculation equality ofof of thecoefficient the intersection intersection signs pointhints point towards was was reduced reduced a possible to to a a intersectionsimplesimple ray-plane ray-plane (see Figure intersection intersection 7). Thus test. test., the actual calculation of the intersection point was reduced to a simple ray-plane intersection test.

Figure 7. Three possible scenarios for a ray-triangle intersection using Plücker coordinates. (a) Ray hits Figure 7. Three possible scenarios for a ray-triangle intersection using Plücker coordinates. (a) Ray exactly one triangle with a dominant edge between. (b) Ray hits two triangles at an adjacent dominant Figurehits exactly 7. Three one possible triangle scenarios with a dominant for a ray-triangle edge between. intersection (b) Ray using hits Pl twoücker triangles coordinates. at an (adjacenta) Ray edge. (c) Ray hits exactly one triangle without a dominant edge. hitsdominant exactly edge.one triangle (c) Ray hitswith exactly a dominant one triangle edge between. without a( bdominant) Ray hits edge. two triangles at an adjacent dominant edge. (c) Ray hits exactly one triangle without a dominant edge. For each of these time steps, discretized by the geometric physically-based simulation system, For each of these time steps, discretized by the geometric physically-based simulation system, the prevalent tool rotation (in accordance with its rotations per minute) was used to determine the the Forprevalent each of tool these rotation time steps, (in accordance discretized with by thitse rotationsgeometric per physically-bas minute) wased used simulation to determine system, the active cutter-workpiece-engagement. The workpiece was modeled as a two-dimensional height ﬁeld theactive prevalent cutter-workpiece-engagement. tool rotation (in accordance The with workpiece its rotations was modeled per minute) as a two-diwas usedmensional to determine height thefield with height values for each sampling position, which are arranged in a grid structure [26]. Since no activewith cutter-workpiece-engagement.height values for each sampling The position, workpiece which was are modeled arranged as ina two-di a gridmensional structure [26]. height Since field no withundercuts height werevalues considered for each sampling for this type position, of face which micromilling are arranged process, in a the grid representation structure [26]. of Since sampling no undercutspositions wereas floating-p consideredoint for values this type is sufficient. of face micromilling These height process, values the can representation be interpreted of samplingas rays of positionsthe form as floating-point values is sufficient. These height values can be interpreted as rays of the form 𝑅𝑎𝑦(𝑝, ℎ =𝑝+ℎ⋅𝑛, (8) 𝑅𝑎𝑦(𝑝, ℎ =𝑝+ℎ⋅𝑛, (8)

J. Manuf. Mater. Process. 2019, 3, 70 8 of 17 J. Manuf. Mater. Process. 2019, 3, x FOR PEER REVIEW 8 of 17 withundercuts p = (x, were y, 0) considered being the x-y-position for this type ofon facethe micromillingheight field, n process, = (0, 0, the1) being representation the normal-vector of sampling in orthogonalpositions as direction floating-point and h values being isthe su ffiactualcient. height These of height the height values field can bepoint. interpreted With this as representation, rays of the form the calculation of the material removal breaks down to a common depth test [27] for each sampling Ray(p, h) = p + h n, (8) point. Instead of calculating the intersection for each of the· mesh triangles, a bounding- box test for each triangle was used to pre-determine potential intersection candidates for each position and with p = (x, y, 0) being the x-y-position on the height field, n = (0, 0, 1) being the normal-vector in rotation (see Figure 8). orthogonal direction and h being the actual height of the height field point. With this representation, The aforementioned intersection operation was subsequently used to determine whether the ray the calculation of the material removal breaks down to a common depth test [27] for each sampling of a height field point is located within one of the potential triangles. The actual intersection was point. Instead of calculating the intersection for each of the mesh triangles, a bounding- box test for calculated by reducing the current value of a height field point to the distance to the currently each triangle was used to pre-determine potential intersection candidates for each position and rotation intersecting triangle. (see Figure8).

FigureFigure 8. Bounding-boxBounding-box test test for for determining determining potential potential intersection intersection candidates.

4.4. BoundaryThe aforementioned Conditions for intersection Sampling Precisions operation was subsequently used to determine whether the ray of a height field point is located within one of the potential triangles. The actual intersection To accurately predict surface topographies using a time step based simulation system, was calculated by reducing the current value of a height field point to the distance to the currently conditions regarding precision have to be met to ensure an optimal discretization of the process. intersecting triangle. The required precision of the discretization of the tool path can be estimated by considering the target surface4.4. Boundary precision Conditions Δp, the for feed Sampling per tooth Precisions fz and the number of cutting teeth zn in combination with its macroscopic shape, i.e., its radius r. The selection of an adequate ratio of discretization stepsTo per accurately rotation predict(spr) and surface minimum topographies workpiece using precision a time step Δp basedfor preventing simulation undersampling system, conditions or moiré-patternsregarding precision [28] in have the resulting to be met surface to ensure topograp an optimalhy was discretization determined by of the the largest process. relative The required motion ofprecision the tool ofduring the discretization one time step. of For the trochoidal tool path pa canths, be the estimated largest and by consideringsmallest movement the target persurface step is locatedprecision orthogonal∆p, the feed to per the tooth feedf z directionand the number due to of the cutting superposition teeth zn in combination of translational with itsand macroscopic rotational movementshape, i.e., (see its radius Figurer 9a).. The At selectionthese positions, of an adequate the angular ratio velocity of discretization is directed in steps and per opposite rotation of feed (spr) direction.and minimum Thus, workpiecethe distance precision Δs between∆p twofor preventing simulation undersamplingsteps using a steps or per moir rotationé-patterns discretization [28] in the isresulting estimated surface by topography was determined by the largest relative motion of the tool during one time step. For trochoidal paths, the largest and smallest movement per step is located orthogonal to 2𝜋𝑟 + 𝑓 ⋅𝑧 the feed direction due to the superposition𝛥𝑠 of translational = and, rotational movement (see Figure9(9)a). 𝑠𝑝𝑟 At these positions, the angular velocity is directed in and opposite of feed direction. Thus, the distance as∆s seenbetween in Figure two simulation 9b. Since Δ stepss should using be a less steps or perequal rotation to thediscretization chosen target isprecision estimated Δp by, the required steps per rotation spr were calculated as 2πr + fz zn ∆s = 2𝜋𝑟 + 𝑓· ⋅𝑧, (9) 𝑠𝑝𝑟 = spr . (10) 𝛥𝑠 as seenTo inpredict Figure surface9b. Since topographies∆s should for be lessmore or complex equal to tool the chosenpaths than target a single precision slot ∆millingp, the requiredprocess, allsteps tool per positions rotation inspr a circularwere calculated area within as a distance of r around a specific height field sampling point are needed, including its prevalent tool rotation and depth of cut. By taking these conditions into 2πr + fz zn consideration, the resulting surface topographiesspr = show· .less deviation to the experiments and (10) ∆s incorporate variations regarding surface quality (see Sections 5 and 6). To predict surface topographies for more complex tool paths than a single slot milling process, all tool positions in a circular area within a distance of r around a specific height field sampling point are needed, including its prevalent tool rotation and depth of cut. By taking these conditions

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J. Manuf. Mater. Process. 2019, 3, x FOR PEER REVIEW 9 of 17 into consideration, the resulting surface topographies show less deviation to the experiments and incorporateJ. Manuf. Mater. variations Process. 2019 regarding, 3, x FOR PEER surface REVIEW quality (see Sections5 and6) 9 of 17

Figure 9. The travelled path of one cutting edge during two consecutive feeds. (a) Relative speed pointing in feed direction during tool rotations. (b) Cutting edge positions based on the spr Figure 9. The travelled path of one cutting edge during two consecutive feeds. (a) Relative speed discretization.Figure 9. The travelled path of one cutting edge during two consecutive feeds. (a) Relative speed pointing in feed direction during tool rotations. (b) Cutting edge positions based on the spr discretization. pointing in feed direction during tool rotations. (b) Cutting edge positions based on the spr 5.5. Resultsdiscretization.

5. ResultsTheThe resultsresults ofof anan applicationapplication ofof thethe describeddescribed approachesapproaches forfor predictingpredicting thethe surfacessurfaces roughnessroughness ofof aa faceface micromillingmicromilling processprocess ofof hardenedhardened HSSHSS areare presentedpresented inin thethe followingfollowing section.section. AfterAfter aa briefbrief evaluationevaluationThe results ofof the the analyticalof analytical an application model, model, theof the resultsthe describedresults of the of models a thepproaches models are compared for are predicting compared to the resultsthe to surfaces the of results the roughness geometric of the simulativegeometricof a face micromilling simula approach.tive approach. process of hardened HSS are presented in the following section. After a brief evaluation of the analytical model, the results of the models are compared to the results of the 5.1.5.1.geometric Validation simula and tive Evaluation approach. ofof thethe AnalyticalAnalytical ModelModel The validation of the analytical model as well as its possibilities and limitations are presented. 5.1. ValidationThe validation and Evaluation of the analytical of the Analytical model asModel well as its possibilities and limitations are presented. FollowingFollowing this,this, aa discussiondiscussion ofof thethe significantsignificant simplificationsimplification ofof complexcomplex interrelationshipsinterrelationships associatedassociated withwith suchThesuch validation modelsmodels isis ofcarriedcarried the analytical out.out. InIn orderorder model toto astestte stwell thethe as presentedpresented its possibilities model,model, and experimentsexperiments limitations werewere are conductedconductedpresented. withwithFollowing aa variationvariation this, a ofof discussion thethe feedfeed per perof the toothtooth significantf fzz. Since simplification the model was of designedcomplex forinterrelationships a singlesingle machiningmachining associated path, singlesinglewith such slots models were milled milled is carried and and afterwardsout. afterwards In order evaluated evaluated to test the regarding regarding presented RmaxR model,max along along experiments a profile a profile in werethe in the center conducted center of the of theslotwith slotin a variationfeed in feed direction. direction. of the Thefeed The results per results tooth of of thefz. the Since experi experiments thements model are are was compared compared designed to to for the the a predictions predictionssingle machining of of the path, newnew analyticalanalyticalsingle slots modelmodel were (Equation milled(Equation and (7)) (7)) afterwards and and those those evaluated introduced introduced regarding by by Bauer Bauer R asmax wellas wellalong as Martellotti as a Martellottiprofile (Equationsin the (Equations center (1) of and the(1) (3),andslot see (3),in Figurefeed see Figuredirection. 10). 10). The results of the experiments are compared to the predictions of the new analytical model (Equation (7)) and those introduced by Bauer as well as Martellotti (Equations (1) and (3), see Figure 10).

FigureFigure 10. 10.Validation Validation of of the the analyticalanalytical modelmodel andand comparisoncomparison to the models of Bauer Bauer and and Martellotti. Martellotti.

TheTheFigure resultsresults 10. Validation showshow an an increase ofincrease the analytical of of roughness roughness model and in in correlation compar correlationison to to to thethe the feedmodels feed per perof tooth Bauer toothin and in the Martellotti.the experiment experiment as wellas well as the as analyticalthe analytical model. model. Comparing Comparing the results the result of thes analyticalof the analytical models models to the experiments, to the experiments, the new contributionthe newThe contribution results (Equation show (Equation (7))an increase shows (7)) a of better shows roughness accordance a better in acoccordance thanrrelation the model tothan the the offeed Bauer model per or tooth of Martellotti. Bauer in the or Martellotti.experiment The offset ofTheas thewell offset higher as theof roughness theanalytical higher value model.roughness of theComparing experimentsvalue of the the isresult experiments dues to of the the simplified analyticalis due to view modelsthe ofsimplified the to analyticalthe experiments, view model.of the Sinceanalyticalthe new the contribution microgeometry model. Since (Equation the of themicrogeometry cutting (7)) shows edge a andofbetter the the acuttingccordance material edge behavior than and the the influence model material of theBauer behavior resulting or Martellotti. influence surface, neglectingtheThe resulting offset suchof surface,the eff ectshigher leadsneglecting roughness to an underestimationsuch value effects of leadthe ofexperimentss to the an roughness underestimation is due values. to the Taking of simplifiedthe theroughness chipping view values. of thethe Takinganalytical the model. chipping Since of thethe cuttingmicrogeometry edges of ofRs the= 1.5 cutting µm into edge account, and the the material offset is behavior within expectable influence deviationthe resulting limits. surface, It is alsoneglecting assumed such that effects the influence leads to ofan deviating underestimation effects decreases of the roughness with increasing values. sizeTaking of thethe chippingmachining of thegrooves. cutting The edges usage of R sof = 1.5analytical µm into models account, allows the offset to showis within characteristic expectable deviation limits. It is also assumed that the influence of deviating effects decreases with increasing size of the machining grooves. The usage of analytical models allows to show characteristic

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µ cuttingJ. Manuf. edges Mater. ofProcess.Rs = 20191.5, 3, mx FOR into PEER account, REVIEW the offset is within expectable deviation limits. It is 10 also of 17 assumed that the influence of deviating effects decreases with increasing size of the machining grooves. Thedependencies usage of analytical regarding models the resulting allows to roughness show characteristic in an efficient dependencies way. To analyze regarding the the effect resulting of the roughnesscorner radius in an r eεffi andcient the way. angle To analyzeof the minor the eff cuttingect of the edge corner ψ on radius the heightrε and of the the angle milling of the marks, minor a cuttingvariation edge ofψ parameteron the height values of the was milling computed marks, based a variation on Equation of parameter (7). The valuescalculated was theoretical computed basedsurface onroughness Equation (7).value The Rth calculated for varying theoretical feeds per surface tooth roughness fz is presented value inRth Figure for varying 11 for feedsvariation per tooth of thef z toolis presentedcorner radius in Figure rε, and 11 forthe variationangle of the of the minor tool cutting corner radius edge. rε, and the angle of the minor cutting edge.

FigureFigure 11. 11.Parameter Parameter variationvariation ofof cornercorner radiusradius andand angle angle of of minor minor cutting cutting edge edge based based on on the the analyticalanalytical model. model.

BasedBased on on these these predicted predicted roughness roughness values, values, an increasean increase of the of the theoretical theoretical roughness roughness depth depth over over an increasingan increasing feed feed per toothper tooth can can be assumed,be assumed, which which corresponds corresponds to to general general experience experience [29 [29,30].,30]. Taking Taking FigureFigure2 into 2 into account, account, this this relationship relationship can can be be explained explained by by the the increase increase of of the the width width of of milling milling marks, marks, whichwhich can can also also be be observed observed in in real real processes processes [31 [31].]. Taking Taking the the variations variations of of corner corner radius radius and and angle angle of of thethe minor minor cutting cutting edge edge into into account, account, the the influence influence of of macrogeometric macrogeometric parameters parameters on on the the theoretical theoretical roughnessroughness depth depth can can be be inferred. inferred. It It can can also also be be seen seen in in Figure Figure 11 11,, that that the the theoretical theoretical roughness roughness depth depth r shouldshould decrease decrease for for increased increased corner corner radius radiusε randε and increase increase for for increased increased angle angleψ ψwhen when multiple multiple cuts cuts areare intersected. intersected. Based Based on on these these observations, observations, a a larger larger corner corner radius radius and and a a lower lower angle angle of of the the minor minor cuttingcutting edge edge are, are, in in theory, theory, beneficial beneficial for for producing producing a higha high surface surface quality. quality. It It should should be be noted noted that that the the effeffectiveective radius radius of of the the tool tool engagement engagement decreases decreases with with increasing increasing corner corner radius, radius, which which can can have have a a negativenegative impact impact on on the the process, process, e. e. g. g. on on the the productivity productivity of of the the process. process. Furthermore, Furthermore, a a decreasing decreasing cuttingcutting speed, speed, which which depends depends on on the the e ffeffectiveective radius radius of of the the tool, tool, could could a ffaffectect the the cutting cutting forces forces and and wearwear progression. progression. These These e ffeffectsects are are not not considered considered in in the the analytical analytical model. model. 5.2. Evaluation of the Geometric Simulation Approach Considering the Tools Microgeometry in the Main Cut 5.2. Evaluation of the Geometric Simulation Approach Considering the Tools Microgeometry in the Main CutTo evaluate the prediction capabilities of the geometric simulation approach, three milling experiments (E1–E3) were conducted and the resulting surface topographies were measured. For the To evaluate the prediction capabilities of the geometric simulation approach, three milling simulation experiment, equal process parameters values were used as well as the modelled cutting experiments (E1–E3) were conducted and the resulting surface topographies were measured. For the edge representation. As can be seen in Figure 12 exemplary, simulated surface topographies show a simulation experiment, equal process parameters values were used as well as the modelled cutting good accordance to the experimental results. Minor deviations regarding plastic deformation can be edge representation. As can be seen in Figure 12 exemplary, simulated surface topographies show a seen throughout the surface. However, the determined deviations of the surface roughness values are good accordance to the experimental results. Minor deviations regarding plastic deformation can be within a range of a few micrometers, which indicates a good prediction capability. seen throughout the surface. However, the determined deviations of the surface roughness values are within a range of a few micrometers, which indicates a good prediction capability.

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Figure 12.12. Exemplary validation of the simulation system using three experiments (E1–E3) for a faceface micromilling process.process.

For aa susufficientfficient considerationconsideration ofof thethe processprocess sequencesequence ofof faceface milling,milling, thethe individualindividual tooltool engagements, main,main, backback and and lateral lateral cut cut were were considered considered step step by by step. step. In addition,In addition, the the influence influence of the of cuttingthe cutting edge edge micro- micro- and macrogeometryand macrogeometry were were examined, examined, e.g., thee.g., geometric the geometric deviations deviations from anfrom ideal an toolideal shape. tool shape. The decisive The decisive portion portion of the of material the materi is removedal is removed during during the main the main cut of cut the of tool the intool feed in direction.feed direction. For a detailedFor a detailed analysis, analysis, only slot only milling slot operations milling operations were considered. were Sinceconsidered. tool engagements Since tool cannotengagements be manipulated cannot be arbitrarily manipulated during arbitrarily an experiment during in an a realexperi machiningment in a process, real machining the discussion process, on thethe maindiscussion cut was on based the main on simulation cut was based results on only.simu Thelation following results only. discussions The following in Sections discussions 5.3 and 5.4 in wereSections carried 5.3 and out 5.4 in directwere comparisoncarried out in to dire experimentalct comparison results. to experimental results. InIn thethe cross-sectioncross-section of of the the path path in thein the feed feed direction, direction, the tool the engagement tool engagement resulted resulted in a mark in pattern,a mark whichpattern, is which characterized is characterized by the tool by shape the tool and shape the tooth and feedthe toothf z. A feed profile fz. lineA profile was extracted line was inextracted the center in ofthe the center machining of the machining path (Profile path I, see(Profile Figure I, see13), Figu whichre 13), has which high similarity has high simi to thelarity engagement to the engagement shown in µ Figureshown2 in. Comparing Figure 2. Comparing the resulting the maximum resulting maximum roughness depthroughness of Profile depth I of ProfileRmax,I I= of0.93 Rmaxm,I to= 0.93 the analyticalµm to the roughness analytical depth roughnessRth = 0.84depthµm, Rth a deviation= 0.84 µm, of a approximately deviation of 9.7%approximately implies a good 9.7% agreement implies a ofgood the agreement analytical modelof the analytical to the simulated model to surface. the simula Dueted to surface. the limited Due significanceto the limited of significance a single profile of a line,single this profile acts asline, a proof-of-concept this acts as a proof-of-concept for the analytical for the model. analyticalStrong model. variations Strong in variations profile depth in profile occur orthogonaldepth occur to orthogonal the feed direction. to the feed As seendirection. in Figure As seen 13, ain profile Figure located 13, a profile in the centerlocated of in the the machining center of trackthe machining (Profile I) track is compared (Profile I) with is compared a profile locatedwith a pr atofile the edgelocated of theat the track edge (Profile of the II).track Profile (Profile II had II). µ aProfile roughness II had valuea roughnessRmax, IIvalue= 0.19 Rmax,m,II which= 0.19 µm, is significantly which is significantly lower than lower for Profile than for I. Profile The varying I. The engagementvarying engagement conditions conditions leads to significantly leads to significantly lower depths lower of the depths machining of marks.the machining Therefore, marks. for a suTherefore,fficient description for a sufficient of the description surface roughness, of the su therface entire roughness, surface area the mustentire be surface considered. area must For this be reason,considered. the simulationFor this reason, results andthe surfacesimulation parameter results values and surface listed were paramete determinedr values over listed the entirewere machineddetermined surface over the considering entire machined multiple surface overlapping consid cuts.ering In multiple this context, overlapping it can also cuts. be In stated, this context, that the significanceit can also be of astated, simulated that topographythe significance for an of entire a simulated surface istopography considerably for higheran entire than surface that of is a singleconsiderably profile. higher than that of a single profile. Since the cutting cutting edges edges of of milling milling tools tools exhibit exhibit a large a large number number of ofdifferent different micro micro defects defects and and are arenot notideally ideally sharp-edged, sharp-edged, such such defects defects should should also also be be taken taken into into account account when when predicting surface typographiestypographies andand roughnessroughness characteristicscharacteristics onon microgeometricmicrogeometric scale.scale. BasedBased onon thethe approachapproach usingusing aa detailed geometric model of the roughrough cutting edge described in SectionSection 4.14.1,, simulationsimulation experimentsexperiments werewere conducted.conducted. AA roughness alongalong thethe cuttingcutting edgesedges ofof RRss == 1.5 µmµm and an axial offset between both minor cutting edges of ∆Δl= = 0.2 µµmm were applied. The resulting topography is presented in Figure 13.13. AsAs seenseen inin extractedextracted ProfileProfile IIIIII withwith aa valuevalue ofofR Rmaxmax,III,III = 1.64 µm,µm, the surface roughness is strongly subjected toto the the deviation deviation of theof tool.the tool. A comparison A comparison between between an experimental an experimental and a simulated and a roughnesssimulated characteristicroughness characteristic considering considering the deviation the of deviation the tool shape of the in tool various shape scales in various is presented scales inis presented Figure 14. in Figure 14.

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Figure 13. InfluenceInfluence of of the the microgeometric microgeometric properties properties of of cutting cutting edges edges on on the surface roughness, Figure 13. Influence of the microgeometric properties of cutting edges on the surface roughness, comparing an ideal shape (left) with a detailed representation ((rightright).). comparing an ideal shape (left) with a detailed representation (right). 5.3. Evaluation Evaluation of the Geometric Simulation Approach Considering the Back Cut 5.3. Evaluation of the Geometric Simulation Approach Considering the Back Cut The influenceinfluence of of the the back back cut cut during during face millingface milling is discussed is discussed in this section.in this Ifsection. the tool If completely the tool The influence of the back cut during face milling is discussed in this section. If the tool completelyoverruns the overruns workpiece the withworkpiece small orwith no small static /ordynamic no static/dynamic deflection ofdeflection the tool, of the the cutting tool, the edges cutting will completely overruns the workpiece with small or no static/dynamic deflection of the tool, the cutting edgesengage will on engage the rear on side the of rear the side tool of in the the tool already in the machined already surfacemachined after surface the main aftercut. the main Based cut. on edges will engage on the rear side of the tool in the already machined surface after the main cut. Basedthe superposition on the superposition of the feed andof the cutting feed movements,and cutting amovements, trochoidal-like a trochoidal-like mark pattern resulted mark pattern on the Based on the superposition of the feed and cutting movements, a trochoidal-like mark pattern resultedsurface, whichon the wassurface, described which in was Section described 4.3. The in Section influence 4.3. of The the backinfluence cuts onof the resultingback cuts surfaceon the resulted on the surface, which was described in Section 4.3. The influence of the back cuts on the patternresulting is illustratedsurface pattern in Figure is illustrated 14, with a comparisonin Figure 14, of with idealized a comparison and modeled of idealized cutting edges and in modeled relation resulting surface pattern is illustrated in Figure 14, with a comparison of idealized and modeled cuttingto experimental edges in measurements.relation to experimental measurements. cutting edges in relation to experimental measurements.

Figure 14. InfluenceInfluence of of back back cut cut on on the the surface surface topography. topography. Figure 14. Influence of back cut on the surface topography. As depicted, the mark pattern on the surface changes significantly due to the back cut in As depicted, the mark pattern on the surface changes significantly due to the back cut in comparisonAs depicted, to Figure the 13 mark. This pattern change on in roughnessthe surface is alsochanges shown significantly by the measured due to roughness the back values.cut in comparison to Figure 13. This change in roughness is also shown by the measured roughness values. AncomparisonRz,ideal = to0.68 Figure0.25 13. Thisµm waschange determined in roughness for theis also idealized shown cuttingby the measured edge without roughness back cut values. and An Rz,ideal = 0.68 ± 0.25± µm was determined for the idealized cutting edge without back cut and value valueAn Rz, ofidealRz, =modeled 0.68 ± 0.25= 1.29 µm was0.33 determinedµm for the detailedfor the idea modellized of cutting the rough edge cutting without edge. back Taking cut and the value back of Rz,modeled = 1.29 ± 0.33 µm± for the detailed model of the rough cutting edge. Taking the back cut into cutof R intoz,modeled account, = 1.29 the± 0.33 values µm forRz, theideal detailed= 0.46 model0.23 µ ofm th ande roughRz,modeled cutting= 0.99edge. Taking0.29 µm the were back obtained. cut into account, the values Rz,ideal = 0.46 ± 0.23 µm± and Rz,modeled = 0.99 ± 0.29 ±µm were obtained. The Theaccount, additional the values engagements Rz,ideal = of0.46 the ±tool 0.23 result µm and in a R significantz,modeled = reduction0.99 ± 0.29 of µm the were average obtained. roughness The additional engagements of the tool result in a significant reduction of the average roughness depth, additional engagements of the tool result in a significant reduction of the average roughness depth, which is attributed to a reduction of the depth of the marks. A strong similarity between the simulated which is attributed to a reduction of the depth of the marks. A strong similarity between the simulated

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J. Manuf. Mater. Process. 2019, 3, x FOR PEER REVIEW 13 of 17 depth, which is attributed to a reduction of the depth of the marks. A strong similarity between the simulatedand the real and milled the real topography milled topography can be observed. can be observed. This was Thisalso wasconfirmed also confirmed by the determined by the determined surface surfaceparameter parameter values. values.A value A valueof Rz, ofmeasuredRz, = 0.83= 0.83± 0.21 0.21µmµ mwas was meas measured,ured, which which suggests suggests an measured ± overestimation ofof the the roughness roughness by by the the simulation simulation of approximatelyof approximately 19.3 19.3 %, see%, Sectionsee Section 5.4. This5.4. This is also is indicatedalso indicated by the by values the values determined determin fored the for arithmetic the arithmetic mean mean roughness roughnessRa values Ra values (see (see Figure Figure 14). 14). A detaileddetailed analysis showed, that the surface quality within a machining path didiffersffers greatly depending onon the measuring positionposition alongalong thethe widthwidth ofof cut.cut. Furthermore, the microgeometry of the cutting edge has anan influenceinfluence onon thethe resultingresulting surfacesurface topographytopography andand allowsallows forfor betterbetter predictionspredictions regardingregarding surfacesurface roughnessroughness values. Since the modeling of the cutting edge is based on aa statisticalstatistical distribution of unevenness and does not reflectreflect the real shapeshape ofof the cutting edge, a certain deviation between simulatedsimulated andand measuredmeasured resultsresults isis toto bebe expected.expected.

5.4. Evaluation of the Geometric Simulation ApproachApproach Considering the Lateral OverlapOverlap ofof ToolTool PathsPaths During faceface millingmilling processes,processes, lateral lateral overlaps overlaps occur, occur, if if subsequent subsequent tool tool paths paths are are off offsetset parallel parallel to previousto previous machining machining paths paths by less by than less the than effective the effective tool diameter tool ddiametereff. Usually, deff these. Usually, lateral these overlapping lateral cutsoverlapping occur in cuts three-axis occur in line three-axis milling processesline milling with proc smallesses orwith no small static/ ordynamic no static/dynamic deflection of deflection the tool. Inof thethe investigatedtool. In the investigated line milling processes, line milling this proce offsetsses, is described this offset by is the described width of cutby theae, whichwidth canof cut be setae, towhich about can 40% be ofset the to about actual 40% tool of diameter the actuald for tool roughing diameter processes, d for roughing as shown processes, in Figure as shown15. The in o ffFigureset of the15. The tool offset path to of previous the tool path engagements to previous leads engagements to additional leads lateral to additional engagements lateral of theengagements tool, influencing of the thetool, previously influencing generated the previously surface generated topography. surface Therefore, topography. the final Therefore, machining the patternfinal machining is influenced pattern by ais multitudeinfluenced of by tool a multitude engagements, of tool as engagements, mentioned in as Section mentioned 4.3. As in seenSection in Figure4.3. As 15 seen, the in machining Figure 15, patternthe machining of both pattern cutting of edges both were cutting significantly edges were di signfferent.ificantly An analysis different. of An the an microalysis end of the mill micro revealed end nomill damage revealed or no break-outs damage onor break- the cuttingouts on edge, the butcutting an axial edge, o ffbutset an of bothaxial cuttingoffset of edges, both cutting which clearlyedges, exceededwhich clearly the modeledexceeded cutting the modeled edge o ffcuttingset of ∆edgel = 0.2 offsetµm. of Δl = 0.2 µm.

Figure 15. InfluenceInﬂuence of lateral overlapping on surface topography.

The lateral overlapping engagements can have a significant effect on the resulting machining pattern. The lateral overlapping engagements can have a significant effect on the resulting machining Areas with different characteristics can be detected on the surface, depending on the width of cut of pattern. Areas with different characteristics can be detected on the surface, depending on the width ae = 0.4 mm. The selected width of cut resulted in areas with double (Section I) and triple (Section II) of cut of ae = 0.4 mm. The selected width of cut resulted in areas with double (Section I) and triple tool engagements. The width of the triple-machined areas depends on the ratio of the effective tool (Section II) tool engagements. The width of the triple-machined areas depends on the ratio of the diameter deff and the width of the radial engagement ae. In the considered scenario, this resulted in a effective tool diameter deff and the width of the radial engagement ae. In the considered scenario, this track width of 160 µm. While the influence of the overlapping engagements on the milling marks can be resulted in a track width of 160 µm. While the influence of the overlapping engagements on the milling marks can be seen from the topography measurements, it can also be identified from the

J. Manuf. Mater. Process. 2019, 3, 70 14 of 17

J. Manuf. Mater. Process. 2019, 3, x FOR PEER REVIEW 14 of 17 seen from the topography measurements, it can also be identified from the related roughness values. related roughness values. While in Section I of the experimentally machined surface a roughness of While in Section I of the experimentally machined surface a roughness of Rz,real.sec.I = 1.08 0.09 µm Rz,real.sec.I = 1.08 ± 0.09 µm was detected, in Section II only Rz,real.sec.II = 0.68 ± 0.07 µm was determined.± was detected, in Section II only Rz, = 0.68 0.07 µm was determined. The analysis of the simulationreal.sec.II results showed± a comparable surface topography with minor The analysis of the simulation results showed a comparable surface topography with minor deviations. The determined surface characteristics also showed a difference in surface quality deviations. The determined surface characteristics also showed a difference in surface quality between between Section I and Section II. The arithmetic mean roughness value Ra of the simulation estimated Section I and Section II. The arithmetic mean roughness value Ra of the simulation estimated the real the real surface characteristics with good correlation and confirmed the previously explained surface characteristics with good correlation and confirmed the previously explained observations observations in the individual sections for an overall consideration of the surface area. A closer in the individual sections for an overall consideration of the surface area. A closer inspection of the inspection of the topographies showed, that they differed in certain aspects. On the real surface, only topographies showed, that they differed in certain aspects. On the real surface, only two different tool two different tool engagement situations can be distinguished clearly. A third situation can be engagement situations can be distinguished clearly. A third situation can be identified in the upper identified in the upper edge of Section I, which is, however, less significant compared to the other edge of Section I, which is, however, less significant compared to the other tool engagements. It is tool engagements. It is assumed that the cutting edge microgeometry was only partially applied to assumed that the cutting edge microgeometry was only partially applied to the surface due to the the surface due to the elastoplastic deformation of the workpiece material. This reduced transfer of elastoplastic deformation of the workpiece material. This reduced transfer of the microgeometry onto the microgeometry onto the surface applied in particular to very filigree tool engagements. In the surface applied in particular to very filigree tool engagements. In addition, plastic deformations of addition, plastic deformations of the material and especially ploughing effects in front of the cutting the material and especially ploughing effects in front of the cutting edge led to smearing of workpiece edge led to smearing of workpiece material and filling of milling marks. This can be seen on the real material and filling of milling marks. This can be seen on the real surfaces shown in Figure 15. While the surfaces shown in Figure 15. While the last engagements can be distinguished, previous engagements last engagements can be distinguished, previous engagements could only be estimated in some areas. could only be estimated in some areas. Such effects could not be reproduced by the presented Such effects could not be reproduced by the presented simulation approach, since this constitutes a simulation approach, since this constitutes a redistribution and not an ideal removal of material. redistribution and not an ideal removal of material. Despite the detected deviation of the topography, Despite the detected deviation of the topography, a sufficient concordance of the calculated a sufficient concordance of the calculated characteristic values could be determined, which confirms characteristic values could be determined, which confirms the potential of the approach. the potential of the approach.

5.5. Evaluation of the Geometric Simulation Approach Considering the Feed Per Tooth The influenceinfluence of the back cut on the maximum roughness depth Rmax and the relevance of the presented approaches analyzed regarding the feed per tooth fz, as seen in Figure 16. The maximum presented approaches analyzed regarding the feed per tooth f z, as seen in Figure 16. The maximum roughness depth values Rmax for the analytical model and for the central profileprofile line of simulated topographies are depicted. Regarding their overall trend, both approaches showed an increasingincreasing roughness over the increasing toothtooth feed,feed, whichwhich correspondscorresponds toto generalgeneral viewview [[29,30].29,30]. The roughness depth is underestimated by the analytical model wi withth a constant offset, offset, which can be explained by the neglect of the cuttingcutting edge microgeometrymicrogeometry in the analytical model. The trend of Rmax for the simulation results was inferred byby linearlinear regressionregression analysisanalysis usingusing aa leastleast squaressquares approach.approach.

Figure 16. InfluenceInﬂuence of of the back cut on the maximum roughnessroughness depth with variation of the tooth feed rate.

A cyclic character can be observed for the roughness values calculated from simulation experiments. A cyclic character can be observed for the roughness values calculated from simulation Maximum and a minimum of Rmax values is reached periodically after an interval of about ∆f = 1.4 µm. experiments. Maximum and a minimum of Rmax values is reached periodically after an intervalz of This could be explained by the inﬂuence of the back cut on the surface topography. Depending on the about Δfz = 1.4 µm. This could be explained by the influence of the back cut on the surface topography. selected feed per tooth, the position of the back cut could coincide within another milling mark peak, Depending on the selected feed per tooth, the position of the back cut could coincide within another milling mark peak, leading to a significant reduction of the roughness depth due to the decreased height of the milling marks. The varying influence of the respective cutting edge can be explained by

J. Manuf. Mater. Process. 2019, 3, 70 15 of 17 leading to a significant reduction of the roughness depth due to the decreased height of the milling marks. The varying influence of the respective cutting edge can be explained by the varying axial positions. It was assumed, that the different cutting edges led to varying influences on the surface topography depending on the respective deviations of the macro- and microgeometry. Furthermore, this general observation led to the assumption that even a minor change in the feed per tooth can have a significant influence on the resulting surface quality due to the different engagements of the back cut. The sensitivity of the roughness value d the back-cut alignment is highly significant in relation to the general trend. This interrelation could be shown by the simulation, but not by the analytical model, since the back cut was not taken into account.

6. Discussion and Outlook The design of machining processes with regard to the resulting surface quality is a challenging optimization goal, especially for precision parts and functional surfaces with high requirements, which is usually accomplished by means of experimental preliminary investigations. In this paper, an analytical model and a geometric simulation approach for predicting the surface roughness resulting from a face micromilling process were presented. It could be shown that the use of an analytical model allows making initial predictions about the influence of certain factors on the resulting roughness with little effort. These include the corner radius of the tool rε, the angle of the secondary cutting edge ψ, and the feed per tooth f z. The results showed a good correlation in comparison to the surface characteristics of a slot milled surface, considering only the main cut of the tool. While a roughness of Rmax = 1.09 µm was determined in a real milling test in the center of the track, the analytical model estimated this selected parameter set with Rmax = 0.84 µm. In this context, however, it was shown that the surface roughness could also deviate significantly within a machining path and that a multitude of influencing factors also had a strong impact on the final surface topography, for instance elastic recovery [32]. Here, the microgeometry of the cutting edge, e.g., its actual roughness along a cutting edge, and deviations of the macrogeometry, such as an axial offset of the cutting edges, are to be mentioned. The usage of a developed analytical model can be used for quick prediction of surface roughness values. In addition, a geometric simulation system was extended and utilized to gradually demonstrate the influence of certain additional factors on the surface topography as well as roughness parameter values. This was done with both, idealized and modeled tool shapes, to take the previously mentioned macro- and microgeometry of a tool into account. It was shown that the consideration of the actual microgeometry of the cutting edges has a significant effect on the determined surface topography, even with a stochastic consideration of the roughness of the cutting edge. In this context, a high consistency with the experimentally determined surface parameter values could be determined with the detailed cutting edge model, although certain deviations were still noticeable in the surface patterns. Furthermore, multiple main, back, and lateral overlapping cuts were examined in detail. The resulting surface topography depends not only on the main cut, but also on a large number of intersecting cuts. In future investigations, it will be necessary to identify, to what extent filigree engagements of microgeometry are transmitted on the workpiece surface, especially regarding elastoplastic behavior of workpiece material. This would entail investigations of different materials in order to evaluate the influence of technical material properties, such as more ductility or ploughing effects. Mechanisms of material redistribution during cutting could be considered for the modeling of burr formation. By using the presented geometrical approach, changes of the cutting edge due to wear mechanisms can be considered to predict surface roughness values. This could be used to analyze whether possible surface roughness values are within given topography specifications in the context of expected tool life cycles. J. Manuf. Mater. Process. 2019, 3, 70 16 of 17

Author Contributions: Conceptualization, J.A.B., A.M., E.K., D.B. and P.W.; methodology, J.A.B., A.M. and E.K.; software, J.A.B.; validation, A.M. and J.A.B.; formal analysis, A.M., E.K. and J.A.B.; investigation, A.M., E.K. and J.A.B.; resources, D.B. and P.W.; data curation, J.A.B. and A.M.; writing–original draft preparation, J.A.B., E.K. and A.M.; writing–review and editing, J.A.B., A.M., E.K. and P.W.; visualization, J.A.B. and A.M.; supervision, D.B. and P.W.; project administration, D.B. and P.W.; funding acquisition, D.B. and P.W. Funding: Gefördert durch die Deutsche Forschungsgemeinschaft (DFG)—Projektnummer 68237143. This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—project number 68237143 within the transregional collaborative research center TR73 “Manufacturing of complex functional components with variants by using a new sheet metal forming process-Sheet-Bulk Metal Forming.” It presents results achieved within the subprojects B2 (“Machining of molds with filigree structures for Sheet-Bulk Metal Forming”). The investigations are based on the research project “Virtual Machining” (PE-216-0024) which is kindly funded by the Stiftung Mercator and the Mercator Research Center Ruhr. Conflicts of Interest: The authors declare no conflict of interest.

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