Analytical force model for drilling out unidirectional carbon fibre reinforced polymers (CFRP)

Journal of Materials Processing Tech. xxx (xxxx) xxxx

Contents lists available at ScienceDirect

Journal of Materials Processing Tech. journal homepage: www.elsevier.com/locate/jmatprotec

Analytical force model for drilling out unidirectional carbon fibre reinforced polymers (CFRP) Lukas Seeholzer*, Daniel Scheuner, Konrad Wegener Institute of Machine Tools and Manufacturing (IWF), ETH Zurich, Leonhardstrasse 21, 8092, Zurich, Switzerland

ARTICLE INFO

ABSTRACT

Associate Editor: E Budak

Carbon fibre reinforced polymer (CFRP) is characterised by a high potential for lightweight constructions due to its high specific strength and stiffness properties. Especially applications in aerospace industries necessitate numerous drilling operations in CFRP with high tolerance requirement for subsequent riveting. The bore quality in drilling CFRP is the result of complex interactions of various geometry parameters of the drilling tool, the process parameters and the material properties. Due to a strong correlation between the process forces and the resulting bore quality, analytical force models are valuable to optimise tool geometries and process strategies. Furthermore, analytical force models enable single parameter variation, which would be feasible in experiments only with excessive effort and costs. This research aims to develop an analytical force model for drilling out predrilled UD CFRP material. Based on detailed chip formation analysis by means of high-speed recordings and scanning electron microscopy (SEM), four intervals with similar chip formation mechanisms and two fundamental failure mechanisms are identified. In the modelling approach, a coordinated structural failure of entire fibre regions by axial compression is considered by micro-buckling. For fibre loading situations dominated by lateral bending deformations, fibre failure by exceeding tensile strength in the contact region is considered. Therefore, the tool-fibre contact situation is simplified as the Hertzian contact between two cylinders. In addition to force components due to an initial fibre separation in front of the cutting edge, additional force components due to elastic spring back effects of the CFRP material on the flank face are taken into account. Combined with the drilling kinematics and under consideration of oblique cutting conditions, the thrust force and the torque values are analytically determined for the entire range of fibre cutting angles during one half tool rotation. Subsequent validation shows a good agreement of simulated and experimental data.

Keywords: Fiber reinforced plastic Drilling Force Modeling

1. Introduction Carbon fibre reinforced polymer (CFRP) is increasingly important in various industrial applications due to its high specific strength and stiffness properties, combined with a low thermal expansion coefficient and a high corrosion and fatigue resistance. According to Sauer and Kühnel (2018), the high potential of CFRP materials for lightweight constructions results in a high demand in aerospace and high-end automotive industries. In accordance with Henerichs et al. (2015), CFRP structural components are often produced in near-net-shape strategy. However, usually additional machining operations are necessary in order to reach the quality requirements for the subsequent manufacturing processes. In this context, Voß (2017) mentioned drilling and milling as the most common machining operations. According to Sheikh-Ahmad (2009), CFRP is characterised by anisotropy and heterogeneity due to the microscopic structure of the

⁎

carbon fibres and their arrangement in the composite material, which result in fibre orientation dependent chip formation mechanisms. The chip formation process and its influence on the process forces and the resulting workpiece quality was extensively analysed by the research community. However, complex tool-workpiece interactions found in drilling and milling operations result in a superposition of various effects and thereby are not suitable for fundamental process analysis. Consequently, many researchers have focussed on the orthogonal machining process due to simplified cutting conditions. In accordance with Sheikh-Ahmad (2009), orthogonal machining enables a differentiated process analysis with a maximum reduction in complexity. As some of the first, Koplev et al. (1983) identified multiple consecutive cracks of brittle carbon fibres in front of the cutting edge resulting in a discontinuous chip formation. Furthermore, the authors detected a strong dependence of the chip formation mechanisms on the fibre orientation, which was subsequently confirmed by various researchers inter alia

Corresponding author. E-mail address: [email protected] (L. Seeholzer).

https://doi.org/10.1016/j.jmatprotec.2019.116489 Received 24 June 2019; Received in revised form 24 October 2019; Accepted 29 October 2019 0924-0136/ © 2019 Elsevier B.V. All rights reserved.

Please cite this article as: Lukas Seeholzer, Daniel Scheuner and Konrad Wegener, Journal of Materials Processing Tech., https://doi.org/10.1016/j.jmatprotec.2019.116489

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

Davim (2013), Sheikh-Ahmad (2009) and Sakuma and Seto (1978). Sheikh-Ahmad (2009) explained this correlation by varying loading conditions of the carbon fibres in the machining process, which subsequently result in different mechanisms of material separation. In order to enable a mechanical description of dominant correlations between the workpiece quality, the process parameters and the tool geometry, numerous analytical force modelling approaches for orthogonal machining were presented by the research community. According to Voss et al. (2018), analytical models offer the opportunity to analyse the influence of a single parameter on the resultant process forces without time consuming numerical simulations. Takeyama and Iijima (1988) presented one of the first analytical force models based on the shear-plane theory of Merchant (1945) in order to predict the process forces in orthogonal machining of glass fibre reinforced polymers. Bhatnagar et al. (1995) identified the shear-plane as a consequence of interlaminar cracks between the carbon fibres and thus upgraded the model of Takeyama and Iijima (1988) by setting the shear angle equal to the fibre cutting angle. In contrast to idealised sharp cutting edges in previous publications, Zhang et al. (2001) divided the contact region of the cutting edge into three individual regions: Rake face, cutting edge radius and flank face. For each region, a representative force component based on elastic contact mechanics was determined. Subsequently, the resultant process force was calculated by superposing these individual force components. Whereas the modelling approach of Zhang et al. (2001) fundamentally based on an equivalent homogeneous material consideration for the tool-workpiece interaction, Jahromi and Bahr (2010) used a more detailed multiphase configuration by means of a representative volume element (RVE). Subsequently, Qi et al. (2015) combined the RVE-consideration according to Jahromi and Bahr (2010) with the subdivision of the tool contact region presented by Zhang et al. (2001). Xu and Zhang (2014a, 2016) published an analytical modelling approach to predict the process forces in conventional and vibration-assisted orthogonal machining of unidirectional (UD) CFRP. Similar to Qi et al. (2015) and Jahromi and Bahr (2010), the authors applied a RVE-simplification of the fibrematrix interaction combined with the minimum potential energy principle (MPEP) in order to determine the bending deflection of an individual fibre in front of the cutting edge. In accordance with Henerichs et al., 2015, CFRP machining is characterised by severe mechanical wear, causing significant changes of the micro-geometry of the cutting edge. As shown in detail by Henerichs et al. (2015), changing contact conditions due to worn cutting edges usually result in significantly higher process forces. Consequently, the actual microgeometry has to be considered in the modelling approach to enable a wear-dependent force prediction. In this context, Voss et al. (2018) presented an analytical modelling approach with consideration of the fibre cutting angle, the tool geometry and for the first time the changing micro-geometry due to tool wear using lookup tables based on experiments. Whereas the analytical prediction of process forces in orthogonal machining is extensively investigated by the research community, only little literature work exists with respect to more complex machining operations e.g. drilling. Chandrasekharan et al. (1995) presented a force prediction model for drilling UD fibre reinforced composite materials, where the chisel edge and the cutting lips are modelled separately. Whereas orthogonal cutting conditions were applied for the chisel edge, oblique cutting was considered for the cutting lips. In the model, an infinitesimal force increment at the cutting edge was determined by the product of the chip load and the corresponding specific cutting pressure. Subsequently, the resulting infinitesimal force components were integrated over the cutting lips. Therefore, the specific cutting pressure was described by a radius dependent power law as a function of the chip thickness, the cutting velocity and the rake angle. However, since the influence of the fibre cutting angle was not included, only averaged process forces per revolution were realisable. Another modelling approach presented by Langella et al. (2005) based on the fundamental

assumption, that an infinitesimal area of the oblique cutting situation can be simplified by two-dimensional orthogonal cutting conditions. Subsequently, two semi-empirical coefficients were required to determine the corresponding averaged three-dimensional process forces similar to Chandrasekharan et al. (1995). Since models, predicting averaged process forces, consider neither fibre cutting angle dependent chip formation nor angular dependent failure mechanisms, the characteristic force fluctuations in drilling UD CFRP due to varying cutting conditions are not representable. One of the first analytical models with consideration of the fibre orientation was presented by Guo et al. (2012). Coinciding with Langella et al. (2005), the authors reduced the complex oblique cutting conditions to the orthogonal machining process. In contrast to idealised sharp cutting edge assumptions in most previous publications with respect to drilling UD CFRP, Guo et al. (2012) adopted the three-part subdivision of the contact area previously presented by Zhang et al. (2001). Although the general trends of thrust force and torque values agreed with experimental data, their model simplifications result in maximum errors of 43% and 45% respectively. Meng et al. (2015) combined the modelling approaches form Langella et al. (2005) and Guo et al. (2012) in order to predict thrust force fluctuations in drilling out predrilled UD CFRP material. In contrast to previous publications, four correction factors were used to consider the influences of cutting velocity, feed rate and varying tool radius on the thrust force. This paper deals with an analytical approach to predict process forces in drilling out predrilled UD CFRP material under consideration of material properties, process parameters and fibre cutting angle dependent cutting conditions. Based on high-speed recordings and scanning electron microscopy (SEM), four intervals with similar chip formation mechanisms and two fundamental failure mechanisms of the carbon fibres in front of the cutting edge are identified and considered in the modelling structure. With respect to the contact situation at the cutting edge micro-geometry, both, an initial fibre separation in front of the cutting edge and an elastic spring back of the CFRP material on the flank face are taken into account. Table 1 represents the nomenclature used in the paper. 2. Experimental setup and CFRP material All drilling experiments are conducted on a Mikron VC1000 threeaxis machining centre, whereby the experimental setup is subdivided into chip formation and force/torque measurement analysis as schematically shown in Fig. 1. The high-speed camera type Vision Research Phantom V12.1 is used to analyse chip formation and failure mechanisms as functions of the fibre orientation in drilling UD CFRP. Therefore, the camera is focussed indirectly on the bottom of the CFRP specimen by using a tilted mirror plane. For the chip formation analysis, the UD CFRP specimen are prepared in the dimensions 35 mm × 35 mm × 6 mm and are subsequently surrounded by two acryl glass layers (type A). In accordance with Fig. 1, all three components are fixed on the lateral faces by adhesive strips to avoid relative shifting. The acryl glass support layers prevent push-out delamination at the bore exit and therefore create cutting conditions in the focal plane of the camera similar to an arbitrary CFRP layer within the bore. Furthermore, the lower acryl glass layer in feed direction ensures a clear view without disturbing chip particles. Additionally, a suction unit Esta Dustomat-16 M/V400 V is applied for evacuation of powderlike CFRP particles at the bore entrance. A serial arrangement of a bright Nikon Micro-Nikkor (105 mm) lens and a teleconverter enables high-quality recordings with a working distance of 20 cm and a reproduction scale of 2:1. The high-speed recordings of the chip formation are conducted with two different cutting velocities with respect to the cutting edge corner, namely vc = 30 m/ min (n = 1504 rev/min) and vc = 90 m/min (n = 4511 rev/min). In combination with a frame rate of 28′000 fps, this choice results in an 2

3

Tool-fibre contact correction factor (micro-buckling), in intervals I-III / interval IV Strain rate correction factor for tensile strength of the fibre

KMB / KMB_t KTf

Fpress_f gax Gm k, s, h Kbc KEm Kij

Force component due to the elastic spring back of the CFRP material (friction) Ratio between Fa and Fl Shear modulus of the matrix material Substitution variables Correction factor bouncing back height Strain rate correction factor for Young’s modulus of matrix Body curvatures

Resultant force of the initial fibre separation Force component due to the elastic spring back of the CFRP material

Fres_init Fpress

Fl Fl_max Fmp

Ffeed_tot

Axial force component (initial fibre separation) Critical (axial) buckling force in intervals I-III / in interval IV (initial fibre separation) Force components due to the initial fibre separation (in cutting and feed direction) Force components due to the elastic spring back of the CFRP material (in cutting, feed and radial direction) Overall thrust force (considering both, initial fibre separation and elastic spring back of the CFRP material) Lateral force component (initial fibre separation) Critical (lateral) force (initial fibre separation) Compressive force considering σmp and ARVE

Fa Fa_max / Fa_max_t Fcut_init, Ffeed_init Fcut_press, Ffeed_press, Frad_pres,

[-] [-]

[N] [-] [MPa] [-] [-] [-] [-]

[N] [N]

[N] [N] [N]

[N]

[N] [N] [N] [N]

[MPa] [MPa] [mm]

Young’s modulus of carbon fibre (axial) Young’s modulus of matrix at actual strain rate / at low strain rate (datasheet) General feed rate / feed rate per revolution

Ec* Ef Em_real / Em f / fz

[μm] [μm] [μm] [μm] [-] [-] [MPa]

bcOM for Φ = 30° bcOM for Φ = 60° bcOM for Φ = 90° bcOM for Φ = 150° Effective cutting direction Eccentricity of the ellipse Young’s modulus of bounced back CFRP material on the flank face

bcOM30° bcOM60° bcOM90° bcOM150° dc e

[μm] [mm] [mm2] [μm] [μm] [μm] [μm] [μm]

Major / minor semi-axes of the contact ellipse Radial extension per half revolution Cross sectional area of the RVE Bouncing back height Corrected bouncing back height Bouncing back height of the previous cutting edge Bouncing back height measured in orthogonal machining bcOM for Φ = 0°

ael /bel ap ARVE bc bc_model bco bcOM bcOM0°

Table 1 Nomenclature.

Δrcont Ω

σ σmp σTf_real / σTf σx, σy, σz, φfib Φ ΦR, Φr

νf, νm

λ µ νXG, νYG, νZG

η

αFP δ ζ1, ζ2 ηMB

αTOP, βTOP, γTOP αWP, βWP, γWP αCD, βCD, γCD

rpd rtool rstep RVE vc vf w

lcut lq Mcut_tot n p0 r, φ R rf / rm

Tool tip angle Representative compressive strength Tensile strength of fibre at actual strain rate / at low strain rate (datasheet) Principal stresses along the z-axis Volume fraction of carbon fibres / matrix in CFRP Fibre cutting angle Fibre cutting angle at the nominal diameter with the radius r, at an arbitrary radius position r Range of the main cutting edge, which is in contact with the material Intermediate angle

Inclination angle Friction coefficient Components of the cutting direction with respect to the global XGYGZG coordinate system Poisson’s ratio of the fibre and matrix material Parameter for matrix slippage in micro-buckling

Effective cutting speed angle

Clearance angle in the fibre plane (FP) Complementary angle of the drilling tool Angles to describe the relative arrangement of X’-axis to the cylinder axis X1 Parameter for interfacial fibre-matrix bonding

Clearance -, wedge- and rake angle in the tool orthogonal plane (TOP) Clearance -, wedge- and rake angle in the working plane (WP) Clearance -, wedge- and rake angle in effective cutting direction

Cutting length (orthogonal machining) Distance from the tool/fibre contact point to the fibre end Overall torque (considering both, initial fibre separation and elastic spring back) Revolution speed Average pressure within the contact region Radial and azimuthal angle coordinate Radius of the nominal bore diameter Radius of carbon fibre / idealised radius of RVE (fibre embedded in matrix material) Radius of the predrilled bore Cutting edge radius tool Radial distance between tool-fibre contact points Representative volume element Cutting speed Feed speed Half of the web thickness

[°] [°]

[°] [MPa] [MPa] [MPa] [-] [°] [°]

[-] [-]

[°] [-] [mm]

[°]

[°] [°] [°] [-]

[°] [°] [°]

[mm] [μm] [μm] [-] [m/min] [m/min] [mm]

[m] [μm] [Ncm] [rev/min] [MPa] [mm], [°] [mm] [μm]

L. Seeholzer, et al.

Journal of Materials Processing Tech. xxx (xxxx) xxxx

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

Fig. 1. Schematic illustration of the experimental test rig on the machine table separated into chip formation and force/torque measurement.

angular resolution of 0.3° and 1° respectively. The tool diameter of 6.35 mm is constant for the optical analysis of chip formation. This work focuses on the analytical prediction of process forces caused by the main cutting edges in drilling UD CFRP. In order to validate the simulation, thrust force and torque measurements are implemented, using a Kistler dynamometer type 9121. Since the influence of the chisel edge is neglected in the modelling approach, the drilling out operation of predrilled UD CFRP material is analysed. In this context, the tool diameters are 4.17 mm for the predrilling and 6.35 mm for the consecutive drilling out operation. Analogously to the chip formation measurements, the CFRP specimen are prepared in the dimensions 35 mm x 35 mm x 6 mm, but without acryl glass layers (type B). Within the scope of the experiments, UD CFRP sheet material type MTM44-1/HTS(12k)-134-35%RW, with a constant fibre orientation in all layers is drilled by cemented carbide drills with a nano-crystalline diamond coating of 8 ± 2 μm thickness. This material contains 35% high performance matrix material MTM44-1 and 65% carbon fibres type HTS(12k). The used drilling tool geometry is characterised by a tip angle of σ = 123° and straight main cutting edges. Whereas the clearance angles of the main cutting edges are constant in the tool orthogonal plane (TOP), the corresponding rake and wedge angles are radius dependent due to the manufacturing process. In accordance with most industrial applications, the feed direction is set perpendicular to the CFRP layer and thus to the fibre orientation.

detailed description of the resulting tool-workpiece contact and related loading situation with respect to individual carbon fibres in front of the cutting edge. Finally, Section 3.3 combines the content of both previous sections with experimental chip formation analysis by means of highspeed recordings and scanning electron microscopy (SEM) in order to formulate material failure mechanisms with respect to space-dependent contact and loading conditions. 3.1. Radius and azimuthal angle dependent cutting conditions In machining UD CFRP, the contact situation of an individual carbon fibre in front of the cutting edge depends on the relative arrangement of the fibre orientation and the direction of the cutting velocity. Their two-dimensional arrangement in the fibre plane is explicitly characterised by the fibre cutting angle Φ, which is schematically shown in Fig. 2 (a). Whereas the fibre cutting angle is constant in orthogonal machining UD CFRP, this value changes during the drilling process due to the rotatory cutting motion and the web thickness 2w of the drilling tool. For each point on the cutting edge, the entire range of the fibre cutting angle 0°≤Φ≤180° is passed twice during one tool rotation. In order to enable a consistent description in the subsequent explanations, two different coordinate systems are introduced. First, a global Cartesian coordinate system XGYGZG with its origin in the bore centre is defined. The ZG-axis is oriented in feed direction and the XGaxis is aligned with the fibre orientation. In accordance with Section 2, the machined CFRP material is characterised by a constant fibre orientation in all layers, whereby the definition of the XG-axis is valid for the entire workpiece. Secondly, a polar coordinate system with the radial coordinate r and the azimuthal coordinate φ is introduced. Due to the predrilling operation, a limited range Δrcont of the main cutting edge is in contact with the CFRP material during the drilling out process. Since the main cutting edges are parallel shifted to the radial centre line gc due to the half web thickness w as schematically shown in Fig. 2 (b), the direction of the cutting velocity and thus the fibre cutting angle change along the cutting edge. Consequently, the fibre cutting angle for an arbitrary point on the cutting edge during the drilling process depends on the radial coordinate r as well as the actual position of the cutting edge and thus the azimuthal angle φ. In the following, the parameter ΦR is used for the fibre cutting angle at the nominal diameter with the radius R. Starting from ΦR = 180° at φ = 0, the fibre cutting angle at the nominal diameter decreases with increasing φ and thus tool rotation as schematically shown Fig. 2 (b). In contrast to ΦR, the

3. Fundamentals in drilling UD CFRP In contrast to orthogonal machining, drilling UD CFRP is characterised by a complex three-dimensional arrangement of the cutting edge and the carbon fibres, whereby the tool-fibre contact situation continuously changes due to the fundamental rotatory cutting motion. Furthermore, the combination of varying tool angles along the cutting edge and a radial changing cutting velocity results in radius dependent cutting conditions. In accordance with Voß (2017), the chip formation and the corresponding process forces significantly depend on the actual tool-workpiece contact situation and the corresponding cutting conditions. Consequently, a detailed description of radius and azimuthal angle dependencies of the cutting conditions in drilling UD CFRP is crucial in order to formulate appropriate loading conditions and thus material failure criteria in the subsequent force modelling approach. In this context, the following Section 3.1 focuses on radius and azimuthal angle dependent cutting conditions with respect to the fibre orientation and the cutting edge angles. Subsequently, Section 3.2 deals with a 4

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

Fig. 2. (a) Definition of the fibre cutting angle Φ, (b) varying fibre cutting angle in drilling UD CFRP (fibre cutting angle at the nominal diameter with the radius R: ΦR, fibre cutting angle at an arbitrary radius position r: Φr).

means of least square method (R2 = 97.5%). Subsequently, the corresponding rake angle can be determined according to (4).

parameter Φr represents the fibre cutting angle at an arbitrary radius position r on the cutting edge. In accordance with Schütte (2014), the fibre cutting angle Φr at an arbitrary radius position r on the cutting edge is a function of the half web thickness w, the radius R of the nominal diameter and the fibre cutting angle at the nominal diameter ΦR. Consequently, each fibre cutting angle on the cutting edge is calculated in relation to the fibre cutting angle at the nominal diameter. r

(r ) =

R

+ (| R|

| r (r )|) =

R

+ sin

1

w R

sin

1

w r

TOP (r )

TOP

0.43

= 9° for rpd

TOP (r )

(1)

= 76.67r

= 90°

for rpd

r

TOP (r )

r

(2)

R

(3)

R

TOP

for rpd

r

R

(4)

In orthogonal machining, the tool orthogonal plane (TOP) coincides with the working plane (WP), which is spanned by the cutting velocity and the feed direction. Hence, the resulting tool-workpiece contact situation at the cutting edge can be described by the cutting edge angles in the tool orthogonal plane determined in (2) - (4). A schematic illustration of the cutting situation in orthogonal machining with an exemplary rake angle is shown in Fig. 4 (a). In drilling, the tool orthogonal plane and the working plane are twisted due to the web thickness of the tool. According to Fig. 4 (b), the resulting twist around the feed axis is radius dependent and corresponds to the actual inclination angle λr. Accordingly, the visible wedge geometry gvis for a carbon fibre, which is in contact with the cutting edge in point L, corresponds to the line of intersection between the working plane and the tool. In order to determine the adjusted clearance angle αWP with respect to gvis, the primary clearance angle αTOP is geometrically transformed form the tool orthogonal plane into the working plane as a function of the tool tip angle σ and the inclination angle λr.

The parameter λ in (1) represents the inclination angle projected in feed direction. In agreement with Fig. 2 (b), the projected inclination angle changes along the cutting edge and thus is a function of the radial coordinate r. Based on (1), the drilling out operation from Ø = 4.17 mm to Ø = 6.35 mm with a half web thickness of w = 0.92 mm results in a shift of the fibre cutting angle of nearly 9° along the cutting edge. The rake and clearance angles change along the cutting edge due to the helix grinding operation in the manufacturing process. In this context, Fig. 3 shows the measured cutting edge angles along the cutting edge with respect to the tool orthogonal plane (TOP). During the drilling out operation, the predrilled bore with a radius rpd is expanded to the nominal bore diameter with the radius R. As already mentioned in Section 2, the used drilling tool is characterised by a constant clearance angle of αTOP = 9° in the tool orthogonal plane. In contrast, the corresponding wedge angle βTOP and rake angle γTOP are radius dependent and thus functions of the radial coordinate r. With respect to the subsequent modelling approach, the radius dependence of the wedge angle is approximated by the power function in (2) by

Fig. 3. Measured cutting edge angles in the tool orthogonal plane (TOP) along the cutting edge and their approximated trend functions. 5

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

Fig. 4. Schematic illustration of the tool orthogonal plane (TOP) and working plane (WP) for (a) orthogonal machining and (b) drilling; (c) compensation of the effective cutting speed angle η.

WP (r )

(6) (7)

ap = fz tan

1

1

= cos

corresponding cutting edge angles in the tool orthogonal plane are approximated by the trend functions (2)-(4) presented in Fig. 3. This section focusses on the consequences of the varying contact conditions on the loading situation of individual carbon fibres in front of the cutting edge. In accordance with Voss et al. (2018), the actual fibre cutting angle is essential for the predominant failure mechanism of the CFRP material in the contact region and thus the corresponding chip formation. Fig. 5 shows a schematic illustration of tool-fibre contact situations at an arbitrary radial coordinate r for exemplary fibre cutting angles: Φr = 180° ( = 0°), 135°, 90° and 45°. The individual contact situations (i)-(iv) are shown within an arbitrary fibre layer and thus in the XGYG-plane. Consequently, the pictured contour lines of the cutting edge in Fig. 5 correspond to the projection of the cutting edge into the fibre plane. In drilling out UD CFRP, the feed motion of the cutting tool results in a continuous radial extension of the bore until the nominal diameter is reached. With respect to a single cutting edge, this radial extension per half revolution ap is a function of the feed per tooth fz and the tool tip angle σ.

(

1 + tan 90°

2

) sin(

2 r

(r )) + tan(

TOP )

cos( r (r ))

(

cos 90°

2

) (5)

In this context, the clearance angle in the working plane is not constant anymore due to the radius dependence of λr. Analogous to (5), the adjusted wedge angle βWP in the working plane is determined as a geometric transformation from the tool orthogonal plane into the working plane. Finally, the corresponding rake angle γWP can be determined as an angular addition of αWP and βWP to 90°.

WP (r )

= cos

1

1

(

1 + tan 90°

2

) sin( r (r )) + tan( TOP +

2 TOP (r ))

cos( r (r ))

(

cos 90°

WP (r )

WP (r )

= 90°

WP

(r )

WP (r )

2

)

The resulting cutting edge angles αWP, βWP and γWP describe the tool-workpiece contact situation in the working plane and thus with respect to gvis. However, since the effective cutting direction dc is represented by a vector addition of the feed rate and the corresponding radius dependent cutting velocity, an additional influence of the effective cutting speed angle η on the cutting edge angles has to be considered. In accordance with Fig. 4 (c), the effective clearance angle is reduced by η, whereas the effective rake angle is enlarged by the same value and the wedge angle remains unchanged.

(r ) = tan

1

f

2r

CD (r )

=

WP (r )

CD (r )

=

WP

CD (r )

=

(r )

WP (r )

(9)

+ (r )

(12)

In accordance with Wang and Zhang (2003), machining CFRP is characterised by elastic spring back effects of the compressed CFRP material after passing underneath the cutting edge. In this context, the elastic recovery of the just cut CFRP material causes additional toolworkpiece contact on the flank face of the drilling tool. According to Voss et al. (2018), the resulting contact region on the flank face is characterised by the actual clearance angle of the cutting edge and the bouncing back height bc. In accordance with Fig. 5, Voss et al. (2018) define the bouncing back height bc as the radial distance between the foremost point A of the cutting edge in direction of the radial extension and the last tool-workpiece contact point C on the flank face due to the elastic spring back of the CFRP material. During the drilling out operation, the CFRP material at the bore edge is re-processed after one half tool rotation, whereby the theoretical cutting depth per cutting edge equals to ap. According to Fig. 5, the elastic spring back of the CFRP material with respect to the previous cutting edge results in a bouncing back height bco and thus in a difference between the programmed and the actual cutting depth. In this modelling approach, the bouncing back height bco of the previous cutting edge is assumed equal to the bouncing back height bc of the current cutting edge after one half tool rotation. Based on orthogonal machining experiments, Voss et al. (2018) and Seeholzer et al. (2018) proved a strong dependence of the bouncing back height on the actual fibre cutting angle. Whereas for Φ = 150° minor spring back effects are detected, machining Φ = 30° and Φ = 60° results in significantly higher values of bc. Furthermore, the authors showed a strong dependence of the process forces on the elastic spring back and thus the bouncing back height. Accordingly, the fibre cutting angle dependent elastic spring back effects have to be considered in the force modelling approach for drilling UD CFRP.

(8)

(r )

( 2)

(10) (11)

3.2. Fibre cutting angle dependent contact and loading situation In accordance with Section 3.1, the fibre cutting angle and the tool micro-geometry are radius and azimuthal angle dependent during the drilling out operation. Up to this point, the fibre cutting angle is described as a function of the radial coordinate r and the fibre cutting angle ΦR at the nominal diameter. Furthermore, the cutting edge angles in the working plane and under consideration of the effective cutting direction are determined by (9)-(11) as functions of the radial coordinate r, the feed rate f, the tool tip angle σ, the half web thickness w and the radius of the nominal diameter R. For this purpose, the 6

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

Fig. 5. Schematic illustration of the tool-workpiece interaction at exemplary fibre cutting angles projected into the fibre plane.

Focussing on an exemplary tool-fibre contact point on the cutting edge, the instantaneous direction vector of the tool motion is represented by the effective cutting direction, which is a vectorial composition of the cutting velocity and the feed rate. Based on an idealised square packing of the carbon fibres, an individual fibre is equally supported in all directions, while it is deformed by the cutting edge in effective cutting direction. In this modelling approach, a temporary fixed tool-fibre contact point is assumed. Furthermore, it is assumed that the direction of fibre deformation and the corresponding loading direction are equal to the effective cutting direction. With respect to the fixed fibre orientation in UD CFRP, the effective cutting direction

always can be divided into a vectorial component axial to the fibre and a vectorial component perpendicular to the fibre. Therefore, the effective cutting direction is initially divided into its XG-, YG- and ZG-components with respect to the global Cartesian coordinate system.

vXG (r ) =

vc (r )cos(

vYG (r ) = vc (r )sin( vZG = vf = n f

r

r

(r )) for rpd

(r )) for rpd

r

r

R

R

(13) (14) (15)

Whereas the combination of vYG and vZG represents a two7

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

dimensional bending deformation in the YGZG-plane, the vector component vXG corresponds to an axial compression (-) or tension (+) deformation of the carbon fibre in front of the cutting edge. In accordance with the schematic illustration of the tool-fibre contact situation for Φr = 180° in Fig. 5 (i), the initial tool-workpiece contact point E equals to the foremost point B of the projected cutting edge contour in cutting velocity direction. With respect to (14), the deformation component vYG vanishes for ΦR = 180°. Consequently, the resulting loading direction is described by an axial compression due to vXG combined with a bending deformation in feed direction due to vZG. However, with respect to the process parameters in the drilling out process and the used drilling tool geometry, the percentage component of the feed motion to the effective cutting velocity is 0.5% at the nominal diameter with the radius R and 0.76% at the predrilled bore diameter with the radius rpd. Hence, the influence of the bending deformation in feed direction is comparatively small, whereby the fibre loading situation for Φr = 180° as well as the corresponding fibre separation are dominated by an axial fibre compression with force transmission at the fibre-end. Starting from Φr = 180°, the effective cutting direction changes with positive tool rotation φ. Since the fibre orientation is constant for UD CFRP, the changing effective cutting direction results in an overall decreasing fibre cutting angle. In accordance with (13)-(15), the vectorial component of the effective cutting direction perpendicular to the fibre increases at the expense of the axial compression component vXG. Accordingly, the loading situation of an individual carbon fibre is characterised by a combined bending and compressive load, whereby the percentage component of bending increases with positive tool rotation φ. According to Voss et al. (2018), machining fibre cutting angles between Φr = 180° and Φr = 90° results in a characteristic saw teeth topography at the bore edge. Similar results are detected for orthogonal machining UD CFRP with a fibre cutting angle Φ = 150° by Henerichs et al. (2015), whereby both, the width and depth of the saw teeth depend on the fibre cutting angle, the cutting edge angles and the process parameters. As a consequence of the saw teeth topography, the cutting edge gets in contact with the oblique flank of a saw tooth, whereby the force transmission point E remains at the fibre-end. A schematic illustration of the resulting contact situation for an exemplary fibre cutting angle of Φr = 135° is shown in Fig. 5 (ii). With further tool rotation and decreasing fibre cutting angle, the vectorial component vXG of the effective cutting direction continuously decreases and is equal to zero at Φr = 90°. Consequently, the carbon fibre in front of the cutting edge is exclusively deformed perpendicular to its longitudinal orientation due to vYG and vZG. Furthermore, the force transmission point E is no longer at the fibre-end, but shifted by the distance lq along the fibre. A schematic illustration of the representative tool-fibre contact situation for a fibre cutting angle of Φr = 90° is shown in Fig. 5 (iii). The distance lq is a function of the radial extension per half revolution ap, the bouncing back height bc and the actual fibre cutting angle Φr.

l q (r ) =

ap + bc sin(

r (r ))

=

fZ tan( sin(

2)

+ bc

r (r ))

for 90°

CD (r )

r (r )

with the chip formation analysis in the following section. 3.3. Chip formation analysis and formulation of failure criteria In order to formulate appropriate and fibre-cutting angle dependent material failure criteria for the subsequent mechanical force modelling approach, a detailed understanding of the occurring chip formation mechanism and its relation to the fibre loading situation is required. In the following, the theoretical considerations from Sections 3.1 and 3.2 are combined with experimental high-speed recordings of the chip formation process and bore edge analysis by means of scanning electron microscopy (SEM). According to Section 2, a clamping device combined with a mirror and a high-speed camera is used to record chip formation for a fixed feed rate of f = 0.1 mm/rev and two cutting velocities vc = 30 m/min (n = 1504 rev/min) and vc = 90 m/min (n = 4511 rev/min). Furthermore, additional detailed high-speed recordings with vc = 90 m/min and a higher feed rate of f = 0.2 mm/rev are used in order to increase the radial extension ap and thus to improve the visibility of the toolworkpiece interaction. Comparison tests have shown that doubling the feed rate has an influence on chip size and process forces but not on fundamental chip formation mechanisms. In accordance with Section 2, the CFRP specimen are surrounded by two acryl glass layers in order to create cutting conditions in the focal plane similar to an arbitrary CFRP layer within the bore. Furthermore, the existence of supporting acryl glass layers enable a conservation of chip fragments next to the bore edge. This allows subsequent chip analysis with respect to the tool rotation and the fibre cutting angle. Fig. 6 shows a snapshot of a highspeed recording, where an exemplary point of the cutting edge with the radial coordinate r (rpd < r < R) is in contact with the CFRP layer in the focal plane. By focussing on the mirror, the rotation direction changes, whereby the coordinate systems are mirror-inverted. As discussed in the following, four intervals of fibre cutting angles (I, II, III and IV) with similar chip formation mechanisms are identified by means of the highspeed recordings and the chip analysis. These four intervals are schematically shown and numbered in Fig. 6. In the following subsections, each interval is discussed separately with respect to the chip formation and the corresponding failure mechanisms. 3.3.1. Interval I (180°≥Φr≥170°) Interval I covers fibre cutting angles in a narrow range of 180°≥Φr≥170°. In accordance with the theoretical considerations in Section 3.2, in this range, the carbon fibres in front of the cutting edge are mainly loaded by axial compression since the main part of the effective cutting direction is oriented in XG-direction. Simultaneously, the superimposed feed motion causes a bending deformation perpendicular to the fibre orientation in feed direction. However, since the percentage component of the feed motion to the effective cutting velocity is small, the influence of the superimposed bending deformation is assumed negligible in comparison to the axial compression. The conducted highspeed recordings show no bending deformations of individual fibres or fibre bundles in front of the cutting edge in feed direction. Instead, the CFRP material in front of the cutting edge abruptly disappears without observable lateral fibre deflections. An exemplary snapshot of the toolworkpiece interaction for a fibre cutting angle of Φr = 180° is shown in Fig. 7 (a). Consequently, the experimental chip formation analysis confirms the theoretical loading condition described in Section 3.2, whereby fibre failure for fibre cutting angles close to Φr = 180° is dominated by axial compression. Fig. 7 (b) shows a detailed SEM image of the bore edge, whereby isolated fibre fractures are detected in the surrounding CFRP material. These fractures are assumed to be the result of a subsequent compression of the previously cut fibres underneath the cutting edge. Based on the subsequent chip analysis in interval II, mainly small chip particles with chip length between 20 μm and 60 μm are produced.

< 0° (16)

Since the fibre cutting angle is radius dependent in accordance with (1), the distance lq varies along the cutting edge. The change of an endloaded, to a not end-loaded fibre depends on the actual rake angle γCD. Starting from Φr = 90° with positive tool rotation, the vectorial component of the effective cutting direction perpendicular to the fibres decreases, whereas the corresponding component parallel to the fibre orientation increases again. However, in contrast to 180°≥Φr≥90°, the vectorial component parallel to the fibre orientation vXG is contrary oriented to the XG-axis and therefore represents a tensile deformation. A schematic illustration of the resulting contact situation for an exemplary fibre cutting angle of Φr = 45° is shown in Fig. 5 (iv). The influence of this loading situation is discussed in detail in combination 8

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

Fig. 6. Exemplary snapshot of the drilling process for the radial coordinate r (rpd < r < R), schematic illustration of the four intervals I, II, III and IV with similar chip formation mechanisms.

Fig. 7. (a) Detailed snapshot of the resulting cutting situation with a fibre cutting angle Φr = 180° (vc = 90 m/min, f = 0.2 mm/rev); (b) SEM image of the bore edge (vc = 90 m/min, f = 0.1 mm/rev).

In accordance with Voss et al. (2018), carbon fibres, which are predominantly loaded by axial compression, fail by micro-buckling in front of the cutting edge. Mayer et al. (2009) describe micro-buckling as a coordinated structural failure of multiple carbon fibres in front of the cutting edge due to a simultaneous buckling action of large portions of adjacent carbon fibres. In the modelling approach, material failure due to micro-buckling is considered by means of a critical buckling load, which is based on a theoretical compressive strength, presented by Xu and Reifsnider (1993).

discontinuous tool-workpiece contact on the process forces is hardly measurable due to the radial shift of the fibre cutting angle along the cutting edge and limitations of the force measuring infrastructure. Furthermore, the saw teeth topography represents a modification of the circular shape of the bore edge, whereby the cutting edge gets in contact with the oblique flank of a saw tooth as schematically shown in Fig. 5 (ii). Thereby, the fibres in the close-up region of contact point E are axially compressed, comparable to the loading situation in interval I. The snapshot of the corresponding high-speed recordings in Fig. 8 (a) shows bending deformations of entire fibre bundles. In this context, several carbon fibres in the close-up region of contact point E are assumed to fail by micro-buckling due to axial compression. Subsequently, this localised fibre separation results in an advancing interlaminar crack with its origin in point E. Furthermore, this interlaminar crack, combined with the cutting motion, causes a bending deformation of the overlying, intact fibre bundle contrary to the direction of the radial extension. With an increasing bending deformation of the fibre bundle, the lowest carbon fibre fails at its outer range. Subsequently, the occurring crack propagates into the adjacent fibres, whereby the entire fibre bundle breaks out. Based on the chip analysis, chip blocks containing several fibre fragments are produced with entire chip length between 60 μm and 110 μm. In accordance with Fig. 9, the partial removal of the saw teeth with respect to the previous bore edge topography corresponds to a displacement of the saw teeth topography contrary to the tool rotation. According to Voss et al. (2014), this chip formation mechanism is characterised by the overall highest surface roughness due to the saw teeth topography. Although the chip formation and the chip size are different in intervals I and II, in both cases, the initial fibre separation and thus the initiation of the chip formation are caused by microbuckling due to axial compression. Additional force components due to the bending deformation of individual fibre bundles are not taken into account in the modelling approach. Consequently, exceeding

3.3.2. Interval II (170° > Φr > 100°) Based on the theoretical considerations in Section 3.2, the loading situation of carbon fibres between Φr = 180° and Φr = 90° is described by a superposition of bending and compressive loads. Starting from Φr = 180°, the decreasing fibre cutting angle results in an increasing bending load perpendicular to the fibre at the expense of the axial compression. A snapshot of the tool-workpiece interaction for an exemplary fibre cutting angle of Φr = 135° is shown in Fig. 8 (a). In addition to the theoretical loading situation of the carbon fibres in front of the cutting edge, the influence of the characteristic saw teeth topography on the chip formation has to be considered. A detailed SEM image of the resulting saw teeth topography is shown in Fig. 8 (b). Based on high-speed recordings, saw teeth are identified for fibre cutting angles in a range of 100° < Φr < 170°. Within this range, the width and the depth of the saw teeth change. Whereas both parameters reach their maximum for a fibre cutting angle close to Φr = 135° and thus in the middle of interval II, they decrease to zero in direction of the adjacent intervals I and III (Φr = 170° and Φr = 100°). Consequently, the transition to interval I is fluently and fundamentally defined by the first existence of saw teeth at the bore edge. In accordance with Fig. 8 (a), the radial depth of occurring saw teeth partially exceeds the radial extension ap. Therefore, the tool-workpiece contact at the bore edge within one fibre layer is discontinuous. However, the influence of a 9

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

Fig. 8. (a) Detailed snapshot of the resulting cutting situation with a fibre cutting angle Φr = 135° (vc = 90 m/min, f = 0.2 mm/rev); (b) SEM image of the bore edge (vc = 90 m/min, f = 0.1 mm/rev).

Fig. 9. Displacement of the saw teeth topography contrary to the tool rotation; (a) before and (b) after machining (vc = 90 m/min, f = 0.2 mm/rev).

compressive strength by a critical buckling load, analogously to interval I, is considered as dominant material failure criterion in interval II.

the dimension of the radial extension ap = 92 μm are produced. This coincides with analysed chip length between 50 μm and 110 μm of isolated fibre fragments. Although the pure bending condition is only valid for the exact fibre cutting angle Φr = 90°, fibre failure by exceeding tensile strength dominates in a wider range. Whereas the transition to the adjacent interval II is defined by the disappearance of the saw teeth topography, the transition to interval IV is not that clear as explained in the following. Starting from Φr = 90° in positive tool rotation, the loading situation of an individual carbon fibre changes again due to the geometrical arrangement of the fibre orientation and the effective cutting direction. In accordance with Section 3.2, the vectorial component of the effective cutting direction axial to the fibre vXG increases at the expense of the lateral component perpendicular to the fibre, which is a vectorial combination of vYG and vZG. In contrast to previous cutting conditions, the vectorial component axial to the fibre vXG is oriented in negative XGdirection and thus corresponds to a tensile deformation. In this context, the maximum transmissible tensile force between the fibre and the cutting tool is theoretically limited by friction. However, occurring “hook-effects” between the cutting edge and the fibres during the cutting process due to micro-deformations are assumed to enable the transmission of significantly larger tensile forces. Based on the highspeed recordings, a decreasing lateral fibre deflection in front of the cutting edge is detected starting from Φr = 90°. For a fibre cutting angle close to Φr = 40°, these lateral fibre deflections are no longer visible with the available optical resolution. As long as the bending deformation dominates fibre failure, the corresponding fibre cutting angles are assigned to interval III and otherwise to interval IV, which is explained in the following subsection.

3.3.3. Interval III (100°≥Φr≥40°) According to the simplifications in Section 3.2, the loading situation of an individual fibre for Φr = 90° is characterised by pure bending. Whereas the bending direction is mainly influenced by the cutting velocity due to the vectorial composition of the effective cutting direction, a comparably small component of the effective cutting direction in feed direction results in a two-dimensional bending deformation. A detailed snapshot of the recorded chip formation for Φr = 90° is shown in Fig. 10 (a). The evaluation of the high-speed recordings confirms lateral fibre deflections in front of the cutting edge. However, in comparison to interval II, the maximum fibre deflections are significantly reduced. This is explained by a changing chip formation from interval II to III as explained in the following. In interval II, an advancing interlaminar crack, in combination with a large bending radius, enables large bending deflections of entire fibre bundles without material failure. In interval III, an individual carbon fibre is bent perpendicular to its longitudinal orientation. In accordance with Fig. 5 (iii), the support effect of the adjacent CFRP material retards the lateral fibre deflection, resulting in high, localised contact stresses at the tool-fibre contact point E. In this context, the detailed SEM image of the bore edge in Fig. 10 (b) shows only marginal visible fibre cracks in the surrounding CFRP material. This supports the assumption that the carbon fibres fail close to the tool-fibre contact point. Furthermore, the surrounding matrix material shows only isolated defects, which implies that the carbon fibres already fail for small lateral deflections. In accordance with Fig. 10 (b), the fibre ends at the bore edge are partially isolated and often characterised by a straight fracture surface perpendicular to the fibre orientation. In accordance with Schmitt-Thomas (2016), this fracture behaviour is typical for fibre failure by tensile loads. In the modelling approach, induced tensile stresses in the tool-fibre contact area are assumed to cause fibre fracture already for small lateral deflections. Consequently, mainly single chip particles with chip length in

3.3.4. Interval IV (40° > Φr > 0°) Interval IV describes the chip formation in the range of fibre cutting angles between approximately Φr = 40° and Φr = 0°. Whereas Fig. 11 (a) shows a snapshot of the high-speed recordings for an exemplary fibre cutting angle of Φr = 30°, detailed SEM images of the bore edge are presented in Fig. 11 (b). 10

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

Fig. 10. (a) Detailed snapshot of the resulting cutting situation with a fibre cutting angle Φr = 90° (vc = 90 m/min, f = 0.2 mm/rev); (b) SEM image of the bore edge (vc = 90 m/min, f = 0.1 mm/rev).

Fig. 11. (a) Detailed snapshot of the resulting cutting situation with a fibre cutting angle Φr = 30° (vc = 90 m/min, f = 0.2 mm/rev); (b) SEM image of the bore edge (vc = 90 m/min, f = 0.1 mm/rev).

4. Modelling approach

Although, the high-speed recordings show no visible lateral fibre deflection for fibre cutting angles in the range 40° > Φr > 0° due to the limited magnification, the elastic foundation is assumed to enable highly localised micro-deformations in the close-up region of the toolfibre contact point E by the cutting edge micro-geometry. The resulting localised temporary tool-fibre fixation is assumed to cause larger contact forces than by pure friction. Assuming this temporary tool-fibre fixation in the close-up region of contact point E, the fibre is predominantly loaded in two different ways. Whereas the fibre section in front of point E is loaded by axial compression, the remaining fibre is loaded by axial tension. According to Sheikh-Ahmad (2009), carbon fibres are characterised by a high tensile strength in longitudinal direction, whereas the corresponding actual compression strength is significantly lower due to stability failure by micro-buckling. It is therefore assumed that deformed fibres in interval IV fail by buckling in front of point E, before the tensile stress in the remaining fibre behind point E exceeds the tensile strength. Consequently, fibre separation due to micro-buckling is considered as failure criterion in interval IV. Fibre separation due to micro-buckling in front of the contact point E, theoretically results in comparatively long remaining carbon fibres, which are still embedded in the remaining CFRP material and therefore are further bent in cutting direction. After passing below the cutting edge, these remaining long fibres show high potential for extensive spring back effects. This agrees well with particularly large bouncing back heights found in orthogonal machining UD CFRP with Φ = 30° (Seeholzer et al., 2018; Voss et al., 2018). In accordance with the detailed high-speed recordings, the tool-workpiece interaction in drilling UD CFRP is characterised by a combination of compressive, tensile and bending loads, whereby their relative composition depends on the actual fibre cutting angle. Based on chip formation analysis by means of high-speed recordings, scanning electron microscopy and chip evaluation, two fundamental failure mechanisms are identified: microbuckling for axial compression and exceeding tensile strength for bending deformation. In combination with the varying fibre cutting angle, these two predominant failure mechanisms result in four different intervals with similar chip formation.

4.1. Modelling concept and assumptions The force prediction model presented in this work enables the determination of the thrust force and torque values in drilling out predrilled UD CFRP during one half tool rotation. In the modelling approach, the composition of the CFRP material is simplified by a representative volume element (RVE) as schematically shown in Fig. 12 (a). The RVE is composed of a carbon fibre, which is surrounded by matrix material. Based on an idealised square packing of the carbon fibres within the CFRP material as shown in Fig. 12 (a), the radius rm of the RVE is determined as a function of the fibre radius rf and the volume fraction of carbon fibres in the CFRP material φfib.

rm =

rf 2

(17)

fib

Based on the RVE consideration, a limited number of carbon fibres is simultaneously in contact with the cutting edge during the drilling out process. For the subsequent force prediction, the fibre cutting angle and the actual tool micro-geometry have to be known for all tool-fibre contact points along the cutting edge. In accordance with (1), the fibre cutting angle is a function of the radial coordinate r, whereas the parallel shift of the cutting edge to the centre line is considered by means of the projected inclination angle. Consequently, the resulting tool-fibre contact points are also determined with respect to the centre axis and are subsequently transformed to the corresponding coordinates on the actual cutting edge. In accordance with Fig. 12 (b), the line of intersection gint between the drilling tool and the centre plane is defined. Subsequently, the radial distance between two adjacent tool-fibre contact points on gint is represented by the radial distance rstep, which is a function of the tool tip angle and the radius of the RVE.

rstep =

11

(

2rm

tan 90°

2

)

(18)

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

Fig. 12. Schematic illustration of the RVE consideration (a) and the tool-fibre contact during the drilling out process (b).

First, the coordinates of all tool-fibre contact points on gint are determined with rstep by starting from the nominal diameter with the radius R. Subsequently, for each tool-fibre contact point, the corresponding fibre cutting angle and cutting edge angles are determined by using (1) and (9)-(11). Later, the thrust and cutting force components are calculated for each of these parameter sets. Subsequently, all force components along the cutting edge are summarised in order to calculate the overall process forces with respect to one cutting edge. By varying ΦR, the overall process forces are determined for different azimuthal angle positions of the cutting edge. In the modelling approach, a rotation increment εrot is used to enable a discrete trend description of the process forces, whereby the azimuthal angle resolution of the simulated trends depends on εrot. During the machining process, the fibre deformation in front of the cutting edge causes fibre fracture of the previously intact CFRP material, whereby the predominant failure mechanism depends on the current tool-workpiece interaction as discussed in detail in Section 3. In the following, the generation of force components at the foremost tool-fibre contact point in cutting velocity direction is defined as the so-called “initial fibre separation”. Consequently, the force components of the initial fibre separation in front of the cutting edge enable material removal and thus are representative for the chip formation. According to Section 3.2, machining CFRP is characterised by an elastic spring back of the CFRP material on the flank face of the drilling tool, resulting in an additional tool-fibre contact with additional force components. In accordance with Voss et al. (2018), the overall process force is a superposition of force components due to the initial fibre separation in front of the cutting edge and to the elastic spring back effects of the CFRP material on the flank face. In the following, both force components are discussed separately. Whereas Section 4.2 focuses on the initial fibre separation in front of the cutting edge, Section 4.3 deals with the influence of elastic spring back effects of the CFRP material on the flank face.

describes a two-dimensional bending deformation perpendicular to the fibre in the YGZG-plane. A schematic illustration of the deformation state with respect to an arbitrary fibre in front of the cutting edge is shown in Fig. 13. In the modelling approach, it is assumed that the resultant force of the initial fibre separation Fres_init is oriented in direction of the fibre deformation in front of the cutting edge and thus in direction of the effective cutting direction. Consequently, the resultant force can be separated into an axial force component Fa and a lateral force component Fl. The relative ratio gax between Fa and Fl can be determined by the vectorial components of the effective cutting direction vXG, vYG and vZG.

Fa = gax (r ) = Fl 1 for + 1 for

vXG _ corr (r ) 2 2 (r ) + vZG vYG r (r ) r (r )

0° > 0°

with

vXG _ corr (r ) = vXG (r )

(19)

With respect to vXG, a cutting angle dependent case-by-case analysis is necessary in order to consider the abruptly changing fibre orientation at Φr = 0° ( = 180°) in the second half of the borehole. In accordance with Section 3, carbon fibres close to Φr = 180° fail by micro-buckling due to axial compression. In the modelling approach, a critical buckling load Fa_max is considered as failure criterion, whereby micro-buckling

4.2. Force components due to the initial fibre separation In accordance with the simplifications described in Section 3, the deformation state of an individual carbon fibre in front of the cutting edge is represented by the vectorial decomposition of the effective cutting direction into the global Cartesian coordinate system: vXG, vYG and vZG (13)-(15). The vectorial component vXG is oriented in XG-direction and thus represents an axial compression (+) or tension (-). Analogously, the combination of the vectorial components vYG and vZG

Fig. 13. Schematic illustration of the tool-fibre contact situation at an exemplary fibre cutting angle. 12

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

occurs as soon as the axial compressive force Fa exceeds the critical buckling load Fa_max. The determination of the critical buckling load is based on the theoretical formulation of the compressive strength of the CFRP material by Xu and Reifsnider (1993). Detailed descriptions of the compressive strength material and the corresponding critical buckling load are explained in Section 4.2.1. For fibre cutting angles close to Φr = 90°, the loading situation of individual carbon fibres is strongly dominated by a bending deformation, which enables a lateral fibre deflection perpendicular to the fibre orientation. In accordance with Section 3, this bending deformation combined with the supporting effect of the adjacent CFRP material results in high, localised contact stresses in the carbon fibre close to the actual tool-fibre contact point. In the modelling approach, exceeding the tensile strength of the carbon fibre for a critical lateral force component Fl_max is considered as failure criterion. In order to determine Fl_max, the tool-fibre contact is modelled as the Hertzian contact between two twisted cylinders. The contact situation between the cylinders strongly depends on their relative arrangement as explained in detail in in Section 4.2.2. For fibre cutting angles in the ranges 180 > Φr > 90° and 90 > Φr > 0°, mixed loading situations occur. In the modelling approach, the incremental force component of the initial fibre separation for an exemplary contact point depends on the critical force component, which is reached first.

is most likely higher than in the uncompressed state, described by φfib. Both, the supporting effect of adjacent fibres and the locally higher volume fraction, have in common to cause an even higher critical buckling load as determined in (21). Therefore, a fitting parameter KMB > 1 is used, whereby the critical buckling load Fa_max for multiple fibres in front of the cutting edge is determined by a multiplication of Fmp and KMB.

Fa _ max = KMB Fmp

Based on parameter fitting to the experiments, KMB = 2 is found to be suitable for the tested cutting conditions and material properties. In accordance with the assumptions in Section 3, the orientation of the effective cutting direction in interval IV represents an axial tensile deformation of an individual carbon fibre. Occurring micro-deformations in the immediate contact region are assumed to cause a localised temporary tool-fibre fixation, whereby the part of the fibre in front of the contact point is axially compressed. Based on this assumption, the carbon fibre initially fails by micro-buckling within the compressed region. The existence of localised micro-deformations is not verified with the conducted high-speed recordings due to a limited feasible magnification. The critical buckling load Fa_max_t in interval IV is determined analogously to (22) but with another fitting parameter KMB_t, in order to consider the changed tool-fibre contact conditions. Based on parameter fitting to the experiments, KMB_t = 3.2 is found to be suitable for the tested cutting conditions and material properties.

4.2.1. Failure criterion I: micro-buckling This failure criterion is based on the structural instability of axially compressed fibres, which according to Mayer et al. (2009), results in a simultaneous, coordinated structural failure of entire fibre regions. In this context, the modelling approach of Xu and Reifsnider (1993) is applied to calculate the representative compressive strength σmp of an individual carbon fibre, which is embedded in the matrix material under consideration of matrix slippage: mp = Gm

fib +

Em (1 Ef

2(1 + m )

Ef

Fa _ max _ t = KMB _ t Fmp

Em fib E + 1 f

fib (1 + fib f + m (1

+1 fib ))

sin( 2

)

(20) In this context, the parameters Gm, Em, Ef, νf and νm represent the shear modulus of the matrix material, the effective Young’s modulus of the matrix material and the fibre and the corresponding Poisson’s ratios. The parameter ηMB describes the fibre-matrix bonding, whereas ηMB = 1 represents a one-sided embedding and ηMB = 2 a both-sided embedding of the carbon fibre (Xu and Reifsnider, 1993). In accordance with Xu and Reifsnider (1993), the parameter ξ describes the share of matrix slippage in relation to the buckling wavelength. Based on experimental validation, Naik and Kumar (1999) found ηMB = 1.98 and ξ = 0.02 to fit well for comparable carbon/epoxy composites and thus are used in this modelling approach. Under consideration of the cross section ARVE of the RVE, the corresponding axial force component Fmp is determined analogously to Voss et al. (2018).

Fmp =

mp

ARVE

=

mp

rm2

(23)

4.2.2. Failure criterion II: contact stresses Whereas failure criterion I considers fibre failure due to an axial compression, failure criterion II focusses on the consequence of a bending deformation of a carbon fibre perpendicular to its longitudinal orientation. In accordance with Section 3.3.3, it is assumed that the bent fibre fails by exceeding tensile strength in the immediate tool-fibre contact region. In order to determine the corresponding critical bending force Fl_max, the tool-fibre contact situation is modelled as Hertzian contact between two cylinders. Whereas the carbon fibre already has a cylindrical shape, the cutting edge rounding of the tool is approximated by a cylinder with a radius equal to the cutting edge radius rtool. In drilling UD CFRP, the carbon fibre and the cutting edge are twisted by an intermediate angle Ω due to the tool tip angle. Fig. 14 (a) shows a schematic illustration of the resulting contact situation for the fibre cutting angle Φr = 90°. In order to enable a consistent description, two body-fixed Cartesian coordinate systems X1Y1Z1 and X2Y2Z2 with their origins in the cylinder contact point T are defined in accordance with Ding et al. (2006). The index 1 is used for the carbon fibre and the index 2 for the cutting edge. The axes Z1 and Z2 are oriented along the common normal of the surface in T and point into the respective bodies. Furthermore, the remaining axes X1, Y1, X2 and Y2 are oriented along the respective principal normal sections of the body surfaces as shown in Fig. 14 (a). During the machining process, the cutting edge is pressed onto the carbon fibre with a bending force Fl, which is oriented in negative Z1direction. The intermediate angle Ω is defined as the angle between the cylinder axes X1 and X2, measured in the XGZG-plane. Since the X2ZGplane and the XGZG-plane are identical for Φr = 90°, the associated intermediate angle Ω corresponds to the complementary angle of the drilling tool δ = 90°- /2 . However, for fibre cutting angles Φr ≠ 90°, the rotation increment Δφ results in a difference between the complementary angle δ and the intermediate angle Ω as schematically shown in Fig. 14 (b). Therefore, the actual intermediate angle Ω corresponds to the projection of δ on the XGZG-plane. Based on the substitutions s, h and k shown in Fig. 14 (b), the actual intermediate angle Ω can be determined as a function of the fibre cutting angle Φr and the complementary angle δ:

fib)

MB r f E 3 m

(22)

(21)

The determination of the critical compressive strength by Xu and Reifsnider (1993) is fundamentally based on a RVE consideration with respect to a single carbon fibre, which is surrounded by matrix material. However, in drilling UD CFRP, multiple carbon fibres are simultaneously in contact with the cutting edge micro-geometry at an arbitrary radial coordinate r. Consequently, the adjacent fibres represent an additional supporting effect and thus have an influence on the critical buckling load, which is necessary to enable micro-buckling. Furthermore, the matrix material in front of the cutting edge is plastically deformed by the tool, whereby the resulting local fibre volume fraction 13

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

Fig. 14. (a) Schematic illustration of the contact situation between two twisted cylinders for Φr = 90°, (b) influence of the rotation increment Δφ on the intermediate angle Ω.

(24)

h = stan( ) k = s cos(

= tan

1

) = s sin(

r

h = tan k

1

(r ))

(25)

tan( ) sin( r (r ))

(26)

is introduced. According to Fig. 15, the X’-axis is oriented along the semi-major axis and the Y’-axis along the semi-minor axis of the elliptical contact area. In accordance with Luré (1964), the angles ζ1 and ζ2 are defined in order to describe the relative arrangement of the X’-axis to the cylinder axis X1 and to the cylinder axis X2 respectively. In order to determine the angles ζ1 and ζ2 as functions of the intermediate angle and the body curvatures of the constituent bodies, the coordinates of both cylinders have to be transformed into the X’Y’Z’-coordinate system (Ding et al., 2006). Therefore, the following transformation rules are used:

According to Hertz (1881), the contact region between two cylinders is characterised by an ellipse, whereby its semi-axes ael and bel strongly depend on the acting contact force, the intermediate angle, the material properties and the body shapes of both constituents. In order to describe the contact situation in contact point T, the body-dependent principal curvatures of the body surfaces Kij have to be known. In this context, the index i represents the body and index j the reference axis in the corresponding coordinate system. In accordance with Ding et al. (2006), the shape of a cylinder in its longitudinal direction corresponds to a radius equal to infinity. Consequently, the corresponding curvatures K11 and K21 become zero. Analogously, K12 and K22 are described with respect to rf and rtool.

K11 = 0

(27)

1 rf

(28)

K21 = 0

(29)

K12 =

K22 =

1 rtool

x1 = x 'cos( 1) + y 'sin( 1) y1 =

x 'sin( 1) + y 'cos( 1)

(33) (34)

x2 = x 'cos( 2) + y 'sin( 2)

(35)

y2 = x 'sin( 2)

(36)

y 'cos( 2)

Subsequently, there is a set of points M1 and M2 in the undeformed state, which are located on the respective body surfaces but with

(30)

Subsequently and in accordance with Ding et al. (2006), the individual body surfaces z1 and z2 are approximated by a second order Taylor’s series in the vicinity of point T.

z1 =

z2 =

K11 x12 + K12 y12 2 K21 x 22 + K22 y 22 2

=

=

y12 2r f y 22 2rtool

(31) (32)

According to Hertz (1881), the elliptical contact area with the semiaxes ael and bel is twisted to the cylinders, whereby its arrangement depends on the intermediate angle Ω and the body curvatures. A schematic illustration is shown in Fig. 15. In order to determine the relative arrangement of the contact ellipse, a new Cartesian coordinate system X’Y’Z’ with its origin in point T

Fig. 15. Schematic illustration of the elliptical contact region with the semiaxes ael and bel. 14

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

identical x’- and y’-coordinates. In accordance with Ding et al. (2006), the vertical distance between these set of points is equal to the addition of the respective z’-coordinates. Under consideration of the mathematical surface approximations in (31)-(32) in combination with the transformation rules in (33)-(36), the vertical distance ztot=z1+z2 is determined as a function of the body curvatures and the angles ζ1 and ζ2 (Ding et al., 2006).

ztot = z1 + z2 = Ax '2 + By '2 + Cx'y' A=

B= C=

K11cos2 ( 1) + K12sin2 ( 1) + K21cos2 ( 2) + K22sin2 ( 2 ) 2

K11sin2 ( 1) + K12 cos2 ( 1) + K21sin2 ( 2) + K22cos2 ( 2 ) 2 (K11

K12)sin(2 1) + (K21

K22)sin(2 2 )

2

K12sin2 ( 1_ 0) + K22sin2 ( 2 _ 0) =

(37) (38)

2_ 0

=

1_0

+

(K21 K22)sin(2 ) K12 + (K21 K22 )cos(2 )

for 0°

90°

e=

bel2 /ael2

1

/2

K(e ) = 0

d

1

e 2sin2 (

1

e 2 sin2 ( ) d

)

0

2 f

By the combination of (48) and (50), a disorganisation with respect to the semi-major axis ael is enabled:

(51)

3 Fl (K (e) e 2 (K12sin2 ( 1_ 0 ) + K22sin2 ( 2 _ 0 )) E *

ael =

E (e ))

1 3

(52)

Based on ael, the semi-minor axis bel is determined by a reorganisation of (45). (53)

e 2)

bel = ael (1

Consequently, there is a mathematical correlation between the force component Fl, the orientation of the contact ellipse described by ζ1 0 and ζ2 0, the body curvatures and the major axis of the elliptical contact region. Once, this set of parameters is known for an individual contact situation, the corresponding stress functions within the contact region have to be determined. Under consideration of Hertzian contact conditions, Johnson (1985) formulates the principal stresses along the zaxis (x’ = 0, y’ = 0 and z’ > 0) based on calculations of Thomas and Hoersch (1930) and Lungberg and Sjövall (1958):

(42) (43)

x

(z ) =

y (z

)=

z (z

)=

(46)

=

' x

=1

y

=

' y

=

(47)

In accordance with (45), the eccentricity e of the ellipse is a function of the semi-axes ael and bel of the contact ellipse. The bold terms K and E describe the complete elliptic integrals of the first and second kind. The elliptical integrals are necessary in order to consider the pressure distribution in the elliptical contact area. Due to the dependence of the complete elliptic integrals on the eccentricity of the ellipse, an analytical disorganisation of (44), with respect to the semi-axes, is not possible. Instead, the software Matlab® is used in order to determine the eccentricity of the ellipse numerically. Once the value of the eccentricity e is known with look-up tables, according to Johnson (1985), the problem to be solved can by reduced to:

2bel p0 e 2ael

2bel p0 e 2ael

x

+

f

' x)

(

y

+

f

' y)

(54) (55)

bel p0 1 Q 2 e 2ael Q

ael2 Q + bel2

(56)

M ( ,e )}

ael2 M ( ,e ) bel2

Qael2 + bel2

1 1 + 2 2Q

1 + Q+ {F ( ,e )

F ( ,e )

ael2 M ( ,e ) bel2

M ( ,e)}

(57) (58)

F ( ,e )

(59) (60)

Q=

bel2 + z'2 ael2 + z'2

(61)

=

z' = cot( ) ael

(62)

1

F(e ) = 0

M(e ) = 0

15

(

1 (1 Q) + {F ( ,e) 2

x

(44)

/2

E(e ) =

(50)

(40)

(45)

1

(49)

E2

1 1 = * E Ef

Up to this point, the relative arrangement of the contact ellipse is described with respect to the intermediate angle Ω and the body curvatures. In a next step, the mechanical correlations between the semiaxes of the contact ellipse, the acting force component Fl and the corresponding pressure distribution in the contact region have to be formulated. Based on potential functions, Johnson (1985) and Luré (1964) formulated the following mathematical correlation between the body curvatures Kij, the angles ζ1 0 and ζ2 0 and the eccentricity e of the ellipse.

K12cos2 ( 1 _ 0) + K22cos2 ( 2_ 0 ) B E (e ) (1 e 2) K (e) = = 2 2 A K12 sin ( 1 _ 0) + K22sin ( 2_ 0 ) (1 e 2)(K (e ) E (e ))

2 2

1

(39)

(41)

K11

(48)

The parameter p0 describes the average pressure within the contact region, which is a function of the acting force and the corresponding semi-axes of the ellipse (Johnson, 1985). Under consideration of a rigid cutting edge, (49) can be simplified:

In order to fulfil this requirement, the mixed term Cx y in (37) has to vanish. By setting (40) equal to zero, a set of ζ1 0 and ζ2 0 is determined, where the mixed term Cx y vanishes and thus the X’- and Y’axes correspond to the principle axes of the contact ellipse as shown in Fig. 15. Accordingly, the respective angles ζ1 0 and ζ2 0 are determined as:

tan(2 1_ 0 ) =

+

E (e ))

3Fl 2 ael bel

p0 =

According to Luré (1964), the fundamental assumption of an elliptical contact area implies that the form of (37) corresponds to the general equation of an ellipse with the semi-axes ael and bel.

y '2 x '2 1= 2 + 2 ael bel

2 1

1 1 = E* E1

p0 bel (K (e ) E * e 2ael2

d

1

e 2sin2 ( )

1

e 2sin2 ( ) d

(63)

(64)

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

Fig. 16. Measured, inter- and extrapolated values of the bouncing back height bcOM for different clearance angles and fibre cutting angles in orthogonal machining (γ = 20°, lcut = 5 m) (Seeholzer et al., 2018).

Whereas the parameter νf represents the Poisson’s ratio of the carbon fibre, the parameters F and M describe the elliptic integrals with respect to ψ and thus represent special cases of the complete elliptic integrals presented in (46)-(47). Based on the stress functions along the z-axis (54)-(56), the equivalent stress value with respect to the maximum normal stress criterion is calculated as a function of z’. Subsequently, fibre failure occurs for a coordinate z’≥0, where the corresponding equivalent stress value exceeds the tensile strength σTf_real of the fibre for the first time. Consequently, the related force component Fl correspond to the critical bending force Fl_max. In contrast to the critical force component in micro-buckling Fa_max, Fl_max depends on the actual fibre cutting angle, since the contact ellipse between the cylinders and thus the corresponding stress distribution are functions of the tool-fibre arrangement described by the intermediate angle. MNS (z

) = max(

MNS_max

= max(

MNS_max (Fl _ max )

x

(z'),

y (z'),

MNS (z'))

=

for z'

Tf _ real

z (z'))

0

bc. In contrast, a fibre cutting angle dependent analysis of bc is possible in orthogonal machining of UD CFRP since the fibre orientation is constant during the machining process. In this context, orthogonal machining experiments with laser marked uncoated cutting inserts are conducted in order to evaluate the bouncing back height bc for individual fibre orientations and tool geometries for a fixed set of process parameters: vc = 90 m/min, f = 0.1 mm/rev. Part of this research as well as detailed information with respect to the experimental setup and the evaluation method are presented by Seeholzer et al. (2018). For these orthogonal machining experiments, UD CFRP material type M21/ 34%/UD194/IMA-12k is used, which is slightly different to the CFRP material used for the drilling experiments. The orthogonal machining experiments are conducted for a constant rake angle of γ = 20°, three different clearance angles (α = 7°, 14°, 21°) and five fibre cutting angles (Φ = 0°, 30°, 60°, 90°, 150°), using a full factorial design of experiments. Subsequently, the bouncing back height bc is evaluated after a cutting length of lcut = 5 m. The corresponding measurements are summarised in the blue area in Fig. 16 with respect to the tested sets of fibre cutting angles and clearance angles. In order to highlight the bouncing back heights measured by orthogonal machining experiments, these are indicated with the parameter bcOM instead of bc. The following explanations bear on the fundamental assumption that the relation of the bouncing back height to the fibre cutting angle, which is found in orthogonal machining, can be transferred to the cutting conditions of the drilling process. Therefore, the compression of the CFRP material as well as its subsequent elastic spring back on the flank face are assumed to remain within the respective fibre layer. Consequently, the complex three-dimensional spring back action is simplified to a two-dimensional situation within one fibre layer, whereby each layer can be considered separately. Under this assumption, the relevant contour line of the cutting edge with respect to the bounced back CFRP material of an exemplary fibre plane is represented by the intersection line of the current fibre plane (FP) and the cutting edge. A schematic illustration is shown in Fig. 17, whereby the contour line of the micro-geometry corresponds to the projection of the cutting edge contour into the fibre plane. The clearance angle αFP at an arbitrary radial coordinate r is determined by a transformation of the corresponding clearance angle in the tool orthogonal plane αTOP into the XGYG-plane as a function of the tool tip angle σ and the inclination angle λr.

(65) (66) (67)

4.3. Force components due to elastic spring back of the CFRP material In accordance with the modelling concept described in Section 4.1, the resultant force at the cutting edge in drilling UD CFRP is a superposition of force components by an initial fibre separation in front of the cutting edge (Section 4.2) and an elastic spring back of the CFRP material on the flank face. Once the carbon fibres are cut in front of the cutting edge with respect to the initial fibre separation, the remaining fibre ends are further loaded in cutting direction since they are still embedded in the CFRP material. Consequently, they are pressed underneath the cutting edge. After passing the lowest point of the cutting edge, the CFRP material tends to spring back elastically. This elastic recovery of the CFRP material, the so-called spring back effect, results in an additional tool-workpiece contact on the flank face of the tool and thus in additional force components. According to Section 3.2, the elastic spring back of the CFRP material is quantified by the bouncing back height bc and the actual clearance angle. In accordance with Section 3, both parameter vary along the cutting edge in drilling UD CFRP. However, the bouncing back height bc is a function of the fibre cutting angle and thus additionally shows an azimuthal angle dependence. A direct measurement of bc during the drilling process of UD CFRP is not possible due to several circumstances: First, the flank face of the drilling tool is located within the bore hole, whereby an optical recording of the spring back effect with subsequent evaluation is not realisable. Secondly, the fibre cutting angle and thus the resulting contact region change during the tool rotation. Consequently, the evaluation of identification marks on the flank face cannot enable a fibre cutting angle dependent analysis of

(

sin 90° FP (r )

= cos

1

)

2 cos( r (r ))

tan( TOP )

1+

(

sin2 90°

sin( r (r )) 2

tan2 ( TOP )

)

(68)

Whereas the clearance angle in the tool orthogonal plane is constant (αTOP = 9°), its projection αFP into the fibre plane is radius dependent due to λr. According to (68), the maximum projected clearance angle is 16

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

Fig. 17. Schematic illustration of the two-dimensional contact situation between the cutting edge and the bounced-back CFRP material for an exemplary fibre cutting angle Φr = 90°.

αFP = 42° and thus clearly exceeds the tested range of clearance angles in the orthogonal machining experiments (Seeholzer et al., 2018). In order to enable an approximated prediction of the fibre cutting angle dependent bouncing back heights for clearance angles αFP > 21°, individual extrapolations for all tested fibre orientations are determined according to the trend functions in the red area in Fig. 16. Whereas a linear dependence of bcOM is suitable for Φ = 150°, logarithmic basic functions are used for the remaining fibre cutting angles.

bcOM 0° (

FP )

=

4.32ln(

FP )

+ 17.99

bcOM 30° (

FP )

=

6.53ln(

bcOM 60° (

FP )

=

10.66ln(

bcOM 90° (

FP )

=

5.64ln(

FP )

0.029

FP

bcOM150° (

FP )

=

FP )

+ 36.86

FP )

+ 47.29

+ 23.45

+ 1.833

Kbc.

bc _ model =

bcOM Kbc

(74)

Consequently, the relative proportion between the bouncing back heights with respect to different fibre cutting angles is preserved, but the corresponding magnitude is reduced by 1/ Kbc. Based on parameter fitting to the experiments, Kbc = 2.5 is suitable and therefore used for the simulation. Based on (69)-(73) in combination with (74), the bouncing back heights for all occurring clearance angles are estimated for the following fibre cutting angles: Φ = 0°, 30°, 60°, 90°, 150°. In drilling out UD CFRP, the fibre cutting angle varies within the entire range of 0° < Φ < 180°, which is considered by means of linear interpolation between the trend functions. Once the bouncing back height and the clearance angle in the fibre plane are known for a specific point on the cutting edge, the corresponding force components have to be determined. In accordance with Zhang et al. (2001), the tool-workpiece contact situation on the flank face, which is schematically shown in Fig. 17, can be approximated by a rigid wedge, which is in contact with an elastic half-space. According to Johnson (1985), the resulting radial force component Fpress is a function of the bouncing back height bc_model, the Young’s modulus of the predamaged CFRP material Ec* and the reference thickness of one fibre layer h. In accordance with Section 4.1, the latter is equal to the diameter of the RVE.

(69) (70) (71) (72) (73)

With respect to the application of the trend functions (69)-(73) in the modelling approach of the drilling process, several points have to be considered: First, the measured bouncing back heights during the orthogonal machining experiments are available at the earliest after a cutting length of lcut = 5 m. After this first machining interval, subsequent microscope analysis of the uncoated cutting inserts shows advanced mechanical wear and thus already changed contact conditions at the cutting edge (Seeholzer et al., 2018). In contrast, the drilling tool used for the drilling out operations shows no significant change of the cutting edge micro-geometry due to the wear resistant diamond coating. In this context, Voss et al. (2018) identified a continuous increase of the bouncing back height with increasing wear state for all tested sets of tool geometry parameters and fibre cutting angles for a maximum cutting length of lcut = 40 m. Consequently, the inter- and extrapolated bouncing back heights based on lcut = 5 m in Fig. 16, are most likely too high due to tool wear. Secondly, the different material properties of the machined CFRP materials in the orthogonal machining and drilling experiments have an influence on the bouncing back height. Thirdly, the transformation of the complex three-dimensional spring back effect to a two-dimensional contact situation represents a significant simplification, whereby the real conditions cannot be reproduced properly but with a first approximation. Furthermore, the trend functions (69)-(73) are based on a constant rake angle of γFP = 20°, whereby the influence of the rake angle on the bouncing back height is neglected. However, according to Seeholzer et al. (2018), the influence of the rake angle is found to be small in comparison to the clearance angle. Finally, the radius dependent cutting velocity is assumed to have an influence on the resulting bouncing back heights, which is not considered in the trend functions (69)-(73). In order to consider these uncertainties in the modelling approach, an additional fitting parameter Kbc > 1 is introduced. In this context, the bouncing back height bc_model, which is used in the modelling approach, is determined by a multiplication of the trend functions (69)-(73) with 1/

Fpress =

bc _ model Ec*h = bc Ec*rm 2

(75)

Based on Fpress, the corresponding force component Fpress_f due to friction is considered by means of the Coulomb friction model as schematically shown in Fig. 17.

Fpress _ f = µFpress cos(

(76)

FP (r ))

Based on cutting process tribometer experiments in orthogonal machining of comparable UD CFRP material by Voss et al. (2016b), a friction coefficient μ = 0.12 is used for the modelling approach. Whereas Fpress represents a pure radial force, the corresponding friction force Fpress_f has force components in radial and in cutting velocity direction. Once Fpress and Fpress_f are known for an arbitrary position on the cutting edge, the corresponding, process-relevant force components in feed, cutting velocity and radial direction are determined.

Frad _ press = Fpress (1 + µcos(

Fcut _ press = µFpress cos (

FP

Ffeed _ press = Frad _ press tan

(r )) 2

(78)

2

= Fpress (1 + µ cos( 17

(77)

FP (r ))sin( FP (r )))

FP (r ))sin( FP (r )))tan

2

(79)

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

5. Modelling results and validation

oblique cutting conditions, the bouncing back height is approximated by the extrapolated trend functions (69)-(73) in combination with the correction factor Kbc as explained in Section 4.3. In accordance with Zhang et al. (2001) and Voß (2017), the bounced back CFRP material is characterised by process-related pre-damages, whereby the corresponding Young’smodulus Ec* of the bounced back composite material is lower than for the undamaged material. In the model, the parameter Ec* is handled as fitting factor and determined by parameter fitting to the experimental data. With respect to fibre failure by micro-buckling in intervals I and II, the critical buckling load according to Xu and Reifsnider (1993) in combination with the fitting parameter KMB = 2 is used. According to Section 4.2.1, the latter considers both, the supporting effect of adjacent fibres and the localised higher volume fraction. In interval IV, an updated fitting parameter KMB_t = 3.2 is introduced, in order to consider changed tool-fibre contact conditions due to micro-deformation as explained in Sections 3.3.4 and 4.2.1. In accordance with Section 4.2, the lateral and axial force components on an arbitrary point on the cutting edge are coupled by the force ratio gax, which is determined by (19) with respect to a vectorial decomposition of the actual effective cutting. Consequently, their relative ratio is known for each azimuthal and radial position on the cutting edge during the drilling out operation. According to Section 4, a critical bending force Fl_max causes fibre failure by induced tensile stresses, whereas a critical axial force Fa_max causes fibre failure by microbuckling. For the force simulation, one half tool rotation is segmented into rotation increments with εrot = 5°. In this context, the fibre cutting angle ΦR at the nominal diameter is used as a control variable. For each rotation increment, the actual tool-fibre contact points along the cutting edge are determined with respect to (18). Subsequently, the fibre cutting angles Φr, the actual cutting edge angles with respect to the effective cutting direction (αCD, βCD, γCD) as well as the projected clearance angles αFP are determined for the entire sets of evaluated tool-fibre contact points. For this purpose, the calculations in (1), (9)-(11) and (68) are used. Subsequently, the actual incremental forces at an exemplary tool-fibre contact point are determined under consideration of the associated force ratio gax and the critical force components Fa_max and Fl_max. Since these values depend on the choice of the coordinates r and φ, they have to be determined individually for each contact point. Fibre separation at an arbitrary contact point occurs as soon as one of the corresponding critical force components is exceeded. Since the fibre cutting angle changes along the cutting edge, the dominant fibre failure criterion may also vary for different tool-fibre contact points. Fig. 18 shows the simulated axial and lateral force components at the nominal diameter with a radius R during one half tool rotation. Therefore, the representation of the fibre cutting angle on the horizontal axis corresponds to the actual cutting sequence during the drilling process. In the following, the trends of both force components are explained in detail. For the fibre cutting angle ΦR = 180°, denoted by U1 in Fig. 18, the fibre orientation and the cutting velocity are oriented in the same direction. Hence, the decomposition of the effective cutting direction into the XGYGZG-coordinates results almost exclusively in an axial compression of the carbon fibres in front of the cutting edge. Simultaneously, the superimposed feed motion in ZG-direction results in a comparatively small bending deformation perpendicular to the fibre. This dependence is described in detail in Section 3.2. Based on the corresponding value of gax, the axial critical buckling load Fa_max and thus fibre failure by micro-buckling occurs before the critical lateral force component Fl_max is reached. Hence, the axial force component in Fig. 18 (b) with respect to U1 is equal to Fa_max, whereas the lateral force component in Fig. 18 (a) is close to zero. Starting from point U1 in positive tool rotation, the changing effective cutting direction with respect to the fibre orientation results in a decreasing value of gax. Consequently, the vectorial share of the lateral force component in the resultant force increases at the expense of the axial force. However, the required axial force component, which would enable a corresponding

In the following, the process forces in drilling out UD CFRP with a predrilled diameter of Ø = 4.17 mm is analytically determined with respect to the modelling concept presented in Section 4. In this context, Table 2 summarises all input variables used in the model. If specific material properties are not available in material datasheets, comparable values from literature are used. In order to validate the simulated process forces, corresponding drilling out experiments with online force and torque measurements are conducted. With respect to the measuring hardware presented in Section 2, the available number of measuring points during one half tool rotation is limited by the measuring infrastructure and depends on the programmed rotational speed and cutting velocity respectively. In order to enable a reasonable angular resolution of the thrust force and the torque during one half tool rotation, the simulation and the experiment are compared with respect to a cutting velocity of vc = 30 m/min (n = 1504 rev/min). In this context, the used sampling frequency of 9111 Hz enables 182 measuring point during one half tool rotation and thus an angular resolution of 1°. Whereas the material properties are usually determined at very low strain rates, significantly higher strain rates have to be considered in drilling operations with a programmed rotational speed of n = 1504 rev/min. Based on experiments with short glass fibre reinforced thermoplastics, Schoßig et al. (2006) proved an increasing tensile strength of the glass fibres with higher strain rates. In accordance with Roos et al. (2017), the stiffness properties of polymers are also characterised by a strong strain rate dependence. Therefore, two correction factors KEm > 1 and KTf > 1 are used in accordance with Voss et al. (2018), in order to consider the influence of higher strain rates on the Young’s modulus of the matrix material and the tensile strength of the carbon fibre. Subsequently, the actual Young’s modulus of the matrix and the tensile strength of the carbon fibre are determined by multiplying the correction factors with the corresponding material properties for low strain rates: Em _ real = KEm Em and Tf _ real = KTf Tf (Voss et al., 2018). Both correction factors are determined by parameter fitting to the experimental data. The elastic spring back of the CFRP material on the flank face is described by the clearance angle αFP in the fibre plane and the fibre cutting angle dependent bouncing back height bc_model. Whereas αFP is geometrically determined by (68) with respect to the Table 2 Material- and process parameters and fitting variables used in the model (Naik and Kumar, 1999; Naya et al., 2017; Voss et al., 2016a, b, 2018; Xu and Zhang, 2016). Carbon fibre – HTS(12k)

Process / tool

Young’s modulus (axial) – Ef Fibre Poisson’s ratio – vf

238 GPa 0.2

Fibre radius – rf

3.5 μm

Fibre volume fraction – Vf Tensile strength of fibre (original) – σTf

65% 4.3 GPa

Matrix – MTM44-1 Young’s modulus (original) – Em

4 GPa

Shear modulus – Gm

1.43 GPa

Poisson’s ratio – vm

0.4

Friction coefficient – μ Parameter matrix slipping – ξ Parameter interfacial bonding – ηMB Cutting velocity – vc Feed rate – f

0.12 0.02

Cutting edge radius (new state) – rtool Web thickness – 2w

14 μm

1.98 30 m/min 0.1 mm/rev

1.84 mm

Material fitting parameters Young’s modulus CFRP region 3 – Ec* Factor Young’s modulus matrix – KEm Factor tensile fibre strength – KTf Factor bouncing back height – Kbc Factor micro-buckling – KMB Factor micro-buckling – KMB_t

4.9 GPa 2.2 3.4 2.5 2 3.2

18

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

Fig. 18. Simulated lateral (a) and axial (b) force components at the nominal diameter during one half tool rotation.

lateral force equal to Fl_max, is still larger than Fa_max. Accordingly, fibre failure is still dominated by micro-buckling. Whereas the axial force in Fig. 18 (b) is equal to Fa_max and thus remains constant, the lateral force component increases due to changed value of gax in accordance with Fig. 18 (a). A change of the predominant failure mechanism happens at point U2 for a fibre cutting angle close to ΦR = 120°. At this point, based on gax, for the first time the critical lateral force component Fl_max is reached before the corresponding axial force component exceeds the axial buckling load Fa_max. Starting from point U2, fibre failure no longer occurs by micro-buckling, but by exceeding tensile strength of the carbon fibres. In agreement with Fig. 18, a further tool rotation results in a decreasing axial force, while the corresponding lateral force increases due to changing contact conditions between the cutting edge and the carbon fibre. For ΦR = 90° in point U3, the lateral force component reaches its maximum, whereas the corresponding axial force component equals to zero due to a pure bending deformation as explained in Section 3. Starting from point U3, the simulated lateral force decreases due to the changing vectorial composition of the effective cutting direction. Simultaneously, the corresponding axial force becomes negative as explained in Section 3.2. In the beginning, the axial component of the effective cutting direction is comparatively small, whereby fibre failure is still dominated by the lateral force component. However, the point U4 close to ΦR = 35° represents the fibre cutting angle, where again the critical buckling load is reached before the corresponding lateral force component enables fibre failure by contact stresses. In accordance with Section 4.2.1, Fa_max_t is used as critical buckling load in order to consider the changed tool-fibre contact conditions. Starting from point U4, the lateral force component further decreases, whereas the axial force is constant and equals to Fa_max_t. Just before ΦR = 0°, the axial force component abruptly changes from negative (tension) to positive (compression) due to a change of the fibre orientation by 180°. If multiple points on the cutting edge are considered simultaneously, the abruptly change from Fa_max_t to Fa_max_t in Fig. 18 (b) is smoothed due to the radial shift of the fibre cutting angle along the cutting edge. In accordance with Section 3.3, the tool-fibre contact situation for fibre cutting angles in the range of 170°≥Φr≥100° (interval II) is characterised by a saw teeth topography, whose influence on the process forces is not yet included in the modelling approach. As explained in Section 3.3.2, the depth of the saw teeth partially exceeds the radial extension and thus causes a discontinuous tool-fibre contact at the bore

edge. In accordance with Voss et al. (2018), the corresponding averaged force components are lower compared to a continuous cutting situation, whereby the reduction depends on the size of the saw teeth. According to Section 3.3.2, the width and the depth of the occurring saw teeth reach their maximum in the middle of interval II at a fibre cutting angle close to ΦR = 135°. For fibre cutting angles ΦR≥170° and ΦR≤100°, no saw teeth are identified during the experiments. Therefore, a reduction of the axial and lateral force components due to a discontinuous toolfibre contact is only valid for the range 170°≥ΦR≥100° (interval II). In order to quantify the discontinuous tool-fibre contact in interval II, Fig. 19 shows an exemplary saw teeth topography during the drilling out process with vc = 30 m/min and f = 0.1 mm/rev. In this context, the exemplary saw teeth topography corresponds to an arbitrary snapshot during one tool rotation. Under consideration of multiple tool rotations, the saw teeth area is extended to 170°≥Φr≥100°, which is additionally indicated with a corresponding range in Fig. 19. With respect to the radial extension per revolution ap, the percentage contact close to ΦR = 135° is reduced by 60% as shown in Fig. 19. Starting from ΦR = 135°, the size of the saw teeth decreases to both sides and thus the associated force reduction. In order to consider a local and thus fibre cutting angle dependent reduction of the process forces in interval II, a specific function TST according to the schematic illustration in Fig. 19 is defined. The value of the function represents the percentage contact and equals to 100 for the entire fibre cutting angles with exception of interval II. For ΦR = 135°, the value of the function is 40, which corresponds to an overall reduction of the process forces by 60%. For the remaining range of fibre cutting angles in interval II, a parabolic distribution of the saw tooth size and the corresponding force reduction is assumed. Subsequently, the updated process forces are calculated by multiplying the previously determined force components with the percentage reduction function TST. Fig. 20 shows the updated trends of axial and lateral force components under consideration of a force reduction in interval II due to the saw teeth topography. In this context, the new points U5 and U6 at ΦR = 170° and ΦR = 100° describe the transition of interval I to II and II to III respectively. In agreement with Section 3.3, the intervals I, II, III and IV are represented by the angular segments between the points U1 to U5, U5 to U6, U6 to U4 and U4 to U1. Whereas micro-buckling is dominant in intervals I, II and IV, interval III is characterised by fibre separation due to exceeding tensile strength of the carbon fibres. Consequently, the 19

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

Fig. 19. Exemplary saw teeth topography close to Φr = 135° (vc = 30 m/min, f = 0.1 mm/rev), schematic illustration of the reduction function TST.

Fig. 20. Simulated lateral (a) and axial (b) force components at the nominal diameter during one half tool rotation; excluding (dotted line) and including (continuous line) saw teeth topography.

predominant failure mechanisms within the intervals of the simulation coincide with the experimental findings in Section 3.3. Based on the simulated lateral and axial forces Fa and Fl of the initial fibre separation and under consideration of the saw teeth topography, the force components FX_init_G, FY_init_G, and FZ_init_G with respect to the global XGYGZG-coordinate system are determined.

FX _ init _ G = Fa

FZ _ init _ G =

vX _ G _ corr |vX _ G _ corr |

FY _ init _ G =

( __ ) vY G vZ G

+1

(82)

Subsequently, the cutting and thrust force components of the initial fibre separation in front of the cutting edge Fcut_init and Ffeed_init are calculated under consideration of the corresponding fibre cutting angle.

Fcut _ init = FX _ init _ Z cos( (80)

Ffeed _ init = FX _ init _ Z sin(

r r

(r )) + FY _ init _ Z sin(

(r ))

FY _ init _ Z cos(

r

(r ))

(83)

r

(r ))

(84)

In order to simulate the process forces during the drilling out operation, the cutting and thrust force components of the initial fibre separation (83)-(84) are combined with the force components of the bounced back CFRP material (77)-(79). The resulting overall cutting

Fl2

2

vY _G Fz _ init _ G vZ _ G

(81) 20

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

and thrust force components are functions of the radial coordinate r and the fibre cutting angle at the nominal diameter. The superimposed thrust force components are summarised along the cutting edge with respect to rstep. Subsequently, the overall thrust force Ffeed_tot is determined by a multiplication with the factor 2 in order to consider both cutting edges. (R rpd )

differences between the simulated and measured thrust force. This effect is significantly lower for the torque values, since the cutting force component is dominated by the initial fibre separation in front of the cutting edge. In accordance with Fig. 21, the thrust force in the intervals 180° > ΦR > 150° and 185° > ΦR > 90° is overestimated, which implies that the linear trend of interpolated bouncing back height within this regions cannot represent the real conditions properly.

rstep

Ffeed _ tot = 2

(Ffeed _ init (rpd + i rstep,

R)

6. Conclusion and outlook

i= 0

+ Ffeed _ press (rpd + i rstep,

(85)

R ))

The force modelling approach presented in this work enables the analytical determination of the thrust force and torque values in drilling out predrilled UD CFRP. With respect to the model validation, the simulated values are in good agreement with the experimental data. In the proposed force prediction model, the resultant force at an arbitrary point on the cutting edge is formulated as a superposition of force components from an initial fibre separation in front of the cutting edge and elastic spring back effects of the CFRP material on the flank face. With respect to the initial fibre separation, two fundamental fibre failure mechanisms are identified by means of high-speed recordings of the chip formation process and subsequent scanning electron microscope analysis. Whereas fibre micro-buckling is considered for loading situations dominated by axial compression, bending deformations of individual fibres perpendicular to their longitudinal orientation cause fibre separation due to induced tensile stresses in the contact region. For micro-buckling, a critical axial buckling load Fa_max is determined with respect to a representative compressive strength based on a single fibre RVE-consideration proposed by Xu and Reifsnider (1993). Analogously, the critical force component Fl_max describes the lateral force component, which causes fibre separation due to induced tensile stresses by exceeding the tensile strength of the fibre. For this purpose, the tool-fibre contact situation is modelled as Hertzian contact between two twisted cylinders, whereby the equivalent stress is determined with respect to the maximum normal stress theory. In order to determine the force components due to elastic spring back effects of the CFRP material, the tool-workpiece contact situation on the flank face is approximated by a rigid wedge, which is in contact with an elastic half-space. In accordance with Voss et al. (2018), the resulting contact region on the flank face is described by the actual clearance angle and the bouncing back height. Both parameter depend on the radial coordinate r as well as on the azimuthal angle position of the cutting edge. In the modelling approach, the bouncing back heights are approximated by linear interpolations of fibre cutting angle dependent trend functions, which are adopted from orthogonal machining operations.

Furthermore, the torque is determined by summarising the multiplications of the superimposed cutting force components and the corresponding radial coordinates r along the cutting edge. (R rpd )

Mcut _ tot =

rstep

(rpd + i rstep )(Fcut _ init (rpd + i rstep,

R)

i= 0

+ Fcut _ press (rpd + i rstep,

R ))

(86)

According to Section 2, online thrust force and torque measurements are conducted in order to validate the force model. In this context, Fig. 21 enables the comparison of simulated and measured thrust force and torque values for several consecutive tool rotations during the drilling out operation. For the experimental raw data, a zero-phase digital filter is used to improve comparability. Especially the torque measurement is characterised by noise effects resulting either from the test rig or from the cutting process. With respect to the latter, short-period force changes due to the initial fibre separation in front of the cutting edge or the formation of saw teeth in interval II may cause high-frequency oscillations with high amplitudes. In contrast, the thrust force shows significantly less noise effects, which is explained by a slow time related change of the bouncing back height during one half tool rotation. Focussing on the filtered data, the simulated trends of thrust force and torque values are in good agreement with the experiments. Both, the oscillation period of 0.02 s and the influence of the fibre cutting angle on the thrust force and torque values can be approximated well. In accordance with Section 4.3, the extrapolated trends of the bouncing back heights are only available for specific fibre cutting angles, namely: Φ = 180°, 150°, 90°, 60° and 30°. Between these specific fibre cutting angles, linear interpolation are used in order to approximate the bouncing back height for an intermediate fibre cutting angle. In accordance with Fig. 21, the linear interpolation of the bouncing back heights results especially for 180° > ΦR > 150° and 185° > ΦR > 90° in

Fig. 21. Comparison of simulated and measured thrust force and torque values in drilling out UD CFRP from rpd to R (vc = 30 m/min and f = 0.1 mm/rev). 21

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al.

During the high-speed recordings of the drilling experiments, a characteristic saw teeth topography in interval II (170°≥Φr≥100°) is identified, whereby the size of the saw teeth changes within the corresponding range. Based on the experiments, a parabolic distribution of the saw tooth size in interval II with its maximum for ΦR = 135° is identified. In the modelling approach, this is considered with a reduction of the axial and lateral force components by means of a parabolic function. The presented model considers the tool geometry, the drilling kinematics as well as the process parameters and describes their analytical interrelations by means of mechanical concepts. The modelled thrust force and torque values are functions of a varying fibre cutting angle and space-dependent geometry and process parameters. Consequently, the model enables to show causalities, e.g. the influence of the tip angle, the cutting edge angles and the web thickness on the resulting process forces. Since the damage pattern is well known to be also influenced by the tool geometry and the fibre cutting angle, a direct causality between the workpiece damages, the process forces and the tool geometry is established. Accordingly, this model represents an important tool for predictive geometry optimisation in drilling processes. In these aspects, the model is superior to experimental measurements and existing models, which do not provide this information easily. In comparison to existing force models, the proposed modelling approach differs by the following particularities:

wear is ongoing and will be presented in future publications. Whereas the proposed model uses the initial new state of the drilling tool, the wear dependent change of the cutting edge micro-geometry has to be considered in future work in order to enable force prediction also for longer machining operations. Wear related changes of the micro-geometry have an influence on the initial fibre separation in front of the cutting edge as well as on the spring back effect of the CFRP material. Furthermore, the modelling approach can be extended by taking into account force components by the chisel- and minor cutting edges. Finally, different CFRP materials with more complex fibre arrangements should be addressed in future work in order to make the modelling approach generally applicable. Declaration of Competing Interest None. Acknowledgements The authors thank the Swiss Innovation Agency (Innosuisse Project 18309.2 PFIW-IW), the companies Dixi Polytool SA, Heule Werkzeuge AG, Oerlikon Surface Solutions AG and Airbus Helicopters Deutschland GmbH for their support. References

• Continuous determination of the thrust force and torque values in • • •

drilling out predrilled UD CFRP as functions of the fibre cutting angle, the process parameters and the tool geometry. Consideration of fibre cutting angle dependent tool-fibre contact situations and fibre failure mechanisms. High degree of details in the modelling approach, e.g. separate consideration of initial fibre separation and spring back effects of the CFRP material, consideration of a saw teeth topography in interval II. The modelling concept is fundamentally based on the principle of drilling out UD fibre reinforced polymers with brittle fibre failure. Consequently, the modelling framework can be used in order to determine the process forces for similar materials by adjusting the corresponding material-, process- and fitting parameters

Bhatnagar, N., Ramakrishnan, N., Naik, N.K., Komanduri, R., 1995. On the machining of fiber reinforced plastic (FRP) composite laminates. Int. J. Mach. Tool Manu. 35, 701–716. Chandrasekharan, V., Kapoor, S., DeVor, R.E., 1995. A mechanistic approach to predicting the cutting forces in drilling: with application to fiber-reinforced composite materials. J. Manuf. Sci. E-T ASME 117, 559–570. Davim, J.P., 2013. Machining Composites Materials. John Wiley & Sons. Ding, H., Chen, W., Zhang, L., 2006. Elasticity of Transversely Isotropic Materials. Springer Science & Business Media, pp. 205–245. Guo, D.-M., Wen, Q., Gao, H., Bao, Y.-J., 2012. Prediction of the cutting forces generated in the drilling of carbon-fibre-reinforced plastic composites using a twist drill. Proc. Inst. Mech. Eng. B 226, 28–42. Henerichs, M., Voss, R., Kuster, F., Wegener, K., 2015. Machining of carbon fiber reinforced plastics: influence of tool geometry and fiber orientation on the machining forces. CIRP J. Manuf. Sci. Tec. 9, 136–145. Hertz, H., 1881. Ueber die Berührung fester elastischer körper. J. Reine Angew. Math. 92, 156–171. Jahromi, A.S., Bahr, B., 2010. An analytical method for predicting cutting forces in orthogonal machining of unidirectional composites. Compos. Sci. Technol. 70, 2290–2297. Johnson, K.L., 1985. Contact Mechanics. Cambridge University Press, Cambridge. Koplev, A., Lystrup, A., Vorm, T., 1983. The cutting process, chips, and cutting forces in machining CFRP. Composites 14, 371–376. Langella, A., Nele, L., Maio, A., 2005. A torque and thrust prediction model for drilling of composite materials. Compos. Part A-Appl. S. 36, 83–93. Lungberg, G., Sjövall, H., 1958. Stress and deformation in elastic solids. Inst. Th. of Elasticity 4, 66–99. Luré, A.I., 1964. Three-dimensional Problems of the Theory of Elasticity. Interscience Publishers, pp. 251–324. Mayer, J., Tognini, R., Widmer, M., Zerlik, H., Wintermantel, E., Ha, S.-W., 2009. Faserverbundwerkstoffe. In: Wintermantel, E., Ha, S.-W. (Eds.), Medizintechnik – Life Science Engineering. Springer Verlag, Berlin, pp. 299–342. Meng, Q., Zhang, K., Cheng, H., Liu, S., Jiang, S., 2015. An analytical method for predicting the fluctuation of thrust force during drilling of unidirectional carbon fiber reinforced plastics. J. Compos. Mater. 49, 699–711. Merchant, M.E., 1945. Mechanics of the metal cutting process. I. Orthogonal cutting and a type 2 chip. J. Appl. Phys. 16, 267–275. Naik, N., Kumar, R.S., 1999. Compressive strength of unidirectional composites: evaluation and comparison of prediction models. Compos. Struct. 46, 299–308. Naya, F., Molina-Aldareguia, J., Lopes, C., González, C., LLorca, J., 2017. Interface characterization in fiber-reinforced polymer–matrix composites. JOM 69, 13–21. Qi, Z., Zhang, K., Cheng, H., Wang, D., Meng, Q., 2015. Microscopic mechanism based force prediction in orthogonal cutting of unidirectional CFRP. Int. J. Adv. Manuf. Tech. 79, 1209–1219. Roos, E., Maile, K., Seidenfuß, M., 2017. Werkstoffkunde Für Ingenieure. Springer-Verlag, Grundlagen, Anwendung, Prüfung. Sakuma, K., Seto, M., 1978. Tool wear in cutting glass-fibre-reinforced plastics: the effect of physical properties of tool materials (Japanese). Bulletin of JSME 44, 1752. Sauer, M., Kühnel, M., 2018. Der Globale CF- Und CC-Markt 2018. Schmitt-Thomas, K.G., 2016. Integrierte Schadenanalyse: Technikgestaltung und das System des Versagens. Springer-Verlag. Schoßig, M., Bierögel, C., Grellmann, W., Bardenheier, R., Mecklenburg, T., 2006.

In addition to various material and process parameters, the presented modelling approach uses six fitting variables (Ec* , KEm, KTf, Kbc, KMB, KMB_t), which are determined simultaneously by parameter fitting to the experimental data. These fitting parameters are needed in order to describe mechanical interdependencies, which either cannot be described analytically or their experimental determination is only possible with excessive effort. Future research should therefore focus on a detailed analysis of these interdependencies in order to reduce the number of fitting parameters and thus to improve the process understanding and the prediction accuracy. The force prediction approach of the initial fibre separation is based on the fundamental assumption that both, the direction of fibre deformation and the corresponding loading direction equal to the effective cutting direction. This assumption and the reduction of the complex three-dimensional spring back action of the CFRP material on the flank face to a two-dimensional description within one fibre layer represent significant simplifications, whereby the real conditions cannot be reproduced properly but with a first approximation. In accordance with Voss et al. (2018), strain rate effects on the Young’s modulus of the matrix material and the tensile strength of the carbon fibre are considered by the fitting parameters KEm and KTf. Whereas a more detailed description of the strain rate properties as functions of the process parameters, the tool geometry and the fibre orientation with respect to the actual position on the cutting edge is part of ongoing research, the application of fitting parameters represents a first approximation of these complex parameter interactions. As a further outlook, additional research with respect to mechanical 22

Journal of Materials Processing Tech. xxx (xxxx) xxxx

L. Seeholzer, et al. Einfluss der Dehnrate auf das mechanische Verhalten von glasfaserverstärkten Polyolefinen, Tagung Werkstoffprüfung, Bad Neuenahr, Germany. pp. 445–450. Schütte, C., 2014. Bohren und Hobeln von kohlenstofffaserverstärkten Kunststoffen unter besonderer Berücksichtigung der Schneide-Faser-Lage. TUHH, Inst. für Produktionsmanagement und-technik. Seeholzer, L., Voss, R., Grossenbacher, F., Kuster, F., Wegener, K., 2018. Fundamental analysis of the cutting edge micro-geometry in orthogonal machining of unidirectional Carbon Fibre Reinforced Plastics (CFRP). Procedia CIRP 77, 379–382. Sheikh-Ahmad, J.Y., 2009. Machining of Polymer Composites. Springer. Takeyama, H., Iijima, N., 1988. Machinability of glassfiber reinforced plastics and application of ultrasonic machining. CIRP Ann.-Manuf. Techn. 37, 93–96. Thomas, H.R., Hoersch, V.A., 1930. Stresses Due to the Pressure of One Elastic Solid upon Another With Special Reference to Railroad Rails: A Report. University of Illinois at Urbana Champaign, College of Engineering. Voß, R., 2017. . Fundamentals of Carbon Fibre Reinforced Polymer (CFRP) Machining. Eidgenössische Technische Hochschule Zurich (ETH). Voss, R., Henerichs, M., Kuster, F., Wegener, K., 2014. Chip Root Analysis After Machining Carbon Fiber Reinforced Plastics (CFRP) at Different Fiber Orientations. CIRP HPC, Berkeley (US), pp. 217–222. Voss, R., Henerichs, M., Walch, C., Kuster, F., Wegener, K., 2016a. Friction analysis between carbon fibre reinforced plastic and Diamond coating with cutting process

tribometer. In: International Conference on Precision Engineering. The Japan Society for Precision Engineering, Hamamatsu, Japan. Voss, R., Seeholzer, L., Kuster, F., Wegener, K., 2016b. Influence of fibre orientation, tool geometry and process parameters on surface quality in milling of CFRP. Cirp J. Manuf. Sci. Technol. Voss, R., Seeholzer, L., Kuster, F., Wegener, K., 2018. Analytical force model for orthogonal machining of unidirectional carbon fibre reinforced polymers (CFRP) as a function of the fibre orientation. J. Mater. Process. Technol. 263, 440–469. Wang, X.M., Zhang, L., 2003. An experimental investigation into the orthogonal cutting of unidirectional fibre reinforced plastics. Int. J. Mach. Tools Manuf. 43, 1015–1022. Xu, W., Zhang, L., 2014a. On the mechanics and material removal mechanisms of vibration-assisted cutting of unidirectional fibre-reinforced polymer composites. Int. J. Mach. Tools Manuf. 80–81, 1–10. Xu, W., Zhang, L., 2016. Mechanics of fibre deformation and fracture in vibration-assisted cutting of unidirectional fibre-reinforced polymer composites. Int. J. Mach. Tools Manuf. 103, 40–52. Xu, Y.L., Reifsnider, K.L., 1993. Micromechanical modeling of composite compressive strength. J. Compos. Mater. 27, 572–588. Zhang, L.C., Zhang, H.J., Wang, X.M., 2001. A force prediction model for cutting unidirectional fibre-reinforced plastics. Mach. Sci. Technol. 5, 293–305.

23