Authors: Christian Hortig and Bob Svendsen Jordan Felkner October 5, 2009

Purpose  Model and simulate shear banding and chip formation during high-speed cutting  Carry out a systematic investigation of size- and orientation-based mesh- dependence of the numerical solution Finite Element Analysis

A Little Vocab Shear Band: region where plastic shear has taken place Adiabatic Shear Banding: shearing with no heat transfer ○ Mechanical dissipation dominates heat conduction Mesh: the size and orientation of the element

Why is this Important?  Cutting forces Shear banding represents the main mechanism of chip formation ○ Results in reduced cutting forces  Tool design  Other technological aspects

References [1] M. B¨aker, J. R¨osier, C. Siemers, A finite element model of high speed metal cutting with adiabatic shearing, Comput. Struct. 80 (2002) 495–513. [2] M. B¨aker, An investigation of the chip segmentation process using finite elements, Tech. Mech. 23 (2003) 1–9. [3] M. Baker, Finite element simulation of high speed cutting forces, J. Mater. Process. Technol. 176 (2006) 117–126. [4] A. Behrens, B. Westhoff, K. Kalisch, Application of the finite element method at the chip forming process under high speed cutting conditions, in: H.K. T¨onshoff, F. Hollmann (Eds.), Hochgeschwindigkeitsspanen,Wileyvch, 2005, ISBN , pp. 112–134. [5] C. Comi, U. Perego, Criteria for mesh refinement in nonlocal damage finite element analyses, Eur. J. Mech. A/Solids 23 (2004) 615– 632. [6] E. El-Magd, C. Treppmann, Mechanical behaviour of Materials at high strain rates, in: H. Schulz (Ed.), Scientific Fundamentals of High-Speed Cutting, Hanser, 2001, ISBN , pp. 113–122. [7] T.I. El-Wardany, M.A. Elbestawi, Effect of material models on the accuracy of highspeed machining simulation, in: H. Schulz (Ed.), Scientific Fundamentals of High-Speed Cutting, Hanser, 2001, ISBN , pp. 77–91. [8] D.P. Flanagan, T. Belytschko, A Uniform Strain Hexahedron and Quadrilateral with Orthogonal Hourglass Control, Int. J. Numer. Methods Eng. 17 (1981) 679–706. [9] G.R. Johnson, W.H. Cook, A constitutive model and data for metals subjected to large strain, high strain-rates and high temperatures, in: Proceedings of the 7th International Symposium on Ballistics, The Hague, The Netherlands, pp. 541–547. [10] T. Mabrouki, J.-F. Rigal, A contribution to a qualitative understanding of thermo- mechanical effects during chip formation in hard turning, J. Mater. Process. Technol. 176 (2006) 214–221. [11] M.E. Merchant, Mechanics of the metal cutting process. I. Orthogonal cutting and a type 2 chip, J. Appl. Phys. 16 (1945) 267– 275.

References [1] M. B¨aker, J. R¨osier, C. Siemers, A finite element model of high speed metal cutting with adiabatic shearing, Comput. Struct. 80 (2002) 495–513. [2] M. B¨aker, An investigation of the chip segmentation process using finite elements, Tech. Mech. 23 (2003) 1–9. [3] M. Baker, Finite element simulation of high speed cutting forces, J. Mater. Process. Technol. 176 (2006) 117–126. [4] A. Behrens, B. Westhoff, K. Kalisch, Application of the finite element method at the chip forming process under high speed cutting conditions, in: H.K. T¨onshoff, F. Hollmann (Eds.), Hochgeschwindigkeitsspanen,Wileyvch, 2005, ISBN , pp. 112–134. [5] C. Comi, U. Perego, Criteria for mesh refinement in nonlocal damage finite element analyses, Eur. J. Mech. A/Solids 23 (2004) 615– 632. [6] E. El-Magd, C. Treppmann, Mechanical behaviour of Materials at high strain rates, in: H. Schulz (Ed.), Scientific Fundamentals of High-Speed Cutting, Hanser, 2001, ISBN , pp. 113–122. [7] T.I. El-Wardany, M.A. Elbestawi, Effect of material models on the accuracy of highspeed machining simulation, in: H. Schulz (Ed.), Scientific Fundamentals of High-Speed Cutting, Hanser, 2001, ISBN , pp. 77–91. [8] D.P. Flanagan, T. Belytschko, A Uniform Strain Hexahedron and Quadrilateral with Orthogonal Hourglass Control, Int. J. Numer. Methods Eng. 17 (1981) 679–706. [9] G.R. Johnson, W.H. Cook, A constitutive model and data for metals subjected to large strain, high strain-rates and high temperatures, in: Proceedings of the 7th International Symposium on Ballistics, The Hague, The Netherlands, pp. 541– 547. [10] T. Mabrouki, J.-F. Rigal, A contribution to a qualitative understanding of thermo- mechanical effects during chip formation in hard turning, J. Mater. Process. Technol. 176 (2006) 214–221. [11] M.E. Merchant, Mechanics of the metal cutting process. I. Orthogonal cutting and a type 2 chip, J. Appl. Phys. 16 (1945) 267– 275.

References [12] E.H. Lee, B.W. Shaffer, The theory of plasticity applied to a problem of machining, J. Appl. Phys. 18 (1951) 405–413. [13] T. O¨ zel, T. Altan, Process simulation using finite element method—prediction of cutting forces, tool stresses and temperatures in high speed flat end milling, J. Mach. Tools Manuf. 40 (2000) 713–783. [14] T. O¨ zel, E. Zeren, Determination of work material flow stress and friction for FEA of machining using orthogonal cutting tests, J. Mater. Process. Technol. 153–154 (2004) 1019–1025. [15] F. Reusch, B. Svendsen, D. Klingbeil, Local and non local gurson based ductile damage and failure modelling at large deformation, Euro. J. Mech. A/Solid 22 (2003) 779–792. [16] P. Rosakis, A.J. Rosakis, G. Ravichandran, J. Hodowany,Athermodynamic internal variable model for the partition of plastic work into heat and stored energy in metals, J. Mech. Phys. Solids 48 (2000) 581–607. [17] R. Sievert, A.-H. Hamann, D. Noack, P. L¨owe, K.N. Singh, G. K¨unecke, R. Clos, U. Schreppel, P. Veit, E. Uhlmann, R. Zettier, Simulation of chip formation with damage during high-speed cutting, Tech. Mech. 23 (2003) 216–233 (in German). [18] R. Sievert, A.-H. Hamann, D. Noack, P. L¨owe, K.N. Singh, G. K¨unecke, Simulation of thermal softening, damage and chip segmentation in a nickel super-alloy, in: H.K. T¨onshoff, F. Hollmann (Eds.), Hochgeschwindigkeitsspa-nen,Wiley-vch, 2005, ISBN , pp. 446–469 (in German). [20] H.K. T¨onshoff, B. Denkena, R. Ben Amor, A. Ostendorf, J. Stein, C. Hollmann, A. Kuhlmann, Chip formation and temperature development at high cutting speeds, in: H.K. T¨onshoff, F. Hollmann (Eds.), Hochgeschwindigkeit-sspanen,Wiley-vch, 2005, ISBN , pp. 1–40 (in German). [21] Q. Yang, A. Mota, M. Ortiz, A class of variational strain-localization finite elements, Int. J. Numer. Methods in Eng. 62 (2005) 1013–1037.

References [12] E.H. Lee, B.W. Shaffer, The theory of plasticity applied to a problem of machining, J. Appl. Phys. 18 (1951) 405–413. [13] T. O¨ zel, T. Altan, Process simulation using finite element method—prediction of cutting forces, tool stresses and temperatures in high speed flat end milling, J. Mach. Tools Manuf. 40 (2000) 713–783. [14] T. O¨ zel, E. Zeren, Determination of work material flow stress and friction for FEA of machining using orthogonal cutting tests, J. Mater. Process. Technol. 153–154 (2004) 1019–1025. [15] F. Reusch, B. Svendsen, D. Klingbeil, Local and non local gurson based ductile damage and failure modelling at large deformation, Euro. J. Mech. A/Solid 22 (2003) 779–792. [16] P. Rosakis, A.J. Rosakis, G. Ravichandran, J. Hodowany,Athermodynamic internal variable model for the partition of plastic work into heat and stored energy in metals, J. Mech. Phys. Solids 48 (2000) 581–607. [17] R. Sievert, A.-H. Hamann, D. Noack, P. L¨owe, K.N. Singh, G. K¨unecke, R. Clos, U. Schreppel, P. Veit, E. Uhlmann, R. Zettier, Simulation of chip formation with damage during high-speed cutting, Tech. Mech. 23 (2003) 216–233 (in German). [18] R. Sievert, A.-H. Hamann, D. Noack, P. L¨owe, K.N. Singh, G. K¨unecke, Simulation of thermal softening, damage and chip segmentation in a nickel super-alloy, in: H.K. T¨onshoff, F. Hollmann (Eds.), Hochgeschwindigkeitsspa-nen,Wiley-vch, 2005, ISBN , pp. 446–469 (in German). [20] H.K. T¨onshoff, B. Denkena, R. Ben Amor, A. Ostendorf, J. Stein, C. Hollmann, A. Kuhlmann, Chip formation and temperature development at high cutting speeds, in: H.K. T¨onshoff, F. Hollmann (Eds.), Hochgeschwindigkeit-sspanen,Wiley-vch, 2005, ISBN , pp. 1–40 (in German). [21] Q. Yang, A. Mota, M. Ortiz, A class of variational strain-localization finite elements, Int. J. Numer. Methods in Eng. 62 (2005) 1013–1037.

Material Assumptions  Inconel 718 Alloy composed of mostly nickel and chromium  Work piece is fundamentally thermoelastic, viscoplastic in nature Thermoelastic ○ Temperature changes induced by stress Viscoplastic ○ permanent deformations under a load but continues to creep (equilibrium is impossible)  Isotropic material behavior

Design Principle  Low cutting speeds Low strain-rates “Fast” heat conduction  High cutting speeds High strain-rates “Slow” heat conduction ○ Thermal softening ○ Shear banding

Design Principle  Johnson-Cook and Hooke Models Plastic deformation results in a temperature increase ○ Temperature increase is a function of strain (left) ○ Temperature increase results in softening At points of maximal mechanical dissipation in the material, softening effects may dominate hardening (right) ○ Results in material instability, deformation localization and shear-band formation

Design Principle  Finite-element simulation of thermal shear-banding Shear angle ○ Φ=40° Cutting tool angle ○ γ=0° Plane strain deformation V c =1000 m/min

FEA: Parallel  Notch represents a geometric inhomogenity Idealized notched structure discretized with bilinear elements oriented in the predicted shear-band direction. Average element edge-length here is mm.

FEA: Parallel  TOP Cutting speed vc=10 m/min Thermal conduction is “fast” ○ No thermal softening ○ NO shear-band formation.  BOTTOM Cutting speed vc=1000 m/min Thermal conduction is “slow” ○ Thermal softening ○ Shear-band formation ○ Chip formation

FEA: Rotated  Restricted to “high” cutting speed Assume adiabatic Idealized structure with elements oriented at 45◦ to the direction of shearing. As before, the average element edge length here is mm

FEA: Rotated  No shear band formation in the expected direction Temperature distribution in the mesh from above after shearing at a rate equivalent to a cutting speed of 1000 m/min

FEA: Rotated  Why? Constant strain elements

FEA: Reduced Parallel  Different element edge lengths Temperature distribution in the notched structure discretized parallel to the shear direction using different element edge lengths: 0.005mm (above), mm (below).

FEA: Reduced Rotated  Different element edge lengths Temperature distribution in the notched structure discretized at a 45◦ angle to the shear direction using different element edge lengths: 0.005mm (above), mm (below).

Results of FEA Shear-band  The coarser mesh in both cases, and the rotated mesh in general, behave more stiffly, resulting in “delayed” shear- band development.

FEA: Chip Formation  δ  discretization angle Finite-element model for the work-piece/tool system used for the cutting simulation. Mesh orientation relative to the cutting plane is represented here by the angle δ.

FEA: Chip Formation  Merchant, Lee and Schaffer Models  φ =π/4 - 1/2(arctanμ − γ) φ  Shear angle γ  Chip angle μ  Coefficient of Friction

FEA: Chip Formation  Chip formation becomes increasingly inhibited and diffuse as δ increases beyond φ. Chip formation and temperature field development for different mesh orientation angles δ:δ=20◦ (left), δ=40◦ (middle), δ=60◦ (right).

FEA: Chip Formation Chip formation with γ =−5◦ and δ=30◦ for different discretizations. Left: 60×10 elements; middle: 150×20 elements; right: 250×30 elements. Note the mesh- dependence of segmentation, i.e., an increase in segmentation frequency with mesh refinement.

Conclusion  Strong dependence on element size and orientation Affects chip geometry and cutting forces Using the mesh to fit the orientation and thickness of simulated shear bands to experimental results is somewhat questionable and in any case must be done with great care.  Better understanding of cutting forces Better efficiency Save money