Available online at ScienceDirect. Procedia CIRP 58 (2017 )

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1 Available online at ScienceDirect Procedia CIRP 58 (207 ) th CIRP Conference on Modelling of Machining Operations Dislocation Density Based Material Model Applied in PFEM-simulation of Metal Cutting Rodríguez, J.M. *,a, Jonsén, P. a, Svoboda, A. a a Division of Mechanics of Solid Materials Department of Engineering Sciences and Mathematics. Luleå University of Technology, Luleå SE-97 87, Sweden Corresponding author. Tel.: ;. address: rodjua@ltu.se Abstract Metal cutting is one of the most common metal-shaping processes. In this process, specified geometrical and surface properties are obtained through the break-up and removal of material by a cutting edge into a chip. The chip formation is associated with large strains, high strain rates and locally high temperatures due to adiabatic heating. These phenomena together with numerical complications make modeling of metal cutting challenging. Material models, which are crucial in metal- cutting simulations, are usually calibrated against data from material testing. Nevertheless, the magnitudes of strains and strain rates involved in metal cutting are several orders of magnitude higher than those generated from conventional material testing. Therefore, a highly desirable feature is a material model that can be extrapolated outside the calibration range. In this study, a physically based plasticity model based on dislocation density and vacancy concentration is used to simulate orthogonal metal cutting of AISI 36L. The material model is implemented into an in-house particle finite-element method software. Numerical simulations are in agreement with experimental results for different cutting speed and feed. c 207 The The Authors. Published by Elsevier by Elsevier B.V. This B.V. is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of the scientific committee of The 6th CIRP Conference on Modelling of Machining Operations. Peer-review under responsibility of the scientific committee of The 6th CIRP Conference on Modelling of Machining Operations Keywords: PFEM, machining, cutting, material model, dislocations ;. Introduction Nowaday, product development puts greater demand on the repeatability and a more predictable product development process. Fast changes in the marked require shorter lead-times, higher degree of innovation and more flexible products and services. As a consequence, fewer mistakes are allowed during the product development process. Numerical modeling and simulation in research and development for the manufacturing industry. Mainly, cutting simulations are used for two purposes: first, they can be used to verify product performance and efficiency within conceptual phase or detail design phase of a product development process. Secondly, instead of the time and resource consuming trial and error approach, they can be used as a test bench to increase the understanding of the physical behavior of both the workpiece as well as cutting tools during machining operations in a shorter time. This makes it possible, within a short period of time, to produce innovative solutions that create greater value in terms of accessibility, quality, productivity and profitability [3]. However, modeling of metal cutting processes has been one of the more challenging research field mainly by two reasons. First, here is a need of a reliable constitutive model that can predict the thermomechanical behavior of materials at very high deformation rates end temperatures. The second challenge is concerned with the modeling and realization of large configuration changes. The purpose of this paper is to combine a physically based constitutive model with the Particle Finite Element Method (PFEM) to solve the problems associated with large configurations changes. An important feature of a physically based plasticity model is the ability to extrapolate the material behavior outside the calibration range, because this models are related to the underlying physics of the deformation coupled to the microstructure evolution. The model is used to predict the chip formation and cutting forces in machining of AISI 36L steel. A comparison of the Johnson Cook and a physically based constitutive model was presented in [2], in this study the physically based model predict better the experimental results. 2. The Particle Finite Element Method The Particle Finite Element Method (PFEM) is a FEM-based method [4], initially developed for the solution of free surface fluid mechanics problems. The main objectives were, on the The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of the scientific committee of The 6th CIRP Conference on Modelling of Machining Operations doi:0.06/j.procir

2 94 J.M. Rodríguez et al. / Procedia CIRP 58 ( 207 ) one hand, to develop a method to eliminate the convective terms in the balance equations. On the other hand, to introduction of a technology, based on the alpha shape method, able to deal with free boundary surfaces is a second objective. The method has evolved from a meshless method, in which the nodes are supposed to be particles that moves according to simples rules of motion, to an improved lagrangian approach in which the advantages of standard FEM are used. PFEM can be characterized by the following ingredients: () the use of a lagrangian formulation to describe the motion. A selected cloud of particles of infinitesimal size are tracked along the motion to describe the continuum properties evolution.(2) Numerical computation are done on the basis of a finite element mesh that is constructed at every time step on the basis of the particle positions.then, Delaunay triangulation [2], allowing the construction of a finite element metsh for a given set of nodes emerges as a suitable meshing procedure. (3) The use of a boundary recognition procedure to identify what particles of the cloud define an external or internal boundary [3]. The PFEM, typically consist of the following steps (see Figure ) Fig. 2. Boundary recognition via the alpha shape []. (a) cloud of points, (b) Delaunay triangulation and the alpha shape The continuous reconnection introduced in step 2 is the key strategy to circumvent the typical mesh distortion generated when a Lagrangian description is used with problems involving large strains. 3. A Physically based plasticity model A plasticity model with a rate-dependent yield limit is used in the present study. The latter is described below [9]. 3.. Dislocations density model The dislocations density consider dislocation glide and climb processes contribution to the plastic straining. The yield limit on this approach is separated into two components according to σ y = σ G + σ () Fig.. Steps in the Particle Finite Element Method.. Fill the solid domain with a set of points refereed to as particles. The accuracy of the numerical solution is clearly dependent on the considered number of particles. 2. Generate the finite element mesh using the particles as nodes. This is achieved using a Delaunay triangulation [2]. 3. Identify the external boundaries to impose the boundary conditions and to compute the domain integrals using the alpha shape method [3] (see Figure 2 ). 4. Solve the non-linear Lagrangian form of the balance equations finding displacement, pressure and temperature (see [6 9]). 5. Update the particle position using the computed values of displacements. 6. Go back to step 2 and repeat for the next time step. where σ G and σ are the long-range component and the shortrange contribution to the flow stress, respectively. The first component,σ G, is the stress needed to overcome the long-range interactions lattice distortions due to the dislocations substructure. The second component, σ, is the stress needed for the dislocations to pass through the lattice and to pass the short-range obstacles. Thermal vibrations will also assist the dislocation when passing an obstacle. The long-range stress component is commonly written as: σ G = mαμb ρ i (2) where m is the Taylor orientation factor, α is a proportionality factor, μ is the temperature dependent shear modulus, b is the Burgers vector and ρ i is the immobile dislocation density. The short-range stress component may be written as, [ ( )] p σ = τμ kt ε q ΔFb 3 μ ln re f ε p where k is Boltzmanns constant, T is the temperature field, ΔF denote the required free energy needed to overcome the lattice resistance or obstacles without assistance from external stress, τ denote the athermal flow strength required to move the dislocation past barriers without assistance of thermal energy, ε re f denote the reference strain rate and ε p the plastic strain rate. The exponent p and q characterize the barrier profiles and usually have values between 0 p and 0 q 2 respectively. (3)

3 J.M. Rodríguez et al. / Procedia CIRP 58 ( 207 ) Table. Cutting data in simulations Table 2. SANMAC36L material parameters Test no Speed (m/min) Feed (mm/rev) Depth (mm) Young Modulus E (200 GPa) Poisson s ratio 0.3 Heat Capacity (445 JKg C ) Expansion Coefficient ( / C) Thermal conductivity (4 W C m ) Density (7900 Kg m 3 ) 3.2. Structure evolution The evolution of the structure is considered to consist of a hardening and a recovery process. The used model assumes that the mobile dislocation density is stress and strain independent and much smaller than the immobile ones. Hence the evolution equation is written as; ρ i = ρ (+) i ρ ( ) i(glide) ρ( ) i(climb) (4) where i index denotes the immobile dislocations. The increase in immobile dislocation density is assumed to be related to the plasticity strain rate and may be therefore be written according to ρ (+) i = m b Λ ε p (5) where Λ denote the mean free path which is a function of the size of the grains and the dislocations sub-cell diameter. The recovery may occur by dislocation glide and/or climb. The former is described by ρ ( ) i(glide) =Ωρ i ε p (6) where Ω is a recovery function which depends on the temperature and strain rate. Recovery by climb is described by ρ ( ) i(climb) = 2c γ D v c eq v μb 3 kt (ρ2 i ρ 2 eq)c v (7) where c v is the vacancy fraction, c eq v is the thermal equilibrium vacancy concentration, D v is the diffusivity and c γ is a calibration parameter and ρ eq is the equilibrium value of the dislocations density. More details about the material model are found in [9,2] 4. Examples, result and discussion An orthogonal cutting operation was employed to mimic 2D plain strain conditions. The depth of cut, used for all the numerical simulations, was equal to 3 mm. The dimension of the workpiece was 8 mm in length and.6 mm in height (see Figure 3). A horizontal velocity corresponding to the cutting speed was applied to the particles at the right side of the tool as is given in Table. The particles along the bottom and the left sides of the workpiece were fixed. Material properties for the workpiece material are available in Tables 2, 3 and 4. The calibration was accomplished via a parameter fitting procedure in conjunction with material data based on uniaxial compression tests at low and high strain rates, at high strain rates a SHPB rig was used (see more details in [2]). Material properties of the tool were Table 3. Known or assumed parameters k ( JK ) m 3.06 b ( m ) ε re f (*0 6 s ) Q vf ( J) χ (0.9) Ω 0 ( (m 3 )) ξ (2/3) g ( m) α (0.4) ρ i0 ( 0 2 (mm 3 )) p (0.333) q (.902) c γ (0.0275) Q vm ( J) D vm m 2 s c v D v0 ( m 2 s ) assumed thermo-elastic. The workpiece was discretized with 05 particles, see Figure 3. The tool geometry was discretized by 2298 tree-node thermo-mechanical elements. Due to adaptive insertion and removal of particles, the average number of particles increased up to The model for the tool-chip interface employed in this study is a generalized Coulumb friction model. The friction coefficient μ = 0.5 was used, according to a estimation obtained using Merchant s theory in a previous work (see [9,2]). 4.. Cutting and Feed forces Average values of the computed forces in the steady state region are compared with the experimental results in Table 5 and Table 6. The error used for the evaluation of the computed results is computed as Computed Measured error = 00% (8) Measured Table 5 and Table 6 show the experimentally measured forces and simulated forces, respectively. Table 6 also shows the calculated errors using equation 8. Table 6 shows that the cutting force was overestimated in all tests by about 5%. Meanwhile, the feed force was underestimated by about 5%. The errors

4 96 J.M. Rodr ıguez et al. / Procedia CIRP 58 (207) Table 4. Temperature dependent parameters. Temperature C Kc Table 6. Simulated cutting forces τ ΔF Ω Fc (N) Test no error (%) F f (N) error (%) Fig. 4. Effective plastic strain rate distribution. Fig. 3. 2D plane strain PFEM model of orthogonal cutting: initial set of particles in Table 6 must be related to the context where they will be used, namely the cutting tool manufacturing industry. Literature overview [2] show that in the industrial production of nominally identical cutting tool as well as variations in material properties of nominally the same material can cause variations around the 0% in forces. Figure 5. Initial value of DD 06 s2 was increased up to 09 s2. In the domains of highest temperature concentrated close to the tool rake face, see Figure 6, the significant generation of vacancies coupled with the dislocation recovery is present, see Figure Material Response All figures presented in this section correspond to the steady state conditions. The results shown are for the cutting velocity of 80 mmin and feed of 0.5 mm. Figure 4 illustrates distribution of plastic strain rates in the primary and the secondary shear zones. Figure 4 presents a maximum plastic strain rate value of s. The dislocation density (DD) and vacancy concentration are shown in Figure 5 and Figure 6, respectively. In the area close to the outer surface of the formed chip with lower temperature level, the increased dislocation density controls the hardening, Table 5. Experimentally measured cutting forces Test no Fc (N) F f (N) Fig. 5. Dislocation Density distribution 5. Conclusions A physically based material model with emphasis on high strain rates was implemented in a in-house PFEM based code and used successfully for the prediction of chip formation in metal cutting. This material model enables to make a more detailed microscopic study of the process zone and build good understanding regarding the interaction between strain hardening, thermal softening and shear localization during the chip formation. Numerical results obtained in this work have been compared with experimental results. In conclusion, the numerical and ex-

5 J.M. Rodríguez et al. / Procedia CIRP 58 ( 207 ) References Fig. 6. Excess of vacancy concentration distribution. perimental results are in agreement for different cutting speed and feed. This shows that the combination of using a physically based models together with PFEM will improve the precision of the numerical results. [] M. Cremonesi, A. Frangi and U. Perego. A Lagrangian finite element approach for the analysis of fluid structure interaction problems. International Journal for Numerical Methods in Engineering, vol. 84, pp , 200. [2] B.N. Delaunay. Sur la Sphère Vide, A la memoire de Georges Voronoi. Otdelenie Matematicheskii i Estestvennyka Nauk, vol. 7, pp , 934. [3] H. Edelsbrunner and E.P. Mucke. Three dimensional alpha shapes. ACM Transaction on Graphics, 3:43-72, 994. [4] S. R. Idelsohn, E. Oñate, and F. D. Pin. The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves. International Journal for Numerical Methods in Engineering, vol. 6, pp , [5] G. H. Johnson and W. H. Cook. A constitutive model and data for metals subjected to large strains high strain rates and high temperatures. Proceedings of the 7th symposium on ballistics, the Hague, the Netherlands, vol. 2, pp , 983. [6] J.M. Rodriguez. Numerical modeling of metal cutting processes using the particle finite element method(pfem). PhD thesis, Universitat Politècnica de Catalunya (UPC), Barcelona, 204. [7] J. M. Rodriguez, J. C. Cante, and J. Oliver. On the numerical modelling of machining processes via the Particle finite Element method (PFEM). CIMNE: Barcelona, vol. 56, pp. 86, (205). [8] J. M. Rodriguez, J. M. Carbonell, J. C. Cante, and J. Oliver. The particle finite element method (PFEM) in thermomechanical problems. International Journal for Numerical Methods in Engineering, DOI 0.002/nme.586,(206). [9] J. M. Rodriguez, P. Jonsén and A. Svoboda. Simulation of metal cutting using the particle finite-element method and a physically based plasticity model. Comp. Part. Mech, DOI:0.007/s [0] M. Sabel, C. Sator and R. Müller. A particle finite element method for machining simulations. Comput Mech, vol. 54, pp. 23-3, 204. [] M. Sabel, C. Sator, TI. Zohdi and R. Müller. Application of the Particle Finite Element Method in Machining Simulation Discussion of the Alpha- Shape Method in the Context of Strengt of Materials. ASME. J. Comput. Inf. Sci. Eng., DOI:0.5/ , 206. [2] A. Svoboda, D. Wedberg, L.E. Lindgren. Simulation of metal cutting using a physically based plasticity model. Modelling and Simulation in Materials Science and Engineering, vol. 8, pp. 23-3, 200. [3] M. Vaz Jr., D. R. J. Owen, V. Kalhori, M. Lundblad and L.E. Lindgren. Modelling and Simulation of Machining Processes. Archives of Computational Methods in Engineering, vol. 4, pp , 2007.