Finite Element Method Simulation of Machining of AISI 1045 Steel With A Round Edge Cutting Tool Tuğrul Özel and Erol Zeren Department of Industrial and Systems Engineering Rutgers, The State University of ew Jersey Piscataway, ew Jersey 08854 USA Abstract In this paper, FEM modeling and simulation of orthogonal cutting of AISI 1045 steel is studied by using dynamics explicit Arbirary Lagrangian Eulerian method. The simulation model utilizes the advantages offered by ALE method in simulating plastic flow around the round edge of the cutting tool and eliminates the need for chip separation criteria. Johnson- Cook work material model is used for elastic plastic work deformations. A methodology developed to determine friction characteristics from orthogonal cutting tests is also utilized for chip-tool interfacial friction modeling. The simulation results include predicted chip formation as well as temperature and stress distributions. These results are highly essential in predicting machining induced residual stresses and other properties on the machined surface. 1. ITRODUCTIO Finite Element Method (FEM) based modeling and simulation of machining processes is continuously attracting researchers for better understanding the chip formation mechanisms, heat generation in cutting zones, tool-chip interfacial frictional characteristics and integrity on the machined surfaces. Predictions of the physical parameters such as temperature and stress distributions accurately play a pivotal role for predictive process engineering of machining processes. Tool edge geometry is particularly important, because its influence on obtaining most desirable tool life and surface integrity is extremely high. Therefore, development of accurate and sound continuum-based FEM models are required in order to study the influence of the tool edge geometry, tool wear mechanisms and cutting conditions on the residual stresses and surface integrity on the machined surfaces. This paper aims to review the FEM modeling studies conducted in the past and to develop FEM models for most satisfying simulation of the physical cutting process and most reasonable predictions for cutting forces, temperatures and residual stresses on the machined surface. In continuum-based FEM modeling, there are two types of analysis in which a continuous medium can be described: Eulerian and Lagrangian. In a Lagrangian analysis, the computational grid deforms with the material where as in a Eulerian analysis it is fixed in space. The Lagrangian calculation embeds a computational mesh in the material domain and solves for the position of the mesh at discrete points in time. In those analyses, two distinct methods, the implicit and explicit time integration techniques can be utilized. The implicit technique is more applicable to solving linear static problems while explicit method is more suitable for nonlinear dynamic problems. A vast majority of research has relied on the Lagrangian formulation [1,2,3,4,5,6], which allows the chip to be modeled from incipient to steady state where as some of the studies also used the Eulerian formulation [7]. However, using the Lagrangian formulation requires a criterion for separation of the undeformed chip from the work piece. Several chip separation criteria (e.g. strain energy density, effective strain criteria) have been developed and implemented [8]. Updated Lagrangian implicit formulation with automatic remeshing without using chip separation criteria has also been used in simulation of continuous and segmented chip formation in machining processes [9,10,11,12,13,14].
Arbitrary Lagrangian Eulerian (ALE) technique combines the features of pure Lagrangian analysis in which the mesh follows the material, and Eulerian analysis in which the mesh is fixed spatially and the material flows through the mesh. ALE formulation is utilized in simulating machining to avoid frequent remeshing for chip separation [15,16]. Explicit dynamic ALE formulation is very efficient for simulating highly non-linear problems involving large localized deformations and changing contact conditions as those experienced in machining. The explicit dynamics procedure performs a large number of small time increments efficiently. The adaptive meshing technique does not alter elements and connectivity of the mesh. This technique allows flow boundary conditions whereby only a small part of the work piece in the vicinity of the tool tip needs to be modeled. Friction in metal cutting plays an important role in thermo-mechanical chip flow and machined work surface formation. Most of the approach in modeling friction is to use an average coefficient of friction. This model consisted of the sticking region for which the friction force is constant, and the sliding region for which the friction force varies linearly according to Coulomb s law. Interfacial friction at the tool-work contacts influences cutting induced residual stresses [17]. FEM simulation of machining using rounded/blunt/worn edge tools is essential in order to predict accurate and realistic residual stress, temperature, strain and strain rate fields. Studies focused on predicting residual stresses on machined surfaces especially on finished machined hardened steels [18,19]. All of the reviewed work contributed to investigate various aspects of fundamental modeling of cutting processes. After conducting the review of the literature, the following deficiencies of the FEM simulations in machining are identified: a) Excessive use of non-commercial FEM codes that makes latest developments highly difficult to apply by end users. b) Lack of reliable work material flow stress models especially those including strain and strain history effects as well as microstructure-based characterization. c) Lack of work material models inclusion of microstructure effects under the processing conditions (strain, strain rate, and temperature) that must be used as input to any material flow simulation program. d) Computational difficulties associated with size of the problems, large solution times and challenges in frequent remeshing of local mesh densities in applying the simulations to realistic 3-D machining operations. e) eed for extensive computer time and engineering effort, making the technique uneconomical to use. 2. WORK MATERIAL MODELIG Accurate and reliable flow stress models are considered highly necessary to represent work material constitutive behavior under high-speed cutting conditions especially for a (new) material. The constitutive model proposed by Johnson and Cook [20] describes the flow stress of a material with the product of strain, strain rate and temperature effects that are individually determined as given in Equation (1). In the Johnson-Cook (JC) model, the parameter A is in fact the initial yield strength of the material at room temperature and a strain rate of 1/s and ε represents the plastic equivalent strain. The strain rate & ε is normalized with a reference strain rate & ε 0. Temperature term in JC model reduces the flow stress to zero at the melting temperature of the work material, leaving the constitutive model with no temperature effect. In general, the parameters A, B, C, n and m of the model are fitted to the data obtained by several material tests conducted at low strains and strain rates and at room temperature as well as split Hopkinson pressure bar (SHPB) tests at strain rates up to 1000/s and at temperatures up to 600 C. JC model provides good fit for strainhardening behavior of metals and it is numerically robust and can easily be used in FEM simulation models. JC shear failure model is based on a strain at fracture criteria given in Equation 2. Many researchers used JC model as constitutive equation for high strain rate, high temperatures deformation behavior of steels (see Table 1). JC shear failure model is utilized in modeling and simulating the segmented and discontinuous chip formations in cutting of AISI 4340 steel [21,22]. m & n ε T T room σ = [ A + B( ε ) ] 1 + C ln 1 (1) & ε 0 Tmelt Troom
f ε = d1 + d2e p d 3 σ eff 1 + d 4 & ε T T ln 1 + & ε0 Tmelt T room room (2) Table 1. Johnson-Cook model constants for various steels Steel AISI 4340 AISI 4340 AISI 1045 Ref. A (MPa) B (MPa) n C m [21] 950.0 725.0 0.375 0.015 0.625 [22] 792.0 510.0 0.26 0.014 1.03 [23] 553.1 600.8 0.234 0.013 1.000 3. FRICTIO MODELIG Several researchers have used Oxley s parallelsided shear zone in which the primary shear zone is assumed to be parallel-sided and the secondary zone is assumed to be of constant thickness, in order to obtain work flow stress data. In order to successfully determine flow stress for JC material model and friction characteristics at the tool- chip interface, Özel and Zeren [24] proposed some modifications and improvements to Oxley s model [25], that includes integration of Johnson-Cook constitutive model as for the flow stress and triangular shaped secondary shear zone as it was confirmed via FEM simulations (see Figure 1). The basic concept of this methodology is the use of orthogonal cutting experiments and inverse solution of Oxley s model in order to determine the flow stress and friction conditions experienced in the range of high-speed cutting. As commonly accepted, in the tool-chip contact area near the cutting edge, sticking friction occurs, and the frictional shearing stress, τ int is equal to average shear flow stress at tool-chip interface in the chip, k chip. Over the remainder of the tool-chip contact area, sliding friction occurs, and the frictional shearing stress can be calculated using the coefficient of friction µ e. The normal stress distribution on the tool rake face can be described as: a σ ( x) = σ 1 ( x/ l ) max c (3) ormal stress distribution over the rake face is fully defined and the coefficient of friction can be computed, once the values of the parameters σ max and a are found. The shear stress distribution on the tool rake face illustrated in Figure 2 can be represented in two distinct regions: a) In the sticking region: τ int ( x) = kchip, µσ e ( x) kchip,0< x lp (4) b) In the sliding region: τ int ( x) = µσ e ( x), µσ ( x) < k, l < x l (5) e chip P C In Equations 4 and 5, k chip is the shear flow stress of the material at tool-chip interface in the chip. Based on the methodology detailed in [26], the constants of the JC material model for AISI 1045 steel are computed as A= 451.6, B= 819.5, C= 0.0000009, n= 0.1736, m= 1.0955 for the extended ranges of strain (0.051-1.07), strainrate (1-17766 1/sec.) and temperature (20-721 C). The calculated friction characteristics include parameters of the normal and frictional stress distributions on the rake face as given in Table 2. α Chip t c t u E A C V Workpiece V c Primary zone Secondary zone M φ δt c F B l p l c D Tool (a) (b) Type I Slip-line field (c) Type II slip-lines field Figure 1. Illustration of the deformation zones and simulated slip-line fields in orthogonal cutting
Table 2. Friction characteristics determined for AISI 1045 steel using carbide cutting tool Material AISI 1045 V c (m/min) 100-400 t u (mm) 0.125-0.5 σ max (MPa) 1312.63 l p (mm) 0.639 l c (mm) 3.122 a 0.183 k chip (MPa) 202.95 µ e 0.64 4. FIITE ELEMET MODEL AD ADAPTIVE MESHIG The essential and desired attributes of the continuum-based FEM models for cutting are: (1) The work material model should satisfactorily represent elastic plastic and thermo-mechanical behavior of the work material deformations observed during machining process, (2) FEM model should not require chip separation criteria that highly deteriorate the physical process simulation around the tool cutting edge especially when there is dominant tool edge geometry such as a round edge or a chamfered edge is in present, (3) Interfacial friction characteristics on the tool-chip and tool-work contacts should be modeling highly accurately in order to account for additional heat generation and stress developments due friction. In this paper, a commercial software code, ABAQUS/Explicit v6.4 and ALE modeling approach is used to conduct the FEM simulation of orthogonal cutting considering round tool edge geometry and all of the above attributes are successfully implemented in the model. The chip formation is simulated via adaptive meshing and plastic flow of work material. Therefore, no chip separation criterion is needed. In the ALE approach, the explicit dynamics procedure performs a large number of small time increments efficiently. The general governing equations are solved both Lagrangian boundaries and Eulerian boundary approaches in same fashion. The adaptive meshing technique does not alter elements and connectivity of the mesh. This technique combines the features of pure Lagrangian analysis in which the mesh follows the material, and Eulerian analysis in which the mesh is fixed spatially and the material flows through the mesh. Explicit dynamic ALE formulation allows flow boundary conditions whereby only a small part of the work piece in the vicinity of the tool tip needs to be modeled. The simulation model is created by including workpiece thermal and mechanical properties, boundary conditions, contact conditions between tool and the workpiece as shown in Figure 2 and given in Table 3. The workpiece and the tool model use four-node bilinear displacement and temperature (CPE4RT) quadrilateral elements and a plane strain assumption for the deformations in orthogonal cutting. The cutting process as a dynamic event causes large deformations in a few numbers of increments resulting in massive mesh distortion and termination of the simulation. It is highly critical to use adaptive meshing with fine tuned parameters in order to simulate the plastic flow over the round edge of the tool. Therefore the intensity, frequency and sweeping of the adaptive meshing is adjusted to most optimum setting for maintaining a successful mesh during cutting. The general equations of motion in explicit dynamics analysis are integrated by using explicit central difference integration rule with diagonal element mass matrices, Equation 6. M u&& + C u& + K u = P (6) [ ]{ } [ ]{ } [ ]{ } { } 1 J J J u() i = ( M ) ( P() i I() i ) && (7) Where M J, P J, I J are the diagonal element mass matrix, the applied load vector, and the internal force vector, respectively. t( i+ 1) + t( i) u &( i ) u& 1 2 ( i 1 2) u&& + = + ( i) (8) 2 = u + t u& (9) u ( i+ 1) ( i) ( i+ 1) ( i+ 1 2) 1 J J J θ() i = C P() i F() i θ ( ) ( ) & (10) = & (11) ( i+ 1) θ ( i) + t ( i+ 1) θ ( i) The expressions for the nodal displacement u, velocities u&, and accelerations u& &, are given in Equations 6 and 7 where t is time increment. u is a degree of freedom and i is the incremental number in the explicit dynamics step
y x CHIP TOOL V = V x c T=T 0 WORKPIECE Lagrangian U x = U y = 0, T= T 0 Figure 2. Simulation model used for ALE with Lagrangian boundary conditions [27]. In this case, we see that the system equations become uncoupled so that each equation can be solved for explicitly. This makes explicit dynamic method highly efficient for nonlinear dynamics problem such as metal cutting. During metal cutting, flow stress is highly dependent on temperature fields as we discussed earlier. Therefore, fully coupled thermal-stress analysis is required for accurate predictions in FEM simulations. In this analysis, the equations of heat transfer are integrated using explicit forward difference time integration with lumped capacitance matrix as given by Equations 10 and 11, where C J is the lumped capacitance matrix, P J is the applied nodal source vector, F J is the internal flux vector, is the temperature θ at node. In summary, the explicit dynamics method is used mainly because it has the advantages of computational efficiency for large deformation and highly non-linear problems as experience in machining. Machining, as a coupled thermalmechanical process, could generate heat to cause mechanical and thermal effects influence each other strongly. In the mean time, work material properties change dramatically as strain rate and temperature changes. Thus, the fully coupled thermal-stress analysis, in which the temperature solution and stress solution are also carried out concurrently, is applied. Table 3. Cutting conditions, work and tool material properties used in the FEM simulation model Orthogonal Cutting Parameters Cutting speed, V c (m/min) 300 Uncut chip thicknes, t u (mm) 0.1 Width of cut, w (mm) 1 Tool rake angle, α (degree) -5 Tool clearance angle (degree) 5 Tool edge radius, ρ (mm) 0.02 AISI 1045 Workpiece Properties Coefficient of thermal expansion 11 (at 20 C) (µm /m C) Density (g/cm 3 ) 7.8 Poisson s ratio 0.3 Specific heat (J/kg/ C) 432.6 Thermal conductivity (W/m C) 47.7 Young s modulus (GPa) 200 Carbide Tool Properties Coefficient of thermal expansion (µm/m C) Density (g/cm 3 ) 15 Poisson s Ratio 0.2 Specific heat (J/kg/ C) 203 Thermal conductivity (W/m C) 46 Young s Modulus (GPa) 800 4.7 (at 20 C) 4.9 (at 1000 C) 5. RESULTS AD DISCUSSIOS The simulations were conducted and the chip formation process from the incipient to the steady state was fully observed as shown in Figure 3.
12 µm 25 µm 50 µm 75 µm Figure 3. FEM Simulation of orthogonal cutting with a round edge tool using ALE with Lagrangian boundary conditions Figure 4. The stress distributions of σ xx and σ yy in orthogonal cutting with a round edge tool (predicted stresses are in 10-4 x MPa)
Figure 3 also shows the temperature predictions at increments of 12, 25, 50 and 75 µm as maximum temperatures of 850, 910, 1090 and 1120 C respectively. The distributions of the predicted the von Mises stress distributions are given in Figure 4. The von Mises stress σ xx and σ yy also represent the residual stress distributions on the machined surface. From the simulation results it was observed that there exist a region of very high deformation rate around the round edge of the cutting tool. The round edge of the cutting tool and the highly deformed region underneath has an dominant influence on the residual stresses of the machined surface. This also signifies the current work when compared the earlier FEM modeling studies that used criteria for chip-workpiece separation [28,29]. The uses of separation criteria undermine the influence of the cutting edge on the residual stress on the machined surface. In this study, the work material is allowed to flow around the round edge of the cutting tool and simulated the physical process. 6. COCLUSIOS In this study we have utilized the dynamics explicit Arbirary Lagrangian Eulerian method and developed a FEM simulation model for orthogonal cutting of AISI 1045 steel using round edge carbide cutting tool. Johnson-Cook work material model and a detailed friction model are also used and work material flow around the round edge of the cutting tool is simulated in conjunction with an adaptive meshing scheme. The simulation of the chip formation, development of temperature distributions as well as predictions of the stress distributions in the chip, tool and on the machined surface are successfully achieved. This study establishes a framework to further study machining induced residual stresses accurately and process design via optimization of cutting conditions, tool edge geometry for high-speed machining applications. 7. REFERECES [1] Strenkowski, J.S., and Carroll, J.T., 1985, A finite element model of orthogonal metal cutting, ASME Journal of Engineering for Industry, 1985, 107, 346-354. [2] Shih, A. J., 1995, Finite element simulation of orthogonal metal cutting, ASME Journal of Engineering for Industry, 117, 84-93. [3] Usui, E. and Shirakashi, T., 1982, Mechanics of machining -from descriptive to predictive theory. In on the art of cutting metals-75 years later, ASME Publication PED, 7, 13-35. [4] Komvopoulos K., and Erpenbeck, S.A., 1991, Finite element modeling of orthogonal metal cutting, ASME Journal of Engineering for Industry, 113, 253-267. [5] Lin, Z. C. and Lin, S. Y., 1992, A couple finite element model of thermo-elastic-plastic large deformation for orthogonal cutting, ASME Journal of Engineering for Industry, 114, 218-226. [6] Zhang, B. and Bagchi, A., 1994, Finite element formation of chip formation and comparison with machining experiment, ASME Journal of Engineering for Industry, 116, 289-297. [7] Strenkowski, J.S., and Carroll, J.T., 1986, Finite element models of orthogonal cutting with application to single point diamond turning, International Journal of Mechanical Science, 30, 899-920. [8] Black, J. T. and Huang, J. M., 1996, An evaluation of chip separation criteria for the fem simulation of machining, Journal of Manufacturing Science and Engineering, 118, 545 553. [9] Marusich, T.D., and Ortiz, M., 1995, Modeling and simulation of high-speed machining, International Journal for umerical Methods in Engineering, 38, 3675-3694. [10] Sekhon, G.S., and Chenot, J.L., 1992, Some Simulation Experiments in Orthogonal Cutting, umerical Methods in Industrial Forming Processes, 901-906. [11] Ceretti, E., Fallböhmer, P., Wu, W.T., and Altan, T., 1996, Application of 2-D FEM to chip formation in orthogonal cutting, Journal of Materials Processing Technology, 59, 169-181.
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