Analysis of metal extrusion by the Finite Volume Method

Transcription

1 Available online at ScienceDirect Procedia Engineering 207 (2017) International Conference on the Technology of Plasticity, ICTP 2017, September 2017, Cambridge, United Kingdom Analysis of metal extrusion by the Finite Volume Method a *, Marcelo Matos Martins b, Sérgio Tonini Button c a Department of Mechanical Engineering, CCT - University of Santa Catarina State (UDESC), CEP , Joinville - SC, Brazil. b Centre University Catholic of Santa Catarina, CEP Joinville - SC, Brazil. c Department of Materials Engineering, Faculty of Mechanical Engineering - (UNICAMP), CEP Campinas - SP, Brazil. Abstract Present work examines and validates the novel numerical scheme to calculate the velocity, stress, strain, pressure and strain rate fields of metal plastic flow in direct extrusion processes by employing the finite volume method, FVM. Traditionally, the classical methods such as the upper-bound, slab, slip-line and more recently the Finite Element Method have been largely applied in metal extrusion analysis. However, recently the FVM has been applied and published by the authors for analysis of metal plastic flow, concluding that direct extrusion of metals could be mathematically modelled by the plastic flow formulation similar to an incompressible non-linear viscous fluid. Tannehill et al. suggested that viscous fluid flow can be numerically simulated by FVM, obeying the mass, momentum and energy conservation equations and boundary conditions. Hence, the governing equations of metal plastic flow in Euler approach were discretized by FVM, using the Explicit MacCormack Method in structured, fixed and collocated mesh. SIMPLE method was applied to attain the necessary pressure-velocity coupling. These new numerical scheme was applied to the analysis of direct hot extrusion process of Al 6351 and Al 6060 aluminium alloys. The velocity and other variables fields achieved fast convergence and a good agreement with experimental results from visioplasticity tests by the grid stripe pattern technique and Forge 2008 software. The MacCormack Method applied to metal extrusion revealed consistent results without the need of artificial viscosity as required by the compressible fluid flow simulation approaches. Therefore, present numerical results confirm that FVM with MacCormack method together with Euler formulation approach and SIMPLE method can be applied satisfactory in the solution of metal forming processes The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the International Conference on the Technology of Plasticity. Keywords: metal extrusion; FVM; stripe technique; aluminum alloy; lead. * Corresponding author. Tel.: ; fax: address: josedivo.bressan@udesc.br The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the International Conference on the Technology of Plasticity /j.proeng

2 426 José D. Bressan et al. / Procedia Engineering 207 (2017) Introduction Generally, extrusion of metals is a thermo-mechanical processing at warm and high temperature by which metallic alloys billets are transformed into simple bar, rod, profiles or bars of complex cross sectional shape. The successful operation, direct or back extrusion, involves the knowledge of the mechanics of metal flow (velocity, pressure, stresses and deformations fields), heat transfer, friction and the metallurgy of microstructure evolution (grain size, phase, porosity, damage, etc.), to manufacture the required quality of product physical and mechanical properties. Nowadays, both analyses are increasingly performed by numerical simulation with the aid of computers and software. Thus, modelling the mechanics of metal flow and microstructure evolution in metal forming is a modern and intense topic of scientific research in industry and academy. Nomenclature r,, z radial, angular and axial coordinates in cylindrical coordinate system (r,,z) respectively v r, v z velocity vector components σ rr, σ θθ, σ zz normal components of Cauchy stress tensor σ σ rz shear stress component of Cauchy stress tensor σ m hydrostatic pressure, m rr zz /3 s ij deviatoric stress, sij ij mij, true strain, strain rate tensor, c, T density, specific heat and temperature respectively q r, q z components of heat flow vector q effective or equivalent stress, 3/ 2s ij sij plasticity multiplier, 3 2 metal equivalent viscosity effective or equivalent plastic strain rate, 2/ 3 ij ij m friction factor, nt mk, where nt is the tangential friction stress t, t time, virtual time step respectively Q, F r, F z and S flow vectors V mn control volume of node mn t1 t1 Q mn, Q t 1 mn, Q mn predictor step, corrector step and current step respectively A, m 1, m 2, m 3, m 4, m 5, m 6, m 7, m 8 material parameters of Hansel-Spittel equation Traditionally, the classical methods such as the slab, slip-line, upper-bound [1,2] and more recently the Finite Element Method have been largely applied in metal extrusion modelling and analysis. In addition, the experimental technique of visioplasticity methods employed to analyses metal flow and validate the theoretical models have been the bi-sectioned billet with scratched grid pattern, stripe pattern grid and colour plasticine. Lead and aluminium have also been employed extensively at room temperature to obtain and analyse the metal flow fields. Materials which are commonly extruded by industries are copper, brass, aluminium, steel, titanium and magnesium alloys. The aim of present work is to employ the FVM to a novel numerical scheme to analyse the mechanics of metal flow in direct extrusion of aluminium alloys in 32.3 o and 13.4 o semi-die angles and compare them with experimental results from bi-sectioned billet with stripe pattern grid technique and Forge software. A secondary aim is to develop a unified approach that could be used in conjunction with microstructure evolution models, based on physical metallurgy laws, to predict the physical and mechanical properties of the final extrusion product. 2. Mathematical modelling of metal extrusion by the Finite Volume Method The analysis of the mechanics of metal flow and microstructure evolution can be treated as separated issues [3]. Recently, a novel numerical scheme has been developed and applied successfully by the authors [4] for analysis of metal plastic flow, concluding that direct extrusion of metals could be mathematically modelled by the plastic flow

3 José D. Bressan et al. / Procedia Engineering 207 (2017) formulation similar to an incompressible non-linear viscous fluid. Hence, in present work, metal plastic flow in direct cold and hot extrusion process was modelled by FVM, using the plastic flow formulation of metals with a fixed Eulerian reference coordinate system. Tannehill et al. [5] suggested that viscous fluid flow can be numerically simulated by FVM, satisfying mass, momentum and energy conservation equations and boundary conditions. Thus, FVM satisfy the conservation laws in discrete and global levels and FEM do not, it do not use control volumes [4] Governing equations of flow formulation According to Martins et al. [4], the governing conservation equations (mass, momentum and energy) in cylindrical coordinates system (r,,z) for axisymmetric case, can be cast in compact matrix structure in the form: Q Q 1 rfr Fz F S t t r r z (1) with F 0and Q, F r, Fz and S are flow vectors which assume the following format in Euler approach: vr Q vz ct F r vr 2 vr rr vrvz rz ctv r q r F z vz vzvr rz 2 vz zz ctv z q z 0 S r (2) 0. Flow vector Q has the field variables to be found. The constitutive equation of metal plasticity behaviour for axisymmetric plastic flow considers the material to be incompressible, isotropic and rigid-viscous-plastic. The relation between the stress tensor σ and strain rate tensor can be written in the following form σ 2 m, where m is hydrostatic pressure inside the solid calculated by m rr zz / 3. Metal plastic flow equivalent viscosity is calculated from the Associated Plastic Flow Rule s ij [3]: The effective strain rate must not assume values less than 10-3 /s to avoid numerical instability The Finite Volume Method For a fixed mesh in space, Eulerian reference coordinate system, integrating Eq. (1) over the control volume V mn, employing Gauss' Theorem, results in the following differential equation [4]: Q t mn 1 1 rfr sm,n rfr sm,n Fz s F s S m,n z m,n mn Vmn r (3) where s is the outward surface vector. Applying the Explicit MacCormack scheme to Eq. (3), considering outward surface vector corrections on the control volume [5], the current incremental step can be calculated. This scheme use explicit Euler scheme on time. MacCormack scheme is a pseudo-transient process, where t is a virtual time increment to obtain the final converged solution in the flow formulation approach with a fixed mesh in space. The MacCormack converged current step is calculated by the average between the predictor and corrector steps, t 1 t 1 Q mn = ( Q mn + Q t 1 t 1 mn )/2, where Q t 1 mn represents the predictor step, Q mn represents the corrector step, t is current time, t 1 t 1 is next time step, t is a virtual time step and Q mn is the final current step to be converged. The flow vector F must be discretized in correct way to ensure the main feature of MacCormack method which is a numerical method of second order accuracy in time and space [5]. The SIMPLE Method was applied to obtain coupling of velocity-pressure variables in present numerical scheme [4]. According to FVM method, "Ghost Volumes" were used at boundary conditions situated at external faces of billet. In these ghost volumes it can be applied boundary condition of "Dirichlet Kind" and "Neumann kind".

4 428 José D. Bressan et al. / Procedia Engineering 207 (2017) However, the boundary conditions for physical consistency applied in present direct extrusion analysis were [4]: Inlet: normalized velocity V z was set 1mm/s and pressure had zero gradient. Outlet: normalized velocity V z was set equal to extrusion ratio R (R=entrance area/exit area) for constancy of volume and pressure had zero gradient. Symmetry line: it was prescribed equal normalized velocity V r, but with opposing signal. Solid Wall: at solid wall interface between material and die, it was considered friction factor model represented by nt mk, where nt is the tangential friction stress, k is the material yield shear stress and m is the friction factor. Considering isotropic material, von Mises yield criteria and shear strain rate nt, the friction shear stress is given by: ( 2 / 3. nt ) 3. Material and Extrusion Modelling by Finite Element Method nt For validating present FVM code, it was employed in numerical simulation of direct extrusion and comparisons with experimental results obtained by performing visioplasticity tests, using the grid stripe pattern technique on EN AW 6060 aluminium alloy billets (details are reported in [6]), and comparison with Forge 2008 software for direct extrusion of Al 6351 aluminium alloy. Al 6060 billets were extruded in controlled conditions of temperature, approximately 400 o C, with different ram stroke (150 and 175 mm over an initial billet length of 293 mm by diameter 138 mm) and extrusion speed (2 and 5 mm/s). Al 6351 billets were extruded in controlled conditions of 450 o C and extrusion speed of 10 mm/s) (details are reported in [7]) and simulated by FEM, Forge 2008 software. Billets with contrast pins were modeled by a structured mesh of tetrahedral elements with variable size, seen in Fig. 1, by flow formulation approach with a fixed mesh in space. In order to simplify calculations, the velocity field was normalized by the inlet velocity V o = 2mm/s, thus, in present FVM simulations it was assumed inlet normalized velocity V o = 1. Consequently, all the numerical simulation velocity field results have to be multiplied by a factor 2. The obtained average material parameters, extrusion process and simulation parameters used in FVM and Forge 2008 simulations of extrusion of Al 6060 and Al 6351 are shown in Table 1. The tools were modeled as rigid. The viscoplastic friction model was adopted with the friction factor equal to 0.5 and 0.8. Table 2 presents the parameters used to model the billet material experimental plasticity behavior with the Hansel-Spittel law [7] which is commonly used in bulk hot metal forming. Flow stress of 6060 aluminium alloy were from torsion tests at 400 o C and strain rate of 0.01, 0.1, 1 and 10/s and for Al 6351 from tensile tests at 450 o C and strain rate of and 0.1/s. The curves were fairly linear with almost constant yield stress which can be represented by a rigid-perfectly-plastic material. Table 1 - Simulation parameters used in the numerical simulations of hot direct extrusion. Parameters EN AW 6060 Al 6351 Density ( ) 2710 (kg.m -3) 2710 (kg.m -3) Yield stress ( Y ) 40 (MPa) 255 (MPa) Area reduction (R) 93.7 % (R=16) 89 % (R=9) Inlet extrusion velocity (V o) Material extrusion temperature (T) 2 (mm/s) 400 o C 10 (mm/s) 450 o C Quantity of control volumes Time step ( t) (s) (s) Die semi-angle (θ) Friction factor parameter (m) Material hardening model, FVM: rigid-perfect-plastic rigid-strain rate sensitive, m 3=0.1 Material hardening model, Forge 2008: - Hansel-Spittel law Table 2 Hansel-Spittel law experimental parameters obtained to model the flow stress of 6060 aluminium alloy in torsion tests at 400 o C and strain rate of 0.01, 0.1, 1 and 10/s and Al 6351 at 450 o C with strain rate of and 0.1/s in tensile tests. Material A m 1 m 2 m 3 m 4 m 5 = m 6 = m 7 EN AW Al Material flow stress behavior was modeled by Hansel-Spittel law which is commonly used in hot forging, m T m 8 m2 m4 m5t m7 m3 m6t 1 Ae T e 1 e m3 and by rigid-strain rate sensitive material: K.

5 José D. Bressan et al. / Procedia Engineering 207 (2017) mm 30 mm 73 mm 26.6 mm 10 mm mm a) b) Fig. 1. Fixed Mesh in space employed in the numerical simulations by present FVM code for direct axisymmetric hot extrusion of aluminium alloys billets at its final length and deformation stage: a) Al 6351 with 1110 volumes and b) Al 6060 with 1580 volumes. 4. Results and Discussions In Fig.1a, mesh geometry with 1110 volumes and Fig.1b mesh with 1580 volumes employed in the present numerical simulations by FVM of hot direct axisymmetric extrusion of Al 6351 and Al 6060 aluminium alloys are shown. About 45,000 iterations were necessary to attain numerical convergence in steady state deformation. Comparisons of axial velocity V z and radial velocity fields from simulation results of hot extrusion of Al 6351 by Forge 2008 and FVM are shown in Fig. 2. Equivalent strain rate results are seen in Fig.3. Correlations of values are quite good despite different hardening law: present FVM simulations assumed strain rate sensitive material. In Fig.4, the comparison of axial velocity V z from experimental stripe pattern results of direct hot extrusion of Al 6060 and numerical simulation results from present FVM code are shown. Only the first 16 contrast rods were captured by the photo of billet stripe pattern grid at longitudinal section. However, in order to improve the experimental results, the deformed contrast rod 18 from a second specimen was added in the photo by copying its exact position and distortion inside the photo of Fig. 4a. Points A18, A, B, C, D, E, F, G, H correspond to the original initial position or distance of equally spaced contrast rods 18 to 9, in relation to the position of billet-ram interfacial contact surface. Thus, these points should have the same axial velocity of V z = 2mm/s which correspond to the inlet extrusion constant velocity V o = 2mm/s. Therefore, V o up to V 6 are estimated experimental isolines of constant axial velocity V z. At the die exit points, the material velocity should be V 6 = 32mm/s due to volume conservation law for extrusion ratio R=16. The dead zone metal of zero velocity is marked in the upper corner. 1.00E E E E E E E-2 a) b) 0.73E E E E E E E E E E E E E E E-2 c) d) 7.45E-2 Fig. 2. Comparison of simulation results of extrusion of Al Axial velocity V z: a) FEM, Forge 2008 and b) FVM, present work. Comparison of radial velocity V r: c) FEM, Forge 2008 with friction m=0.5 and d) FVM, present work with friction m=0.8.

6 430 José D. Bressan et al. / Procedia Engineering 207 (2017) a) b) Fig. 3. Comparison of simulation results of equivalent strain rate in extrusion of Al 6351: a) FEM, Forge 2008 with friction m=0.5 and b) FVM, present work with friction m=0.8. a) b) 5. Conclusions Fig. 4. Axial velocity field V z for axisymmetric hot extrusion of 6060 aluminium alloy: a) from experimental stripe pattern results b) normalized axial velocity field from numerical simulations by present FVM code, assuming friction factor m=0.5. From experimental and numerical simulations with FVM and FEM codes analysis of hot direct axisymmetric extrusion of Al 6351 and Al 6060 aluminium alloys, the following conclusions can be drawn, - Velocity and other variables fields achieved fast convergence and a fairly good agreement with experimental results from visioplasticity tests by the grid stripe pattern technique and FEM by Forge 2008 software, - Present numerical results confirm that FVM with MacCormack method, Euler formulation and SIMPLE method can be applied satisfactory in the solution of metal forming processes without the need of artificial viscosity. Acknowledgements The authors would like to thank the support of CNPq- Brazil, University of Santa Catarina State (UDESC)-Brazil, University Centre- Catholic of Santa Catarina, Joinville/SC - Brazil and UNICAMP/SP - Brazil. References [1] R. Hill, The Mathematical Theory of Plasticity. Oxford University Press, [2] W. Johnson, R. Sowerby and R.D. Venter, Plane Strain Slip-Line Fields for Metal Deformation Processes. Pergamon Press, Oxford, [3] B. Verlinden, J. Driver, I. Samajdar and Roger D. Doherty, Thermo-Mechanical Processing of Metallic Materials, R.W. Cahn (Ed.), Elsevier, Oxford, [4] M.M. Martins,J.D. Bressan and S.T. Button, Lead extrusion analysis by finite volume method, in: E. Oñate, D.R.J. Owen, D. Peric and B. Suárez (Eds), XII International Conference on Computational Plasticity. Fundamentals and Applications. COMPLAS 2013, Barcelona, [5] J.C.Tannehill, D.A Anderson and R.H. Pletcher, Computational Fluid Mechanics and Heat Transfer. Taylor&Francis, London, [6] C. Bandini, B. Reggiani, L. Donati, L. Tomesani, Code validation and development of user routines for microstructural prediction with Qform. in: Aluminium Two Thousand World Congress and International Conference on Extrusion and Benchmark ICEB Materials Today: Proceedings 2 ( 2015) [7] J.D. Bressan, M.M. Martins and S.T. Button, Analysis of aluminium hot extrusion by finite volume method, in: Aluminium Two Thousand World Congress and International Conference on Extrusion and Benchmark ICEB Materials Today: Proceedings 2 ( 2015)