Total Lagrangian Smooth Particle Hydrodynamics (TLSPH) has been applied to a set of non-trivial, commercially interesting forging examples. Being a mesh-free method, TLSPH can conveniently simulate processes having large deformation and material separation. Test cases were designed that were characterized by large material ﬂows having large changes in grain connectivity. The implementation used, Smooth Mach Dynamics (SMD), provided tunable simulation parameters that enabled the simulation to optimally match each case. The results showed that the TLSPH/SMD has the potential to model the metal forging process efﬁciently without numerical instabilities. Each case studied required adaptation of the simulation parameters to optimize the results. 1 Introduction 1.1 Motivation Forging is an important manufacturing operation because it is economical and produces parts having high strength, toughness, and ductility as a result of the control of grain formation and orientation. The motivation for forging simulation is to predict the outcome of each stage of the forging workﬂow in order to mitigate problems such as: incomplete die ﬁlling production and cracking of ﬂash undesirable grain structure cold shuts fracture during forging excessive required forging pressures high residual stress poor billet design undesirable material ﬂow excess operations needed to create a part Ability to predict these outcomes can help optimize the design of preform dies, forging operations and process changes such as alterations of billet temperature and sub- sequent heat treatment operations. 1.2 Background Understanding material deformation and ﬂow patterns during forging can provide key insights to the design offorging operations. Forging defects can lead to premature failure of structures. Predicting the generation of defects will allow forging operations to be optimized. Forging processes are characterised by rapid deformations under large forces that result in large plastic deformations. When the conventional ﬁnite element methods (FEM) are used to model large strain problems, high deformation in- evitably leads to mesh distortion and possible instability in the simulation because the topology and shape of the elements have profound effects on the quality and accu- racy of FE solutions [1]. Severely distorted elements may even lead to a singular stiffness matrix (non-positive Jaco- bian) and consequent analysis failure [2]. These issues are compounded in forging simulations, where very high dis- placements may produce mesh inter-penetration and en- tanglement resulting in the loss of connectivity and topo- logical integrity of the ﬁnite element network. In such cases, re-meshing part or all of the geometry is the only alternative solution. A mesh-less method, such as spherical particle hydrody- namics (SPH), offers many advantages for modelling con- tinuous high deformation processes such as forging. SPH is a particle based numerical method for solving partial differential equations. The problem geometry is idealized into ‘particles’ that represent speciﬁc material volumes. The method uses a local interpolation from the neigh- boring particles to construct continuous ﬁeld approxima- tions that can be used to discretise the problem space. Even though SPH has been traditionally used for mod- Modelling Closed-Die Forging Operations Using Total Lagrangian Smooth Particle Hydrodynamics elling ﬂuid ﬂows, SPH has been adapted to a wide va- riety of solid mechanics problems [3]. Since SPH does not require an explicit grid or mesh structure to repre- sent the problem geometry it eliminates time-consuming effort of mesh generation and reﬁnement through an im- plicit mechanism. It also avoids the need to address the difﬁculties associated with traditional mesh-based meth- ods in maintaining the integrity and quality of the mesh under large deformations thereby reducing associated er- rors. The grid-free nature of SPH makes this method ide- ally suited to modelling material forging processes that involve large deformations and changes in topological connectivity. Moreover, SPH intrinsically handles sur- face contacts during forging and naturally predicts self- contacts. To address the speciﬁc needs of solid mechanics, the total- Lagrangian SPH (TLSPH) was devised [4, 5]. In order for TLSPH to be viable, several problems had to be solved [6] to ensure its convergence and stability in order that large deformations having plastic ﬂow could be accurately sim- ulated [6]. In lieu of explicity remeshing, TLSPH han- dles remeshing implicitly by resetting of the material ref- erence frame in response to a displacement metric. The implementation used in this study was implemented and packaged within the Lammps simulation platform [7] as the package USER-SMD. 1.3 Outline The paper will review the theory associated with the TLSPH method, discuss the simulation method and re- lated parameters, introduce the four forging case studies, present the results of these simulations and ﬁnally discuss issues found during the study. 2 Materials and Methods 2.1 Theory The conservation equations will be solved by means of a total-Lagrangian approach which uses a reference con- ﬁguration to compute stresses and accelerations. Both conservation and constitutive equations are expressed in terms of the reference coordinates X, which are taken to be the coordinates of the initial, undeformed conﬁgura- tion. A mapping between the two systems describes the motion at time t: x= (X;t): (1) The displacement u (relative to the reference conﬁgura- tionX) is given by: u=xX: (2) The associated deformation gradient is given by: F=dx dX=du dX+I: (3) which is the transformation matrix that describes the rota- tion and stretch relative to the reference conﬁguration.The equations for mass, impulse, and energy conservation are given by: J=0 u=1 0r0PT _e=1 0_F:P (4) here,is the mass density, Pthe ﬁrst Piola–Kirchhoff stress tensor, eis the internal energy, and ris the gradi- ent or divergence operator and symbol T denotes the trans- pose of a matrix. The subscript 0 refers to a quantity eval- uated in the reference conﬁguration, while the absence of this subscript means that the current conﬁguration is to be used. J is the determinant of the deformation gradient F. 2.1.1 Total Lagrangian SPH Formulation In the SPH numerical method, the interpolated value of a functionAat any position xcan be calculated from the discrete values Ajon the surrounding particles by: A(x) =X j:jxxjj<2hVjAjW(xxj;h) (5) Here, the sum is over all particles jwithin a radius 2hof x.W(x;h)is a spline based interpolation or smoothing kernel with radius 2handVjis the volume fraction of the support particle j. At the beginning of each time step, a shape matrix Ki=X j2SiV0 jrWi(Xij) (XjXi): (6) is computed as the sum of the dyadic product of two vec- tors,rWi(Xij), the gradient of the kernel function and XjXi, the displacement of support particle jrelative to particleiin the reference (ie. material) conﬁguration. Here,Wi(Xij)is a kernel function which depends only on the scalar distance between particles i and j. Vjdenotes the supporting particle volume, and Sithe support region of the particle i. The shape matrix reﬂects the distribu- tion of the particles in the support region surrounding the particleiwithin the reference conﬁguration and is used to normalize the deformation to a spherical shape so that ﬁrst order consistency is maintained. This constraint is expressed: X j2SiV0 j(XjXi) rWi(Xij) =I (7) The deformation gradient and its time derivative are then obtained as: Fi=X j2SiV0 j(xjxi) K1rWi(Xij): (8) _Fi=X j2SiV0 j(vjvi) K1rWi(Xij): (9) A. C. Manson 2 Modelling Closed-Die Forging Operations Using Total Lagrangian Smooth Particle Hydrodynamics where xandvare the coordinates and velocities in the current conﬁguration. The velocity gradient in the cur- rent conﬁguration is then obtained as Li _FiF1 i. The Cauchy stress is ultimately obtained by time integration of a constitutive model which relates the Cauchy stress rate to the strain rate tensor Di 1 2(Li+LT i). To obtain the Cauchy stress, one ﬁrst needs the polar de- composition of the deformation gradient F: RiUi=Fi (10) which is used to remove the rigid rotation from the strain rate tensor: di=RT iDiRi (11) From there, one can then use the constitutive relations to calculate the Cauchy stress i. The ﬁrst Piola–Kirchhoff stress, Pi=JiFT iis used to link the material frame to the current frame where the deformation related forces are calculated as: fd i=X j2SiV0 jV0 i(PiK1 irWi(Xij) PjK1 jrWj(Xji)):(12) In addition to the deformation related force, there is a vis- cous force which becomes signiﬁcant at higher velocities. This is given as: fv i=X j2SimjmiijIrWi(Xij) (13) where ij=( cijij+ 2 ij ij;(vivj)(xixj)<0; 0; else: (14) whereij=h(vivj)(xixj) jxixjj2 ,ij=i+j 2,cijthe speed of sound between particles iandj, and , the shear and bulk viscosities respectively. Finally, the net force acting on particleiis: fi=fd i+fv i (15) 2.1.2 Hourglass Control Within the region neighboring any given particle, a de- generacy of zero energy motions can occur, the so-called hourglass effect. To eliminate this effect, an ideal particle separation based on the initial conﬁguration is proposed: hxijii=FiXij (16) As the particles deviate from this ideal separation, a cor- rective force can be applied. This corrective force is de- signed to minimize the error metric: "ij=1 2 hxijiixij+hxijijxij =1 2(Fi+Fj)Xijxij (17)To enforce the hourglass condition and maintain proper angular momentum, the corrective force: fHG i=X j2SiV0 jV0 iW(Xij)"ij"ijxij x2 ijxij xij:(18) is applied. 2.1.3 Constitutive Model The stress is broken into its pressure pand deviatoric d components. =pI+d: (19) The shear modulus for an isotropic material is given as: S=E 2(1 +): (20) In this study, a simple piecewise linear constitutive model is used where stress varies linearly with strain up to the yield point (in proportion with the elastic modulus) and varies linearly past the yield point (in proportion to the strain hardening modulus). 2.1.4 Plasticity Model The radial return algorithm of Wilkins [8] is used to calcu- late the elastoplastic deformation. A trial deviatoric stress is ﬁrst calculated assuming an initial elastic response. This gives an increment in plastic strain: p=vmy 3G+H(21) Here,vmis the von Mises stress and yis the current yield stress. The increment in the plastic strain is: p(t+ t) =p(t) + p(22) The yield stress increment is given as: y=Hp(23) whereHis the hardening modulus. The updated devia- toric stressdis given by: d(t+ t) =rsd(t) (24) wherersis the radial scale factor given as: rs=y vm(25) 2.1.5 Algorithm The algorithm for describing the stress driven motion of SMD particles given in [9] is given in Appendix A con- taining the steps to reset the material reference frame along with the radial return algorithm used to handle plas- tic deformation [9]. 2.2 Forging Cases The test cases have been chosen to address select prob- lem areas mentioned in the introduction such as die ﬁll- ing and fracture. The simplest billet geometries (blocks, cylinders) were chosen to simplify the interpretation of material ﬂow patterns. A. C. Manson 3 Modelling Closed-Die Forging Operations Using Total Lagrangian Smooth Particle Hydrodynamics (a) Part (b) Die (c) Mesh Lower Die Figure 1: Yoke: a: part, b: die, c: lower die mesh (a) Finish die (b) Preform die Figure 2: Smoothing applied to yoke die mesh central sec- tion. (a) Part (b) Die (c) Die Mesh Figure 3: Reverse Extrusion of Seamless Pipe: a: Part, b: Die, c: Die mesh 2.2.1 Sliding Yoke The sliding yoke was chosen due to its non-trivial topol- ogy and variation in depth and breadth along its long di- mension. The regions between the legs of the yoke and around the slider pin are expected to be regions of high shear. Here, the particle topology will change and require a reset of the reference state. The part is forged normal to its thickness (the smallest di- mension), similar to an engine connecting rod and forged from a rectangular billet with a volume slightly larger than the part itself to allow for complete die ﬁlling. The billet is oriented lengthwise to the part and has a width slightly smaller than the part itself. The center of the ﬁnish die has a raised region that forms the space between the legs of the yoke ﬁg. 2a. Preliminary simulations showed that this feature would cause an excessive separation (ie. fracture) of the billet away from the die cavity toward the ﬂash re- gion of the die. To address this, the hollowed section was smoothed to create a preform die ﬁg. 2b. 2.2.2 Seamless Pipe (Reverse Extrusion) The seamless pipe is awkward to model using FEM tech- niques due to the continuous changes in grain topology at the shear zone near the outer edge of the punch. The meshless approach eliminates these remeshing steps. The die was modeled as two concentric cylinders, each open at one end ﬁg. 3. The billet is cylindrical and the forging path along the axis of the two die cylinders. As the forging progresses, material is compressed and ini- tially displaced outward followed by ﬂow in the opposite direction of the punch motion. Preliminary simulations showed that a preform die was not neccesary. A. C. Manson 4 Modelling Closed-Die Forging Operations Using Total Lagrangian Smooth Particle Hydrodynamics (a) Part (b) Die (c) Lower Mesh Die Figure 4: 4arm part: a: part, b: die, c: die mesh 2.2.3 Four arm offset joint This part, ﬁg. 4, is similar to a part modeled in previously [10] and will be reproduced for consistency. It is expected to exhibit frequent particle topology changes along with non-trivial material ﬂows. As the material navigates the constricted and expanding regions, one expects perturba- tions in velocity and ﬂowrate. The combined two punch motion is collinear with the two aligned arms of the part ﬁg. 4b. The operation requires the closure of the upper and lower die sections followed by the two cylindrical punches sliding toward the center. In this case one is expected to initially see compression along the long axis of the cylindrical billet. As the two punches move toward each other, the billet will buckle and ﬁll the two orthogonal arms of the die. (a) Part (b) Compound Die (c) Crank Unit Die Mesh Figure 5: crossplane crankshaft part: a: part, b: die, c: die mesh 2.2.4 Crankshaft Unit This case has similarities to the yoke case, but with a wider variation of thickness in the direction of the punch motion. As a result, it can test the extent of die ﬁlling as a function of initial billet placement. A cross plane crankshaft can be forged using a compound die with two forging planes oriented at right angles ﬁg. 5b. Each plane has two die units each shaping a single crank throw. To handle the cross plane case, a full length billet stradling each crank across the main bearings is needed, otherwise one would rely on the fusion of pairs of oxi- dized surfaces. The construction of this billet will be ad- dressed in later work. Each crankshaft throw has a large depth variation and is expected to produce large material deformations ﬁg. 6. The action of the forging press is along the shortest di- mension of the part ie. normal to the plane of the crank A. C. Manson 5 Modelling Closed-Die Forging Operations Using Total Lagrangian Smooth Particle Hydrodynamics (a) Finish die (b) Preform die (c) Finish die (side) (d) Preform die (side) Figure 6: Comparison of ﬁnish die and preform die. throw. Due to the large hollowed out section in the center of the crank throw, a preform die, similar to the yoke case, is employed ﬁg. 6d. 2.3 Simulation Strategy Each part and and its corresponding die geometry was modeled and meshed using SALOME [11] and the die meshs exported as .stl ﬁles describing a 2D triangular surface mesh. The forging simulation was done using Lammps [7] with the SMD user package [9]. Each sim- ulation instance had a corresponding Lammps script ﬁle where the material, simulation parameters and simulation control logic were deﬁned. The die geometry was im- ported to Lammps and the billet geometry deﬁned within Lammps itself. The material used was steel at different temperatures: three cases at room temperature and the 4arm case at 1000C. During each trajectory, the po- sitions, velocities and relevant SMD properties (such as stress and strain) were written to a Lammps dump ﬁle. Other properties such as punch displacement and punch reaction force were computed and written out to data ﬁles to be graphed. Initial parameter settings and billet sizes were adjusted according to any issues that occured dur- ing the simulation. The volume of the part was used to estimate the initial billet size. The billet geometry was matched to the die geometry as closely as possible to pre- vent excess ﬂash. In two cases, preform dies were mod- eled to address problematic material ﬂows observed when ﬁnish dies were directly applied to the raw billet. This work will report the simulation of the preform dies since this is where most of the deformation occured.Material Parameters Parameter (units) Description E (GPa) Youngs Modulus y(GPa) Yield Stress 0(kg/mm3) Initial density  Poissons ratio H(GPa) Strain hardening modulus Table 1: Deﬁnition of Material Parameters. Material Parameters by Case Parameter Yoke Pipe 4arm crank E 210 210 9 210 y 0.5 0.5 0.023 0.5 0 7.7e-6 7.7e-6 7.7e-6 7.7e-6  0.3 0.3 0.3 0.3 H 0.002 0.002 0.002 0.002 Table 2: Material Parameters used in each forging case. 2.4 Simulation Platform and Parameters 2.4.1 Material Parameters Table 1 identiﬁes the relevant material parameters. Here, steel was used where the modulus of elasticity and the yield stress varied with temperature according to [12]. Ta- ble 2 provides the material parameters by case. 2.4.2 Simulation Parameters A set of tunable SMD parameters were used to control aspects of the simulation. These parameters are deﬁned in tab. 3 and the settings given, by case, in tab. 4. 2.5 Presentation of results The results are generated from the simulation trajectories using the Open Visualization Toolkit (OVITO) [13] and typically show the evolution of deformation and other pa- rameters in space and time for each case. In the ﬁgures to follow, rainbow color coding is used where blue hues indicate a lower value while red hues indicate a higher value. For instance, strain values will vary within the range [1:0!1:0]with the negative values being blue SMD Simulation Parameters Parameter (units) Description q1 Artiﬁcial Viscosity l0(mm) Particle Size Hg Hourglass Coefﬁcient skr Smoothing Kernel Radius Multiplier up Update Parameter Ec(GPa) Contact Stiffness Hp Hertz Parameter Table 3: Simulation Parameter Deﬁnitions. A. C. Manson 6 Modelling Closed-Die Forging Operations Using Total Lagrangian Smooth Particle Hydrodynamics Simulation Parameters by Case Parameter Yoke Pipe 4arm crank q1 0.6 0.6 1.0 0.6 l0 0.3 0.2 0.3 0.4 Hg 10 10 10 10 skr 2.01 2.01 2.01 2.01 up 0.3 0.5 0.4 NA Ec 80.0 80.0 80.0 80.0 Hp 1.0 1.0 0.8 1.0 Table 4: Simulation Parameters by Case. (a) Step 00 (b) Step 10 (c) Step 20 (d) Step 30 Figure 7: Evolution of rectangular layer through mid- section of yoke. and the positive values being red. Unless the coloring is obvious, the range of values for the hues is given for each ﬁgure or subﬁgure. Data is typically dumped from Lammps every 200 time steps. The notation m/n indicates snapshot m out of n total snapshots. 3 Results 3.1 Sliding Yoke 3.1.1 Preform Die The yoke system showed good die ﬁlling with a reason- able amount of ﬂash. A lateral cross section about mid- way along the legs of the yoke ﬁg. 1a was featured with respect to particle deformation and other properties. The evolution of a diagonal layer of particles was followed in ﬁg. 7. The trajectory showed that the outer diagonal would align with the legs of the yoke and the center di- agonal would ﬂatten out ﬁg. 7d. As the die closes, the orientation/elongation of the particles at the die surface becomes apparent. (a)xx (b)yy (c)xy Figure 8: Sectional strains time 14/31: [-0.1 !0.1]. (a) step 18 (b) step 22 (c) step 29 Figure 9: Evolution of plastic strain rate: [0 !18]. A. C. Manson 7 Modelling Closed-Die Forging Operations Using Total Lagrangian Smooth Particle Hydrodynamics (a)xx: [-0.1!0.1] (b)yy: [-0.1!0.1] (c) Plastic ﬂow: [0 !20] Figure 10: Properties at ﬁnal state. (a) Time 2/31 (b) Time 10/31 Figure 11: V on Mises stress: [0 !0.5]. The sectional strain is shown in ﬁg. 8 at an intermedi- ate stage of the forging process. The lateral strain xxis maximized near the midline of the billet where the cen- tral region is being carved out. The vertical strain yyis minimized near the midline of the billet. The shear yx zone appears at the corners where the central die section displaces the region between the yoke legs. The plastic strain rate ﬁg. 9 is also maximized in this central region where material is being displaced toward the sides. Initially it is concentrated at the surface and then spreads ubiquitously throughout the part as the die reaches the bottom of its stroke ﬁg. 9. As the upper die descends, the strain rate increases on the lower billet as it is pressed against the lower die ﬁg. 9a. From there the strain rate spreads throughout the billet and becomes conﬁned to the midline 9b. As the base of the forging stroke is reached, the strain rate distributes throughout as the billet completely ﬁlls the die cavity ﬁg. 9c. The global distribution of properties is shown in ﬁg. 10. The strainxx, normal to the major axis, given in ﬁg. 10a, shows highest strain in the region between the legs of the yoke as the material is displaced to ﬁll the legs of the part. The strainyy, in the direction of the punch motion, shows (a) Step 00 (b) Step 10 (c) Step 20 (d) Step 50 Figure 12: Evolution of cylindrical layer during reverse extrusion. (a) Neighbors: [0 !55] (b) Plastic Strain: [0 !5] (c)vz: [-10!10] Figure 13: Properties at 30% extrusion. a maximum at the center of each leg. Plastic ﬂow is max- imized in the region between the yoke legs and in the re- gion outward from the hanger pin where the ﬂash is ex- pected to form in the ﬁnal stages of the punch motion ﬁg. 10c. Figure 11 shows the V on Mises stress in the early stages of the forging process. The red hue corresponds to the yield stress of the material. Initially, yield stress values are conﬁned to the region near the pin ﬁg. 11a but expand to the entire part ﬁg. 11b as forging progresses. A. C. Manson 8 Modelling Closed-Die Forging Operations Using Total Lagrangian Smooth Particle Hydrodynamics Figure 14: Grain orientation indicated by orientation of ellipsoid. Grain structure colored by zz. Clipping plane normal to x axis. Figure 15: Force vs Displacement Forging Stages 3.2 Reverse Extrusion of Seamless Pipe The evolution of a cylindrical material layer in the seam- less pipe is shown in ﬁg. 12. The layer is compressed and squeezed outward and then upward along the outer part of the die cylinder ﬁg. 12d. The divergence of the grains from the center can be easily seen, ﬁg. 12, as the material is displaced outward. Figure 13 shows the evolution of selected properties at about 30 per cent of the punch motion. The nearest neigh- bor plot depicts the expected higher number of neighbors toward the center and the lower number of neighbors in the extruded pipe region 13a. The plastic strain plot ﬁg. 13b shows maximum plasticity on the inner and outer sur- faces of the extruded pipe as expected. The velocity vzin ﬁg. 13c shows the particles below the punch moving in a downward direction and those above the punch, on the outer extruded section, moving upward at nearly the same (a) Initial (b)1 3 (c)2 3 (d) Final Figure 16: Evolution of cylindrical slice in one quadrant of the 4arm die in thirds of trajectory. speed. The regions well below and to the outside of the punch motion show neutral motion. The SMD implementation computes the shape and orien- tation of each particle ﬁg. 14. This information allows the structure of the particle grains and their orientation to be rendered as an ellipse. In ﬁg. 14 one sees the grains below the punch being compressed while those on the extruded pipe walls being elongated and aligned with the die wall. Force vs displacement is shown in ﬁg. 15. In the early stages there is a spike in force as the material is upset. As the punch descends the force increases and near the bottom there is a notable upward inﬂection where the last bit of material is squeezed out. The force becomes slightly negative as the punch rebounds. 3.3 Four Armed Extrusion To visualize the results for the four arm part, two clipping planes (at right angles) are used to carve out a quadrant of the extrusion process ﬁg. 16, showing the axial and lateral cross sectional evolution of the billet. The die is shown in a purple color to indicate the boundary of material ﬂow ﬁg. 16. The initial stages of extrusion exhibit a buck- ling phase as the billet is longitudinally compressed (not shown). This compression gives way to the lateral plastic ﬂow of material into the lateral arms of the die ﬁg. 16b. As the material ﬂows through the constricted regions, the billet narrows ﬁg. 16c. Eventually the material ﬁlls the die cavity ﬁg. 16d. The spatial-temporal distributions of selected billet prop- erties are shown in ﬁg. 17. The strain along the punch axis ﬁg. 17a and the associated particle velocity ﬁg. 17b indicate peak values in the constricted regions and lower A. C. Manson 9 Modelling Closed-Die Forging Operations Using Total Lagrangian Smooth Particle Hydrodynamics (a)yy: [-0.1!0.1] step 193 (b)vy: [0!0.1] step 199 (c) Neighbors: [0 !55] step 199 (d) Plasticity rate: [0 !3] step 204 (e) Shape Y: [0.1 !0.3] step 206 (f) Plasticity: [0 !10] step 187 Figure 17: Relative properties in quadrant of 4arm die. Figure 18: Force vs Displacement 4arm part (a)xx: [-0.1!0.1] Step 125 (b)yy: [-0.1!0.1] Step 125 Figure 19: Final xx,yyof crank throw shown within preform die. values where the radius of the part increases. In the lateral arms,vyandyybecome more uniform in the direction of ﬂow normal to the puch axis. The nearest neighbor plot ﬁg. 17c shows highest values at the center of the die as expected, but also shows a decrease in the center regions where the die has a larger radius, probably due to the expansion as material ﬂows into these regions. The plasticity rate ﬁg. 17d is maximized at the entrance to the die and at its outer surface, areas of ﬂow expansion and high shear respectively. This is expected as the snapshot is taken late in the forging process as the material is ﬁlling out the die. The y-axis aligned grain orientation ﬁg. 17e depicts how the particles are aligned with the punch axis. Grain alignment is maximized in the constricted regions and near the outer surface of the die. The plasticity ﬁg. 17f correlates well with the plasticity rate. It is seen to be more coherent along the punch axis and conversely in the lateral directed die arm. The force vs displacement is shown in ﬁg. 18. In compar- ison to the seamless pipe extrusion ﬁg. 15, it appears to be more noisy. The forging process is characterized by stages that result in non-uniform ﬂows such as when material is expanding into thicker regions or where it is buckling just prior to changing direction. Once these bottlenecks were overcome, the punch force decreased. In the latter extru- sion phases, the force increases to allow the material to ﬁll the die cavity fully. 3.4 Cross Plane Crankshaft (Unit) 3.4.1 Preform Die The forged crankunit is shown in the context of the die mesh surface ﬁg. 19. The strain in the direction of die motion is shown in ﬁg. 19a and the strain in the direction of the longest part dimension in ﬁg. 19b. The strain, xx, is maximized at the counterweights opposite to the con- necting rod bearing whereas the strain, yy, is maximized toward the rod bearing. This is expected because the ma- terial will tend to ﬂow in the direction normal to the punch motion in the shallower part of the die cavity. The evolution of a billet layer (normal to the punch mo- tion) is shown in ﬁg. 20. The billet is shown with a A. C. Manson 10 Modelling Closed-Die Forging Operations Using Total Lagrangian Smooth Particle Hydrodynamics (a) Step 00 (b) Step 70 (c) Step 100 (d) Step 125 Figure 20: Evolution of layer of billet in thickest cross section (slight oblique view angle). (a) Step 00 (b) Step 40 (c) Step 80 (d) Step 125 Figure 21: Oblique top view of the evolution of zz: [-0.1!0.1]. (a)zzstep 80 (b)zzstep 95 (c)zzstep 110 (d)zzstep 125 Figure 22: Relative zz: [-0.1!0.1] along crankshaft axis z in thickest cross section (view along y axis). Figure 23: Relative xx: [-0.1!0.1] at step 125 in thick- est cross section. A. C. Manson 11 Modelling Closed-Die Forging Operations Using Total Lagrangian Smooth Particle Hydrodynamics Figure 24: Relative yy: [-0.1!0.1] at step 125 (top view). clipping plane through the counterweight section, cho- sen to include the main (central) bearing. There is global deformation throughout the billet with the greatest ma- terial ﬂow occuring in the deepest part of the die ﬁg. 20d. A similar view of the process is given in ﬁg. 21 where the surface is painted with the strain along the main axis. The increase in the strain around the central bearing and throughout the region between the counterweights is clearly visible ﬁg. 21d. The evolution of zzthroughout the interior of the crankunit, in its thickest part, is given in ﬁg. 22. Here the effects around the central bearing and near the surface between the counterweights are further emphasized 22d. The effect of the forging process on the thickest features of the part are highlighted in ﬁg. 23 where xxis shown. One sees that the strain in this region is aligned with the axis of die motion. Finally the strain along the longest part dimension (direction of crank offset) is shown ﬁg. 24. The strain is maximized near the parting line and toward the connecting rod bearing where the die is most shallow. 4 Discussion Overall the simulations provided stable and credible re- sults for the cases under study. The predicted deforma- tions helped in identifying problem areas such as billet fracturing and incomplete die ﬁlling (data not shown) so that the test cases could be optimized. In some cases, how- ever, the simulation parameters needed to be modiﬁed to adapt to the nuances of each case. 4.1 Effect of Simulation Parameters Update frequency and contact stiffness had the greatest impact on the trajectories. The contact stiffness was in- creased to compensate for the stiffness of steel so the billet would not penetrate the die surface. The reference frame update and smoothing kernel radius inﬂuenced the out-put of the highest plasticity cases, namely the seamless pipe extrusion and the four arm extrusion (discussed be- low). Decreased particle size was also investigated (data not shown), but not surprisingly, was found to have a cu- bic impact the on simulation time. 4.1.1 Effect of Update Frequency In order to handle large plastic deformations and main- tain simulation stability [14], the initial conﬁguration of the system was reset when the displacement of any one particle exceeded a tunable displacement criteria. The re- set criteria occured when the displacement between any two particles exceeded 50 percent of the smoothing ra- dius. This was notable in regions of high shear where the particle registry (contact topology) would change. When this threshold was exceeded, the initial conﬁguration for allbillet particles was reset to the current positions. The update would affect the cumulative strain but (by design) leave the stress and plastic deformation history intact. The part geometry and conﬁguration of the forging system also had an impact on these updates. For instance, the yoke and crank-throw experienced few updates, while the seamless pipe and four arm part experienced several up- dates; the material ﬂow in these latter cases being more complex. The reverse extrusion had shear bands where particles were disconnected from their neighbors by the punch action, similar to a cutting tool. Here, the effects were most apparent on the inner wall of the extruded pipe. The ﬂow pattern within the four arm die exhibited diverted ﬂows and ﬂow through constrictions. Within these regions the particles would shear past each other so the conﬁgura- tion would frequently be reset. When the particles were reset, the strain would also be re- set. To get a sense for the overall strain, the Lammps [7] platform and the SMD implementation [6] were altered to compute a cumulative strain based on the initial state of the trajectory. Even though this strain was invalid around the areas where the reset criteria triggered, it nonetheless gave an indication of the overall deformation. To facilitate the display, the determinant of the deformation matrix F was output (in the dump data) and used to ﬁlter out par- ticles [13] where the determinant was either negative or assumed excessive values. 4.2 Effect of Physical Parameters The higher temperature case required lower average die forces as expected, compare ﬁgs. 15, 18. This can be un- derstood from the data given in [12], which shows a 20 fold reduction in billet stiffness for the higher tempera- ture. Also apparent was the greater material ﬂow due to the lower yield stress. 4.3 Summary of Effects of Die Geometry The simulations where the range of punch motion relative to the maximum billet extent was smallest seemed to be the best behaved. In these cases, the number of reference A. C. Manson 12 Modelling Closed-Die Forging Operations Using Total Lagrangian Smooth Particle Hydrodynamics state resets was minimal. When the punch motion was along the longest axis of the billet, however, the reference state had to be reset many times. Also, it was noted in the 4arm case, that the billet would undergo large compres- sive deformation before actually ﬂowing into the lateral die spaces. This was surprising considering the relative incompressibility of steel, even at high temperatures. This anomaly is probably due to a simplistic constitutive model and could be remedied by modeling incompressibility. 5 Conclusion The Lammps implementation of Smooth Mach Dynamics was used to simulate four representative non-trivial metal forging operations. The cases covered a wide spectrum of material deformation and forging die geometry. The Total Lagrangian method provided credible predictions of the deformation process and proved useful in identifying problems occuring in the design of forging processes.