On the numerical simulation of machining processes

On the Numerical Simulation of Machining Processes

M. Vaz Jr.

Departamento de Engenharia Mecânical

Universidade do Estado de Santa Catarina

89223-100 Joinville,SC Brazil

[e-mail]

Numerical simulation of machining processes can be traced back to the early seventies when finite element models for continuous chip formation were proposed. The advent of fast computers and development of new techniques to model large plastic deformations have favoured machining simulation. Relevant aspects of finite element simulation of machining processes are discussed in this paper, such as solution methods, material models, thermo-mechanical coupling, friction models, chip separation and breakage strategies and meshing/re-meshing strategies.

Keywords: Machining Simulation, Orthogonal Metal Cutting, Finite Elements.

Introduction

Machining processes involve a wide range of both mechanical and thermal physical phenomena, as depicted in Fig. 1, thereby constituting a problem of difficult numerical modelling. In recent years, the advent of fast computers, robust mathematical and numerical modelling for elasto-(visco)plasticity at finite strains and algorithms able to handle contact and friction of multiple/fragmented bodies have provided useful tools for more realistic analyses of metal cutting processes. On the other hand, an extensive survey shows that, despite the variety of models upon which machining simulation are based, there were only few attempts to address the problem with a higher degree of generality. The present paper discusses modelling strategies adopted in numerical simulation of machining processes (mechanical and/or thermo-mechanical solutions only), in which aspects of finite element approach, solution methods, material models, thermo-mechanical coupling, friction models, chip separation and breakage strategies and meshing/re-meshing strategies are examined.

Nomenclature

Finite Element Modelling

In general, finite element formulations are based on either quasi-static implicit or dynamic explicit schemes. The former requires convergence at every time step or load increment and the latter solves an uncoupled equation system based on information from previous time steps.

Quasi-static implicit time integration schemes have been largely used in metal cutting simulations. The finite element equations for Lagrangian formulations can be expressed, in general, as

(1)

in which K_n+1is the stiffness matrix, ?u_n+1is the vector of unknown incremental displacements, F_n+1is the load vector and u_n+1and u_nare the current and previous total nodal displacements. The general form of the finite element equations for Eulerian formulations can be represented as follows

(2)

where

_n+1 are the nodal velocities.

In implicit algorithms, the requirement of convergence at every solution increment provides better accuracy, however, difficulties in dealing with discontinuous chip formation and restrictive contact conditions are the main drawbacks of these schemes.

Dynamic explicit time integration schemes have been employed in metal forming problems which involve high non-linearity, complex friction-contact conditions and fragmentation (Hashemi et al., 1994; Marusich and Ortiz, 1995; Vaz Jr. et al., 1998a; Vaz Jr. et al., 1998b; Owen and Vaz Jr., 1999). The finite element equations can be represented as

(3)

where

_n, _nand u_nnare the nodal acceleration, velocity and displacement at time t_n, M and C are mass and damping matrices and P and F are internal and external forces. Although no iteration procedure is required, the stability of explicit algorithms depends strongly on the time step size, which is invariably much smaller then its implicit counterpart.

Implicit schemes can be used for simulation of continuous chip formation due to simple requirements of frictional contact. On the other hand, complex geometry and contact detection/interaction of discontinuous chip formation recommends use of explicit schemes.

Solution Methods

Both Lagrangian and Eulerian approaches have been used to model machining simulation. The former assumes that the particles are fixed to the finite element mesh and stresses and strains are computed incrementally updating the nodal coordinates at the end of each solution increment, as represented by Eq. (1) (Shirakashi and Usui, 1974; Usui and Shirakashi, 1982; Iwata et al., 1984; Strenkowski and Carrol III, 1985; Carrol III and J.S. Strenkowski, 1988; Yang et al., 1989; Shih et al., 1990; Komvopoulos and Erpenbeck, 1991; Shih and Yang, 1991; Lin and Lin, 1992; Lin and Pan, 1993a; Lin and Pan, 1993b; Lin and Pan, 1994; Shih and Yang, 1993; Hashemi et al., 1994; Lin et al., 1994; Lin and Lee, 1995; Shih, 1995; Marusich and Ortiz, 1995; Xie et al., 1994; Obikawa and Usui, 1996; Shih, 1996a, 1996b, Lin and Liu, 1996; Vaz Jr. et al., 1998a, Vaz Jr. et al., 1998b, Owen and Vaz Jr., 1999). In Eulerian formulations the mesh is fixed in space and the particles are allowed to cross its boundaries, whose corresponding finite element approximation is represented in Eq. (2) (Strenkowski and Carrol III, 1985; Carrol III and J.S. Strenkowski, 1988; Eldridge et al., 1991; Tyan and Yang, 1992; Joshi et al., 1994; Wu et al., 1996; Kim and Sin, 1996; Strenkowski and Athavale, 1997).

The main advantages of Lagrangian formulations are: the chip geometry is the result of the simulation and simpler schemes to simulate transient processes and discontinuous chip formation. However, element distortion has been always matter of concern which limited the analysis to incipient chip formation in some studies (Yang et al., 1989; Shih et al., 1990; Lin and Lin, 1992; Lin and Pan, 1993a, Lin and Pan, 1993b, Xie et al., 1994). Pre-distorted meshes (Strenkowski and Carrol III, 1985; Komvopoulos and Erpenbeck, 1991; Shih et al., 1990; Shih and Yang, 1993; Shih, 1995; Shih, 1996a; Shih, 1996b; Obikawa and Usui, 1996; Obikawa et al., 1997) and re-meshing techniques (Marusich and Ortiz, 1995; Skhon and Chenot, 1993; Vaz Jr. et al., 1998a; Vaz Jr. et al. 1998b; Owen and Vaz Jr., 1999) have been adopted to minimize the problem.

Eulerian formulations are not affected by element distortion once the mesh is fixed. Moreover, this strategy allows steady-state machining to be easily simulated by requiring fewer elements and by avoiding use of chip separation criteria. However, Eulerian problems require the knowledge of the chip geometry in advance, which, undoubtedly, restricts the range of cutting conditions capable of being analysed. In order to overcome this drawback, some works have adopted iterative procedures to adjust the chip geometry and/or chip/tool contact length.

A mixed Eulerian and Lagrangian formulation has also been proposed (Sekhon and Chenot, 1993) in order to add to the Eulerian formulation the generality of the Lagrangian capability of dealing with transient machining. In this case, the solution computes nodal velocities under an Eulerian approach which are used to update the nodal coordinates.

Material Models

A wide range of constitutive models have been employed for the workpiece, such as rigid-plastic, rigid-viscoplastic, elasto-perfectly-plastic, elasto-plastic and elasto-viscoplastic. Elasto-plastic and elasto-viscoplastic are the most commonly used materials and account for elastic deformations. The former is rate-independent whereas the latter incorporates a plastic strain-rate dependency in the constitutive equation. Both approaches are able to capture the spring-back effect, variation of the residual stresses and thermal strains.

The elasto-viscoplastic model has been adopted only by Shih (1990, 1995, 1996a, 1996b). Rigid-plastic (Iwata et al., 1994) and rigid-viscoplastic (Kim and Sin, 1996; Joshi et al., 1994; Skhon and Chenot, 1993, Eldridge et al., 1991; Strenkowski and Moon, 1990; Carrol III and J.S. Strenkowski, 1988; Wu et al., 1996; Strenkowski and Athavale, 1997) materials can simplify the analysis (no elastic deformation is allowed), however, thermal strains, residual stresses and spring-back effects cannot be evaluated.

Special material modelling has been employed to simulate adiabatic shear-banding and fracture when machining Ti-6Al-4V titanium alloy (Xie et al., 1994; Vaz Jr. et al., 1998a; Vaz Jr. et al., 1998b; Owen and Vaz Jr., 1999; Obikawa and Usui, 1996). Adiabatic shear localization can cause material failure and is caused by thermal softening in materials of low thermal diffusivity or in high-speed processes. Vaz Jr. et al. (1998b) introduced use of a two-parameter model to describe material failure due to shear banding, i.e., a failure indicator and an energy release factor. The former indicates failure onset and the latter defines the amount of energy released by the element during the softening process before the actual element failure.

Thermo-Mechanical Coupling

In cutting processes, energy is generated due to dissipation of both inelastic work and frictional work being transferred through the workpiece/chip and tool and lost to the surrounding environment/coolant by convection and radiation. Temperature rise causes thermal strains and affects material properties.

Adiabatic heating and complete coupling have been largely used to model the thermal behaviour of the workpiece in machining simulations. The former assumes that the heat generated remains localized whereas the latter account for heat conduction within the workpiece and tool.

In adiabatic processes no heat transfer takes place, i.e., the heat generated due to inelastic deformation and friction is kept inside the element causing its temperature to rise (Strenkowski and Carrol III, 1985; Eldridge et al., 1991; Vaz Jr. et al., 1998a; Vaz Jr. et al. 1998b; Owen and Vaz Jr., 1999). This approximation can be safely adopted for low-diffusivity materials or in high-speed processes. It is worth mentioning that Marusich and Ortiz (1995), using the complete coupling model, showed that heat conduction plays a small role in workpiece analysis of high-speed machining.

Thermal contact is the numerical technique to model heat transfer between chip and tool. Several approaches have been adopted, such as those presented in Table 1. Heat conduction continuity along the chip/tool interface was used by Usui and Shirakashi (1982) which assumes an equal heat flux at the chip and tool sides of the interface.

For simulations in which the tool is assumed to be isothermal, an estimation of the fraction of the frictional energy, q_f, absorbed by the chip is performed. In its simplest form, a constant value, f_c, is assumed. Alternatively, the so-called two semi-infinite bodies in contact model has been used to estimate the frictional energy transferred to the chip, q_w, and to the tool, q_t.

Obikawa and Usui (1996) used concepts of thermal equilibrium (First Law of Thermodynamics) to estimate the interface temperature. Firstly, the frictional energy is assumed to increase instantaneously; the chip temperature is determined by a simple thermal balance comprising only the elements along the chip/tool interface. The thermal analysis of the tool is performed separately after computing the new interface temperatures.

Although temperature affects friction parameters, temperature-independent friction has been assumed by most authors. The tool has been largely neglected as far as thermal analysis is concerned. Few studies offer a proper transient analysis (Lin and Lin, 1992; Lin and Pan, 1993a; Lin and Pan, 1993b; Obikawa and Usui, 1996). Finally, it is worth mentioning that evaluation of the energy lost to the environment (natural or forced convection) has also been largely ignored.

Friction Models

Friction between chip and tool constitutes one of most important aspects of machining processes. Table 2 presents the friction models used in the simulations. The simple temperature-independent Coulomb's law has been the most commonly used friction model. The model assumes that the frictional stresses, t_f, are directly proportional to the normal stresses, s_n.

Experimental models were introduced by Shirakashi and Usui (1974) who used an exponential law to relate frictional stress to normal stresses and maximum shear flow stresses, t_e. The model was initially conceived to non-ferrous metals and was derived from the concept of apparent/effective contact area. The model was later applied to metals in general.

Based on the experiment where a bar-shaped tool slids over the inner surface of a ring specimen, Iwata et al. (1984) proposed an expression for frictional stresses dependent on Coulomb's friction coefficient, normal stress and Vickers hardness of the workpiece material, H_?.

Eldridge et al. (1991) used an experimental curve which relates shear stress and yield stress in shear. The temperature dependency is accounted for by an exponential function so that the friction stresses and temperature are inversely proportional. Later, Wu et al. (1996) assumed that the friction stress is directly proportional to the equivalent stress.

Sekhon and Chenot (1993) adopted Norton's friction law which assumes that frictional stresses are proportional to the relative sliding velocity between the chip and tool.

Chip Separation and Breakage

Chip separation has always been a matter of controversy among researchers on experimental analysis of metal cutting. The discussions are reflected by the numerical simulations where no clear, definitive or general direction has been agreed. Table 3 shows the variety of chip separation and breakage criteria available in the literature.

Usui and Shirakashi (1982) pioneered the discussion by proposing a chip separation criterion based on the distance between the tool tip and the nearest node along a pre-defined cutting direction, as illustrated in Fig. 2(a). The criterion is purely based on geometrical considerations and does not account for possible chip breakage outside the cutting line. On the other hand chip separation can be easily controlled (Usui and Shirakashi, 1982; Shih, 1995; Obikawa and Usui, 1996; Yang et al., 1989; Shih et al., 1990; Shih and Yang, 1993; Shih, 1996a; Shih, 1996b; Komvopoulos and Erpenbeck, 1991).

In simulations using this strategy, as the tool advances, the distance between the node F_w,c and the tool tip decreases and, at a critical distance, d_cr, either a new node is created or a restriction in superimposed nodes are removed, which makes it possible for the material to separate.

Equivalent plastic strain, advocated by Strenkowski and co-workers (1985, 1988), Xie et al. (1994) and Hashemi et al. (1994), has also been employed as a chip separation criterion. In this case, the chip is said to separate when e_p, calculated at the node nearest the cutting edge, reaches a critical value, e_p,cr. This criterion has been frequently criticized due to the fact that node separation propagates faster then the cutting speed forming a large open crack ahead of the tool tip. The process is illustrated in Fig. 2(b), in which the chip separation indicator, I_cr, represents the equivalent plastic strain. Furthermore, the value of e_p,crwas found to affect the magnitude of the residual stresses (Lin and Lin, 1992).

A chip separation criterion based on the total strain energy density is said to overcome the previous shortcomings (Lin and Lin, 1992; Lin and Pan, 1993a; Lin and Pan, 1993b). A modification of the criterion based on e_p,crto account for effects of temperature, hydrostatic stress and plastic strain-rate was suggested as a chip breakage criterion when machining titanium alloys (Obikawa and Usui,1996). Similar principles have also been adopted for Cu-40 brasses (Obikawa et al., 1997).

In all the previous cases the chip separates along a pre-defined cutting plane. It is worth mentioning that the first systematic analysis of material separation in orthogonal machining was presented by Huang and Black (1996), who evaluate chip separation criteria based on geometrical (nodal distance) and physical (equivalent plastic strain, energy density and stresses) considerations. The analyses are restricted to material separation along a pre-defined parting line. The authors acknowledge that the physics of chip separation is not yet completely understood and conclude that neither a geometrical nor a physical criterion simulate incipient cutting correctly.

A combination of equivalent plastic strain, to model chip separation, and maximum principal stress, to simulate chip breakage, has been used by Hashemi et al. (1994) to model high-speed machining. The authors pioneer use of explicit finite element schemes in association with multi-fracturing materials.

The use of ductile fracture concepts was first introduced by Iwata et al. (1984), who suggested a version of Cockroft and Latham (1968) and Osakada et al. (1984) as possible chip separation criteria. Although the initial consideration had been quite tentative, the model simulates steady-state machining and, therefore, no actual chip separation takes place. The fracture criteria are computed a posteriori based on the final stress-strain state. According to the authors, Osakada's criterion is said to be the best chip separation indicator once it produces the largest fractured area (the total area where the indicator reaches its critical value).

Marusich and Ortiz (1995) proposed use of either brittle or ductile fracture criteria depending on the machining conditions. The former is formulated in terms of the toughness K_IC, being used in conjunction with a multi-fracturing algorithm. The latter is based on void growth and coalescence and uses a version of Rice and Tracey's (1969) criterion.

Owen and Vaz Jr. (1999), using a combined finite/discrete element algorithm and multi-fracturing materials, adopted a chip breakage criterion based on damage considerations in conjunction with an adaptive re-meshing scheme and element erosion. Chip separation is accomplished by multiple re-meshing.

It is worth noting that most simulations assume a pre-defined parting line, irrespective of potential fracture elsewhere. Exceptions are due to Hashemi et al. (1994), Marusich and Ortiz (1995) and Owen and Vaz Jr. (1999). Finally, the physics of chip separation and breakage has not been fully understood, which, allied to algorithmic complexity of producing fractured chips, has hindered more realistic simulations of discontinuous chip formation.

Meshing Strategy

Element distortion, caused by large plastic deformation, and efforts to capture more accurately chip separation and breakage paths have been directing recent works towards the use of adaptive re-meshing procedures. The literature shows a variety of methods conceived to improve numerical solutions, such as pre-distorted meshes and local refinement. Table 4 summarizes the meshing techniques used in numerical simulations.

Local re-meshing was employed by Yang et al. (1989) and Shih et al. (1990), who defined a 12-element cell ahead of the tool tip in order to capture more efficiently stress and strain concentration. The technique was later abandoned by the authors.

Element distortion was firstly tackled using pre-distorted meshes Strenkowski and Carrol (1985). The strategy has frequently been used in conjunction with a pre-defined parting line and consists of designing an initial mesh with pre-distorted elements over the upper surface of the workpiece, as illustrated in Fig. 3, in anticipation of plastic shear deformation (Strenkowski and Carrol, 1985; Komvopoulos and Erpenbeck, 1991; Obikawa and Usui, 1996; Obikawa et al.,1997; Shih et al., 1990; Shih, 1995; Shih and Yang, 1993; Shih, 1996a; Shih, 1996b).

An alternative strategy was adopted by Sekhon and Chenot (1993), who employed an automatic re-meshing technique based on element distortion to simulate continuous chip formation. The authors emphasize that this strategy presents the advantage of avoiding use of any criteria to simulate chip separation.

Marusich and Ortiz (1995) employed an adaptive procedure using plastic work rate as a re-meshing indicator. A local re-meshing technique, in which the crack tip was surrounding with a rosette of elements, has also been used to increase angular resolution when computing the crack direction.

An error estimator based on material failure considerations was proposed by Vaz Jr. et al. (1998b) (also described in Owen and Vaz Jr., 1999) to simulate high-speed machining. The authors point out the necessity of capturing more efficiently chip breakage under plastic strain localization conditions. Figure 4 shows typical finite element meshes at two different cutting stages. It is important to note that, in this example, material failure is inhibited otherwise the chip would have separated before reaching the stage shown in Fig. 4(b).

Concluding Remarks

The present paper summarizes some of the most relevant issues associated with numerical simulation of machining processes. The first simulations attempted to model continuous chip formation by using implicit finite element schemes in conjunction with a pre-defined cutting plane. Recent studies have adopted adaptive mesh refinement in conjunction with Lagrangian formulations to account for large deformations. Furthermore, explicit algorithms have proved more robust to simulate discontinuous chip formation. Despite the number of solutions available, important aspects deserve further investigation, such as heat transfer between the chip and environment, temperature-dependent friction, thermo-mechanical coupled analysis of the tool, chip separation and breakage, non-orthogonal cutting and tool wearing and breakage amongst many others.

Manuscript received: May 1999, Technical Editor: Alisson Rocha Machado.

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Publication Dates

• Publication in this collection
  18 Dec 2000
• Date of issue
  2000

History

• Received
  May 1999