Metal Machining Episode 8 pptx

In addition to the choice of finite element method based on computational criteria,particular softwares for metal machining should be able to model the variation of flowstress with strai

with time, so the coefficients of the [B] matrix, which depend on element shape (for

exam-ple equation (7.3), for a triangular element), need only be computed once However, in aproblem such as machining, in which determining the location of the free surface of thechip is part of the problem to be solved, it is not clear where the elements should be drawn

It is necessary to develop the free surface boundaries of the element mesh by iteration Amore general problem is how to describe the convection of material property changes, likestrain hardening, from element to element (Eulerian analyses are more common in fluidmechanics than in solid mechanics because fluid properties vary less with deformationthan do those of solids.) In steady flow problems, it is assumed that material propertiesconvect along the streamlines

The Lagrangian view has no problem with convection of material properties The state

of a material is fixed in an element However, the element changes shape during a flow: the

[B] matrix requires continued updating This leads to geometrical non-linearities in addition

to material non-linearities in the finite element equations In extreme cases it may becomenecessary to simplify a distorted element shape by remeshing (see the next section) There

is a further complication An element most likely rotates as well as distorts as it passes

through a flow (as shown in Figure 7.2) After a while, its local x and y directions will differ

from those of other elements However, a common set of axes is required for the mation of individual element equations to a global assembly Counter-rotating the local

transfor-element coordinate system, as well as updating the [B] matrix, is repeatedly required.

Structured or adaptive meshing – and other matters

It is common sense that a finer mesh is needed where problem variables (velocity, ature) vary strongly with position than where they do not In metal machining, fine detail isneeded to model the primary and secondary shear zones This poses no problem forEulerian meshes: a choice is made where to refine the mesh and by how much However,for computing efficiency with a Lagrangian mesh, there is a need to refine and then coarsenhow the material is divided into elements as it flows into and out of plastic shear zones.The need to refine Lagrangian meshes is particularly accute near the cutting edge of atool, where the work material flow splits into flow under the cutting edge and flow into thechip A range of approaches to separation at the cutting edge has been developed, fromintroducing an artificial crack in the work, to highly adaptive remeshing, to developingspecial elements with singularities in them These are not needed in Eulerian analyses

temper-Finite element background 203

Fig 7.2 Eulerian and Lagrangian views of a plastic flow

In addition to the choice of finite element method based on computational criteria,particular softwares for metal machining should be able to model the variation of flowstress with strain, strain rate and temperature (Section 6.3) and the variation of rake facefriction conditions from high load to low load conditions (Chapter 2, Section 2.4)

Summary

The choice of finite element methods for machining problems involves rigid-plastic orelastic–plastic material models; Eulerian or updated Lagrangian flow treatments; struc-tured or adaptive meshes; chip/work separation criteria needed or not needed; and coupling

to thermal calculation models or not Some of the achievements of these approaches, andmethods of overcoming computational problems, are chronicled in the next section Onbalance, the updated Lagrangian analyses’ advantage of easily tracking material propertychanges outweighs the disadvantages of computational complexity The simplicity ofEulerian computations is not fully realized in the large free surface movement conditions

of a chip forming process

Fig 7.3 Shear zone development, loading a pre-formed chip (Zienkiewicz, 1971)

example, it neglects friction between the chip and tool, and strain rate and temperature ial flow stress variations are not considered either More fundamentally, it assumes the shape

mater-of the chip in the first place: the main purpose mater-of chip forming analyses is to predict the shape.The limitations of this initial work were removed by Shirakashi and Usui (1976) Whilekeeping the computational advantages of supposing the tool to move into a pre-formedchip, they developed an iterative way of changing the shape of the pre-form until the gener-ated plastic flow was consistent with the assumed shape They also included realisticchip/tool friction conditions (from split-tool experiments), a temperature as well as amechanical calculation, and material flow stress variations with strain, strain rate andtemperature, measured from high strain rate Hopkinson bar tests (see Section 7.4) Theiriterative convergence method (ICM) is shown in Figure 7.4

The first step of the ICM is to assume a steady state chip shape (similar to Figure 7.3,except for supposing there to be a small crack at the cutting edge to enable the chip to

Historical developments 205

Fig 7.4 The iterative convergence method (ICM) – Shirakashi and Usui (1976)

separate from the work) and (for plane strain modelling) to create a three-node triangular meshfollowing the streamlines of the flow In the first iteration, the tool is moved against the chip:the development of nodal velocities is followed with an updated Lagrangian elastic–plasticanalysis When it is judged that the plastic flow is fully developed, the nodal velocity field isused to calculate the element strain rates along the streamlines; strains are obtained by inte-grating the strain rates with respect to time along the streamlines (as if material had reachedits current position by flowing along a streamline) Temperatures are calculated from the inter-nal and friction work rates and the work and tool materials’ thermal properties (in the firstapplication of the ICM, temperature was calculated by a finite difference method, but later thefinite element method was used) Material flow stress is then set according to its strain, strainrate and temperature, the tool and chip are unloaded and the cycle of moving the tool into thechip repeated This is continued until converged strain rates and temperatures are achieved Atthat stage, the flow field is used to modify the initially assumed streamlines to be closer to thecalculated flow The complete cycle is then repeated, and repeated again until the assumed andcalculated flow fields agree The displacement of the tool needed to establish the flow field issufficiently small that the need to reform the crack at the cutting edge does not arise Withinlimits, the crack size does not influence the predicted chip flow.

Figure 7.5 shows chip shape, equivalent plastic strain rate and temperature fields

Fig 7.5 (a) Strain rates and (b) temperatures predicted by the ICM method for dry machining α-brass, cutting speed

48 m/min, rake angle 30˚, feed 0.3 mm

calculated by Shirakashi and Usui for machining an a-brass Chip shape agrees with

experiment, as does the temperature field (which was studied experimentally with infraredmicroscopy)

The procedure of loading a tool against an already formed chip greatly reduces ing capacity requirements and, in the 1970s, made elastic–plastic analysis possible However,

comput-it does not follow the actual path by which a chip is formed and, as outlined in Section 7.1and Appendix 1, the development of elastic–plastic flows is path dependent The justification

of the method is that it gives good agreement with experiment The ICM has been developed

further, in analyses of cutting fluid action (Usui et al., 1977), built-up-edge formation (Usui

et al., 1981) and more recently in studies of low alloy semi free-machining steels (Childs and

Maekawa, 1990) It is given further consideration in Section 7.3 and Chapter 8

The 1980s

Rigid–plastic modelling does not require the actual loading path to be followed (alsodiscussed in Section 7.1 and Appendix 1) Steady state rigid–plastic modelling, within aEulerian framework, also adjusting an initially assumed flow field to bring it into agreement

with the computed field, was first applied to machining by Iwata et al (1984), using

soft-ware developed from metal forming analyses They included friction and work hardeningand also a consideration of whether the chip would fracture, but not heating (and obviouslynot elastic effects) Experiments at low cutting speed (0.15 mm/min in a scanning electronmicroscope) supported their predictions It was not necessary with the Eulerian frame tointroduce a crack at the cutting edge, but it was necessary, to avoid computational difficul-ties, to give the cutting edge a small radius (about one tenth of the feed)

The mid-1980s, with a growth in available computer power, saw the first non-steadychip formation analyses, following the development of a chip from first contact of a cuttingedge with a workpiece, as in practical conditions (Figure 7.6(a)) Updated Lagrangian elas-tic–plastic analysis was used, and the chip/work separation criterion at the cutting edge

Historical developments 207

Fig 7.6 Non-steady state analysis: (a) initial model and (b) separation of nodes at the cutting edge

became an issue (Figure 7.6(b)): should the connection between elements be broken by alimiting strain, limiting energy or limiting displacement condition? Figure 7.7 shows theearliest example (Strenkowski and Carrol, 1985), which used a strain-based separationcriterion At that time, neither a realistic friction model nor coupling of the elastic–plastic

to thermal analysis (and hence nor a realistic flow stress variation with cutting conditions)was included

At the same time as plastic flow finite element methods were being developed for metalmachining, linear fracture mechanics methods were being developed for the machining ofbrittle ceramics (Ueda and Sugita, 1983)

The 1990s

The 1990s have seen the development of non-steady analysis, from transient to uous chip formation, the first three-dimensional analyses and the introduction of adaptivemeshing techniques particularly to cope with flow around the cutting edge of a tool.Figure 7.8 shows an updated Lagrangian elastic–plastic simulation of discontinuous

discontin-chip formation in b-brass at low cutting speed To obtain this result a geometrical

Fig 7.7 An example of non-steady state analysis (Strenkowski and Carrol, 1985)

Fig 7.8 Discontinuous chip formation in β-brass (Obikawa et al 1997): (1–6) element deformation and (7) lent plastic strain distribution, at different cut distances l

equiva-(7)

(displacement controlled) parting criterion at the cutting edge was combined with anempirical crack nucleation and growth criterion, considered further in Section 7.3 andChapter 8 Other authors have taken different approaches to crack growth during chip

formation (Ueda et al 1991).

Figures 7.9 and 7.10 are the earliest examples of elastic–plastic steady and non-steadythree-dimensional analyses The steady state example is an extension of the ICM to three

Historical developments 209

Fig 7.9 Three-dimensional steady state chip formation by the ICM (Maekawa and Maeda, 1993): (a) initial model and

(b) equivalent strain rate distribution

dimensions The non-steady example employs a geometrical parting criterion at both theprimary and secondary cutting edges In both these cases, temperature and strain rateeffects are ignored, to reduce the computing requirements This restriction was soonremoved: three-dimensional elastic–plastic, thermally coupled, ICM simulation soon

became used for cutting tool design, also considered further in Chapter 8 (Maekawa et al.

1994)

In parallel with the extension of elastic–plastic methods to non-steady and

three-dimen-sional conditions, the rigid–plastic method (Iwata et al., 1984) was similarly being oped (Ueda and Manabe, 1993; Ueda et al., 1996), with a shift from Eulerian to

devel-Lagrangian modelling Figure 7.11 shows the simulation of spirally curled chip formationduring milling with a non-zero axial rake tool A simple form of remeshing at the cuttingedge, instead of a geometrical crack, was introduced to accommodate the separation of thechip from the work

Adaptive mesh refinement in non-steady flows, whereby during an increment of flow (atime step) the mesh is fixed to the work material in a Lagrangian manner – but between stepsthe mesh connectivity and size is changed according to rules based on local severities ofdeformation – offers the advantage over fixed Lagrangian approaches of concentrating themesh where it is needed most, in the primary shear zone, at the cutting edge and along therake face Concentration at the cutting edge provides an alternative to introducing a crack forfollowing the separation of the chip from the work Both rigid–plastic (Sekhon and Chenot,

Fig 7.10 Three-dimensional non-steady chip formation (Sasahara et al., 1994): (a) element deformations and (b) equivalent plastic strain distribution

1993; Ceretti et al., 1996) and elastic–plastic (Marusich and Ortiz, 1995) adaptive

remesh-ing softwares have been developed and are beremesh-ing applied to chip formation simulation Theyseem more effective than arbitrary Lagrangian–Eulerian (ALE) methods in which the mesh

is neither fixed in space nor in the workpiece (for example Rakotomolala et al., 1993).

Summary

The 1970s to the 1990s has seen the development and testing of finite element techniquesfor chip formation processes Many of the researches have been more concerned with thedevelopment of methods than their immediate application value: the limited availability of

Historical developments 211

Fig 7.11 Three-dimensional non-steady chip formation by rigid plastic finite element method (Ueda et al., 1996): (a) initial model and (b) spiral chips

reliable friction and high strain, strain rate and temperature material flow properties did nothold back this work The ICM approach is the exception: from the start it has beenconcerned with supporting machining applications Now that all methods are approachingmaturity, attention is shifting to the provision of appropriate friction and material flowproperty data (see Section 7.4).

In the future there are likely to be three main avenues of finite element modelling ofchip formation: (1) the ICM method for steady state processes, because of its extremelyhigh computing efficiency; (2) Lagrangian adaptive mesh refinement methods for unsteadyprocesses, both elastic–plastic as the most complete treatment and rigid–plastic for itsfewer computing requirements if elastic effects are not needed; and (3) fixed meshLagrangian methods (with chip separation criteria) to support educational studies ofunsteady processes in a time effective manner Chapter 8 will concentrate on the first andthe last of these, but a future edition may well include more of the second

7.3 The Iterative Convergence Method (ICM)

Sections 7.3.1 and 7.3.2 give more details of the ICM method (which was introduced inthe previous section), as background to the examples of its use presented in Chapter 8.Section 7.3.3 introduces a treatment of unsteady processes (case (3) above)

7.3.1 Principles and implementation

As has already been described, the ICM method is an updated Lagrangian elastic–plasticfinite element analysis for predicting steady state chip flows Such analyses normally mustfollow the development of strain along a material’s load path and are computationally veryintensive The ICM method replaces the real path by a shorter one: loading the tool onto

an already formed chip It provides a way, by iteration, of finding the formed chip shapethat is consistent with the material’s flow properties and friction interaction with the tool

A key point is that its finite elements are structured to follow the stream lines of the steadystate chip flow (as will be seen in Figure 7.13)

The flow chart of the ICM procedure as it was originally introduced, is shown in Figure7.4 Figure 7.12 shows its developed form An initial guess of the chip flow or stream lines(usually of the simple straight shear plane type considered in Chapter 2) is made and thetool is placed so that its rake face just touches the back surface of the chip Calculationproceeds by incrementally displacing the workpiece towards the tool so that a load devel-ops between the chip and tool At each increment, it is checked if the plastic flow is fullydeveloped (saturated): if it is not, a further increment is applied (loop I) Once the flow isdeveloped, the initial guess is systematically and automatically reformed to bring it intocloser agreement with the calculated flow The strain rate in each element of the reformedflow is calculated; and the strain distribution is obtained by integrating strain rate along thestreamlines The element flow stress associated with the reformed flow is then estimated;but this requires temperature as well as strain and strain rate to be known A second loop(loop II), a thermal finite element analysis, is entered to determine the temperature field.Finally, it is checked whether the derived material flow stress, temperature and flow fieldshave converged: if they have not, the whole iteration is repeated (loop III) The next para-graphs give some details that are special to the calculations

Judgement of saturation of the plastic flow is made either on the basis of the tool loadreaching a maximum value or of conservation of volume – i.e that the computed flow ofmaterial out of the plastic zone into the chip balances that of the work into the plastic zone.Reformation of the flow field supposes that the separation between nodes along astreamline is unchanged by reformation, but that the direction from one node to the next

is altered to bring it more closely tangential to the calculated flow For each flow line

consisting of a node sequence j – 1, j, j + 1 , the updated (x, y) coordinates of node j are

given by

u˘ x, j–1 + u˘ x, j u˘ y, j–1 + u˘ y, j

x j = x j–1 + ————— L j, y j = y j–1 + ————— L j (7.10)

The Iterative Convergence Method (ICM) 213

Fig 7.12 Developed flow-chart of the iterative convergence method

where (u˘ x, j , u˘ y, j ) are the calculated velocities at node j,
jis the resultant average velocity

of nodes j–1 and j, and L j is the separation between nodes j–1 and j:

j= ((u˘ x, j–1 + u˘ x, j)2+ (u˘ y, j–1 + u˘ y, j)2)1/2 (7.11a)

L j= ((x j–1 – x j)2+ (y j–1 – y j)2)1/2 (7.11b)The reformation using equations (7.10) and (7.11) is implemented from the beginning tothe end of a flow line so that the coordinates (xj–1, yj–1) have already been revised

The equivalent plastic strain e— in each element is evaluated by the integration of its rate e˘— along the reformed flow lines:

e—˘

v˘ewhere v˘ethe element velocity, obtained from the average of an element’s nodal velocities.Relations between flow stress, strain, strain rate and temperature are considered in Section7.4

Figure 7.13 shows an ICM mesh for two-dimensional machining with a single point

tool, in which the x- and y-axes are taken respectively parallel and perpendicular to the

cutting direction, in a rectangular Cartesian coordinate system The tool is assumed to bestationary and rigid, while the workpiece moves towards it at the specified cutting speed

Fig 7.13 Two-dimensional finite element assemblage with boundary conditions

The mesh is highly refined in the primary and secondary shear zones, in line with theconsiderations of Chapter 6.

The friction boundary at the tool–chip interface is treated as follows For the nodescontacting the rake face, the conditions imposed on the finite element equation (equation

7.9(b)) with respect to the nodal force rate F˘ and the nodal velocity u˘ are:

dt F˘ x′= (——)F˘ y′, u˘ y′= 0 (7.13)

dsnwhere x ′ and y′ are the local coordinate systems parallel and perpendicular to the rake face (as shown in Figure 7.13) and (dt/dsn) is the local slope of the friction characteristic curve

(for example the inset in Figure 2.23) at the value of snassociated with the nodal force F y′

In the course of the elastic–plastic analysis, loop I of Figure 7.12, the chip contact lengthmay increase or decrease A chip surface node in contact with the rake face is judged to leave

contact if its F y′force becomes tensile; and a node out of contact is judged to come into

contact if its reformed y′ becomes negative (penetrates the tool) Thus, the ICM method matically determines the chip-tool contact length as one aspect of determining the chip flow.The separation of material at the cutting edge is taken into account geometrically Thestreamline at the cutting edge bifurcates both onto the rake face and onto the clearancesurface of the work In the ICM calculation, the relative displacement between the worknear the cutting edge and the tool is only about 1/20 of the uncut chip thickness A smallcrack imposed on the mesh, of that length, is sufficient to cope with separation withoutadditional treatments, such as reconstruction of node and element sequences and specialprocedures to ensure a force balance at the crack tip (This is not the case when the actualloading path of an element has to be followed, as in the analysis of unsteady or discontin-uous chip flows, to be considered in Section 7.3.3.)

auto-Finally, Figure 7.13 shows the boundary conditions for the temperature analysis (loopII) The forward and bottom surfaces of the work are fixed at room temperature No heat

is conducted across the chip and work exit surfaces (adiabatic condition), although there

is of course convection Heat loss by convection is allowed at those surfaces surrounded

by atmosphere Heat loss by radiation is negligible in the analysis

7.3.2 ICM simulation examples

The following is an example of the application of the ICM scheme to the two-dimensional

machining of an 18%Mn–5%Cr high-hardness steel (Maekawa et al., 1988) The cutting

conditions used were a cutting speed of 30 m/min, an uncut chip thickness of 0.3 mm, unitcutting width, a P20 grade carbide tool with a zero rake angle and dry cutting Figure 7.14shows the predicted chip shape and nodal displacement vectors Material separation at thetool tip and chip curl are successfully simulated Figure 7.15 gives the distribution ofequivalent plastic strain rate, showing where severe plastic deformation takes place Thedeformation concentrates at the so-called shear plane, but is widely distributed around thatplane The secondary plastic zone is also clearly visible along the rake face, although thedeformation is not as severe as in the primary zone

These features are reflected in the temperature distribution in the chip and workpiece,

as shown in Figure 7.16 A maximum temperature of more than 800˚C appears on the rakeface at up to two feed distances from the tool tip

The Iterative Convergence Method (ICM) 215

Fig 7.14 Chip shape and velocity vectors in machining high manganese steel: cutting speed = 30 m/min, undeformed

chip thickness = 0.3 mm, width of cut =1 mm, rake angle = 0º, no coolant

Fig 7.15 Distribution of equivalent plastic strain rate, showing concentration of plastic deformation: cutting

condi-tions as Figure 7.14

Experimental verification has also been performed Figure 7.17 compares the predictedand measured specific cutting forces under the same conditions (but varying speed) Theobserved force–velocity characteristics are well simulated Similar agreement wasconfirmed in other quantities such as chip curl, rake temperature, stresses on the rake faceand tool wear For tool wear, a diffusive wear law as described in equation (4.1) was

assumed (Maekawa et al., 1988).

The calculation time for the ICM method depends both on the computer hardware and

on the number of finite elements In the above case, it takes only a few minutes from ICMexecution to graphical presentations, using a recent high-specification PC (Pentium II, 400MHz CPU) and an assemblage of 390 nodes and 780 triangular elements However, a pre-processor to prepare the finite element assemblage and a post-processor to handle a largeamount of data for visualization are required

Further ICM steady flow examples will be presented in Chapter 8, together with thefinite element analysis of unsteady and discontinuous chip formation The latter requiresmore consideration of the chip separation criterion

7.3.3 A treatment of unsteady chip flows

As has been written above, the ICM scheme cannot be applied to the analysis of steady metal machining The iteration around an incremental small strain plastic loading

non-The Iterative Convergence Method (ICM) 217

Fig 7.16 Isotherms near the cutting tip, cutting conditions as Figure 7.14