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18 V.P. Astakhov and J.C. Outeiro

The adaptive mesh procedure can improve the accuracy and efficiency of the

calculation/simulation in some cases, for example those with complicated geo-

metry and large gradients [41]. For this reason, the adaptive mesh procedure

is increasingly being used together with the ALE formulation to simulate the

metal forming and metal cutting processes. An example of this procedire is

shown in Figure 1.12. As seen, the solution begins with a relatively coarse pri-

mary mesh and, after obtaining the solution on this primary mesh, the mesh is

modified in some regions in order to increase the mesh density in the areas of

strong gradients.

1.2.4 Work Material Modelling

Several material constitutive models are used in FEM of metal cutting, including

rigid-plastic, elasto-plastic, viscoplastic, elasto-viscoplastic, etc. These models

take into account the high strains and temperatures reportedly found in metal cut-

ting. Among others, the most widely used is the Johnson and Cook model [42].

This is a thermo-elasto-visco-plastic material constitutive model represented as

()

σε

⎡

⎤

⎞

⎛

⎞

⎛

−

⎞

⎛

=+ + −

⎟

⎢

⎥

⎜

⎟

⎜

⎜⎟

−

⎝

⎠

⎢

⎥

⎝

⎠

⎝

⎠

⎣

⎦



eq eq

AB Cln

(1.24)

where

is the equivalent plastic strain,



is the equivalent plastic strain rate,



is the reference equivalent plastic strain rate (normally

−



), T is the

temperature, T

is the room temperature, T

is the melting temperature and A, B, C,

n and m are constants that depend on the material determined through material tests.

Because the strain rates in conventional material tests (tensile, compressive or

torsion) are in the range of

10 10

−−

− s

–1

, non-conventional material tests, re-

ferred to as dynamic tests, are usually preferred by many researchers. Split Hop-

kinson pressure bar (SHPB) impact testing is the most common. This test is used

to determine the material behaviour at strain rates up to 10

–1

[42]. This test can

be conducted at an elevated temperature (500°C or even more).

There are two major problems with the use of the discussed model and its

method of the determination of its constants. First, only few laboratories and spe-

cialist in the world can conduct SHPB testing properly, assuring the condition of

dynamic equilibrium. None of the known tests in metal cutting was carried out in

these laboratories. Second, the high strain rate in metal cutting is rather a myth

than reality, as discussed by Astakhov [7]. Third, the temperature in the so-called

primarily deformation zone where the complete plastic deformation of the work

materials takes place can hardly exceed 250

C. It is understood that the mechani-

cal properties of the work material obtained at room temperature are not affected

by this temperature so metal cutting is a cold working process [10], although the

chip appearance can be cherry-red. Fourth, it is completely unclear how to corre-

late the properties of the work materials obtained in SHPB uniaxial impact testing

with those in metal cutting with a strong degree of stress triaxiality.

Metal Cutting Mechanics, Finite Element Modelling 19

1.2.5 Modelling of Contact Conditions

Although the contact conditions at the tool–chip and tool–workpiece interfaces are

complicated [7], many known FE metal cutting simulations use simple Coulomb

friction condition. Moreover, the friction coefficient is frequently assumed to be

constant over these interfaces.

As discussed by Astakhov [7], if the friction coefficient

0577

≥

. , no rela-

tive motion can occur at the tool–chip interface, as follows from the fundamental

properties of work materials. The experiment data are in direct contradiction with

this limiting value: Zorev [3] obtained

06 18

=−

.., Kronenberg [43] 0.77−1.46,

Armarego and Brown [44] 0.8−2.0, Finnie and Shaw [45] 0.88−1.85, Usui and

Takeyama [46] 0.4−2.0 etc. In FEM of metal cutting, Stenkowsky and Moon [29]

used

. , Komvopoulos and Erpenbeck [16] 0.0−0.5, Lin et al. [47] 0.074,

Lin and Lin [22] 0.001, Carroll and Stenkowski [28] 0.3, Olovsson et al. [48] 0.1

etc. As seen, the reported values of

obtained in metal cutting tests are well

above 0.577. On the other hand, the values of

used in FEM modelling are

always below the limiting value to suit the sliding condition at the interface. Inter-

estingly, the results of FEM modelling were always found to be in a good agree-

ment with the experimental results regardless of the particular value of the friction

coefficient selected for such a modelling.

It is true that, in general, the coefficient of friction for sliding surfaces remains

constant within wide ranges of the relative velocity, apparent contact area and

normal load. In contrast, in metal cutting the coefficient of friction varies with

respect to the normal load, the relative velocity and the apparent contact area. The

coefficient of friction in metal cutting was found to be so variable that Hahn [49]

doubted whether this term served any useful purpose. Moreover, Finnie and Shaw

[45] concluded that the concept of the coefficient of friction is inadequate to char-

acterize the sliding between chip and tool and thus recommended to discontinue

using the concept of the coefficient of friction in metal cutting. An extensive

analysis of the inadequacy of the concept of the friction coefficient in metal cut-

ting was presented by Kronenberg (pp. 18−25 in [43]) who stated “I do not agree

with the commonly accepted concept of coefficient of friction in metal cutting and

I am using the term ‘apparent coefficient of friction’ wherever feasible until this

problem has been resolved.” Unfortunately, it has never been resolved, although

more than 40 years passed since Kronenberg made this statement.

The proper modelling of contact conditions can be achieved if the relevant tri-

bological parameters at the tool–chip and tool–workpiece interfaces are considered

and the proper boundary conditions that adequately describe these conditions are

part of the FEM [7].

1.2.6 Numerical Integration

Because the metal cutting process is highly non-linear and dynamic, its simulation

requires a robust time integration method. Direct methods of time integration,

which includes the implicit and explicit direct methods of time integration, are

used for non-linear problems [50]. The explicit method presents some advantages

20 V.P. Astakhov and J.C. Outeiro

when applied to dynamic problems. For both the explicit and the implicit time

integration method, equilibrium is defined in terms of external applied forces, P,

the internal forces, I, and the nodal accelerations

⋅= −



uPI (1.25)

where M is the mass matrix and



u is the acceleration. Both methods use the same

element calculations to determine the internal element forces. The significant differ-

ence between the two methods lies in the manner in which the nodal accelerations

are computed [51]. The implicit method solves a set of linear equations using a direct

solution method (such as the Newton–Raphson method), performing iterations per

increment of time unttil a converged solution is reached within a given tolerance.

The explicit method determines the solution without evoking iterations, but uses

a central difference rule to integrate the equations of motion explicitly through time.

The explicit method has a number of advantages when used for highly dynamic

problems, especially for transient problems involving complex models. It is par-

ticularly suitable for problems involving complex contact interactions between

many independent bodies, as it happens in metal cutting simulations. For such

contact conditions, the convergence of the implicit method becomes questionable

as it needs a large number of iterations [51].

1.2.7 Errors

Modelling of cutting process using sophisticated FEM software has become quite

popular and many researchers try to refer to the results of such modelling as some-

thing conclusively proven. There are a thousand points of doubt in every complex

computer program. It should be very clear that FEM software incorporates many as-

sumptions that cannot be easily detected by its users but that affect the validity of the

results. There are several sources of errors which can lead to improper results [41]: (1)

poor input data due to the lack of information about the process, (2) unreasonable

simplifications, idealization and assumptions of metal cutting process, (3) improper

modelling of the boundary conditions, (4) numerical round-off (in solving the simul-

taneous equations), (5) discretisation error and (6) errors associated with re-mapping.

1.2.8 Example

Figure 1.12 shows an example of the results of FEM modelling. The following

conditions were considered in this modelling:

•

Tool: normal rake angle 0

, normal flank angle 7

, inclination

angle

, radius of the cutting edge 0005

. mm, tool material, K15.

•

Work material: AISI steel 316L, 517

UTS

MPa and 218

MPa.

•

Cutting regime: cutting speed 75=v m/min, uncut chip thickness

02=t.mm, width of cut 2=

d mm.

Figures 1.13(a) and (b) show the state of the deformation zone for two distinctive

phases of chip formation of the saw-toothed chip. Figures 1.13(c) and (d) show the

temperature distribution in the deformation zone for these two stages. As can be

Metal Cutting Mechanics, Finite Element Modelling 21

Figure 1.13. FEM modelling of the formation of saw-toothed continuous fragmentary chip:

(a,b) shape of the chip, (b,c) temperature distribution, (e,f) stress distribution

seen, the temperatures in the deformation zone vary within a chip formation cycle

[10]. Figures 1.13(e) and (f) present the stress distributing in the deformation zone

for the discussed phases.

1.2.9 Advanced Numerical Modelling

Standard FEM works well when it is applied to problems which do not involve

severe plastic deformations of the work material. In metal cutting, however, the

maximum deformation equal to the strain at fracture has to be achieved to accom-

plish the process. This presents a challenging problem for computational methods

as it is necessary to deal with extremely large deformations and thus distortion of

(a) (b)

Tem perature (C)

500

466

432

398

364

330

296

262

229

195

161

127

Temperature (C)

500

466

432

398

364

330

296

262

229

195

161

127

Stress-XX (MPa)

3.00E+02

1.71E+02

4.29E+01

-8.57E+01

-2.14E+02

-3.43E+02

-4.71E+02

-6.00E+02

-7.29E+02

-8.57E+02

-9.86E+02

-1.11E+03

-1.24E+03

-1.37E+03

-1.50E+03

Stress-XX (MPa)

3.00E+02

1.71E+02

4.29E+01

-8.57E+01

-2.14E+02

-3.43E+02

-4.71E+02

-6.00E+02

-7.29E+02

-8.57E+02

-9.86E+02

-1.11E+03

-1.24E+03

-1.37E+03

-1.50E+03

(e) (f)

22 V.P. Astakhov and J.C. Outeiro

the mesh. Moreover, because metal cutting is the purposeful fracture of the layer

being removed [7,

10], it is necessary to model the propagation of cracks with

arbitrary and complex path and track the growth and phase boundaries and exten-

sive microcracking.

These models are not well suited for conventional computational methods such

as FE, finite volume or finite difference methods. The underlying structure of

these methods, which originates from their reliance on a mesh, is not well suited to

the treatment of discontinuities which do not coincide with the original mesh lines.

To deal with moving discontinuities in methods based on meshes is to re-mesh in

each step of the evolution so that the mesh lines remain coincident with the dis-

continuities throughout evolution of the problem. Being the most viable, this strat-

egy introduces numerous difficulties such as the need to project between meshes

in successive stages of the evolution, which leads to degradation of accuracy of

the solution in unpredictable manner and greatly increases the complexity of the

computer programs [52−54].

An emerging new technique is the element-free Galerkin (EFG) method, also

called the meshless or meshfree method. The objective of EFG methods is to

eliminate at least part of this structure by construction the approximation function

entirely in terms of nodes. Thus, it becomes possible to solve large classes of

problems which are very awkward with mesh-based methods.

1.2.10 Model Validation

FEM analysis has proved to be a powerful investigative tool capable of encompass-

ing various aspects of the metal cutting process. However, despite its great poten-

tial, there are a number of limitations and approximations in the commercially

available FE programs. Although many papers have been published on FEA of the

metal cutting process, the issue of accuracy and adequacy of the modelled results

remains widely open due to the inherent drawbacks of the traditional FEA, highly

idealized models involved and inadequate representation of boundary conditions.

The highly structured nature of FE approximations imposes severe penalties in

the solution of metal cutting problems, and it is the decrease in structure that

makes meshless methods appealing. Meshless methods, however, require consid-

erable more central processing unit (CPU) time [52].

The results obtained using FEA strongly depend on the experience and judg-

ment of the engineers involved in the analysis of the problem and definition of

a simulation model. It should be clear that FEM makes sense if and only if a phys-

ically sound and adequate model of the metal cutting process is included at the

very foundation of this modelling. Therefore, the construction of such a model,

careful selection and justification of the proper boundary conditions and then

model validation should be the cornerstone of FEM. The market is overpopulated

with FE codes and tools for metal cutting simulations, but the issue of model

validity is silently set aside. How can one seriously propose to eliminate a physi-

cal test with a digital simulation if he cannot say that his model has, say, a 95%

level of confidence, or credibility, attached to it?

Metal Cutting Mechanics, Finite Element Modelling 23

The best way to validate the model used in FE simulation is through relevant

testing conducted under the same conditions assumed in the model construction.

However, this is rarely done because there are a number of problems with ade-

quate metal cutting tests. The existence of scatter in metal cutting experiments, it

appears, has now been widely accepted and is no longer under discussion. The

uncertainty behind work and tool materials properties, working loads, boundary

conditions at the tool–chip and tool–workpiece interfaces and other particularities

of the machining system is the main reason why two supposedly identical machin-

ing operations will yield different results in terms of tool life, quality of the ma-

chined part and achievable production rate. This fact is well known and yet, sur-

prisingly, engineers are involved in building very detailed models with the

intention of capturing the results of one particular test. It is difficult to understand

what exactly is detailed in these models, given the fact that the real world is nei-

ther detailed, nor accurate, nor precise. Now, since tests are, for very good physi-

cal reasons, nonrepeatable, this race for accuracy is lost from the very beginning.

The simplest yet most accurate way to validate a FEM in metal cutting is

through the chip compression ratio

[7,

10], defined as

(1.26)

where ν is the cutting velocity (speed), ν

the chip velocity, t

is the thickness of

the layer being removed (uncut chip thickness) and t

is the chip thickness.

Astakhov showed that the chip compression ratio is the one of the most impor-

tant process output (in terms of process evaluation and optimization). Besides the

cutting force and energy considered in this chapter, it directly correlates with the

tool–chip contact length, contact stresses and temperatures [7,

10]. Using this

parameter, one can easily validate a FE model.

Such a validation can be as simple as that: (a) adjust the test setup to the same

conditions assumed in the construction of the model including boundary condi-

tions, i.e., use the corresponding work material, cutting tool and machining re-

gime, (b) collect a typical chip and determine the chip compression ratio using the

simple known methods [10], (c) using the determined chip compression ratio,

calculate the tool–chip contact area and (d) compare the chip shape, chip compres-

sion ratio and the tool–chip contact length obtained using the test with those ob-

tained in FEM. Figure 1.14 presents an example of such a comparison. Although

the shape of the chips in both cases are rather similar, the experimentally obtained

value is

342

= .

while, as follows from Figure 1.14(b)

165

= .

. The chip–tool

contact length, calculated using the experimentally obtained chip compression

ratio is

158=

mm while that in Figure 1.13(b) is

065=

mm. Therefore, the

FE model cannot be considered as valid.

The FEM of thermal process in metal cutting may need more sophisticated

validation. Commercially available and specially designed infrared imaging

equipment, which includes both hardware and software, are available for tempera-

ture distribution measurements in the deformation zone in three-dimensional cut-

ting process. The high flexibility and low cost of this equipment make it suitable

to evaluate high temperature gradients, such as those produced by metal cutting

24 V.P. Astakhov and J.C. Outeiro

[8,

55]. Figure 1.15 shows an example of such measurements. These results should

be compared with those obtained using FEM. Then the model used in the model-

ling should be fine tuned to mach the experimentally obtained result.

Figure 1.14. Validation of a FE model by comparison the chips obtained experimentally (a)

and through FEM (b). Orthogonal cutting of the AISI 316L; tool material ISO M10

tungsten carbide; tool geometry: normal rake angle,

−4.29°; normal relief (flank) angle,

4.29°; normal wedge angle,

90°; inclination angle of the cutting edge,

−14°;

tool cutting edge angle,

72°; nose radius, r

0.8 mm. The cutting speed was 145

m/min; the cutting feed was 0.25 mm/rev; the depth of cut was 2.5 mm. No cutting fluid

was used in the tests

Figure 1.15. Distribution of the temperatures on the: (a) chip free surface (b) chip contact

surface and minor second tool flank face. Cutting conditions: work material AISI 1045

steel; uncoated tool; V

125 m/min; f

0.05 mm/rev; p

2.5 mm (b) coated tool; V

125

m/min; f

0.05 mm/rev; p

5 mm

(a) (b)

Temperature (C)

350

325

300

275

250

225

200

175

150

125

100

(a) (b)

Metal Cutting Mechanics, Finite Element Modelling 25

The foregoing considerations show that the model used in FEM of metal cut-

ting process should be adequate to the process. Moreover, the concept of FE

model must be broadened in order to embrace important facets of physics includ-

ing uncertainty, which unfortunately has been axiomatized out of modern metal

cutting research. Model validation should be considered as the mandatory step in

FEM. Breakthroughs in these directions will have considerable impact by making

metal cutting numerical simulation useful for practical optimization of various

metalworking operations including the cutting and machine tools, the metal work-

ing fluids and fixtures, adaptive CNC controllers and programs.

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