Astakhov V. Tribology of Metal Cutting

Generalized Model of Chip Formation 17

1.2.3 Comparison of the known solutions for the single-shear plane model with

experimental results

The next logical question is: how good is the single-shear plane model? In other words,

how far is this model from reality? Naturally, during the period of 1950–1960, when

decent dynamometer and metallographic equipment became widely available, a number

of fundamental studies were carried out to answer this question. The results of these

extensive researches are well summarized by Pugh [66] and Chisholm [67]. In the

author’s opinion, the best research study and a detailed description of the experimental

methodology were presented by Pugh [66]. The results obtained by Pugh were discussed

by Bailey and Boothroyd ten years later [68]. In his study, all the possible “excuses” for

“inadequate” experimental technique were eliminated. The experimental results conclu-

sively proved that for every work material tested, there is a marked disagreement in the

“ϕ vs.

(

µ − γ

)

” relation between the experiment and the prediction of the Ernest

and Merchant, the Merchants, and the Lee and Shafer theories (Eqs. (1.5), (1.7) and

(1.8), respectively). The examples of the experimental results obtained are shown in

Figs. 1.9–1.12.

Figure 1.9 shows the experimental results for lead as the work material. Although such

a choice of the work material might seem strange, one should realize that lead deﬁ-

nitely has a signiﬁcant advantage in cutting tests. This is because lead is chemically

passive so it does form neither solid-state solutions nor chemical compositions with

common cutting tool materials. Therefore, the use of lead as the work material allows

to carry out much more “pure” cutting tests. In Fig 1.9, line 1 graphically represents the

Ernst and Merchant solution, line 2 represents the Lee and Shafer solution, and line 3

approximates the experimental results. Figure 1.10 shows the results for various tested

Shear angle, ϕ(°)

Dry

−30−40 −20

Lubricated

0−10 10

5060 40

−101020 0 −20

30 5040 60

j + m − g =π/4

m − g (°)

2j + m − g =π/2

g (°):

Fig. 1.9. Relation between ϕ and

(

µ −γ

)

for lead: 1 – Ernst and Merchant solution, 2 – Lee and

Shafer solution and 3 – experimental results.

18 Tribology of Metal Cutting

Mild

Steel

Copper

Aluminum

Tin

Lead

−30 −20 −100 102030405060

Shear angle, j(°)

j + m − g =π/4

m − g (°)

2j + m − g =π/2

Fig. 1.10. Comparison between calculated and experimental results for tin, aluminum, mild steel,

lead and copper.

work materials. As seen, the experimental results are not even close to those pre-

dicted theoretically. Similar conclusive results were presented by Creveling, Jordon and

Thomsen [69] (an example is shown in Fig. 1.11 for steel 1113, where various cutting

ﬂuids were used) and by Chisholm [67].

The modiﬁed Merchant solution in which the shear stress is assumed to be linearly

dependent on the normal stress through a factor k



c = cot

−1



τ = τ

σ (1.19)

(according to Merchant, τ

and k

are work material constants) has also been examined

for a wide variety of work materials. Equation (1.19) is shown plotted in Figs. 1.12(a)

and (b) for copper and mild steel, respectively, together with the experimentally obtained

values [66]. As shown, the shear stress does not increase with the normal stress at the rate

required by the modiﬁed Merchant solution, i.e. to ﬁt the experimental results. In fact,

it would appear that the shear stress is almost independent of the normal stress on the

single shear plane.

The above conclusions were conﬁrmed by Bisacre [66] who conducted similar cutting

experiments. The results of these experiments enabled Bisacre to conclude that if the

Merchant solution (theory) was correct, there would be a marked effect of the normal

stress on the shear stress acting along the shear plane. To support his point, Bisacre

noted that the results of tests carried out, in which the same material was subjected

simultaneously to torsion and axial compression, showed that the shear strength of the

Generalized Model of Chip Formation 19

Air

CCl

Lusol

20 25 4035

−20 −100 102030

g (°):

Shear angle, j(°)

j + m − g = π/4

m − g (°)

2j + m − g = π/2

Fig. 1.11. Relation between ϕ and

(

µ −γ

)

for steel SAE 1113.

280 350210

0 140

140

g,(°)

Nornal Stess, s (MPa)

Shear stress, t (MPa)

210

280

350

Copper

140

210

0 280

250

280

420

560420 700 840

Mild Steel

490

560

(a)

Nornal Stess, s (MPa)

(b)

Dry Lubricated

g,(°)

Dry Lubricated

−20

t = t

=Cot60°

t = t

ot4

Fig. 1.12. Comparison between the estimated and experimentally obtained relationship “shear

stress–normal stress” for copper (a) and steel (b).

material was almost independent of normal stress. As a result, the difference between

the theoretical and experimental results cannot be attributed to the effect of the normal

stress on the shear strength of the work material, as suggested by Merchant.

Zorev also presented clear experimental evidences that the solutions discussed are inade-

quate [70]. He showed that the Merchant solution is not valid even in the simplest case of

cutting at low cutting speeds. Reading this, one may wonder why Zorev did not mention

20 Tribology of Metal Cutting

his ﬁndings about the single-shear plane model in his book [29] published ﬁve years

later. In the author’s opinion, if he had done so, he would have recognized that there

was no model of metal cutting available. As a result, he included the above-discussed

“general solution” for the single-shear plane model “forgetting” to mention that none of

the possible particular solutions to this model is in any reasonable agreement with the

experimental results.

1.2.4 Conclusions

It is conclusively proven that there is a marked disagreement between the solutions

available for the single-shear plane model and the experimental results. Hill, one of the

founders of engineering plasticity [71], noticed [72] that “it is notorious that the extent

theories of mechanics of machining do not agree well with experiment.” Other prominent

researchers in the ﬁeld conclusively proved that the experimental results are not even

close to those predicted theoretically [66,67,69,70]. Recent researchers further clariﬁed

this issue presenting more theoretical and experimental evidences [38,73,74].

As one might expect, knowing these results, the single-shear plane model would become

history. In reality, however, this is not the case and the single-shear plane model managed

to “survive” all these conclusive facts and is still the ﬁrst choice for practically all the

textbooks on metal cutting used today [10–12,42,55,62,75]. In contradiction, all the

excellent works showing complete disagreement of this model with reality are practically

forgotten and not even mentioned in modern metal cutting books, which still discuss the

single-shear plane model as the very core of the metal cutting theory. Moreover, the book

“Application of Metal Cutting Theory” [11] is entirely based on this model showing how

to apply it in practical calculations although other research works complain about the

absence of “predictive theory or analytical system which enables us, without any cutting

experiment, to predict cutting performance such as chip formation, cutting force, cutting

temperature, tool wear and surface ﬁnish” [2]. It should become clear that any progress in

the prediction ability of the metal cutting theory could not be achieved if the single-shear

plane model is still used.

It is necessary to list the major drawbacks of the single-shear plane model:

Inherent drawbacks

• Inﬁnite strain rate. Inﬁnite deceleration and thus strain rate of a microvolume

of the work material passing through the shear plane.

• Unrealistically high shear strain. The calculated shear strain in metal cutting

is much greater than the strain at fracture achieved in the mechanical testing of

materials under various conditions. Moreover, when the chip compression ratio

ζ = 1, i.e. the uncut chip thickness is equal to the chip thickness, no plastic

deformation occurs in metal cutting [49], the shear strain, calculated by the

model remains very signiﬁcant without any apparent reason for that.

• Unrealistic behavior of the work material. Perfectly rigid plastic work material

is assumed which is not the case in practice.

Generalized Model of Chip Formation 21

• Improper accounting for the resistance of the work material to cut. The

shear strength or the ﬂow shear stress cannot be considered as an adequate

characteristic feature with respect to this because, considered alone, the stress

does not account for the energy spent in cutting.

• Unrealistic representation of the tool–workpiece contact. The cutting edge is

perfectly sharp and no contact takes place on the tool ﬂank surface. This is in

obvious contradiction to the practice of machining where the ﬂank wear (due

to the tool ﬂank–workpiece contact) is a common criterion of tool life [76].

• Inapplicability for cutting brittle work materials. This model is not applicable

in the case of cutting of brittle materials, which exhibit no or very little plastic

deformation by shear. Nevertheless, the single-shear model is still applied to

model the machining of gray cast iron [77], cryogenic water ice [78], etc.

Ernst and Merchant-induced drawbacks

• Incorrect velocity diagram. In the known considerations of velocities in metal

cutting, the common coordinate system is not set, hence the known velocity

diagram consists of velocity components from different coordinate systems. As

a result, unrealistic velocity components are considered.

• Incorrect force diagram. The bending moment due to the parallel shift of the

resultant cutting force is missed in the force diagram. As shown [25], this

missed moment is the prime cause for chip formation and thus it distinguishes

the cutting process among other deforming processes. Moreover, the state of

stress imposed by this moment in the chip root causes chip curling.

• Constant friction coefﬁcient. Because the friction coefﬁcient at the tool–

chip interface can be thought as the ratio of the shear and normal force on

this interface, the distributions of the normal and shear stresses should be

equidistant over this interface. The available theoretical and experimental data

[12,29,43,52–56] do not conform to this assumption.

1.3 What is the Model of Chip Formation?

In metal cutting, the term “chip formation” has been used since the ninteenth century.

Its initial meaning is the formation of the chip in the primary and secondary deforma-

tion zones. Primary attention was devoted to the kinematic relationships, cutting force

and contact processes at the tool–chip interface. Later on, the chip-breaking problem

became increasingly important with increasing cutting speed and the development of

new difﬁcult-to-machine materials. Even though the term “chip formation” is still in use,

its original meaning has been transformed. The modern sense of this term implies the

chip, which just left the tool–chip interface, is yet to be broken [79,80].

The ﬁrst widely known classiﬁcation of the chip was presented by Ernst [61]. Accord-

ing to this classiﬁcation, there are three basic types of chips found in metal cutting:

Type 1 – discontinuous chip (segmental chip) in which initially compressed layer passes

22 Tribology of Metal Cutting

off with each chip segment. According to Ernst, this type of chip is most easily dis-

posed off; ﬁnish of the machined surface is good, when pitch of the segments is small;

Type 2 – continuous chip with continuously escaping compressed layer adjacent to the

tool face. According to Ernst, this chip type is the ideal chip form from the stand-

point of quality of ﬁnish of the machined surface, temperature of the tool point and

power consumption; and Type 3 – continuous chip with built-up edge adjacent to

the tool face. According to Ernst, this chip type is commonly encountered in ductile

materials. Finish is rough due to fragments of built-up edge escaping with the workpiece

[61]. Although these chip types were identiﬁed as “classical” [51] and this classiﬁ-

cation is still widely used today in many books on metal cutting [11,42,81], no one

pays attention to either the way these chip types were obtained (cutting regime, tool

and work materials, tool geometry, etc.) or to the physical characteristics of these chip

types.

As is well known [6,12,25,29], the shape of the chip depends primarily on the work

material, cutting regime, and tool material and geometry. According to Ernst [61], these

chip types were obtained in pure orthogonal cutting at extremely low cutting speed

(2 in/min = 0.05 m/min) using very speciﬁc work materials (high lead bronze and low

carbon, medium nickel chromium steel SAE 3115) and cutting tool (rake angle 23

◦

From the results of numerous experiments presented by Zorev [29], it is conclusively

proven that cutting physics and mechanics of the machining are entirely different at low

and at high cutting speeds as well as in the appearance, shape and metallurgy of the

chip formed.

Using the results of comparison of cutting at low and high cutting speeds obtained by

Zorev [29], one can conclude that the classiﬁcation discussed cannot satisfy growing

theoretical and practical requirements to understand the nature of chip formation. As a

result, the national industries of the developed countries have adopted more practical

classiﬁcations of chip type. For example, in Japan, the Subcommittee “Chip Disposal”

of the Japan, Society for Precision Engineering (JSPE) adopted a revised system of chip

forms, which includes nine chip types, basically classiﬁed according to the length of

the chip. Standard ISO 3685-1977 gives a comprehensive chip-form classiﬁcation based

on the size and shape of various chips generally produced in metal machining. Other

available classiﬁcations are discussed in detail by Jawahir and Luttervelt [82].

Unfortunately, the known classiﬁcations of the chip formed in machining originate only

from the differences in chip appearance, but pay no attention to the physical state of the

chip, including its state of stress and strain, hardness, texture, etc. Moreover, neither the

complete set of tool geometry parameters nor the cutting regime (for example, the true

uncut chip thickness and its width) is taken into consideration [83]. Thus, the known

classiﬁcations are of a post-process nature rather than being of help in making pre-process

intelligent decisions in process optimization and in understanding the tool–chip contact

phenomena.

A need is felt to develop a model of chip formation that can be used to analyze actual

tribological conditions at the tool–chip interface. As such, in addition to the system

concept in metal cutting, time dependence of cutting system parameters and their dynamic

interactions [25,83] and the time axis will be added to this model.

Generalized Model of Chip Formation 23

1.4 System Concept in Metal Cutting

The system concept in metal cutting was ﬁrst introduced by Astakhov and Shvets [83].

According to this concept, metal cutting is considered to be taking place in a system

consisting of the following components: the tool, the workpiece and the chip. The process

of metal cutting is deﬁned as a deforming process, which takes place in the components

of the cutting that system are so arranged through which the external energy applied

causes the purposeful fracture of the layer being removed. This fracture occurs due to

the combined stress including the continuously changing bending stress causing a cyclic

nature of this process.

The most important property in metal cutting studies is the system time. The system time

was introduced as a new variable in the analysis of the metal cutting system and it was

conclusively proven that the relevant properties of the cutting system’s components are

time dependent [83]. The dynamic interactions of these components take place in the

cutting process causing a cyclic nature of this process.

1.5 Generalized Model of Chip Formation

Although the basics of the cutting tool geometry are discussed in Appendix A, it is

necessary to set here proper deﬁnitions for the terms “cutting edge” and “cutting wedge”

that will be used in further considerations because these two are mostly misunderstood

and thus misused in the literature on metal cutting. One should clearly realize that the

cutting edge is the line of intersection of the face and the ﬂank surfaces. Therefore,

the basic characteristics of the cutting edge should be thought of as those that can be

attributed to a line. When one talks about cutting edge geometry, it should be understood

only as the geometric characteristics of a line in terms of its length, 3D shape, etc. It is

highly improper to speak about cutting edge wear because a line cannot have any wear.

The cutting tool geometry can be discussed as attributed to the point of the cutting edge

considered. In this case, however, this line should be considered as belonging to the

corresponding rake or ﬂank surface so that the geometry of the corresponding surface

should be meant.

The working part of the cutting tool is referred to as the cutting wedge, which is enclosed

between the tool rake and ﬂank contact surfaces intersecting to form the cutting edge. This

cutting wedge is under the action of stresses applied on the tool–chip and tool–workpiece

contact surfaces. Moreover, due to the heat that ﬂows into the cutting tool, this wedge

has high temperature. Therefore, when it is said that the cutting tool penetrates into the

workpiece, the penetration of the cutting wedge is meant because the conﬁguration of

the cutting tool outside the cutting wedge does not affect the metal cutting mechanics

and physics of the cutting process (at least theoretically when the tool is considered to

be inﬁnitely rigid as in any model of chip formation).

Consider the cutting wedge (tool) starting to advance into the workpiece under the action

of the applied penetration force P, as shown in Fig. 1.13(a). As a result, the stresses grow

in the workpiece and, as might be expected, the maximum stress occurs in front of the

24 Tribology of Metal Cutting

Cutting tool

Workpiece

(a)

(b) (c)

Fig. 1.13. Cutting tool starting to advance into the workpiece.

cutting edge due to stress singularity at the point. When this maximum reaches a certain

limit, depending on the properties of the work material the following happens:

• If the work material is brittle, a crack appears in front of the cutting edge

(Fig. 1.13(b)). Later on, this crack leads to the ﬁnal fracture of the layer being

removed.

• If the work material is ductile, a visible crack would not be readily observed because

of the “healing” effect of plastic deformation. Instead, a certain elastoplastic zone

forms in the workpiece in front of the tool rake face, as shown in Fig. 1.13(c). The

dimensions of the plastic and elastic parts of this zone depend on ductility of the

work material. It is understood that for a perfectly plastic work material, the elastic

zone would not form at all, while for a perfectly brittle material the plastic zone

would never form.

This simple qualitative consideration illustrates that the properties of the work material

play an extremely important role from the beginning of chip formation. Because the

general behavior of work materials in metal cutting can be classiﬁed as ductile or brittle

depending upon the degree of plastic deformation that a material exhibits in its defor-

mation, this fact should be organically incorporated in the chip formation model. As

such, a certain criterion (or criteria) should be established to classify a given material

as brittle or ductile because no perfectly brittle or perfectly ductile work materials exist

in reality.

In mechanical metallurgy, elongation or strain at fracture is usually used for such a

purpose [81]. It would be necessary to emphasize here, however, that none of the known

criteria of ductility (brittleness) can be regarded as a mechanical property of the work

material because the same material may exhibit considerably different elongations and

strain at fracture depending upon the loading conditions which include the state of stress,

strain rate, temperature, etc. Unfortunately, today the decision about ductility of the work

material is routinely made on the basis of its mechanical properties obtained in one of

the standard material tests under uniaxial stress state, room temperature and moderated

strain rate (4–5 orders lower than that found in metal cutting).

Generalized Model of Chip Formation 25

(a) (b) (c)

Fig. 1.14. Difference in deformation pattern in compression and in cutting: (a) specimen with the

scribed grid, (b) distortion of the initial grid in compression and (c) distortion of the initial grid in

cutting.

Another important issue follows from Fig. 1.14. Figure 1.14(a) shows a specimen made

of a ductile material with the grid scribed on its cylindrical surface. Figure 1.14(b) shows

grid distortion occurred in compression where simple shearing is the prime deformation

mode. Figure 1.14(c) shows grid distortion occurred in cutting. If one compares deforma-

tion patterns due to compression and cutting, one observes signiﬁcant difference because

simple shearing is not the prime deformation mode in metal cutting, as suggested by the

single-shear plane model discussed earlier. This simple fact is known from mechanics of

materials and could be easily conﬁrmed by anyone conducting a simple test similar to

that shown in Fig. 1.14. Unfortunately, the known works on metal cutting do not account

for this simple result.

1.5.1 Brittle work materials

Consider the machining of a brittle work material using the system approach in metal

cutting. Two basically different cases are possible here depending on the arrangement

of the components of the cutting system. The ﬁrst one takes place when the compo-

nents of the cutting system are arranged so that the resultant force R acting on the chip

from the tool rake face intersects the conditional axis of the partially formed chip, as

illustrated in Fig. 1.15. As such, the bending moment that rose in the root of the chip

(Section 1.1) due to the action of this force, causes chip fracture along the path where

the combined stress (compression and bending) exits the fracture stress for this work

material. Figure 1.16 illustrates a system consideration of the model shown in Fig. 1.15.

Phase 1 illustrates the initial stage when the tool comes in contact with the workpiece.

26 Tribology of Metal Cutting

Conditional axis of the chip

Fig. 1.15. Resultant force R intersects the conditional axis of the partially formed chip (after

Astakhov [18]).

234

5678

PPPP

System Time

Fig. 1.16. System consideration of the model shown in Fig. 1.15.

To advance the tool into the workpiece, the penetration force P has to be applied to the

tool, provided the workpiece is ﬁxed rigidly. The interaction between the tool and the

workpiece results in the formation of a stressed zone ahead of the tool (Phase 2). Because

there are absolutely no brittle work materials, this zone consists of the elastic and plastic

parts although its plastic part is small compared to the elastic part. When the maximum

stress in the stressed zone reaches a certain limit, a crack forms in front of the cutting

edge (Phase 3). Further increase in the penetration force P leads to the development of

this crack formed (Phase 4). At this stage, cutting completely resembles splitting. Due to

Astakhov V. Tribology of Metal Cutting

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