Klocke F. Manufacturing Processes 1: Cutting

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3.8 Cutting Theory 91

The amount of resultant force can now be calculated from Eqs. (3.20) and (3.21):





sin ·cos

(

 + ρ − γ

)

· b ·h (3.22)

The functional relations of the values of the resultant force components cutting

force and feed force can be taken from the circle of Thales (Fig. 3.50).



|=|



|·cos

(

ρ − γ

)

(3.23)



|=|



|·sin

(

ρ − γ

)

(3.24)

If we now insert Eq. (3.22) into Eqs. (3.23) and (3.24), the following is valid for

the values of the resultant force components:



cos

(

ρ − γ

)

· τ



sin · cos

(

 + ρ − γ

)

· b · h (3.25)



sin

(

ρ − γ

)

· τ



sin · cos

(

 + ρ − γ

)

· b · h (3.26)

3.8.1.3 Calculation of the Shear Angle

RYSTOF developed a model for the simple calculation of the shear angle [Krys39].

He made the following assumptions:

• The theory of the shear plane is valid.

• The maximum shear stress leads to material collapse.

In the main axial system, maximum shear stresses occur at an angle of 45

◦

For calculations of the cutting process, the position of the main axial system

is assumed to be approximately in the direction of the resulting contact stress

(Fig. 3.51).

For purposes of simplicity, the principle stress is not represented by the stress

tensor but simply by vector addition. In this case, the friction angle approximates

the orientation of the main axial system.

The function relation between the shear angle, the tool orthogonal rake angle and

the friction angle can be derived from Fig. 3.51. The following is valid for the shear

angle:

 =

−

(

ρ − γ

)

(3.27)

ERCHANT has suggested a possibility of calculating the shear angle that makes

the following assumptions [Merc45, Merc45a]:

• The theory of the shear plane is valid.

• The position of the shear plane is determined by the minimum of cutting energy.

The value of cutting force can be read off the circle of T

HALES in Fig. 3.50:

92 3 Fundamentals of Cutting

Trace of shear plane P

Trace of cutting

edge plane

Trace of tool reference plane

≡

Tool

γn

−

Fig. 3.51 Principle of the maximum shear stress, acc. to KRYSTOF [Krys39]



cos

(

ρ − γ

)

cos

(

 + ρ − γ

)

b · h

sin

· τ



(3.28)

The necessary and the sufﬁcient conditions for calculating the position of the

energy minimum are

nec.:

∂E

∂

= 0suff.:

∂

∂

= 0 (3.29)

Since the cutting energy is calculated as the product of the cutting force and

the cutting path, and the cutting path is not a function of the shear angle, we can

simplify to:

nec.:

· ∂|



∂

= 0suff.:

∂



∂

= 0 (3.30)

From which follows for the shear angle:

 =

−

(

ρ − γ

)

(3.31)

3.8.2 Application of Plasticity Theory to Cutting

HUCKS and OPITZ developed a shear angle equation by applying MOHR’s slip the-

ory to the orthogonal cutting process [Huck51, Opit53, Mohr06]. As opposed to

the models of K

RYSTOF and MERCHANT, this theory makes it possible to take a

material-speciﬁc slip angle into consideration instead of an idealized 45

◦

. It should

3.8 Cutting Theory 93

Tool

≡

Slip direction

Trace of tool reference plane P

Trace of cutting

edge plane

≡ P

Trace of shear plane P

1,3: Principle normal stress directions

Fig. 3.52 Surface element at the cutting part, acc. to HUCKS [Huck51]

again be pointed out that the previously known shear angle need not necessarily be

equal to the angle given by the actual position of the slip surface, since the direction

of the shear plane was previously only purely geometrically determined by ideal

considerations. In this case the directional angle of the slip surface towards the tool

cutting edge plane is designated as the shear angle . First a square surface element

is considered on the rake face that experiences both shear and compressive stresses

from the tool (Fig. 3.52).

The shear stresses also appear on all remaining faces of the quadratic element

as a result of the law of equality of shear stress. Due to the unhindered ﬂow of the

chip over the face there is a very small amount of compressive stress in this direc-

tion, which will be neglected in the following. There are therefore two compressive

stresses and four shear stresses on the surface element under consideration.

If we now determine the principle normal stress with the help of M

OHR’s stress

circle, we establish the validity of the following equation:

ω =

arctan(2 · μ) (3.32)

The angle between the rake face and the principle stress direction is thus deter-

mined by the friction coefﬁcient between the rake face and the chip. This relation

makes it clear, for example, why different cutting speeds yield different shear angles.

The friction coefﬁcient is dependent on the cutting speed and consequently alters the

principle stress direction. The slip direction on the other hand is inclined towards

the principle stress directions by a materially dependent angle ϕ. This angle can be

94 3 Fundamentals of Cutting

calculated from the compressive ﬂow stress σ

and the shear ﬂow stress τ

with the

help of ﬂow tests and M

OHR’s stress circle:

ϕ = 45 −

arcsin



− 2τ



(3.33)

For an ideal plastic body which is approximately realized by steel at the yielding

point, the ﬂow plane is under 45

◦

in relation to the principle stress direction, since

there the shear stress is at a maximum [Huck51].

With this, we obtain the principle stress and slip directions shown in Fig. 3.52

relative to the rake face. With the help of the angles γ , ϕ, and ω, the slip angle 

can now be calculated:



= ϕ −

arctan(2μ) + γ

(3.34)

Slip angle 

is in the still undeformed area, which is why only slip angle 

(which is identical with the shear angle) is signiﬁcant. The shear angle calculated

here presumes an extrapolation of the stress conditions via the shear plane and the

width of ﬂank wear land as well as the formation of a continuous chip. This approx-

imation was proven to be in accordance with experiments made on materials with

envelope lines that are inclined and parallel to the σ-axis. H

UCKS did not experiment

with materials with bent envelope curve [Huck51].

With the help of the shear angle, the resultant forces can now also be calculated.

To do this, the stresses that are active in the shear plane are multiplied with shear

plane surface, resulting in a normal and tangential force in the shear plane. We

will in this context proceed without the derivation of H

UCKS’s formulae. For steel,

which has an envelope line that is parallel to the σ -axis, the following formulae for

cutting and feed force result with width of undeformed chip b and chip thickness h

according to H

UCKS:

= τ

· b ·h ·

⎡

⎣



+ cot g(φ)

⎤

⎦

(3.35)

= τ

· b ·h ·

⎡

⎣

cot g(φ)



− 1

⎤

⎦

(3.36)

To sum up, we can say that the highly numerous experimental relations in chip

formation depend only on a few numbers of materials and the friction coefﬁcient.

UCKS’s work made the fundamentally new discovery that the stress ﬁeld in the

chip on the cutting edge depends, besides its intensity, only on the friction coefﬁcient

and the normal direction of the envelope line [Huck51].

Chapter 4

Cutting Tool Materials and Tools

Tool change times, and with them both manufacturing times and tool, machine and

labour costs, are affected by wear. Wear is affected in turn by the properties of the

cutting tool materials. Development in the cutting tool material sector is therefore far

from ﬁnished, but is constantly aiming both to improve cutting tool materials that are

already established as well as to discover new materials for use in the manufacture

of cutting tools.

Cutting tool materials should have the following properties in order to do justice

to the stresses placed on them:

• hardness and pressure resistance,

• bending strength and toughness,

• edge strength,

• inner bonding strength,

• high temperature strength,

• oxidation resistance,

• small propensity to diffusion and adhesion,

• abrasion resistance,

• reproducible wear behaviour.

Should we consider all these characteristics simultaneously, we are then faced

with the requirement for the “ideal” cutting tool material (Fig. 4.1). However, there

is no one cutting tool material that uniﬁes all the required properties. One rea-

son for this, for example, is the physical opposition of hardness and toughness.

Developments i n the cutting tool material sector are concentrating on optimizing

and modifying chemical composition, manufacturing methods, geometry and coat-

ing in order to broaden the areas of application of cutting tool materials and tools in

accordance with the requirements of modern production.

4.1 Overview of Cutting Tool Materials

The cutting tool materials used in cutting can be summarized as (Fig. 4.2):

• tool steels,

• cemented carbides,

F. Klocke, Manufacturing Processes 1, RWTH edition,

DOI 10.1007/978-3-642-11979-8_4,



Springer-Verlag Berlin Heidelberg 2011

96 4 Cutting Tool Materials and Tools

Ideal cutting tool material

Cutting speed

Wear-resistance and heat-resistance

1 Considering the hardness

2 Considering the hardness

and the temperature

DP:

Durability and bending-strength

Feed

HSS

Coated HSS

Fine grain

and submicron

grain cemented

carbide

+ TiC

Coated

cermet

Coated

cemented-

carbide

ceramic

3 -

ceramic

Cemented-carbide

on tungsten-carbide-basis

Cermet

Fig. 4.1 Schematic classiﬁcation of several cutting tool materials

Cold work steels

High speed steels

WC-Co

TiC/TiN - Co,Ni

WC-(Ti,Ta,Nb)C-Co

Oxide-ceramic

Silicon-nitride-ceramic

Mixed-ceramic

Boron-nitride

Diamond

Tool steels

Cemented carbide

Cutting - ceramics

Super-hard cutting

tool materials

Cutting tool materials for chipping

with geometrical defined cutting edge

Fig. 4.2 Classiﬁcation of cutting tool materials for machining

4.1 Overview of Cutting Tool Materials 97

• cutting ceramics and

• super-hard cutting tool materials made of boron nitride and diamond.

The hardness and wear resistance of the cutting tool materials increases in this

order. On the other hand, their bending strength decreases in this order.

Tool steels include unalloyed and alloyed cold work tool steels as well as high

speed steel. The term cemented carbide refers to conventional cemented carbides

based on tungsten carbide as well as “cermets” based on titanium boron nitride.

Ceramics is the generic term for cutting tool materials made of oxide, mixed and

non-oxide ceramic.

4.1.1 Classiﬁcation of Hard Cutting Tool Materials

According to DIN ISO 513, the notation and application of hard cutting tool mate-

rials of cemented carbide, ceramic, diamond and boron nitride are classiﬁed in

accordance with Fig. 4.3. According to it, uncoated cemented carbides based on

tungsten carbide with a WC grain size of ≥ 1 μm take the abbreviation HW and

with grain sizes of < 1 μm the notation HF. “Cermets” – cemented carbides based on

titanium boron nitride – are notated with HT, coated cemented carbides and coated

cermets with HC. Corresponding abbreviations from Fig. 4.3 are valid for other hard

cutting tool materials based on ceramic, diamond or boron nitride.

The goal of DIN ISO 513, besides that of identifying cutting tool materials, is

above all their assignment to materials, for which they are most suitable for cutting.

Expanding earlier norms, DIN ISO 513 provides six main application groups and

thus six classes of workpiece-materials, which are classiﬁed with the code letters

P, M, K, N, S and H as well as by colour (Fig. 4.4). DIN ISO 513 thus maintains

the established code letters P, M and K but now only designate with them the main

application groups that comprise the material groups steel (P), stainless steel (M),

and cast iron (K). The major machining group K included according to the old stan-

dard not only cast iron materials but also a number of other materials. In DIN ISO

513, these now new major application groups were provided with the labels N for

nonferrous metals, S for special alloys and H for hard materials.

Every main application group is subdivided into particular application groups

(Fig. 4.4). These are notated with a code letter for the main application group they

belong to and with an index. The index refers to the toughness and wear resis-

tance of the cutting tool material. The higher it is within each application group,

the lower is the material’s wear resistance and the higher its toughness. The indices

are merely reference numbers indicating a certain sequence. They provide no infor-

mation about the extent of the wear resistance or the toughness of a cutting tool

material. Manufacturers of cutting tool materials should assign their cutting tool

materials to the appropriate application group, depending on wear resistance and

toughness. Examples are HW-P10, HC-K20, and CA-K10. An application group

thus contains comparable cutting tool materials of different manufacturers, although

they may differ in wear behaviour and performance. It is also possible for a cutting

tool material from one manufacturer to be categorized under multiple application

groups if suitable to them.

98 4 Cutting Tool Materials and Tools

Ceramic

Code-letter Group of materials

Oxide-ceramic,

predominantly made from

aluminium-oxide (Al

)

Mixed-ceramic, based on

aluminium-oxide (Al

but also with other

components than oxides

Silicon-nitride-ceramic,

predominantly made from

silicon-nitride (Si

)

Code-letter

Cutting-ceramic like above,

but coated

These cemented carbides are also called

"Cermet"

Cemented carbide

Group of materials

Uncoated cemented

carbide, predominantly made

from tungsten-carbide (WC)

with a grain size

µm

Uncoated cemented carbide,

predominant

titan-carbide (TiC) or

titan-nitride (TiN) or

both

Cemented carbide like above,

but coated

Uncoated cemented

carbide, predominantly made

from tungsten-carbide (WC)

with a grain size <

µm

Cutting-ceramic,

predominantly made of

Aluminium-oxide (Al

reinforced

Diamond

Code-letter Group of materials

DP Polycrystalline diamond

DM Monocrystalline diamond

Boron-nitride

Code-letter Group of materials