Jackson M.J. Micro and Nanomanufacturing

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Diamond Nanogrinding 523

Fig. 10.1. Nanoscale grinding detritus showing blunt and sharp abrasive grains of

diamond and metal chips

Figure 10.1 shows abrasive grains with blunted cutting edges

(wear flats), and abrasive grains with sharp cutting edges that are re-

leased from the surface of the piezoelectric crystal before they have

chance to grind nanoscale chips from the surface of the workpiece.

The process suffers with a loss of diamond grains even when the in-

terfacial adhesion between diamond and piezoelectric material is

very good. A more closely related process that has been reported

widely is that of the wear of probes used in atomic force microscopy

[1,2].

However, these observations were purely experimental with

no explanation of how to design probes that inhibit, or retard, wear.

A performance index used to characterize diamond wear re-

sistance is the grinding ratio, or G-ratio, and is expressed as the ratio

of the change in volume of the workpiece removed,

to the

change in the volume of the diamond abrasive grain removed,

and is shown in Eq. (10.1),

AvjAv

(10.1)

524 Micro-and Nanomanufacturing

Grinding ratios for processes at the nanoscale have not yet

been characterized. However, the complexities of wear of abrasive

materials at any scale lead us to believe that the variety of different

and interacting wear mechanisms involved, namely, plastic flow of

abrasive, crumbling of the abrasive, chemical wear etc., makes the

wear of diamond at the nanoscale too complicated to be explained

using a single theoretical model [3].

The following analysis of diamond grains represented by loaded

wedges assumes that grain fracture is the dominant wear mechanism

when grinding at the nanoscale using the piezoelectric nanogrinding

process.

10.3 Stress Analysis in a Nanogrinding Grain

10.3.1 Nomenclature

<i>

Wedge half-space

Constant

Tangential force

Equivalent tangential grinding force

Equivalent normal grinding force

Grinding ratio

Force multiplier, or coefficient

Point load

Radial polar co-ordinate

Thickness of wedge

Volume of grinding wheel removed

Volume of workpiece material removed

Rake angle of wedge (abrasive grain)

Stress function

Angular polar co-ordinate

Principal stress

Compressive strength of abrasive

[mm]

[N]

[mm /mm ]

[N]

[mm]

[mm

]

[N/mm

]

[N/ram

]

[N/mm

]

Diamond Nanogrinding 525

Radial stress [N/mm

]

Tensile strength of

the

abrasive [N/mm ]

0 Circumferential, or hoop stress [N/mm

]

6 Shear stress [N/mm

]

co Angle between point load and apex of wedge

10.3.2 Analysis of Loaded Nanogrinding Grains

Diamond grains are blocky in nature and possess sharp cutting

points prior to nanogrinding workpiece materials. Figure 10.2

shows a collection of diamond abrasive grains that have well defined

cutting points that form a wedge at their apex. When bonded into a

strong matrix, these grains can be considered to be representative in-

finite wedges.

Fig. 10.2. A collection of diamond grains showing cutting points located at their

apex, and locations of {100} and {110} planes. Diamond grains are approxi-

mately 60 |j,m in diameter.

526 Micro-and Nanomanufacturing

An infinite wedge represents the cutting point of an abrasive

grain in contact with the workpiece material (Fig. 10.3).

Fig 10.3. The single-point, loaded infinite wedge

The wedge is loaded at the apex by a load P in an arbitrary

direction at angle

<x>

to the axis of symmetry of

the

wedge. Resolving

the force into components P.cosco in the direction of the axis, and

P.sinco perpendicular to that the stresses due to each of these forces

can be evaluated from two-dimensional elastic theory [4]. The state

of stress in the wedge, due to force P.cos co, can be obtained from

the stress function,

<f)

C.r.0.

sin 6

(10.2)

Where r and 6 are polar coordinates, or the point N in Fig. 10.4,

and C is a constant.

Diamond Nanogrinding 527

Fig. 10.4. The single-point, loaded infinite wedge showing force components, and

the point N within the wedge at polar co-ordinates, r and 9

The stress function yields the following radial, tangential, and

shear stress components,

-2C

COS0

(10.3)

(10.4)

(10.5)

To determine the constant, C, the equilibrium of forces along the

y-axis is:

Pcosa- \ a cos0dA

= O

J-a

(10.6)

528 Micro-and Nanomanufacturing

Where dA is an element of cross sectional area within the wedge.

If, t, is the thickness of wedge, then,

COSCDP= f 2C^-.t.r.cos0dO =2Ct ^cos

OdO

Ct[2a + sin2a] (10.7)

Therefore, c

Pc0SC

° (10.8)

t(2a + sin

la)

And,

IPcosO.Qosco

<?.. = —

r.t(2a + sin 2a)

(10.9)

Note that the negative sign denotes that the stress is compressive.

The state of stress in the wedge, due to force P sin co, can be ob-

tained from the stress function,

4 = Crj9'.sin0' (10.10)

Therefore,

Gr=

-2C^-

(10.11)

=0 (10.12)

r„=0 (10.13)

Equilibrium offerees along the x-axis (Fig. 10.4) yields the fol-

lowing solution for the constant, C,

Diamond Nanogrinding 529

Psinco-

<r„J.r.cos0V0'=O

(10.14)

Psinco =

-C.t(2a-sin2a)

2C?^-Jj>.cose'dO'

=2Ct.ff

cos

e'd0' =

{-a

%-a

(10.15)

C-

Psinco

t(la - sin la)

(10.16)

Thus,

_ 2Pcos#\sin#>

r.t(2a

- sin 2d)

(10.17)

Expressing in terms of the angle 9 (where 0' is negative), yields,

2Pcos#.sin&>

r.t(2a

- sin 2d)

(10.18)

Therefore, the combined stresses are:

2P_

r.t

cos

cos 0 sin&>cos#

+ •

2a + sin la la - sin la

(10.19)

It follows that a

vanishes for angle 9

defined using the expres-

sion:

tan<9 =

1 2a - sin 2a

tan co la

sin la

(10.20)

This equation corresponds to a straight line through the apex

as shown in Fig. 10.5. This natural axis separates the regions of

530 Micro-and Nanomanufacturing

compressive and tensile stresses. It can be seen that for values of an-

gle co which gives, |9

| >

\a\

the neutral axis lies outside the included

angle of the wedge. This means that the whole area of the wedge

will be under stresses of uniform sign. Expressing Eq. (10.19) in

terms of the rake angle of the abrasive grain, (3, and force compo-

nents F

and nF

(Fig. 10.6), yields,

J[«.cos^-/^+sin(a-/7)]cos# [{cos^-/?)-«.sin(<2-/7)}cos#]

rt 1 2a+sir\2a la-sinla

(10.21)

It can be observed that:

tan&> =

cos(a - P) - «.sin(a - /?)

n.cos(a - P) + sin(a - /?)

(10.22)

Neutral

axis

a*- = constant

c%= O

---^Jie= O

Fig. 10.5. Stress analysis of a single-point loaded wedge

Diamond Nanogrinding 531

In the simple case of a wedge with the normal force nF

along the

wedge axis, a is equal to p, hence,

tan (o = 1 / n

(10.23)

It is interesting to examine the radial stresses on the left-hand

face of the wedge, which corresponds to the leading face of ideal-

ized wedge. Thus, for the left-hand face, 0 is equal to -a, and from

Eq. (10.19),

0", = —

2P_

r.t

cos

cos a -cosa.sm^y

2a + sin 2a 2a - sin 2a

(10.24)

Workpiece

Fig. 10.6. Ideal wedge-shaped cutting point and grinding force diagram

532 Micro-and Nanomanufacturing

This stress is zero, i.e., the neutral axis coincides with the left-

hand limit of the wedge, when,

1 _ sin a(2a + sin 2a)

tan&> cos a{2a - sm2a)

Thus if,

(a) a = p, then,

sin

sin 2a

(10.25)

n =

cos a 2a-sm2a

(10.26)

(b)

a-(3=*_„

(as is the case when F

is parallel to the right-hand face of the

wedge). From Eq. (10.23),

sina-n.cosa sma-ncosa r-\mn\

tanco= = (lU.z/;

n.sina

cosa n.sina

cosa

And substituting in (10.25), yields,

n sin a + cos a _ sin a 2a + sin 2a

sina-ncosa cos a 2a -sin 2a

(10.28)

-^ 2

—

.MRU! 7/7

OCT

.*sin2a

cos *

2a + s[n2a (10

.29)

sin

a — .n.sinla