Jackson M.J. Micro and Nanomanufacturing

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Diamond Nanogrinding 533

—./?.sin 2a(2a - sin 2a) + 2a cos

a - cos

a.sin 2a =

= 2a.sin

a + sin

a.sin 2a .n.sin 2a(2a + sin 2a)

—jz.sin

2a(2<2

- sin

2a + 2# sin 2d)

- 2a(sin

a - cos

+ sin 2a

n.2a.sin2a = - 2a.cos2a + sin2a

n =

cot 2a

2a (10.30)

Equation (10.25) expresses the condition for the whole of the

wedge's cross-sectional area to be under compressive stress. It can

be seen that this depends not only upon the rake angle, P, but also

upon the ratio, n. In general the relative size of the region of com-

pressive to the region of tensile stresses depends upon P and n as Eq.

(10.20) and (10.22) indicate. Also, from Eq. (10.21), the magnitude

of the stress on the left-hand face of the wedge is found to be de-

pendent upon the tangential force component, F

, and the force com-

ponent ratio, n. Referring to Eq. (10.19), it can be seen that for con-

stant stress, a

= constant,

r. Cj =

.cos

.sin

0 (10.31)

Where Ci, C2, C3 are constants. Eq. (10.31) represents, in polar

co-ordinates, the circumference of a circle tangent to the line,

O =

.cos0+C

.sin 0 (10.32)

that is to the neutral axis at the point r

0. However, the point r =

0 must be considered separately because the stress at that point ap-

proaches infinity, since by definition P is a point load. The central

point of these circles are of constant radial stress, and so the point of

constant maximum shear stress must lie on a line perpendicular to

the neutral axis at the point where r is equal to zero. The radius of

each of those circles depends upon the magnitude of the radial

stress,

534 Micro-and Nanomanufacturing

10.4 Fracture Dominated Wear Model

Brittle materials exhibit high strength properties when loaded in

compression than in tension. The ratio of rupture strengths is usually

between 3:1 and 10:1. The existence of relatively low tensile stresses

in the abrasive grains may cause failure by fracture to occur. To

model the action of diamonds bonded to piezoelectric ceramics, one

must consider a single active cutting grain to be classed as a wedge

of constant width loaded at its inverted apex with point loads, F

and, nF, which represent the radial and tangential force components

with reference to the grinding wheel in which the grain is supported,

and P is the resultant force (Fig. 10.6).

The stress distributions within point-loaded wedges can be

determined analytically, and the results of such an analysis indicate

that if tensile stresses exist within the wedge then it will occur at its

maximum along the rake face. The existence of a tensile stress de-

pends on the magnitude of the force ratio, n. If the ratio is especially

small that a tensile stress exists in the wedge, then for a specific

force ratio the tensile stress is proportional to the tangential grinding

force, F. Stresses of this nature would extend to and beyond the

abrasive grain-bonding interface. The fracture of abrasive grain,

bonding phase, or the interface between the two, depends on the par-

ticular type of piezoelectric ceramic material used and the magnitude

the

tensile stress induced during nanogrinding.

Grains of diamond are 10 times stronger in compression than

in tension. The probability of grain fracture is likely to increase with

an increase in tensile stress exerted in the grain although the magni-

tude of the stress may be slightly higher than one-fifth the magnitude

of the maximum compressive stress in the grain. A significant bar-

rier to the acceptance of stress patterns evaluated for such situations

arises because point loads applied to perfectly sharp wedges produce

infinitely high stresses at, and about, the point of contact. Loads

must be applied over a finite area.

It seems likely that higher tensile stresses are associated with

higher grain fracture probability resulting in rapid loss of diamond

grains and, consequently, lower grinding ratios. The wear model

should incorporate the fact that the loads are spread over a finite

area. This implies that point loads are applied along the rake face.

Diamond Nanogrinding 535

The model should allow the examination between the wear rate of a

diamond coated piezoelectric ceramic material and the general na-

ture of stresses established in active cutting grains subjected to

nanogrinding forces. This means that it is necessary to estimate the

force components of grinding on each active cutting grain.

10.5 Nanogrinding

10.5.1 Nanogrinding Apparatus

The experimental apparatus consisted of holding a polished

specimen between the jaws of a vice so that a piezoelectric crystal

oscillator traverses back-and-forth the specimen thus machining the

specimen by creating a depth of cut between the diamonds adhered

on the piezoelectric crystal and the workpiece material. Workpiece

materials were polished with a 100 nm sized polishing compound.

All samples were divided into four sections and each section was

analyzed prior to machining and after machining occurred. The

workpieces were mounted in a vice that was attached to a x-y-z lin-

ear slide in order to achieve accurate positioning of the workpiece.

The piezoelectric crystal was mounted on a steel framework that was

orthogonal to the workpiece. The whole unit was located within a

tetrahedral space frame to dampen excess vibrations (Fig. 10.7).

10.5.2 Nanogrinding Procedure

When the crystal and workpiece were aligned, the depth of cut

was incremented in stages of 10 nm. The motion of the diamonds

attached to the piezoelectric crystal generates a machining effect that

is caused by the action of diamonds grinding into the workpiece ma-

terial. Tracks or trenches are created by the diamond grain gouging

the surface of the material when an electric current is applied. The

material is removed until the end of the oscillating motion creates

the material to plow.

536 Micro-and Nanomanufacturing

Fig. 10.7. Piezoelectric machining center capable of nanogrinding

The mechanism of oscillation can be described as a restricted

bending mode that simulates a shear displacement of the crystal. At

this point it is normal procedure to know how to estimate the num-

ber of grains contacting the surface of the workpiece.

The estimation of the number of active cutting grains is esti-

mated quite simply by driving the diamond coated piezoelectric ce-

ramic at the prescribed specific metal removal rate into a piece of

lead. The impression that the grinding wheel produces in the length

of lead is equal to the number of cutting points that are active during

the grinding stroke at that particular depth of cut. The effect of os-

cillating the diamond-coated crystal is shown in Fig. 10.8.

Diamond Nanogrinding 537

Fig. 10.8. Material removal by oscillating a diamond-coated piezoelectric ceramic

crystal

The motion of the diamonds imparted by oscillating the crys-

tal in the bending mode that causes a shear displacement to occur,

which contributes to plowing of the material at the end of the

nanogrinding stroke, is shown in Fig. 10.9.

538 Micro-and Nanomanufacturing

Fig. 10.9. Plowed material at the end of

nanomachined track

The effect of using the piezoelectric to machine tracks, or

trenches, in engineering materials opens up the prospect of nano-

manufacturing products that require geometric features such as

channels so that fluids and mixed phase flows can be manipulated in

devices such as micro-and nanofluidic "lab-on-a-chip" products.

Figure 10.10 clearly shows such a channel produced using the pie-

zoelectric nanogrinding process.

Diamond Nanogrinding 539

Fig. 10.10. Nanomachined tracks in steel

The measured force components of the nanogrinding opera-

tion are measured using a dynamometer. These components of force

are then applied to a "model abrasive grain by dividing the grinding

force data into the number of active cutting grains over an area that

simulates the abrasive grain-workpiece contact area. Stresses estab-

lished in this area are calculated using finite elements. The wear of

the piezoelectric material by diamond loss, expressed in terms of a

grinding ratio, and its relationship to the stress levels set up in the

model grain is investigated using a stress analysis method.

540 Micro-and Nanomanufacturing

10.5.3 Stress Analysis

The assumed geometry of an ideal grain in the vicinity of its cutting

edge is a simple symmetrical wedge of constant width with an in-

cluded angle of 70° that results in a rake angle of-35°. There is no

wear flat on the model cutting grain. In order that a finite element

method is used to evaluate stresses in the wedge, the wedge was

subdivided into 210 diamond-shaped elements with a total of 251

nodes.

Forty-one nodes were constrained at the boundary of the

wedge and the leading five nodes on the rake face were loaded (Fig.

11.11).

The tangential and normal grinding forces were replaced by

equivalent forces acting perpendicular to (normal load) and along

(shear load) the rake face of the wedge.

The concentrated loads at the five nodes are representative of

the distributed and normal loads acting on the rake face over the

abrasive grain-chip contact length. The normal force distribution on

the rake face was taken to be maximum at the cutting edge and de-

creasing linearly to zero at the end of the abrasive grain-chip contact

length. The shear force was taken to be constant over the first half

of the contact length, decreasing linearly to zero over contact length.

Grinding loads were also applied directly to the rake face and

at the tip of the grain without calculating equivalent forces. This

was performed in order to compare and contrast the effect of differ-

ent force distributions on the stresses generated within the wedge.

To measure the value of using the maximum tensile stress as

a way to estimate grain fracture tendency, the correlation between

the two sets of data were calculated for each set of

data.

The region

of fracture initiation was also located using Griffith's criterion of

fracture. For,

^.(7

+cT3>0 (10.33)

Then,

Diamond Nanogrinding

541

But

for,

(10.34)

—.o-j +cr

(10.35)

Then,

(hH^+M'

»0\

1(7.

- (7

,|)=o

(10.36)

AAAAAAAAAAAAAAAA

V V V

V Y

Y Y

YYV

WxyYxxSoooooo<XXVV

XXX

XAAAAAA

X X

x v x

x x x

x y

v YVYv v

YvYVVYYYV

vyVvYvyWy

v y

A A A

^x>

/YYVVYV

YYSAAAAA

vvXVvYvv

\1YV

/vy

Fig.

10.11.

Finite element assemblage with grinding loads applied

at the

rake face

nodes

542 Micro-and Nanomanufacturing

Where a \ and a

are the principal stresses, assuming that

a \ > J3, o

is the ultimate tensile strength of the abrasive grain,

and a

is the ultimate compressive strength. For diamond grain ma-

terial, the ratio of o

and a

is 0.1.

The results of the two-dimensional stress analyses were con-

sistent with the experimentally determined stress distribution ob-

tained by Loladze [5] when cutting soft metal with photoelastic

tools.

The maximum tensile stress always occurs at the rake face at

a distance from the cutting edge ranging from 1.5 to 4 times the

abrasive grain-chip contact length, the exact magnitude of the

coef-

ficient depends on the loading conditions for a particular machining

event. For a given value of the tangential force component, F, the

higher the force ratio, F/nF, the greater the distance the maximum

tensile stress is away from the cutting edge.

a,+ 0.2a = 0

a+0.2a>0

Zone of dangerous

points

(tf!>

tensile

fracture)

a+0.2a<0

Sample size of zone of

dangerous

points

for

compressive failure,

Fig. 10.12. Griffith's criterion applied to the idealized wedge showing tensile and

compressive fracture initiation zones