53

7

Nanoindentation Techniques

Manjima Bhattacharya, Arjun Dey, and Anoop Kumar Mukhopadhyay

7.1 Introduction

7.1.1 Hardness Analysis

Based on the load versus depth-of-penetration data generated, the parameters

involved in a nanoindentation experiment (Figures7.1a and 7.1b) are given

by h

f

, the nal depth of penetration; h

max

, the maximum depth of penetration

of the indenter when the load P = P

max

; and h

c

, the contact depth, i.e., the dis-

placement where the indenter has maximum contact with the surface while

unloading [1]. Any nanoindentation hardness calculation is also just like the

conventional hardness calculation, i.e., load/load-bearing contact area. Now,

there can be two area concepts used in nanoindentation evaluation. If the area

is calculated from the contact depth, h

c

, it is generally denoted by the contact

area, A

s

. There is another area called A

p

, which is the projected contact area

of an ideal nanoindenter. Since the nanoindenters have very small tip radius,

typically about 40–200 nm, and the load may be ultralow, from a few microne-

wtons to a few millinewtons, it is expected that the indentation imprints also

will be very small. Therefore, to measure the contact area and the contact

height directly experimentally is rather difcult because of the elastic recov-

ery. Therefore, these are calculated based on the projected area, A

p

. Assuming

that a given nanoindenter (e.g., Vickers or Berkovich) has an ideal shape and

the surfaces that it indents are perfectly at, the contact depth (h

c

) is given by

=h

A

24.56

c

p

and when h

c

is known, the contact area A

c

is given by, e.g., A

c

= 26.44h

c

2

,

while the corresponding projected contact area A

p

is given by, e.g.,

A

p

(h

c

)=24.56h

c

2

. The quantity A

p

can be measured with sufcient accuracy

from the images ofthe nanoindent, provided that the tangent height is set

at typically about 90% below the surface height. Once A

p

is known, h

c

can

be calculated. When h

c

is calculated, the value of A

s

can be easily predicted,

54 Nanoindentation of Brittle Solids

asmentionedpreviously. Because of the repeated contact events and friction

with the surface being indented, the tip of the nanoindenter gets worn out

and, as a result, it becomes rounded. Thereby, it physically shifts from its

ideal shape. Hence, the calibration of the indenter area function is done to

account for the deviation from the ideal shape.

Projected area: …

()

=+++++Ah ah ah ah ah ah

pc 0c

2

1c 2c

1

2

3c

1

4

8c

1

128

(7.1)

Surface area:

…

()

=+++++Ah bh bh bh bh bh

sc 0c

2

1c 2c

1

2

3c

1

4

8c

1

128

(7.2)

The quantity a

0

in equation (7.1) is approximately 24.56, whereas the quantity

b

0

in equation (7.2) is approximately 26.44 [2]. The calculations of the pro-

jected area functions are done through dedicated software packages (which

may vary from machine to machine) [3, 4] by analyzing the load–unload

curves generated from a series of nanoindents made over a range of prespec-

ied depths. Further, h

c

is related to h

f

by h

c

= h

f

(1 + e

c

), where e

c

physically is

about 10%–30% for most metals and reects the amount of elastic recovery

following the unloading of the nanoindenter. Thus, from the experimentally

measured value of h

f

using standard software packages, h

c

is derived to digi-

tally reconstruct the fully loaded nanoindentation cavity surface from the

residual nanoindentation cavity surface (Figure7.2). To improve the accuracy

Residual Indent

Surface

Reconstructed

Indent Surface

FIGURE 7.2

Reconstructed nanoindent cavity surface and the residual nanoindent cavity surface.

Force

Pile-Ups

Displacement

(a) (b)

h

f

h

f

h

c

h

c

h

max

FIGURE 7.1

Denition of typical physical parameters in nanoindentation: (a) load (P) versus depth (h) plot;

(b) concept of contact depth (h

c

) and nal depth of penetration (h

f

). (Adapted from Fischer-

Cripps [4].)

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