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Materials Science and Engineering A 460461 (2007) 95100

Single crystal bulk material micro/nano indentation hardnesstesting by nanoindentation instrument and AFM

Liang Zhou a,b,, Yingxue Yao a

a School of Mechanical and Electrical Engineering, Harbin Institute of Technology, Harbin 150001,PR Chinab School of Material Science and Engineering, Harbin Institute of Technology, Harbin 150001,PR China

Received 4 June 2006; received in revised form 2 January 2007; accepted 11 January 2007

Abstract

The micro/nano indentation hardness of single crystal aluminium and single crystal silicon are investigated. Loaddepth curves can be obtainedby nanoindentation instrument, and materials indentation hardness can be calculated by OliverPharr method and work of indentation method

directly from these curves. The hardness that obtained by OliverPharr method is overestimate because of material pile-up effect, and the hardness

that obtained by work of indentation method is not very correct because of its empirical equations inaccurate. The true hardness can be calculated

by plastic work of indentation and plastic volume that obtained by integrating fitted polynomial according to loaddepth curves and atomic force

microscopy, respectively. Comparison and analysis of the results that obtained by these methods are made.

2007 Elsevier B.V. All rights reserved.

Keywords: Nanoindentation; Hardness; OliverPharr method; Work of indentation method; Atomic force microscopy

1. Introduction

As an indicator of the materials ability against deformation,

hardness has been studied for more than 100 years [1,2]. Dur-

ing the last 20 years, researches have become more and more

interested in mechanical properties of smaller volumes, and the

mechanical properties under micro/nano scale may differ from

the macro scale properties due to the size effect and surface

effect, etc. [1].

Nanoindentation as an instrumented indentation method is

widely used to determine the mechanical properties of both bulk

solids and thin films. The loaddepth curves including loading

andunloading process canbe obtained by nanoindentation tech-

nology. It hasbeen shownthat theOliverPharr method[3,4] and

work-of-indentation method [5,6], which are commonly used to

measure hardness from loaddepth curves.Fig. 1 shows the typical loaddepth relationship curves from

nanoindentation experiments, where Pmax is the peak indenta-

tion load, hmax the indenter depth at peak load, and hr is the final

depth of contact impression after unloading [3].

Corresponding author at: School of Mechanical and Electrical Engineering,

Harbin Institute of Technology, Harbin 150001,PR China.

Tel.: +86 451 86402543/608; fax: +86 451 86413810.

E-mail address: [email protected] (L. Zhou).

The hardness usually defined as the ratio ofPmax to project

area of hardness impression Ac (Eq. (1)), and perhaps the most

widely used method is OliverPharr method [3].

Hop =Pmax

Ac(1)

The OliverPharr method analysisprocedurebegins by fitting

the unloading curve to an empirical power-law relation [3,7].

P= (h hr)m (2)

where P is the indentation load, h the indenter depth, and m

are empirically determined fitting parameters. Once the param-

eters and m are obtained by curve fitting, the initial unlading

stiffness, S, can be established by differentiating Eq. (2) at themaximum depth of penetration, h = hmax. The contact depth, hc,

is estimated from the loaddepth data [3,7].

hc = hmax Pmax

S(3)

where is a constant dependent on the indenter geometry. For

the generally employed Berkovich indenter, it has been shown

that has an empirical value of 0.75 [3,7]. The project area of

hardness impression function is made by fitting the Ac versus hc

0921-5093/$ see front matter 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.msea.2007.01.029
mailto:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/10.1016/j.msea.2007.01.029http://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/10.1016/j.msea.2007.01.029mailto:[email protected]


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96 L. Zhou, Y. Yao / Materials Science and Engineering A 460461 (2007) 95100

Fig. 1. Typical loaddepth curves from nanoindentation experiments.

data to the relationship:

Ac = 24.56h2c +

7i=0

Cih1/2i

c (4)

where Ci are constants, the lead term describes a perfect

Berkovich indenter, the other parameters describe deviations

from the Berkovich geometry due to blunting at the tip [3].

Work of indentation method describes indentation experi-

ments process as the use of the energy dissipated or work done

during the indentation. The energies are based on the integral of

the loading and unloading curves. The area under the loading

curve gives the total workWt (gray region in Fig. 1) done during

indentation, whilethe elastic contribution,We (weak gray region

in Fig. 1), is given by the area under the unloading curve. Thus,

the plastic workWp (deep gray region in Fig. 1) is the difference

between Wt and We (Fig. 1) [5,6,8].

Wp = Wt We (5)

Theworkof indentationmethodto estimatematerialhardness

was first proposed by Stilwell and Tabor [8]. It was shown that

the conventional representation of hardness, indentation load

divided by the projected area of permanent impression, is equiv-

alent to the ratio of plastic work to plastically deformed volume

[5,6,8]:

load, P(N)plastic area, Ap(m2)

= plastic work,Wp(J)plastic volume, Vp(m3)

(6)

The total workWt can be obtained by integrating the load-

ing curve. In general, it is found that for sharp indentation of

an elasticplastic material the loading response is governed by

P = Ch2, where Cis a constant and h is the penetration depth [6]

(Eq. (7)).

Wt =

hmax0

P(h) dh =

hmax0

Ch2 dh =Pmaxhmax

3(7)

It has been shown that the ratio ofhr/hmax is equivalent to

that ofWp/Wt. Thus, the plastic workWp can be obtained by Eq.

(8) [6,8].

We

Wt= 1

Wp

Wt= 1

hr

hmax(8)

Tuck etal. [5] suggested that thehardness could be calculated

on the basis work-of-indentation alone, and can be represented

by

HWt =P3max

9W2t(9)

where is a constant equal to 0.0408 for Berkovich indenter.

Alternatively, by takingthehardness to be concernedwith the

plastic work of indentation, then the total work term is replaced

with the plastic work and the hardness is obtained also [5] (Eq.

(10)).

HWp =P3max

9W2p(10)

Although nanoindentation technology give significant infor-mation concerning the mechanical response to indentation and

without the need for indent image, the effects of material pile-

up that often was found at the indent edges [4,6,9] can seriously

affect the calculated values.

In addition to atomic force microscopy (AFM) high resolu-

tion, three dimensional imaging capability, the use of AFM for

imaging residual indentationshasalready proved to be oneof the

only methods presently available for obtaining accurate dimen-

sional information from an image area of only a few microns

[9]. Combining Matlab soft, the real plastic deformation vol-

ume of micro/nano indentation can be obtained according to

AFM image.

In this paper, the indentation hardness of single crystal alu-

minium and single crystal silicon are investigated. The hardness

of these materials is initially determined using OliverPharr

method and work-of-indentation method. In order to determine

the actual deformation volume and area, the indentations are

directlymeasured viaAFM andMatlab soft.Theeffectofpile-up

on hardness values is investigated also.

2. Experimental

Bulk materials such as single crystal silicon (10mm10mm 1 mm) and single crystal aluminium ( 25mm

5 mm) were used for this experiment. Single crystal silicon sur-face roughnessis less than 5 nmafterelectropolishing, andsingle

crystal aluminium surface roughness is less than 10 nm after

turning by diamond cutting tool in ultraprecise machine tool.

The indentation experiments were conducted with a

Berkovich indenter using Nano II (manufactured by MTS

nanoinstruments). It has load and displacement resolutions of

75 nN and0.04 nm, respectively.

In each testing run, the indenter was driven into the specimen

surface under a load gradually increased to the predetermined

depth, then unloaded after being hold at peak load for 10 s.

Such a procedure repeated with different predetermined depths

as 500nm, 800 nm, 1000 nm, 1200 nm and 1500 nm for single


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L. Zhou, Y. Yao / Materials Science and Engineering A 460461 (2007) 95100 97

Fig. 2. Loaddepth curves of single crystal silicon.

Fig. 3. Loaddepth curves of single crystal aluminium.

crystal silicon and 400 nm, 500 nm, 800 nm, 1000 nm, 1500 nm

and 1700nm for single crystal aluminium. Figs. 2 and 3 show

the loaddepth curves of the single crystals silicon and single

crystal aluminium, respectively.

AFM is Dimension 3100 (manufactured by Digital Instru-

ments). Its integral nonlinearity (X, Y) is 1%, XYimaging area

is 90m, Z range is 6m, and RMS vertical noise floor is


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Fig. 8. Workdepth curves of single crystal silicon.

We can see that all hardness values increasing with the depth

decreasing from Figs. 6 and 7. This phenomenon knows as the

indentation size effect (ISE). And it is clear that the HWt


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L. Zhou, Y. Yao / Materials Science and Engineering A 460461 (2007) 95100 99

Fig. 13. Plots of hardness vs. depth of single crystal silicon.

In Figs. 11 and 12, we can see that the work either obtained

by Eqs. (7) and (8) or obtained by integrating fitted polynomial

have a good linearity with volume, but the slopes of these fitting

lines are different, i.e. the hardness values are different because

of work of indentation obtained by different method.

Thus, wecanobtain thehardnessof crystals byworkof inden-tation divided by plastic volume. When discussing the hardness

results, the following nomenclature is used to distinguish the

hardness values (Hop, HWt and HWp) that have been described

in this paper. The hardness that obtained by Wt1/Vt is named

HWt1, the hardness obtained by Wp1/Vt is named HWp1, and the

hardness obtained by Wp1/Vb is named HWp2.

Plots of hardness, Hop, HWt, HWt1, HWp1 and HWp2 versus

depth are drawn for crystals examined in the present study and

are now shown in Figs. 13 and 14.

Whether the hardness calculated by OliverPharr method or

work-of-indentation method, and whether the work of indenta-

tion directly obtained by empiricalequations or integrating fittedpolynomial, the hardness values existing ISE phenomenon. But

the hardness values that obtained by the ratio of plastic work to

total plastic deformation volume, i.e. HWp1 = Wp1/Vt variation

is smoother than others.

Fig. 13 shows that single crystal silicon indentation hardness

HWt1 in excess of two times ofHWp1, this means the large elastic

work mainly contribution to the total work of indentation that

calculated by integrating fittedpolynomial. Reversely, in Fig.14,

HWt1 is almost similar with HWp1 of single crystal aluminium

because of the deformation in indentation processing almost

only plastic deformation happening.

Fig. 14. Plots of hardness vs. depth of single crystal aluminium.

Further analysis, the hardness values HWp2, i.e. plastic work

dividedby indentplastic volume only, is verynearly thehardness

values Hop that obtained by OliverPharr method for all exam-

ined crystals in present study. Hence, the OliverPharr method

hardness yields values have not considering the material pile-up

plastic volume that would seriously affect the results. Due to the

plastic contact depth is greatly underestimated in OliverPharr

method because of ignoring materials pile-up, lead to overesti-

mate hardness values.

To HWt, the hardness values that obtained by work of inden-

tation method (Eqs. (7) and (9)) are similar with Hop of single

crystal aluminium, but similar with HWp1 of single crystal sili-

con, so, the hardness value HWt is more suitable elasticplastic

materials not plastic materials, the reason is under the under-

standing that Hop is overestimate the indentation hardness and

HWp1 is nearest the true hardness of materials.

The constant = 0.0408 of Eq. (9) using in work of indenta-

tion methodand =0.75ofEq. (3) using in OliverPharrmethod

are dependent on the indenter geometry, but under micro/nano

scale, the indenter geometry did not self-similar. So, the twoconstant will generate errors under different indentation depths.

And the project area Ac (Eq. (4)) of OliverPharr method and

the hardness HWt (Eq. (9)) of work-of-indentation method are

empiricalequationor semi-empiricalequation thatwill seriously

affects by testing condition, testing instruments, irregularities of

material, initial state of material surface and so on. To HWp1,

the plastic work is calculated by integrating fitted polynomial

according to loaddepth curves, the plastic volume including

material pile-up volume and indent volume are obtained by real

indent AFM images, so, the results ofHWp1 perhaps are more

accurate andbelievable than other results, andtheeffects of pile-

up are eliminated because of considering the pile-up volume incalculation.

4. Conclusions

Under micro/nano scale, loaddepth curve can be obtained

by nanoindentation instrument, and real indent image can be

obtained by AFM. Materials indentation hardness can be calcu-

lated directly from loaddepth curves by OliverPharr method

and work of indentation method. But the hardness values are

overestimatedby OliverPharrmethod because of pile-up effect,

and the hardness values are inaccurate by work of indentation

method because of its suggestions are not completely correctly.

Plastic work can be obtained integrating fitted polynomialbased on loaddepth curve and real plastic volume including

pile-up volume and indent volume obtained by AFM images.

The hardness values that using plastic work divided by plastic

volume are more smoother than other results, and also perhaps

more accurate and believable.

Acknowledgements

Support for this work by the Scientific Research Founda-

tion of Harbin Institute of Technology (Project: HIT.2003.23)

and Doctor Research Startup Foundation of Harbin Institute of

Technology.


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single crystal bulk material micro-nano indentation hardness testing by nanoindentation instrument...

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