ibis handbook of nanoindentation · part 1. theory of nanoindentation 1.1 mechanical properties of...

Handbook of Nanoindentation

The

byA.C. Fischer-Cripps

P.O. Box 9, Forestville NSW 2087 Australia. www.fclabs.com.au

The IBIS Handbook of Nanoindentation

Anthony C. Fischer-Cripps

ISBN 0 9585525 4 1

Copyright © Fischer-Cripps Laboratories Pty Ltd 2013

You may download, print and reproduce this handbook in unaltered form only for your personal, non-commercial use. Apart from any use as permitted under the Copyright Act 1968, all other rights are reserved.

Information provided in this book is believed to be accurate and reliable, however, no responsibility is assumed by Fischer-Cripps Laboratories Pty Ltd for its use; nor for any infringements of patents or other rights of third parties which may result from its use.

IBIS is a registered trademark of Fischer-Cripps Laboratories Pty Ltd.

Published by: Fischer-Cripps Laboratories Pty LtdP.O. Box 9, Forestville NSW 2087 AustraliaEmail: [email protected]

Part 1. Theory of Nanoindentation1.1 Mechanical Properties of Materials 6

Stress and strain, combined stresses, elasticity, Poisson's ratio, plastic deformation, fracture mechanics, visco-elasticity.

1.2 Contact Mechanics 14Hertzian contact, spherical indenter, conical indenter, other indenters, contact stiffness, Hertzian fracture, plastic zone, constraint factor, hardness testing,.

1.3 Theoretical Analysis 23Load-displacement curve, theoretical analysis, Oliver and Pharr method, Field and Swain method, indenter or arbitrary shape, energy method, theoretical modelling, finite element analysis, creep, dynamic nanoindentation, multiple-frequency nanoindentation, residual stress.

Part 2. Practical Nanoindentation2.1 Nanoindentation Testing 38

Specimen preparation, indenters, nanoindentation testing, 10% rule, step-height, scratch testing, fracture mechanics, .

2.2 Applying Corrections 47Thermal drift, contact determination, instrument compliance, indenter shape correction, piling-up.

2.3 Analysing the Data 53Fitting the unloading curve, image analysis, nanoindentation results, do's and don’t's.

Part 3. More Information 3.1 Books 583.2 Literature 593.3 Manufacturers 60

Contents

The IBIS Handbook of Nanoindentation 3

Fischer-Cripps Laboratories Pty Ltd

The IBIS Handbook of Nanoindentation4


Part 1Theory of Nanoindentation

1.1 Mechanical Properties of Materials1.2 Contact Mechanics

1.3 Theoretical Analysis



1.1 Mechanical Properties


5 m

Precise positioning capability of a nanoindenter can allow one to measure the properties of the individual constituent phases of complex microstructures.


1.1.1 Stress and Strain

Normal stress:

A

F

Force

Area (normal to direction of force)

A

P

Shear stress: Force

Area (parallel to direction of force)

l

l

Normal strain:

Change in length

Original length

h

xh

x

tan

Shear strain:

Change in length (opposite)

Height (adjacent)

P

x

h

A

F

P

P

P

AP

P

P

P

l

l



2sin2cos 2

1

2

1

cossin2sincos 22

xyyxyx

xyyx

2cos2sin 2

1

cossin cossin 22

xyyx

xyyx

22

2,1 22 xyyxyx

21

22

max

2

1

2

xy

yx

1.1.2 Combined Stresses

22

3,1 22 rzzrzr

2

31max 2

1

Principal stresses

Stresses at an angle

Cartesian Axis-symmetric

Hydrostatic stress

3

3

321

zyx

m

md

md

md

33

22

11

21221

213

2323

1oct

x

y

z



P

x

2P

2x

k depends on the type of material and the dimensions of the specimen.

Hooke’s law (1676)

kxP

Young’s modulus (1807)

l

xE

A

PA

klE

l

x

A

klA

kx

A

P

kxP

Let

Start with Hooke’s law

Divide both sides by A, the cross-sectional area of the specimen

Multiply and divide by l, the length of the specimen

Stress Strain

The elastic modulus (or Young's modulus) E is a material property which describes the elasticity, or stiffness of a material. It is not the same thing as "hardness".

1.1.3 Elasticity

12

EG

213

EK

Shear modulus

Bulk modulusxzxz

yzyz

xyxy

G

G

G

1

1

1

In shear, we have:



Poisson’s ratio is the ratio of the fractional change in one dimension to the fractional change of the other dimension.

ww

l

l

P

l

lw

w

1.1.4 Poisson’s Ratio

Poisson’s ratio is a measure of how much a material tries to maintain a constant volume under compression or tension.

When the material contracts inwards (plane stress) under an applied tensile stress T, there is no sideways stress induced in the material. If the sides of the material are held in position by external forces or restraints (plane strain), then there is a stress induced, the value of which is given by:

TIn terms of stresses and strains, in plane strain conditions (sides held in position), there is an effective increase in the stiffness of the specimen due to the induced sideways stresses. Hooke’s law becomes:

21

E

Plane stress

x

Plane strain

yxzz

zxyy

zyxx

E

E

E

1

1

1

yxz yxz E

1

0 ;0 ;0 yzxzz0 ;0 ;0 yzxzz

rzz

rz

zrr

E

E

E

1

1

1Cartesian Axis-symmetric

y



1.1.5 Plastic Deformation

Simple Compression

Specimen fractures

Large confining pressure

Specimen contains large

number of microcracks and looks "ductile"

The deformed material is

“constrained”

Small confining pressure

Specimen fractures along a

shear plane

Ductile

Brittle

Compression

Tension

Slip across shear planes

Y2

12

131max

213

232

2212

1Y

Tresca

Von Mises

Plastic flow in ductile materials is dominated by slippage across

planes in shear. Two commonly used plastic flow criterion are the Tresca and Von Mises conditions

for plastic flow. For the Tresca criterion, plastic flow occurs

when the maximum shear stress becomes equal to half of the

material's uniaxial yield stress.



a

a

c Plastic energy absorbed here

Strain energy is released here

Surface energy is created here

1.1.6 Fracture Mechanics

One of the most important concepts of fracture is the Griffith energy balance criterion.

This condition for crack extension is equivalently expressed in terms of Irwin's stress intensity factor K. For Mode I loading, K1 is given by:

dc

dU

dc

dU s

For an increment of crack extension, the amount of strain energy released G per crack tip must be greater than or equal to that required for the surface energy of the two new crack faces 2.

22

E

ca

cYK a 1

E

KG

21

The strain energy release rate G is related to K by:

E

K

E

K

E

KG IIIIII

1

**

222

or more generally:

E

cU a

s

22 cU 4

thus



1.1.7 Visco-elasticity

Elastic solid Viscous fluid Visco-elastic material

dt

dxkxP

k

Voigt

Maxwell

dt

dF

kP

dt

dx 11

k

Consider the response of a parallel spring and dashpot (Voigt) to an sinusoidal force. The resulting displacement x is also sinusoidal but is out of phase with the force.

:

ix

iexdt

dx

exx

ePP

tio

tio

tio

Thus:

xikdt

dxkxP

The product is a loss term and is related to viscosity.

The spring stiffness is a storage term and is related to elastic modulus.

The damping coefficient has the same relationship to viscosity as the spring stiffness k does to the elastic modulus E. The damping coefficient and stiffness apply to a particular specimen geometry where the modulus and viscosity apply to the material (i.e. E and are material properties).

Units: N s m1k

kxP dt

dxP

NewtonHooke

22221

21

2212121

kk

kikkkkkhP

22221

21

22221

2212121

sin

cos

kk

k

h

PTF

kk

kkkkk

h

PTF

o

oim

o

ore

Transfer function

1

12

11

kikhP

k1

k2

P(t)

Three-element combined model



1.2 Contact Mechanics


Example of elastic-plastic contact. The picture shows the residual impression made by a Berkovich indenter into bulk aluminium. Note the raised material at the edge of the contact. The plastic zone extends about twice the contact diameter.


1.2.1 Hertzian Contact – Spherical Indenter

An understanding of nanoindentation begins with a study of contact mechanics between solid bodies. The most well-known system is that of a spherical indenter being pressed against a flat specimen and was first studied by Hertz in the late 19th century. Hertz determined that the radius of the circle of contact acould be calculated in terms of the load P, the radius, and the elastic modulus by:

*3

4

3

E

PRa

'

'111 22

* EEE

21

111

RRR

ara

r

a

P

Eh

2

42

312

2

*

R

P

E

22

*3

4

3

R

ah

2

2321*

3

4hREP

2a

Ppm

R

aEpm

3

4 *

where E* is the combined elastic modulus of the indenter and specimen expressed as:

and R is the combined radius of the indenter and the specimen (where for a flat specimen, R1 –):

Another important relationship arising from these equations is the profile of the deformed surface where for an equivalent rigid spherical indenter, the displacement of the surface (z = 0) as a function of the distance from the axis of symmetry r is given by:

Combining the above two equations gives the distance of mutual approach of the indenter and the specimen (which is equal to the total penetration depth for a perfectly rigid indenter) at r = 0.

and hence

Expressed in terms of the penetration depth h, the load is thus:

Normalising the load by the contact area, we obtain the mean contact pressure:

where

R1

R2

a

R



1.2.2 Conical Indenter

cot2

*aEa

P

a raa

rh

cot2

tan2 2*hEP

Similar equations where developed for the case of the elastic contact between a rigid conical indenter and an elastic, semi-infinite, half-space by Sneddon in the mid 20th century. The relationship between the load and the contact radius is expressed in terms of the indenter cone half-angle as:

The displacement of the specimen free-surface beneath the indenter is given by:

And hence the load and the penetration depth (r = 0) is found from:

a

h

The equations of contact here apply to a completely elastic contact between the indenter and the specimen. The (theoretically) infinitely sharp tip radius of the cone would ensure that plastic deformation would occur as soon as contact were made and thus the singularity of stresses at the tip of the cone predicted by these equations would be avoided. In practice, an indenter would not have an infinitely sharp tip radius and so the initial stages of contact would be similar to that involving a spherical indenter.



1.2.3 Other Indenters

ccc RhhRhA 22 2

2

22

49.24

tan33

c

c

h

hA

2

22

504.24

tan4

c

c

h

hA

2

212

4.65

tantan2

c

c

h

hA

2

22

60.2

tan33

c

c

h

hA

22 tanchA

Of particular interest is the relationship between the contact area and the depth of the circle of contact (as measured from the tip of the indenter), and also, that between the load and the total penetration depth for popular indenter shapes.

Spherical indenter

Berkovich indenter

Vickers indenter

Knoop indenter

Cube corner indenter

Conical indenter

= 65.27

= 68

1 = 86.25, 2 = 65

= 35.26

2321*

3

4hREP

2* tan2

hE

P

= 42.278

= 70.296

= 70.3

= 77.64

Pyramidal indenters are analysed in terms of an equivalent conical indenter angle that gives the same area-to-depth ratio as the actual indenter.

(Note: Berkovich's original design was for = 65 02')

Note, this is a basic assumption of small displacements which carries through into all Hertz's equations of contact

Cylindrical punch indenter

2aA

haEP *2



hEdh

dP tan

22 *

1.2.4 Contact Stiffness

Adh

dPE

2

1*

aEdh

dP *2

A very important quantity in the analysis of nanoindentation test data is the contact stiffness. Consider the case of the elastic contact for a cone and a flat specimen. The force and the indenter displacement are given by Sneddon's equation:

2* tan2

hEP

Taking the derivative of P with respect to h we obtain:

and substituting back to the first equation we can see that:

hdh

dPP

2

1

That is, the slope of the load displacement curve for an elastic loading at any particular point on the curve is twice that given by the ratio P/h.

Now, at r = 0, the displacement of the indenter and the radius of the circle of contact are related by:

0 cot2

rah

and since A = a2, we obtain:

Or:

These last two equations apply to the elastic contact of any axis-symmetric indenter (such as a sphere, cone, cylindrical punch) and allow the combined elastic modulus of the indenter and the specimen to be calculated from the contact stiffness and the contact radius. The contact stiffness can be measured experimentally. The contact radius can be found from the depth of the circle of contact hc and the geometry of the indenter.



R

a

P Surface flaws

Cone

crack

1.2.5 Hertzian Fracture

Brittle

Ductile

Maximum tensile stress

0 1 2 3 4r/a

-2

-1.5

-1

-0.5

0

0.5

r/

p m

In a brittle material, surface cracks extend under the influence of surface radial stresses (maximum at the edge of the circle of contact) and form a cone crack. Due to the diminishing nature of the stress field, the crack stops at a certain depth and is stable.

In a ductile material, plasticity occurs at the location of greatest shear stress and a plastic zone forms.

Plastic zone

0 1 2 3 4

-4

-3

-2

-1

0

z/a

Maximum shear stress



1.2.6 Plastic Zone

Plasticity usually occurs as a result of the initiation of slip under shear. The maximum shear stress in the elastic contact stress field for a spherical indenter occurs beneath the surface on the axis of symmetry and this is where the plastic zone begins to develop. When the load is increased, the plastic zone grows in size until the size of the plastic zone scales with the geometry of the contact and the mean contact pressure approaches a constant value.

p m

a/R

H

1.

2. 3.

Mea

n co

ntac

t pr

essu

re

Indentation strain

Spherical indenter

R

a

1.

2.

3.

Plastic zone

pm < 1.1Y — full elastic response, no permanent or residual impression left in the test specimen after removal of load.

1.1Y < pm < CY — plastic deformation exists beneath the surface but is constrained by the surrounding elastic material, where C is a constant whose value depends on the material and the indenter geometry.

pm = CY — plastic region extends to the surface of the specimen and continues to grow in size such that the indentation contact area increases at a rate that gives little or no increase in the mean contact pressure for further increases in indenter load.



1.2.7 Constraint Factor

The indentation test can be used to measure the hardness of what are nominally brittle materials because of the confined nature of the indentation stress field. The stresses beneath the indenter are confined by elastic strains in the surrounding material, and this enables the shear stress to reach a level at which plastic flow may occur without fracturing the specimen (as would happen in uniaxial tension or compression test). The constraint factor relates the hardness to the yield stress of the material.

CYH

Tabor showed in the 1950's that C 3 for most metals (high value of E/H). Experiments show that C < 3 for brittle materials (e.g. ceramics – low value E/H). Some feeling for the value of C can be found from Johnson's formulation of the expanding cavity model. In this well-known model, particles within the specimen beneath the indenter are assumed to form part of a hydrostatic core of material which pushes outwards onto the surrounding elastic matrix.

16

214cotln2

3

2 YE

Y

pm

where we let the mean pressure be equal to the hardness H. For example, if Eand H are known, then Y can be estimated from the above equation. Such a formulation assumes that the plastic zone is hemispherical. This is usually true for high value of E/H. For low values of E/H, the shape of the plastic zone becomes more spherical.

The mean contact pressure is related to the core pressure and is expressed:

plastic

elastic

Hydrostatic core

P



1.2.8 Hardness Measurements

Meyer

Brinell

Vickers

Knoop

22

2

dDDD

PBHN

228544.1

2

136sin

2

d

P

d

PHV

2

130tan

2

5.172cot

2

2d

PKHN

A

PH

Actual area

Actual area

Projected area

Projected area

HHV 094495.0 P = kgfd = Length of diagonal mm1 kgf = 9.806 N

Universal (Martens)

h 426

F =

h

F = HM

23.cossin4 22 Vickers indenter

h

P= HM

costan33 2Berkovich indenter

Actual area

d = Length of long diagonal of residual impression

D = Diameter of indenterd = diameter of residual

impression

h measured from specimen free surface



1.3 Theoretical Analysis


h

ha

P

hmax

dPdh

Pmax

he

hc

hr

Analysis of the load-displacement curve allows calculation of elastic modulus and hardness without a direct measurement of the contact area. It is the elastic unloading response that is the basis of the technique.


loaded

1.3.1 Load-Displacement Curve

Nanoindentation is conventionally performed using instrumented indentation whereby the load and the indenter displacement are recorded during the indentation process. Both loading and unloading responses are recorded in the form of a load-displacement curve.

Although most nanoindenters are load-controlled machines, it is conventional practice to plot the load on the vertical axis and the displacement on the horizontal axis. The load displacement curve must be corrected for contact depth determination, instrument compliance, and indenter tip shape (area function) before during analysis for the determination of E and H.

0 0.2 0.4 0.6 0.8

h (m)

0

10

20

30

40

50

P (

mN

)

Power law fitting:B: 193.1hr: 0.308m: 1.278Unloading analysis:hi : 0.0080 mPmax : 49.7 mNhmax : 0.656 mhc : 0.441 mdP/dh : 183.9 mN/ma : 1.233 mA : 4.774 m2H : 10.42 GPaE*: 72.3 GPaE : 75.3 GPa

hmax

hehr

hahc

Load-displacement curve for Berkovich indenter on fused silica. The curve shown has been corrected for initial penetration and instrument compliance.

unloaded

he

hr

a

hmaxhc

ha

Elastic-plastic loading

Elastic unloading



The load-displacement curve is used to determine the depth of contact by using the elastic unloading data (even if the contact involves plastic deformation). The actual indenter is conveniently modelled as an equivalent conical indenter. The equations of contact are:

a raa

rh

' cot2

'tan2 2*

ehEP

At r = 0, h = he:

At r = a, h = ha:

'cot2

ahe

'cot12

aha

ea hh

2

and: act hhh

ect hhh

2

hc

A

Cone

a

3.70

Berkovich

hc

A

3.65

unloaded

loadedhe

hr

a

hmaxhc

ha

From which we obtain:

Thus:

'tan2

2 *

ehEdh

dP

dhdPPh

hdh

dPP

e

e

22

1

(1)

From Eqn. 1:

dhdP

Phh c

22

max

A

PH

Adh

dPE

aE

aEdh

dP

ah

hEdh

dP

hEP

e

e

e

2

1

2

2

22

cot2

tan2

2

tan2

*

*

*

*

2*

Calculation of E*

1.3.2 Theoretical Analysis

Thus:

' is the combined angle of the cone and the sides of the residual impression

hmax, Pmax and dP/dh are all obtained from measurements and so hc can be determined.

Calculation of H

2

22

5.24

tan33

c

c

h

hA



1.3.3 Oliver and Pharr Method

Two of the most important physical properties of materials are elastic modulus and hardness. In nanoindentation, the depth of penetration of a diamond indenter is measured along with the prescribed load. The resulting load-displacement response typically shows an elastic-plastic loading followed by an elastic unloading. The elastic equations of contact are then used in conjunction with the unloading data to determine the elastic modulus and hardness of the specimen material.

Berkovich geometry

hs

he

hr

hmaxhc

a

A

PH Hardness is calculated from:

The three-sided Berkovich indenter is the most popular choice of geometry for nanoindentation testing. The contact depth (as measured from the total depth of penetration) is given by:

dhdP

Phhc

maxmax

Elastic modulus is given by:Adh

dPE

2

1*

The factor depends upon the indenter shape. Once hc is known, the area of contact A is found from the geometry of the indenter.

ha

P

h

dPdh

Pt

he

hc

hr

hmax


Elastic unloading

Residual impression of Berkovich indenter in hardened steel.

Oliver and Pharr method



1.3.4 Field and Swain Method

The Oliver and Pharr method uses the slope of the tangent to the unloading data at maximum load in conjunction with the derivative of the elastic equations of contact for an equivalent conical indenter to determine the depth of the circle of contact. An alternative method uses the unloading data directly with the elastic equations of contact and this method was first proposed by Field and Swain for spherical indenters although the method can also be used with other indenter shapes.

he

a

hc

R

hmax

hr

loaded

unloadedha = he/2

a ra

r

a

P

Eh

21

4

312

2

*

At r = 0, h = he: At r = a, h = ha:

thus:

a

P

Ehe 4

31*

ea hh

2

1and ac hhh max

2321*

3

4ehREP

2

1

4

31* a

P

Eha

The Hertzian elastic equations of contact for a sphere are:

Elastic unloading


h

Pt

hehr

hc

P

0.5he

Ps

ht

ha

2

2

2

rtc

rtt

et

atc

hhh

hhh

hh

hhh

1

1

4

3

1

4

3

32

32

3

2

323

2

*

323

2

*

31

31

st

tstsr

s

t

rs

rt

srs

trte

PP

hPPhh

P

P

hh

hh

PRE

hh

PRE

hhh

For Pt

For Ps

Take the ratio

Solve for hr

Once we have hc, then H and E* can be determined as before

Take two points at loads Pt and Ps

Now:

hr can be found from experimental quantities.

Thus:



1.3.5 Indenter of Arbitrary Shape

The form of the unloading slope, for a pyramidal (or equivalent cone) indenter is theoretically expected to be a square law in accordance with Sneddon's equations of elastic contact. However, this assumption relies on the contact between a conical indenter and a conical impression (the residual impression) in which the sides of the residual impression are straight. Experiment shows that the sides of the residual impression are actually curved upwards and this gives the unloading slope a power law dependence according to:

mre hhCP

nBrz

nn

nh

n

n

n

n

B

EP 11

1

1

*

12

212

1

2

nm

11

nn

e n

n

n

n

C

EB

1*

12

212

1

21

hdh

dP

mP

1

n

n

n

n

1

12

21211

ea hm

h

dhdP

Phhc

maxmax

The elastic contact is thus seen to take place between an indenter of a smooth arbitrary shape and a flat half-space (i.e. the shape of the residual impression is transferred conceptually to the indenter). The indenter is thus described as:

The elastic equation of contact is expressed:

(1) = 1, (3/2) = 1/21/2, (2) = 1, (1/2) = 1/2

Differentiating (1) with respect to h, we obtain:

Letting we can show that in (1):

Since dP/dh = 2Ea, and ha = he – hc, the parameter can be expressed

where h = he

Letting:

We have the familiar equation: where evaluates to approximately 0.75.

nBrz

Brz



1.3.6 Energy Method

The area under the unloading curve represents the release in elastic strain energy (i.e. energy recovered) Ue. The area enclosed by the load-displacement curve represents the energy lost to plastic deformation Up. The sum of the two is the total energy of the penetration Ut.

2hCP p

mre hhCP max

mrep hhChCP max2

maxmax

3

3maxh

CU pt

1

1max

max

m

hhCdhhhCU

mr

e

h

h

mree

r

13

1max

3max

m

hhC

hCUUU

mr

ep

etp

max

11

31

h

h

mU

Ur

t

p

27.027.1max

h

h

U

Ur

t

p

t

e

U

U

E

H

*

max

22*

4Pdh

dP

H

E

max

222

max

2*

222

max2

1

4

1

4

41

PU

U

dh

dPH

PU

U

dh

dPE

U

U

dP

dhPA

t

e

t

e

e

t

The elastic-plastic loading follows a square law:

The shape of the unloading curve follows a power law with index m.

At maximum load, the two curves meet, so:

Thus, the total energy can be expressed:

The elastically recovered energy is given by the integral:

and so:

Taking the ratio

Experiments and modelling show that: m = 1.36

and also: where 5.3 for a conical indenter of half-angle 70.3

A

PH

dh

dP

AE

2

1*since

dP/dh, Ue, Ut, and Pmax

are all experimental values



1.3.7 Theoretical Modelling

2

1

2max

tan33

H

Phc

*max

2E

HPha

*2maxmax

2tan33

1

E

H

HPh

rrs

s hhhP

Ph

max

21

max

cr hhh

221 max

One of the more interesting aspects of the equations of contact is the ability of predict the shape of the load-displacement curve, even for elastic-plastic deformation. Assuming a fully plastic contact, where H = P/A, we have working backwards:

The tangent to the unloading curve gives a triangle with base ha/ and height Pmax and so:

and so:ca hhh max

Elastic-plastic loading curve.

The unloading curve can be found by considering an arbitrary unloading point hs, Ps and forming the ratio Ps/Pmax:

where:

The equations above assume a square-law dependence on the elastic unloading. A power law dependence can easily be incorporated. The unloading curve for a power law is found in a similar manner to the above where the residual depth hr is now expressed:

cr hm

hm

h

max1

nm

11where

With a suitable choice for m, the power law method gives a much closer agreement to the experimental results for the unloading than the standard square law treatment.



1.3.8 Finite Element Analysis

P

A very powerful technique in modelling elastic-plastic contacts is the finite-element method. Due to the number of elements involved, it is usual to model non-axis-symmetric indenters by an axis-symmetric equivalent cone. The chief advantage of the method is for the determination of the load-displacement curve and also the interior stresses for thin film systems where analytical solutions are not available.

To obtain a result with the method, it is necessary to enter the elastic-plastic properties of the specimen material. This is usually done in the form of a uniaxial stress-strain curve. An elastic, perfectly-plastic (bi-linear) approach works well, but strain-hardening can easily be incorporated as desired.

It is necessary to have a fore-knowledge of the material's modulus and yield stress to define the stress-strain curve. These values in turn are used in conjunction with the selected failure criterion (usually Tresca or von Mises) to model plastic flow. Usually, it is also possible to specify a friction coefficient for the contact elements between the indenter and the specimen.

Y

E



1.3.9 Creep

Many materials such as metals and plastics exhibit creep under steady load conditions. In an indentation test, creep often manifests itself as a bowing out or “nose” in the unloading portion of the load-displacement curve. This makes it impossible to obtain a measurement of hardness and modulus of the material since the slope of the unloading response, which is ultimately used to determine the contact area, is affected by creep in the material.

The most common method of measuring creep is to apply a constant load to the indenter and measure the change in depth of the indenter as a function of time. The resulting “creep curve” can then be analysed using conventional spring and dashpot mechanical models.

E*1

E*2

E*1

E*2

2

E*1

1

tE

Pth o11

cot2 1

*2

2*

111

cot2 2

*1

*2

Et

o eEE

Pth

te

EEPth

Et

o12

*1

*2 1

111

cot2

2

2*

P

P

P

The higher the number of adjustable parameters the better the fit to the data but the longer and more unstable is the convergence to a solution.



1.3.10 Dynamic Nanoindentation

The quantity dP/dh quantity can also be measured by applying a sinusoidal oscillation to the indenter shaft. The advantage of this is that measurements of dP/dh can be obtained more quickly (e.g. during the loading sequence) and also, the storage and loss modulus of the specimen material may be measured.

tio

tio

ehth

ePtP

2222 mSh

PTF

o

o2

tan

mS

himS

dt

hdm

dt

dhShP

2

2

2

sin"

cos' 2

o

oim

o

ore

h

PTFG

mSh

PTFG

reo

o TFAh

P

AE

2cos

2*

imo

o TFAh

P

AE

2sin

2"

'

"tan

E

E

"'* iGGG

'

";"

'

"'*

GG

i

S

m

The relationship between P and h is called the transfer function. The magnitude and phase angle of the transfer function are given by:

The real and imaginary parts of the transfer function represent the storage G' and loss G" modulus of the system.

For a simple spring, the storage modulus corresponds to the spring stiffness. In an indentation test, the spring is "nonlinear" and the reduced or combined modulus E* must be obtained from the storage modulus through the equations of contact:

The real and imaginary parts of the transfer function represent the storage G' and loss G" modulus of the system.

Viewed from the point of view of viscosity, the real part of the complex viscosity (the dynamic viscosity) represents the fluid nature of the system the imaginary component is representative of the elastic response.

Complex modulus



1.3.11 Multiple-Frequency Nanoindentation

Multiple-frequency nanoindentation testing is a new technique which is similar to presently available dynamic indentation test methods but with one important difference. In conventional methods, a single sinusoidal frequency of oscillation is applied to the indenter shaft and the resulting storage and loss moduli are calculated from the dynamic load and displacement data. In the multiple-frequency mode of operation, a superposition of frequencies is employed and a Fourier analysis is used to separate out the frequency dependent storage and loss moduli, and also the complex viscosity, of the specimen material in the one indentation test.

t

P(t)

Conventional dynamic nanoindentation single-frequency drive signal

Multiple-frequency dynamic nanoindentation drive signal

50

t

P(t)

thF

tPFTF

The method relies on the fact that the transfer function between load and displacement is given by the ratio of the Fourier transform of the component signals.

ferenceInstrument

InstrumentSpecimenEq TFTF

TFTFTF

Re

ferenceEqSpecimen TFTFTF Re

The Fourier transforms F(P(t)) and F(h(t)) of the experimental quantities P and hare themselves complex quantities, the real part of the ratio of them thus gives the storage and the imaginary part the loss modulus of the system.

The actual specimen properties can be separated from the combined specimen/instrument dynamic properties by the use of a reference spectrum whose transfer function TFReference is known. The force and displacement signals obtained when testing a specimen contain a convolution of the force and displacement responses in the time domain of the specimen and the instrument. Convolutions in the time domain become multiplications in the frequency domain, thus:

and so: gives TF of the specimen alone.



One of the most important influences on the results from nanoindentation testing is the presence of residual stress in the specimen surface. This usually occurs in thin films, but may also result from grinding and polishing of bulk specimens. It is generally observed that the contact stiffness increases with increasing compressive residual stress but this trend is not observed in all materials. While the presence of residual stress may affect the values of E and H determined with the usual methods of analysis, the effect is very much reduced if the area of contact is measured independently (such as with an AFM).

2

*

16.1 c

PP oR

Chaudhri and Phillips (1990)

Measurement of size of median cracks.

Load for same size crack in annealed glass

Measurement of difference in loads to produce the same depth of penetration for stressed and stress-free states.

A

PP os

*Re

Lee and Kwon (2002)

Load for same size crack in stressed glass

R

aEYR

3.3

4 *

Measurement of change in yield point in the presence of residual stress.

In some cases, nanoindentation may be used to estimate the value of residual stress although it is acknowledged that no one method is suitable for all applications.

Taljat and Pharr (2000)

Indentation strain at onset of yielding

of

s

s

sf RRt

tE 11

61

2

Wafer curvature method (Stoney equation)

Stoney (1909)Compressive stress

Tensile stress

Initial radius of curvature

1.3.12 Residual Stress



Part 2Practical

Nanoindentation2.1 Nanoindentation Testing

2.2 Applying Corrections2.3Analysing the Data



2.1 Nanoindentation Testing

Nanoindentation test instruments have been on the market for about 15 years yet only now is the technique becoming widespread. A typical instrument consists of a load frame, indenter shaft, load actuator, force and displacement sensors, XY positioning table and optical microscope for precise positioning.



2.1.1 Specimen Preparation

One of the largest influence on the validity or quality of nanoindentation test data is the condition of the surface of the specimen and the way in which it is mounted for testing.

Cleaning and polishing will of course influence the final value of the surface roughness of the specimen. A typical indentation into a bulk material is usually of the order of 200-500 nm. The theoretical basis of the contact equations assumes a perfectly flat surface so any irregularity in the surface profile will cause scatter in the readings. The table below shows the maximum penetration depth and the modulus and hardness calculated from tests on optical grade fused silica, and an as-deposited TiN-BN film. The increased scatter in the result for the film is almost entirely due to the greater amount of surface roughness.

hmax (um) E(s) H0.653 71.60 9.6600.654 71.80 9.7400.648 72.50 9.9900.652 70.60 10.000.650 72.50 9.7700.652 70.70 9.990

hmax (um) E (GPa) H (GPa)0.439 371.0 40.900.429 450.0 44.200.414 500.0 47.800.421 461.0 45.300.424 579.0 62.40

Fused silica TiN-BN

10 m 10 m

Surface roughness can be quantified by the parameter :

2a

Rs

Indenter radius

Maximum asperity height

Contact radius

< 0.05 for minimum effect on validity of contact equations



2.1.2 Indenters

Indenters for nanoindentation testing are usually made from diamond which has been ground to shape and sintered in a stainless steel chuck. These indenters are useable in air up to approximately 700 C.

Berkovich indenter

Vickers indenter

Knoop indenter


Sphero-Conical indenter

= 65.27

= 68

1 = 86.25, 2 = 65

= 35.26 = 42.278

= 70.296

= 70.3

= 77.64

Cone angle 60 or 90. Tip radius 0.7, 1, 2, 5, 10, 20, 50, 100, 200 m.

Typical radius when new 150 nm. Typical radius after 12 months use 250 nm.

Line of conjunction at tip limits the sharpness of tip for determination of hardness for very shallow indentations.

Notes

Originally designed for hard metals but can be used to probe anisotropy in specimen surface.

Designed to provide greater amount of deformation for fracture toughness determination from induced cracks. Very fragile and easily broken.



2.1.3 Nanoindentation Testing

Nanoindentation testing requires a judicial choice of indenter geometry and load as well as many other test variables such as number of data points, % unloading, maximum load, and the way in which the load is applied:

Constant load rate:

Constant depth rate:

Constant strain rate:

Cdt

dh

Cdt

dP

Chdt

dh

Pdt

dP

11

All nanoindentation is a trade off between the total penetration and:

• The effect of thermal drift. If maximum penetration depth is say 50 nm and thermal drift is 1 nm per second, then if the test takes one minute, error due to thermal drift will completely swamp the true penetration reading.

• Initial penetration. If the initial penetration depth is too large a fraction of the total penetration then the estimation of the initial penetration must be done extremely well. As well, if the initial penetration is too large, one may penetrate the film being tested before the tests begins and so the substrate properties will be measured instead.

• The thickness of the sample and influence from substrate. If the substrate is soft (low H), then plasticity may occur in the substrate before the film. If the substrate is compliant (low E) then the value of modulus for film may be too much influenced by deformation of the substrate.

• Indenter tip radius. If radius is too large, then may not achieve a fully-developed plastic zone. If specimen is soft, then plastic zone may form but may extend into substrate. Objective is to have fully-developed plastic zone in the material whose properties are to be measured.

• Surface roughness. If the surface roughness is too high relative to the maximum penetration and also tip radius, then the assumption of a semi-infinite specimen embodied in the contact equations will be invalid.

• Creep. If the specimen material exhibits creep, then the displacements recorded will contain both creep and penetration. It may be possible to wait at maximum load for creep to subside before unloading. If creep continues, then it is generally not possible to do a static indentation test.



Any indentation will result in some influence from the substrate since the elastic deflections of both the substrate and the film contribute to support the indenter load. However, because of the localized nature of the indentation stress fields, more support comes from the film than the substrate. Hence, the 10% rule does not apply to modulus determinations although it is a reasonable place to begin. It is best to perform a series of indentations from a very low load to a reasonably high load and then plotting modulus vs indentation depth. Extrapolation of the data back to zero depth should result in a value of modulus for the film only.

2.1.4 The 10% Rule and Thin Films

Hardness

Modulus

0 0.2 0.4 0.6 0.8 1

ht (m)

200

300

400

500

600

E (

GP

a)

ES

0 0.2 0.4 0.6 0.8 1

ht (m)

20

30

40

50

60H

(G

Pa)

tHF

HS(b)

(a)

EF

ES

t

For measurement of hardness, the objective is to obtain a fully developed plastic zone within the film. Depending on the film/substrate properties and the radius of the indenter and the thickness of the film, this may not be possible. Usually, a plot of H vs penetration depth is made and if there is a plateau in the values of H, then this is taken as the hardness of the film only. It is generally observed that an influence from the substrate is negligible if the penetration depth is less than 10% of the film thickness.

(a) Film on hard substrate(b) Film on soft substrate

(a)

(b)

Hf

Ef



2.1.5 Step-Height Measurement

(1) (2)

d

h

An inspection of the data shows a total slope of 834 nm over 450 um of distance which is 1.8 nm/um. The distance between data point 003 and 004 is 50 um, of which 0.0018*50 = 92 nm is due to specimen slope. The corrected step height is thus 263 – 92 = 171 nm.

The displacement sensitivity and the positional accuracy of a typical nanoindenter can be used to determine the height of a step, such as the thickness of a thin film. In this type of test, a series of very low load indentations is made at regular spacing intervals that span the step. The height readings, when corrected for specimen slope, allow the step height to be computed.

Copper film (left) on glass (right)

To collect meaningful data for this type of test, the instrument used must have a means of zeroing or taring the depth sensor at the first indentation and thereafter not re-zeroing for each subsequent indentation. The recorded depths thus are referenced to the initial contact. Since this test might take several minutes, it is important to make sure there is no thermal drift.

The following results were collected on the film shown above:

The above example is a series of static indentation tests. Similar data may be obtained by a "scratch" method whereby displacement readings are taken as the sample is translated under the indenter with the indenter in contact with the surface at a small contact force. In this method, force feedback is required to ensure that the force applied to the indenter remains constant during the translation.



0 -20 -40 -60 -80

Lateral Distance (um)

0

0.02

0.04

0.06

0.08

0.1

Fric

tion

2.1.6 Scratch Testing

Depending on the instrument capabilities, scratch testing can be done by moving the sample relative to the indenter tip while load is applied (either ramped or steady) to the indenter. At the same time, the lateral force needed to move the sample is typically measured. The ratio of the lateral force to the normal force is the coefficient of friction at the contact.

The results for a scratch test depend very much on the test variables and the sample. Typically, it may take several tests (perhaps 10 or more) to establish a suitable test regime. The AFM scan across the scratch above shows piling-up of material at the edge and characteristic tensile cracking as arcs opening in the direction of travel of the indenter.

A constant load scratch test may be done as a series of scratches at ever-increasing loads at regularly spaced intervals.

When the scratch is repeated along the same path, then it is a wear test. For scratches on thin films, delamination can be observed by visual inspection and the load at that position along the scratch is the critical load for failure of the film. This is a form of adhesion testing. The scratch resistance or scratch hardness of a surface is a measure of its ability to resist permanent deformation by a moving indenter (as opposed to a stationary indenter in indentation hardness).

Parallel scratches on gold film on glass with 5 um spherical indenter with loads 5, 10, 15, 20… mN at 5 m intervals.

75 m scratch made with a 20 m radius sphero-conical indenter with a ramped load (1 to 30 mN) on a bond pad inside an integrated circuit. The scratch speed was 5 m/sec.



2.1.7 Fracture Toughness

One of the less well-known applications of indentation testing is the determination of fracture toughness. The procedure has some advantages over more conventional methods since the test can be performed on the surface and does not require large-scale specimen preparation and pre-cracks or notches to be introduced. The method is particularly suitable for brittle materials.

l

c

a

The method relies upon an optical measurement of crack size. Cracks usually appear at the corners of the residual impression in brittle materials.

2/3

3/22/1

1 015.0073.1c

P

H

ElaK

2/3

3/22/1

1 015.0c

P

H

ElaK

Radial crack pattern from Berkovich indenter

Vickers indenter

Berkovich indenter

M.T. Laugier, J. Mater. Sci. Lett. 6, 1987, 897.

R. Dukino and M.V. Swain, J. Am. Ceram. Soc. 75 12, 1992, 3299.

2/3

2/1

1 036.0c

P

H

EK


D. S. Harding, W. C. Oliver and G. M. Pharr: Mat. Res. Soc. Symp. Proc., 356, 1995, 663.

The figure shows an indentation made in fused silica with a cube corner indenter at 50 mN.

4.07 m

al

21

236

62

1

2/3

2/1

1

m MPa 60.0

1007.4

1005.0

5.9

5.72036.0

036.0

c

P

H

EK

10 m



2.2 Applying Corrections

One of the most important corrections to apply to nanoindentation test data is to correct for non-ideal tip shape. The picture shows a nominal 1 um radius sphero-conical indenter. It is evident that the tip is not a perfect sphere.



2.2.1 Thermal Drift

Thermal drift refers to the variations in the depth measurement signal resulting from thermal expansion or contraction of the sample or the indenter during an indentation test. These variations, unless minimised or corrected for, will be interpreted at real displacements of the indenter into the specimen and so have an adverse effect on the accuracy of the results.

tThh d'

Correcteddepth

Measured depth (from instrument)

Drift rate um/sec

The displacement sensor of a typical nanoindenter has a very high resolution and a very small range. nm scale changes in physical dimensions of the indenter and specimen can readily be detected. For this reason, one cannot simply handle a specimen or an indenter and then perform an indentation test. A waiting period of at least 20 minutes is required for everything to reach thermal equilibrium.

Thermal drift, expressed as nm/sec, is usually increasing, decreasing but can also be oscillating. It may be measured by indenting the specimen with a known, constant force and monitoring the displacement signal.

If the thermal drift rate is a constant, then it is possible to correct experimental data by the drift rate to compensate. However, the thermal drift rate is usually exponential in character and this correction procedure will generally not be satisfactory.

Sample/tip is heating up (expansion) so that the indenter shaft is being pushed upwards (more negative) hence a reduction in penetration depth. Long term variations are due to thermal drift. Local variations (arrow) are due to mechanical disturbance.

Example of displacement reading for condition of no thermal drift.

If possible, it is best to wait until thermal equilibrium is established before beginning the test rather than try to correct for thermal drift afterwards.



2.2.2 Contact Determination

One of the most important events in a nanoindentation test is the point of initial contact of the indenter and the specimen. It is this position of the indenter that is the reference, or "zero" datum for all the depth measurements. There are a variety of methods used to determine the point of contact. Some instruments approach the surface and wait until a very low contact force is registered. At this point, the depth sensor is bought to zero or tared, and then the test proceeds. If this taring operation is done properly, then it is possible to utilise the full resolution of the analog to digital converter in the instrument for the subsequent depth measurements.

One must recognise, however, that any contact with the specimen, no matter how small, will result in an initial penetration (perhaps only a few nm) that will not be accounted for since it is tared off.

To correct for this, a smooth curve is fitted to the initial data and extrapolated back to the zero force position. The initial penetration hi is then taken as the displacement offset and is added on to all the raw displacement data to give the total indenter penetration depth.

hi

(Pi,0)

ihhh '

Correcteddepth


Initial depth hi

The initial penetration depth correction has the effect of shifting the load displacement data to the right (since a constant value of hi is added onto all the depth measurements). For testing very thin films, the initial contact force Pi has to be made as small as possible so that the initial penetration depth correction is the smallest possible proportion of the total penetration depth. This initial penetration often limits the thickness of film that can be tested with a nanoindenter. Is this correction important? It is if your total penetration depth is less than say 200 nm.

For an initial contact force of 0.015 mN, the initial penetration was estimated to be 4.3 nm. (Berkovich indenter, fused silica)



2.2.3 Instrument Compliance

In a nanoindenter, at any application of load to the indenter, the reaction force is taken up by deflection of the load frame and may be registered by the depth sensor. If not accounted for, then an error proportional to the load will be introduced into the displacement readings. The compliance correction seeks to minimise this error by correcting the displacement readings by the deflection of the load frame. It is assumed that the compliance of the load frame is known beforehand.

PCc f

Specimen contact stiffness S = dP/dh

1/Cf

fCSdP

dh

1

PChh f'

Correcteddepth


Deflection of load frame under load P

The compliance correction has the effect of shifting the load-displacement data to the left an amount which depends upon the value of the load at each data point.

Spherical indenter Berkovich indenter

fci

ChREdP

dh

2121*

1

2

1f

c

ChEdP

dh

1

2

1

5.24 *

The compliance of a nanoindenter can be estimated by direct measurement, or by performing a series of tests at increasing loads on a standard specimen. In this case, the compliance can be expressed:

dh/d

P

1/hc

Cf = 6 10-4 m/mN



2.2.4 Indenter Shape Correction

The equations of contact that are used to determine the contact stiffness and hence the contact area assume that the geometry of the indenter is ideal. In practice, this is not the case, especially at the tip of the indenter where there is a radius instead of an infinitely sharp tip.

A

A

A

PH i

A

A

Adh

dPE i

2*

The correction for indenter shape takes the form of a ratio of the actual area over the ideal area A/Ai. The correction applies as the square root for the modulus and is directly proportional in the case of the hardness.

An area correction function is usually obtained by performing indentations on a specimen of known modulus (in some cases, hardness is used as the reference). The most commonly used reference material is fused silica. However, in some cases it is beneficial to use a reference material with a value of E/H that is similar to the proposed test specimen. Doing so may also compensate errors in the results from piling-up in the material.

Usually, a Berkovich indenter is blunter rather than sharper than the ideal geometry which means at a given contact depth hc, the actual area of contact Ais larger than that calculated upon the basis of a perfect geometry Ai.

The area correction is the perhaps the single most important correction which will have the greatest effect on the accuracy of the final results. The correction becomes less important as the penetration depth increases since most indenters are usually made with a very precise face angle. At smaller penetration depths, the area ratio may be as high as 50:1 due to tip rounding.

Area function approaches 1 at larger contact depths

Area function rises steeply at smaller contact depths due to tip rounding.

iA

A

ch

A

hc

hc

Ai

a



2.2.5 Piling-up

The elastic equations of contact assume that the contact circle is beneath the specimen surface (the surface "sinks-in"). Depending on the ratio of E/H of the specimen material, instead of sinking-in, a material may be pushed upwards and be "piled-up" around the edges of the indentation. When this happens, more material is supporting the indenter load than is assumed by the contact equations. As a result, the specimen appears (from the point of view of the equations) stiffer (higher modulus) and harder than it really is.

hchhc

Sinking-in Piling-upa

When piling-up occurs, the depth of penetration for a given contact area is less than what it would be (due to the extra support). Since the equations take the depth of penetration as the input, then the calculated contact area is less than it really is and so H = P/A is over-estimated. Similarly, the modulus E* is also over estimated since:

2

1*

Adh

dPE

There is no one method for accounting for piling-up. The best way to avoid the effect is to measure the contact area with an AFM or SEM. However, there are other methods of instrumented data analysis emerging that appear not to be so influenced by the effect.

Piling-up is not restricted to ductile materials. The figure shown here shows clear evidence of piling up in fused silica when loaded with a 70.3conical indenter at 100 mN.



2.3 Analysing the Data

Analysing nanoindentation data is a specialist activity with many corrections and fitting procedures to choose from. Interpretation of the results also requires care, especially when testing structures such as thin films.



2.3.1 Fitting the Unloading Curve

The tangent to the unloading curve at Pmax is drawn and extended downward to cross the h axis. The contact depth hc is then found from:

dhdP

Phh c

maxmax

0 0.2 0.4 0.6 0.8

0

10

20

30

40

50P

(m

N)

hmax

hc

To perform this procedure by computer, it is required to fit an equation to the unloading data and then take the derivative at Pmax

and form the equation of a straight line and determine the value of h where this line crosses the P axis.

What equation to fit?

1st choice: Power law fit to the unloading data.

mre hhCP where Ce, hr and m are unknowns. Determining these unknowns from the data requires an iterative procedure. Convergence may be difficult to achieve.

2nd choice: Polynomial fit (m = 2):

22 2 reree hChhChCP Usually provides a reasonable fit to the upper half of the unloading data. Little computational effort.

3rd choice: Linear fit (m = 1):

re hhCP The initial unloading data may be fairly linear and so a simple linear fit can be used. WARNING! If you use a linear fit to the upper unloading data, then the factor should still be set to 0.75 and not 1 for highly elastic materials (low value E/H).



2.3.2 Image Analysis

One of the most accurate ways to determine the contact area is to image the residual impression with a surface probe such as an AFM. The procedure is somewhat time-consuming and difficult, and the actual measurement of the area is not so straight-forward. After obtaining the required image, one of the most difficult choices is to establish where the edges of the residual impression might be. For example, consider the image below (Berkovich indenter, 50 mN, fused silica) and the choices regarding the outline:

CAUTION: It is very important that the imaging device be calibrated. A typical AFM instrument is usually supplied with a rough calibration which may be correct to within 10-20%. Changing the tip or probe can alter the calibration. Errors in the linear dimension of the image are squared when determining the contact area. Calibration must be performed using a standard specimen, usually a photo-lithographic mask made to precise dimensions.

A is under-estimated Best estimate A is over-estimated

5 m

The example above shows the sensitivity of results on the variations in estimation of the contact area. Different operators may as well choose slightly different paths for the edges of the indentation. It is important to know that it is the contact area under load that is required, even if some of the material within that area fully recovers upon unloading. Some practitioners coat the surface with a thin layer of gold in the expectation of more reliably estimating where the contact actually took place.

2.795.6

50

95.6

H

A

9.1358.3

50

58.3

H

A

8.1060.4

50

60.4

H

Am2

GPa

m2

GPa

m2

GPa



2.3.3 Nanoindentation Results

dhdP

Phhc

maxmax

'

'111 22

* EEE

ccc RhhRhA 22 2

2

22

49.24

tan33

c

c

h

hA

By far the most popular and robust method of analysing nanoindentation data is the Oliver and Pharr method. In this method, the contact depth is found from:

and the area of contact A determined by the shape of the indenter:

Spherical indenter

Berkovich indenter

Including the area correction function A/Ai, we have:

A

A

A

PH i

A

A

Adh

dPE i

2* and

75.0

A very good test of the bona-fides of your test method and the instrument being used is to test three known samples of different values of E and H. Fused silica, silicon and sapphire are commonly used. Consistent results for both E and Hshould be obtained an all three specimens before quoting results for tests on specimens of unknown material properties. Reasonable values are given below:

Fused silica:

Silicon:

Sapphire:

E = 72.5 GPa, H = 8 – 10 GPa, = 0.17

E = 170 - 180 GPa, H = 10 – 12 GPa, = 0.28

E = 420-520 GPa, H = 25 - 30 GPa, = 0.25

Results should be very repeatable and fairly independent of depth although some rise in H may be observed at low loads due to partially developed plastic zone.

Results should be very repeatable and fairly independent of depth. Values for E should be > 170 GPa. Results can depend on presence of surface layer (eg. some samples are coated with SiN).

Results for hardness may increase at low loads with increasing penetration depth due to partially developed plastic zone. This may be offset by surface hardening from polishing. Values for H should be quite repeatable. Values for E can vary quite a lot due to anisotropy in crystalline properties.

E Diamond 1050 – 1150 GPa diamond 0.07



2.3.6 Do's and Don't's

It is hoped that this handbook provides a handy outline of the issues involved so that you may perform and present reliable, precise and noteworthy results using this very versatile testing technique.

Do

Don't

• Wait for thermal equilibrium before starting your test rather than to try and correct for thermal drift later.

• Be aware of the thickness of your sample and select an appropriate minimum and maximum load.

• Mount your sample firmly with the minimum of adhesive and ensure a good solid contact.

• Apply corrections, they are important, don't ignore them.

• Be aware that creep in the specimen, piling-up, indentation size effects, etc may affect your results.

• Calibrate your tip regularly, especially if you are not the only user. Who knows what damage a previous user has inflicted without notifying anyone.

• Remember to calculate E for the specimen from the reduced modulus E*

before quoting results for E.

• Make a brief mention of the corrections applied and the test conditions when you quote your results in the literature.

• Blindly analyse data without examining if the fit to the unloading is reasonable.

• Try to fit an equation to your indenter area function, better to capture local variations in indenter shape than try and impose an artificial mathematical function on it.

• Accept the manufacturer's values for instrument compliance implicitly, measure it yourself on a known sample so that you can get a feeling for the compliance of your sample mounting method.

• Rush. Nanoindentation requires time and patience.



Part 3More Information



There are a very small number of books available which provide essential and useful information for nanoindentation testing.

Nanoindentation (3rd Edition) A.C. Fischer-Cripps Springer-Verlag N.Y. 2007. Comprehensive reference book containing background theory, history, analysis techniques, testing techniques, test standards, review of instruments, frequently asked questions and more.

Introduction to Contact Mechanics (2nd Edition)A.C. Fischer-Cripps Springer-Verlag N.Y. 2000. Companion books to the above dealing with mechanical properties of materials, general fracture mechanics, and the fracture of brittle solids, and detailed description of indentation stress fields for both elastic and elastic-plastic contact. Brittle cracking, the meaning of hardness, and practical methods of indentation testing. This book is designed to make contact mechanics accessible to those entering the field for the first time.

3.1 Books

The Hardness of Metals, D. Tabor, Clarendon Press Oxford 1951. Long out of print, but a classic book worth getting hold of. There's always something new in this book.

Contact Mechanics, K.L. Johnson, Cambridge University Press, 1985. Classic must-have book which presents the field of contact mechanics clearly and covers an extensive range of applications.

Mechanical Behaviour of Materials, D. François, A. Pineau, and A. Zaoui, Kluwer Academic Publishers, The Netherlands, 1998.

Contact Problems in the Classical Theory of Elasticity, G.M.L. Gladwell, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1980.

Fracture of Brittle Solids B.R. Lawn,, 2nd Ed., Cambridge University Press, Cambridge, 1993.

Mechanics of Elastic Contacts D.A. Hills, D. Nowell, A. Sackfield, Butterworth Heinemann, Oxford, 1993.

Ceramic Hardness I.J. McColm, Plenum press, NY, 1990.

Intermolecular & Surface Forces IJ. Israelachvili, 2nd Ed. Academic Press, London, 1997 .

Fundamentals of Surface Mechanics with Applications F. Ling, Frederick, M.W. Lai, D.A. Lucca, 2nd Ed., Springer-Verlag NY, 2002



3.2 Literature

Approximately 50-60 papers are published each year in the scientific literature and presented at conferences which deal with nanoindentation testing. Some of the more important classic papers are listed below.

H. Hertz, “On the contact of elastic solids,” J. Reine Angew. Math. 92, 1881, pp. 156–171. Translated and reprinted in English in Hertz’s Miscellaneous Papers, Macmillan & Co., London, 1896, Ch. 5.

H. Hertz, “On hardness,” Verh. Ver. Beförderung Gewerbe Fleisses 61, 1882, p. 410. Translated and reprinted in English in Hertz’s Miscellaneous Papers, Macmillan & Co, London, 1896, Ch. 6.

M.T. Huber, “Contact of solid elastic bodies,” Ann. D. Physik, 14 1, 1904, pp. 153–163.E. Meyer, “Untersuchungen über Harteprufung und Harte,” Phys. Z. 9, 1908, pp. 66–74.

I.N. Sneddon, “Boussinesq‘s problem for a rigid cone,” Proc. Cambridge Philos. Soc. 44, 1948, pp. 492–507.

E.S. Berkovich, “Three-faceted diamond pyramid for micro-hardness testing,” Ind. Diamond Rev. 11 127, 1951, pp. 129–133.

K.L. Johnson, “The correlation of indentation experiments,” J. Mech. Phys. Sol. 18, 1970, pp. 115–126.

S.I. Bulychev, V.P. Alekhin, M. Kh. Shorshorov, and A.P. Ternorskii, “Determining Young’s modulus from the indenter penetration diagram,” Zavod. Lab. 41 9, 1975, pp. 11137–11140.

F. Frölich, P. Grau, and W. Grellmann, “Performance and analysis of recording microhardness tests,” Phys. Stat. Sol. (a), 42, 1977, pp. 79–89.

J.B. Pethica, “Microhardness tests with penetration depths less than ion implanted layer thickness in ion implantation into metals,” Third International Conference on Modification of Surface Properties of Metals by Ion-Implantation, Manchester, England, 23-26, 1981, V. Ashworth et al. eds., Pergammon Press, Oxford, 1982, pp. 147–157.

M.F. Doerner and W.D. Nix, “A method for interpreting the data from depth-sensing indentation instruments,” J. Mater. Res. 1 4, 1986, pp. 601–609.

J.S. Field, “Understanding the penetration resistance of modified surface layers,” Surface and Coatings Technology, 36, 1988, pp. 817–827.

W.C. Oliver and G.M. Pharr, “An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments,” J. Mater. Res. 7 4, 1992, pp. 1564–1583.

J.S. Field and M.V. Swain, “A simple predictive model for spherical indentation,” J. Mater. Res. 8 2, 1993, pp. 297–306.

A. Bolshakov and G.M. Pharr, “Understanding nanoindentation unloading curves,” J. Mater. Res. 17 10, 2002, pp. 2660–2671.



3.3 The IBIS nanoindentation system

Accurate, reliable, affordable.

Specifications

• Closed loop force/depth feedback

• Robust sensor design

• Easy indenter changeover

• Traceable calibration

• Reliable (3 year warranty)

Significant features

Accessories/Options• Lateral force (scratch)

• Finite element interface

• Fluid cell, AFM and more.

• Choice of staging

• Surface reference option

• Quasi-static nanoindentation

• Partial unload technique

• Multiple frequency dynamic testing

• Creep

• Scratch

• Batch (unattended) operation

Operating modes

Load0‐50 & 0‐500mN

Resolution 0.07 uN

Noise floor <1uN

Minimum load 5 uN

Depth 0‐2 um, 0‐20um

Resolution 0.003 nm

Noise floor 0.1 nm

Note: IBIS is a dual range instrument with automatic switching from low to high ranges on force during testing. Resolutions shown are for the lower range setting.

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