Czichos H., Saito T., Smith L.E. (Eds.) Handbook of Metrology and Testing

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378 Part C Materials Properties Measurement

•

The reproducibility of a hardness machine can be

thought of as how well the hardness values agree un-

der changing testing conditions. Inﬂuences such as

different machine operators and changes in the test

environment often affect the performance of a hard-

ness machine.

•

The ﬁnite resolution of devices that display the hard-

ness value or that measure the size of indentations

prevents the determination of an absolutely accurate

hardness value. Devices such as a dial display on

a Rockwell hardness machine or a portable hand-

scope for measuring Brinell hardness indents may

have a low resolution which can contribute a signif-

icant level of uncertainty to the hardness value.

•

When reported measurement values are based on the

average of multiple hardness tests, the nonunifor-

mity of the material under test contributes a com-

ponent of uncertainty in the hardness value, such as

would occur in the calibration of a reference block.

•

Reference test blocks provide the link to the hard-

ness standard to which traceability is claimed. All

reference hardness blocks should have a reported

uncertainty in the certiﬁed hardness value. This

uncertainty contributes to the measurement uncer-

tainty of hardness machines calibrated or veriﬁed

with the reference test blocks.

The general approach of this procedure is to calculate

a combined standard uncertainty u

by combining the

contributing components of uncertainty u

, u

,...,u

for the applicable sources of error discussed above, such

that



+···+u

. (7.26)

The appropriate contributing components of uncer-

tainty to combine for this calculation are dependent

on what the uncertainty value represents. For example,

a combined uncertainty may be calculated for a single

hardness value, an average of multiple hardness values,

a hardness machine’s error determined as part of an in-

direct veriﬁcation, or the certiﬁed value resulting from

a reference block calibration.

Measurement uncertainty is usually expressed and

reported as a combined expanded uncertainty U

,which

is calculated by multiplying the combined standard un-

certainty u

by a numerical coverage factor k, such that

=ku

. (7.27)

A coverage factor is chosen that depends on how well

the standard uncertainty was estimated (number of mea-

surements), and the level of uncertainty that is desired.

The measurement bias b of the hardness machine is the

difference between the expected hardness measurement

results and the true hardness of a material. When test

systems are not corrected for measurement bias, the bias

then contributes to the overall uncertainty in a measure-

ment. Ideally, measurement biases should be corrected;

however, in practice, this is commonly not done for tra-

ditional types of hardness tests. There are a number of

possible methods for incorporating uncorrected biases

into an uncertainty calculation, each of which has both

advantages and disadvantages. A simple and conserva-

tive method is to combine the bias with the expanded

uncertainty as

U = ku

, (7.28)

where |b| is the absolute value of the bias.

Discussions of the calculations for each of the un-

certainty components, which uncertainty components

should be combined, and how they should be combined

for all types of hardness measurements are too involved

to be described in detail here. Guidance on this ap-

proach can be found in newly developed recommended

procedures recently added to the international hardness

test method standards of ISO and ASTM for the speciﬁc

conventional hardness tests.

Reporting Uncertainty

Ideally, an individual measurement uncertainty should

be determined for each hardness scale and hardness

level of interest since the contributing components of

uncertainty may vary depending on the scale and hard-

ness level. In practice, this may not be practical. In

many cases, a single uncertainty value may be applied

to a range of hardness levels based on the laboratory’s

experience and knowledge of the operation of the hard-

ness machine. Also, because several approaches may be

used to evaluate and express measurement uncertainty,

a brief description of what the reported uncertainty

value represents should be included with the reported

uncertainty value.

7.3.4 Instrumented Indentation Test (IIT)

Principle of the Instrumented Indentation Test

[7.183]

The continuous monitoring of the force and the depth

of indentation can allow the determination of hardness

values equivalent to traditional hardness values. More

signiﬁcantly, additional properties of the material, such

as its indentation modulus and elasto–plastic hardness,

can also be determined. All these values can be calcu-

Part C 7.3

Mechanical Properties 7.3 Hardness 379

max

Fig. 7.32 Schematic representation of test procedure (Key:

1 – Application of the test force; 2 – Removal of the test

force; 3 – Tangent to curve 2 at F

max

)

lated without the need to measure the indent optically

(Figs. 7.32, 7.33). An indenter consisting of a material

harder than the material tested with the following shapes

can be used.

1. Diamond indenter shaped as an orthogonal pyra-

mid with a square base and with an angle α =136

◦

between the opposite faces at the vertex (Vickers

pyramid)

2. Diamond pyramid with triangular base (e.g.,

Berkovich pyramid)

3. Hardmetal ball (especially for the determination of

the elastic behavior of materials)

4. Diamond spherical indenter.

For tests at indentation depths of less than 1 μm, the me-

chanical deformation strongly depends on the real shape

of indenter tip and the calculated materials parameters

are signiﬁcantly inﬂuenced by the contact area function

of the indenter used in the testing machine. There-

fore careful calibration of both instrument and indenter

shape is required in order to achieve an acceptable

reproducibility of the material parameters determined

with different machines.

At high contact pressures, damage to the indenter is

possible. For this reason in the force range higher than

2 N, hardmetal indenters are often used. For test pieces

with very high hardness and modulus of elasticity the

inﬂuence of indenter deformation on test result should

be taken into account.

max

Fig. 7.33 Schematic representation of a cross section through in-

dentation (Key: 1 – Indenter; 2 – Surface of residual plastic

indentation in test piece; 3 – Surface of test piece at maximum

indentation depth and test force)

The test procedure can either be force-controlled

or displacement-controlled. The test force F,thecor-

responding indentation depth h and time are recorded

during the whole test procedure. The result of the test

is the data set of the test force and the relevant in-

dentation depth as a function of time (Fig. 7.32). For

a reproducible determination of the force and cor-

responding indentation depth, the zero point for the

force/indentation depth measurement must be assigned

individually for each test.

Where time-dependent effects are being measured

1. using the force-controlled method, the test force is

kept constant over a speciﬁed period and the change

of the indentation depth is measured as a function of

the holding time of the test force,

2. using the indentation-depth-controlled method the

indentation depth is kept constant over a speciﬁed

period and the change of the test force is measured

as a function of the holding time of the indentation

depth.

For references for the application of the instrumented

indentation test, see [7.184–202].

Table 7.12 gives the symbols and designations used

for the instrumented indentation test.

Table 7.13 gives an overview about the parameters

of the instrumented indentation test and additional in-

ﬂuence factors on the test result.

Martens Hardness (HM)

Determination of Martens Hardness. Martens hard-

ness is measured under an applied test force. Martens

hardness is determined from the values given by the

force/indentation depth curve during the increasing of

the test force, preferably after reaching the speciﬁed test

force. Martens hardness includes the plastic and elastic

Part C 7.3

380 Part C Materials Properties Measurement

Table 7.12 Symbols and designations

Symbol Designation Unit

a,b

α Angle, speciﬁc to the shape of the pyramidal indenter

◦

r Radius of spherical indenter mm

F Test force N

max

Maximum test force N

h Indentation depth under applied test force mm

max

Maximum indentation depth at F

max

Point of intersection of the tangent to curve 2 at F

max

with the indentation depth axis (Fig. 7.32) mm

Permanent indentation depth after removal of the test force mm

Depth of the contact of the indenter with the test piece at F

max

(h) Surface area of the indenter at distance h from the tip mm

) Projected area of contact of the indenter at distance h

from the tip mm

HM Martens hardness N/mm

Martens hardness, determined from the slope of the increasing force/indentation depth curve N/mm

Indentation hardness N/mm

Indentation modulus N/mm

Indentation creep %

Indentation relaxation %

total

Total mechanical work of indentation Nm

elast

Elastic reverse deformation work of indentation Nm

Relation W

elast

total

To avoid very long numbers multiples or sub-multiples of the units may be used

1N/mm

=1MPa

deformation, thus this hardness value can be calculated

for all materials.

Martens hardness is deﬁned for both pyramidal in-

denters. It is not deﬁned for the Knoop indenter or for

ball indenters.

Martens hardness is deﬁned as the test force F

divided by A

(h) the surface area of the indenter pen-

etrating beyond the zero point of the contact and is

expressed in N/mm

The relationship between Martens hardness, inden-

tation depth and test force is given in Fig. 7.34.

(a) Vickers indenter (b) Berkovich indenter

HM =

(h)

26.43h

HM=

(h)

26.44h

(7.29)

(h) =

4 sin





cos





(h) =

√

3tanα

cos α

(7.30)

Designation of the Martens Hardness [7.184]. The

Martens hardness is denoted by the symbol HM.

Example. HM 0.5/20/20 =8700 N/mm

HM 0.5 Test force in N

/20 Application time of test force in s

/20 Duration time of test force in s

=8700 N/mm

Hardness value

Indentation Hardness (H

)

Determination of the Indentation Hardness. The in-

dentation hardness H

is a measure of the resistance to

permanent deformation or damage.

max

, (7.31)

where F

max

is the maximum applied force, A

the projected (cross-sectional) area of contact between

the indenter and the test piece determined from the

force–displacement curve and a knowledge of the area

function of the indenter, see 4.5.2 of ISO 14577-2.

Equation (7.31) deﬁnes hardness as the maximum

applied force, divided by the projected (cross-sectional)

contact area of the indenter with the test piece. This def-

Part C 7.3

Mechanical Properties 7.3 Hardness 381

Table 7.13 Parameters of the instrumented indentation test and additional inﬂuence factors on the test result

Martens hardness HM

Indentation hardness H

Indentation modul. E

Indentation creep C

Indentation relax. R

Indentation work η

Applied test force × × × × × ×

Applied indentation depth × 

Duration of test force × × × × × ×

Application time of test force ×

Holding time at constant test force × × × 

Holding time at constant indentation depth × 

Time of removal of test force in the curve-ﬁtting range × × 

Material and shape of the indenter       

Mode of control (test force or indentation depth)       

Approach speed of the indenter   

Speed of application of the test force    

Speed of application of the indentation depth    

Position and duration of additional holding periods    

Temperature       

× Expressed with the test result

Additional factors that inﬂuences the test result

inition is in accord with that generally agreed and ﬁrst

proposed by Meyer [7.185].

For indentation depths < 6 μm the area function of

the indenter cannot be assumed to be that of the theoret-

ical shape, since all pointed indenters will have some

degree of rounding at the tip and spherically ended

indenters (spherical and conical) are unlikely to have

a uniform radius. The determination of the exact area

function for a given indenter is required for indentation

depths < 6 μm, but is beneﬁcial for larger indentation

depths (see 4.2.1 and 4.6 of ISO 14577-2).

For indentation depths > 6 μm a ﬁrst approximation

to the projected area A

is given by the theoretical shape

of the indenter.

For a perfect Vickers indenter

=24.5h

Fig. 7.34 Relationship between Martens hardness, inden-

tation depth and test force (Key: 1 – Macro range; 2 –

Micro range; 3 – Nano range; 4 – Rubber; 5 – Plastics; 6 –

Nonferrous metals; 7 – Steel; 8 – Hardmetals, ceramics)



For a perfect Berkovich indenter

=23.96h

For a modiﬁed Berkovich indenter

=24.5h

Part C 7.3

382 Part C Materials Properties Measurement

where h

is the depth of contact of the indenter with the

test piece calculated as

= h

max

−ε(h

max

−h

) .

Figures 7.32 and 7.33 schematically show the different

depths monitored during an indentation test. The the-

oretical basis of the method for the determination of

contact depth is given in [7.203]. The contact depth is

derived from the force removal curve using the tangent

depth h

and the maximum displacement h

max

correct-

ing for elastic displacement of the surface according

to Sneddon’s analysis [7.186], where ε depends on the

indenter geometry (Table 7.14).

is derived from the force–displacement curve and

is the intercept of the tangent to the unloading cycle at

max

with the displacement axis.

Designation of Indentation Hardness H

. The inden-

tation hardness is denoted by the symbol H

Example. H

0.5/10/20/30 =11 300 N/mm

0.5/ Test force in N

10/ Application time of test force in s

20 Duration time of test force in s

/30 Time taken to remove the test force

during the ﬁtted portion of the test

force removal curve in s

=11 300 N/mm

Hardness value

Indentation Modulus (E

)

Determination of Indentation Modulus. The indenta-

tion modulus E

can be calculated from the slope of

the tangent for the calculation of indentation hardness

and is comparable with the Young’s modulus of the

material

1 −(ν

)

−

1−(ν

)

, (7.32)

√



, (7.33)

Table 7.14 Correction factor ε for different indenter ge-

ometries

Indenter geometry ε

Flat punch 1

Conical 2(π −2)/π =0.73

Paraboloid of revolution

(includes spherical)

3/4

Berkovich, Vickers 3/4

where

is the Poisson’s ratio of the test piece;

is the Poisson’s ratio of the indenter (for diamond

0.07) [7.187];

is the reduced modulus of the indentation contact;

is the modulus of the indenter (for diamond 1.14 ×

N/mm

)[7.187];

is the compliance of the contact, i. e. dh/dF of the

(frame stiffness corrected) test force removal curve

evaluated at the maximum test force;

β is the correction factor for different tip geometry

(Table 7.15).

Designation of Indentation Modulus. The indentation

modulus is denoted by the symbol E

Example. E

0.5/10/20/30 =222 000 N/mm

0.5/ Test force in N

10/ Application time of test force in s

20 Duration time of test force in s

/30 Time taken to remove the test force

during the ﬁtted portion of the test

force removal curve in s

=222 000 N/mm

Indentation modulus

Nanoindentation Method

The elastic and plastic properties of a coating are crit-

ical factors determining the performance of the coated

product. Indeed many coatings are speciﬁcally devel-

oped to provide wear resistance that is usually conferred

by their high hardness. Measurement of coating hard-

ness is often used as a quality-control tool. Young’s

modulus becomes important when calculation of the

stress in a coating is required in the design of coated

components. For example, the extent to which coated

components can withstand external applied forces is an

important property in the capability of any coated sys-

tem.

It is relatively straightforward to determine the hard-

ness and indentation modulus of bulk materials using

instrumented indentation. However, when measure-

ments are made normal to a coated surface, depending

Table 7.15 Correction factor β for different indenter ge-

ometries [7.195]

Indenter geometry β

Axis symmetric (e.g. conical, spherical) 1

Berkovich 1.034

Vickers 1.012

Part C 7.3

Mechanical Properties 7.3 Hardness 383

on the force applied and the thickness of the coating, the

substrate properties inﬂuence the result.

The purpose of this part of handbook is to provide

guidelines for conditions where there is no signiﬁcant

inﬂuence of the substrate and, where such inﬂuence

is detected, to provide possible analytical methods to

enable the coating properties to be extracted from the

composite measurement.

The analysis used here does not make any al-

lowances for pile-up of indents. Use of atomic force

microscopy (AFM) to assess the indent shape allows the

determination of possible pile-up of the surface around

the indent. This surface effect results in an underesti-

mate of the contact area in the analysis and hence may

inﬂuence the measured results. Pile-up generally occurs

for fully workhardened materials. Pile-up of soft, duc-

tile materials is more likely for thinner coatings due to

the constraint of the stresses in the zone of plastic de-

formation in the coating. It has been reported that the

piled-up material results in an effective increase of the

contact area for the determination of hardness, while the

effect is less pronounced for the determination of inden-

tation modulus, since the piled-up material behaves less

rigidly [7.197].

Veriﬁcation and Calibration of Testing Machines. The

instrument shall be calibrated according to the proce-

dures set out in ISO 14577-2 [7.204].

Indirect veriﬁcation using a reference material shall

be made to ensure the direct calibration is valid and that

no damage or contamination has occurred to the inden-

ter tip. If the results of these initial indentations indicate

the presence of contamination or damage then the inden-

ter should be cleaned using the procedure recommended

in ISO 14577-1 before further trial indents are made. If

indentation into the reference material still indicates the

presence of contamination or damage after cleaning, in-

spection with an optical microscope at a magniﬁcation

of 400× is recommended. Detection of submicroscopic

damage or contamination is possible using appropriate

microscopy of indents or the indenter. Where damage is

detected the indenter shall be replaced.

The machine compliance C

and the area function

) calibration/veriﬁcation shall be implemented be-

fore a new indenter is used, see subclause Procedures

for determination of machine compliance and indenter

area function.

The instrumented indentation instrument shall

achieve the required mechanical and thermal stability

before starting an indentation cycle, see subclause Mea-

surement procedure.

Indentation experiments may be performed with

a variety of differently shaped indenters, which should

be chosen to optimize the plastic and elastic defor-

mation required for a given coating substrate system.

Typical indenter shapes are Vickers, Berkovich, coni-

cal, spherical and corner cube. For the determination

of coating plastic properties, pointed indenters are

recommended. The thinner the coating, the sharper

the indenter should be. For the determination of

coating elastic properties, spherical indenters are rec-

ommended. Ideally, a large-radius sphere is used, which

enables indentation in the fully elastic regime over a rea-

sonable indentation displacement. If the radius is too

large, the surface effects (roughness, surface layers etc.)

will dominate the uncertainties. And if the radius is too

small, the maximum force or displacement before plas-

tic deformation begins will be very low. The optimum

radius can be identiﬁed by preliminary experiments or

modeling.

Test Piece. Generally surface preparation of the test

piece should be kept to a minimum, and if possible,

the test piece should be used in the as-received state if

surface ﬂatness is consistent with the criteria given in

ISO 14577-1.

The test piece shall be mounted using the same

methods as employed for determination/veriﬁcation of

the instrument frame compliance, and shall be such that

the test surface is normal to the axis of the indenter and

such that the local surface at the proposed indentation

site is less than ±5

◦

from perpendicular to the inden-

tation axis. Perpendicularity can be checked in practice

by imaging the indent made by a nonaxisymmetric in-

denter.

Indentation into rough surfaces will lead to an in-

creased scatter in the results with decreasing indentation

depth. Clearly when the roughness value R

approaches

the same value as the indentation depth the contact area

will vary greatly from indent to indent, depending on

its position relative to peaks and valleys at the surface.

Thus it is recommended that the ﬁnal surface ﬁnish

shall be as smooth as available experience and facili-

ties permit. The R

value should be less than 5% of the

maximum penetration depth whenever possible.

It has been shown that, for a Berkovich indenter,

the angle that the surface normal presents to the axis

of indentation has to be greater than 7

◦

for signiﬁcant

errors to result [7.198]. The important angle is that be-

tween the indentation axis and the local surface normal

at the point of contact. This angle may be signiﬁcantly

different to the average surface plane for rough surfaces.

Part C 7.3

384 Part C Materials Properties Measurement

While R

has been recommended as a practical

and easily understood roughness parameter, it should

be borne in mind that this is an average, and thus

single peaks and valleys may be greater than this, as

deﬁned by the R

value, although the likelihood of

encountering the maximum peak, for example, on the

surface is small. Modeling to investigate the roughness

of the coating surface has concluded that there are two

limiting situations for any R

value. When the wave-

length of the roughness (in the plane of the coating

surface) is much greater than the indenter tip radius,

the force–penetration response is determined by the lo-

cal coating-surface curvature, but when the wavelength

is much less than the tip radius, asperity contact oc-

curs and the effect is similar to having an additional

lower-modulus coating on the surface.

In cases where coatings are used in the as-received

condition, nevertheless, random defects occur such as

nodular growths or scratches and where an optical

system is included in the testing machine, it is rec-

ommended that ﬂat areas away from these defects are

selected for measurement.

The roughness proﬁlometer probe radius should be

comparable to the indenter radius. If the roughness pa-

rameter R

is determined with an AFM on a scan area,

the size of this area should be agreed upon between the

customer and the measurement laboratory. A scan area

of 10 μmby10μm is recommended.

It should be appreciated that mechanical polishing

of surfaces may result in a change in the workhardening

and/or the residual stress state of the surface and conse-

quently the measured hardness. For ceramics this is less

of a concern than for metals, although surface damage

may occur. Grinding and polishing shall be carried out

such that any stress induced by the previous stage is re-

moved by the subsequent stage and the ﬁnal stage shall

be with a grade of polishing medium appropriate to the

displacement scale being used in the test.

Many coatings replicate the surface ﬁnish of the

substrate. If it is acceptable to do so, surface-preparation

problems can be reduced by ensuring that the substrate

has an appropriate surface ﬁnish, thus eliminating the

need to prepare the surface of the coating. In some

cases, however, changing the substrate surface rough-

ness may affect other coating properties; therefore, care

should be taken when using this approach.

In coatings it is common for there to be relatively

large residual stresses e.g. arising from thermal expan-

sion coefﬁcient mismatch between the coating and the

substrate and/or stress induced by the coating deposition

process. Thus, a stress-free surface would not normally

be expected. Furthermore, stress gradients in coatings

are not uncommon, so that removal of excessive ma-

terial during a remedial surface preparation stage may

result in a signiﬁcant departure from the original surface

state.

Polishing reduces the coating thickness and so the

effects of the substrate will be enhanced. Where the data

analysis requires an accurate knowledge of the coating

thickness indented, polishing will require remeasure-

ment of the coating thickness. This again emphasizes

the need to carry out minimum preparation.

Generally, provided the surface is free from obvious

surface contamination, cleaning procedures should be

avoided. If cleaning is required, it shall be limited to

methods that minimize damage, e.g.

•

application of a dry oil-free ﬁltered gas stream

•

application of subliming particle stream of CO

(avoiding surface temperatures below the dew point)

•

application of ultrasonic methods

•

rinse with a solvent (which is chemically inert to the

test piece) and then dry.

If these methods fail and the surface is sufﬁciently ro-

bust, the surface may be wiped with a lintless tissue

soaked in solvent to remove trapped dust particles and

the surface shall be rinsed in a solvent as above. Ultra-

sonic methods may not be used as these are known to

create or increase damage to coatings.

Test Conditions. Indenter geometry, maximum force

and/or displacement and force displacement cycle (with

suitable hold periods) shall be selected by the operator

to be appropriate to the coating to be measured and the

operating parameters of the instrument used.

It is important that the test results are not affected by

the presence of an interface, free surface or by any plas-

tic deformation introduced by a previous indentation in

a series. The effect of any of these depends on the in-

denter geometry and the materials properties of the test

piece. Indentations shall be at least three times their in-

dentation diameter away from interfaces or free surfaces

and, where multiple indentations are planned, the mini-

mum distance between indentations shall be at least ﬁve

times the largest indentation diameter.

The indentation diameter is the in-plane diameter at

the surface of the test piece of the circular impression of

an indent created by a spherical indenter. For noncircu-

lar impressions, the indentation diameter is the diameter

of the smallest circle capable of enclosing the inden-

tation. Occasional cracking may occur at the corners

Part C 7.3

Mechanical Properties 7.3 Hardness 385

of the indentation. When this occurs, the indentation

diameter should enclose the crack.

The minimum distances speciﬁed are best applica-

ble to ceramic materials and metals such as iron and its

alloys. For other materials it is recommended that sepa-

rations of at least ten indentation diameters be used. If in

doubt, it is recommended that the values from the ﬁrst

indentation are compared with those from subsequent

indentations in a series. If there is a signiﬁcant differ-

ence, the indentations may be too close and the distance

should be increased. A twofold increase is suggested.

The following parameters of coating/substrate inﬂu-

encing the measurement result should be considered.

1. Substrate hardness, Young’s modulus and Poisson’s

ratio

2. Coating thickness

3. Surface roughness

4. Adhesion of the coating to the substrate.

All these parameters should be kept constant if a di-

rect comparison is to be made between two or more test

pieces.

The time dependence of the materials parameter

should be taken into account.

Variations in test-piece parameters other than hard-

ness or modulus can affect measurement of these

quantities. If the indentation depth is a sufﬁciently

small fraction of the coating thickness, or the coating

Use hold to

estimate drift

Hold at contact

Elastic limit

exceeded

Take into account:

Surface layers,

vibration

Linear correction of

displacement

Take into account:

Creep,

viscoelasticity,

cracking,

surface layers,

vibration

Drift removed using

reference surface

before and after

Significant

Possible

Not possible

Yes

Noise is

high

Not

significant

Don't need to

correct

Hold at 90% unload

Take into account:

Creep (recovery),

viscoelasticity,

cracking (tensile),

stiff contacts

Fig. 7.35 Decision tree to assist in estimating the drift during a force-controlled experiment

thickness is well known, it is possible to compare coat-

ings of different thickness. The exact limits depend on

the ratio of properties of coating and substrate. It is

recommended that methods for normalizing results to

determine coating properties from coatings of different

thickness always be used.

Measurement Procedure. Introduce the prepared test

piece and position it so that testing can be undertaken

at the desired location. Carry out the predetermined

number of indentation cycles using the selected test

conditions.

A single force application and removal cycle shall

be used. A decision tree to assist in estimating the ther-

mal drift during the experiment is shown in Fig. 7.35.If

the drift rate is signiﬁcant the displacement data shall be

corrected by measuring the drift rate during a hold at as

close to zero force as is practicable or at 90% removed

force. The hold period shall be sufﬁcient to allow deter-

mination of the drift in displacement due to temperature

ﬂuctuations. Drift (as opposed to noise) in displacement

values determined during the hold period at 90% force

removal or at close to zero is believed to result from

temperature changes, and, a linear correction should be

applied such that

(corrected displacement)

=(displacement) −(thermal drift rate) × time .

Part C 7.3

386 Part C Materials Properties Measurement

If a contact in the fully elastic regime can be obtained,

a hold at initial contact is preferred. In this way, ma-

terial inﬂuences (creep, viscoelasticity, cracking) can

be avoided. If no elastic contact can be obtained, de-

pending on the material under investigation, a hold at

initial force (e.g. viscoelastic material) or at 90% re-

moved force (e.g. soft material) may be preferred. For

difﬁcult materials, a hold period at both ends of the in-

dentation cycle may be included. It is recommended that

the ﬁrst 10–20 s of the hold data should be discarded

for the analysis since these initial data may be signif-

icantly inﬂuenced by time-dependent effects (material

time-dependent deformation, formation of capillary sur-

face layers) [7.199,200].

A further hold period shall be performed at maxi-

mum force to allow for completion of any time-depen-

dent deformation. In all cases this hold period shall be

long enough to reduce the error in the slope of the force

removal curve to less than 1%. The minimum hold-

period length is therefore dependent on the instrument

capability and the material being tested. The hold period

shall be long enough and/or the time to remove force

shall be short enough such that

(creep rate)× (time to remove force)

< F

max

/S/100 .

Force application and removal rates may be the same

but it is recommended that the removal rate should be

higher than the application rate (if possible) to minimize

the inﬂuence of creep. Slower force application reduces

the hold period length required at F

max

to achieve the

necessary reduction in creep rate.

The inﬂuence of the material creep behavior on

hardness and modulus results has been reported [7.199].

The results show that, especially for materials with low

hardness-to-modulus ratio (among them most metals),

the modulus results are not reliable if the hold period

is too short. A modulus error, due to creep, of more

than 50% may arise. The variation of the hold period

produced a hardness change of up to 18%. Reference

[7.199] proposes hold periods dependent on the mater-

ial type that range from 8 s for fused quartz to 187 s for

aluminum. The criterion used was that the creep rate

should have decayed to a value where the depth increase

in one minute is less than 1% of the indentation depth.

It is recommended that the creep rate be assessed in

preliminary experiments.

Data Analysis and Evaluation of Results. The hardness

and indentation modulus of the test piece can be calcu-

lated using (7.31)–(7.33). The properties thus calculated

are composite properties for the coating/substrate com-

bination. This clause provide methods for extracting the

indentation modulus of the coating from the composite

properties measured, assuming that the coating proper-

ties are constant with depth.

For hardness measurement of electroplated coatings

on steels [7.201], it is recommended that the indenta-

tion depth does not exceed one tenth the thickness of

the coating, while for paint ﬁlms [7.202] penetration of

up to one third the coating thickness may be allowed.

These approximations can be unsatisfactory in many

cases. Test parameters for ductile and brittle coatings

need to be considered separately.

For in-plane indentation, elastic deformation of the

substrate will mostly occur for all coatings, even though

this could be negligibly small for a thick compliant coat-

ing on a stiff substrate. Thus the measured modulus will

be the composite modulus of the coating and substrate

and the value obtained will be a function of indentation

depth.

For hardness measurement it is recommended to use

as small a radius indenter (i. e. as sharp) as possible

to limit the plastic deformation to be within the coat-

ing. A measurement of the uncoated substrate hardness

is a useful guide to the appropriate choice of analysis

(soft versus hard). In some circumstances it is possi-

ble to identify a range of indentation depth over which

the measured hardness is constant (i. e. before the on-

set of substrate plastic deformation) and then carry out

indentation experiments within this range.

Estimates of coating hardness and modulus may

be extracted from the composite values E

, H

ob-

tained from in-plane indentation by expressing those

composite values as a function of the contact radius

a or indentation depth h

normalized to the coating

thickness. Measurement of the coating thickness t

recommended for reproducible measurement of coating

properties. For indenters of different geometries (e.g.,

Berkovich, Vickers, spherical, cone, etc.), a is approxi-

mated by the radius of a circle having the same area as

the projected area of contact with the indenter

a =



. (7.34)

This value has exact equivalence for a spherical or con-

ical indenter but becomes increasingly less physically

meaningful as the axial symmetry of the indenter re-

duces, i. e. cone = sphere > Vickers > Berkovich.

It is relatively easy to measure the hardness of duc-

tile coatings or the elastic modulus of brittle coatings. It

is more difﬁcult to determine the hardness of brittle or

Part C 7.3

Mechanical Properties 7.3 Hardness 387

hard coatings or the elastic modulus of ductile coatings.

Where t

is not measured, nominal values of t

may

be used but comparison between coatings of different

thicknesses will be less accurate.

In the case of force-controlled cycles and test pieces

of unknown indentation response, a set of trial inden-

tations shall be performed (e.g. at two widely spaced

forces) and analyzed to enable estimates of the test force

required for the range of a/t

speciﬁed below.

In the case of soft/ductile coatings indentation force

or displacement and indenter geometry shall be chosen

such that data shall be obtained in the region where

a/t

< 1.5. The coating indentation modulus E

obtained by taking a series of measurements at dif-

ferent indentation depths and extrapolating a linear

ﬁt to indentation elastic modulus versus a/t

to zero

(Fig. 7.36).

A linear ﬁt to indentation elastic modulus ver-

sus a/t

to zero is a ﬁrst approximation in the range

a/t

< 2. However, in general, a nonlinear relationship

appears and can be reproduced by ﬁnite element anal-

ysis (FEA). The exact nature of this nonlinear relation

is not known and so a linear ﬁt over the restricted range

indicated is a robust ﬁrst approximation but is not appli-

cable over a range wider than this.

For the determination of coating indentation hard-

ness, it is recommended that an elastic stress analysis

of the coating/substrate system be undertaken using

the approximation of a spherical indenter of a radius

equivalent to the tip radius of the self-similar-geometry

indenter. This will determine whether the coating or the

(GPa)

300

250

200

150

100

0 0.5 1 1.5 2 2.5 3

a/t

Fig. 7.36 Indentation reduced elastic modulus versus nor-

malized contact radius Au on Ni, selected data for spherical

(ﬁlled circle), Berkovich (ﬁlled triangle) and Vickers (ﬁlled

square) indenter

substrate will yield ﬁrst during indentation and therefore

whether it is possible to determine the coating hard-

ness at all. It is recommended that hardness values for

the substrate be obtained for comparison, by testing if

necessary. Delamination or fracture of the coating can

be recognized by the hardness values obtained, clus-

tering at the substrate value, even at low h

.Note

that sharper indenters may cause fracture at lower forces

than blunter indenters.

In the case of hard coatings on a softer substrate,

the indentation force or displacement and indenter ge-

ometry shall be chosen such that h

(or ≈ a/t

)is

in a range where H

is a maximum. Commonly, the

range is 0 < h

< 0.5. If a constant maximum value

of H

(a plateau) is observed over this range, this

is the coating indentation hardness. If only a maxi-

mum in H

occurs and indentation of a thicker coating

yields the same value, then this is a strong indicator

that this is the value for the coating. Otherwise this is

only the minimum estimate of the coating indentation

hardness.

The extent of substrate plastic deformation will de-

pend upon a number of factors, including the relative

difference in hardness and modulus between the coating

and the substrate, adhesion, the coating thickness, the

indenter radius of curvature (sharpness) and the maxi-

mum force.

To measure the coating hardness the coating must

yield sufﬁciently before the substrate yields. The best

conditions for this are when the maximum of the von

Mises stress is inside the thickness of the coating and

causes plastic deformation whilst the stress in the sub-

strate below does not exceed the substrate yield stress.

In a spherical contact, the maximum of the von Mises

stress is approximately 0.47× (mean pressure) at 0.5a

below the surface.

Procedures for Determination of Machine Compli-

ance and Indenter Area Function. The calibration

procedures detailed below require the use of refer-

ence materials (see ISO 14577-3 [7.205]) which shall

be isotropic and homogeneous. The Young’s modulus

and Poisson’s ratio are assumed to be independent of

the indentation depth. Some procedures need reference

materials of known Young’s modulus and Poisson’s

ratio.

Currently recommended materials are freshly pol-

ished tungsten and fused silica. The forces used should

not exceed the threshold at which cracking occurs

(≈80–100 mN for an average radius Berkovich tip in-

denting fused silica).

Part C 7.3