review article indentation depth dependent mechanical behavior in...

Review ArticleIndentation Depth Dependent Mechanical Behavior in Polymers

Farid Alisafaei and Chung-Souk Han

Department of Mechanical Engineering University of Wyoming 1000 E University Avenue Laramie WY 82071 USA

Correspondence should be addressed to Chung-Souk Han chan1uwyoedu

Received 20 April 2015 Accepted 9 July 2015

Academic Editor Victor V Moshchalkov

Copyright copy 2015 F Alisafaei and C-S Han This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Various experimental studies have revealed size dependent deformation of materials at micro and submicron length scales Amongdifferent experimental methods nanoindentation testing is arguably the most commonly applied method of studying size effect invarious materials where increases in the hardness with decreasing indentation depth are usually related to indentation size effectsSuch indentation size effects have been observed in bothmetals and polymersWhile the indentation size effects inmetals arewidelydiscussed in the literature and are commonly attributed to geometrically necessary dislocations for polymer the experimentalresults are far sparser and there does not seem to be a common ground for their rationalesThe indentation size effects of polymersare addressed in this paper where their depth dependent deformation is reviewed alongwith the rationale provided in the literature

1 Introduction

Nanoindentation is one of the most commonly appliedmethods for determining of material properties and charac-terization of deformation behavior at nano- to micrometerlength scales Performing nanoindentation experiments themechanical behavior of various materials has been found tobe dependent on the indentation depth indicating changingdeformation mechanisms with the length scale Length scaledependent deformation has been experimentally observed indifferent materials including metals and polymers In metalssize dependent deformation has been reported in micro-and nanoindentation tests of bulkmaterials [1ndash7] microsizedbeam bending experiments [8] microtorsion of thin wires[9] tension of perforated plates [10 11] nanoindentation ofthin films [12] particle-reinforced metal matrix composites[13 14] and so forth Similar to metals length scale depen-dent deformation has been also experimentally observed inpolymers at micro to submicron length scales by nanoinden-tation [15 16] microsized beam bending experiment [17 18]and bending test of nanofibers [19 20] aswell as in particulatereinforced composites (a relatively recent review can befound in [21]) The length scale dependent deformationhas been experimentally observed in various polymers suchas epoxy [16 17 22ndash27] polycarbonate (PC) [15 16 28]polystyrene (PS) [15 28] polymethyl methacrylate (PMMA)

[15 28 29] polydimethylsiloxane (PDMS) [30ndash35] siliconrubberelastomer [36ndash38] nylon (PA66) [39] polypropylene(PP) [18] polyimide (PI) [40] polyvinyl chloride (PVC) [28]polyamide 6 (PA6) [41] and polystyrene acrylonitrile (SAN)[28]

In metals length scale dependent deformation is usuallyassociated with geometrically necessary dislocations [42 43]resulting in an increase of strain gradients with decreasinglength scale [44ndash48] In polymers however such a notion isnot applicable and different explanations have been suggestedin the literature as a rationale for such size effects includinghigher order displacement gradient models [16 17 22ndash25 2933 49ndash51] surface effects [36 52] changes in the materialproperties through thickness (heterogeneity of the material)[15 31] and friction [34] Due to the complex and variedmolecular structures of polymers the mechanisms of theabovementioned length scale dependent deformations inpolymers are arguably more complicated than in metalsFurthermore as polymers are known to be rate and tem-perature sensitive such length scale dependent deformationsin polymers should be more affected by ratetemperaturechanges than in metals

While length scale dependent deformation in polymershas been observed in different experiments including bend-ing of microbeams nanoindentation bending of nanofiberand particulate-reinforced composites in this paper the

Hindawi Publishing CorporationAdvances in Condensed Matter PhysicsVolume 2015 Article ID 391579 20 pageshttpdxdoiorg1011552015391579

2 Advances in Condensed Matter Physics

F

FmaxB

D

C

A hf hl hmax h

WineI

S

WeI

(a)

Initial surfaceSurface profile after load removal

Indenter

Surface profile

F

hfhmaxhc

hs

a

under Fmax

(b)

Figure 1 (a) Schematic representation of load-displacement curve in nanoindentation testing (b) Schematic representation of cross sectionthrough indentation of a material sample

Table 1 Projected contact area surface area angles geometric constants and geometry correction factors for common indenter geometries[55] where ℎc is the contact depth (shown in Figure 1(b)) ℎ is the indentation depth under applied force 119865 (see Figure 1(a)) 119877 is the radiusof spherical tip 120579 is the semiangle of indenter tip (for pyramidal tips 120579 is the face angle with the central axis of the tip [55]) 120572 is the halfangle defining an equivalent cone 120598 is the geometric constant (for the pyramidal tips 120598 is theoretically 072 however a value of 075 hasbeen experimentally found to better represent experimental data [55]) and 120573 is the geometry correction factor used for the determinationof elastic modulus [53] Note that 119860c and 119860 s given in Table 1 for Berkovich Vicker Knoop cube corner and cone indenters are respectivelythe projected and surface areas of a perfectly sharp indenter without imperfectionbluntness (see eg [53 55] for the projected contact areaof a blunt indenter)

Indenter tip Projectedcontact area 119860c

Surface area 119860 s Semiangle 120579 (deg) Effective coneangle 120572 (deg)

Geometricconstant 120598

Geometrycorrection factor 120573

Sphere 120587(2ℎc119877 minus ℎ2

c) NA NA NA 075 1Berkovich (perfect) 3radic3ℎ

2

c tan2120579 3radic3ℎ

2 tan 120579 cos 120579 6503∘ 703∘ 075 1034Berkovich (modified) 3radic3ℎ

2

c tan2120579 NA 6527∘ 703∘ 075 1034

Vicker 4ℎ2

c tan2120579 4ℎ

2 sin 120579cos2 120579 68∘ 703∘ 075 1012

Knoop 2ℎ2

c tan 1205791 tan 1205792 NA 1205791= 8625∘1205792= 65∘ 7764∘ 075 1012

Cube corner 3radic3ℎ2

c tan2120579 NA 3526∘ 4228∘ 075 1034

Cone 120587ℎ2

c tan2120572 NA 120572 120572 0727 1

length scale dependent deformation in polymers observedwith indentation experiments is reviewed along with thenotions on its physical origin as well as theoretical modelsthat predict such length scale dependent deformations inpolymers In Section 2 the length scale dependent defor-mation of polymers observed in nanoindentation testing iscritically reviewed followed by discussions on the possibleexplanations for the origin of length scale dependent defor-mation in polymers in Section 3

2 Size Dependent Deformation of PolymersObserved in Nanoindentation Experiments

To set the stage and clarify the notation Figure 1(a) shows aschematic load-displacement curve of nanoindentation con-sisting of loading (section A-B) holding (section B-C) andunloading (section C-D) sequences [25] A correspondingschematic cross section of such an indentation is depicted

in Figure 1(b) identifying parameters used in the analysis[53 54] which will be discussed later

21 Hardness and Elastic Modulus Determination Similar tometals [2] the hardness parameter (indentation hardness [53]and universal hardness [54]) is usually used to investigate theindentation size effect (ISE) in polymers [15 22] Indentationhardness119867I can be obtained as

119867I =119865max119860c

(1)

where 119865max is the maximum applied force (shown inFigure 1(a)) and 119860c is the projected contact area at themaximum applied force 119865max The projected contact area 119860cas a function of contact depth ℎc (see Figure 1(b)) for somecommon indenter geometries is given in Table 1 [55] As canbe seen in Figure 1(b) the contact depth ℎc is determined by

ℎc = ℎmax minus ℎs (2)

Advances in Condensed Matter Physics 3

where ℎmax is the indentation depth ℎ at the maximumapplied force 119865max before the unloading section (see Figure 1)and ℎs is determined by

ℎs = 120598119865max119878

(3)

in which 119878 is the initial slope of unloading curve (seeFigure 1(a)) and 120598 is a geometric constant given in Table 1for different tip geometries To determine the stiffness 119878 bythe Oliver and Pharr approach [53] the unloading curve ofFigure 1(a) is described as

119865 = 119861 (ℎ minus ℎf)119898

(4)

where the parameters 119861 119898 and ℎf are obtained by fittingthe upper portion of unloading curve Thus 119878 is determinedby taking the derivative of 119865 with respect to ℎ in (4)at the maximum applied force 119865max shown in Figure 1(a)In addition to the indentation hardness 119867I the universalhardness 119867U (also known as Martens hardness) can bealso determined from nanoindentation tests The universalhardness for a perfect Berkovich (unmodified Berkovich) andVicker tips is defined as follows

119867U =119865

119860 s (5)

where 119860 s is the surface area under the initial surface (seeFigure 1(b)) given in Table 1 as a function of indentationdepth ℎ As mentioned above the indentation size effect ofmaterials is usually described by measuring the hardness(119867I andor119867U) at different indentation depths ranging frommicrons to few nanometers However in contrast to metals[56 57] in polymers length scale dependent deformationhas been also experimentally observed by monitoring theelastic modulus determined from indentation experimentsby various approaches with sharp indenter tips which willbe denoted as indentation elastic modulus in the followingThe indentation elasticmodulus ofmaterials is often obtainedfollowing Oliver and Pharr (OampP) [53] as

119864OampPr =

radic120587119878

2120573radic119860c (6)

where120573 is the tip geometry correction factor (see Table 1) and119864OampPr is the reduced elastic modulus [53] which is related to

the elastic modulus of polymer 119864OampP via

1119864OampPr

=1 minus V2

119864OampP +1 minus V2i119864i

(7)

in which 119864i and Vi are respectively the elastic modulus andPoissonrsquos ratio of indenter tip with V being the Poissonrsquos ratioof the polymer

22 Indentation Size Effect in Polymers The indentation sizeeffects of various polymers are depicted in Figure 2 wherethe indentation hardness 119867I increases with decreasing ℎmax[15 16 26 36 39] Therein it can be seen that there is a

10 100 1000 100000

01

02

03

04

05

06

07

UHMWPEPA66nylonSilicon elastomerPC (a)PC (b)

PMMAEpoxy (a)PSEpoxy (b)

hmax (nm)

HI

(GPa

)Figure 2 Indentation hardness119867I versus ℎmax for various polymersredrawn from [15] (UHMWPE PC (b) PMMA PS) [16] (PC(a) epoxy (a)) [26] (epoxy (b)) [36] (silicon elastomer) and [39](PA66nylon)

significant rise in the indentation hardness of epoxy [16 26]polycarbonate (PC) [15 16] polystyrene (PS) [15] polymethylmethacrylate (PMMA) [15] silicon elastomer [36] and nylon(PA66) [39] as ℎmax decreases However 119867I of ultra-highmolecular weight polyethylene (UHMWPE) [15] remainsalmost constant for ℎmax gt 100 nm where there is onlya slight increase in the indentation hardness of UHMWPEat very shallow indentation depths ℎmax lt 100 nm whichwill be discussed later Similar to the trend observed for theindentation hardness of polymers in Figure 2 length scaledependent deformation of polymers can be also observed inFigure 3 where the universal hardness 119867U increases withdecreasing ℎmax [15 25 28] Comparing 119867I of Figure 2 with119867U of Figure 3 the indentation hardness is found to beslightly higher than the universal hardness as the projectedcontact area 119860c (see (1)) is usually smaller than the surfacearea 119860 s used in (5) [25]

Generally due to the complex structures of polymers andtheir rate temperature and pressure dependence behavior inaddition to the higher order displacement gradient effectsindentation size effects can be caused by various sourcesThese sources include but are not restricted to surface effect[36 52] change in the glass transition temperature [41]adhesion [59 60] tip bluntness [61] material inhomogeneity[15] and confined molecular motion interfacial region (for-mation of an interfacial region of confined molecular motionbetween the surface and probe by contact loading) [62]

It can be seen in Figures 2 and 3 that the characteristics oflength scale dependent deformation in polymers significantlydiffer from polymer to polymer As shown in Figures 2 and3 the depth ℎmax at which the hardness starts to noticeably


10 100 1000 1000001

015

02

025

03

035

04

045

PCPS (a)PS (b)

PMMA (a)PMMA (b)Epoxy

hmax (nm)

HU

(GPa

)

Figure 3 Universal hardness 119867U versus ℎmax for various polymersredrawn from [15] (PC PS (a) PMMA (a)) [28] (PS (b) PMMA(b)) and [25] (epoxy with 5 s loading time)

increase can be remarkably different for different polymers[67] For instance the hardness of epoxy starts to increasebelow about ℎmax = 3 120583m [16 25 26] while for PC PS andPMMA hardness starts to rise below about 1 120583m [15 16]Such an increase of hardness in PA66nylon is only observedfor ℎmax lt 200 nm [39] where there is no increase in thehardness of UHMWPE for ℎmax gt 100 nm [15] indicatingthat the characteristics of length scale dependent deformationcan strongly vary in different polymers [68] Furthermore theincrease in the hardness is significantly different in differentpolymers In epoxy [16] PC [15 16] PS [15] and PMMA [15]119867I increases by a factor of about 13 and in silicon elastomer[36] by a factor of about 67 for the indentation depth rangeof around 2 to 02 120583m while no size effect is observed inUHMWPE [15] and polytetrafluoroethylene (PTFE) [69] forthe same indentation depth range of about 2 to 02120583mIn the following the indentation size effect of polymersin the literature is reviewed in three polymer categories ofthermosets thermoplastics and elastomers

221 Thermoset Polymers In addition to the microsizedbeam bending experiment [17] length scale dependentdeformation of epoxy has been experimentally observedby nanoindentation testing in various studies [16 22ndash27]Length scale dependent deformation of thermoset epoxy wasfirst observed in [16 22] where the indentation hardness119867Iof epoxy with 10 parts per hundred resin (phr) was found toincrease with decreasing ℎmax as shown in Figure 2 Nanoin-dentation tests of [16] were performed by a sharp Berkovichtip in load-controlled mode with a constant loading andunloading rate of 001119865maxThe indentation hardness of epoxy(see (1)) was determined according to Oliver and Pharr [53]where a 10 s holding time (sections B-C in Figure 1(a)) was

0 1000 2000 3000100

200

300

400

500

600

700

Reference [25]Reference [23]

hmax (nm)

HI

(MPa

)

Figure 4 Indentation hardness of epoxy with 20 phr from [25](with 20 s loading holding and unloading times) and [23] (with aconstant loading and unloading rate of 1 of the maximum appliedforce with a 10 s holding time) obtained by the Oliver and Pharrapproach [53]

applied to minimize viscous effects To ensure the accuracyof the hardness data119867I the projected contact area119860c at eachtest was measured by atomic force microscopy (AFM) andcompared with the one determined by the unloading curve(see (2)) where good agreement was achieved between themeasured (AFM method) and determined (OampP method)contact areas

Such length scale dependent deformation of epoxy wasalso experimentally observed in [26] with a Berkovich inden-ter tip at an indentation depth range between 4500 and500 nm Nanoindentation tests in [26] were performed inload-controlled mode with the following sequences loadingsegment with a constant strain rate of 005 sminus1 until themaximum applied force 119865max followed by a holding time of10 s at the maximum applied force unloading segment with aconstant unloading rate equal to 70 of the loading rate until80 of themaximum applied force119865max is removed a secondholding period of 2 s followed by a final unloading segmentuntil 119865max is completely removed Indentation hardness ofepoxy determined according to [53] as a function of ℎmax isplotted in Figure 2 where each point represents an averagevalue of ten indents

Performing nanoindentation tests with a sharp Berkovichindenter tip the indentation size effect in epoxy has beenalso experimentally observed in [25] where119867I increases withdecreasing ℎmax as shown in Figure 4 Taking the effects oftip bluntnessimperfection into account it was shown thatfor a sharp Berkovich tip with a tip roundness of about40 nm tip bluntness becomes significant only at ℎmax lt

250 nm as illustrated in Figure 5 (see [25] for tip roundnessdetermination) As the indentation size effect is observed atlarge depths far above ℎmax asymp 250 nm it was concluded in


10 100 10000

100

200

300

400

500

600

700

hmax (nm)

HuncorrectedU

HcorrectedU

HU

(MPa

)

Figure 5 Effect of tip bluntness on the indentation size effect inepoxy with 20 phr (119867corrected

U and119867uncorrectedU represent the universal

hardness with and without considering the effect of tip bluntnessresp) [25]

6 8 10 12 14 16 18 20 2200005

0001

00015

0002

00025

0003

00035

(phr)

120588x

(mol

m3)

Figure 6 Cross-link density of epoxy as a function of hardenerpercentage [23]

[25] that the tip bluntness cannot explain such size effect inepoxy

The effect of cross-link density on the length scale depen-dent deformation of epoxywas investigated in [23] by varyingthe hardener content To this end the cross-link density120588x of epoxy resin cured with varying amount of hardenercontent was first determined by dynamic mechanical tests InFigure 6 the cross-link density of epoxy determined from thedynamicmechanical tests is plotted for the hardener contents(phr) of 75 9 11 15 and 20 The highest cross-link densitywas observed at 11 phr where 120588x decreases from its peakvalue on both sides [23] Performing nanoindentation testswith the same experimental details of [16] mentioned abovethe indentation hardness of epoxy with different hardener

8 10 12 14 16 18 20 22210

220

230

240

250

260

270

(phr)

Ho

(MPa

)

Figure 7 Indentation hardness of epoxy at large indentation depthℎmax as a function of hardener percentage [23]

0 500 1000 1500 2000 2500200

250

300

350

108720

15115

hmax (nm)

HI

(MPa

)

Figure 8 Indentation hardness of epoxy with different hardenercontents as a function of indentation depth ℎmax [23]

percentages was determined as a function of indentationdepth Figure 7 depicts the indentation hardness of epoxy atlarge indentation depths119867o for different hardener contents(phr) of 87 10 11 15 and 20 The maximum value of 119867owas observed at 11 phr where cross-link density 120588x reaches itsmaximum value [23] As 119867o (indentation hardness at largedepths where size effect is negligible) is directly proportionalto the yield stress Figure 7 implies that the yield stress ofepoxy decreases with 120588x Using a sharp three-sided Berkovichtip the indentation hardness 119867I as a function of ℎmax isdepicted in Figure 8 for the hardener contents (phr) of 87 1011 15 and 20 Length scale dependent deformation of epoxyis observed in Figure 8 where indentation hardness of epoxyincreases with decreasing ℎmax for all phr hardener valuesAnalyzing the hardness data of Figure 8 it was observed thatthe length scale dependent deformation of epoxy becomes


0 002 004 006 0080

20

40

60

80

100

True strain

True

stre

ss (M

Pa)

Figure 9 Compressive true stress-strain curve of 20 phr epoxysamples [24]

more significant with increasing cross-link densityThe effectof cross-link density on the size effect of epoxy observed in[23] is consistent with results presented in [34] which will belater discussed It should be mentioned that epoxy samplesof [23] were prepared in the same way as in [16] where bothliquid resin (bisphenol A epichlorohydrin) and curing agent(diethylenetriamine) were mixed and cast into a rectangularTeflon crucible As the curing agent of epoxy is hydrophilicwhile Teflon surface and open air are hydrophobic suchopposing tendencies may potentially cause depletion of cur-ing agent from the surface of epoxy samples and segregationof resin and curing agent [23] If curing agent is segregatedfrom resin then the surface region should contain less curingagent than the interior region of the sample Subsequently thehardness should be lower at small indentation depths closeto the surface of the sample However an opposite trend isobserved in Figure 8 where indentation hardness of epoxyincreases with decreasing indentation depth ℎmax indicatingthat the length scale dependent deformation of epoxy is notassociatedwith the segregation of curing agent during samplepreparation [23]

The effect of plastic straindeformation on the length scaledependent deformation of epoxy was studied by Lam andChong [24] Prismatic epoxy specimens with 20 phr werefirst loaded between two parallel glass platens (uniaxial com-pression) to induce plastic strain Figure 9 shows the com-pressive stress-strain behavior of unprestrained epoxy wherematerial hardening is observed with increasing compressionAfter compressive prestraining nanoindentation testing wasperformed on the optically smooth side of epoxy samplesorthogonal to the loading direction [24] All nanoindentationtests were conducted in a displacement controlled mode witha Berkovich tip at an indentation range of 32 to 01120583m A10 s holding time was employed at themaximum applied load119865max to minimize time dependent effects Results obtainedfrom the nanoindentation tests are depicted in Figure 10where indentation hardness 119867I (obtained by the Oliver andPharr approach [53]) is plotted as a function of indentation

0 500 1000 1500 2000 2500 3000 3500180

200

220

240

260

280

300

320

340

HI

(MPa

)

120576120576o = 1

120576120576o = 1025

120576120576o = 1045

120576120576o = 1070

hmax (nm)

Figure 10 Indentation hardness of prestrained epoxy (20 phr)where 120576 and 120576o are respectively prestrain level and initial uniaxialyield strain [24]

depth ℎmax for plastically prestrained epoxy samples Incontrast to Figure 9 the hardness of epoxy (depicted inFigure 10) decreases as prestraining increases The decreasein the hardness of prestrained epoxy samples was attributedto the Bauschinger effect [24] The Bauschinger effect hasbeen experimentally observed in polymers when a prismaticspecimen compressively prestrained in the uniaxial directionexhibits softening when it is compressed in the orthogonaldirection [70] Similarly indentation hardness 119867I in theorthogonal direction decreases with inducing prestrain asa result of Bauschinger effect [24] Taking the Bauschingereffect into account an energy based methodology was pre-sented in [24] to separate the size effect from the prestrainsoftening effect (Bauschinger effect) As shown in Figure 11after correcting for the Bauschinger effect the indentationhardness 119867I increases with increasing plastic strain whichis in agreement with the hardening behavior observed inFigure 9 It is seen in Figure 11 that size effect in epoxydiminishes with increasing plastic deformation

In addition to epoxy indentation size effect of thermosetPMR-15 polyimide has been also studied in the literatureRedrawn from [40] the indentation hardness of PMR-15polyimide as a function of indentation depth is depicted inFigure 12 A sharp Berkovich tip with a nominal tip radius of119903 lt 20 nm was used to determine the indentation hardnessof Figure 12 where a 2 s loading time was applied for allindentation tests Creep effects were also taken into accountin [40] for the determination of119867I of Figure 12 where ℎmax in(2) was replaced by ℎmax minus ℎcreep with ℎcreep being the changein the indentation depth during the holding time and 119878 in (3)was replaced by 119878e which is the elastic contact stiffness withconsidering the creep effect

1119878e=1119878minusℎv

(8)


0 500 1000 1500 2000 2500 3000 3500200

220

240

260

280

300

320

340

120576120576o = 1120576120576o = 1025

120576120576o = 1045

120576120576o = 1070

hmax (nm)

HI

(MPa

)

Figure 11 Indentation hardness of 20 phr epoxy after correction forthe Bauschinger effect where 120576 and 120576o are respectively prestrain leveland initial uniaxial yield strain [24]

0 1000 2000 3000 4000 5000

04

05

06

07

08

hmax (nm)

HI

(GPa

)

Figure 12 Indentation hardness of PMR-15 polyimide with aBerkovich tip [40]

where ℎv is the creep rate at the end of the holding time and is the unloading time Thus the contact depth ℎc of (2) isdetermined as

ℎc = (ℎmax minus ℎcreep) minus 120598119865max (1119878minusℎv

) (9)

It is seen in Figure 12 that indentation size effect in PMR-15 polyimide is less pronounced than in epoxy and becomesnoticeable only at shallow indentation depths of ℎmax lt

500 nm [40]

222 Thermoplastic Polymers Performing nanoindentationtests with a Berkovich indenter tip indentation size effect offour thermoplastic polymers PC PS PMMA andUHMWPEwas investigated by Briscoe et al [15] The effect of tip

10 100 1000 10000200

300

400

500

600

700

800

hmax (nm)

HcorrectedI

HuncorrectedI

HI

(MPa

)

Figure 13 Effect of tip bluntness on the indentation size effect inPMMA(119867corrected

I and119867uncorrectedI represent the indentationhardness

with andwithout considering the effect of tip bluntness resp) where10 s loading holding and unloading times together with strain rateof 01 sminus1 were imposed [15]

bluntnessimperfection on the size effect of PMMA isdepicted in Figure 13 where 119867corrected

I (indentation hardness119867I obtained by a calibrated area function) starts to deviatefrom 119867

uncorrectedI (indentation hardness 119867I assuming a per-

fectly sharp Berkovich tip) at ℎmax of below about 400 nm Itis seen in Figure 13 that tip bluntness becomes important atshallow indentation depths however length scale dependentdeformation in PMMA is still significant even after tipbluntness correction Taking the effect of tip bluntness intoaccount indentation hardness of amorphous polymers PCPS and PMMA (including 119867corrected

I of Figure 13) is shownin Figure 14 where 10 s loading holding and unloadingtimes together with strain rate of 01 sminus1 were imposed inall indentation tests Size effects are observed for all threepolymers as indentation hardness (after correction for thetip bluntness) increases with decreasing indentation depthHigh scatter in the experimental data is observed for all threepolymers at shallow indentation depths below 100 nm indi-cating the presence of potential inaccuracies in the hardnessdetermination at these small depths Ignoring hardness databelow 100 nm due to possible inaccuracies such as surfaceroughness remarkable length scale dependent deformationsare still observed at ℎmax gt 100 nm in all three polymersSimilarly size effects in PC PS and PMMA can be observedin Figure 15 where the universal hardness (after correctionfor the tip bluntness) of these polymers determined from thesame experiments of Figure 13 is plotted over the indentationdepth Higher scatter in the data is again observed at smallindentation depths ℎmax gt 100 nm It should be notedthat while size effects are clearly observed in PC PS andPMMAat depths of above 100 nm the indentation hardness ofUHMWPEobtained from the same experiments of Figures 14


10 100 1000 10000150

200

250

300

350

400

450

500

PCPSPMMA

hmax (nm)

HI

(MPa

)

Figure 14 Indentation hardness of thermoplastic polymersobtained with 10 s loading holding and unloading times togetherwith 01 sminus1 strain rate [15]

10 100 1000 10000100

150

200

250

300

350

400

450

PCPSPMMA

HU

(MPa

)

hmax (nm)

Figure 15 Universal hardness of thermoplastic polymers [15]obtained from the same experiments of Figure 14 (10 s loadingholding and unloading times together with 01 sminus1 strain rate)

and 15 does not show size effect for the same depth range (seeFigure 2 where indentation hardness of UHMWPE remainsalmost constant)

Length scale dependent deformation of thermoplasticpolymers is also observed in [28]where nanoindentation testswere performed by a Vicker tip at constant tip velocity duringloading and unloading together with a 6 s holding time Themaximum applied force 119865max varies from 02 to 300mNwhere loading rate also increases from 014 to 7061mNswith increasing 119865max Figure 16 shows indentation size effect

0 2000 4000 6000 8000 10000150

200

250

300

350

PSPMMA

HU

(MPa

)

hmax (nm)

Figure 16 Universal hardness of thermoplastic polymers [28]

Table 2 11986702mNU 119867

bulkU ratio where 11986702mN

U and 119867bulkU are respec-

tively the universal hardness at 119865max = 02mN and large indentationdepths [28]

Sample 11986702mNU 119867

bulkU

PS 184SAN 1 179SAN 2 173PMMA 1 178PMMA 2 177PVC 1 160PVC 2 157PC 171

in PS and PMMA where universal hardness increases withdecreasing ℎmax [28] In addition to PS and PMMA sizeeffects were also observed in other amorphous thermoplasticpolymers such as PC PVC and SAN where the hardnessat shallow indentation depth (universal hardness at 119865max =02mN) was found to be significantly higher than the bulkhardness (universal hardness at large indentation depths)[28] The ratio of the universal hardness at 119865max = 02mN(11986702mN

U ) to the bulk microhardness (119867bulkU ) is shown in

Table 2 for different amorphous thermoplastic polymers ofPS PMMA PC PVC and SAN This ratio is around 18 forall glassy polymers of Table 2 however for the more ductilepolymer of PVC this ratio is slightly lower (about 16) [28]The effect of material inhomogeneity (change in the materialproperties through thickness) and surface roughness on thesize effect of polymers was also experimentally shown in [28]In Figure 17 the universal hardness of PS is depicted as afunction of indentation depth after 05 1 and 5 120583m of the topsurface are removed A significant decrease is observed in thehardness of PS after removing the top layers of the surfaceindicating a softening after surface removal The softeningwas speculated in [28] to be attributed to a rough surface


100 1000 10000160

180

200

220

240

260

280

300

320

HU

(MPa

)

hmax (nm)

50 120583m10 120583m

05 120583m00 120583m

Figure 17 Universal hardness of PS when the first 05 1 and 5 120583mlayers of the top surface are removed [28]

profile (compared to the smooth original surface) due to thecutting process

Size effects were also observed in nylon 66 (PA66) bynanoindentation experiments with a Berkovich tip [39 7172] Nanoindentation tests were performedwith a continuousstiffness measurement (CSM) approach as follows loadingsegment with a constant strain rate of 005 sminus1 to a depthof 5000 nm a holding time of 60 s at the maximum appliedforce followed by unloading segment with the same rate ofloading until 10 of the maximum applied force 119865max isreached The indentation size effect of PA66 is illustrated inFigure 2 where 119867I increases with decreasing ℎmax at shallowindentation depths As observed in [39] polishing doesnot significantly affect the hardness of PA66 depicted inFigure 2 The indentation size effect of PA66 was found tobe unaffected by polishing as almost similar trends in thehardness were obtained for both polished and unpolishedsamples [39] Performing nanoindentation tests with thesame experimental details of PA66 [39 71 72] size effectswere also experimentally observed in nylon 6 (PA6) [73ndash75] which exhibited a similar characteristic as observed inFigure 2 for PA66

In addition to nylon 66 (PA66) [39 71 72] and nylon 6(PA6) [73ndash75] length scale dependent deformation was alsoexperimentally observed in nylon 11 (PA11) by nanoindenta-tion testing with a Berkovich indenter tip [63] As shown inFigure 18 indentation hardness of PA11 remarkably increaseswith decreasing indentation depth The same experimentaldetails of PA66 [39 71 72]mentioned abovewere also appliedin the nanoindentation tests of PA11 However comparedto PA66 where size effect becomes significant at depthsof around 200 nm size effects in PA11 appear at largerindentation depths (at around ℎmax asymp 2000 nm) as shown inFigure 18

0 1000 2000 3000 4000 5000008

009

01

011

012

013

014

015

016

HI

(GPa

)

hmax (nm)

Figure 18 Indentation hardness of nylon 11 (PA11) obtained by aBerkovich tip and CSM method (a 005 sminus1 strain rate is imposedduring loading and unloading with a 60 s holding time) [63]

0 5 10 15 20 25 30 35

100

150

200

250H

ardn

ess (

MPa

)

hmax (120583m)

Universal hardness HU

Indentation hardness HI

Figure 19 Indentation hardness 119867I and universal hardness 119867U ofPAI obtained with a Berkovich tip with a 20 s loading and unloadingtimes (without holding time) [64]

Load-controlled indentations on polyamide-imide (PAI)with a Berkovich tip showed that size effect becomes sig-nificant at ℎmax below 10 120583m [64] Indentation hardness 119867Iand universal hardness 119867U of PAI for an indentation depthrange of about 30 to 1 120583m are depicted in Figure 19 where 20 sloading andunloading times (without holding)were imposedin all tests [64]

Indentation hardness of glassy poly(ethylene terephtha-late) (PET) obtained with a Vicker tip is shown in Figure 20where a 6 s holding time was applied in all nanoindentationtests [61] It is seen in Figure 20 that indentation hardness119867I is independent of indentation depth ℎmax at ℎmax gt

2000 nm However 119867I of PET significantly increases withdecreasing ℎmax at small indentation depths (see 119867uncorrected

I


0 2000 4000 6000 8000 10000140

150

160

170

180

190

200

210

220

HI

(MPa

)

HuncorrectedU

HcorrectedU

hmax (nm)

Figure 20 Indentation hardness of glassy poly(ethylene terephtha-late) (PET) obtained with a Vicker tip (119867corrected

I and 119867uncorrectedI

represent the indentation hardnesswith andwithout considering theeffect of tip bluntness resp) where a holding period of 6 s was usedat the maximum applied force [61]

in Figure 20) This increase in the hardness of PET withdecreasing indentation depth was attributed to indenter tipdefects in [61] which will be later discussed in Section 3Replacing the contact depth ℎc of (2) with the corrected con-tact depth ℎ1015840c which includes tip bluntness indentation hard-ness of PET is depicted in Figure 20 where hardness remainsalmost constant with indentation depth (see 119867corrected

I inFigure 20)

As mentioned above indentation size effects becomemore pronounced in thermoset epoxy with increasing cross-link density [23] Nanoindentation tests on thermoplasticPS with a Berkovich tip (with a tip curvature radius 119903 =50 nm) revealed that the indentation size effects increase withincreasing molecular weight [58] Six different polystyrenestandards withmolecular weights (119872p) of 4920 10050 1988052220 96000 and 189300 gmol were studied by nanoinden-tation testing [58] where 12 120583mmin loading and unloadingrates were imposed without holding time at peak loadFigure 21 demonstrates the indentation size effect of PS with119872p of 52220 gmol as indentation hardness 119867I determinedby OampP method [53] increases with decreasing depth Theexperimental hardness data of Figure 21 has been fitted withthe theoretical hardness model of Nix and Gao [1]

119867 = 119867lowastradicℎlowast

ℎ+ 1 (10)

where 119867lowast is the hardness at infinitemacroscopic depthand ℎlowast is a length scale parameter As shown in Figure 22and Table 3 the length scale parameter ℎlowast increases withincreasing molecular weights [58] Such an increase in thelength scale parameter ℎlowast indicates that size effect in PS

Table 3 Molecular length 119871 end-to-end distance 119877 length scaleparameter ℎlowast and ℎlowast119877 ratio of PS with different molecular weights119872p [58]

119872p (gmol) 119871 (120583m) 119877 (120583m) ℎlowast (120583m) ℎ

lowast119877

4920 0015 00047 0082 1710050 0030 00068 0100 1519880 0059 00095 0150 1652220 0150 00150 0170 11

becomes more pronounced by increasing molecular weightThis change in the length scale parameter ℎlowast with molecularweight is depicted in Figure 22 for 119872p of 4920 1005019880 and 52220 gmol Asmolecular length 119871 increases withincreasing molecular weights (see Table 3) it was concludedthat ℎlowastand subsequently the size effect in PS increase withincreasingmolecular length [58] Note that in Figure 22 thereis a sudden change in the slope of ℎlowast minus 119872p curve between19880 and 52220 gmol A possible explanation for such asudden change in Figure 22 might be associated with thedegree of entanglement [58] As shown in Figure 22 a criticalmolecular weight119872c = 36000 gmol is reported for PS whichcan be associated with the minimum length of chains neededfor generating chain entanglements Polystyrene sampleswith molecular weights lower than 119872c are characterizedby few or no chain entanglement (PS samples with 119872p of4920 10050 and 19880 gmol) while polystyrene sampleswith molecular weights higher than119872c are characterized byconsiderable number of chain entanglement (PS sample with119872p of 52220 gmol) which may influence the length scaledependent deformation of PS [58] It should be mentionedthat indentation tests on PS samples with high molecularweights (PS samples with 119872p of 19880 and 52220 gmol)provided highly scattered hardness data for which no lengthscale parameter (ℎlowast) could be deduced [58] As shown inTable 3 analogous to size effect in metals where ℎlowast119887 isconstant (119887 is the Burgers vector) [58] the ratio ℎlowast119877 is almostconstant in PS where 119877 is the end-to-end distance (see [58]and references therein for further information about 119877) Inaddition to the Oliver and Pharrrsquos method [53] a direct mea-surementmethod was also used to determine the indentationhardness of PS samples where the contact area of the sampleat the maximum applied force was calculated by an atomicforce microscope (AFM) The hardness data obtained fromthe direct measurement method 119867d

I was found to be lowerthan the ones determined by Oliver and Pharr [53] Howeversimilar to the trend of 119867I in Figure 21 a significant increasewas observed in119867d

I (hardness of PS determined by the directmeasurement method) with decreasing ℎmax illustrating theexistence of length scale dependent deformation in PS [58]

223 ElastomerRubber Redrawn from [36] the indentationhardness of a silicon elastomer as a function of indentationdepth is depicted in Figure 2 Comparing with the size effectsobserved in glassy polymers (both thermoset and thermo-plastic) presented above the size effects in the silicon elas-tomer [36] are more pronounced as the hardness increases


0 05 1 15 2 2501

015

02

025

03

035

04

HI

(GPa

)

hmax (120583m)

Figure 21 Indentation hardness of PS with molecular weights (119872p)of 52220 according to Oliver and Pharr method [53] obtained by aBerkovich tip with 12 120583mmin loading and unloading rates and zeroholding time [58]

0 10 20 30 40 50 60008

01

012

014

016

018

Disentangled Entangled

hlowast

(120583m

)

Mc

Mp (kgmol)

Figure 22 The length scale parameter ℎlowast of (10) for PS withmolecular weights of 4920 10050 19880 and 52220 gmol [58]

by a factor of about seven with decreasing indentation depthfrom about 1500 to 200 nm Such an amazing increase inthe indentation hardness of silicon elastomer with decreasingdepth is also reported in [37] which is redrawn in Figure 23Nanoindentation tests were carried out by a Berkovich tipwith constant loading and unloading rates of 150 120583Ns whereOliver and Pharrrsquos method [53] was adopted to determine theindentation hardness of Figure 23 [37]

Other nanoindentation studies reported in the literatureon filled silicon rubber [38] natural rubber [76] and PDMS[30ndash35] also show significant indentation size effects whichare also considerably more pronounced than in thermosetand thermoplastic polymers The size effects in PDMS canbe observed in Figure 24 where 119867I of two different samplesof PDMS (prepared in the same way with 1 1 weight ratioof base agent to curing agent) is plotted with respect toℎmax [32] Each point in Figure 24 represents ten indentationscarried out by a Berkovich tipwith a curvature radius of about

0 200 400 600 800 10000

02

04

06

08

1

12

hmax (nm)

HI

(GPa

)

Figure 23 Indentation hardness of silicon elastomer determinedwith a Berkovich tip and the same loading and unloading rates of150 120583Ns [37]

0 100 200 300 400 5000

5

10

15

20

25

Sample 1Sample 2

HI

(MPa

)

hmax (nm)

Figure 24 Indentation hardness of PDMS from two differentsamples obtained by a Berkovich tip with the same loading andunloading times of 40 s and a holding time of 3 s (each pointrepresents ten indentations) [32]

100 nm [32] Nanoindentation tests were performed with thesame loading and unloading times of 40 s together with a 3 sholding time

Similarly indentation size effect in PDMS was observedin [33] where universal hardness 119867U obtained by a sharpBerkovich tip was found to significantly increase withdecreasing indentation depth from 5000 to 100 nm Figure 25illustrates the length scale dependent deformation of PDMSwith 10 cross-link density by volume where 119867U is deter-mined with three loading times of 5 20 and 80 s [33] Itis seen that size effect in PDMS is time independent forℎmax gt 1000 nm However 119867U increases with decreasingloading time at shallow indentation depths ℎmax lt 1000 nm


10 100 1000 100000

5

10

15

20

25

30

35

40

hmax (nm)

HU

(MPa

)

80 s20 s5 s

Figure 25 Universal hardness of PDMS determined by a Berkovichtip with different loading times of 5 20 and 80 s [33]

0 10 20 30 400

02

04

06

08

1

80 s20 s

HU

(MPa

)

hmax (120583m)

Figure 26 Universal hardness of PDMS determined by a Berkovichtip with different loading times of 20 and 80 s [59]

indicating time dependence of size effect in PDMS at smallindentation depths [33] In a recent study indentation sizeeffect of PDMS (with 10 cross-link density by volume) wasinvestigated for an indentation depth of about 5 to 30120583m[59]Universal hardness of PDMS determined with a Berkovichtip is shown in Figure 26 for two different loading times of20 and 80 s [59] It should be here mentioned that similar tothermoset epoxy [23] size effect in PDMS was observed toincrease with increasing cross-link density [34] Performingindentation tests with aVicker indenter tip on PDMS samples(with three different cross-link densities of 25 5 and10 by volume) showed that size effect in PDMS becomes

more pronounced with increasing cross-link density [34] Asshown in [30 33 34 59] and depicted in Figures 24ndash26in addition to being significant size effect in PDMS can beobserved at very high indentation depths (indentation depthin which size effect in PDMS becomes significant was foundto be dependent on the cross-link density of PDMS [34])However in contrast to [30 33 34 59 77] size effect in PDMSwas only observed at small indentation depths ℎmax lt 200 nm[78 79] indicating that size effect in soft polymers is morecomplicated as it can be caused by various sources (such assurface detection or adhesion) which will be discussed later

3 Origins of Indentation SizeEffect in Polymers

In metals size effect is usually considered to be associatedwith geometrically necessary dislocations arising from anincrease in strain gradients related to nonuniform plasticdeformation [1] However size effect characteristics in poly-mers are more complicated than metals as size effect inpolymers has been also observed in elastic deformation [1718 30 33] Also in some polymers as in UHMWPE [15] sizeeffects are essentially absent Various explanations suggestedas the origin of size effect in polymers will be reviewed in thefollowing

31 Higher Order Displacement Gradients Similar to metalswhere indentation size effect is considered to be associatedwith an increase in strain gradients with decreasing indenta-tion depth [1 4] size effects in polymers are also attributedto higher order gradients in the displacement [16 17 22ndash25 29 33 49ndash51 68]The effects of higher order displacementgradients on the length scale dependent deformation ofpolymers were experimentally studied in [25] where theelastic modulus of epoxy was obtained by two indenter tipswith different geometries a Berkovich tip with a curvatureradius 119903 lt 40 nm and a spherical tip with a radius of119877 = 250120583m Figure 27 depicts the elastic modulus of epoxydetermined by the Berkovich tip 119864OampP (using Oliver andPharr method [53] see (7)) and the spherical tip 119864Hertz

(using Hertz theory [80]) where the elastic modulus 119864Hertz

is obtained by the Hertz contact theory

119865 =43(ℎ

119877)

321198772119864Hertzr (11)

in which 119865 is the applied force ℎ the probing depth 119877 theradius of the sphere and 119864Hertz

r the reduced elastic modulusobtained from Hertz theory which is related to the elasticmodulus of polymer 119864Hertz via

1119864Hertzr

=1 minus V2

119864Hertz+1 minus V2i119864i

(12)

where similar to (7) 119864i and Vi are respectively the elasticmodulus and Poissonrsquos ratio of indenter tip with V being thePoissonrsquos ratio of the polymer As it is seen in Figure 27 the


0 1000 2000 3000 40000

2

4

6

8

10

Elas

tic m

odul

us (G

Pa)

Berkovich tip (Oliver and Pharr)Spherical tip (Hertz)

hmax (nm)

Figure 27 Elastic modulus of epoxy [25] determined with aBerkovich tip 119864OampP (using Oliver and Pharr method [53] with 20 sloading holding and unloading times) and with a spherical tip119864Hertz (using Hertz theory with 20 s loading time)

elastic modulus of epoxy obtained with the Berkovich tip119864OampP increases with decreasing ℎmax while 119864Hertz obtained

by the spherical tip remains constant with ℎmax [25] Such adifferent trend in the elastic modulus of epoxy determinedby the Berkovich and spherical tips over the probing depths isattributed to the effect of higher order displacement gradients[25] As shown in Figure 28 applying a sharp indenter tipthe higher order displacement gradients (strain gradients120594 inFigure 28) increase inmagnitudewith decreasing indentationdepth while these gradients remain essentially constant withℎmax when a spherical tip is used [65 66] Note that for aspherical tip the strain gradient 120594 is only dependent on theradius of tip 119877 and is essentially independent of indentationdepth as shown in Figure 28 Similar trends have been alsoobserved in PDMS [59] where elastic modulus of PDMSdetermined by a Berkovich tip increases with ℎmax while119864Hertz obtained from spherical tests remains almost constant

with indentation depth As illustrated in Figure 29 the elasticmodulus of PDMS determined with a Berkovich tip (119864OampP

and 119864Sneddon using OampPmethod and Sneddonrsquos theory resp)increases with decreasing depth while 119864Hertz (applying Hertztheory) obtainedwith a spherical tip (with a radius of 250120583m)does not change with indentation depth For elastic materialsand a conical tip 119864Sneddon [81] for frictionless indentationcan be obtained by Sneddonrsquos theory via the following load-displacement relation

119865 =2 tan (120572) 119864Sneddon

120587 (1 minus V2)ℎ2 (13)

where 120572 = 703∘ is the half angle defining a cone equivalentto a Berkovich tip It should be here mentioned that similarto Figures 27 and 29 Swadener et al [2] observed different

trends for the hardness of metals determined by Berkovichand spherical tips Similar to the present finding in epoxy andPDMS it was shown that for metals hardness determinedwith a Berkovich tip increases with decreasing ℎmax whereashardness is independent of indentation depth when a spher-ical tip is used [2]

As classical elastoplastic theories do not take the effectsof higher order displacement gradients into account lengthscale dependent deformation of polymers presented abovecannot be predicted by these local classical theories Tocapture the abovementioned size effect in polymers fewtheoretical models have been developed where size effect ofpolymers is attributed to the effect of higher order displace-ment gradients at micro to submicro scales

Based on amolecular kinkingmechanism and themolec-ular theory of yielding of glassy polymers a strain gradientplasticity lawwas developed in [16 22] where amolecular andtemperature dependent strain gradient plasticity parameterwas proposed and related to indentation hardness A theoret-ical indentation hardness model

119867 = 119867lowast(1+radicℎlowast

ℎ) (14)

was proposed in [16 22] which shows good agreementwith the experimental hardness data of the thermoset epoxyand the thermoplastic PC [16] depicted in Figure 2 where119867lowastis the hardness at infinite depth and ℎ

lowastis a constant

dependent on material behavior and temperature [16 22] Astrain gradient plasticity was suggested in [16 22] which wasincorporated into a viscoelastic-plastic constitutive modelof [82] to predict size effect of glassy polymers observedin nanoindentation experiments [51] Finite element simu-lation results demonstrated that higher order gradients ofdisplacement should be incorporated into constitutive modelto capture the length scale dependent deformation of glassypolymers in micronanoindentations [51]

Another higher order displacement gradients basedmodel was developed in [49] where size effect in elasticdeformation of polymers was attributed to rotation gradi-ents effects arising from finite bending stiffness of polymerchains and their interactions To account for the effect ofhigher order gradients of displacement on the deformationmechanism of polymers the local elastic deformation energydensity 119882e

120576is augmented by a nonlocal rotation gradients

term119882120594yielding the total elastic deformation energy density

119882 [49]

119882 = 119882e120576+119882120594=12120576119894119895119862119894119895119896119897120598119896119897+

3120594S119894119895120594S119894119895 (15)

where 120576119894119895is the infinitesimal strain tensor119862

119894119895119896119897is the elasticity

tensor and is a material constant (see [49] where is micromechanically motivated) Following couple stresselasticity theory of [83] it is seen in (15) that only thesymmetric part of the rotation gradient

120594S119894119895=12(120594119894119895+120594119895119894) =

12(120596119894119895+120596119895119894) (16)


hmaxhmax

1205791L

120594 =1205791L

prop1

hmax

1205792

R1205792

R

120594 =1205792R1205792

=1

R

Figure 28 Dependence of strain gradient 120594 on ℎmax for different tip geometries [65 66]

0 5 10 15 20 25 30 350

1

2

3

4

5

6

Elas

tic m

odul

us (M

Pa)

Spherical tip (Hertz)Berkovich tip (Sneddon)Berkovich tip (Oliver and Pharr)

hmax (120583m)

Figure 29 Elastic modulus of PDMS [59] determined with loadingand unloading times of 80 s (without holding time)

is assumed to contribute to the total elastic deformationenergy density119882 (more specifically to119882

120594in (15)) where the

symmetric part of the rotation gradient tensor 120594S119894119895in (16) can

be either defined by the rotation gradient tensor

120594119894119895=12119890119894119899119898119906119898119895119899

(17)

or the infinitesimal rotation vector

120596119894=12119890119894119898119899119906119899119898 (18)

where 119906119894 119906119894119895 and 119890

119894119898119899are the displacement vector displace-

ment gradients and permutation symbol respectively [49]The abovementioned rotation gradient based elastic formu-lation was extended in [50] to include inelastic deformationand a theoretical hardness model was developed

119867 = 1198670 (1+119888ℓ

ℎ) (19)

where similar to 119867lowastin (14) 119867

0is the hardness at infi-

nitemacroscopic depth and 119888ℓis the material length scale

parameter which has been micromechanically motivatedin [50] The theoretical hardness model (19) showed goodagreement with the experimental hardness data of PC [68]epoxy [25 68] PS [68] PAI [68] PA66 [68] PMMA [68] andPDMS [33] Based on the hardness model (19) a theoreticalstiffness model

119870 = 1198670119862 (119888ℓ + 2ℎ) (20)

was proposed by Alisafaei et al [25] where the parameters1198670and 119888ℓare obtained by fitting the hardness model (19)

to the experimental hardness data and 119862 is a constant equalto 2643 for Berkovich tip The theoretical stiffness model(20) was compared with the experimental stiffness 119870m ofepoxy [25] and PDMS [33] obtained from nanoindentationtests 119870m is obtained by calculating the derivative of force119865 with respect to indentation depth ℎ at the last datapoint of loading curve (point B in Figure 1(a)) [25 33] InFigures 30 and 31 the theoretical stiffness model (20) iscompared with the experimental data of 119870m obtained forPDMS [33] and epoxy [25] respectively As demonstratedin Figures 30 and 31 good agreement is found between theexperimental stiffness data of PDMS and epoxy with thetheoretical stiffness model (20) corroborating the validity ofthe hardnessmodel (19) to predict the size effects in elastomerPDMS and thermoset epoxy It is seen in Figures 30 and31 that unlike materials which do not exhibit size effect119870m experimentally determined for PDMS and epoxy do notapproach zero when ℎmax rarr 0 This trend observed in theexperimental stiffness 119870m is consistent with the theoreticalstiffness model (20) which predicts a nonzero value of thestiffness (119870 = 119888

ℓ1198670119862) when ℎmax rarr 0 For materials which

do not show size effect (119888ℓ= 0) the theoretical stiffness

model (20) predicts a linear relation between the stiffness 119870and indentation depth ℎ (119870 = 2119867

0119862ℎ) which is consistent

with experimental stiffness data of fused silica in [25 33] (sizeeffect in fused silica is known to be not pronounced [84])

In addition to the two higher order displacement gradientbased theories [22 49] mentioned above extending themechanism based theories of crystalline metals Voyiadjis


100 100010

100

1000

Inde

ntat

ion

stiffn

ess (

Nm

)

hmax (nm)

80 s experimental results80 s stiffness model (20)

Figure 30 Comparison of the experimental stiffness 119870m of PDMSobtained with a Berkovich tip with the indentation stiffness model(20) where 80 s loading time was used in nanoindentation experi-ments [33]

10 100 1000

1

10

Inde

ntat

ion

stiffn

ess (

kNm

)


hmax (nm)

Figure 31 Comparison of the experimental stiffness 119870m of epoxyobtained with a Berkovich tip with the indentation stiffness model(20) where 80 s loading time was used in nanoindentation experi-ments [25]

et al [29] recently developed a strain gradient plasticityformulation for semicrystalline polymers inwhich size effectsin semicrystalline polymers were attributed to crystallinesubphase embedded within an amorphous matrix Twokink mechanisms of grounded kink and active kink wereconsidered to respectively capture nonlocalized effects (eguniformplastic deformation) and localized effects (viscoplas-tic deformation effects induced by nonuniform slippagemechanisms eg localized plastic deformation which is

found in indentation tests around a sharp indenter tip)Utilizing a Cosserat continuum formulation and adoptingconsiderations in [49] size effects in amorphous polymerswere also elaborately modeled in [29] where a flow rulewas proposed for amorphous polymers Furthermore atheoretical hardness model was developed

(119867

1198671015840)

120573

= 1+(ℎ1015840

ℎ)

1205732

(21)

where similar to (10) (14) and (19)1198671015840 is the hardness at infi-nite depth (microhardness of material which is determinedin the absence of size effect) 120573 is the tip geometry correctionfactor represented in (6) and defined in Table 1 and ℎ1015840 is amaterial constant which is related to the material length scale[29]

It may be worth noting that for the material model of(15) to (18) a finite element procedure has been developed in[85 86] with which two-dimensional numerical simulationshave been performed in [87] to study stiffening effects in rein-forced polymer composites with decreasing reinforcementsizes [21] Similar to the experimental results [21] numericalsimulations of [87] showed stiffening of reinforced polymercomposites with decreasing reinforcement size at high fibervolume fractions

311 Influence of Molecular Structure on ISE As discussedin Section 2 size effects in elastomers are considerablymore pronounced than in plastic polymers (thermoset andthermoplastic polymers) However as shown in Figure 2the characteristics of size effects even in thermoset andthermoplastic polymers were found to differ for differentpolymers The characteristics of size effect in thermosetand thermoplastic polymers were attributed to molecularstructures of these polymers [68] which are discussed here

Depth Dependent Hardness of Polymers with Aromatic Com-ponents Indentation size effect of PC [16] shown in Figure 2is redrawn in Figure 32 together with a sequence of itsmolecular structure [68] and the theoretical hardness model(19) where 119867

0= 0165GPa and 119888

ℓ= 00442 120583m It is seen

that there is excellent agreement between the experimentalhardness data of PC and the theoretical hardness model(19) Such a strong indentation size effect observed at largeindentation depths is also observed in epoxy PS and PAIwhere all these polymers contain aromatic components (see[68] for their molecular structures) and their experimentalhardness data show excellent agreement with the hardnessmodel (19)

Depth Dependent Hardness of Polymers without AromaticComponents Indentation hardness of UHMWPE as a func-tion of ℎmax depicted in Figure 2 is redrawn in Figure 33as well as a sequence of its molecular structure [68] Asmentioned in Section 2 and shown in Figure 33 UHMWPEdoes not exhibit any significant indentation size effect evenat shallow indentation depths ℎmax lt 50 nm As demon-strated in Figure 33 unlike the abovementioned polymerswith aromatic components UHMWPE has a homogenous


01 1 1001

015

02

025

03

035

04

Experiments [16]Hardness model (19)

O

C

H HH

H HH

C

C

C

OO

HI

(GPa

)

hmax (120583m)

Figure 32 Indentation hardness of PC [16] and its molecularstructure [68]

10 100 1000 100000

005

01

015

02

H

C

H

H

C

H

H

C

H

H

C

H

hmax (nm)

HI

(GPa

)

Figure 33 Indentation hardness of UHMWPE [15] and its molecu-lar structure [68]

and highly flexible molecular structure Similar trend canbe observed in PTFE [69] where indentation hardness 119867Iremains constant with indentation depth indicating thatPTFE does not show any indentation size effects at anindentation depth range of about 2000 to 20 nm Notethat similar to UHMWPE shown in Figure 33 PTFE has ahomogenous and highly flexible molecular structure withoutany aromatic components (see [68]) Indentation size effect ofPA66 [39] presented in Figure 2 is also depicted in Figure 34alongwith the hardnessmodel (19) with119867

0= 00905GPa and

119888ℓ= 00276120583m Compared to UHMWPE and PTFE PA66

has a more complex molecular structure (see Figures 33 and34) and shows indentation size effect at smaller indentationdepths (compared to those polymers which contain aromaticcomponents) [68]

10 100 1000 10000005

01

015

02

025

03

PA66nylonHardness model (19)

C

O

C N

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

O

C

H

H

C

H

H

C

H

H

C

H

H

N

H

HI

(GPa

)

hmax (nm)

Figure 34 Indentation size effect of PA66 [39] and its molecularstructure [68]

32 Surface Effect Friction Tip Imperfection and MaterialInhomogeneity In addition to the abovementioned higherorder displacement gradient based models there are alsoother explanations suggested as the origin of size effectin polymers including change in the material propertiesthrough thickness [15] friction [34] surface effects [3652] tip imperfection [61] adhesion and others Unlike inindentation size effects in metals where the elastic modulusremains constant with depth elastic moduli of polymerswere found to increase with decreasing indentation depthAs illustrated in Figure 27 elastic modulus 119864OampP of ther-moset epoxy determined by a Berkovich tip increases withdecreasing indentation depth Corresponding to Figures 14and 15 elastic modulus 119864OampP of thermoplastic polymersPC PS and PMMA obtained from the same experimentsof Figures 14 and 15 increases with decreasing indentationdepth (see Figure 35) In addition to thermoplastic polymersPC PS and PMMA mentioned above elastic moduli ofother thermoplastic polymers PA66 [39] PMR-15 [40] andPA6 [41] were also found to increase with decreasing ℎmax(however opposite to the abovementioned trend for 119864OampP

of thermoplastic polymers elastic modulus 119864OampP of PS in[58] was reported to be independent of indentation depth)In addition to thermoset and thermoplastic polymers elasticmodulus of PDMS elastomer was also shown to increase withdecreasing indentation depth [30 59] As demonstrated inFigure 29 elastic modulus of PDMS obtained by a Berkovichtip (119864OampP and 119864Sneddon using OampP method and Sneddonrsquostheory resp) increases as ℎmax decreases [59] Similarlyin a careful elaborated nanoindentation experiments witha Berkovich tip [30] elastic modulus of PDMS shown inFigure 36 was found to significantly increase with decreasingindentation depth indicating the existence of size effect inelastomers A straightforward possible explanation for sucha change in the elastic modulus can be changes in thematerial properties through the depth of the polymer [15]


10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)

Figure 35 Elastic modulus of thermoplastic polymers according toOliver and Pharr method [53] obtained with 10 s loading holdingand unloading times together with 01 sminus1 strain rate [15]

However in [25 59 76] it has been argued that such materialinhomogeneities (changes in the material properties throughthe depth) cannot be the only source of indentation size effectin polymers as the measured elastic moduli of epoxy andPDMS obtained by a spherical tip (119864Hertz resp presentedin Figures 27 and 29) stay constant with ℎmax while theelastic moduli determined by a sharp Berkovich tip increasewith decreasing indentation depthThe influences ofmaterialinhomogeneity and surface roughness on the size effect ofpolymers have been investigated in [28] where a significantdecrease in the hardness of PS was observed by removingthe top surface as shown in Figure 17 However the hardnessof PS still increases with decreasing ℎmax even after remov-ing the top layers of the surface indicating that materialinhomogeneity and surface roughness cannot be the onlysource of length scale dependent deformation in polymersFurthermore studying the effects of material inhomogeneityon the size effect of epoxy [17] revealed that the significantincrease in the bending rigidity of epoxy microbeams is notassociated with the presence of stiff surface layers Consid-ering material inhomogeneity it should be mentioned thatcertain sample fabricationprocessing methods may inducestructural inhomogeneity that may even result in anisotropicbehavior of the polymer sample [88]

In addition to the material inhomogeneity surfaceeffectstress and surface roughness are also known to beamong the factors that cause indentation size effect at shallowindentation depths [36 52]There exists a critical indentationdepth below which surface effect plays an important role onthe length scale dependent deformation of polymers [36]Adhesion [89ndash94] is also another factor that can cause inden-tation size effect particularly at small indentation depths ofsoft polymers [59 60]

0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)

Figure 36 Elastic modulus of PDMS (with 15 cross-link densityby weight) determined by a Berkovich tip [30]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Thematerial of this paper is based on work supported by theUS National Science Foundation under Grants no 1102764and no 1126860

References

[1] W D Nix and H Gao ldquoIndentation size effects in crystallinematerials a law for strain gradient plasticityrdquo Journal of theMechanics and Physics of Solids vol 46 no 3 pp 411ndash425 1998

[2] J G Swadener E P George and G M Pharr ldquoThe correlationof the indentation size effectmeasuredwith indenters of variousshapesrdquo Journal of the Mechanics and Physics of Solids vol 50no 4 pp 681ndash694 2002

[3] R K Abu Al-Rub ldquoPrediction of micro and nanoindentationsize effect from conical or pyramidal indentationrdquoMechanics ofMaterials vol 39 no 8 pp 787ndash802 2007

[4] G Feng andW D Nix ldquoIndentation size effect inMgOrdquo ScriptaMaterialia vol 51 no 6 pp 599ndash603 2004

[5] A A Elmustafa and D S Stone ldquoIndentation size effect inpolycrystalline FCC metalsrdquo Acta Materialia vol 50 no 14pp 3641ndash3650 2002

[6] R K Abu Al-Rub and G Z Voyiadjis ldquoAnalytical and exper-imental determination of the material intrinsic length scaleof strain gradient plasticity theory from micro- and nano-indentation experimentsrdquo International Journal of Plasticity vol20 no 6 pp 1139ndash1182 2004

[7] Q Ma and D R Clarke ldquoSize dependent hardness of silversingle crystalsrdquo Journal of Materials Research vol 10 no 4 pp853ndash863 1995

[8] C Motz T Schoberl and R Pippan ldquoMechanical properties ofmicro-sized copper bending beams machined by the focusedion beam techniquerdquo Acta Materialia vol 53 no 15 pp 4269ndash4279 2005


[9] N A Fleck G M Muller M F Ashby and J W HutchinsonldquoStrain gradient plasticity theory and experimentrdquo Acta Metal-lurgica et Materialia vol 42 no 2 pp 475ndash487 1994

[10] L Tsagrakis and E C Aifantis ldquoRecent developments ingradient plasticity Part I Formulation and size Effectsrdquo Journalof EngineeringMaterials and Technology vol 124 no 3 pp 352ndash357 2002

[11] M B Taylor H M Zbib and M A Khaleel ldquoDamage and sizeeffect during superplastic deformationrdquo International Journal ofPlasticity vol 18 no 3 pp 415ndash442 2002

[12] Y Cao S Allameh D Nankivil S Sethiaraj T Otiti and WSoboyejo ldquoNanoindentation measurements of the mechanicalproperties of polycrystalline Au and Ag thin films on siliconsubstrates effects of grain size and film thicknessrdquo MaterialsScience and Engineering A vol 427 no 1-2 pp 232ndash240 2006

[13] Y W Yan L Geng and A B Li ldquoExperimental and numericalstudies of the effect of particle size on the deformation behaviorof themetalmatrix compositesrdquoMaterials Science andEngineer-ing A vol 448 no 1-2 pp 315ndash325 2007

[14] Z Xue Y Huang and M Li ldquoParticle size effect in metallicmaterials a study by the theory of mechanism-based straingradient plasticityrdquo Acta Materialia vol 50 no 1 pp 149ndash1602002

[15] B J Briscoe L Fiori and E Pelillo ldquoNano-indentation ofpolymeric surfacesrdquo Journal of Physics D Applied Physics vol31 no 19 pp 2395ndash2405 1998

[16] A C M Chong and D C C Lam ldquoStrain gradient plasticityeffect in indentation hardness of polymersrdquo Journal of MaterialsResearch vol 14 no 10 pp 4103ndash4110 1999

[17] D C C Lam F Yang A C M Chong J Wang and P TongldquoExperiments and theory in strain gradient elasticityrdquo Journalof the Mechanics and Physics of Solids vol 51 no 8 pp 1477ndash1508 2003

[18] AWMcFarland and J S Colton ldquoRole of material microstruc-ture in plate stiffness with relevance tomicrocantilever sensorsrdquoJournal of Micromechanics and Microengineering vol 15 no 5pp 1060ndash1067 2005

[19] A Arinstein M Burman O Gendelman and E ZussmanldquoEffect of supramolecular structure on polymer nanofibreelasticityrdquo Nature Nanotechnology vol 2 no 1 pp 59ndash62 2007

[20] L Sun R P S Han J Wang and C T Lim ldquoModelingthe size-dependent elastic properties of polymeric nanofibersrdquoNanotechnology vol 19 no 45 Article ID 455706 2008

[21] S-Y Fu X-Q Feng B Lauke and Y-WMai ldquoEffects of particlesize particlematrix interface adhesion and particle loadingon mechanical properties of particulate-polymer compositesrdquoComposites Part B Engineering vol 39 no 6 pp 933ndash961 2008

[22] D C C Lam and A C M Chong ldquoIndentation model andstrain gradient plasticity law for glassy polymersrdquo Journal ofMaterials Research vol 14 no 9 pp 3784ndash3788 1999

[23] D C C Lam and A C M Chong ldquoEffect of cross-link densityon strain gradient plasticity in epoxyrdquo Materials Science andEngineering A vol 281 no 1-2 pp 156ndash161 2000

[24] D C C Lam and A C M Chong ldquoCharacterization andmodeling of specific strain gradient modulus of epoxyrdquo Journalof Materials Research vol 16 no 2 pp 558ndash563 2001

[25] F Alisafaei C-S Han and N Lakhera ldquoCharacterization ofindentation size effects in epoxyrdquo Polymer Testing vol 40 pp70ndash75 2014

[26] A K Dutta D Penumadu and B Files ldquoNanoindentationtesting for evaluating modulus and hardness of single-walled

carbon nanotubendashreinforced epoxy compositesrdquo Journal ofMaterials Research vol 19 no 1 pp 158ndash164 2004

[27] M Sanchez J Rams M Campo A Jimenez-Suarez and AUrena ldquoCharacterization of carbon nanofiberepoxy nanocom-posites by the nanoindentation techniquerdquo Composites Part BEngineering vol 42 no 4 pp 638ndash644 2011

[28] F J B Calleja A Flores and G H Michler ldquoMicroindentationstudies at the near surface of glassy polymers influence ofmolecular weightrdquo Journal of Applied Polymer Science vol 93no 4 pp 1951ndash1956 2004

[29] G Z Voyiadjis A Shojaei and N Mozaffari ldquoStrain gradientplasticity for amorphous and crystalline polymers with applica-tion to micro- and nano-scale deformation analysisrdquo Polymervol 55 no 16 pp 4182ndash4198 2014

[30] C A Charitidis ldquoNanoscale deformation and nanomechanicalproperties of polydimethylsiloxane (PDMS)rdquo Industrial amp Engi-neering Chemistry Research vol 50 no 2 pp 565ndash570 2011

[31] C Charitidis ldquoNanoscale deformation and nanomechanicalproperties of softmatter study cases polydimethylsiloxane cellsand tissuesrdquo ISRN Nanotechnology vol 2011 Article ID 71951213 pages 2011

[32] E P Koumoulos P Jagdale I A Kartsonakis M Giorcelli ATagliaferro and C A Charitidis ldquoCarbon nanotubepolymernanocomposites a study on mechanical integrity throughnanoindentationrdquo Polymer Composites 2014

[33] F Alisafaei C-S Han and S H R Sanei ldquoOn the time andindentation depth dependence of hardness dissipation andstiffness in polydimethylsiloxanerdquo Polymer Testing vol 32 no7 pp 1220ndash1228 2013

[34] Y Y Lim and M M Chaudhri ldquoIndentation of elastic solidswith a rigid Vickers pyramidal indenterrdquo Mechanics of Materi-als vol 38 no 12 pp 1213ndash1228 2006

[35] A J Wrucke C-S Han and P Majumdar ldquoIndentation sizeeffect ofmultiple orders ofmagnitude in polydimethylsiloxanerdquoJournal of Applied Polymer Science vol 128 no 1 pp 258ndash2642013

[36] T-Y Zhang andW-HXu ldquoSurface effects on nanoindentationrdquoJournal of Materials Research vol 17 no 7 pp 1715ndash1720 2002

[37] W-H Xu Z-Y Xiao and T-Y Zhang ldquoMechanical propertiesof silicone elastomer on temperature in biomaterial applica-tionrdquoMaterials Letters vol 59 no 17 pp 2153ndash2155 2005

[38] R V S Tatiraju andC-S Han ldquoRate dependence of indentationsize effects in filled silicone rubberrdquo Journal of Mechanics ofMaterials and Structures vol 5 no 2 pp 277ndash288 2010

[39] L Shen T Liu and P Lv ldquoPolishing effect on nanoindentationbehavior of nylon 66 and its nanocompositesrdquo Polymer Testingvol 24 no 6 pp 746ndash749 2005

[40] Y C Lu D C Jones G P Tandon S Putthanarat and GA Schoeppner ldquoHigh temperature nanoindentation of PMR-15polyimiderdquo Experimental Mechanics vol 50 no 4 pp 491ndash4992010

[41] R Seltzer J-K Kim and Y-W Mai ldquoElevated temperaturenanoindentation behaviour of polyamide 6rdquo Polymer Interna-tional vol 60 no 12 pp 1753ndash1761 2011

[42] J F Nye ldquoSome geometrical relations in dislocated crystalsrdquoActa Metallurgica vol 1 no 2 pp 153ndash162 1953

[43] A Arsenlis and D M Parks ldquoCrystallographic aspectsof geometrically-necessary and statistically-stored dislocationdensityrdquo Acta Materialia vol 47 no 5 pp 1597ndash1611 1999

[44] N A Fleck and J W Hutchinson ldquoA phenomenological theoryfor strain gradient effects in plasticityrdquo Journal of the Mechanicsand Physics of Solids vol 41 no 12 pp 1825ndash1857 1993


[45] H Gao Y Huang W D Nix and J W HutchinsonldquoMechanism-based strain gradient plasticitymdashI Theoryrdquo Jour-nal of theMechanics and Physics of Solids vol 47 no 6 pp 1239ndash1263 1999

[46] C-S Han H Gao Y Huang and W D Nix ldquoMechanism-based strain gradient crystal plasticitymdashI Theoryrdquo Journal ofthe Mechanics and Physics of Solids vol 53 no 5 pp 1188ndash12032005

[47] Y Huang S Qu K C Hwang M Li and H Gao ldquoA con-ventional theory ofmechanism-based strain gradient plasticityrdquoInternational Journal of Plasticity vol 20 no 4-5 pp 753ndash7822004

[48] Y Huang Z Xue H Gao W D Nix and Z C Xia ldquoA studyof microindentation hardness tests by mechanism-based straingradient plasticityrdquo Journal of Materials Research vol 15 no 8pp 1786ndash1796 2000

[49] S Nikolov C-S Han and D Raabe ldquoOn the origin of sizeeffects in small-strain elasticity of solid polymersrdquo InternationalJournal of Solids and Structures vol 44 no 5 pp 1582ndash15922007

[50] C-S Han and S Nikolov ldquoIndentation size effects in polymersand related rotation gradientsrdquo Journal of Materials Researchvol 22 no 6 pp 1662ndash1672 2007

[51] S Swaddiwudhipong L H Poh J Hua Z S Liu and K KTho ldquoModeling nano-indentation tests of glassy polymers usingfinite elements with strain gradient plasticityrdquoMaterials Scienceand Engineering A vol 404 no 1-2 pp 179ndash187 2005

[52] T-Y Zhang W-H Xu and M-H Zhao ldquoThe role of plasticdeformation of rough surfaces in the size-dependent hardnessrdquoActa Materialia vol 52 no 1 pp 57ndash68 2004

[53] W C Oliver and G M Pharr ldquoImproved technique for deter-mining hardness and elastic modulus using load and displace-ment sensing indentation experimentsrdquo Journal of MaterialsResearch vol 7 no 6 pp 1564ndash1583 1992

[54] ISO ldquoMetallic materialsmdashinstrumented indentation test forhardness and materials parameters Part 1 test methodrdquo TechRep 14577-1 International Organization for Standardization2002

[55] A C Fischer-CrippsNanoindentation Springer NewYorkNYUSA 3rd edition 2011

[56] R Saha andW D Nix ldquoEffects of the substrate on the determi-nation of thin film mechanical properties by nanoindentationrdquoActa Materialia vol 50 no 1 pp 23ndash38 2002

[57] R Rodrıguez and I Gutierrez ldquoCorrelation between nanoin-dentation and tensile properties influence of the indentationsize effectrdquoMaterials Science and Engineering A vol 361 no 1-2pp 377ndash384 2003

[58] J A Tjernlund E K Gamstedt and Z-H Xu ldquoInfluence ofmolecular weight on strain-gradient yielding in polystyrenerdquoPolymer Engineering amp Science vol 44 no 10 pp 1987ndash19972004

[59] C-S Han S H R Sanei and F Alisafaei ldquoOn the originof indentation size effect and depth dependent mechanicalproperties of elastic polymersrdquo Journal of Polymer Engineering2015

[60] J K Deuschle G Buerki H M Deuschle S Enders J Michlerand E Arzt ldquoIn situ indentation testing of elastomersrdquo ActaMaterialia vol 56 no 16 pp 4390ndash4401 2008

[61] A Flores and F J Balta Calleja ldquoMechanical properties ofpoly(ethylene terephthalate) at the near surface from depth-sensing experimentsrdquo Philosophical Magazine A vol 78 no 6pp 1283ndash1297 1998

[62] C A Tweedie G Constantinides K E Lehman D J BrillG S Blackman and K J Van Vliet ldquoEnhanced stiffness ofamorphous polymer surfaces under confinement of localizedcontact loadsrdquo Advanced Materials vol 19 no 18 pp 2540ndash2546 2007

[63] Y Hu L Shen H Yang et al ldquoNanoindentation studies onNylon 11clay nanocompositesrdquo Polymer Testing vol 25 no 4pp 492ndash497 2006

[64] R V S Tatiraju C-S Han and S Nikolov ldquoSize dependenthardness of polyamideimiderdquoTheOpenMechanics Journal vol2 no 1 pp 89ndash92 2008

[65] W W Gerberich N I Tymiak J C Grunlan M F Horste-meyer and M I Baskes ldquoInterpretations of indentation sizeeffectsrdquo ASME Journal of Applied Mechanics vol 69 no 4 pp433ndash442 2002

[66] N I Tymiak D E Kramer D F Bahr T J Wyrobek and WW Gerberich ldquoPlastic strain and strain gradients at very smallindentation depthsrdquo Acta Materialia vol 49 no 6 pp 1021ndash1034 2001

[67] C-S Han ldquoIndentation size effect in polymersrdquo in Advances inMaterials Science Research M C Wythers Ed vol 10 chapter15 pp 393ndash413 Nova Science New York NY USA 2011

[68] C-S Han ldquoInfluence of the molecular structure on indentationsize effect in polymersrdquo Materials Science and Engineering Avol 527 no 3 pp 619ndash624 2010

[69] X Li and B Bhushan ldquoContinuous stiffness measurement andcreep behavior of composite magnetic tapesrdquo Thin Solid Filmsvol 377-378 pp 401ndash406 2000

[70] T K Chaki and J C M Li ldquoLatent hardening in highminusdensitypolyethylenerdquo Journal of Applied Physics vol 56 no 9 pp 2392ndash2395 1984

[71] L Shen I Y Phang L Chen T Liu and K Zeng ldquoNanoinden-tation and morphological studies on nylon 66 nanocompositesI Effect of clay loadingrdquo Polymer vol 45 no 10 pp 3341ndash33492004

[72] L Shen I Y Phang T Liu and K Zeng ldquoNanoindentation andmorphological studies on nylon 66organoclay nanocompos-ites II Effect of strain raterdquo Polymer vol 45 no 24 pp 8221ndash8229 2004

[73] T Liu I Y Phang L Shen S Y Chow and W-D ZhangldquoMorphology andmechanical properties of multiwalled carbonnanotubes reinforced nylon-6 compositesrdquo Macromoleculesvol 37 no 19 pp 7214ndash7222 2004

[74] L Shen I Y Phang and T Liu ldquoNanoindentation studies onpolymorphism of nylon 6rdquo Polymer Testing vol 25 no 2 pp249ndash253 2006

[75] W D Zhang L Shen I Y Phang and T Liu ldquoCarbon nano-tubes reinforced nylon-6 composite prepared by simple melt-compoundingrdquo Macromolecules vol 37 no 2 pp 256ndash2592004

[76] G Chandrashekar F Alisafaei and C-S Han ldquoLength scaledependent deformation in natural rubberrdquo Journal of AppliedPolymer Science 2015

[77] J K Deuschle HM Deuschle S Enders and E Arzt ldquoContactarea determination in indentation testing of elastomersrdquo Jour-nal of Materials Research vol 24 no 3 pp 736ndash748 2009

[78] J Deuschle S Enders and E Arzt ldquoSurface detectionin nanoindentation of soft polymersrdquo Journal of MaterialsResearch vol 22 no 11 pp 3107ndash3119 2007

[79] J K Deuschle E J de Souza E Arzt and S Enders ldquoNanoin-dentation studies on cross linking and curing effects of PDMSrdquo


International Journal of Materials Research vol 101 no 8 pp1014ndash1023 2010

[80] K L Johnson Contact Mechanics Cambridge University Press1985

[81] I N Sneddon ldquoThe relation between load and penetration inthe axisymmetric boussinesq problem for a punch of arbitraryprofilerdquo International Journal of Engineering Science vol 3 no1 pp 47ndash57 1965

[82] L E Govaert P HM Timmermans andW AM BrekelmansldquoThe influence of intrinsic strain softening on strain localizationin polycarbonate modeling and experimental validationrdquo Jour-nal of Engineering Materials and Technology Transactions of theASME vol 122 no 2 pp 177ndash185 2000

[83] F Yang A C M Chong D C C Lam and P Tong ldquoCouplestress based strain gradient theory for elasticityrdquo InternationalJournal of Solids and Structures vol 39 no 10 pp 2731ndash27432002

[84] G Kaupp Atomic Force Microscopy Scanning Nearfield OpticalMicroscopy and Nanoscratching Application to Rough andNatural Surfaces Springer 2006

[85] N Garg and C-S Han ldquoA penalty finite element approach forcouple stress elasticityrdquo Computational Mechanics vol 52 no 3pp 709ndash720 2013

[86] N Garg and C-S Han ldquoAxisymmetric couple stress elasticityand its finite element formulation with penalty termsrdquo Archiveof Applied Mechanics vol 85 no 5 pp 587ndash600 2015

[87] N Garg F Alisafaei G Chandrashekar and C-S Han ldquoFiberdiameter dependent deformation in polymer compositesmdashanumerical studyrdquo Submitted

[88] L Shen W C Tjiu and T Liu ldquoNanoindentation and morpho-logical studies on injection-molded nylon-6 nanocompositesrdquoPolymer vol 46 no 25 pp 11969ndash11977 2005

[89] D M Ebenstein ldquoNano-JKR force curve method overcomeschallenges of surface detection and adhesion for nanoinden-tation of a compliant polymer in air and waterrdquo Journal ofMaterials Research vol 26 no 8 pp 1026ndash1035 2011

[90] F Carrillo S Gupta M Balooch et al ldquoNanoindentation ofpolydimethylsiloxane elastomers effect of crosslinking work ofadhesion and fluid environment on elastic modulusrdquo Journal ofMaterials Research vol 20 no 10 pp 2820ndash2830 2005

[91] Y Cao D Yang and W Soboyejoy ldquoNanoindentation methodfor determining the initial contact and adhesion characteristicsof soft polydimethylsiloxanerdquo Journal ofMaterials Research vol20 no 8 pp 2004ndash2011 2005

[92] D M Ebenstein and K J Wahl ldquoA comparison of JKR-basedmethods to analyze quasi-static and dynamic indentation forcecurvesrdquo Journal of Colloid and Interface Science vol 298 no 2pp 652ndash662 2006

[93] S Gupta F Carrillo C Li L Pruitt and C Puttlitz ldquoAdhesiveforces significantly affect elastic modulus determination of softpolymeric materials in nanoindentationrdquoMaterials Letters vol61 no 2 pp 448ndash451 2007

[94] J D Kaufman and C M Klapperich ldquoSurface detection errorscause overestimation of the modulus in nanoindentation onsoftmaterialsrdquo Journal of theMechanical Behavior of BiomedicalMaterials vol 2 no 4 pp 312ndash317 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


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Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


F

FmaxB

D

C

A hf hl hmax h

WineI

S

WeI

(a)

Initial surfaceSurface profile after load removal

Indenter

Surface profile

F

hfhmaxhc

hs

a

under Fmax

(b)

Figure 1 (a) Schematic representation of load-displacement curve in nanoindentation testing (b) Schematic representation of cross sectionthrough indentation of a material sample

Table 1 Projected contact area surface area angles geometric constants and geometry correction factors for common indenter geometries[55] where ℎc is the contact depth (shown in Figure 1(b)) ℎ is the indentation depth under applied force 119865 (see Figure 1(a)) 119877 is the radiusof spherical tip 120579 is the semiangle of indenter tip (for pyramidal tips 120579 is the face angle with the central axis of the tip [55]) 120572 is the halfangle defining an equivalent cone 120598 is the geometric constant (for the pyramidal tips 120598 is theoretically 072 however a value of 075 hasbeen experimentally found to better represent experimental data [55]) and 120573 is the geometry correction factor used for the determinationof elastic modulus [53] Note that 119860c and 119860 s given in Table 1 for Berkovich Vicker Knoop cube corner and cone indenters are respectivelythe projected and surface areas of a perfectly sharp indenter without imperfectionbluntness (see eg [53 55] for the projected contact areaof a blunt indenter)

Indenter tip Projectedcontact area 119860c

Surface area 119860 s Semiangle 120579 (deg) Effective coneangle 120572 (deg)

Geometricconstant 120598

Geometrycorrection factor 120573

Sphere 120587(2ℎc119877 minus ℎ2

c) NA NA NA 075 1Berkovich (perfect) 3radic3ℎ

2

c tan2120579 3radic3ℎ

2 tan 120579 cos 120579 6503∘ 703∘ 075 1034Berkovich (modified) 3radic3ℎ

2

c tan2120579 NA 6527∘ 703∘ 075 1034

Vicker 4ℎ2

c tan2120579 4ℎ

2 sin 120579cos2 120579 68∘ 703∘ 075 1012

Knoop 2ℎ2

c tan 1205791 tan 1205792 NA 1205791= 8625∘1205792= 65∘ 7764∘ 075 1012

Cube corner 3radic3ℎ2

c tan2120579 NA 3526∘ 4228∘ 075 1034

Cone 120587ℎ2

c tan2120572 NA 120572 120572 0727 1

length scale dependent deformation in polymers observedwith indentation experiments is reviewed along with thenotions on its physical origin as well as theoretical modelsthat predict such length scale dependent deformations inpolymers In Section 2 the length scale dependent defor-mation of polymers observed in nanoindentation testing iscritically reviewed followed by discussions on the possibleexplanations for the origin of length scale dependent defor-mation in polymers in Section 3

2 Size Dependent Deformation of PolymersObserved in Nanoindentation Experiments

To set the stage and clarify the notation Figure 1(a) shows aschematic load-displacement curve of nanoindentation con-sisting of loading (section A-B) holding (section B-C) andunloading (section C-D) sequences [25] A correspondingschematic cross section of such an indentation is depicted

in Figure 1(b) identifying parameters used in the analysis[53 54] which will be discussed later

21 Hardness and Elastic Modulus Determination Similar tometals [2] the hardness parameter (indentation hardness [53]and universal hardness [54]) is usually used to investigate theindentation size effect (ISE) in polymers [15 22] Indentationhardness119867I can be obtained as

119867I =119865max119860c

(1)

where 119865max is the maximum applied force (shown inFigure 1(a)) and 119860c is the projected contact area at themaximum applied force 119865max The projected contact area 119860cas a function of contact depth ℎc (see Figure 1(b)) for somecommon indenter geometries is given in Table 1 [55] As canbe seen in Figure 1(b) the contact depth ℎc is determined by

ℎc = ℎmax minus ℎs (2)



ℎs = 120598119865max119878

(3)


119865 = 119861 (ℎ minus ℎf)119898

(4)


119867U =119865

119860 s (5)


119864OampPr =

radic120587119878

2120573radic119860c (6)



1119864OampPr

=1 minus V2


(7)



10 100 1000 100000

01

02

03

04

05

06

07



hmax (nm)

HI

(GPa






10 100 1000 1000001

015

02

025

03

035

04

045

PCPS (a)PS (b)


hmax (nm)

HU

(GPa

)




0 1000 2000 3000100

200

300

400

500

600

700


hmax (nm)

HI

(MPa

)







10 100 10000

100

200

300

400

500

600

700

hmax (nm)

HuncorrectedU

HcorrectedU

HU

(MPa

)




6 8 10 12 14 16 18 20 2200005

0001

00015

0002

00025

0003

00035

(phr)

120588x

(mol

m3)




8 10 12 14 16 18 20 22210

220

230

240

250

260

270

(phr)

Ho

(MPa

)


0 500 1000 1500 2000 2500200

250

300

350

108720

15115

hmax (nm)

HI

(MPa

)




0 002 004 006 0080

20

40

60

80

100

True strain

True

stre

ss (M

Pa)




0 500 1000 1500 2000 2500 3000 3500180

200

220

240

260

280

300

320

340

HI

(MPa

)

120576120576o = 1

120576120576o = 1025

120576120576o = 1045

120576120576o = 1070

hmax (nm)




1119878e=1119878minusℎv

(8)


0 500 1000 1500 2000 2500 3000 3500200

220

240

260

280

300

320

340

120576120576o = 1120576120576o = 1025

120576120576o = 1045

120576120576o = 1070

hmax (nm)

HI

(MPa

)


0 1000 2000 3000 4000 5000

04

05

06

07

08

hmax (nm)

HI

(GPa

)




) (9)


500 nm [40]


10 100 1000 10000200

300

400

500

600

700

800

hmax (nm)

HcorrectedI

HuncorrectedI

HI

(MPa

)










10 100 1000 10000150

200

250

300

350

400

450

500

PCPSPMMA

hmax (nm)

HI

(MPa

)


10 100 1000 10000100

150

200

250

300

350

400

450

PCPSPMMA

HU

(MPa

)

hmax (nm)




0 2000 4000 6000 8000 10000150

200

250

300

350

PSPMMA

HU

(MPa

)

hmax (nm)


Table 2 11986702mNU 119867




Sample 11986702mNU 119867

bulkU






100 1000 10000160

180

200

220

240

260

280

300

320

HU

(MPa

)

hmax (nm)

50 120583m10 120583m

05 120583m00 120583m





0 1000 2000 3000 4000 5000008

009

01

011

012

013

014

015

016

HI

(GPa

)

hmax (nm)


0 5 10 15 20 25 30 35

100

150

200

250H

ardn

ess (

MPa

)

hmax (120583m)







I


0 2000 4000 6000 8000 10000140

150

160

170

180

190

200

210

220

HI

(MPa

)

HuncorrectedU

HcorrectedU

hmax (nm)





I inFigure 20)



ℎ+ 1 (10)




lowast119877

4920 0015 00047 0082 1710050 0030 00068 0100 1519880 0059 00095 0150 1652220 0150 00150 0170 11






0 05 1 15 2 2501

015

02

025

03

035

04

HI

(GPa

)

hmax (120583m)


0 10 20 30 40 50 60008

01

012

014

016

018


hlowast

(120583m

)

Mc

Mp (kgmol)




0 200 400 600 800 10000

02

04

06

08

1

12

hmax (nm)

HI

(GPa

)


0 100 200 300 400 5000

5

10

15

20

25

Sample 1Sample 2

HI

(MPa

)

hmax (nm)





10 100 1000 100000

5

10

15

20

25

30

35

40

hmax (nm)

HU

(MPa

)

80 s20 s5 s


0 10 20 30 400

02

04

06

08

1

80 s20 s

HU

(MPa

)

hmax (120583m)









119865 =43(ℎ

119877)

321198772119864Hertzr (11)



1119864Hertzr

=1 minus V2


(12)



0 1000 2000 3000 40000

2

4

6

8

10

Elas

tic m

odul

us (G

Pa)


hmax (nm)






119865 =2 tan (120572) 119864Sneddon

120587 (1 minus V2)ℎ2 (13)






ℎ) (14)


lowastis a constant





119882 [49]

119882 = 119882e120576+119882120594=12120576119894119895119862119894119895119896119897120598119896119897+

3120594S119894119895120594S119894119895 (15)




120594S119894119895=12(120594119894119895+120594119895119894) =

12(120596119894119895+120596119895119894) (16)


hmaxhmax

1205791L

120594 =1205791L

prop1

hmax

1205792

R1205792

R

120594 =1205792R1205792

=1

R


0 5 10 15 20 25 30 350

1

2

3

4

5

6

Elas

tic m

odul

us (M

Pa)


hmax (120583m)






120594119894119895=12119890119894119899119898119906119898119895119899

(17)


120596119894=12119890119894119898119899119906119899119898 (18)

where 119906119894 119906119894119895 and 119890



119867 = 1198670 (1+119888ℓ

ℎ) (19)





119870 = 1198670119862 (119888ℓ + 2ℎ) (20)










100 100010

100

1000

Inde

ntat

ion

stiffn

ess (

Nm

)

hmax (nm)



10 100 1000

1

10

Inde

ntat

ion

stiffn

ess (

kNm

)


hmax (nm)




(119867

1198671015840)

120573

= 1+(ℎ1015840

ℎ)

1205732

(21)





0= 0165GPa and 119888

ℓ= 00442 120583m It is seen




01 1 1001

015

02

025

03

035

04


O

C

H HH

H HH

C

C

C

OO

HI

(GPa

)

hmax (120583m)


10 100 1000 100000

005

01

015

02

H

C

H

H

C

H

H

C

H

H

C

H

hmax (nm)

HI

(GPa

)



0= 00905GPa and



10 100 1000 10000005

01

015

02

025

03


C

O

C N

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

O

C

H

H

C

H

H

C

H

H

C

H

H

N

H

HI

(GPa

)

hmax (nm)





10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)




0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)




Acknowledgment


References









































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





ℎs = 120598119865max119878

(3)


119865 = 119861 (ℎ minus ℎf)119898

(4)


119867U =119865

119860 s (5)


119864OampPr =

radic120587119878

2120573radic119860c (6)



1119864OampPr

=1 minus V2


(7)



10 100 1000 100000

01

02

03

04

05

06

07



hmax (nm)

HI

(GPa






10 100 1000 1000001

015

02

025

03

035

04

045

PCPS (a)PS (b)


hmax (nm)

HU

(GPa

)




0 1000 2000 3000100

200

300

400

500

600

700


hmax (nm)

HI

(MPa

)







10 100 10000

100

200

300

400

500

600

700

hmax (nm)

HuncorrectedU

HcorrectedU

HU

(MPa

)




6 8 10 12 14 16 18 20 2200005

0001

00015

0002

00025

0003

00035

(phr)

120588x

(mol

m3)




8 10 12 14 16 18 20 22210

220

230

240

250

260

270

(phr)

Ho

(MPa

)


0 500 1000 1500 2000 2500200

250

300

350

108720

15115

hmax (nm)

HI

(MPa

)




0 002 004 006 0080

20

40

60

80

100

True strain

True

stre

ss (M

Pa)




0 500 1000 1500 2000 2500 3000 3500180

200

220

240

260

280

300

320

340

HI

(MPa

)

120576120576o = 1

120576120576o = 1025

120576120576o = 1045

120576120576o = 1070

hmax (nm)




1119878e=1119878minusℎv

(8)


0 500 1000 1500 2000 2500 3000 3500200

220

240

260

280

300

320

340

120576120576o = 1120576120576o = 1025

120576120576o = 1045

120576120576o = 1070

hmax (nm)

HI

(MPa

)


0 1000 2000 3000 4000 5000

04

05

06

07

08

hmax (nm)

HI

(GPa

)




) (9)


500 nm [40]


10 100 1000 10000200

300

400

500

600

700

800

hmax (nm)

HcorrectedI

HuncorrectedI

HI

(MPa

)










10 100 1000 10000150

200

250

300

350

400

450

500

PCPSPMMA

hmax (nm)

HI

(MPa

)


10 100 1000 10000100

150

200

250

300

350

400

450

PCPSPMMA

HU

(MPa

)

hmax (nm)




0 2000 4000 6000 8000 10000150

200

250

300

350

PSPMMA

HU

(MPa

)

hmax (nm)


Table 2 11986702mNU 119867




Sample 11986702mNU 119867

bulkU






100 1000 10000160

180

200

220

240

260

280

300

320

HU

(MPa

)

hmax (nm)

50 120583m10 120583m

05 120583m00 120583m





0 1000 2000 3000 4000 5000008

009

01

011

012

013

014

015

016

HI

(GPa

)

hmax (nm)


0 5 10 15 20 25 30 35

100

150

200

250H

ardn

ess (

MPa

)

hmax (120583m)







I


0 2000 4000 6000 8000 10000140

150

160

170

180

190

200

210

220

HI

(MPa

)

HuncorrectedU

HcorrectedU

hmax (nm)





I inFigure 20)



ℎ+ 1 (10)




lowast119877

4920 0015 00047 0082 1710050 0030 00068 0100 1519880 0059 00095 0150 1652220 0150 00150 0170 11






0 05 1 15 2 2501

015

02

025

03

035

04

HI

(GPa

)

hmax (120583m)


0 10 20 30 40 50 60008

01

012

014

016

018


hlowast

(120583m

)

Mc

Mp (kgmol)




0 200 400 600 800 10000

02

04

06

08

1

12

hmax (nm)

HI

(GPa

)


0 100 200 300 400 5000

5

10

15

20

25

Sample 1Sample 2

HI

(MPa

)

hmax (nm)





10 100 1000 100000

5

10

15

20

25

30

35

40

hmax (nm)

HU

(MPa

)

80 s20 s5 s


0 10 20 30 400

02

04

06

08

1

80 s20 s

HU

(MPa

)

hmax (120583m)









119865 =43(ℎ

119877)

321198772119864Hertzr (11)



1119864Hertzr

=1 minus V2


(12)



0 1000 2000 3000 40000

2

4

6

8

10

Elas

tic m

odul

us (G

Pa)


hmax (nm)






119865 =2 tan (120572) 119864Sneddon

120587 (1 minus V2)ℎ2 (13)






ℎ) (14)


lowastis a constant





119882 [49]

119882 = 119882e120576+119882120594=12120576119894119895119862119894119895119896119897120598119896119897+

3120594S119894119895120594S119894119895 (15)




120594S119894119895=12(120594119894119895+120594119895119894) =

12(120596119894119895+120596119895119894) (16)


hmaxhmax

1205791L

120594 =1205791L

prop1

hmax

1205792

R1205792

R

120594 =1205792R1205792

=1

R


0 5 10 15 20 25 30 350

1

2

3

4

5

6

Elas

tic m

odul

us (M

Pa)


hmax (120583m)






120594119894119895=12119890119894119899119898119906119898119895119899

(17)


120596119894=12119890119894119898119899119906119899119898 (18)

where 119906119894 119906119894119895 and 119890



119867 = 1198670 (1+119888ℓ

ℎ) (19)





119870 = 1198670119862 (119888ℓ + 2ℎ) (20)










100 100010

100

1000

Inde

ntat

ion

stiffn

ess (

Nm

)

hmax (nm)



10 100 1000

1

10

Inde

ntat

ion

stiffn

ess (

kNm

)


hmax (nm)




(119867

1198671015840)

120573

= 1+(ℎ1015840

ℎ)

1205732

(21)





0= 0165GPa and 119888

ℓ= 00442 120583m It is seen




01 1 1001

015

02

025

03

035

04


O

C

H HH

H HH

C

C

C

OO

HI

(GPa

)

hmax (120583m)


10 100 1000 100000

005

01

015

02

H

C

H

H

C

H

H

C

H

H

C

H

hmax (nm)

HI

(GPa

)



0= 00905GPa and



10 100 1000 10000005

01

015

02

025

03


C

O

C N

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

O

C

H

H

C

H

H

C

H

H

C

H

H

N

H

HI

(GPa

)

hmax (nm)





10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)




0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)




Acknowledgment


References









































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




10 100 1000 1000001

015

02

025

03

035

04

045

PCPS (a)PS (b)


hmax (nm)

HU

(GPa

)




0 1000 2000 3000100

200

300

400

500

600

700


hmax (nm)

HI

(MPa

)







10 100 10000

100

200

300

400

500

600

700

hmax (nm)

HuncorrectedU

HcorrectedU

HU

(MPa

)




6 8 10 12 14 16 18 20 2200005

0001

00015

0002

00025

0003

00035

(phr)

120588x

(mol

m3)




8 10 12 14 16 18 20 22210

220

230

240

250

260

270

(phr)

Ho

(MPa

)


0 500 1000 1500 2000 2500200

250

300

350

108720

15115

hmax (nm)

HI

(MPa

)




0 002 004 006 0080

20

40

60

80

100

True strain

True

stre

ss (M

Pa)




0 500 1000 1500 2000 2500 3000 3500180

200

220

240

260

280

300

320

340

HI

(MPa

)

120576120576o = 1

120576120576o = 1025

120576120576o = 1045

120576120576o = 1070

hmax (nm)




1119878e=1119878minusℎv

(8)


0 500 1000 1500 2000 2500 3000 3500200

220

240

260

280

300

320

340

120576120576o = 1120576120576o = 1025

120576120576o = 1045

120576120576o = 1070

hmax (nm)

HI

(MPa

)


0 1000 2000 3000 4000 5000

04

05

06

07

08

hmax (nm)

HI

(GPa

)




) (9)


500 nm [40]


10 100 1000 10000200

300

400

500

600

700

800

hmax (nm)

HcorrectedI

HuncorrectedI

HI

(MPa

)










10 100 1000 10000150

200

250

300

350

400

450

500

PCPSPMMA

hmax (nm)

HI

(MPa

)


10 100 1000 10000100

150

200

250

300

350

400

450

PCPSPMMA

HU

(MPa

)

hmax (nm)




0 2000 4000 6000 8000 10000150

200

250

300

350

PSPMMA

HU

(MPa

)

hmax (nm)


Table 2 11986702mNU 119867




Sample 11986702mNU 119867

bulkU






100 1000 10000160

180

200

220

240

260

280

300

320

HU

(MPa

)

hmax (nm)

50 120583m10 120583m

05 120583m00 120583m





0 1000 2000 3000 4000 5000008

009

01

011

012

013

014

015

016

HI

(GPa

)

hmax (nm)


0 5 10 15 20 25 30 35

100

150

200

250H

ardn

ess (

MPa

)

hmax (120583m)







I


0 2000 4000 6000 8000 10000140

150

160

170

180

190

200

210

220

HI

(MPa

)

HuncorrectedU

HcorrectedU

hmax (nm)





I inFigure 20)



ℎ+ 1 (10)




lowast119877

4920 0015 00047 0082 1710050 0030 00068 0100 1519880 0059 00095 0150 1652220 0150 00150 0170 11






0 05 1 15 2 2501

015

02

025

03

035

04

HI

(GPa

)

hmax (120583m)


0 10 20 30 40 50 60008

01

012

014

016

018


hlowast

(120583m

)

Mc

Mp (kgmol)




0 200 400 600 800 10000

02

04

06

08

1

12

hmax (nm)

HI

(GPa

)


0 100 200 300 400 5000

5

10

15

20

25

Sample 1Sample 2

HI

(MPa

)

hmax (nm)





10 100 1000 100000

5

10

15

20

25

30

35

40

hmax (nm)

HU

(MPa

)

80 s20 s5 s


0 10 20 30 400

02

04

06

08

1

80 s20 s

HU

(MPa

)

hmax (120583m)









119865 =43(ℎ

119877)

321198772119864Hertzr (11)



1119864Hertzr

=1 minus V2


(12)



0 1000 2000 3000 40000

2

4

6

8

10

Elas

tic m

odul

us (G

Pa)


hmax (nm)






119865 =2 tan (120572) 119864Sneddon

120587 (1 minus V2)ℎ2 (13)






ℎ) (14)


lowastis a constant





119882 [49]

119882 = 119882e120576+119882120594=12120576119894119895119862119894119895119896119897120598119896119897+

3120594S119894119895120594S119894119895 (15)




120594S119894119895=12(120594119894119895+120594119895119894) =

12(120596119894119895+120596119895119894) (16)


hmaxhmax

1205791L

120594 =1205791L

prop1

hmax

1205792

R1205792

R

120594 =1205792R1205792

=1

R


0 5 10 15 20 25 30 350

1

2

3

4

5

6

Elas

tic m

odul

us (M

Pa)


hmax (120583m)






120594119894119895=12119890119894119899119898119906119898119895119899

(17)


120596119894=12119890119894119898119899119906119899119898 (18)

where 119906119894 119906119894119895 and 119890



119867 = 1198670 (1+119888ℓ

ℎ) (19)





119870 = 1198670119862 (119888ℓ + 2ℎ) (20)










100 100010

100

1000

Inde

ntat

ion

stiffn

ess (

Nm

)

hmax (nm)



10 100 1000

1

10

Inde

ntat

ion

stiffn

ess (

kNm

)


hmax (nm)




(119867

1198671015840)

120573

= 1+(ℎ1015840

ℎ)

1205732

(21)





0= 0165GPa and 119888

ℓ= 00442 120583m It is seen




01 1 1001

015

02

025

03

035

04


O

C

H HH

H HH

C

C

C

OO

HI

(GPa

)

hmax (120583m)


10 100 1000 100000

005

01

015

02

H

C

H

H

C

H

H

C

H

H

C

H

hmax (nm)

HI

(GPa

)



0= 00905GPa and



10 100 1000 10000005

01

015

02

025

03


C

O

C N

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

O

C

H

H

C

H

H

C

H

H

C

H

H

N

H

HI

(GPa

)

hmax (nm)





10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)




0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)




Acknowledgment


References









































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




10 100 10000

100

200

300

400

500

600

700

hmax (nm)

HuncorrectedU

HcorrectedU

HU

(MPa

)




6 8 10 12 14 16 18 20 2200005

0001

00015

0002

00025

0003

00035

(phr)

120588x

(mol

m3)




8 10 12 14 16 18 20 22210

220

230

240

250

260

270

(phr)

Ho

(MPa

)


0 500 1000 1500 2000 2500200

250

300

350

108720

15115

hmax (nm)

HI

(MPa

)




0 002 004 006 0080

20

40

60

80

100

True strain

True

stre

ss (M

Pa)




0 500 1000 1500 2000 2500 3000 3500180

200

220

240

260

280

300

320

340

HI

(MPa

)

120576120576o = 1

120576120576o = 1025

120576120576o = 1045

120576120576o = 1070

hmax (nm)




1119878e=1119878minusℎv

(8)


0 500 1000 1500 2000 2500 3000 3500200

220

240

260

280

300

320

340

120576120576o = 1120576120576o = 1025

120576120576o = 1045

120576120576o = 1070

hmax (nm)

HI

(MPa

)


0 1000 2000 3000 4000 5000

04

05

06

07

08

hmax (nm)

HI

(GPa

)




) (9)


500 nm [40]


10 100 1000 10000200

300

400

500

600

700

800

hmax (nm)

HcorrectedI

HuncorrectedI

HI

(MPa

)










10 100 1000 10000150

200

250

300

350

400

450

500

PCPSPMMA

hmax (nm)

HI

(MPa

)


10 100 1000 10000100

150

200

250

300

350

400

450

PCPSPMMA

HU

(MPa

)

hmax (nm)




0 2000 4000 6000 8000 10000150

200

250

300

350

PSPMMA

HU

(MPa

)

hmax (nm)


Table 2 11986702mNU 119867




Sample 11986702mNU 119867

bulkU






100 1000 10000160

180

200

220

240

260

280

300

320

HU

(MPa

)

hmax (nm)

50 120583m10 120583m

05 120583m00 120583m





0 1000 2000 3000 4000 5000008

009

01

011

012

013

014

015

016

HI

(GPa

)

hmax (nm)


0 5 10 15 20 25 30 35

100

150

200

250H

ardn

ess (

MPa

)

hmax (120583m)







I


0 2000 4000 6000 8000 10000140

150

160

170

180

190

200

210

220

HI

(MPa

)

HuncorrectedU

HcorrectedU

hmax (nm)





I inFigure 20)



ℎ+ 1 (10)




lowast119877

4920 0015 00047 0082 1710050 0030 00068 0100 1519880 0059 00095 0150 1652220 0150 00150 0170 11






0 05 1 15 2 2501

015

02

025

03

035

04

HI

(GPa

)

hmax (120583m)


0 10 20 30 40 50 60008

01

012

014

016

018


hlowast

(120583m

)

Mc

Mp (kgmol)




0 200 400 600 800 10000

02

04

06

08

1

12

hmax (nm)

HI

(GPa

)


0 100 200 300 400 5000

5

10

15

20

25

Sample 1Sample 2

HI

(MPa

)

hmax (nm)





10 100 1000 100000

5

10

15

20

25

30

35

40

hmax (nm)

HU

(MPa

)

80 s20 s5 s


0 10 20 30 400

02

04

06

08

1

80 s20 s

HU

(MPa

)

hmax (120583m)









119865 =43(ℎ

119877)

321198772119864Hertzr (11)



1119864Hertzr

=1 minus V2


(12)



0 1000 2000 3000 40000

2

4

6

8

10

Elas

tic m

odul

us (G

Pa)


hmax (nm)






119865 =2 tan (120572) 119864Sneddon

120587 (1 minus V2)ℎ2 (13)






ℎ) (14)


lowastis a constant





119882 [49]

119882 = 119882e120576+119882120594=12120576119894119895119862119894119895119896119897120598119896119897+

3120594S119894119895120594S119894119895 (15)




120594S119894119895=12(120594119894119895+120594119895119894) =

12(120596119894119895+120596119895119894) (16)


hmaxhmax

1205791L

120594 =1205791L

prop1

hmax

1205792

R1205792

R

120594 =1205792R1205792

=1

R


0 5 10 15 20 25 30 350

1

2

3

4

5

6

Elas

tic m

odul

us (M

Pa)


hmax (120583m)






120594119894119895=12119890119894119899119898119906119898119895119899

(17)


120596119894=12119890119894119898119899119906119899119898 (18)

where 119906119894 119906119894119895 and 119890



119867 = 1198670 (1+119888ℓ

ℎ) (19)





119870 = 1198670119862 (119888ℓ + 2ℎ) (20)










100 100010

100

1000

Inde

ntat

ion

stiffn

ess (

Nm

)

hmax (nm)



10 100 1000

1

10

Inde

ntat

ion

stiffn

ess (

kNm

)


hmax (nm)




(119867

1198671015840)

120573

= 1+(ℎ1015840

ℎ)

1205732

(21)





0= 0165GPa and 119888

ℓ= 00442 120583m It is seen




01 1 1001

015

02

025

03

035

04


O

C

H HH

H HH

C

C

C

OO

HI

(GPa

)

hmax (120583m)


10 100 1000 100000

005

01

015

02

H

C

H

H

C

H

H

C

H

H

C

H

hmax (nm)

HI

(GPa

)



0= 00905GPa and



10 100 1000 10000005

01

015

02

025

03


C

O

C N

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

O

C

H

H

C

H

H

C

H

H

C

H

H

N

H

HI

(GPa

)

hmax (nm)





10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)




0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)




Acknowledgment


References









































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 002 004 006 0080

20

40

60

80

100

True strain

True

stre

ss (M

Pa)




0 500 1000 1500 2000 2500 3000 3500180

200

220

240

260

280

300

320

340

HI

(MPa

)

120576120576o = 1

120576120576o = 1025

120576120576o = 1045

120576120576o = 1070

hmax (nm)




1119878e=1119878minusℎv

(8)


0 500 1000 1500 2000 2500 3000 3500200

220

240

260

280

300

320

340

120576120576o = 1120576120576o = 1025

120576120576o = 1045

120576120576o = 1070

hmax (nm)

HI

(MPa

)


0 1000 2000 3000 4000 5000

04

05

06

07

08

hmax (nm)

HI

(GPa

)




) (9)


500 nm [40]


10 100 1000 10000200

300

400

500

600

700

800

hmax (nm)

HcorrectedI

HuncorrectedI

HI

(MPa

)










10 100 1000 10000150

200

250

300

350

400

450

500

PCPSPMMA

hmax (nm)

HI

(MPa

)


10 100 1000 10000100

150

200

250

300

350

400

450

PCPSPMMA

HU

(MPa

)

hmax (nm)




0 2000 4000 6000 8000 10000150

200

250

300

350

PSPMMA

HU

(MPa

)

hmax (nm)


Table 2 11986702mNU 119867




Sample 11986702mNU 119867

bulkU






100 1000 10000160

180

200

220

240

260

280

300

320

HU

(MPa

)

hmax (nm)

50 120583m10 120583m

05 120583m00 120583m





0 1000 2000 3000 4000 5000008

009

01

011

012

013

014

015

016

HI

(GPa

)

hmax (nm)


0 5 10 15 20 25 30 35

100

150

200

250H

ardn

ess (

MPa

)

hmax (120583m)







I


0 2000 4000 6000 8000 10000140

150

160

170

180

190

200

210

220

HI

(MPa

)

HuncorrectedU

HcorrectedU

hmax (nm)





I inFigure 20)



ℎ+ 1 (10)




lowast119877

4920 0015 00047 0082 1710050 0030 00068 0100 1519880 0059 00095 0150 1652220 0150 00150 0170 11






0 05 1 15 2 2501

015

02

025

03

035

04

HI

(GPa

)

hmax (120583m)


0 10 20 30 40 50 60008

01

012

014

016

018


hlowast

(120583m

)

Mc

Mp (kgmol)




0 200 400 600 800 10000

02

04

06

08

1

12

hmax (nm)

HI

(GPa

)


0 100 200 300 400 5000

5

10

15

20

25

Sample 1Sample 2

HI

(MPa

)

hmax (nm)





10 100 1000 100000

5

10

15

20

25

30

35

40

hmax (nm)

HU

(MPa

)

80 s20 s5 s


0 10 20 30 400

02

04

06

08

1

80 s20 s

HU

(MPa

)

hmax (120583m)









119865 =43(ℎ

119877)

321198772119864Hertzr (11)



1119864Hertzr

=1 minus V2


(12)



0 1000 2000 3000 40000

2

4

6

8

10

Elas

tic m

odul

us (G

Pa)


hmax (nm)






119865 =2 tan (120572) 119864Sneddon

120587 (1 minus V2)ℎ2 (13)






ℎ) (14)


lowastis a constant





119882 [49]

119882 = 119882e120576+119882120594=12120576119894119895119862119894119895119896119897120598119896119897+

3120594S119894119895120594S119894119895 (15)




120594S119894119895=12(120594119894119895+120594119895119894) =

12(120596119894119895+120596119895119894) (16)


hmaxhmax

1205791L

120594 =1205791L

prop1

hmax

1205792

R1205792

R

120594 =1205792R1205792

=1

R


0 5 10 15 20 25 30 350

1

2

3

4

5

6

Elas

tic m

odul

us (M

Pa)


hmax (120583m)






120594119894119895=12119890119894119899119898119906119898119895119899

(17)


120596119894=12119890119894119898119899119906119899119898 (18)

where 119906119894 119906119894119895 and 119890



119867 = 1198670 (1+119888ℓ

ℎ) (19)





119870 = 1198670119862 (119888ℓ + 2ℎ) (20)










100 100010

100

1000

Inde

ntat

ion

stiffn

ess (

Nm

)

hmax (nm)



10 100 1000

1

10

Inde

ntat

ion

stiffn

ess (

kNm

)


hmax (nm)




(119867

1198671015840)

120573

= 1+(ℎ1015840

ℎ)

1205732

(21)





0= 0165GPa and 119888

ℓ= 00442 120583m It is seen




01 1 1001

015

02

025

03

035

04


O

C

H HH

H HH

C

C

C

OO

HI

(GPa

)

hmax (120583m)


10 100 1000 100000

005

01

015

02

H

C

H

H

C

H

H

C

H

H

C

H

hmax (nm)

HI

(GPa

)



0= 00905GPa and



10 100 1000 10000005

01

015

02

025

03


C

O

C N

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

O

C

H

H

C

H

H

C

H

H

C

H

H

N

H

HI

(GPa

)

hmax (nm)





10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)




0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)




Acknowledgment


References









































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 500 1000 1500 2000 2500 3000 3500200

220

240

260

280

300

320

340

120576120576o = 1120576120576o = 1025

120576120576o = 1045

120576120576o = 1070

hmax (nm)

HI

(MPa

)


0 1000 2000 3000 4000 5000

04

05

06

07

08

hmax (nm)

HI

(GPa

)




) (9)


500 nm [40]


10 100 1000 10000200

300

400

500

600

700

800

hmax (nm)

HcorrectedI

HuncorrectedI

HI

(MPa

)










10 100 1000 10000150

200

250

300

350

400

450

500

PCPSPMMA

hmax (nm)

HI

(MPa

)


10 100 1000 10000100

150

200

250

300

350

400

450

PCPSPMMA

HU

(MPa

)

hmax (nm)




0 2000 4000 6000 8000 10000150

200

250

300

350

PSPMMA

HU

(MPa

)

hmax (nm)


Table 2 11986702mNU 119867




Sample 11986702mNU 119867

bulkU






100 1000 10000160

180

200

220

240

260

280

300

320

HU

(MPa

)

hmax (nm)

50 120583m10 120583m

05 120583m00 120583m





0 1000 2000 3000 4000 5000008

009

01

011

012

013

014

015

016

HI

(GPa

)

hmax (nm)


0 5 10 15 20 25 30 35

100

150

200

250H

ardn

ess (

MPa

)

hmax (120583m)







I


0 2000 4000 6000 8000 10000140

150

160

170

180

190

200

210

220

HI

(MPa

)

HuncorrectedU

HcorrectedU

hmax (nm)





I inFigure 20)



ℎ+ 1 (10)




lowast119877

4920 0015 00047 0082 1710050 0030 00068 0100 1519880 0059 00095 0150 1652220 0150 00150 0170 11






0 05 1 15 2 2501

015

02

025

03

035

04

HI

(GPa

)

hmax (120583m)


0 10 20 30 40 50 60008

01

012

014

016

018


hlowast

(120583m

)

Mc

Mp (kgmol)




0 200 400 600 800 10000

02

04

06

08

1

12

hmax (nm)

HI

(GPa

)


0 100 200 300 400 5000

5

10

15

20

25

Sample 1Sample 2

HI

(MPa

)

hmax (nm)





10 100 1000 100000

5

10

15

20

25

30

35

40

hmax (nm)

HU

(MPa

)

80 s20 s5 s


0 10 20 30 400

02

04

06

08

1

80 s20 s

HU

(MPa

)

hmax (120583m)









119865 =43(ℎ

119877)

321198772119864Hertzr (11)



1119864Hertzr

=1 minus V2


(12)



0 1000 2000 3000 40000

2

4

6

8

10

Elas

tic m

odul

us (G

Pa)


hmax (nm)






119865 =2 tan (120572) 119864Sneddon

120587 (1 minus V2)ℎ2 (13)






ℎ) (14)


lowastis a constant





119882 [49]

119882 = 119882e120576+119882120594=12120576119894119895119862119894119895119896119897120598119896119897+

3120594S119894119895120594S119894119895 (15)




120594S119894119895=12(120594119894119895+120594119895119894) =

12(120596119894119895+120596119895119894) (16)


hmaxhmax

1205791L

120594 =1205791L

prop1

hmax

1205792

R1205792

R

120594 =1205792R1205792

=1

R


0 5 10 15 20 25 30 350

1

2

3

4

5

6

Elas

tic m

odul

us (M

Pa)


hmax (120583m)






120594119894119895=12119890119894119899119898119906119898119895119899

(17)


120596119894=12119890119894119898119899119906119899119898 (18)

where 119906119894 119906119894119895 and 119890



119867 = 1198670 (1+119888ℓ

ℎ) (19)





119870 = 1198670119862 (119888ℓ + 2ℎ) (20)










100 100010

100

1000

Inde

ntat

ion

stiffn

ess (

Nm

)

hmax (nm)



10 100 1000

1

10

Inde

ntat

ion

stiffn

ess (

kNm

)


hmax (nm)




(119867

1198671015840)

120573

= 1+(ℎ1015840

ℎ)

1205732

(21)





0= 0165GPa and 119888

ℓ= 00442 120583m It is seen




01 1 1001

015

02

025

03

035

04


O

C

H HH

H HH

C

C

C

OO

HI

(GPa

)

hmax (120583m)


10 100 1000 100000

005

01

015

02

H

C

H

H

C

H

H

C

H

H

C

H

hmax (nm)

HI

(GPa

)



0= 00905GPa and



10 100 1000 10000005

01

015

02

025

03


C

O

C N

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

O

C

H

H

C

H

H

C

H

H

C

H

H

N

H

HI

(GPa

)

hmax (nm)





10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)




0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)




Acknowledgment


References









































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




10 100 1000 10000150

200

250

300

350

400

450

500

PCPSPMMA

hmax (nm)

HI

(MPa

)


10 100 1000 10000100

150

200

250

300

350

400

450

PCPSPMMA

HU

(MPa

)

hmax (nm)




0 2000 4000 6000 8000 10000150

200

250

300

350

PSPMMA

HU

(MPa

)

hmax (nm)


Table 2 11986702mNU 119867




Sample 11986702mNU 119867

bulkU






100 1000 10000160

180

200

220

240

260

280

300

320

HU

(MPa

)

hmax (nm)

50 120583m10 120583m

05 120583m00 120583m





0 1000 2000 3000 4000 5000008

009

01

011

012

013

014

015

016

HI

(GPa

)

hmax (nm)


0 5 10 15 20 25 30 35

100

150

200

250H

ardn

ess (

MPa

)

hmax (120583m)







I


0 2000 4000 6000 8000 10000140

150

160

170

180

190

200

210

220

HI

(MPa

)

HuncorrectedU

HcorrectedU

hmax (nm)





I inFigure 20)



ℎ+ 1 (10)




lowast119877

4920 0015 00047 0082 1710050 0030 00068 0100 1519880 0059 00095 0150 1652220 0150 00150 0170 11






0 05 1 15 2 2501

015

02

025

03

035

04

HI

(GPa

)

hmax (120583m)


0 10 20 30 40 50 60008

01

012

014

016

018


hlowast

(120583m

)

Mc

Mp (kgmol)




0 200 400 600 800 10000

02

04

06

08

1

12

hmax (nm)

HI

(GPa

)


0 100 200 300 400 5000

5

10

15

20

25

Sample 1Sample 2

HI

(MPa

)

hmax (nm)





10 100 1000 100000

5

10

15

20

25

30

35

40

hmax (nm)

HU

(MPa

)

80 s20 s5 s


0 10 20 30 400

02

04

06

08

1

80 s20 s

HU

(MPa

)

hmax (120583m)









119865 =43(ℎ

119877)

321198772119864Hertzr (11)



1119864Hertzr

=1 minus V2


(12)



0 1000 2000 3000 40000

2

4

6

8

10

Elas

tic m

odul

us (G

Pa)


hmax (nm)






119865 =2 tan (120572) 119864Sneddon

120587 (1 minus V2)ℎ2 (13)






ℎ) (14)


lowastis a constant





119882 [49]

119882 = 119882e120576+119882120594=12120576119894119895119862119894119895119896119897120598119896119897+

3120594S119894119895120594S119894119895 (15)




120594S119894119895=12(120594119894119895+120594119895119894) =

12(120596119894119895+120596119895119894) (16)


hmaxhmax

1205791L

120594 =1205791L

prop1

hmax

1205792

R1205792

R

120594 =1205792R1205792

=1

R


0 5 10 15 20 25 30 350

1

2

3

4

5

6

Elas

tic m

odul

us (M

Pa)


hmax (120583m)






120594119894119895=12119890119894119899119898119906119898119895119899

(17)


120596119894=12119890119894119898119899119906119899119898 (18)

where 119906119894 119906119894119895 and 119890



119867 = 1198670 (1+119888ℓ

ℎ) (19)





119870 = 1198670119862 (119888ℓ + 2ℎ) (20)










100 100010

100

1000

Inde

ntat

ion

stiffn

ess (

Nm

)

hmax (nm)



10 100 1000

1

10

Inde

ntat

ion

stiffn

ess (

kNm

)


hmax (nm)




(119867

1198671015840)

120573

= 1+(ℎ1015840

ℎ)

1205732

(21)





0= 0165GPa and 119888

ℓ= 00442 120583m It is seen




01 1 1001

015

02

025

03

035

04


O

C

H HH

H HH

C

C

C

OO

HI

(GPa

)

hmax (120583m)


10 100 1000 100000

005

01

015

02

H

C

H

H

C

H

H

C

H

H

C

H

hmax (nm)

HI

(GPa

)



0= 00905GPa and



10 100 1000 10000005

01

015

02

025

03


C

O

C N

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

O

C

H

H

C

H

H

C

H

H

C

H

H

N

H

HI

(GPa

)

hmax (nm)





10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)




0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)




Acknowledgment


References









































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




100 1000 10000160

180

200

220

240

260

280

300

320

HU

(MPa

)

hmax (nm)

50 120583m10 120583m

05 120583m00 120583m





0 1000 2000 3000 4000 5000008

009

01

011

012

013

014

015

016

HI

(GPa

)

hmax (nm)


0 5 10 15 20 25 30 35

100

150

200

250H

ardn

ess (

MPa

)

hmax (120583m)







I


0 2000 4000 6000 8000 10000140

150

160

170

180

190

200

210

220

HI

(MPa

)

HuncorrectedU

HcorrectedU

hmax (nm)





I inFigure 20)



ℎ+ 1 (10)




lowast119877

4920 0015 00047 0082 1710050 0030 00068 0100 1519880 0059 00095 0150 1652220 0150 00150 0170 11






0 05 1 15 2 2501

015

02

025

03

035

04

HI

(GPa

)

hmax (120583m)


0 10 20 30 40 50 60008

01

012

014

016

018


hlowast

(120583m

)

Mc

Mp (kgmol)




0 200 400 600 800 10000

02

04

06

08

1

12

hmax (nm)

HI

(GPa

)


0 100 200 300 400 5000

5

10

15

20

25

Sample 1Sample 2

HI

(MPa

)

hmax (nm)





10 100 1000 100000

5

10

15

20

25

30

35

40

hmax (nm)

HU

(MPa

)

80 s20 s5 s


0 10 20 30 400

02

04

06

08

1

80 s20 s

HU

(MPa

)

hmax (120583m)









119865 =43(ℎ

119877)

321198772119864Hertzr (11)



1119864Hertzr

=1 minus V2


(12)



0 1000 2000 3000 40000

2

4

6

8

10

Elas

tic m

odul

us (G

Pa)


hmax (nm)






119865 =2 tan (120572) 119864Sneddon

120587 (1 minus V2)ℎ2 (13)






ℎ) (14)


lowastis a constant





119882 [49]

119882 = 119882e120576+119882120594=12120576119894119895119862119894119895119896119897120598119896119897+

3120594S119894119895120594S119894119895 (15)




120594S119894119895=12(120594119894119895+120594119895119894) =

12(120596119894119895+120596119895119894) (16)


hmaxhmax

1205791L

120594 =1205791L

prop1

hmax

1205792

R1205792

R

120594 =1205792R1205792

=1

R


0 5 10 15 20 25 30 350

1

2

3

4

5

6

Elas

tic m

odul

us (M

Pa)


hmax (120583m)






120594119894119895=12119890119894119899119898119906119898119895119899

(17)


120596119894=12119890119894119898119899119906119899119898 (18)

where 119906119894 119906119894119895 and 119890



119867 = 1198670 (1+119888ℓ

ℎ) (19)





119870 = 1198670119862 (119888ℓ + 2ℎ) (20)










100 100010

100

1000

Inde

ntat

ion

stiffn

ess (

Nm

)

hmax (nm)



10 100 1000

1

10

Inde

ntat

ion

stiffn

ess (

kNm

)


hmax (nm)




(119867

1198671015840)

120573

= 1+(ℎ1015840

ℎ)

1205732

(21)





0= 0165GPa and 119888

ℓ= 00442 120583m It is seen




01 1 1001

015

02

025

03

035

04


O

C

H HH

H HH

C

C

C

OO

HI

(GPa

)

hmax (120583m)


10 100 1000 100000

005

01

015

02

H

C

H

H

C

H

H

C

H

H

C

H

hmax (nm)

HI

(GPa

)



0= 00905GPa and



10 100 1000 10000005

01

015

02

025

03


C

O

C N

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

O

C

H

H

C

H

H

C

H

H

C

H

H

N

H

HI

(GPa

)

hmax (nm)





10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)




0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)




Acknowledgment


References









































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 2000 4000 6000 8000 10000140

150

160

170

180

190

200

210

220

HI

(MPa

)

HuncorrectedU

HcorrectedU

hmax (nm)





I inFigure 20)



ℎ+ 1 (10)




lowast119877

4920 0015 00047 0082 1710050 0030 00068 0100 1519880 0059 00095 0150 1652220 0150 00150 0170 11






0 05 1 15 2 2501

015

02

025

03

035

04

HI

(GPa

)

hmax (120583m)


0 10 20 30 40 50 60008

01

012

014

016

018


hlowast

(120583m

)

Mc

Mp (kgmol)




0 200 400 600 800 10000

02

04

06

08

1

12

hmax (nm)

HI

(GPa

)


0 100 200 300 400 5000

5

10

15

20

25

Sample 1Sample 2

HI

(MPa

)

hmax (nm)





10 100 1000 100000

5

10

15

20

25

30

35

40

hmax (nm)

HU

(MPa

)

80 s20 s5 s


0 10 20 30 400

02

04

06

08

1

80 s20 s

HU

(MPa

)

hmax (120583m)









119865 =43(ℎ

119877)

321198772119864Hertzr (11)



1119864Hertzr

=1 minus V2


(12)



0 1000 2000 3000 40000

2

4

6

8

10

Elas

tic m

odul

us (G

Pa)


hmax (nm)






119865 =2 tan (120572) 119864Sneddon

120587 (1 minus V2)ℎ2 (13)






ℎ) (14)


lowastis a constant





119882 [49]

119882 = 119882e120576+119882120594=12120576119894119895119862119894119895119896119897120598119896119897+

3120594S119894119895120594S119894119895 (15)




120594S119894119895=12(120594119894119895+120594119895119894) =

12(120596119894119895+120596119895119894) (16)


hmaxhmax

1205791L

120594 =1205791L

prop1

hmax

1205792

R1205792

R

120594 =1205792R1205792

=1

R


0 5 10 15 20 25 30 350

1

2

3

4

5

6

Elas

tic m

odul

us (M

Pa)


hmax (120583m)






120594119894119895=12119890119894119899119898119906119898119895119899

(17)


120596119894=12119890119894119898119899119906119899119898 (18)

where 119906119894 119906119894119895 and 119890



119867 = 1198670 (1+119888ℓ

ℎ) (19)





119870 = 1198670119862 (119888ℓ + 2ℎ) (20)










100 100010

100

1000

Inde

ntat

ion

stiffn

ess (

Nm

)

hmax (nm)



10 100 1000

1

10

Inde

ntat

ion

stiffn

ess (

kNm

)


hmax (nm)




(119867

1198671015840)

120573

= 1+(ℎ1015840

ℎ)

1205732

(21)





0= 0165GPa and 119888

ℓ= 00442 120583m It is seen




01 1 1001

015

02

025

03

035

04


O

C

H HH

H HH

C

C

C

OO

HI

(GPa

)

hmax (120583m)


10 100 1000 100000

005

01

015

02

H

C

H

H

C

H

H

C

H

H

C

H

hmax (nm)

HI

(GPa

)



0= 00905GPa and



10 100 1000 10000005

01

015

02

025

03


C

O

C N

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

O

C

H

H

C

H

H

C

H

H

C

H

H

N

H

HI

(GPa

)

hmax (nm)





10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)




0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)




Acknowledgment


References









































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 05 1 15 2 2501

015

02

025

03

035

04

HI

(GPa

)

hmax (120583m)


0 10 20 30 40 50 60008

01

012

014

016

018


hlowast

(120583m

)

Mc

Mp (kgmol)




0 200 400 600 800 10000

02

04

06

08

1

12

hmax (nm)

HI

(GPa

)


0 100 200 300 400 5000

5

10

15

20

25

Sample 1Sample 2

HI

(MPa

)

hmax (nm)





10 100 1000 100000

5

10

15

20

25

30

35

40

hmax (nm)

HU

(MPa

)

80 s20 s5 s


0 10 20 30 400

02

04

06

08

1

80 s20 s

HU

(MPa

)

hmax (120583m)









119865 =43(ℎ

119877)

321198772119864Hertzr (11)



1119864Hertzr

=1 minus V2


(12)



0 1000 2000 3000 40000

2

4

6

8

10

Elas

tic m

odul

us (G

Pa)


hmax (nm)






119865 =2 tan (120572) 119864Sneddon

120587 (1 minus V2)ℎ2 (13)






ℎ) (14)


lowastis a constant





119882 [49]

119882 = 119882e120576+119882120594=12120576119894119895119862119894119895119896119897120598119896119897+

3120594S119894119895120594S119894119895 (15)




120594S119894119895=12(120594119894119895+120594119895119894) =

12(120596119894119895+120596119895119894) (16)


hmaxhmax

1205791L

120594 =1205791L

prop1

hmax

1205792

R1205792

R

120594 =1205792R1205792

=1

R


0 5 10 15 20 25 30 350

1

2

3

4

5

6

Elas

tic m

odul

us (M

Pa)


hmax (120583m)






120594119894119895=12119890119894119899119898119906119898119895119899

(17)


120596119894=12119890119894119898119899119906119899119898 (18)

where 119906119894 119906119894119895 and 119890



119867 = 1198670 (1+119888ℓ

ℎ) (19)





119870 = 1198670119862 (119888ℓ + 2ℎ) (20)










100 100010

100

1000

Inde

ntat

ion

stiffn

ess (

Nm

)

hmax (nm)



10 100 1000

1

10

Inde

ntat

ion

stiffn

ess (

kNm

)


hmax (nm)




(119867

1198671015840)

120573

= 1+(ℎ1015840

ℎ)

1205732

(21)





0= 0165GPa and 119888

ℓ= 00442 120583m It is seen




01 1 1001

015

02

025

03

035

04


O

C

H HH

H HH

C

C

C

OO

HI

(GPa

)

hmax (120583m)


10 100 1000 100000

005

01

015

02

H

C

H

H

C

H

H

C

H

H

C

H

hmax (nm)

HI

(GPa

)



0= 00905GPa and



10 100 1000 10000005

01

015

02

025

03


C

O

C N

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

O

C

H

H

C

H

H

C

H

H

C

H

H

N

H

HI

(GPa

)

hmax (nm)





10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)




0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)




Acknowledgment


References









































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




10 100 1000 100000

5

10

15

20

25

30

35

40

hmax (nm)

HU

(MPa

)

80 s20 s5 s


0 10 20 30 400

02

04

06

08

1

80 s20 s

HU

(MPa

)

hmax (120583m)









119865 =43(ℎ

119877)

321198772119864Hertzr (11)



1119864Hertzr

=1 minus V2


(12)



0 1000 2000 3000 40000

2

4

6

8

10

Elas

tic m

odul

us (G

Pa)


hmax (nm)






119865 =2 tan (120572) 119864Sneddon

120587 (1 minus V2)ℎ2 (13)






ℎ) (14)


lowastis a constant





119882 [49]

119882 = 119882e120576+119882120594=12120576119894119895119862119894119895119896119897120598119896119897+

3120594S119894119895120594S119894119895 (15)




120594S119894119895=12(120594119894119895+120594119895119894) =

12(120596119894119895+120596119895119894) (16)


hmaxhmax

1205791L

120594 =1205791L

prop1

hmax

1205792

R1205792

R

120594 =1205792R1205792

=1

R


0 5 10 15 20 25 30 350

1

2

3

4

5

6

Elas

tic m

odul

us (M

Pa)


hmax (120583m)






120594119894119895=12119890119894119899119898119906119898119895119899

(17)


120596119894=12119890119894119898119899119906119899119898 (18)

where 119906119894 119906119894119895 and 119890



119867 = 1198670 (1+119888ℓ

ℎ) (19)





119870 = 1198670119862 (119888ℓ + 2ℎ) (20)










100 100010

100

1000

Inde

ntat

ion

stiffn

ess (

Nm

)

hmax (nm)



10 100 1000

1

10

Inde

ntat

ion

stiffn

ess (

kNm

)


hmax (nm)




(119867

1198671015840)

120573

= 1+(ℎ1015840

ℎ)

1205732

(21)





0= 0165GPa and 119888

ℓ= 00442 120583m It is seen




01 1 1001

015

02

025

03

035

04


O

C

H HH

H HH

C

C

C

OO

HI

(GPa

)

hmax (120583m)


10 100 1000 100000

005

01

015

02

H

C

H

H

C

H

H

C

H

H

C

H

hmax (nm)

HI

(GPa

)



0= 00905GPa and



10 100 1000 10000005

01

015

02

025

03


C

O

C N

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

O

C

H

H

C

H

H

C

H

H

C

H

H

N

H

HI

(GPa

)

hmax (nm)





10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)




0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)




Acknowledgment


References









































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 1000 2000 3000 40000

2

4

6

8

10

Elas

tic m

odul

us (G

Pa)


hmax (nm)






119865 =2 tan (120572) 119864Sneddon

120587 (1 minus V2)ℎ2 (13)






ℎ) (14)


lowastis a constant





119882 [49]

119882 = 119882e120576+119882120594=12120576119894119895119862119894119895119896119897120598119896119897+

3120594S119894119895120594S119894119895 (15)




120594S119894119895=12(120594119894119895+120594119895119894) =

12(120596119894119895+120596119895119894) (16)


hmaxhmax

1205791L

120594 =1205791L

prop1

hmax

1205792

R1205792

R

120594 =1205792R1205792

=1

R


0 5 10 15 20 25 30 350

1

2

3

4

5

6

Elas

tic m

odul

us (M

Pa)


hmax (120583m)






120594119894119895=12119890119894119899119898119906119898119895119899

(17)


120596119894=12119890119894119898119899119906119899119898 (18)

where 119906119894 119906119894119895 and 119890



119867 = 1198670 (1+119888ℓ

ℎ) (19)





119870 = 1198670119862 (119888ℓ + 2ℎ) (20)










100 100010

100

1000

Inde

ntat

ion

stiffn

ess (

Nm

)

hmax (nm)



10 100 1000

1

10

Inde

ntat

ion

stiffn

ess (

kNm

)


hmax (nm)




(119867

1198671015840)

120573

= 1+(ℎ1015840

ℎ)

1205732

(21)





0= 0165GPa and 119888

ℓ= 00442 120583m It is seen




01 1 1001

015

02

025

03

035

04


O

C

H HH

H HH

C

C

C

OO

HI

(GPa

)

hmax (120583m)


10 100 1000 100000

005

01

015

02

H

C

H

H

C

H

H

C

H

H

C

H

hmax (nm)

HI

(GPa

)



0= 00905GPa and



10 100 1000 10000005

01

015

02

025

03


C

O

C N

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

O

C

H

H

C

H

H

C

H

H

C

H

H

N

H

HI

(GPa

)

hmax (nm)





10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)




0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)




Acknowledgment


References









































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




hmaxhmax

1205791L

120594 =1205791L

prop1

hmax

1205792

R1205792

R

120594 =1205792R1205792

=1

R


0 5 10 15 20 25 30 350

1

2

3

4

5

6

Elas

tic m

odul

us (M

Pa)


hmax (120583m)






120594119894119895=12119890119894119899119898119906119898119895119899

(17)


120596119894=12119890119894119898119899119906119899119898 (18)

where 119906119894 119906119894119895 and 119890



119867 = 1198670 (1+119888ℓ

ℎ) (19)





119870 = 1198670119862 (119888ℓ + 2ℎ) (20)










100 100010

100

1000

Inde

ntat

ion

stiffn

ess (

Nm

)

hmax (nm)



10 100 1000

1

10

Inde

ntat

ion

stiffn

ess (

kNm

)


hmax (nm)




(119867

1198671015840)

120573

= 1+(ℎ1015840

ℎ)

1205732

(21)





0= 0165GPa and 119888

ℓ= 00442 120583m It is seen




01 1 1001

015

02

025

03

035

04


O

C

H HH

H HH

C

C

C

OO

HI

(GPa

)

hmax (120583m)


10 100 1000 100000

005

01

015

02

H

C

H

H

C

H

H

C

H

H

C

H

hmax (nm)

HI

(GPa

)



0= 00905GPa and



10 100 1000 10000005

01

015

02

025

03


C

O

C N

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

O

C

H

H

C

H

H

C

H

H

C

H

H

N

H

HI

(GPa

)

hmax (nm)





10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)




0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)




Acknowledgment


References









































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




100 100010

100

1000

Inde

ntat

ion

stiffn

ess (

Nm

)

hmax (nm)



10 100 1000

1

10

Inde

ntat

ion

stiffn

ess (

kNm

)


hmax (nm)




(119867

1198671015840)

120573

= 1+(ℎ1015840

ℎ)

1205732

(21)





0= 0165GPa and 119888

ℓ= 00442 120583m It is seen




01 1 1001

015

02

025

03

035

04


O

C

H HH

H HH

C

C

C

OO

HI

(GPa

)

hmax (120583m)


10 100 1000 100000

005

01

015

02

H

C

H

H

C

H

H

C

H

H

C

H

hmax (nm)

HI

(GPa

)



0= 00905GPa and



10 100 1000 10000005

01

015

02

025

03


C

O

C N

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

O

C

H

H

C

H

H

C

H

H

C

H

H

N

H

HI

(GPa

)

hmax (nm)





10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)




0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)




Acknowledgment


References









































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




01 1 1001

015

02

025

03

035

04


O

C

H HH

H HH

C

C

C

OO

HI

(GPa

)

hmax (120583m)


10 100 1000 100000

005

01

015

02

H

C

H

H

C

H

H

C

H

H

C

H

hmax (nm)

HI

(GPa

)



0= 00905GPa and



10 100 1000 10000005

01

015

02

025

03


C

O

C N

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

O

C

H

H

C

H

H

C

H

H

C

H

H

N

H

HI

(GPa

)

hmax (nm)





10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)




0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)




Acknowledgment


References









































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




10 100 1000 100002

3

4

5

6

7

8

9

10

PCPSPMMA

hmax (nm)

EO

ampP

(GPa

)




0 1000 2000 3000 40000

10

20

30

40

50

60

EO

ampP

(MPa

)

hmax (nm)




Acknowledgment


References









































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



review article indentation depth dependent mechanical behavior in...

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