International Journal of Solids and Structures 47 (2010) 969–977

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International Journal of Solids and Structures

journal homepage: www.elsevier .com/locate / i jsols t r

Analytical and experimental study of a circular membrane in Hertzian contactwith a rigid substrate

Dewei Xu, Kenneth M. Liechti *

Research Center, Mechanics of Solids, Structures and Materials, Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin,Austin, TX 78712, USA

a r t i c l e i n f o

Article history:Received 31 October 2009Received in revised form 8 December 2009Available online 21 December 2009

Keywords:Contact mechanicsMembranesMoiré deflectometryHertzCell contact and adhesionMicro-electro-mechanical systems (MEMS)Micro-opto-electro-mechanical systems(MOEMS)

0020-7683/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijsolstr.2009.12.013

* Corresponding author. Fax: +1 512 471 5500.E-mail address: kml@mail.utexas.edu (K.M. Liecht

a b s t r a c t

The problem that is addressed here is that of a pressurized circular membrane in contact with a rigid sub-strate. A closed-form membrane analysis with Hertz-type contact is developed to describe the relation-ship between pressure, contact radius and contact force. Both the variation in the slope of the deflectionprofile of the portion of the membrane outside the contact zone and the contact radius itself are mea-sured by an apparatus based on moiré deflectometry. Contact experiments with a 3 lm PET film and aglass substrate show that this analysis predicts both the slope field and contact radius well.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The adhesion, contact and deformation of thin membranes haveplayed important roles in many fields. For instance, in biologicalscience, cell membrane and substratum adhesion is vital in normalcell functioning and locomotion (Fisher, 1993) and vesicle mem-brane fusion is of practical importance for targeted drug delivery(Bakowsky et al., 2008). In micro- or micro-opto-electro-mechani-cal systems (MEMS or MOEMS), electrostatically driven bridges ordiaphragm membranes operate over trillions of cycles in their lifespan and the study of reliability and durability of such MEMS/MOEMS devices relies on a quantitative understanding and deter-mination of change in adhesion and contact over time (Rebeiz,2003). Furthermore, an accurate determination of contact size isnecessary to evaluate contact resistance, heat dissipation and con-tact temperature in DC-contact-switch MEMS (Hyman and Meh-regany, 1999; Rebeiz, 2003).

The contact mechanics of two elastic solids has been well estab-lished by the advent of the Hertz (1881), JKR (Johnson et al., 1971),DMT (Derjaguin et al., 1975; Maugis, 1992) and Maugis (Maugis,1992) theories. The application conditions and selection guidelinesof these theories are summarized in the Johnson-Greenwood map(Johnson and Greenwood, 1997), which is based on the dimension-

ll rights reserved.

i).

less parameter initiated by Tabor (1977). However, these theoriesfor elastic solids are not applicable to thin membranes in contact.This is due to the fact that the elastic strain energy is determinedby the membrane stresses which result from the large out-of-planedeflections of the thin membrane. Consequently, geometrical non-linearity has to be considered and exact closed-form solutions arenot possible. The first configuration studied by membrane contactmechanics was a cellular membrane compressed between two par-allel plates (Cole, 1932; Harvey, 1938; Hiramoto, 1963), which wasused to characterize the mechanical properties of cellular mem-branes. The other extensively studied configuration is a sphericalcapsule adhered to a substrate (Shanahan, 1997; Wan and Liu,2001). This configuration is widely used to explore cell/vesicle/liposome/microcapsule–substrate contact and adhesion whichplay critical roles in biological and biomedical science. The thirdclass of problems includes one-dimensional strips or axisymmetricmembranes contacting a rigid substrate or punch under adhesivesurface forces (Plaut et al., 1999, 2001; Wan and Julien, 2009;Wong et al., 2007; Yang, 2004). These configurations have beenused to study the contact and adhesion between thin membranesand substrates and stiction and adhesion in MEMS structures.

In this paper, a pressurized circular membrane clamped periph-erally and contacting a rigid substrate is studied, which can be re-garded as a contact configuration in the third category mentionedabove. This geometry is also reminiscent of the constrained blister

(a)

2a

2c g hp w(r)

F

(b) r

Tr Tr

2c

w θ

Fig. 1. (a) Schematic of a clamped membrane contacting with a rigid plate placedabove with a gap g. (b) The free body diagram of the membrane.

970 D. Xu, K.M. Liechti / International Journal of Solids and Structures 47 (2010) 969–977

test (Chang et al., 1989; Napolitano et al., 1988) but instead theedge of the blister is clamped and the contact and adhesion be-tween the membrane and the constrained plate is of interest. Plautet al. (2003) obtained extensive numerical solutions for the contactmechanics of this configuration under linear plate, nonlinear plateand membrane assumptions, with or without adhesion. The sameauthor has also studied the mechanical response of axisymmetricmembranes with or without contact under various loading condi-tions and theories (Plaut, 2009b). However, residual stresses werenot considered in these analyses and so far, no experimental veri-fication of these analyses has been reported. Furthermore, the useof uniform pressure over the contact region makes this configura-tion attractive for the study of contact and adhesion between thinmembranes and substrates and in MEMS structures. Therefore, it isdesirable to develop closed-form analytical solutions to predict therelationships between contact radius, contact force and pressurefor simplicity, as the Hertz, DMT, JKR and Maugis contact theoriesfor elastic bodies did.

In the following, a combined analytical and experimental studyis pursued. An approximate closed-form analysis is used to predictthe relationship between the pressure, the contact radius and thecontact force without considering adhesion, which is reminiscentof the Hertz contact theory for two elastic bodies. Furthermore,residual stress is included in the analysis and this analysis is veri-fied by contact experiments. An apparatus modified from a previ-ous bulge tester (Xu and Liechti, 2009) based on moirédeflectometry (Kafri, 1980) is used to simultaneously measureboth the contact radius and the slope of the deflection profile ofthe membrane outside the contact zone. The direct measurementof this slope, which has not been reported in the literature, is crit-ical for thin membrane contact because the membrane strain andthe corresponding stress are best determined from the slope ofthe deflection rather than the deflection itself.

This paper is organized as follows: the Hertz-type contact the-ory is developed in Section 2; this is followed by an introductionof the principles and formulation for measuring the contact areaand the slope of the deflection profile using moiré deflectometryin Section 3; sample preparation, a description of the apparatusand resulting moiré patterns are presented in Section 4; finally,experimental and analytical results are presented and discussedin Section 5.

2. Hertz theory

Fig. 1 shows a schematic of the contact configuration for thisstudy. A smooth thin film with a thickness h, Young’s modulus Eand Poisson’s ratio v is clamped peripherally to a substrate witha circular opening with a diameter of 2a. As a pressure p is appliedon one side, the thin film bulges with a profile wðrÞ and contacts asmooth, parallel and rigid surface which is placed above the sub-strate with a prescribed gap g when the pressure approaches a cer-tain value. The goal of the present analysis is to find therelationship between the pressure p, the contact radius c and thecontact force F. The assumptions made in the analysis are as fol-lows: (i) the thin film has negligible flexural rigidity and onlymembrane stresses are considered, i.e., a; g � h; (ii) the gapg � a and sin h � dw=dr, where h is defined in Fig. 1b, which, aswill be seen later, places a restriction on the extent of the contactradius; (iii) a constant radial stress, i.e., rr ¼ r, is assumed; (iv) thecontact between the film and the rigid surface is frictionless.

The free body diagram of the thin membrane and the frame ofreference are shown in Fig. 1b. Equilibrium in the r-direction is gi-ven by (Timoshenko and Woinowsky-Krieger, 1959)

Tr � Tt þ rdTr

dr¼ 0; ð1Þ

where Tr and Tt are the radial force and tangential forces per unitlength. According to assumption (iii) ðTr ¼ rrh ¼ const:Þ and anequi-biaxial stress state, Tr ¼ Tt ¼ rh, is obtained. Outside the con-tact region ðc < r 6 aÞ, the equilibrium equation in the w-direction is

Tr �dwdr� 2pr þ pðr2 � c2Þ � p ¼ 0: ð2Þ

Considering the boundary conditions wðcÞ ¼ g; w0ðcÞ ¼ 0 andwðaÞ ¼ 0; r and w are, respectively, obtained as

r ¼pa2 1� ðc=aÞ2 þ ðc=aÞ2 lnðc=aÞ2� �

4gh; ð3Þ

and

w ¼g; 0 < r 6 c;

gð1�ðr=aÞ2þðc=aÞ2 lnðr=aÞ2Þ1�ðc=aÞ2þðc=aÞ2 lnðc=aÞ2

; c < r 6 a:

(ð4Þ

Accordingly, the slope dw=dr is given by

dwdr¼

0; 0 < r 6 c;�2gððr=aÞ�ðc=aÞ2ða=rÞÞ

að1�ðc=aÞ2þðc=aÞ2 lnðc=aÞ2Þ; c < r 6 a:

(ð5Þ

The resulting equi-biaxial stress state from assumption (iii) cannotbe satisfied over the entire domain. For example, the tangentialstrain et varies from the equi-biaxial state over the domain0 < r 6 c to et ¼ 0 at r ¼ a. Considering that er ¼ du=drþðdw=drÞ2=2 and et ¼ u=r, where u is the radial displacement, a meanstrain (Williams, 1997) is defined as

e ¼ 12ðer þ etÞ ¼

12

dudrþ u

rþ 1

2dwdr

� �2 !

¼ rð1� vÞE

: ð6Þ

Consequently, the constitutive relationship in integral form underthe resulting equi-biaxial stress state from assumption (iii) is repre-sented asZ a

0dðurÞ þ

Z a

0

12

dwdr

� �2

r dr ¼ 2rð1� vÞE

Z a

0r dr: ð7Þ

Noting that uð0Þ ¼ uðaÞ ¼ 0 and dw=dr ¼ 0 for 0 < r 6 c and dw=dr ¼ �pðr2 � c2Þ=2rhr for c < r 6 a, Eq. (7) becomes

D. Xu, K.M. Liechti / International Journal of Solids and Structures 47 (2010) 969–977 971

r3 ¼ Ep2a2

32ð1� vÞh2 1� 4ca

� �2þ 3

ca

� �4� c

a

� �4ln

ca

� �4� �

: ð8Þ

Comparing Eqs. (3) and (8), the relationship between the prescribedpressure p and the corresponding contact radius c is

p ¼2Eg3h 1� 4ðc=aÞ2 þ 3ðc=aÞ4 � ðc=aÞ4 lnðc=aÞ4

� �ð1� vÞa4 1� ðc=aÞ2 þ ðc=aÞ2 lnðc=aÞ2

� �3 : ð9Þ

If an equi-biaxial residual stress r0 exists before pressurization andnoting that the strain in Eq. (6) is referred to the deformed state, thequantity r in Eq. (7) should be substituted with r� r0. As a result,the relationship between the prescribed pressure p and the corre-sponding contact radius c with an equi-biaxial residual stress r0

is obtained as

p¼ 2Eg3hð1�4ðc=aÞ2þ3ðc=aÞ4�ðc=aÞ4 lnðc=aÞ4Þþ4ð1�vÞr0gha2C20

ð1�vÞa4C30

;

ð10Þ

where C0 ¼ 1� ðc=aÞ2 þ ðc=aÞ2 lnðc=aÞ2. Note that from Eq. (10)

limc!0

p ¼ 2Eg3hþ 4ð1� vÞr0gha2

ð1� vÞa4 ; ð11Þ

which is consistent with the governing equation that was used forcircular bulge tests (Xu and Liechti, 2009).

The contact force exerted on the rigid plate can be representedas

F ¼ pc2p; ð12Þ

where c can be obtained by solving Eq. (9) or (10) for a known pres-sure p. Alternatively, the direct relationship between the contactforce F and the pressure p with or without residual stress r0 canbe obtained by combining Eq. (12) with Eq. (10)

p ¼2Eg3h 1� 4F

ppa2 þ 3 Fppa2

� �2� F

ppa2

� �2ln F

ppa2

� �2� �

ð1� vÞa4C31

: ð13Þ

Alternatively, combining Eq. (12) with Eq. (9) yields

p¼2Eg3h 1� 4F

ppa2þ3 Fppa2

� �2� F

ppa2

� �2ln F

ppa2

� �2� �

þ4ð1�vÞr0gha2C21

ð1�vÞa4C31

;

ð14Þ

where C1 ¼ 1� Fppa2 þ F

ppa2

� �ln F

ppa2

� �.

3. Moiré deflectometry for contact radius and slopemeasurements

The deflection profile of the membrane outside the contact re-gion, its corresponding slope and the relationship between thepressure and the contact radius were derived above in Section 2.In this study, the contact radius and the slope are also measuredsimultaneously by moiré deflectometry, which was introduced byKafri (1980) as an incoherent light technique that measures raydeflection of a collimated beam instead of measuring differencesin optical path length. Therefore, compared to interferometry, itssetup is much simpler and is superior with respect to mechanicalstability. Furthermore, the fringes generated from moiré deflec-tometry map the slope field of the deflected surface and this un-ique technique has found numerous applications in opticalmapping, fluid density analysis, studies of transient phenomena,etc. (Kafri and Glatt, 1990).

A schematic of the moiré deflectometry setup for measuring thecontact size and the slope of the deflection profile of the thin mem-

brane is shown in Fig. 2a. This arrangement is very similar to thatused in the earlier bulge tester to determine the mechanical prop-erties of transparent thin films by measuring the focal length of thelens structure formed by the bulged film and the pressurizing med-ium (Xu and Liechti, 2009). A collimated beam passes through thelens structure L formed by the bulged film and the pressurizationmedium with a refractive index n and two identical Ronchi grat-ings G1 and G2 with a pitch p, which are placed behind the samplewith a separation D and with a small rotation h (<10�) betweenthem. The grating G1 is placed at a distance D0 from the sample.The y-axis is defined such that the angle between it and G1 or G2

is h=2. A diffuse screen S is attached to the rear of the grating G2.Before pressurization, the resulting moiré fringes on S (Fig. 2b)are shown in gray. The black fringes in Fig. 2b result when the pres-surized thin membrane contacts the transparent rigid plate with acontact diameter 2c. It can be seen that these black fringes insidethe contact zone remain in the same direction as the gray fringesand they deviate from the gray ones outside the contact zone.The contact diameter is determined to be the distance betweentwo turning points on the central black fringe from straight toslant. For example, C and C0 shown in Fig. 2b are the two turningpoints on the central black fringe and CC0 ¼ 2c.

The slope at A on the sample can be determined as (Karny andKafri, 1982; Xu and Liechti, 2009)

dwdr

����A

� un� 1

� hyA0

ðn� 1ÞD ; ð15Þ

where u is the angle between the original beam AA00 and the de-flected beam AA0; yA0 is the y-coordinate (Fig. 2c) of A0 which isthe corresponding point on the moiré pattern to A on the sample;the refractive index of the pressurizing fluid is n. The radial positionof A can be represented in terms of coordinates rA0 and yA0 of A0

rA ¼ rA0 þ ðDþ D0Þu ¼ rA0 þ ðDþ D0ÞhyA0

ðn� 1ÞD ; ð16Þ

where rA0 is the r-coordinate (Fig. 2c) of A0. Because rA0 and yA0 canbe directly measured from the moiré pattern recorded during con-tact experiments, the pointwise slope of the membrane can be ob-tained using Eqs. (16) and (15).

4. Experimental

In this section, we describe the specimen fabrication, the appa-ratus and the procedures for conducting the contact experiments.The resulting pressure history and moiré patterns are presentedand examined.

4.1. Sample preparation and apparatus

A polyethylene terephthalate (PET) film with a thickness of3 lm was used in the contact experiments of a circular membraneon a rigid substrate. The PET films were first stretched and fixed toa 127 mm diameter ring. The films were then bonded to smalleraluminum frames with an aperture of 20.32 mm using NOA 68optical adhesive (Norland Products Inc., Cranbury, New Jersey).After being cured under ultraviolet (UV) light for 20 min, sampleswere harvested from the 127 mm ring by cutting the films offaround the corresponding aluminum frame. This frame wasmounted onto the pressure manifold described in the followingand the freestanding film over the central aperture was the samplearea used for contact experiments.

A schematic view of the apparatus used in this study is shownin Fig. 3. It consists of a pressurization device along with videorecording and data acquisition equipment and the componentsfor moiré deflectometry. The collimated beams were produced by

(c)

A' yA'

O

rA' C

(b)

C

C'O

A'

(a)

ry

w

D

G1 G2 SLϕ

D0

rA

A A'’

A'

2cC

C'

OO

Fig. 2. (a) A moiré deflectometry setup for measuring contact radius and the slope of the deflection of a pressurized thin sample (not to scale). (b) The resulting moirépatterns: gray ones are formed by the beam without the sample and black fringes (partially shown) are formed by the beam through the sample which contacts the rigid platewith a contact radius c. (c) A detailed view of the part boxed in (b).

972 D. Xu, K.M. Liechti / International Journal of Solids and Structures 47 (2010) 969–977

an 8mW He-Ne laser source and a beam expander. The aluminumframe with the PET sample was fixed to the manifold and sealed bya rubber o-ring. A transparent glass plate was fixed on steel shimstrips (Precision Brand Products, Inc.) which were placed betweenthe glass plate and the aluminum frame and acted as a spacer toprescribe the gap between them. Two different shim strips withthicknesses of 203 lm and 305 lm were used. The pressurizingmedium was deionized water and a syringe pump (NE500, NewEra Pump Systems Inc., Wantagh, New York), which can operatein both infusion and withdrawal, was used to pressurize the sys-tem. The pressure was measured with a pressure transducer (Sen-sotec Z/0761-09ZG, Columbus, Ohio) with a capacity of 103.4 kPa.It was connected to a data acquisition board (National InstrumentsPCI-MIO-16XE-50, Austin, Texas) which was installed in a PC withLabVIEW software. The pitch of the gratings used for moiré deflec-tometry was 0.254 mm. The separation D between the two grat-ings was 12.7 mm and the rotation h between them was about7.3�. The value of h=D was calibrated as 0.102 (Xu and Liechti,2009).

CollimatedLightsource

Video

Camera

Pressuretransducer

Data& videoacquisition

O-ring

SpacerGlass

Fig. 3. A schematic view

4.2. Contact experiments, pressure history and moiré patterns

Before contact experiments, bulge tests with both rectangularand circular specimens were used to determine the mechanicalproperties of the PET sample (Xu and Liechti, 2009). A Young’smodulus of 4.65 GPa, Poisson’ ratio of 0.34 and an equi-biaxialresidual stress of 7.50 MPa were obtained.

After the circular bulge test, the shim strips and the glassplate were fixed above the sample. The glass plate was sonicatedin deionized water and dried in a dry nitrogen flow. Before pres-surization, trapped air bubbles were carefully eliminated fromthe chamber in order to obtain satisfactory moiré patterns. Thevolume rate during both infusion and withdrawal was 3 ml/h.The acquisition of the changing pressure and moiré patterns dur-ing the experiment was synchronized and their history was thenrecorded.

Fig. 4 shows the pressure history during a contact experimentwith a gap of 203 lm. The PET film first deflected under pressuri-zation and behaved as a membrane until A (200 s, 218 Pa) where

Syringepump

Glass window

Gratings and screenSpecimenNOA68adhesive

of the apparatus.

2c

y

1 cm

Fig. 5. The moiré pattern obtained at 547 Pa from the contact experiment with agap of 203 lm. The determination of the contact radius and the slope of themembrane are illustrated.

D. Xu, K.M. Liechti / International Journal of Solids and Structures 47 (2010) 969–977 973

the film started to contact the glass plate. Note that the pressure atA can be used to check the mechanical properties using Eq. (11).After A, the contact size increased as the syringe pump continuedto pressurize to B (340 s, 1036 Pa). Then the syringe was reversedand consequently the pressure decreased. The pressure decreasedas the syringe pump continued to withdraw until D (870 s,210 Pa) when the PET film lost contact with the glass plate. Thepressure was then released. The consistency of the pressures at Aand D demonstrates that there was no adhesion and that the con-tact was fully reversible. Note that from B to C, the pressure variedmuch more slowly than from C to D in spite of the fact that thepumping rate was constant at 3 ml/h. This is due to mechanicalhysteresis (e.g. backlash, viscous effects) in the pumping system.

The recorded moiré fringes were used to determine both thecontact size and the slope of the membrane deflection outsidethe contact zone. Fig. 5 shows the moiré pattern that was obtainedat 547 Pa. Note that the recorded fringes rotated in the oppositedirection to that shown in Fig. 2b because they were recorded frombehind the diffuser S. The central fringe was identified by LabVIEWsoftware using an intensity threshold to establish the fringe widthand then taking the middle of the fringe as its location. The diam-eter of the contact area was measured as the distance between thetwo turning points on the central fringe. The r- and y-coordinatesof an arbitrary point on the middle of the central fringe with re-spect to the center were used to obtain the real position on thesample using Eq. (16) and the corresponding slope using Eq. (15),respectively.

The contact experiment and analysis were repeated with a gapof 305 lm and the results will be presented in next section.

5. Results and discussion

Fig. 6 shows the variation of the slope of the deflection profilewith normalized radius obtained from the contact experimentswith a 203 lm gap. Two sets of data that were obtained whenthe normalized contact radius was 0.247 and 0.428 are shown inthe figure. These slope values were determined from the medianline on the central fringe, as described in Section 4. The error barsrepresent the uncertainty of slopes which is equal to the slope

0

200

400

600

800

1000

1200

0 200 400

Pres

sure

(Pa)

Tim

A

B

Fig. 4. The pressure history for a contact experimen

value resulting from one quarter of the fringe width. Finer fringesand more sophisticated image processing would lead to more accu-rate measurements of the slope. The predictions shown in Fig. 6 areevaluated from Eq. (5), which predicts the slope of the membraneover c < r 6 a quite satisfactorily. Note that both the deflectionprofile (Eq. (4)) and the derivative (Eq. (5)) of the membrane areindependent of mechanical properties of and residual stress inthe film. This is due to the membrane assumption where themechanical properties do not appear in the equilibrium equations(Eqs. (1) and (2)). In addition, we compare our data and approxi-mate solution (Fig. 6) with the numerical solution (Lai and Dillard,1996; Plaut, 2009a; Plaut et al., 2003). The results are veryconsistent.

Figs. 7 and 8 show the variation of contact radius with pressureduring both loading and unloading for gaps of 203 lm and 305 lm,respectively. As shown in Fig. 7, contact in the experiments withthe gap of 203 lm was made at 218 Pa, corresponding to A inFig. 4. The minimum contact radius is limited by the slope and spa-tial resolution of the moiré deflectometry. The solid line representsthe prediction obtained from Eq. (10). The data from both loadingand unloading are consistent with the analysis and no hysteresis

600 800 1000

e (s)

C

D

t involving PET on glass with a gap of 203 lm.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-dw

/dr

r/a

DataEq. (5)

0.247

0.428

c/a Plaut and Lai

Fig. 6. A comparison of the slopes of the deflection profile between the approximate solution (Eq. (5)), numerical solutions (Lai and Dillard, 1996; Plaut, 2009a; Plaut et al.,2003) and measurements from the contact experiment with the 203 lm gap at two different normalized contact radii c=a ¼ 0:247 (220 s, 274 Pa) and 0.428 (260 s, 492 Pa).

974 D. Xu, K.M. Liechti / International Journal of Solids and Structures 47 (2010) 969–977

was observed in either experiment. The numerical solutions (Laiand Dillard, 1996; Plaut, 2009a; Plaut et al., 2003) (dashed lines)are also added to Figs. 7 and 8 and are very consistent with ourdata and approximate solution.

The data indicates that the approximate analysis in Section 2predicts both the slope of the deflection profile of the membraneand the contact area quite accurately. Assumption (i), which is re-lated to classical membrane assumptions, was easily satisfied. Forassumption (ii), the condition g � a does not necessarily implythat sinh � dw=dr, which is used to derive Eq. (2). For example,even if g � a, when c/a is large, so is dw=dr. Consequently, an addi-

0

1

2

3

4

5

6

7

0 200 400 6

Eq. (10)

LoadingUnloading

Plaut and Lai

Con

tact

radi

us (m

m)

Pressu

g=2

200

400

600

800

1000

1200

1

Pres

sure

(Pa)

Fig. 7. The comparison of contact radii between the prediction (Eq. (10)), numerical solutthe contact experiment with the 203 lm gap. The error in contact radius measurement wobtain the Young’s modulus and residual stress.

tional constraint on the contact radius should be considered as partof assumption (ii). According to Eq. (5), dw=dr is a monotonicallyincreasing function of c/a and r in the region c < r 6 a. Table 1 liststhe maximum ratios c

a

��max that satisfy assumption (ii) for various

values of a/g. This maximum ratio was obtained from Eq. (5) byassuming that hmax ¼ 10� at r ¼ a. The ratios of a/g correspondingto the data shown in Figs. 7 and 8 were 33 and 50, respectively.Thus it can be seen that the corresponding ratios of c/a in Figs. 7and 8 are well within the limits shown in Table 1.

We focus on assumptions (iii) and (iv) in the following. Theassumption (iii) of a constant radial stress and the introduction

00 800 1000 1200

re (Pa)

03 μm

2 3 4 5 6 7Contact radius (mm)

Fit by Eq. (10)

ions (Lai and Dillard, 1996; Plaut, 2009a; Plaut et al., 2003) and measurements fromas less than 0.1 mm. The inset shows the curve-fit of data by Eq. (10) which is used to

0

1

2

3

4

5

6

0 500 1000 1500 2000

Eq. (10)

LoadingUnloading

Plaut and Lai

Con

tact

radi

us (m

m)

Pressure (Pa)

g=305 μm

400

800

1200

1600

2000

1 2 3 4 5 6

Pres

sure

(Pa)

Contact radius (mm)

Fit by Eq. (10)

Fig. 8. The comparison of contact radii between the prediction (Eq. (10)), numerical solutions (Lai and Dillard, 1996; Plaut, 2009a; Plaut et al., 2003) and measurements fromthe contact experiment with the 305 lm gap. The error in contact radius measurement was less than 0.1 mm. The inset shows the curve-fit of data by Eq. (10) which is used toobtain the Young’s modulus and residual stress.

Table 1The maximum ratio c

a

��max in terms of a/g.

ag 20 30 40 50 100ca

��max

0.474 0.642 0.728 0.781 0.889

1 The authors are grateful to Uni-Pixel for providing this data.

D. Xu, K.M. Liechti / International Journal of Solids and Structures 47 (2010) 969–977 975

of a mean strain over the entire domain resulted in the integralform of the constitutive equation (Eq. (7)). When the contact radiusc ¼ 0 the resulting stress obtained from assumption (iii) is 11%lower than the true radial mean stress (Kelkar et al., 1985; Wil-liams, 1997). However, when contact is made, the stresses areequi-biaxial inside the contact zone and therefore, the deviationbetween the resulting stress from assumption (iii) and the corre-sponding true radial mean stress is expected to be less than 11%.This can be seen by considering the state of stress when c ¼ a. Inthis situation, the stress over the entire membrane is equi-biaxial.Therefore, assumption (iii) is exact and the resulting error is zero.For the intermediate case ð0 < c < aÞ, the error must lie between11% and zero. In addition, if the Poisson’s ratio is greater than 0.3the deviation decreases (Williams, 1997) in the noncontact caseand lowers the upper bound when contact is made. Furthermore,the existence of residual stresses can significantly alleviate thedeviation of the assumed average stress from true radial meanstress because it can be represented as 11=ðaþ 1Þ%, where a isthe ratio of the pre-existing residual stress to the true radial meanstress developed by the large deformation. In this study, the resid-ual stress was 7.5 MPa and the corresponding ratio a was approx-imately 6, making the maximum deviation approximately 2%. Aquantitative error estimate based on contact radius was provided(Plaut, 2009a) via a direction comparison (Figs. 7 and 8) of thenumerical solution to the problem (Lai and Dillard, 1996; Plautet al., 2003) and the approximate solution presented here. The er-ror increased with contact radius but was less than 2%. Therefore,assumption (iii) is quite reasonable here, where the residual stres-ses were relatively high, and explains why the agreement betweenthe analysis (Eq. (10)) and the measurements was so good.

Finally, we consider the assumption (iv) of frictionless contact.In other words, it is assumed that there is no adhesive interactionbetween the PET film and the glass plate due to the close relation-

ship between friction and adhesion (Frisbie et al., 1994; Park andThiel, 2008; Xu et al., 2008). The very weak adhesion betweenthe PET film and the glass plate can be seen from Fig. 4 wherethe pressures corresponding to A and D were almost the sameand from Figs. 7 and 8 where no adhesive hysteresis was observedbetween experimental data obtained during loading and unload-ing, making the assumption of frictionless contact appear quitereasonable. The true surface energy of the PET is on the order of100 mJ/m2 (Pandiyaraj et al., 2008) which would lead to a largecontact hysteresis. The observed ultralow adhesion is due to thesurface roughness of the PET film that was used in the contactexperiments. Fig. 9 shows the surface topography of the PET filmas imaged by an optical profiler (Veeco, NT1100)1. It can be seenthat the surface of the PET film was quite rough and the correspond-ing RMS (root-mean-square) roughness was about 65 nm. The largesurface roughness of the PET film was due to embedded particleswith heights up to hundreds of nm. These small particles were addedto the PET film surface during the manufacturing process to preventself-adhesion of the film when stored as a roll. Furthermore, the elas-tic energy in thin membrane contact is stored by membrane forces,which are not sensitive to the surface roughness. This is quite differ-ent from the contact between two elastic bodies where the surfaceroughness plays a critical role (Greenwood and Williamson, 1966;Persson, 2001).

In the following, two issues related to the Hertzian contact anal-ysis that was developed in Section 2 are discussed. First, this anal-ysis was developed on the basis of displacement control, i.e., thegap between the substrate and the constrained plate was pre-scribed a priori. For contact under load control with a prescribedcontact force F, a combination of Eq. (12) and either Eq. (9) or(10) can be used to determine the contact radius c and the separa-tion g without or with residual stresses, respectively. Second, it isalso interesting to consider the use of Eq. (10) to curve-fit the con-tact radius-pressure responses as shown in Figs. 7 and 8 and ex-tract the Young’s modulus and residual stress. A similar schemeis widely used in nanoindentation tests (Kiely and Houston,1998; Wang et al., 2004) to obtain the mechanical properties of

Fig. 9. An image of the surface topography of the PET film scanned by a Veeco optical profiler (courtesy of Uni-Pixel Inc.).

976 D. Xu, K.M. Liechti / International Journal of Solids and Structures 47 (2010) 969–977

elastic bodies. The corresponding curve-fits of pressure–contact ra-dius responses for both loading and unloading are shown in the in-sets of Figs. 7 and 8. Assuming a Poisson’s ratio of 0.34, from thedata shown in Fig. 7 with the 203 lm gap, the resulting Young’smodulus and residual stress were 3.89 GPa and 7.83 MPa, respec-tively. Similarly, for the data shown in Fig. 8 with the 305 lmgap, the resulting Young’s modulus and residual stress were4.42 GPa and 7.86 MPa, respectively. These values are quite consis-tent with the Young’s modulus 4.65 GPa and residual stress7.50 MPa that were obtained from bulge testing and that wereused in the analysis. This curve-fitting scheme for obtaining themechanical properties can be applied to various situations, wheretwo parameters among contact force F, contact radius c and pres-sure p under a gap g may be used. This is very useful because thecontact force F and pressure p, which are relatively easy to measurecan be employed for property extraction. In a complementarymanner, under gap control and measuring the contact force F,the contact radius c and pressure p can be determined along withthe material properties.

6. Conclusions

An approximate closed-form Hertz-type contact theory hasbeen developed to describe the relationship between pressure,contact radius, contact force and gap. Both the slope of the deflec-tion profile of the membrane outside the contact zone and contactradius itself were measured by an apparatus based on moirédeflectometry. Note that the slope field of the deflection profile israther difficult to measure directly by other techniques. Contactexperiments with a 3 lm PET film showed that this analysis pre-dicts both the slope field and contact radius quite accurately. Itturned out that the surface roughness of the PET film resulted ina Hertz-type contact, without any surface interactions. In addition,the validity of the approximate analysis increases with residualstress. Finally, using the analysis to curve-fit the response betweenany two parameters among contact force F, contact radius c, pres-sure p and gap g can be used to obtain the Young’s modulus of andresidual stress in the thin membrane.

Acknowledgements

The authors thank Uni-Pixel Inc. for financial and technicalassistance in the course of this study. The authors also thank Pro-fessor Ray Plaut for his help on error estimation, via private com-

munication. One of the reviewers was also extremely helpfulwith comments.

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