identiﬁcation of elastic–plastic anisotropic parameters...

Mechanics of Materials 39 (2007) 340–356

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Identification of elastic–plastic anisotropic parametersusing instrumented indentation and inverse analysis

Toshio Nakamura *, Yu Gu

Department of Mechanical Engineering, State University of New York at Stony Brook, Stony Brook, NY 11794, United States

Received 11 May 2006; received in revised form 22 May 2006

Abstract

Mechanical responses of thin films or coatings often display anisotropic behaviors because of their unique microstruc-tures. However, their small size scales can also make determination of material properties difficult. The present paper intro-duces a simple yet versatile procedure with advanced data interpretation scheme to identify key anisotropic parameters.This procedure utilizes instrumented indentations and an inverse analysis to extract unknown parameters of elastic–plastictransversely isotropic materials. In particular, it post-processes load–displacement records of depth-sensing indentations toobtain best estimates of Young’s moduli and yield stresses along longitudinal and transverse directions, respectively. Majoradvantages of this method are the minimal specimen preparations and the straightforward testing procedure. To enhancethe accuracy, the method utilizes two differently profiled indenter heads, spherical and Berkovich. Prior to actual testing,detailed simulations were performed to verify the method’s applicability and robustness. In the experiment, a thermallysprayed NiAl coating which possesses process-induced anisotropic features is considered. The load–displacement recordsof spherical and Berkovich nano-indentations are post-processed with the proposed inverse analysis scheme. The estimatedresults predict dissimilar responses along the longitudinal and transverse directions. Separate tests are also conducted withmicro-indenter heads under larger loads. They demonstrate lesser anisotropic effects but with more compliant responses.These results are attributed to the unique morphology of thermally sprayed coatings, which inherently exhibit size andanisotropic effects.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Elastic–plastic transversely isotropic; Inverse analysis; Kalman filter; Thin films; Thermally sprayed coatings; Instrumentednano-indenter; Spherical and Berkovich indenters

1. Introduction

Thin films and coatings have been widely usedin various applications such as large-scale inte-

0167-6636/$ - see front matter � 2006 Elsevier Ltd. All rights reserved

doi:10.1016/j.mechmat.2006.06.004

* Corresponding author. Fax: +1 631 6328544.E-mail address: [email protected] (T. Naka-

mura).

grated circuits, electronic packaging, sensors, opti-cal films as well as protective and decorativecoatings (e.g., Elshabini-Riad and Barlow, 1976).For effective designing and accurate evaluation oftheir structural integrity, determinations of theirmechanical properties are vital for product devel-opment. Fabrication techniques for thin film/coat-ing include physical vapor deposition, chemical

.

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T. Nakamura, Y. Gu / Mechanics of Materials 39 (2007) 340–356 341

vapor deposition, chemical methods (e.g., electro-plating), and thermal spray (e.g., plasma spray,combustion spray). Due to these fabrication pro-cesses and resulting microstructures, thin films/coatings often exhibit anisotropic mechanicalresponses. Crystalline/columnar microstructuresobserved in vapor deposited materials and lamellarmicrostructures by solidified splats in thermallysprayed coatings are some good examples. Fig. 1shows a cross-section of atmospheric-plasmasprayed Ni–5wt.%Al coating containing splat andpore/crack morphology. These geometrical featuresare predominantly aligned along the transverse/horizontal direction (i.e., perpendicular to spraydirection) to induce different responses along thelongitudinal and transverse directions. Here thelongitudinal is defined as the normal to the coatingsurface or the spray direction while the transverserefers to the two in-plane directions. Althoughthese anisotropic responses of thin films/coatingshave been well known, due to their small scales,there are limited experimental procedures to mea-sure their inelastic responses.

For isotropic characterizations of thin materials,several notable experimental procedures have beenintroduced in recent years. Many of them utilizedepth-sensing indentations (Fischer-Cripps, 2000).For Young’s modulus and hardness, Oliver andPharr (1992) and Doerner and Nix (1986) proposedmethods based on maximum loads and unloadingslopes. Field and Swain (1993) also investigatedyield stress and strain hardening characteristics.Suresh and Giannakopoulos (1998) and Giannako-

Fig. 1. SEM micrograph showing cross-section of air plasmasprayed Ni–5wt.% Al. The vertical direction corresponds to spraydirection (longitudinal). Very dark regions correspond to cracksand pores.

poulos and Suresh (1999) introduced effective pro-cedures to quantify residual stresses and elastic–plastic properties of homogeneous materials.Guidelines and assumptions needed for accuratemeasurements were also discussed by Cheng andCheng (1999) and Venkatesh et al. (2000). Nakam-ura et al. (2000) proposed an inverse analysis proce-dure and instrumented micro-indentation toestimate properties of graded medium using two dif-ferently sized spherical heads. Dao et al. (2001) uti-lized a forward–reverse method and instrumentedsharp indentation to characterize elastic–plastic iso-tropic properties. Chollacoop et al. (2003) improvedthis method by employing two different indenterswith Berkovich and cone heads. In addition, special-ized experimental techniques such as bulge testing,substrate curvature, X-ray diffraction, micro-Raman spectroscopy, and electron diffraction con-trast imaging, have been developed to determinemechanical properties of both free-standing filmsand films bonded to substrates (Vinci and Vlassak,1996). While most of them did not consider theanisotropy, some have considered such effects (Vlas-sak and Nix, 1994; Vlassaka et al., 2003; Meng andEesley, 1995; Wang and Lu, 2002). These, however,investigated only elastic properties of anisotropicbased on indentations along different orientations.For significantly larger specimens, such as rolled-aluminum alloy sheets, there are several studieswhich considered elastic–plastic anisotropic proper-ties (e.g., Barlat et al., 1997; Wu et al., 2003).

Clearly, a simple examination of load–displace-ment records from instrumented indentations doesnot reveal any anisotropic effects. In fact, no expli-cit correlation between such measurements andanisotropic properties exists. This is the reason aninverse analysis is needed to extract the propertiesthrough suitable post-processing of measuredrecords. In general, such processing schemes arenecessary when direct relations between measure-ments and unknown variables are not availableor obvious. In the past, various inverse approacheshave been utilized in mechanics of material prob-lems. For examples, it was used to detect andquantify critical damage mechanisms like cracking,delamination, and corrosion in aerospace and civilstructures (e.g., Chang, 2000). However, an inverseanalysis technique is not valuable unless it satisfiesconvergence and consistency conditions. In manycases, direct implementations of inverse analysistechniques lead to inaccurate solutions due to ill-posed conditions (i.e., non-convergence to unique

Permanent magnet Coil

Limit stop

Frictionlesspivot

Capacitor plates

Sample stub

Indenter

342 T. Nakamura, Y. Gu / Mechanics of Materials 39 (2007) 340–356

solutions). A robust procedure requires suitabletechnique and proper tailoring. Furthermore, adetailed verification analysis is always necessaryprior to its application in actual tests. Here, finiteelement simulations are performed to evaluate itsaccuracy as well as robustness. The main advan-tage of proposed method is the simplicity of speci-men preparation as well as measurement process.Since indentations are used for the measurement,a free-standing specimen that requires a separationof film/coating from its substrate is not necessary.The separation process can be difficult and oftenalters the properties of film/coatings.

Balance weight

Optional Impulseassemby

Damping plate

Fig. 2. Schematic of the nano-indentation platform.

2. Nano-indentation and material models

2.1. Dual indentations

In order to obtain as much information on mate-rial responses as possible without resorting to amore complex measurement process, two separateindenter heads, Berkovich and spherical are utilized.These differently shaped heads, one is sharp whilethe other is round, generate different deformationfields (discussed in Section 3.4), and thus offer addi-tional information on material behaviors. The useof dual indenters only adds small testing effort. Noteboth heads only indent onto a top free-surface ofspecimen. Recently, some attempts were made toindent onto a side lateral surface to quantify theanisotropy (e.g., Sampath et al., 2004). Howeverthe lateral indentation introduces two complexities.First, indentations on (thinner) lateral planes of thinfilms are usually difficult because of small scale. Infact, to avoid boundary effects, the specimen mustbe sufficiently thick, which excludes very thin films(t < �5 lm) from testing. Second, correct interpre-tations of material response requires full 3D analy-sis for anisotropic materials which can be costly.Thus, in view of keeping the measurements andpost-analysis straightforward, indentations ontotransverse planes are not considered here.

In the present test, indentations are carried with anano-indenter from Micro-Materials, Ltd., whoseschematic is shown in Fig. 2. This nano-scale inden-ter has maximum loading range of 350 mN anddepth resolution of 15 nm. The schematics of Ber-kovich and spherical indentations are illustrated inFig. 3. The Berkovich indenter has a diamond tip(E = 1140 GPa, m = 0.07) while the spherical inden-ter has a sapphire tip (E = 335 GPa, m = 0.25) and

the radius of curvature is 270 lm (measured frommicrograph). Prior to the test, the machine compli-ance was determined by calibrated indentationsonto a fused quartz sample and the compliance val-ues were adjusted from the load–displacementrecords. Note that both indenters have much highermodulus and hardness than those of test specimenand they are capable of generating large plasticdeformation in the specimen.

2.2. Specimens

The test specimen chosen for our analysis wasatmospheric-plasma sprayed (APS) Ni–5wt.% Al(t = 290 lm) coating. This material is often usedas bond coats for thermal barrier coatings on tur-bine components because of its good adhesion char-acteristics. The material selection was made basedon well known anisotropic characteristics of APSmaterials. We have also tested EB-PVD coppercoating. However it exhibited limited plasticityand was not suitable to evaluate the feasibility ofproposed method. Due to inhomogeneous micro-structures of thermally sprayed coatings (as shownin Fig. 1), interpretation of indented measurements

substrate

t

P

R

thin film/coating

x2 (L)

x1 (T )

x3 (T )

substrate

t

P

thin film/coating

x2 (L)

x1 (T )

x3 (T )

a

b

Fig. 3. Schematics of dual indentations, (a) Berkovich and (b)spherical. Anisotropic directions are noted with L (longitudinal)and T (transverse).


requires a special care. Depending upon the inden-tation location, measured load–displacement recordmay vary substantially. In fact the morphologyunderneath the contact (e.g., center of splat orsplat–splat interfaces) dictates the deformationcharacteristics. With NiAl tested here, the displace-ment variability among different indentations wasabout ±10% at the constant peak load. Note, thesizes or microstructural inhomogeneity can exceed10 lm while the indentation depths are less than1 lm. This means, even with polished surfaces, theirregularities underneath indentation can affect themeasurements. Since the purpose of current analysisis to study the feasibility of proposed method, wedid not probe into the variability effect in details.Instead, an average of load–displacement curvesgenerated from 10 separate indentations was usedas the input in the data analysis. Therefore theresults reported here should be treated as the aver-aged properties. Future studies are needed to eluci-date this aspect of thermally sprayed materials.

In order to further demonstrate the applicabilityof proposed method, micro-indentation tests, whichhave much larger indented foot-prints, are also per-formed. Since a single indentation covers manysplats, the measured variability (about ±3%) amongdifferent indentation was less than that of nano-indentations. The indentations by micro-indentersoffer additional confirmation of the procedure andalso reveal the size effects of thermally sprayed coat-ings. Due to unique microstructures, APS coatingsexhibit different responses at various size scales.The size effects on material responses are describedin Section 5.3. These tests should prove the versatil-ity of the proposed method and its applicability tothin films/coating systems possessing transverselyisotropic properties.

2.3. Elastic–plastic transversely isotropic materials

Many thin films and coatings exhibit isotropicresponse on one plane and rotational symmetrywith respect to one axis (usually along the thicknessdirection), and often their anisotropic responses aremodeled as transversely isotropic. In a transverselyisotropic material, the elastic properties aredescribed by five independent constants. Supposethe x1–x3 plane is the transverse plane of isotropy(see Fig. 3(a)), the stress–strain relations are repre-sented as

e11

e22

e33

c12

c23

c31

2666666664

3777777775¼

1=ET �mLT=EL �mT=ET 0 0 0

�mTL=ET 1=EL �mTL=ET 0 0 0

�mT=ET �mLT=EL 1=ET 0 0 0

0 0 0 1=GL 0 0

0 0 0 0 1=GL 0

0 0 0 0 0 1=GT

2666666664

3777777775

r11

r22

r33

r12

r23

r31

2666666664

3777777775:

ð1ÞHere, the subscripts ‘L’ and ‘T’ denote for thelongitudinal and transverse directions, respectively.The Poisson’s ratios mLT and mTL are related as

mTL

ET

¼ mLT

EL

: ð2Þ

The in-plane (x1–x3 plane) shear modulus can bealso expressed as

GT ¼ET

2 1þ mTð Þ : ð3Þ

For the plastic yielding of anisotropic body,Hill’s criterion (1948) for orthotropic materials isadopted. His yield function is modified for trans-versely isotropic materials as

f ðrÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAðr11 � r22Þ2 þ Aðr22 � r33Þ2 þ Bðr33 � r11Þ2 þ 2Cs2

12 þ 2Cs223 þ 2Ds2

31

q� r0 ¼ 0: ð4Þ


Here r0 is the reference stress, and A � D are thedimensionless constants related to r0 as

A ¼ 1

2

r0

r0L

� �2

; B ¼ 1

2

2r20

r20T

� r20

r20L

� �;

C ¼ 1

2

r0

s0L

� �2

; D ¼ 1

2

r0

s0T

� �2

:

ð5Þ

In the above, r0L, r0T, s0L and s0T are the yieldstresses along the respective directions. In the cur-rent model, the post-yielding behavior was assumedto be represented by a power-law hardening with aconstant hardening exponent. In terms of uniaxialstress and strain, the constitutive relations alonglongitudinal and transverse directions are

eL ¼r0L

EL

rL

r0L

� �n

for rL > r0L and

eT ¼r0T

ET

rT

r0T

� �n

for rT > r0T: ð6Þ

where eL and eT are anisotropic strains and n is thestrain hardening exponent. For simplicity, the sameexponent was assumed for the longitudinal andtransverse directions since separate estimations with

reasonable accuracy would not be possible fromindentation measurements. However any differencein actual post-yielding hardening behaviors isimplicitly approximated in the estimated values ofyield stresses along L and T. In other words, if theexponents are different, the estimated yield stressesare adjusted to offset the error. In summary, thepresent transversely isotropic material is describedby five elastic constants, four yielding stresses, andone hardening exponent.

2.4. Unknown material parameters

The present procedure is designed to estimate thematerial parameters of anisotropic materials. How-ever, it is very difficult and perhaps not practical todetermine the entire 10 unknown constants neededto define the elastic–plastic behavior of transverselyisotropic materials. Instead we concentrate onextracting key parameters of such materials. Theyare the Young’s moduli and yield stresses alongthe longitudinal and transverse directions, respec-tively, and the hardening exponent n. In the processto reduce the total number of independent materialvariables, some assumptions are made a priori.

First, all the Poisson’s ratios are presumed.Although the measurements of Poisson’s ratios ofisotropic materials have been carried out with tech-niques like ultrasound (e.g., Hardwick, 1987; Liet al., 2001; Lasaygues and Pithioux, 2002), it wouldbe difficult to obtain accurate values for anisotropicfilms/coatings. Here, the Poisson’s ratio in the trans-verse plane mT was assumed as 0.25, which is close tothe bulk NiAl system. Second, the sum of two otherPoisson’s ratios is defined as twice of mT, i.e., mTL +mLT = 2mT (note mTL and mLT are not constant asmodulus changes). Third, the out-of-plane shearmodulus was approximated as

GL ¼ðET þ ELÞ=2

2½1þ ðmTL þ mLTÞ=2� : ð7Þ

This follows the definition of GT shown in (3). Theseapproximations should be valid (within a reason-able accuracy) since the differences in propertiesalong the transverse and longitudinal directions ofmaterials considered are not as large as those of fi-ber-reinforced composites materials (where ET andEL are more than a magnitude different). Theseassumptions reduce the independent elastic parame-ters to ET and EL.

For the plastic parameters, shear yield stressesare set according to the von Mises criterion (i.e.,s0 ¼ r0=

ffiffiffi3pÞ as

s0T ¼r0ffiffiffi

3p and s0L ¼

r0ffiffiffi3p

ffiffiffiffiffiffiffir0L

r0T

r: ð8Þ

Here, the reference stress r0 is defined as the meantensile yield stress, r0 = (r0T + r0L)/2. The aboveequations leave the unknown plastic parameters tobe r0T, r0L and n. Together with the elastic con-stants, five unknown values are estimated from theproposed procedure.

3. Parameter identification procedure

3.1. Estimation process

Even with the reduction of unknown parameters,simultaneous estimations of five parameters are notrealistic. An optimum approach is to separate theestimation process in steps. First, the mean Young’smodulus is defined as Em = (ET + EL)/2, and itsvalue is estimated. Due to pile-ups and other phys-ical factors, measured Young’s modulus from


indentation often contains a substantial error (evenin isotropic solids), especially when the material isductile (Hay and Pharr, 1999; Giannakopoulos andSuresh, 1999). Thus, it makes more sense to estimatethe mean modulus and then the modulus ratio(ET/EL), instead of directly determining individualET and EL. The mean modulus is measured usingthe well-established Oliver and Pharr method(1992) which uses the unloading portion of load–dis-placement curve. Since the present procedure utilizesboth spherical and Berkovich indentations, Em iscomputed from the average of two measurements.This process should limit a possible error in the esti-mated ratio ET/EL even if a large error exists in theestimated Em. In fact our sensitivity study shows thatsmall change in Em does not translate to a larger devi-ation in ET/EL (discussed in Section 5.1). Since thedetermination of Em is made from the unloading partof indentations and this data is not used to estimatethe other parameters, reuse of same data is avoided.

Next, the strain hardening exponent n is estimatedfrom the method introduced by Dao et al. (2001) forisotropic elastic–plastic material. As noted by Cholla-coop et al. (2003), accurate determination of n is dif-ficult with indentations although the problem can beovercome by multiple indenters. However, for aniso-tropic materials such an approach is not feasible sinceit would require development of very complex for-ward solutions. Furthermore, a small perturbationin n does not significantly alter the overall stress–

strain relation since any deviation of n is compensatedby the estimated value of yield stress r0. Suppose anestimate of n is larger (less hardening) than the actualvalue, then the estimate of r0 (subsequently made)would be larger than the correct value to offset theerror in n. This means the actual and estimatedstress–strain curves would still look similar exceptnear the yield point. Although the present methodutilized the technique developed for isotropic materi-als with Berkovich indentation (Dao et al., 2001), onemay opt to use a bulk material’s hardening value forn. Either approach should not cause significant errorin the estimated stress–strain relation at least withinsome range of plastic strains. After Em and n are esti-mated, the remaining unknown parameters ET/EL,r0T, and r0L are estimated using the inverse analysisdescribed next.

3.2. Inverse analysis

In the inverse analysis technique, Kalman filter(Kalman, 1960) is utilized. This technique is widely

used in many areas of engineering such as sensorcalibration, tracking applications, trajectory deter-mination and signal processing. The Kalman filteralgorithm has an advantage over other adaptivealgorithms through its fast convergence to optimalsolutions in nonlinear problems. Here the algorithmwas tailored to estimate three unknown parametersfrom two measurement records.

In the formulation, the three unknown parame-ters are represented in a state vector form as xt =[(ET/EL)t, (r0T/r0L)t, (r0)t]

T. Here t may representactual or load step/increment (e.g., from Pmin toPmax). The successive estimates for xt are indirectlymade from the indented displacements DBerk

t andDSphe

t , measured with Berkovich and spherical heads,respectively. Since they have different load ranges,actual loading level at a given t is different whilethe total numbers of increments are kept the same.At t = 0, the initial estimates x0 = [(ET/EL)0,(r0T/r0L)0, (r0)0]T are assigned and the subsequentestimates (at t = 1,2,3, . . .) are made according tothe following equation:

xt ¼ xt�1 þ K t½Dmeast �Dtðxt�1Þ�: ð9Þ

Here, Kt is the ‘Kalman gain matrix’ and Dmeast is a

vector containing measured displacements (i.e.,Dmeas

t ¼ ðDBerkt ;DSphe

t ÞTÞ at t. Also Dt(xt�1) is a vectorcontaining measured parameters computed withestimated unknown state parameters at the previousincrement. Note that these computations requiresolutions of measured parameters for a given setof state parameters. Such solutions are sometimesreferred as the ‘forward’ solutions. In the aboveequation, the Kalman gain matrix multiplies the dif-ference between the measured and computed dis-placements to make corrections to the unknownstate parameters. The Kalman gain matrix is com-puted as

K t ¼ PtdTt R�1

t where

Pt ¼ Pt�1 � Pt�1dTt ðd tPt�1dT

t þ RtÞ�1d tPt�1: ð10Þ

With three state and two measured parameters, thesize of Kalman gain matrix is 3 · 2. Also dt is a 2 · 3matrix that contains the gradients of Dt with respectto the state parameters as

d t ¼oDt

oxt¼

oDBerkt

oðET=ELÞoDBerk

t

oðr0T=r0LÞoDBerk

t

or0

oDSphet

oðET=ELÞoDSphe

t

oðr0T=r0LÞoDSphe

t

or0

0BBB@

1CCCA:ð11Þ

Set t = 0 andassign initial estimates for state vector

containing unknown parametersx0 = [(ET/EL)0, (σ

0T/σ 0L)0, (σ

0)0]T.

Set t = t + 1

Compute Kalman Gain MatrixKt = Pt dt Rt

-1

Here covariance matrix (of estimated xt) is

Pt = Pt−1 – P t−1 d tT(d t P t−1d t−1

T + Rt )-1dt P t−1,

gradient of displacement is dt =t

t

xD

∂∂

and

vector for measurable parameters is Dt.

Read in measured Dtmeas and

update as xt = x t−1 + kt (Dtmeas – Dt)

Check t = tmax

Set xest = xt

Yes

No

Fig. 4. Flow chart of Kalman filter procedure to determine theunknown parameters using instrumented indentation records.


In addition, Pt is the ‘measurement covariance ma-trix’, related to the range of unknown state param-eters at increment t, and, Rt is the ‘error covariancematrix’, related to the size of measurement error.Once the initial values are imposed, Pt is updatedevery step, while Rt is must be prescribed at eachstep. However, in many cases, fixed values can beassigned to the components of Rt as long as mea-surement error bounds do not vary substantiallyduring the duration of measurements. Since the con-vergence rate is sensitive to the values of Pt and Rt,proper assignments for these two matrices are essen-tial. The initial measurement covariance matrix P0

and the error covariance matrix Rt are set as

P0 ¼ðDðET=ELÞÞ2 0 0

0 ðDðr0T=r0LÞÞ2 0

0 0 ðDr0Þ2

0B@

1CA

and Rt ¼ðRBerkÞ2 0

0 ðRSpheÞ2

!: ð12Þ

Here D(ET/EL),D(r0T/r0L) and Dr0 denote the pre-dicted ranges of the unknown parameters, respec-tively. While P0 is diagonal, the procedure resultsin a filled Pt matrix during subsequent increments.In the current analysis, the diagonal componentsof Rt are chosen based on the estimated measure-ment error for the weight and the strain measure-ments. The values of RBerk and RSphe are setapproximately 1% of the respective displacementsat the maximum loads. The Kalman filter proce-dure, which is summarized in Fig. 4, is implementedin a computational code.

3.3. Reference data via finite element model

Any inverse analyses require at least some knowl-edge of relations between unknown state parametersand measurable parameters. Here the Kalman filteralgorithm requires load-dependent solutions ofindented displacements as functions of given stateparameters. As no simple solutions, which explicitlyaccount for the anisotropic plastic flow, are avail-able, detailed finite element simulations are carriedout to construct the reference data or so-calledforward solutions.

The reference data comprise of displacementsand their gradients for various values of ET/EL,r0T/r0L and r0 for the load range of interest andthey were computed with axisymmetric finite ele-ment models. Here a conical model was assumed

to represent the Berkovich indenter. This is acommonly used approach to circumvent costly 3Dcalculations. Here the cone angle is set to 140.6�to represent the same projected area-to-depth ratioas the Berkovich indenter (Fischer-Cripps, 2000).During the mesh construction, careful steps weretaken to minimize the instability which can occurnear the perimeter of indentation (due to suddenrotation of element surfaces). An enlarged finite ele-ment mesh of the coating near the contact region isshown in Fig. 5. The finite element model containsapproximately 8200 axisymmetric four-noded ele-ments. A contact condition is prescribed along theboundary between the indenter and coating. Forthe materials and loading conditions consideredhere, there were essentially no effects of friction as

Fig. 5. Local axisymmetric finite element mesh of thin film nearindentation. Profiles of conical (approximates Berkovich withcone angle 140.6�) and spherical indenters are shown.

(

P

(mN

)

Berkovich (conical) indenter

sphericalindenter

Simulated Indentations

0

100

200

300

400

0 0.5 2.51.51 2

stre

ss (

MP

a)

strain (%)

ET = 52GPa, EL = 48GPa

oT = 214MPa, oL = 186MPan = 5

0 0.5 1.51 2

transverse

0

100

200

400

300

longitudinal

Sample Model

μm)Δ

σσ

a

b

Fig. 6. (a) Uniaxial stress–strain curves along longitudinal andtransverse directions of sample transversely isotropic model. (b)Simulated load–displacement indentations curves.


the sliding between the indenter and the surfaces isvery limited. Although the actual specimen isattached to a steel substrate, the dimensions of coat-ing are large enough with respect to the indentationsizes to avoid any substrate and outer boundaryeffects.

3.4. Simulated indentation of transversely isotropic

materials

Prior to generating reference data, a samplestress–strain relation was used to evaluate indenta-tion responses of a transversely isotropic material.The prescribed uniaxial stress–strain curves alonglongitudinal and transverse directions of a fictitiousmaterial are shown in Fig. 6(a). These material val-ues are arbitrary chosen although they are similar inmagnitudes with those of actual specimen. Simu-lated load–displacement records from the dualindentations are shown in Fig. 6(b) with the unload-ing portions represented in dashed lines. In the pres-ent analysis, the choice of peak or maximum load isvital. Since the elastic–plastic properties are sought,the load must be large enough to cause sufficientplastic deformation underneath the indentations.A few trial simulations were conducted to determinethe suitable maximum loads. They are chosen as100 mN and 350 mN for the Berkovich and spheri-cal indenters, respectively. These loads within the

range of optimal load for the present indenter sys-tem. Simulations were also performed to evaluatethe frictional effects between the heads and speci-men surfaces. For the models and materials consid-ered here, basically no effects on load–displacementcurves were observed for any frictional coefficients(0 < ls < 0.9). Although stress states near the con-tact were slightly different, the relative energy lossdue to friction was minimal.

Some observations can be made from the simu-lated results. First, substantial differences are dis-played in the stress fields under two indentations.The stress contours of rz, rr, srz are shown inFig. 7 for respective indentations. They clearly exhi-bit higher stress concentration under the conical(Berkovich) indentation. In contrast, the sphericalindentation causes larger spread of high stresses.Its difference with the conical case is most noticeablein the rr contours. As noted earlier, these different

characteristics are valuable in obtaining accurate

Fig. 7. Axial (rz), radial (rr), and shear (srz) stresses near contact region. Conical and spherical indentations display different stress fields.


estimates in the inverse analysis. We also observedthe load–displacement curves tend to shift in theopposite direction when the ratio of Young’s mod-ulus is changed as illustrated in Fig. 8. With thespherical indentation, larger displacements areobserved with larger ET/EL (for a given load) whilesmaller displacements are observed with the conicalindentation. The geometrical shapes of indenters areresponsible of these phenomena. The sphericalindenter has flatter contact surface and its responseis more influenced by the longitudinal modulus EL.On the other hand, the sharper Berkovich indentergets influenced by the transverse modulus ET more.These conflicting behaviors exhibited by the two

indenters are helpful in estimating the propertiessince they offer more information to the inverseanalysis. In summary, this simulation study sup-ports the utilization of two differently shapedindenters.

3.5. Domain of unknown and reference data

The inverse analysis to determine the unknownparameters requires the following steps: (1) setdomain/ranges of unknown state parameters; (2)construct reference data; (3) choose optimal loadincrements for measurement inputs in KalmanFilter process. The domain or bounds of unknowns

P conical indenter

with fixed Em, OT, OL

spherical indenter

Increasing ET/EL

Increasing ET/EL

Δ

σ σ

Fig. 8. Schematic of load–displacement curves from conical andspherical indentations. They display opposite behaviors as themodulus ratio changes. ET /EL

T / L

0 (MPa)

100

1.4

0.6

250

1.40.6

domain of unknown parameters

σ

σ σ

Fig. 9. Domain of three unknown state parameters ET/EL, r0T/r0L, and r.


must be chosen carefully since the analysis cannotestimate the unknowns if they fall outside. How-ever, if the convergence is not achieved, the domainsize can be always adjusted. In our current studies,known values and earlier studies are used to setthe suitable domain size.

For the ratio of elastic moduli in anisotropicmaterials, there are some experimentally identifiedresults (e.g., Vlassak and Nix, 1994; Meng and Ees-ley, 1995; Wang and Lu, 2002). They reported thatthe difference between elastic moduli of different ori-entations in single crystals can be up to 30%. Thetransverse modulus ET of thermal sprayed coatingcan be significantly higher than longitudinal modu-lus EL (Khor et al., 2003). On the other hand, mea-sured anisotropic yield stresses of rolled-aluminumalloy sheets were less than 5% (Barlat et al., 1997;Wu et al., 2003). Based on these references, thedomains for ET/EL and r0T/r0L are both set from0.6 to 1.4 in this analysis. These ranges should allowsufficient variations in the transversely isotropicparameters.

For the range of yield stress, r0 was set from100 MPa to 250 MPa based on previous analysison sprayed NiAl coatings (Sampath et al., 2004).The domain of three unknown parameters is illus-trated in Fig. 9. Using these ranges, the referencedata source was constructed. Since many computa-tions with slightly different values of state parame-ters are not practical, we utilize cubic Lagrangianinterpolation functions to compute the load–dis-placement relations. In this approach, 64 (=4 ·4 · 4) based points are chosen within the domain,and finite element computation is carried out with

each set of state parameters. During each calcula-tion, indented displacements at about 100 load stepsare recorded. Since both Berkovich and sphericalindentation solutions are required, total of 128 sep-arate finite element simulations are carried out.Once these base solutions are made, displacementsas well as displacement gradients, for any combina-tions of state parameters are interpolated at anyload level.

4. Verification study

Prior to implementing the proposed procedure toestimate unknown properties of real specimens, theinverse analysis was carried out with simulatedmeasurements. Since exact solutions are known insimulations, they can be used to evaluate the accu-racy and robustness of the procedure. Because firsttwo steps to estimate the mean modulus and thehardening parameter are already established meth-ods, only the remaining three parameters are esti-mated here. In the calculation, the materialparameters as shown in Fig. 6(a) (ET/EL = 1.08,r0T/r0L = 1.16, r0 = 200 MPa with Em = 50 GPaand n = 5) are assumed, and the simulated load–dis-placement records in Fig. 6(b) are used to backtrackthe parameters. For the input, indented displace-ments at 20 load increments were chosen. Forconical/Berkovich indentations, the load range ofP = 20 mN to 96 mN with intervals of 4 mN, and40–344 mN with 16 mN intervals for the sphericalindentation are chosen.


To obtain unknown parameters with the Kalmanfilter, initial estimates of ET/EL, r0T/r0L, and r0

must be assigned so that they are updated as themeasured parameters are supplied at each loadincrement. However, final estimates are generallynot identical for different initial estimates. In ourunique procedure to identify the best estimates,the Kalman filter is carried out with many differentinitial estimates with their values spread evenly (at41 · 41 · 41 = 68,921 points) in their domain ofunknowns shown in Fig. 9. For each set of initialestimates, the Kalman filter computes the final esti-mates. In order to illustrate the different values offinal estimates, a contour plot is constructed. Thisintensity of convergence plot can then be used toidentify the most likely values or the best estimates

of unknowns. A location with large intensity signi-fies more initial estimates converged near these val-ues of unknowns. A location with high intensity ofconvergence is chosen for the best estimates.Although three unknowns are estimated simulta-neously, their intensity of convergence is shown in

Fig. 10. Intensity of convergence plot generated by the inverse anarepresents convergence of many initial estimates and likely location oflocation of exact solutions (i.e., input values) is also noted.

three separate 2D plots (for clarity) in Fig. 10. Notethat the scale represents a relative intensity and itdoes not have any physical significance.

In inverse analyses, good convergence characteris-tics prevail when more measurements are availableand unknown state parameters are sensitive tomeasurement parameters. Furthermore, for a goodconvergence, the number of measurements is ideallyequal to the number of unknowns. Although thereare three unknowns with only two measurements(i.e., P–D data from dual indentations), a relativelygood convergence is obtained because of sufficientsensitivity to the measurements here. Based tothe weighted average of converged locations shownin Fig. 10, the best estimates are chosen asET/EL = 1.06, r0T/r0L = 1.18, r0 = 199 MPa. Thesevalues are very close to the exact solutions used togenerate the simulated measurements (i.e., ET/EL =1.08, r0T/r0L = 1.16, r0 = 200 MPa). Thus the verifi-cation study supports the effectiveness of proposedapproach although the actual implementation stillrequires additional work as described next.

lysis from simulated dual indentation results. A high intensitybest estimates. The scale of intensity (i.e., 0–100) is relative. The


5. Anisotropic properties of Ni–5wt.%Al coating

The test specimen, Ni–5wt.%Al coating on steelsubstrate was fabricated by plasma spray facilityat the Center for Thermal Spray Research at SUNYStony Brook. The feedstock is Metco 4008NSpowder with sizes range in 11–45 lm. The substratetemperature was pre-heated at 150 �C. Prior to theindentations, the coating surface was polished usingfine emery papers (240, 320 and 600 grit) and dia-mond suspensions (9, 3 and 1 lm). The final coatingthickness was measured as t = 290 lm with thedimensions of substrate as 50 mm · 25 mm ·2.5 mm.

5.1. Nano-indentation test

First nano-indentations were carried out on theNiAl coating. Similar to the simulation describedin Section 4, the maximum load was set at100 mN for the Berkovich indenter while it was setat 350 mN for the spherical head. For each case,10 indentations are carried out and the averagedP–D curve is determined are shown in Fig. 11. Asdiscussed in Section 2.2, indented displacements atthe maximum load exhibited about ±10% variationdue to the inhomogeneous morphology of sprayedmaterial. Prior to performing the Kalman filter,the mean values of Young’s modulus Em and thehardening coefficient n are estimated. First the mod-ulus is estimated from the unloading curves ofspherical and Berkovich indentations shown inFig. 11. Using Oliver–Pharr approach (1992) the

0 0.5 1.510

100

200

300

400

( m)

P (

mN

)

Berkovich indentation

sphericalindentation

Dual Nano-Indentations

μΔ

Fig. 11. Measured nano-indentations of plasma sprayed Ni–5wt.%Al coating. Each curve represents average of 10indentations.

spherical head predicted 87 GPa and the Berkovichresulted 108 GPa. Here the average of the two ischosen and the mean modulus is set as Em =98 GPa. As previously noted, this rather large differ-ence is common in indentations of ductile materials.In fact, this is the reason that the modulus ratioinstead of individual modulus is chosen as aunknown. The accuracy of Em estimate will be dis-cussed later. Second, the post-yield hardeningparameter was approximated using the forwardsolutions of Dao et al. (2001). Through an iterativeprocess, the best fit was determined as n = 2.1.Again, the preciseness of n is not so critical sinceany difference is compensated in the subsequent esti-mate of r0.

With estimated Em and n, the Kalman filer is car-ried out using the loading portions of the measuredP–D records shown in Fig. 11. A similar procedureas described in the verification analysis is followed.The convergence behavior at the final increment isillustrated in the intensity of convergence plots inFig. 12. At first glance, relatively large domains ofconvergence (i.e., shaded regions) are observed,especially as compared to the ones shown inFig. 10 from the numerical simulations. The deteri-oration is expected since the actual material doesnot exactly follow the assumed constitutive modeland the measurements contain greater errors. How-ever if one concentrates on inspecting regions withhigher intensity (e.g., >50), they are sufficientlywell-contained. This suggests a reasonable accuracyof the estimates. Based on these plots, the best esti-mates are identified as ET/EL = 1.07, r0T/r0L =0.77, and r0 = 204 MPa. The estimates implyslightly stiffer elastic responses along the transversedirection but with lower yielding stress. Theseresults are consistent with preliminary studies onanisotropy of thermally sprayed coatings (e.g., Sam-path et al., 2004). Additional measurements areneeded to relate the elastic–plastic anisotropicbehavior to the microstructures of these materials.Since they are beyond the scope of present study,this aspect will be investigated more closely in afuture study. The anisotropic stress–strain curvescorresponding to these estimates are shown inFig. 13(a).

In most inverse analyses, there is no independentway to prove their best estimates are indeed corrector near-correct solutions. However, there are twoways to validate the likeliness of accuracy. One isfrom the sizes of converged regions shown in theintensity of convergence plots. A small region

Fig. 12. Intensity of convergence plot generated by the inverse analysis from measured P–D records of Berkovich and spherical nano-indentations. The location of best estimates is determined from weighted averages of convergence.


implies that the inverse method is robust since manyinitial estimates converged near the same location(i.e., similar estimates). The converged regionsshown in Fig. 12 are reasonably contained althoughother analyses (e.g., Gu et al., 2003) have shownbetter convergences. An additional confirmationon the accuracy can be made from re-simulationof indentation. Using the best estimates identifiedin the inverse analysis as the inputs, both sphericaland Berkovich indentations are simulated again.The computed load–displacement curves are thencompared with the actual measurements. As shownin Fig. 13(b), throughout the loading range, the sim-ulated results are close to the measured results(shown with circles).

We have further examined the sensitivity of solu-tions by assigning slightly different values to ET/EL

and r0T/r0L (e.g., ET/EL = 1.04 instead of 1.07),and re-simulated the indentations. Although notshown here, the resulting P–D curves consistentlyexhibited poorer agreements with the measured datathan those shown in Fig. 13(b). Although theseresults still do not prove the uniqueness of solutions,at least they support the likeliness of correct solu-

tions. Furthermore, since small changes in the mod-ulus ratio always produced inferior match with themeasurements, the P–D curves appear to be suffi-ciently sensitive to ET/EL. Essentially it implies(though not proven) that no other values ofunknown parameters would produce better fit withthe measured data for given material model.

5.2. Micro-indentation test

Similar to the nano-scale measurements, dualmicro-indentations re also performed to test theapplicability of proposed procedure and to probethe size effects of thermally sprayed coatings (seeSection 5.3). The tests are conducted with Micro-Material’s larger load indenter whose load scaleranges 1–20 N. For the Berkovich test, a standardpyramid diamond tip was used while the sphericalindenter is made of carbide-cobalt tungsten whoseradius is 795 lm with E = 614 GPa and m = 0.22,respectively. For the micro-indentation, suitableload ranges were also sought prior to actual tests.They were chosen as 3 N and 10 N for the Berko-vich and spherical indenters, respectively. The

( m)

P(m

N)

0 0.5 1.510

100

200

300

400



Dual Nano-Indentation

Experimental measurements

FE simulations

stre

ss (

MP

a)

strain (%)

APS NiAl coating

transverse

longitudinal

From nano-indentation

0.0 0.2 0.4 0.6 0.8 1.00

100

200

300

400

500

μΔ

a

b

Fig. 13. (a) Transversely isotropic constitutive relation of NiAlcoating estimated by the dual nano-indentation and inverseanalysis. (b) Comparison between measured and simulatedresults. The latter solutions are obtained by assigning the bestestimates as input properties.

0 2 4 6 10 1280

4

6

8

10

12

2

( m)

P (

N)



Dual Micro-Indentations

μΔ

Fig. 14. Measured micro-indentations of air plasma sprayed Ni–5wt.%Al coatings. Each curve represents average of 10indentations.


load–displacement measurements are shown inFig. 14. Prior to performing the Kalman filter, themean values of Young’s modulus Em and the hard-ening coefficient n are estimated as well. Using theOliver–Pharr approach (1992), the spherical casepredicted 66 GPa and the Berkovich resulted70 GPa with the averaged modulus of Em =68 GPa. The smaller difference obtained here, ascompared to that of nano-indentations, is probablydue to smaller pile-ups (in relative sizes) with thelarger micro-indenters. Again with the forwardsolutions (Dao et al., 2001), the hardening exponentwas estimated as n = 2.2, which was close to the onefrom the nano-test.

For the Kalman filter, 20 load increments arechosen with the load increments ranging P = 1.0–2.9 N at equal intervals of 0.1 N for the Berkovichindentation. For the spherical indentation, it is setP = 2.0–9.6 N with 0.4 N equal intervals. As in thecase of nano-indentations, the domain of unknown

parameters as well as corresponding reference dataor forward solutions are constructed with separatefinite element simulations. The range of mean yieldstress is lowered while the ranges for the ratios arekept the same with those of the nano-indentations.The resulting intensity of convergence from theKalman filter is shown in Fig. 15. The sizes ofconverged areas are similar to those of nano-inden-tations which imply a reasonable convergence.Following the same procedure, the best estimatesare identified as ET/EL = 1.31, r0T/r0L = 0.82, andr0 = 98 MPa, respectively. Using these parameters,the stress–strain relations based on micro-indenta-tions are constructed as shown in Fig. 16. Althoughnot shown here, simulated P–D results with the esti-mated parameters showed similar agreements withthe measured data as those shown for the nano-indentations in Fig. 13(b).

5.3. Summary of results

The estimated parameters from both indentationsare listed in Table 1. The combined anisotropicstress–strain relations from nano- and micro-measurements shown in Fig. 16 clearly exhibit thesize-scale effect. This result is consistent with thewell-known mechanical responses of thermallysprayed coatings. With the nano-indenters, the sizeof region (underneath the indenters) which influencesthe responses is less than single splat size. Since a splatitself contains limited defects, the its response is stiff-ener and closer to that of bulk material. For themicro-indentation, more compliant response isobtained since the influencing region is much larger

Fig. 15. Intensity of convergence plot generated by the inverse analysis from measured P–D records of Berkovich and spherical micro-indentations. The location of best estimates is determined from weighted averages of convergence.

strain (%)

stre

ss (

MP

a)

APS NiAl coating

longitudinal direction transverse direction

Nano-indentation

Micro-indentation

0.0 0.2 0.4 0.6 0.8 1.00

100

200

300

400

500

Fig. 16. Transversely isotropic constitutive relation of NiAlcoating estimated by nano- and micro-indentations. The differ-ence in responses is attributed to size effect exhibited by plasmasprayed coating.

Table 1Estimated anisotropic material properties obtained from instrumented

Ni–5wt.%Al ET (GPa) EL (GPa)

Nano-indentation 101 ± 11 95 ± 10Micro-indentation 77 ± 6 59 ± 5


which encompasses many splats and defects. Alsomany pores and cracks exist along splat boundarieswhich make the response to be more compliant.

Detailed investigations of mechanisms causing tothe elastic–plastic anisotropic behaviors in plasma-sprayed coatings require additional studies and theyare beyond the scope of current study. However,some explanations can be made. The nano-inden-tation estimates higher yield stress along the longi-tudinal or spray direction. Due to predominantsolidification and cooling along the spray direction,columnar microstructures are often observed withinsplats. Such geometrical feature may explain thehigher yield stress along the longitudinal direction.In the micro-indentations, splat interfaces andcracks/pores also define the indentation responses.Since they are mostly aligned perpendicular to the

indentations with the inverse analysis

rT (MPa) rL (MPa) n

177 ± 21 231 ± 25 2.1 ± 0.388 ± 8 108 ± 9 2.2 ± 0.3


spray direction (longitudinal), the elastic modulusalong the transverse direction is greater as observedin Fig. 16. However, in the micro-indentation, theanisotropic effect appears to decrease for higher stres-ses (i.e., similar strains for a given stress) and alsoslightly less anisotropy in yield stresses. These phe-nomena may be caused by larger or more averagedmaterials are included in its response. Again thephysical significance of this behavior is not clear atthis point.

6. Discussions

In the present study, a novel method to estimatethe elastic–plastic anisotropic parameters of thinmaterials is proposed and examined. The main goalis to establish a method which requires minimalspecimen preparation as well as testing effort. Ourverification study as well as its initial applicationsto actual thermally sprayed NiAl coatings demon-strated promising results. The method relies ontwo common types of indenters, Berkovich andspherical, to gain sufficient information to extractbest estimates of unknown properties of trans-versely isotropic material. Here the five constantsare chosen as two Young’s moduli EL and ET, twoyield stresses r0L and r0T, and post-yielding harden-ing exponent n. First the procedure estimates themean modulus Em and the exponent n using thewell-established methods for isotropic materials(e.g., Oliver and Pharr, 1992; Dao et al., 2001).Then the remaining three parameters are simulta-neously estimated by the Kalman filter technique.

This technique extracts best estimates of un-knowns. In the present study, estimated valuesappeared to be reasonably accurate although an inde-pendent confirmation was not made due to lack ofavailable materials with well-known elastic–plasticanisotropic properties. Other approaches to measureelastic–plastic anisotropic properties (even qualita-tively) of thin films and coatings are at best limited.In fact, this was the motivation for the present study.Note that there are some MEMS based methods toconduct tensile test of thin specimens. However theyare extremely complex and expensive. Furthermore,anisotropic property measurements (including out-of-plane direction) of thin films would be nearlyimpossible with free-standing specimens. These diffi-culties suggest the only viable measurement to be theone from instrumented indentations.

It is important to note that many indentationmeasurements and their interpreted data contain

innate errors due to pile-ups and other physical phe-nomena associated with their deformation process.The present procedure does not remove these poten-tial errors arising from the indentation process. Inreality, this may make the accuracies of estimatedanisotropic parameters less than desirable. None-theless, the present method can still offer valuableinsights to anisotropic characteristics of thin filmsand coatings, which are very difficult to obtainotherwise. Furthermore, our sensitivity analysis sug-gests that the estimated ratios of modulus and yieldstress are not magnified by the potential errors. Infact, it is likely that the errors in estimated ratiosare smaller than maximum variations possible withthe error bounds noted in Table 1 for the modulusand the yield stress. In other words, though not con-firmed, the estimated ratios ET/EL and r0T/r0L aremore accurate that the individual values.

Acknowledgements

The authors gratefully acknowledge the supportsof the US Army Research Office under thegrants DAAD19-02-1-0333, DAAD19-01-1-0709(equipment) and the National Science FoundationMRSEC program at the SUNY at Stony Brookunder the grant # DMR-0080021. We also appre-ciate S. Singh, G. Bancke and L. Li at StonyBrook for providing nano-indentation and SEMmicrograph of NiAl coating. The finite elementcomputations were carried out using the code ABA-QUS, which was made available under academiclicense.

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