Materials Science and Engineering A 416 (2006) 139–149

On the determination of residual stress and mechanicalproperties by indentation

Xi Chena,∗, Jin Yanb, Anette M. Karlssonb

a Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027-6699, USAb Department of Mechanical Engineering, University of Delaware, Newark, DE 19716-3140, USA

Accepted 5 October 2005

Abstract

Residual stresses are of practical importance in bulk materials and coatings, which critically affects their mechanical integrity and reliability.Comparing with traditional techniques, the depth-sensing indentation technique provides a quick and effective method of measuring the residualstress field. In this study, we have used the finite element method to investigate the effect of in-plane residual stress on hardness and stiffnessmeasurements of a bulk material/thick coating. It is found that the contact hardness, stiffness, and indentation work are sensitive to the residualstress, in particular for materials with a relatively high yield strain. Based on the reverse analysis, a new technique is proposed for measuringt n test. Thee tic zone, ands the surfacep©

K

1

frmssetadtbs[bse

ents,own.prop-ion isng’s

e-d to

uringdata

tedith aus

andn-

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he yield strength, Young’s modulus, and in-plane residual stress in bulk material and thick coatings through one simple indentatioffectiveness of this method is demonstrated through numerical examples. The effect of residual stress on indentation stress, plasurface profile are also investigated. It is found that when the indentation is dominated by elastic deformation, the large recovery makesrofile measured after unloading an unreliable parameter in characterizing the properties.2005 Elsevier B.V. All rights reserved.

eywords: Indentation; Residual stress; Finite element analysis

. Introduction

The mechanical reliability of bulk materials and coatings (e.g.atigue, fracture, corrosion and wear) is strongly affected by theiresidual stress, commonly introduced by thermal mismatch, orechanical and thermal processing. The most critical residual

tress component in coatings is the in-plane equi-biaxial residualtress typically caused by lattice spacing mismatch and thermalxpansion mismatch between the film and substrate. Moreover,he mechanical properties and residual stress of all materialsre temperature dependent. For thin coatings and substrates, theependency becomes exacerbated with a change in tempera-

ure. This difference in behavior due to temperature change cane critical in service, since mismatch between coating and sub-trate leads to high residual stress and ultimately coating failure1]. For example, coatings used for thermal protection in gas tur-ine engines, experience local compressive stresses as high aseveral GPa at ambient[2–4]. Thus, in order to achieve a reliablengineering design, the actual level and sign of residual stress in

∗ Corresponding author. Tel.: +1 212 854 3787; fax: +1 212 854 6267.

the specimen must be determined, frequently by measuremsince associated material properties are many times unkn

There are many challenges associated with measuringerties of small scale structures, but instrumented indentatwidely used to probe mechanical properties such as Youmodulus,E, and yield strength,σy, for most engineering matrials. Therefore, we will extend the usefulness of this methomeasure the residual stress,σres.

1.1. A brief review of the indentation technique

A variety of techniques have been developed for measmechanical properties from indentation load-displacementof an elastic-perfectly plastic,stress-free bulk material[5–8],which will be review briefly in this subsection. Instrumenindentation is characterized by a sharp rigid indenter (whalf apex angleα) penetrating normally into a homogeneosolid where the indentation load,P, and displacement,δ, arecontinuously recorded during one complete cycle of loadingunloading (Fig. 1a and b). To simplify the analysis, the indeter is usually modeled as a rigid cone withα = 70.3, so tha

E-mail address: xichen@civil.columbia.edu (X. Chen). the ratio of cross-sectional area to depth is the same as for a

921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2005.10.034

140 X. Chen et al. / Materials Science and Engineering A 416 (2006) 139–149

Fig. 1. Schematic of instrumented indentation with a sharp indentation: (a) indentation on a homogeneous, isotropic semi-infinite substrate; (b) typical load-displacement curves obtained from an indentation experiment; (c) conical indentation on a specimen with equi-biaxial in-plane residual stress.

Berkovich or Vickers indenter[9]. By neglecting friction andthe finite compliance of the measuring system and the indentertip, the equations used to extract the hardnessH and indentationmodulusM are

H = P/A = cbσy, (1)

and

S = γβ2√π

M√

A. (2)

Here, the hardnessH is defined as the ratio between indenta-tion load P, projected contact areaA and yield stressσ. Theindentation modulusM is given by the plane-strain modulus,E ≡ E/1 − ν2, for isotropic materials and by a more compli-cated weighted average of the elastic constants for anisotropicmaterials[10], whereE and v are the Young’s modulus andPoisson’s ratio of the bulk material, respectively. The contactstiffnessS = dP/dδ is obtained from the slope of initial por-tion of the elastic unloading curve (Fig. 1b). The constantcbin Eq. (1) is a constraint factor that depends on indenter shapeand material properties:cb increases withE/σ and approaches aconstant (≈3) whenE tan α/σy > 20 [9,11]. β is a shape fac-tor, with β = 1 for axisymmetric indenters andβ = 1.03–1.05for indenters with square or rectangular cross-sections[12].γ = π

π/4+0.155 cotα(1−2ν/4(1−ν))(π/2−0.831 cotα(1−2ν/4(1−ν)))2

is a correction factor for the

c en-d t isi

indet wn as tationd

As the indenter penetrates the specimen, the materials eitherproduce plastic pile-up at the crater rim (when the yield strain,σy/E, is small), or exhibit the elastic sink-in effect (whenσy/Eis large)[11]. The amount of pile-up/sink-in is denoted asδp(Fig. 1a). For conical indenters, the projected contact areaA isgiven by

A = πa2 = π(tan α)2δ2c = 24.5δ2

c, (3)

where the contact depth:

δc = δ + δp. (4)

Eq.(4) contains contributions of both plastic pile-up around theindenter and elastic sink-in, which is counted negative. It is obvi-ous from Eqs.(3) and (4), that the projected contact area dependson δp. Oliver and Pharr[5] proposed an elastic model for deter-mining the contact area in which plastic pile-up is neglected:

∣∣δp∣∣ = εPmax

Smax(5)

whereε = 0.75 for a conical indenter. Alternatively, the pile-upand contact area can be measured experimentally or determinedfrom the numerical analysis (such as the finite element methodor molecular dynamics). OnceA is determined properly, thehardness is then obtained from Eq.(1) and the stiffness can bederived from Eq.(2), which allows one to measureE andσy byu

1

siduals i et al.1 ves-t sses

onical indenter[13]. Both hardness and stiffness are indepent of the indentation depth if the strain gradient effec

gnored1 [14].

1 For metals it is observed that the hardness increases with decreasingation depth, when the penetration is in the sub-micron regime. This is knotrain gradient plasticity. We ignore such effect by assuming that the indenepth is sufficiently deep.

n-s

sing the indentation technique.

.2. The effect of in-plane residual stress

Several attempts have been made to incorporate retress measurements from instrumented indentation. Tsu996[15] used standard nanoindentation techniques to in

igate the influence of uniaxial and biaxial in-plane stre

X. Chen et al. / Materials Science and Engineering A 416 (2006) 139–149 141

(applied to the specimen by bending) on the hardness and elas-tic modulus measurements of aluminum. The projected contactareas were measured by optical techniques. It was found that thehardness and elastic modulus thus calculated were essentiallyindependent of the residual stress level, which was verified bya parallel finite element simulation[16]. Based on the assump-tion that the hardness is unaffected by tensile or compressiveelastic residual stress, several general methods was proposedfor determining the in-plane equi-biaxial residual stress by con-ical or spherical indentation. Most approaches compare thecontact depth (or contact area, or load-displacement curve) ofthe stressed and unstressed specimens, from whichσres can beestimated[17–20]. In some other studies[21,22], the residualcompression was seen to enhance the pile-up and residual ten-sion reducedδp. The ratio between the projected and nominalcontact areas, i.e. (δc/δ)2, was calculated as a function ofσresbyusing finite element analysis. Therefore, the authors suggests,the in-plane residual stress can be obtained ifδp can be mea-sured properly for a given material[21,22].

The above techniques have several disadvantages: they eitherrequire testing on a stress-free specimen as a reference, or deter-mining the contact depth accurately, which is often not feasible.In addition, only few materials were tested which may lead toincomplete conclusions,2 and a complete examination of all pos-sible combinations of material properties and residual stressesremains yet to be developed.

s s anm ctedc ismsT sep ntaca on oh ater h tho rialsu thod( s one ngs.

studb ecti ant essT strae e, thi ss;a thickc tiona on ah laner pliedt tingt

sitivet

2. Model definition

2.1. Dimensional analysis

The axisymmetric model for indentation on a bulk materialor thick coating with in-plane residual stress is shown inFig. 1aand c. The substrate is taken to be infinitely deep. The halfapex angle of the rigid conical indenter isα = 70.3. The coat-ing is elastic-perfectly plastic, applicable to most high-strengthalloys and ceramic coatings (the effect of work hardening willbe explored elsewhere). Recent work by Mesarovic and Fleck[23] has shown that the Poisson’s ratio is a minor factor forstatic indentation. Dimensional analysis dictates the followingrelationship for hardness and stiffness:

P

πa2σy= f

[σres

σy,σy

E

], (6)

S

2aE= g

[σres

σy,σy

E

]. (7)

As discussed above, for conical indentation in absent of resid-ual stress,S/2aE = γβ ≈ 1.1 is a constant; andP/πa2σy = cb,which varies withσy/E. Inspired by the fact that the pile-up(or contact area) is significantly affected by the residual stress(discussed below), the third independent equation governing thethree unknown variables is

I ev ep thatt

a

T∫

I tionw

B alf le-m

Can we measure all three unknowns, (E, σy, σres) from oneimple indentation test? How do the different residual stresechanical property combinations affect the pile-up, proje

ontact area, plastic zone, and other indentation mechanhe finite element method is useful for this purpose becaurovides a convenient way of measuring the projected corea between indenter and material needed for calculatiardness and stiffness. It is also straightforward to vary mials properties and residual stress over a wide range, sucur numerical analysis would cover most engineering matesed in practice. Thus, in this study, the finite element meFEM) is used to investigate the effect of residual streslastic–plastic indentation of bulk materials and thick coati

For indentation on a coating/substrate system, recenty Chen and Vlassak[11] has shown that the substrate eff

s negligible when the coating is softer than the substratehe indentation depth is less than half of the coating thickno simply the present study, we assume there is no subffect. That is, for soft coating deposited on a hard substrat

ndentation depthδ is smaller than 50% of the coating thicknend for hard coating on a soft substrate, the coating is veryompare withδ (e.g. thermal barrier coatings). This assumpllows us to only focus on the mechanics of indentationomogeneous, isotropic bulk material with equi-biaxial in-pesidual stress. The findings of this study can be readily apo thin film/coatings when the above criterions for negleche substrate effect are met.

2 In fact, it will be show later in this study that hardness becomes very seno residual stress whenσy/E gets large.

d

?ittf-at

y

d.tee

δp

δ= H

[σres

σy,σy

E

]. (8)

t should be noted that althoughδp/δ is very sensitive to thariations ofσy/E andσres/σy, it is very difficult to measure thile-up or sink-in amount during the experiment. We note

he contact radius can be represented by

= (δp + δ) tan α = δ tan α

(1 + H

[σres

σy,σy

E

]). (9)

hus, the work done by indentation is

δmax

0Pdδ =

∫ δmax

0πa2σyf

[σres

σy,σy

E

]dδ

= π tan2 α

3f

[σres

σy,σy

E

]

×(

1 + H

[σres

σy,σy

E

])2

σyδ3max. (10)

t is therefore convenient to define a normalized indentaork, which implicitly embeds the variation of pile-up:∫ δmax0 Pdδ

σyδ3max

= h

[σres

σy,σy

E

]≡ π tan2 α

3f

[σres

σy,σy

E

]

×(

1 + H

[σres

σy,σy

E

])2

. (11)

y varying σres/σy and σy/E in a wide range, the functionorms off, g, andh can be determined from extensive finite eent analysis. The hardnessP/(πa2), contact stiffnessS, and

142 X. Chen et al. / Materials Science and Engineering A 416 (2006) 139–149

indentation work∫ δmax

0 Pdδ can be readily obtained from anexperimental indentation load-displacement curve (c.f.Fig. 1b),as long as the contact radius a (or the projected contact areaA) can be measured properly by take into account the pile-upor sink-in effect.3 Finally, the mechanical properties (E, σy) andin-plane residual stress (σres) of the specimen can then be solvedfrom the reverse analysis:

P

πa2 = σyf

[σres

σy,σy

E

],

S

2a= Eg

[σres

σy,σy

E

],

∫ δmax

0Pdδ

δ3max

= σyh

[σres

σy,σy

E

].

(12)

2.2. Finite element model

Finite element calculations were performed using the com-mercial code ABAQUS[24] on Dell workstations. The rigidcontact surface option was used to simulate the rigid indenter,and the option for finite deformation and strain was employed.A typical mesh for the axisymmetric indentation model com-prises more than 5000 8-node elements. As already mentioned,the substrate material is taken to be elastic-perfectly plastic, witha tionl cieni nta-t d tot conic reac thenσ

is os-s trese atingt oni e (i.ep

3

3

intera theh ctio3 ard-n ince

rmint a higy thod(

the indentation pressure is compressive and perpendicular to theapplied surface, the existence of tensile surface stress in the spec-imen increases the magnitude of the shear stresses beneath theindenter compared with an unstressed specimen. Consequently,the plastic deformation in the specimen is enhanced, which leadsto a reduction of contact hardness H and normalized pile-upδp/δwhen in-plane residual tensile stress presents.

Some additional insight to the mechanics behind this can begained by studying the in-plane stress component,σrr, undermaximum indentation load for various residual stress,σres/σy,Fig. 2. When the residual stress is tensile,σres/σy = 0.8,Fig. 2a,the region of indentation compressive stress is smaller thanfor the initial stress-free case,σres/σy = 0, Fig. 2b. This givesrise to an apparent lower hardness material. The opposite sit-uation holds for specimen with residual compression (Fig. 2c,σres/σy =−0.8): the maximum compressive stress is larger thanthat in the unstressed sample, and the resulting overall compres-sion zone is much larger.

Additional insight to the change in apparent hardness canbe gained by studying the size of the plastic zone at maximumindentation load for various combinations ofE/σ andσres/σy,Fig. 3. For E/σy = 100 (Fig. 3b), the plastic strain is enlargedwith increasing level of residual stress. Since the plastic regionis larger and the deformation beneath the indenter is finite plas-tic without work hardening, when in-plane residual tensile stresspresents, the surface is allowed to “sink-in” during the indenta-t ssedb esultsi mu-l tion( etra-tF isr las-t withi rs ticr asticz a sig-n ss ist g ina exts

3

rac-t lcu-l iablep mate-r ise rater( e atm esid-u ionm thats t still

Von Mises surface to specify yielding. The Coulomb’s fricaw is used between contact surfaces, and the friction coeffis taken to be 0.1, which is also a minor factor for nanoindeion [23]. The equi-biaxial in-plane residual stress is appliehe specimen by means of thermal expansion, followed byal indentation on the free surface. The projected contact aalculated directly from the numerical results by analyzingodes in contact with the indenter. All three parameters (E, σy,res) are varied over a large range, such that the yield strainσy/E

s varied between 0.001 to 0.1, and the residual stressσres/σypans from−1 to 1. Such a wide range covers almost all pible combinations of mechanical properties and residual sncountered in engineering materials. In addition to calcul

he functional formsf, g, andh, the effects of residual stressndentation residual stress, plastic zone, and surface profilile-up) are also examined by FEM and elaborated below.

. Numerical results

.1. The effect of residual stress on indentation stress

The stress and strain fields caused by the indentationct strongly with the pre-existing fields, which in turn affectardness, stiffness, and indentation work, discussed in Se.3. The effect of in-plane residual (or applied) stress on hess can be explicated from the view of shear plasticity. S

3 Optical measurements or AFM surface scan are typically used to detehe pile-up (or sink-in) and true projected contact area. If the material hasield strain or if the indentation is nearly elastic, then the Oliver-Pharr meEq.(5)) can be applied to obtainδp anda accurately.

t

-is

s

.

-

n

eh

ion, which leads to the reduction of contact hardness (discuelow). On the other hand, the residual compressive stress r

n a smaller plastic region, and with a larger local accuation. This results in a significant pile-up during indentaFig. 4), more resistance to plastic deformation during penion, and an increase of normalized hardness (see Section3.3).or hard materials (Fig. 3a with E/σy = 10) the plastic regionelatively small, and a significant part of the deformation is eic. Nevertheless, the trend that the plastic zone enlargesncreasing tensile residual stressσres/σy is obvious. A softepecimen withE/σy = 1000 shows a more significant plasegion (Fig. 3c). For compressive residual stresses the plone is more confined near the impression crater, whereificant pile-up is observed. When the in-plane residual stre

ensile, the plastic strain is distributed more evenly, resultinsmallerδp/δ. The details of pile-up are presented in the n

ection.

.2. The effect of residual stress on surface profile

A common output from indentation testing is the chaerization of the deformed surface, commonly used to caated material properties. However, this may not be a relarameter when residual stresses are present. For hardials (E/σy = 10), a significant amount of the deformationlastic, and thus only sink-in presents at the indentation cFig. 4a). In addition, for hard materials the surface profilaximum penetration is essentially independent of the ral stress,σres/σy. However, upon unloading the deformatechanism is dominated by a significant elastic recovery

trongly depends on the residual stress level. A smaller bu

X. Chen et al. / Materials Science and Engineering A 416 (2006) 139–149 143

Fig. 2. Contour plots of indentation radial stress (σrr/σy) with different resid-ual stress level, all withE/σy = 100: (a) residual tension (σres/σy = 0.8); (b) noresidual stress (σres/σy = 0); (c) residual compression (σres/σy =−0.8).

noticeable recovery is also observed forE/σy = 100 (Fig. 4b). Formany high-strength alloys and coatings which haveE/σy ∼ 100,the effect of residual stress on pile-up/sink-in is obvious throughFig. 4b: comparing with the base case whereσres/σy = 0, thepresence of residual compression significantly enhances the pile-up, while residual tension leads to larger plastic zone and moresink-in effect. When the specimen is even softer (E/σy = 1000,Fig. 4c), indentation only develops pile-up around the crater rimand the amount of pile-up increases with increasing residualcompression. The deformation is finite plastic and the plastic

Fig. 3. Plastic zone as a function of residual stress showing the increase of plasticdeformation with residual tension: (a)E/σy = 10; (b)E/σy = 100; (c)E/σy = 1000.

zone is very large beneath the indenter, hence, the elastic recov-ery is almost negligible.

The pile-up/sink-in amount (δp/δ) is thus dependent on bothrelative softness (E/σy, in consistent with literature[5,9,11]) andnormalized residual stress (σres/σy, similar to the trend found in[21,22]). However, caution must be taken since the large dif-ference between the surface profiles before and after unloadingmay lead to incorrect measurements ofδp/δ, as shown inFig. 4a.As discussed in Section2, it is therefore more reliable to rep-resent the indentation work in the 2D space of (E/σy, σres/σy),numerated below.

144 X. Chen et al. / Materials Science and Engineering A 416 (2006) 139–149

Fig. 4. The variation of surface profile near the indentation crater as a functionof residual stress. Impression geometries for both maximum penetration anunloading are shown: (a)E/σy = 10; (b)E/σy = 100; (c)E/σy = 1000.

3.3. Functional forms determined from indentation testing

The functional forms off, g, andh (with reference to Eqs.(6), (7), and (11)) are shown inFig. 5a–c, respectively. Thenormalized hardness is essentially a constant (about 2.9) whenE/σy > 300, and the normalized stiffness is invariant (about 1.1)when E/σy > 30. This suggests that for soft materials, due tothe finite plastic deformation induced by indentation, both hard-ness and stiffness are essentially insensitive to the residual stress.Consequently, they cannot be used to measure the residual stressin the substrate unless new equations (e.g. the indentation work)are taken into account. Tsui et al.[15] have found that for a softaluminum alloy, both hardness and stiffness determined with theoptically measured true contact area are essentially independentof the in-plane residual stress. This is in consistent with the cur-rent approach. Both limits of rigid plastic deformation,cb = 2.9(Fig. 5a) andγ = 1.1 (Fig. 5b), agree with the values found inliterature[7,9,13].

When the substrate is not too soft (E/σy < 300, which isapplicable to many high-strength alloys and ceramic coatings),the existence of a tensile in-plane residual stress significantlyenhances the plastic flow in the specimen, which producesa larger hardness impression and a lower measured hardness(Fig. 5a). The normalized hardness also decreases with increas-ing yield strain of the substrate, thanks to the increasing impor-tance of elasticity. In absent of residual stress, the dependenceo h thec

nessw -u sidualc hards n willf ffecta -s rk isrss insaa nsitivet ields

aw ep (c.f.F sesw res-s sub-s cili-t ationw

thatt sses,i r-t tw s and

d

f normalized hardness versus yield strain agrees well witlassic analysis by Johnson[9].

The residual stress significantly affects the contact stiffhen E/σy < 30. It is found fromFig. 5b that tensile residal stress tends to increase the contact stiffness and reompression will decrease the contact stiffness. Within aubstrate, the presence of residual compression or tensioacilitate or impede the unloading process, and such emplifies with increasing elastic componentE/σy. The analyis of the unloading curves shows that more elastic woecovered (bothS and residual indentation depthδf /δmax aremaller) with decreasingE/σy and decreasingσres/σy, in con-istent withFig. 4. The normalized contact stiffness remat constantγ in absent of residual stress. Overall,Fig. 5and b suggests that both hardness and stiffness are se

o residual stress for high-strength materials with large ytrain.

The normalized indentation work (Fig. 5c) appears to haveide spread for bothE/σy andσres/σy. This is partly because thile-upδp/δ varies with both yield strain and residual stressig. 4). In general, the normalized indentation work increaith increasingE/σy: that is, in order to create the same impion, larger load (or plastic work) is needed to indent thetrate with higher modulus. Since the residual tension “faates” the penetration of indenter tip, the normalized indentork decreases with increasingσres/σy.The results presented in the previous section indicate

he indentation testing is highly dependent on residual stren particular for materials withE/σy < 300 and thus have impoant applications for high-strength materials.Fig. 5suggests thahen combined together, the normalized hardness, stiffnes

X. Chen et al. / Materials Science and Engineering A 416 (2006) 139–149 145

Fig. 5. (a) Normalized hardnessP/π2σy, (b) normalized contact stiffnessS/2aE, and (c) normalized indentation work∫ δmax

0Pdδ/δ3

maxσy, as a function of materialelastic–plastic propertyE/σy and normalized yield stressσres/σy.

146 X. Chen et al. / Materials Science and Engineering A 416 (2006) 139–149

Fig. 6. Comparison between the finite element results (Fig. 5) and fitting function(13): (a) normalized hardness, (b) normalized contact stiffness, and (c) normalizedindentation work.

X. Chen et al. / Materials Science and Engineering A 416 (2006) 139–149 147

indentation work plots should yield sufficient information forthe reverse analysis, from which the three unknowns (E, σy,σres) can be measured by indentation. Over the entire region ofparameters considered in this study, the three functional formsf, g, andh can be fitted to the following functions. The com-parisons between fitted surfaces (Eq.(13)) with original dataobtained from FEM (Fig. 5a–c) are shown inFig. 6a–c. Thefitting has an uncertainty less than 3% over the entire (E/σy,σres/σy) space:

Ω = a1 + a2τ + a3τ2 + a4

√τ + a5

3√

τ + a6ξ + a7ξτ

+a8ξτ2 + a9ξ

√τ + a10ξ

3√

τ + a11ξ2 + a12ξ

2τ

+a13ξ2τ2 + a14ξ

2√τ + a15ξ2 3√

τ + a16ξ3

+a17ξ3τ + a18ξ

3τ2 + a19ξ3√τa20 + ξ3 3

√τ. (13)

In Eq. (13), The functionΩ represents normalized hardness(P/πa2σy, or f), normalized contact stiffness (S/2aE, or g),or normalized indentation work (

∫ δmax0 Pdδ/δ3

maxσy, or h), and

τ ≡ E/σy and ξ ≡ σres/σy are two variables. The coefficientsai(i = 1–20) are listed inTable 1.

4. Reverse analysis and examples

In order to solve the three unknowns (E, σy, σres) from thethree fitting functions (f, g, h), the optimization method is usedto minimize the total error of Eq.(12). A flow chart of the reverseanalysis based on such algorithm is given inFig. 7. The hard-ness, contact stiffness, and indentation work are measured froman indentation load-displacement curve after the experiment. Foreach numerical iteration, the variables (E, σy, σres) are substi-tuted in Eq.(13)and the error betweenΩ and the correspondingmeasured values are found. Atotal error function is definedas the summation of errors fromf, g, andh. The roots (E, σy,σres) are found by searching a minimum of the total error func-tion of those three variables on a fixed interval. The algorithmis based on golden section search and parabolic interpolation.The initial boundary are given by a rough range of the Young’s

Fig. 7. Schematic of the proc

ess flow of reverse analysis.

148 X. Chen et al. / Materials Science and Engineering A 416 (2006) 139–149

Table 1Parameters of three fitting functional forms

Parameters Normalizedhardness

Normalized contactstiffness

Normalized work ofindentation

a1 −1.4943 1.1041 −3.7438a2 0.0032 3.7528× 10−4 −0.0771a3 4.1178× 10−7 −8.5558× 10−8 1.7110× 10−5

a4 −0.7500 −0.0235 2.9685a5 2.4572 0.0476 0.4208a6 2.5012 0.8105 8.9619a7 −0.0244 −0.0032 0.0507a8 4.9842× 10−6 6.1966× 10−7 −1.9059× 10−5

a9 1.8116 0.2781 0.0963a10 −4.0314 −0.7044 −4.6897a11 −1.4904 −0.0783 −8.1284a12 0.0095 2.8227× 10−4 0.0270a13 −2.4728× 10−6 −5.2211× 10−8 −2.0045× 10−6

a14 −0.6405 −0.0258 −2.7352a15 1.4709 0.0649 6.8228a16 −1.0050 −0.1904 5.9693a17 0.0089 0.0011 −0.0691a18 −1.9143× 10−6 −2.3340× 10−7 1.6321× 10−5

a19 −0.6334 −0.0786 4.5106a20 1.4098 0.1864 −9.5823

Fig. 8. Comparison between the material properties predicted from reverse anaysis and the input parameters used in numerical indentation experiments.

modulusE, and an initial guess of bothσy/E andσres/σy. Basedon the searching results obtained in the last numerical step, thboundary is updated automatically to give faster convergencand more accurate local solutions. An accurate solution withtotal error less than 5% can be obtained after several numericaiterations.

In order to examine the accuracy of reverse analysis, a totaof 33 numerical experiments of indentation are performed, withσy/E andσres/σy varying in a large range.4 The input materialparameters used in FEM are shown as open squares inFig. 8.For each experiment, the contact hardness, stiffness, indentatio

4 Note that some of these parametric combinations where not used in geneatingFigs. 5 and 6.

work and projected contact area are measured from the finiteelement analysis. These numbers are then fed into the reverseanalysis to predictσy/E andσres/σy. The results obtained fromnumerical analysis are plotted as solid circles inFig. 8. Excellentagreements between the original input data and reverse analysisare found for all possible combinations of residual stress andmaterial parameters.

It would be ideal to compare the model with existing experi-mental data. Unfortunately, a complete set of experimental datawhich involves Berkovich indentation load-displacement curvesfor biaxially stressed elastic-perfectly plastic bulk material couldnot be found in literature. For example, Tsui et al.[15] andSuresh Giannakopoulos[17] did not show the variation of inden-tation load-displacement curves with residual stress (which isrequired for analyzing the indentation work in this study); whileSwadener et al.[18] have used spherical indenters. Some mate-rials that have been used in experiments have either significantstrain hardening[20] (for low-carbon steel) or strain gradienteffect at small indentation depth[19] (for single crystal tung-sten), which leads to a normalized hardness far above 3 and thusdoes not apply to the present investigation. Some others onlyinvolve uniaxial residual stress instead of biaxial stress[22,25],or excessive substrate effect for indentation on biaxially stressedthin films[26]. Therefore, although bulk materials and coatingswith equi-biaxial residual stress are very common in applica-tions, to our knowledge, the existing indentation data in literatured sentm etero laterd

5

ed toi s onh a bulkm me-t bothh lly forh sticp bev s thed sible.B ness,s entedw dulus,a com-p versea tatione

lasticz com-p leadst ness.O umi plas-t the

l-

ee

l

l

n

r-

oes not yield sufficient information to compare with the preodel. Indentation experiments that include desired paramutputs are currently in progress, and will be presented at aate.

. Conclusion

In this study, the finite element analysis has been usnvestigate the effect of in-plane equi-biaxial residual stresardness and stiffness measured from indentation tests onaterial (or thick coating). By varying the material para

er and residual stress over a large range, it is found thatardness and stiffness vary with residual stress, especiaigh-strength materials with large yield strain. Both the plaile-up (or elastic sink-in) and indentation work are found toery sensitive to the level of residual stress, which makeetermination of residual stress from indentation test posased on the functional forms of normalized contact hardtiffness, and indentation work, a reverse algorithm is preshich may be used to measure the material yield stress, mond residual stress from one simple indentation test. Thearison between the material properties predicted from renalysis and the input parameters used in numerical indenxperiments shows good agreement.

The effects of residual stress on indentation stress and pone are also investigated. It is found that the indentationressive stress diminishes with residual tension, which

o an enlarged plastic zone and gives apparent lower hardn the other hand, with residual compression the maxim

ndentation compressive stress is increased, causing lessic deformation (with smaller plastic zone) and gives rise to

X. Chen et al. / Materials Science and Engineering A 416 (2006) 139–149 149

hardness. It is also found that the elastic recovery upon unloadingis very large for materials with high yield strain, and the elasticrecovery increases with increasing residual compression. There-fore, caution must be taken when measuring the contact depthafter unloading.

Unlike many of the previous studies[17–20], the new indenta-tion technique proposed in this paper does not require a referencestress-free material for comparison purposes. Thus, this methodhas the potential to map the residual stress field on the surfaceof a specimen quickly and effectively.

Acknowledgements

The work of XC is supported in part by NSF CMS-0407743,and in part by the Department of Civil Engineering and Engi-neering Mechanics, Columbia University. The work of JY andAMK is supported by NSF DMR-0346664 and ONR N00014-04-1-0498.

References

[1] J.W. Hutchinson, Z. Suo, Adv. Appl. Mech. 29 (1992) 63–191.[2] V.K. Tolpygo, D.R. Clarke, K.S. Murphy, Surf. Coatings Technol. 146

(2001) 124–131.[3] V. Teixeira, M. Andritschky, W. Fischer, H.P. Buchkremer, D. Stover, J.

Mater. Process. Technol. 92 (1999) 209–216.

[4] J. Shi, S. Darzens, A.M. Karlsson, Mater. Sci. Eng. A 392 (2005)301–312.

[5] W.C. Oliver, G.M. Pharr, J. Mater. Res. 7 (1992) 1564–1583.[6] F. Doerner, W.D. Nix, J. Mater. Res. 1 (1986) 601–609.[7] G.M. Pharr, Mater. Sci. Eng. A 253 (1998) 151–159.[8] G.M. Pharr, A. Bolshakov, J. Mater. Res. 17 (2002) 2660–2671.[9] K.L. Johnson, Contact Mechanics, Cambridge University Press, 1985.

[10] J.J. Vlassak, W.D. Nix, J. Mech. Phys. Solids 42 (1994) 1223–1245.[11] X. Chen, J.J. Vlassak, J. Mater. Res. 16 (2001) 2974–2982.[12] R.B. King, Int. J. Solids Struct. 23 (1987) 1657–1664.[13] J.C. Hay, A. Bolshakov, G.M. Pharr, J. Mater. Res. 14 (1999)

2296–2305.[14] N.A. Fleck, J.W. Hutchinson, Adv. Appl. Mech. 33 (1997) 295–361.[15] T.Y. Tsui, W.C. Oliver, G.M. Pharr, J. Mater. Res. 11 (1996) 752–

759.[16] A. Bolshakov, W.C. Oliver, G.M. Pharr, J. Mater. Res. 11 (1996)

760–768.[17] S. Suresh, A.E. Giannakopoulos, Acta Mater. 46 (1998) 5755–5767.[18] J.G. Swadener, B. Taljat, G.M. Pharr, J. Mater. Res. 16 (2001)

2091–2102.[19] Y.H. Lee, D. Kwon, Scripta Mater. 49 (2003) 459–465.[20] Y.H. Lee, D. Kwon, Acta Mater. 52 (2004) 1555–1563.[21] S. Carlsson, P.L. Larsson, Acta Mater. 49 (2001) 2179–2191.[22] S. Carlsson, P.L. Larsson, Acta Mater. 49 (2001) 2193–2203.[23] S.D. Mesarovic, N.A. Fleck, Proc. Roy. Soc. London A455 (1999)

2707–2728.[24] ABAQUS, ABAQUS 5.8 User’s Manual, ABAQUS Inc., 1998.[25] A.V. Zagrebelny, C.B. Carter, Scripta Mater. 37 (1997) 1869–1875.[26] G. Wei, B. Bhushan, S.J. Jacobs, Ultramicroscopy 100 (2004) 375–

389.