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Acta Materialia 56 (2008) 5345–5354

Measurement of scratch-induced residual stress within SiC grainsin ZrB2–SiC composite using micro-Raman spectroscopy

Dipankar Ghosh a, Ghatu Subhash a,*, Nina Orlovskaya b

a Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USAb Department of Mechanical, Materials and Aerospace Engineering, University of Central Florida, Orlando, FL 32816, USA

Received 2 April 2008; received in revised form 13 July 2008; accepted 14 July 2008Available online 26 August 2008

Abstract

An analytical framework for determination of scratch-induced residual stress within SiC grains of ZrB2–SiC composite is developed.Using a ‘‘secular equation” that relates strain to Raman-peak shift for zinc-blende structures and the concept of sliding blister field modelfor scratch-induced residual stress, explicit expressions are derived for residual stress calculation in terms of phonon deformation poten-tials and Raman peak shift. It is determined that, in the as-processed composite, thermal expansion coefficient mismatch between ZrB2

and SiC induces compressive residual stress of 1.731 GPa within the SiC grains and a tensile tangential stress of 1.126 GPa at the ZrB2–SiC interfaces. With increasing scratch loads, the residual stress within the SiC grains becomes tensile and increases in magnitude withscratch load. At a scratch load of 250 mN, the calculated residual stress in SiC was 2.6 GPa. Despite this high value, no fracture wasobserved in SiC grains, which has been rationalized based on fracture strength calculations from Griffith theory.� 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Residual stress; Raman spectroscopy; Borides; Scratch test; Ceramic matrix composite

1. Introduction

Ultra-high-temperature materials with low density,good mechanical strength and high oxidation resistanceat elevated temperatures (>2000 �C) are potential candi-dates for applications in future hypersonic vehicles, kineticenergy interceptor missiles, reusable launch vehicles, etc.[1–6]. Specific examples include wing leading edges, enginecowl inlets, nose-caps, etc. These components have sharpaero-surfaces that are subjected to reactive environmentsat temperatures >2000 �C. Currently, materials used inhigh-temperature aerospace structural components aremostly carbon–carbon composites and silicon carbide-based composites [4]. Although, these composites havehigh-temperature structural capabilities, their oxidationresistance is known to be poor. Ultra-high-temperature

1359-6454/$34.00 � 2008 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2008.07.031

* Corresponding author. Tel.: +1 352 392 7005; fax: +1 352 392 7303.E-mail address: subhash@ufl.edu (G. Subhash).

ceramics (UHTC) such as borides of transition metals(e.g., zirconium (Zr) and hafnium (Hf)) and their compos-ites are identified as the next generation materials for high-temperature aerospace applications [1–4]. Among UHTC,zirconium diboride–silicon carbide (ZrB2–SiC) compositeshave been receiving significant attention in recent years[3,7–19] owing to their low density (�6.0 g cm�3), highmelting point (>3000 �C) and good oxidation resistance>1500 �C.

In the above-mentioned aerospace applications, theUHTC may be subjected to impact by atmospheric debristhat can lead to wear of the structural components. Thisabrasive action of the atmospheric debris particles causesinelastic deformation within the material. Therefore, it isimportant to study the wear behavior and the associateddamage mechanisms in UHTC subjected to such loads.However, most of the available literature on ZrB2–SiCcomposites is focused on processing and their oxidationbehavior [3,7–19]; knowledge on wear behavior of thesematerials is literally non-existent.

rights reserved.

5346 D. Ghosh et al. / Acta Materialia 56 (2008) 5345–5354

Indentation and scratch experiments have been usedeffectively to model deformation and damage mechanismsthat evolve in ceramics during an abrasion process [20–29]. In scratch experiments, both normal and tangentialpoint loads are applied whereas, in indentation experi-ments, only a normal point load is applied on the surface[24,26–29]. Experimental and theoretical studies haveshown that the loading phase of the scratch process (orindentation) causes elastic–plastic deformation within abrittle material, resulting in accumulation of mechanicalresidual stress upon unloading [24,27,28]. The loadingphase also causes evolution of radial and median cracksduring indentation and scratch experiments [24,27,28].Upon unloading, the accumulated deformation-inducedresidual stress causes lateral cracking. Initiation of suchdamage and consequent material removal from the surfacedue to these contact processes may result in lower mechan-ical reliability of the components made out of UHTC.Therefore, quantification of mechanical residual stressdue to scratch or abrasion phenomena is necessary for afundamental understanding of the material removal pro-cess and for design of abrasion-resistant materials.

The authors have recently investigated scratch-induceddeformation and damage patterns in a ZrB2–5 wt.% SiCcomposite in the load range 50–250 mN [29]. Microstruc-tural studies of the scratch grooves revealed microplasticity(in the form of slip lines) and microcracks within the ZrB2

phase. The present paper describes the results of scratch-induced mechanical residual stress measurements withinthe SiC grains of the ZrB2–SiC composite using micro-Raman spectroscopy (MRS).

Raman spectroscopy has emerged as a useful non-destructive tool for residual stress measurements. The sensi-tivity of a Raman peak to mechanical stress (or more pre-cisely the strain) was first reported by Anastassakis et al.[30]. In particular, MRS is useful in determining local resid-ual stress owing to its high spatial resolution (of the order of1 lm) [30–37]. For example, in thin films and micro electro-mechanical systems devices, MRS is largely used for residualstress determination arising as a result of coefficient of ther-mal expansion (CTE) mismatch between the substrate andthe film materials [31,33,36,37]. Indentation- and scratch-induced residual stress analyses are also conducted usingMRS [34,35]. In crystalline materials, the atomic vibrationalfrequencies depend on the interatomic force constants [36].In strain-free crystalline materials, interatomic force con-stants as well as the vibrational frequencies correspond tothe equilibrium atomic spacing. Residual stress resultingfrom either thermal process (such as sintering) or mechani-cal deformation (such as indentation, scratch, etc.) causes adefinitive residual strain, which in turn changes the equilib-rium atomic spacing within a material and thus the inter-atomic force constants. As a result, Raman scatteringwave numbers are also perturbed. Depending upon the ten-sile or compressive nature of the residual stress, bondlengths and force constants either increase or decrease com-pared with the equilibrium values. Accordingly, a Raman

peak shifts to lower or higher frequency for tensile or com-pressive residual stress, respectively [30,31,36,37], thoughthere is no unique general relationship between the Ramanspectrum parameters (particularly the wave number shift)and the residual stress state. One way to establish such rela-tionships is by the calibration procedure, which correlatesRaman peak shift in a material with the known appliedstress. Then, using the calibration curve, unknown residualstresses can be determined from the changes in Raman peakpositions for that particular material. However, the resul-tant calibration curve also depends on the applied stressstate and, therefore, there is no unique relationship betweenresidual stress and Raman peak shift for a given material. Incontrast, explicit expressions relating Raman peak shift andresidual stress have been derived for simple situations suchas uniaxial stress state, hydrostatic or equi-biaxial stressstate which can also approximate the residual stresses[31,36,38]. Therefore, expressions for appropriate scenariosmust be developed to estimate residual stress within Ramanactive materials using MRS.

Although ZrB2 is not Raman active, SiC is known to beRaman active, and its characteristic Raman peaks, partic-ularly for cubic SiC, are sensitive to residual stress [38–40]. Therefore, it is possible to determine the magnitudeof the residual stresses from the stress/strain sensitiveRaman peaks of SiC material. In the current study, thechanges in Raman peak positions of SiC grains locatedwithin the scratch grooves of a ZrB2–5 wt.% SiC compositewere measured as a function of scratch load. Then, fromthese Raman measurements a mechanics-based expressionwas derived to estimate scratch-induced mechanical resid-ual stress within the SiC grains in the composite.

2. Experimental

A ZrB2–5 wt.% SiC composite, processed by the plasmapressure compaction P2C� method [29,41,42] was used inthis investigation. From X-ray diffraction studies, it wasconfirmed that the sintered composite consisted of onlytwo crystalline phases: hexagonal (H) ZrB2 and cubic(3C) SiC [29]. Scratch experiments at constant loads inthe range 50–250 mN were conducted on the polished sur-faces of the composite using a MTS nanoindenter XPS sys-tem employing a Berkovich nanoindenter (tip radius�100 nm). Each of the scratches was conducted at anindenter translational speed of 5 lm s�1 for a length of200 lm. A scanning electron microscope (SEM, JEOLJSM-6335F) was used to analyze the microstructural fea-tures of the residual scratch tracks.

A micro-Raman spectrometer, Renishaw inVia RamanMicroscope, was used in the current study. The Ramanspectrometer consisted of a Si laser (532 nm) to excite thespecimen, a single spectrograph fitted with holographicnotch filters, and an optical microscope (a Leica micro-scope with a motorized XYZ mapping stage) rigidlymounted and optically coupled to the spectrograph. Thespectrometer was initially calibrated with a Si standard

650 750 850 950 1050Raman shift (cm-1)

Inte

nsity

(a.u

.)

TO-peak LO-peak

Away from scratch

250 mN

150 mN

50 mN

Fig. 2. Raman spectra collected from the SiC grains present within andaway from the scratch grooves.

D. Ghosh et al. / Acta Materialia 56 (2008) 5345–5354 5347

using a Si band position at 520.3 cm�1. A 100� objectivelens was used to focus the incident beam (spot size�1.5 lm) on the desired SiC grains and to collect the scat-tered beam from the specimen. A maximum 25 mW of thelaser power was used. Raman spectra were collected fromseveral SiC grains located within the scratch grooves. Forcomparison purposes, Raman spectra were also collectedfrom several SiC grains located outside the scratch grooves.All the Raman measurements were performed at roomtemperature.

3. Results and discussion

3.1. Results of scratch experiments and MRS

Fig. 1 shows a residual scratch groove at 250 mN loadon a polished surface of the ZrB2–SiC composite. The con-tinuous gray phase is the ZrB2 matrix in which the SiC par-ticulate phase (dark phase) is distributed. In the ZrB2–SiCcomposite, the average size of the ZrB2 grains was approx-imated to be 5 lm, whereas the SiC particulate phase was�1 lm. Microstructural observations indicated that theSiC particles, distributed within the composite, were mostlysingle crystals. Occasionally, multiple grains were observedto exist. The scratch process induces microcracks and slip-lines in and around the scratch grooves, as indicated inFig. 1 (see the inset). Detailed discussion about the inelasticdeformation features observed within the scratch groovesat various load levels can be found elsewhere [29]. Dam-aged regions due to limited lateral cracks extending beyondthe scratch groove are also observed in Fig. 1.

Fig. 2 shows the Raman spectra collected from the SiCgrains located outside the groove and within the scratchgrooves resulting from constant loads at 50, 150 and250 mN. The Raman spectrum from SiC grains outsidethe scratch groove as shown in Fig. 2 is typical of 3C–SiC which consists of one transverse optical (TO) peak at

Fig. 1. Scratch pattern at a load of 250 mN revealing slip lines and microcrack

796 cm�1 and one longitudinal optical (LO) phonon peakat 972 cm�1 [38–40]. These two peak positions have beenconsistently reported for many annealed 3C–SiC thin filmsand thus can be adopted to correspond to Raman peaks instress-free 3C–SiC [38–40]. However, in the current study,SiC grains away from the scratch groves showed both theTO and LO peaks at higher wave numbers (�802.3 cm�1

and 978.9 cm�1, respectively) compared with stress-free3C–SiC. Therefore, it was inferred that compressive resid-ual stress is present within these SiC grains in the as-pro-cessed composite. The origin of compressive residualstress within the as-processed composite is due to the mis-match in Young’s moduli (E) and CTE (a) between H-ZrB2

(E = 489 GPa, aavg �5.9�10�6 K�1) [3] and 3C–SiC(E = 694 GPa, a �3.5�10�6 K�1) [43,44], as well as the dif-ference between room temperature (25 �C) and sinteringtemperature (1750 �C) of the composite. Because the CTEof polycrystalline ZrB2 is greater than that of 3C–SiC, cool-

s (shown clearly in the inset), and damaged regions due to lateral cracking.

5348 D. Ghosh et al. / Acta Materialia 56 (2008) 5345–5354

ing of the consolidated compact from sintering temperaturewill result in compressive residual stresses within SiCgrains.

With increasing scratch load, both the TO and LO peaksin the Raman spectra collected from the SiC grains withinthe scratch grooves progressively shifted to lower wavenumbers compared with the Raman spectra collected fromthe SiC away from the scratches (see Fig. 2). With increas-ing scratch load, greater peak widening as well as asymme-try in peaks can be seen in these spectra. Raman peak shiftto lower wave numbers indicates development of tensileresidual stress in these SiC grains. The magnitude of thepeak shift (and hence tensile residual stress) increased withscratch load. Fig. 3a shows the measured LO-peak and

945

950

955

960

965

970

975

980

985

0 50 100 150 200 250 300

780

785

790

795

800

805

810

0 50 100 150 200 250 300Scratch load (mN)

0 50 100 150 200 250 300Scratch load (mN)

≈ ≈

Stress-free TO-peak

Stress-free LO-peak

LO-peak

TO-peak

Ram

an s

hif

t (cm

-1)

-20

-15

-10

-5

0

5

Pea

k sh

ift (

cm-1

)

a

b

Fig. 3. (a) LO- and TO-Raman peak positions of SiC grains within thescratch grooves at 50, 100, 150 and 250 mN loads. LO- and TO-peakpositions at 0 mN correspond to the Raman spectra collected from SiCgrains outside the scratch groove. (b) Changes in peak positions to stress-free LO- and TO-positions.

TO-peak positions for scratch loads of 50, 100, 150 and250 mN, and Fig. 3b shows the shift in TO-peak andLO-peak positions compared with the same peaks instress-free 3C–SiC materials (796 cm�1 for TO peak and972 cm�1 for LO peak). At 50 mN scratch load, both theTO peak and LO peak were lower than the stress-free posi-tion, indicating the presence of compressive residual stressin SiC grains. As the scratch load increased beyond 50 mN,both the peaks further shifted to lower wave numbers indi-cating development of a tensile stress state within thesegrains. In SiC grains, present within the scratch grooves,the total residual stress-state (rR) is the sum of process-ing-induced thermal residual stress (rR

t ) and scratch-induced mechanical residual stress (rR

s ), i.e.,

rR ¼ rRt þ rR

s ð1ÞIn the SiC grains that exist within unscratched regions of

the composite, rR ¼ rRt , which is compressive, as indicated

by the Raman spectroscopy. However, the scratch processinduces rR

s within the SiC grains, and its magnitudeincreases with load.

Peak broadening and peak asymmetry of the Ramanspectra due to the scratch process, observed in Fig. 2,indicated structural disorder inside the SiC grains [39].The extent of disorder increased with increasing scratchload. Stacking faults have been identified as the primarydefects in SiC whiskers [45] and in polycrystalline SiC[46]. It has been reported that an increase in numberof stacking faults caused a Raman peak shift to lowerwave numbers as well as peak broadening in 3C–SiCceramics [39]. Based on the bond polarizability model,Rohmfeld et al. [39] simulated the influence of stackingfault distance on the TO-phonon mode in 3C–SiC. Itwas shown that as the average stacking fault distancedecreased (i.e., structural disorder increased with increasein stacking fault density), the peak shifted to lower fre-quencies and line-broadening was observed in the TO-phonon peak. In the current study, similar observationswith increasing scratch load were noted in the Ramanspectra collected from the SiC grains residing withinthe grooves. Therefore, it is argued that, as the scratchload increased, a greater number of stacking faults (i.e.,structural disorder) were generated within the SiC grains,which in turn caused an increase in the residual tensilestress. In the following, the shift in the Raman peakpositions with scratch load will be quantified in termsof residual stress within the SiC grains. The TO peakswere relatively more symmetrical compared with theLO peak and therefore, only the TO-peak shifts will beconsidered for residual stress calculation.

3.2. Evolution of residual stress field

Before embarking on the determination of residualstress from Raman spectra, this section first discusses theprocessing-induced residual stress state within the compos-ite [47] and residual stress field that evolves in a brittle

z

y

x

a b Plastic zone

Median crack

Scratch groove

Lateral crack

Radial crack Fn

FtScratch direction

Indenter

Fig. 4. Schematic of the scratch process and different crack systems thatevolve during the scratch process are shown.

D. Ghosh et al. / Acta Materialia 56 (2008) 5345–5354 5349

material due to a scratch process [24,27]. Then, Ramanpeak shift will be related to the appropriate stress compo-nent to determine the scratch-induced residual stress.

3.2.1. Residual stress in SiC grains of as-processed composite

In the ZrB2–SiC composite, the SiC grains (which areassumed equiaxed and almost spherical, as can be seen fromFig. 1) can be assumed as elastic spheres of uniform size dis-tributed in an infinite elastic continuum of ZrB2 matrix [47].This results in axially symmetric stress distribution aroundSiC grains. Let us consider that the grain has an effectiveradius a, and the surrounding matrix has radius b. It hasbeen shown that the ceramic grain will be under a uni-form/hydrostatic pressure P which can be expressed as [47]

P ¼ ðam � apÞDT0:5ð1þ mmÞ þ ð1� 2mmÞV p

Emð1� V pÞþ 1� 2mp

Ep

� � ð2Þ

where m is Poisson’s ratio, DT is temperature change, V p isvolume fraction of grains, and subscripts m and p stand formatrix and grain. Radial (rrad) and tangential (rtan) stres-ses within the ZrB2 matrix and at a distance r from the cen-ter of grain are expressed, respectively, as [47]

rrad ¼P

1� V p½a

3

r3� V p� and

rtan ¼ �P

1� V p½ a

3

2r3þ V p�

ð3Þ

Thus, from the above equations, it can be seen that,when am > ap, as is the case for ZrB2–SiC composites, cool-ing of the composite from the sintering temperature willinduce a uniform compressive stress within the SiC grains.The radial and tangential stresses within ZrB2 will be com-pressive and tensile, respectively, and both the stresses willbe maximum at the ZrB2–SiC interface.

3.2.2. Residual stress in SiC grains located within the scratch

grooves

The mechanical residual stress induced by the scratchprocess needs more in-depth analysis. Single-pass scratchprocesses have been widely modeled [24,26,27] as a slidingmicroindentation event, where point loads are simulta-neously applied in normal (Fn) and tangential (Ft) directionson the surface of a specimen, as shown schematically inFig. 4. Ahn et al. [24] and Jing et al [27]. extended Yoffe’sblister field model [48], developed for indentation, to a slid-ing blister field model (SBFM) to describe the elastic stressfield due to a scratch process. The complete elastic stressfield during the scratch process is constructed from thesuperposition of the Boussinesq field, Cerruti field[24,26,27,49] and Yoffe’s blister field [48]. The Boussinesqfield and Cerruti field are elastic stress fields, and ariseowing to the application of point normal (Fn) and point tan-gential (Ft) loads, respectively, whereas the blister field is aninelastic stress field related to the residual stress due to load-ing and unloading of the point normal loads.

During the sliding process, a moving scratch tool on thespecimen surface is assumed to create a semi-cylindricalscratch groove surrounded by a semi-cylindrical inelasticzone (see Fig. 4) [24,27]. The material directly in contactwith the scratch tool undergoes elastic–plastic deformationduring loading of the indenter. Owing to the misfit strain ofdeformation between elastic and plastic regions, residualstresses are induced within the material during unloadingof the scratch tool. According to SBFM [24,27] the normalresidual elastic-stress components (rR) along the x, y and z

directions are expressed as

rRx

2B¼ � 2mðy2 � z2Þ

ðy2 þ z2Þ2þ x

ðy2 þ z2Þ2q5� ð2mx4y2 � 2x2y4

þ 6mx2y4 � 2y6 þ 4my6 � 2mx4z2 � 4x2y2z2 þ 2mx2y2z2

� 3y4z2 þ 6my4z2 � 2mx2z4 � 4mx2z4 þ z6 � 2mz6Þð4Þ

rRy

2B¼ � 2y2ðy2 � 3z2Þ

ðy2 þ z2Þ3þ x

ðy2 þ z2Þ3q5� ð2x4y4 þ 6x2y6

� 2mx2y6 þ 4y8 � 2my8 � 6x4y2z2 � 7x4y2z2

� 6mx2y4z2 � 2y6z2 � 8my6z2 � 12x2y2z4 � 6mx2y2z4

� 15y4z4 � 12my4z4 þ x2z6 � 2mx2z6 � 8y2z6

� 8my2z6 þ z8 � 2mz8Þð5Þ

rRz

2B¼ � 2z2ðy2 � 3y2Þ

ðy2 þ z2Þ3þ xz2

ðy2 þ z2Þ3q5� ð6x4y2 þ 15x2y4 þ 9y6

� 2x4z2 þ 10x2y2z2 þ 12y4z2 � 5x2z4 � 3y2z4 � 6z6Þð6Þ

where q2 ¼ x2 þ y2 þ z2, and B is the blister field strengthper unit length [24,27].

Fig. 5 shows the 3-D distribution of the three normalresidual stress components rR

x , rRy and rR

z (all normalizedwith P

2c2) for B/P=0.005 on the x–y plane in the vicinityof the indenter tip during a scratch process. These normalresidual stress components arise during a scratch process asthe indenter moves from �x ¼ �1 (�x ¼ x

c) to its present loca-tion at �x ¼ 0. It is seen from Fig. 5c that rR

z is highly tensile

Fig. 5. Distribution of normalized residual stress components: (a) rRx , (b)

rRy and (c) rR

z on the x–y plane (B/P=0.005). The black triangle indicatesthe position of the indenter whereas the black arrow indicates the scratchdirection.

5350 D. Ghosh et al. / Acta Materialia 56 (2008) 5345–5354

in the wake of the scratch tool, whereas rRx and rR

y are com-pressive directly behind the indenter along the �x -axis. rR

y isslightly tensile away from the �x -axis behind the indenter.rR

z reaches its maximum value just behind the indenter(�x < 0) (see Fig. 5c), and its magnitude is significantly lar-ger than rR

x and rRy . From these 3-D scratch-induced resid-

ual stress distributions, it is clear that rRz is the most

dominant residual stress component and is known to con-tribute to subsurface lateral crack initiation [24], which wasalso observed in the present investigation (see Fig. 1). With

increasing scratch load, the strength B of the sliding blisterfield also increases and, therefore, the magnitude of rR

z willalso increase causing greater lateral cracking and materialremoval in brittle materials. For these reasons, the follow-ing section will correlate the above residual stress compo-nent rR

z to the Raman peak shifts observed in Figs. 2 and3 and quantify the residual stress magnitude within theSiC grains which reside within the scratch grooves.

3.3. Relationship between mechanical residual stress and

Raman spectroscopy

Recall that Raman peak shift is related to induced strainin a deformed specimen. If the Raman wave numbers ofoptical phonons, in the absence and the presence of strain,are denoted by woj and wj (j ¼ 1� 3), respectively, thestrain-induced Raman shift Dwj is expressed as [31,36]

Dwj ¼ wj� wjo � kj2wjo

ð7Þ

where kj are the eigen values of the well-known ‘‘secularequation”, relating strain to Raman peak shift for diamondand zinc-blende structures [31,36,40,50], as

peRx þqðeR

y þ eRz Þ�k rcR

xy rcRxz

rcRxy peR

y þqðeRx þ eR

z Þ�k rcRyz

rcRxz rcR

yz peRz þqðeR

x þ eRy Þ�k

�������

�������¼ 0

ð8Þwhere p, q and r are the phonon deformation potentialswhich describe the change in effective spring constants in-duced by the strain, and eR and cR are the residual normaland shear strain tensor components, respectively. 3C-SiChas a zincblende crystal structure [40] and the relation be-tween residual strain and stress tensor components, for thisstructure, can be expressed according to Hooke’s law as[37,51]

eRx

eRy

eRz

cRxy

cRxz

cRyz

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

¼

S11 S12 S13 0 0 0

S21 S22 S23 0 0 0

S31 S32 S33 0 0 0

0 0 0 S44 0 0

0 0 0 0 S55 0

0 0 0 0 0 S66

2666666664

3777777775

rRx

rRy

rRz

sRxy

sRxz

sRyz

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

ð9Þ

where the S terms represent the compliances, and rR and sR

are the residual normal and shear stress components,respectively. For cubic structures, S11 ¼ S22 ¼ S33,S12 ¼ S13 ¼ S21 ¼ S23 ¼ S31 ¼ S32 and S44 ¼ S55 ¼ S66.

As discussed above, the magnitude of rRz is significantly

greater compared with the other two normal stress compo-nents and, therefore, to simplify the residual stress calcula-tions, all other residual normal (rR

x and rRy ) and shear stress

(cRxy ;c

Ryz and cR

xz) components are neglected in the subsequentdiscussions. Accordingly, solving Eq. (9), gives

eRx ¼ S13r

Rz ; e

Ry ¼ S23r

Rz ; e

Rz ¼ S33r

Rz and cR

xy ¼ cRxz ¼ cR

yz ¼ 0

ð10Þ

Table 1

Material E (GPa) a (K�1) m Volume fraction (%)

ZrB2 489 [3] 5.9�10–6 [3] 0.16 [3] �91SiC 694 [43] 3.5�10–6 [44] 0.17 [65] �9

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300Scratch load (mN)

Res

idu

al s

tres

s (G

Pa)

Tensile

Compressive

Fig. 6. Evolution of residual stress within SiC grains as a function ofscratch load.

D. Ghosh et al. / Acta Materialia 56 (2008) 5345–5354 5351

Noting that S33 ¼ S11 and S13 ¼ S23 ¼ S12, one can rewritethe normal strain components in Eq. (10) as

eRx ¼ S12r

Rz ; e

Ry ¼ S12r

Rz and eR

z ¼ S11rRz ð11Þ

Thus, Eq. (8) reduces to

peRx þqðeR

y þ eRz Þ�k 0 0

0 peRy þqðeR

x þ eRz Þ�k 0

0 0 peRz þqðeR

x þ eRy Þ�k

�������

�������¼ 0

ð12Þ

Solving for k and substituting eRx , eR

y and eRz from Eq. (11),

one gets

k1 ¼ rRz fS12ðp þ qÞ þ S11qg;

k2 ¼ rRz fS12ðp þ qÞ þ S11qg;

and

k3 ¼ rRz ðS11p þ 2S12qÞ: ð13Þ

Substituting the above k1, k2 and k3 expressions in Eq.(7), the relation between Raman shift and residual stressis derived as

Dw1 ¼1

2wo1

fS12ðp þ qÞ þ S11qgrRz ; ð14Þ

Dw2 ¼1

2wo2

fS12ðp þ qÞ þ S11qgrRz ; ð15Þ

and

Dw3 ¼1

2wo3

fS11p þ 2S12qgrRz : ð16Þ

The above equations will now be used to calculate theresidual stress from the observed Raman peak shifts inFig. 3. In strained diamond cubic and zinc-blende crystalstructures, a maximum of three Raman modes are possible,two TO vibrations and one LO vibration [50]. The first twoRaman modes (w1 and w2) correspond to TO peaks,whereas the third Raman mode (w3) is associated withthe LO peak. Therefore, Eqs. (14) and (15) can be usedto calculate residual stress from TO-peak shift, whereasEq. (16) can be used to calculate residual stress from LO-peak shift. The values of p and q for 3C-SiC are not avail-able directly in the literature. Therefore, these values havebeen calculated in the following way. The mode Gruneisenparameters for hydrostatic stress (co) and uniaxial stress(cs) are defined as [40]

co ¼ �ðp þ 2qÞ

6w2o

and cs ¼ðp � qÞ

2w2o

ð17Þ

where wo is the Raman peak in strain-free condition. Themode Gruneisen parameter for the TO-phonon (cTO

o ) in3C–SiC is 1.56 [40], but the value of cTO

s is not readily avail-able. Although, for uniaxial and biaxial stresses, cTO

s andcTO

o may differ, it is assumed here that cTOs ¼ cTO

o ¼ 1:56.For the TO-peak, assuming wo ¼ 796 cm�1, one obtainspTO ¼ �0:623� 106=cm2 and qTO ¼ �2:634� 106=cm2.Also, S11 ¼ 3:7� 10�13 cm2=dyn and S12 ¼ �1:05�

10�13cm2=dyn were used for residual stress calculation[52]. The final expression for residual stress calculationfrom the TO peak, using Eq. (14), is expressed as

rRz ¼

2wo1

fS12ðp þ qÞ þ S11qgDw1

¼ �251:66DwTO ðMPaÞ ð18Þ

where DwTO is the shift in TO-peak position. From Eq.(18), it is clear that shift of TO peaks to lower or higherwave number will result in tensile or compressive residualstress, respectively.

3.4. Determination of residual stress measurements

Thermal residual stresses in the as-processed compositewere calculated from Eqs. (2) and (3) using the materialproperties and volume fractions of ZrB2 and SiC phasesas given in Table 1, and assuming an average radius of1 lm for SiC phase. The temperature difference, DT , usedin Eq. (2) was 1725 �C. The calculation revealed that theSiC grains were under a uniform compressive residualstress of 1.731 GPa, whereas the compressive radial andtensile tangential stresses at ZrB2–SiC interfaces were1.731 GPa and 1.126 GPa, respectively. The magnitudesof these stresses decrease sharply away from the interface.The high tensile tangential stress at the interface can causeradial microcracking in the ZrB2 matrix surrounding theSiC grains and eventually act as potential sites for thedevelopment of larger cracks or critical flaws.

Fig. 6 shows the magnitude of mechanical residual stresswithin SiC grains (calculated from Eq. (18)) due to thescratch process as a function of applied scratch load. At50 mN, residual stress was still compressive whereas,

5352 D. Ghosh et al. / Acta Materialia 56 (2008) 5345–5354

>50 mN, tensile residual stress is generated within the SiCgrains and increases almost linearly in magnitude withscratch load. The highest mechanically induced tensileresidual stress at 250 mN within SiC grains was estimatedto be �2.6 GPa. Such large magnitude of residual stresshas been reported on many material systems in the litera-ture. Raman spectroscopic measurements have revealedtensile residual stress as high as 2.1 GPa in diamond films[53,54], up to 2 GPa in carbon thin films [55] and up to1.2 GPa in porous Si films [31].

The calculated tensile residual stress (2.6 GPa) at250 mN in the SiC grains was well above the reportedtensile fracture stress of 0.9 GPa for polycrystalline 3C–SiC [56]. However, it was observed from SEM micro-graphs that the SiC grains present within the groovesafter the scratch, from which Raman spectra were col-lected were crack free at this scale. For example, Fig. 7revealed that a SiC grain within a groove formed at250 mN load was completely intact, whereas the sur-rounding ZrB2 matrix was heavily microcracked owingto the scratch process. Similar features were alsoobserved at several locations within the scratch grooves.These observations indicate unusually high fracturestrength of the SiC grains in the composite comparedwith a typical polycrystalline SiC ceramic. In this sce-nario, it may not be appropriate to compare directlythe available fracture strength values of polycrystallineSiC ceramics with SiC grains present within the particu-late phase distributed in the ZrB2–SiC composite. Inaddition, typical polycrystalline 3C–SiC ceramics containlarge elongated grains with a high aspect ratio in therange 2–6 [56], whereas the SiC grains present in thecomposite were almost equiaxed. Microstructural investi-gations did not reveal any porosity within the SiC phase.Also, no sintering additive was used during the process-ing, and hence no grain boundary glassy phase isexpected. Polycrystalline SiC ceramics are known to exhi-bit predominantly intergranular fracture mode as

SiC grain

Scratch direction

ZrB2

Microcracks

Scratchgroove

Fig. 7. SEM micrograph of a scratch groove at 250 mN revealing theuncracked SiC grain surrounded by the heavily microcracked ZrB2 matrix.

opposed to transgranular fracture [56,57]. The densityof critical flaws or defects that can act as nucleation sitesfor crack propagation is expected to be much higher in amonolithic ceramic compared with the defect densitywithin the isolated grains of the same size in a composite.Therefore, in the absence of pores, secondary glassyphase and limited SiC–SiC grain boundary areas, trans-granular fracture is expected to be the predominant modeto relieve the accumulated residual stress within the indi-vidual SiC grains. Possible sources of transgranular crackinitiation will be the internal flaws present within theseindividual SiC grains. Fracture stress of a polycrystallineceramic is limited by the critical flaw size, which scalesdown with grain size. This in turn increases the macro-scopic fracture strength. Based on this fact, a simple frac-ture mechanics based (using the Griffith criterion)calculation was made to estimate the minimum fracturestrength of these individual grains. Thus, if one considerscrack initiation/propagation from a pre-existing criticalflaw, its maximum size will be equivalent to the grain sizewhich will result in minimum fracture strength. For acritical flaw size of c, assuming semielliptical surfaceflaws, the fracture strength (rf ) and mode I fracturetoughness (KC) are related by [58]

rf ¼KC

1:35ffiffifficp : ð19Þ

Clearly, estimation of rf is dependent on the chosen KC

and the c values which are not readily known for this par-ticular SiC phase present within the composite. Dependingon the assumptions made on the nature of SiC phase(whether monocrystalline or polycrystalline) distributedwithin the ZrB2 matrix, the estimated rf can differ consid-erably. If one assumes the SiC phase in the composite to bemade of several grains, it is appropriate to consider a crit-ical flaw size of the order of average grain size (�1 lm) forthe estimation of minimum rf . The available fracturetoughness values for polycrystalline SiC ceramics vary overa broad range (3.4–6.8 MPa m1/2) and depend on processconditions as well as microstructural features such as aver-age grain size, shape, aspect ratio, etc. [57,58]. For thesevalues, Eq. (19) results in a minimum fracture strengthbetween 2.5 and 5 GPa. These values provide a conserva-tive estimate of rf . However, if one assumes the SiC phaseto be a single crystal, a more complicated situation arises.In a single crystal, typical processing- or deformation-induced defects are vacancies, interstitials, dislocations,stacking faults, twinning, etc., which are significantly smal-ler in size than a grain level microcrack. The current studyhas related the measured Raman peak shift to the develop-ment of deformation-induced stacking faults within the SiCphase. Therefore, stacking faults are conjectured as themain source for crack nucleation. Based on the micro-graphs of stacking faults presented in Shih et al. [59] andongoing TEM studies [60], the width of the stacking faultsare taken as 50 nm. Using a KC value of 3.2 MPa m1/2 for asingle crystal 3C–SiC thin film [61], a fracture strength of

D. Ghosh et al. / Acta Materialia 56 (2008) 5345–5354 5353

10.6 GPa was obtained. This value is only two to four timesthe conservative estimate made earlier based on micro-scopic flaw size. Therefore, it is argued that, owing to smallsize and high fracture strength, SiC grains dispersed withinZrB2 phase were able to sustain the large scratch-inducedtensile residual stress without any fracture.

Raman stress coefficients (defined as the change inRaman peak position per unit residual stress) for diamond,SiC and Si are in the range 0.4–4 cm�1 GPa�1 [31,35–38,40,50,62–64]. This indicates that small changes inRaman peak positions may result in large residual stress.In the current investigation, large peak shifts to lower wavenumbers, particularly at 250 mN, were observed (seeFig. 3). From the calculated tensile residual stress, theRaman stress coefficient was estimated to be3.9 cm�1 GPa�1, which is in the above range of thereported Raman stress coefficients for various materials.Also, residual stress measurements employing MRS arehighly dependent on spatial location within the grains(due to the small spot size of the incident beam) and henceprovide only localized residual stress values within smallregions. Therefore, it should be noted that the estimatedhigh tensile residual stress is only a localized value andnot necessarily the average over the entire SiC grain(s).

From the above discussion, it is clear that the SiCgrains in ZrB2–SiC composite were initially in a state ofresidual compressive stress, and the scratch processinduces large residual tensile stress. Similarly to the SiC,tensile residual stress is also expected to develop withinthe ZrB2 matrix, particularly at higher scratch loads.Although the magnitude of residual stress was unknownin the ZrB2 phase as it is not Raman active, evolutionof residual tensile stress state within ZrB2 matrix is evi-denced from the onset of lateral crack formation at250 mN (see Fig. 1). Although unknown, scratch-inducedtensile residual stress within ZrB2 at 250 mN is predictedto be close to its fracture strength (�500 MPa) [3]. Sincethe ZrB2 phase is continuous and coarse grained, it can-not sustain high tensile residual stress similar to the indi-vidual SiC grains. It is argued that, up to 200 mN scratchload, the magnitude of tensile residual stress within theZrB2 phase was probably below the fracture stress tocause any lateral cracking. But >250 mN, tensile residualstress can locally become comparable with or higher thanthe fracture stress of ZrB2, resulting in lateral cracking, asseen in the experiments.

The above model for scratch-induced residual stress esti-mation within SiC grains in a ZrB2–SiC composite is basedon the SBFM [24,27]. This analytical model was derived topredict the scratch-induced residual stress state in a homo-geneous brittle material. But the scratch-induced residualstress state in a composite could be more complex and,therefore, application of the sliding blister model for resid-ual stress determination may not yield exact values.Another major assumption was that Raman peak shift isrelated to the dominant stress component (rR

z ) in thescratch-induced residual stress field. Other stress compo-

nents may also have some correlation to the observedRaman peak shifts in SiC grains. Nevertheless, the calcu-lated values appear reasonable. Therefore, it is argued thatthe proposed model can be used for scratch-induced resid-ual stress determination in Raman active materials withdiamond cubic or zinc-blende crystal structures.

4. Conclusions

(1) MRS measurements were conducted on 3C-SiCgrains of as-processed ZrB2–SiC composite as wellas on grains that lie within the scratch grooves. It isfound that the TO peak and LO-Raman peak in3C–SiC were shifted to increasingly lower wave num-bers with increasing scratch loads.

(2) An analytical model was developed to relate thescratch-induced mechanical residual stresses withinthe SiC grains to the Raman peak shift in terms ofphonon deformation potentials of 3C–SiC.

(3) Residual stress measurements using MRS on SiCgrains of ZrB2–SiC composite revealed that thesegrains were under compressive residual stress in theas-processed composite and then experience tensileresidual stress due to scratch-induced deformation.The magnitude of residual compressive stress in theas-processed composite is �1.731 GPa, and thescratch-induced tensile residual stress increases line-arly with load. At 250 mN scratch load, the magni-tude of tensile residual stress can be as high as2.6 GPa.

Acknowledgements

This work was funded by a grant from the US NSF(Grant CMS-0324461) with Dr. Ken Chong as the pro-gram manager.

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