J O U R N A L O F M A T E R I A L S S C I E N C E 3 9 (2 0 0 4 ) 1891 – 1893

Shear strength of ceramics

XIAOTONG WANG, N. P. PADTURE∗Department of Metallurgy and Materials Engineering, Institute of Materials Science, University of Connecticut,Storrs, CT 06269-3136, USAE-mail: nitin.padture@uconn.edu

Nanoindentation experiments [1, 2], Bragg-Nyebubble-raft simulation experiments [3], and atomistic/finite-element modeling [4] have revealed that incip-ient plasticity in ductile metals occurs at shear stressvalues approaching the theoretical shear strength. Thisincipient plasticity, characterized by a sudden burst ofdisplacement in the nanoindentation load-depth (P-h)response at sub-milliNewton loads (0.01 to 0.03 mNfor single-crystal Al) [2], represents homogeneous nu-cleation of defects such as dislocations. In order to in-vestigate if incipient plasticity can be detected quan-titatively in brittle ceramics using displacement burstsin nanoindentation P-h responses, we have performedlow-load nanoindentation experiments on two single-crystals: Y2O3 and SrTiO3, cubic ceramics whose shearmoduli differ by about a factor of two. We have foundthat displacement bursts in the nanoindentation P-h re-sponses occur at average indentation loads of 0.15 and0.65 mN in Y2O3 and SrTiO3, respectively, and thatthese loads are consistent with the calculated theoreti-cal shear strengths of the two ceramics.

Single-crystal SrTiO3 (100) was obtained com-mercially (Material-Technologie & Kristalle Gmbh,Germany), whereas a polished large-grain size(100 µm) Y2O3 polycrystalline specimen was obtainedfrom Dr. W. H. Rhodes. While the (100) surface ofthe SrTiO3 single-crystal was indented, the orientationof the indented grain in the Y2O3 was unknown. Ap-proximately 50 nanoindentation experiments were per-formed on each material at different surface locations,using Nanoindenter XP (MTS/Nanoinstrument, OakRidge, TN), in Dynamic Contact ModuleTM (DCM)mode, equipped with a diamond Berkovich pyramid.The peak load ranged from 0.1 to 1.0 mN, while thetotal loading time required to reach the peak load wasmaintained constant at 30 s.

Fig. 1A and B show representative P-h curves forthe nanoindentation of Y2O3 and SrTiO3, respectively.Note the displacement bursts at indentation loads (P∗)of 0.145 and 0.625 mN in Y2O3 and SrTiO3, respec-tively. The indentation loads at P-h discontinuitiesfor Y2O3 ranged from 0.1 to 0.2 mN, with an av-erage of P∗ = 0.15 mN. In the case of SrTiO3, P∗ranged between 0.5 and 0.8 mN, with an average ofP∗ = 0.65 mN. A hard “dummy” specimen (SiC) wasnanoindented several times (at different locations) un-der the same conditions, where no displacement burstswere observed. This confirmed that the observed dis-

∗Author to whom all correspondence should be addressed.

placements bursts in Y2O3 and SrTiO3 are not an arti-fact of the instrument, and that those bursts arise frommaterial response alone.

The maximum shear stress under the Berkovich in-denter at indentation load P∗, where the P-h discon-tinuity occurs, can be determined using the followingrelation [2, 5]:

τMax = 0.31

(6P ∗ E∗2

π3 R2

)0.33

, (1)

where R is the radius of the tip of the indenter, andE* is defined in terms of the Young’s moduli and thePoisson’s ratios of the diamond indenter (Ei, νi) and thespecimen (Es, νs) as follows [2, 5]:

E∗ =(

1 − ν2s

Es+ 1 − ν2

i

Ei

)−1

. (2)

Although the Berkovich pyramid is a “sharp” indenter,it has a finite radius, which is assumed to be R ∼ 50nm for a new indenter [2]. Using the values for elasticproperties of diamond, Y2O3, and SrTiO3 (Table I), themaximum shear stress values τMax at P-h discontinuitiesfor Y2O3 and SrTiO3 are calculated to be 16.1 and 34.1GPa, respectively (Table I). A rough estimate for thetheoretical shear strength of crystals is given by [6]:

τTh = G/2π, (3)

where G is the shear modulus of the specimen, whichis given by [6]:

G = Es

2(1 + νs). (4)

Using Equations 3 and 4, the theoretical shear strengthsfor Y2O3 and SrTiO3 are calculated, and they are alsogiven in Table I, along with τMax/τTh ratios. Althoughthe estimate of the theoretical shear strength is onlyan approximation (Equations 3) [6], these ratios arenot significantly different compared with unity (1.5 and1.7), indicating that τMax represents the theoretical shearstrength of the ceramic in question. This notion is fur-ther reinforced by the fact that the τMax/τTh ratios forthe two ceramics are not very different. The local break-down of the material could be manifest as fracture or

0022–2461 C© 2004 Kluwer Academic Publishers 1891

T ABL E I Relevant mechanical properties of ceramics, and calculated values of τMax (Equation 1) and τTh (Equation 3)

Ceramic P∗ (GPa) E (GPa) ν G (GPa) τMax (GPa) τTh (GPa) τMax/τTh

Y2O3 0.15 172 [12] 0.31 [12] 66 [12] 16.1 10.5 1.5SrTiO3 0.65 300 [13] 0.20 [13] 125 [13] 34.1 19.9 1.7Diamond – 1000 [2] 0.07 [2] – – – –MgO 1.0 [7] 290 [11] 0.19 [11] 130 [11] 38.5 19.4 2.0Al2O3 15 [9] 400 [10] 0.22 [10] 164 110.8 26.1 4.2

Figure 1 Representative nanoindentation P-h responses (loading) forsingle-crystals of: (A) Y2O3 and (B) SrTiO3.

dislocation creation [4], the exact nature of which is notclear at this time.

In this context, Gaillard et al. [7] have recently ob-served similar displacement bursts during the nanoin-dentation of another cubic ceramic, MgO. For single-crystal MgO (001) specimens, the indentation load atwhich this burst occurs (P∗) is 1.0 mN. However, thedisplacement burst in terms of indentation depth (h)in MgO (25 nm) [7] was found to be significantlygreater than that observed for Y2O3 (3 nm) and SrTiO3(7 nm). Using Equations 1 to 4 and elastic-property val-ues (Table I) in the case of MgO, the τMax/τTh ratio wasfound to be 2.0, which, once again, is consistent with

τMax identifying with the theoretical shear strength ofMgO.

Gaillard et al. [7] have also performed careful etchingexperiments on nanoindented MgO, which is a way toreveal dislocations that intersect the free surface [8], andthey have characterized the resulting etch-pits using theatomic force microscope (AFM). For nanoindentationloads greater than P∗, they have found well-definedpatterns of etch-pits around the nanoindentation site[7]. For P < P∗, the etching results were inconclusive[7]. These results suggest that the displacement bursts innanoindentation of cubic ceramics are associated withthe nucleation of dislocations.

Page et al. [9] were the first to report nanoinden-tation displacement bursts in ceramics. However, theyfound the burst to occur at much higher loads (P∗ forsingle-crystal Al2O3 ∼ 15 mN). Using Equations 1 to4 and elastic-property values (Table I) in the case ofrhomohedral Al2O3, the τMax/τTh ratio is found to be4.2, which is higher than that for cubic ceramics. Thiscould be possibly due to the reduced crystal symme-try in Al2O3. Page et al. [9] have also attributed thesebursts to the formation of incipient cracks in Al2O3.

In the case of metals, multiple displacement burstshave been observed, which occur at successively highernanoindentaion loads P, and over larger h ranges [2].It has been shown that this is due to the motion ofthe already formed dislocations, which requires ordersof magnitude lower shear stresses (identifies with thePeierls stress) relative to theoretical shear strengths. Inthe nanoindentation of brittle ceramics, such multipledisplacement bursts are not common, possibly due tothe very high values of the Peierls stress relative toductile metals [10, 11].

In summary, the displacement bursts we have ob-served during the nanoindentation of ceramics arise asa result of material response, and are not an instrumentartifact. In the case of cubic ceramics, the maximumshear stress (τMax) associated with the nanoindentationload at which these bursts occur identifies with the re-spective theoretical shear strength (τTh) of the ceramicin question. It appears that homogenous nucleation ofdislocations is responsible for the displacement burst,however, further work is needed to elucidate this issue.

AcknowledgmentsThe authors thank Prof. S. Suresh (MIT) and Dr. D. T.Smith (NIST) for fruitful discussions, and Dr. R. H.Rhodes for providing the Y2O3 specimen. Partial fund-ing for this work was provided by the Office of NavalResearch through a subcontract from MIT (prime grantDURINT N00014-01-1-0808).

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13. K . H . H E L L W E G E and A. M. H E L L W E G E , “Landolt-Bornstein New Series: Group III,” Vol. 16, Oxides (Springer-Verlag,New York, NY, 1981) p. 64.

Received 13 Augustand accepted 16 September 2003

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