Indentation strength of a piezoelectric ceramic ...materials.iisc.ernet.in/~ramu/publications/paper84.pdf · Indentation strength of a piezoelectric ceramic: Experiments and simulations

Indentation strength of a piezoelectric ceramic:Experiments and simulations

S.N. KambleDepartment of Materials Engineering, Indian Institute of Science, Bangalore 560012, India

D.V. KubairDepartment of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India

U. Ramamurtya)

Department of Materials Engineering, Indian Institute of Science, Bangalore 560012, India

(Received 15 August 2008; accepted 27 October 2008)

The spherical indentation strength of a lead zirconate titanate (PZT) piezoelectric ceramicwas investigated under poled and unpoled conditions and with different electricalboundary conditions (arising through the use of insulating or conducting indenters).Experimental results show that the indentation strength of the poled PZT is higher thanthat of the unpoled PZT. The strength of a poled PZT under a conducting indenter ishigher than that under an insulating indenter. Poling direction (with respect to thedirection of indentation loading) did not significantly affect the strength of material.Complementary finite element analysis (FEA) of spherical indentation of an elastic,linearly coupled piezoelectric half-space is conducted for rationalizing the experimentalobservations. Simulations show marked dependency of the contact stress on the boundaryconditions. In particular, contact stress redistribution in the coupled problem leads to achange in the fracture initiation, from Hertzian cracking in the unpoled material tosubsurface damage initiation in poled PZT. These observations help explain theexperimental ranking of strength the PZT in different material conditions or underdifferent boundary conditions.

I. INTRODUCTION

Piezoelectric materials, wherein a first-order couplingbetween mechanical and electrical fields exists, are ext-ensively used in many branches of modern technology,including aerospace, automotive, medical, and electronicindustries, as actuators as well as sensors. However, amajor drawback of the piezoelectric ceramics such as thewidely used lead zirconate titanate (PZT) is their brittle-ness. Significant advances have been made in the pro-cessing and manufacturing of these materials in therecent past. However, mechanical failure as well as elec-trical degradation caused by the presence and growth ofvarious defects such as cracks, holes, and inclusions arestill of major concern from the reliability viewpoint.Consequently, role of defects on the strength of thesematerials has been extensively investigated.1–11

Generally piezoelectric actuators, sensors, and othercomponents are in the form of thin films, beams, andplates. With increasing miniaturization (and the adventof technologies such as MEMS), the dimensions of thesecomponents are getting smaller. Naturally, property

assessment of such small-volume materials is a chal-lenge and using the indentation technique for such mea-surements is a viable and attractive option.12–14 Further,indentation provides a method for assessing a given pie-zoelectric material’s response to the contact loading,which also plays an important role in the reliability ofthe piezoelectric ceramics. While indentation is a simpleas well as an easy experimental method to perform,deciphering the results obtained is complicated becauseof the inhomogeneous stress and strain distribution un-derneath the indenter. This gets even more complicatedin piezoelectric materials because of the intrinsic cou-pling between the applied mechanical and the resultantelectrical fields. Therefore, contact problems in coupledmaterials have attracted considerable attention.Matysiak15 examined the surface deformation and

tractions at the contact perimeter of rigid and perfectlyconducting flat and spherical punches acting on piezo-electric half-space. The plane strain (Flamant type) prob-lem of a line force and a line charge acting on thepiezoelectric half-space was solved by Sosa and Cas-tro.16 Stroh’s formulation has been used by Fan et al.17

to solve two-dimensional contact on a piezoelectric half-space with nonslipping and slipping contacts. Chen18

has performed three-dimensional investigation of rigid

a)Address all correspondence to this author.e-mail: [email protected]

DOI: 10.1557/JMR.2009.0115

J. Mater. Res., Vol. 24, No. 3, Mar 2009 © 2009 Materials Research Society926

smooth punch on a transversely isotropic piezoelectrichalf-space by employing potential theory method. Hisanalysis reveals that stress and electric displacementintensity factor at the edge of punch have the same formas that of pure elastic problem.

Giannakopoulos and Suresh19 have developed a gen-eral theory for axisymmetric indentation of piezoelectricsolids within the context of fully coupled, transverselyisotropic elasticity. This work shows that the functionalrelationship between the load, P, and the depth of pene-tration, h, does not get altered because of electromechan-ical coupling; i.e., P / h1.5 for spherical indentation ofelastic solid remains valid. However, the indentationstiffness (the proportionality constant, S, in P = Sh1.5)depends on the condition of material (poled versusunpoled) as well as the electrical nature of the indenter(conducting versus insulating), which was experiment-ally verified by Ramamurty et al.20 Similarly, predic-tions on the electrical response were confirmed bySridhar et al.21 Makagon et al.10 have performed indenta-tion experiments using spherical and conical indenters.Their study indicates that the contact stiffness is weaklyaffected by the piezoelectric coupling. Zarnik et al.9 haveperformed nanoindentation experiments and finite ele-ment analysis to characterize the mechanical complianceof PZT thick films. Zhang and Gao8 have performedtheoretical indentation fracture analysis of piezoelectricmaterials. Their analysis showed that fracture mechanicsformalism is applicable in determining the strengthof piezoceramics. Axisymmetric analyses of sphericalindentation were performed recently by Yang,11 and heconcluded that the electric field could either increase orsuppress indentation deformation depending on the direc-tion of electric field. This indicates that the indentationstrength could be altered depending on the type of indent-er (insulating or conducting) used.

However, the strength of piezoelectric ceramics underindentation loading, which can be an important designparameter, has not yet been assessed. In particular, therole of coupled contact stresses and electrical fields indetermining the indentation strength needs detailed un-derstanding. Accordingly, the objective of this work ison the experimental assessment of the spherical indenta-tion strength of PZT under different material (poled ver-sus unpoled) and boundary (conducting versus insulatingindenter) conditions. The results, which show strongdependence of strength on the experimental variables,have been rationalized by recourse to finite element anal-ysis of quasi-static normal spherical indentation of a

transversely isotropic, linear-elastic, fully coupled pie-zoelectric half-space. Spherical indentation has beenchosen since it allows for two-dimensional (2D) axisym-metric analysis rather than computationally costly andcomplicated three-dimensional (3D) analysis as is thecase with Vickers or Berkovich indentation.

II. MATERIAL AND EXPERIMENTS

Spherical indentation experiments were conductedon a polycrystalline PZT-4 [chemical composition(in mol%) of 47PbTiO3, 47PbZrO3, 6SrZiO3] piezoelec-tric ceramic, which has high coupling constants. Poled aswell as unpoled disks (3 mm in thickness and 16 mm indiameter) were obtained from Sensor Technology Limit-ed, Collingwood, Ontario, Canada. The relevant mech-anical, dielectric, and piezoelectric properties of PZT-4are listed in Table I. Before testing, all the specimenswere polished only on the side on which indentationexperiments were conducted. Polishing was done in agentle manner, so that it would not cause any heavydeformation that may introduce a depoled surface layer.

Figure 1 shows the schematic of the indentation setup.To enforce the experimental conditions similar to theboundary conditions of zero electric potential far awayfrom the indenter (see Sec. III), the back surface (the sideopposite to that being indented) as well as the side sur-face of the specimen were coated with gold. Further, thelower crosshead of the testing device on which specimenwas placed and the upper crosshead of the testing deviceto which the indenter was attached were both electricallygrounded, ensuring zero electric potential far away fromthe point of contact. Experiments were conducted usingan Instron testing machine along with in-house devel-oped indentation attachment. A 10 kN load cell (preci-sion of 1 N) and a linear variable differential transformer(LVDT) (1 mm rating and 1 mm precision) attached to theindentation device were used to measure the load and thecorresponding depth of penetration.

The relation between the contact radius, a, and thedisplacement, hr, corresponding to a rigid sphere ofdiameter D, is given at fo = 0 by linear elastic contactmechanics condition:

hr ¼ 2a2

D: ð1Þ

The resilience of the spherical indenter is accounted bymodifying the ideally rigid displacement with the actualmeasured displacement ha and correction factor Kc.

TABLE I. Material properties of PZT-4 ceramic (SI units).

Elastic stiffness coefficients (�1010 N/m2) Piezoelectric coefficients (C/m2) Dielectric coefficients (�10�9 C/Vm)

C11 C12 C13 C33 C44 e31 e33 e15 l11 l3313.9 7.78 7.43 11.3 2.56 �6.98 13.84 13.44 6.00 5.47

S.N. Kamble et al.: Indentation strength of a piezoelectric ceramic: Experiments and simulations

J. Mater. Res., Vol. 24, No. 3, Mar 2009 927

h3=2a

h3=2r

¼ 1þ Es 1� n 2i

� �Ei 1� n 2

s

� � ¼ Kc ; ð2Þ

where ha = vt, v is constant indentation displacementrate; t is time; Es and ns are Young’s modulus and Pois-son’s ratio, respectively, of the specimen; and Ei and niare corresponding properties of the indenter. To mini-mize deformation at the back face of indenter, a tungstencarbide (WC) back plate (Eb = 495 GPa, n = 0.22) wasintroduced between the indenter and the back surface.Further, applied load was ensured to be lower than criti-cal load, which otherwise may cause yielding of ind-enter. Consequently, total compliance S should take intoaccount contact compliance S2 between indenter andback surface as well as compliance S1 between indenterand specimen. The load–displacement and total compli-ance relation is:

P ¼ Sh3=2 : ð3aÞ

S ¼ 1

S1

� �2=3

þ 1

S2

� �2=3" #�3=2

: ð3bÞ

S1 ¼ 2ffiffiffi2

p

3

1� nsEs

þ 1� niEi

� � ffiffiffiffiD

p: ð3cÞ

The expression for S2 is similar to that in Eq. (3c) withEs and ns being replaced by that of back surface plateintroduced between indenter and upper side of Instron.

Experiments were conducted with tungsten carbide(WC with 10 wt% Co, E = 475 GPa and n = 0.25) andsapphire (E = 378 GPa and n = 0.25) balls. While theformer serves as a conducting indenter, the latter is aninsulating indenter. Two sizes of WC balls were used,9.52 and 3 mm in diameter, to assess the size effectson the indentation strength. The sapphire ball’s dia-meter was 9.52 mm. All the tests were conducted indisplacement control mode with a rate of 8 mm/min.

Initial cracking of the specimen leads to a sudden dropin the load, which is considered the failure load Pf ofthe ceramic. Each indentation experiment was repeatedat least five times, and only the average values arereported. We are cognizant of the fact that the strengthof ceramics is stochastic in nature and hence capturingthe Weibull parameters is important. However, we didnot pursue this because of the limitation in the numberof specimens. Nevertheless, the average strengths,obtained experimentally and reported in this paper,should not get altered significantly by increasing thenumber of samples.

III. FINITE ELEMENT ANALYSIS

Complementary finite element simulations were per-formed to work out the contact stress and electrical fielddistributions. For this purpose, a two-way coupled finiteelement scheme was used, wherein the electric chargesaffect the displacements and elastic strains and viceversa. A simplified axisymmetric analysis of the spheri-cal indentation on the piezoelectric ceramics was per-formed. The schematic of the boundary value problemis shown in Fig. 2. The axis of indenter is the axis ofrevolution. A coupled piezoelectric finite element codethat can handle complex constitutive material models(functionally graded materials) as well as multisite dam-age scenarios (multiple cohesive cracks) was developedfor this specific purpose. In our coupled piezoelectroe-lastic finite element analysis, we modeled the indenta-tion problem using 5625 four-noded axisymmetricisoparametric quadrilateral elements (Fig. 3). For thesake of simplicity we have assumed the material to behomogeneous. The stress and electric field beneath theindenter have large gradients, and we modeled the con-tact radius by 625 square (aspect ratio 1) elements tocapture the singularity effect. The electrical and mechan-ical boundary conditions described earlier were pre-scribed (Fig. 2).

FIG. 1. Schematic representation of experimental setup.


J. Mater. Res., Vol. 24, No. 3, Mar 2009928

We start our formulation of the finite element schemefrom writing the conservation of linear momentum andMaxwell’s equation for an axisymmetric problem, whichcan be written as

@srr

@rþ @srz

@zþ srr � syy

r¼ 0

@srz

@rþ @szz

@zþ srz

r¼ 0

9>>>=>>>;

: ð4Þ

@Dr

@rþ Dr

rþ @Dz

@z¼ 0 : ð5Þ

The kinematic relations between the infinitesimal strainsand displacements can be written as

Err ¼ @ur@r

; Eyy ¼ urr; Ezz ¼ @uz

@z; grz ¼

@ur@z

þ @uz@r

:

ð6ÞThe components of the electric flux vector (Er, Ez) aregiven by

Er ¼ � @f@r

; Ez ¼ � @f@z

: ð7Þ

To relate the mechanical and electrical stresses, we as-sume that the loading axis (oz) is aligned with the direc-tion of polarization. The constitutive relation in theabsence of any residual polarization strains for the axi-symmetric problem can be written as

srr ¼ C11Err þ C12Eyy þ C13Ezz � e31Ez

syy ¼ C12Err þ C11Eyy þ C13Ezz � e31Ez

szz ¼ C13 Err þ Eyyð Þ þ C13Ezz � e33Ez

srz ¼ C44grz � e15Er

Dr ¼ e15grz þ l11Er

Dz ¼ e31 Err þ Eyyð Þ þ e33Ezz þ l33Ez

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

: ð8Þ

Here, C11, C12, C13, C33, and C44 are the nonzeroelastic constants whose numerical values from lite-rature,1,2 are listed in Table I. For an out-of-phasepolarization (180�), i.e., polarization switching, the pie-zoelectric constants in the above equation reverse theirsign, while the elastic and dielectric constants remainunaltered.

The indenter was assumed to be frictionless, rigid andperfectly spherical (diameter D) as illustrated in Fig. 2.The indenter transfers a normal load P due to whichan indentation of depth, h, results with a circular contactarea of radius a. The above statements of the experi-mental details translate to the following mechanicalboundary conditions of the axisymmetric finite elementanalysis:

uz r; 0ð Þ ¼ h� r2

D; 0� r < a

srz r; 0ð Þ ¼ 0; r� 0

szz r; 0ð Þ ¼ 0; r > a

9>>=>>; : ð9Þ

In the experiments, both conducting (tungsten carbide)and insulated (sapphire) indenters were used. The choiceof the indenter alters the electrical boundary conditions.For the conducting indenter it can be written as

f r; 0ð Þ ¼ f0; 0� r < a

Dz r; 0ð Þ ¼ 0; r > a

�; ð10Þ

whereas for an insulating indenter the electrical bound-ary conditions take the following form

FIG. 2. Schematic representation of indentation boundary value

problem.

FIG. 3. Schematic representation of axisymmetric finite element

model.



Dz r; 0ð Þ ¼ 0; r� 0 : ð11ÞTo overcome any edge effects, the outer boundary of

the finite element problem was chosen to be 20a. Also,we have assumed that the second-order derivatives of themechanical and electrical displacements vanish at theouter edge,

ur; uz;fð Þ ! 0; 0; 0ð Þ asffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2 þ r2ð Þ

p! 1 : ð12Þ

IV. RESULTS

A. Strength measurements

Typical experimentally measured P–h curves obtainedwith two different indenter diameters (3 and 9.52 mm)are shown in Figs. 4(a) and 4(b), respectively. In bothcases, the curves follow the analytical expression, P =Sh1.5 predicted by Giannakopoulos and Suresh.19 Valuesof S, obtained through least-square fits, are listed inTable II. As expected, S depends on the indenter–polingcondition combination. While the absolute values of Sobtained by Ramamurty et al.20 on PZT-4 differ fromthose measured in this study (as a result of the relativelymore compliant loading frame of the present study), thetrends are generally consistent. Specifically, S decreasesin the following order of material condition/electricalboundary condition combinations: unpoled, poled/conducting, and poled/insulating.

The relative direction of poling (with respect to that ofthe indentation) was found to be inconsequential, withpositive (i.e., poling and indentation directions being the

same) and negative (i.e., poling and indentation direc-tions being antiparallel) poling directions giving essen-tially the same results (within the experimental scatter).Hence, the role of poling direction on the indentationresponse is ignored hereafter.The P–h curves culminate at the breaking point of the

PZT specimen. This load, Pmax, which corresponds to thefracture load of the specimen for a given indenter diame-ter, was converted into indentation strength using the con-tact radius, a, that has been calculated using Eqs. (1) and(2). Values of the Pmax and the indentation strengths calcu-lated are summarized in Table II. The following are someof the key observations that can be made from this table.(i) Indentation strength of the poled PZT is higher thanthat of the unpoled PZT, irrespective of the electricalnature of the indenter.(ii) The strength of the poled indenter is sensitive tothe electrical nature of the indenter. For the 9.52 mmdiameter indenter, the strength of the poled PZT is about17% higher when measured with a conducting indentervis-a-vis that measured with an insulating indenter.(iii) The strengths are sensitive to the indenter size.Indentation strengths recorded with the larger indenterare significantly smaller than the respective strengthsmeasured with the smaller diameter (3 mm) indenters.The above results suggest that electromechanical cou-

pling in the piezoelectric solids alters the indentationstrength considerably. Further, the experimental resultsclearly indicate that the type and size of indenter hasa major influence on the measured strength. The PZTused in this study is a ceramic and hence sensitive to

TABLE II. Indentation strengths of a fresh PZT samples for different test combinations.

Indenter size

(mm)

Condition of

the PZT

Electric nature of

the indenter

Indentation stiffness

S (N/mm1.5)

Maximum load,

Pmax (N)

Indentation strength,

sf (MPa)

9.52 Unpoled Conducting 2.71 685.0 � 12.33 318.0 � 22.05

Poled Conducting 2.44 931.7 � 35.19 403.4 � 15.24

Poled Insulating 2.21 758.5 � 16.50 346.0 � 7.53

3.00 Unpoled Conducting 1.53 585.0 � 20.00 695.0 � 23.62

Poled Conducting 1.49 701.7 � 20.53 722.5 � 21.14

FIG. 4. Representative load, P, versus depth of penetration, h, experimental curves obtained with indenter diameters, f, of (a) 9.52 and

(b) 3.00 mm.



strength-limiting flaw population and distribution. There-fore, a reduced strength at higher indenter diameters canbe anticipated. This is because of the larger volume of thematerial that is subjected to deformation with the largerindenter. The experimental results clearly imply complexinterplay between defects in the materials and the state ofstress, the latter being affected by the first-order couplingthat exists in the piezoelectric ceramics. These results areexplained with the help of finite element results.

B. Numerical analysis

The quantitative behavior of the electromechanicalvariables beneath the spherical indenter when thePZT half-space subjected to constant average pressure,(P/pa2) of 980 MPa is illustrated in Figs. 5 and 6. The

maximum principal stress, s1 contours within the regionthat is twice the contact radius, a, are shown in Fig. 5(a)for the unpoled case. The contours for the case of thepoled PZT indented by a conducting (WC) indenter areshown in Fig. 5(b) whereas Fig. 5(c) shows those withthe use of an insulating indenter. Respective contours ofresultant electric field E ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Er þ Ezp

, induced by theindentation of the poled PZT are shown in Figs. 6(a)and 6(b). Table IV summarizes the maximum normal-ized principal tensile stress and maximum resultant elec-tric field for different combinations of indentation tests.

These simulation results show that, for the sameapplied constant pressure, the unpoled and poled PZTexhibit differences in the distribution as well as magni-tude of maximum principal stress and resultant electricfield induced, which are fundamental parameters govern-ing the strength of brittle materials. The stress redistribu-tion is caused by the induced electric displacement in thepoled case [Figs. 5(b) and 5(c)]. The maximum principalstresses are tensile beneath the indenter, whereas theunpoled PZT exhibits classical stress distribution givenby Hertzian contact; i.e., the maximum principal stressbecomes tensile at the contact perimeter along the radialdirection [Fig. 5(a)]. For the case when the poled PZT isindented with a conducting indenter [Fig. 5(b)], the re-gion where the stresses become tensile is larger in areacompared with the unpoled case. The position of the highstress region is found to be �1.25a beneath the indenterfrom the specimen surface. This indicates that the elec-tromechanical coupling in the poled PZT results in ashift of the location of maximum tensile stress to aregion beneath the indenter, which is constrained bythe surrounding material. It can also be noticed fromFig. 5(c) that the stress distribution beneath the indenteris relatively uneven for the case of indentation of poledPZT with an insulating indenter. Further, the magnitudeof maximum principal tensile stresses are �20% lowervis-a-vis that induced in the unpoled material whereasthey are much higher (nearly 50%) in the case ofpoled PZT indented with insulating indenter (Table IV).The latter occurs because the conducting indenter acts as anelectrode and opposite charges are attracted toward it;therefore, the electric field is more concentrated within thecontact region, as can be seen in Fig. 6(a). With an insulat-ing indenter, charges are neither attracted nor free to move.From Fig. 6(b), it is seen that the electric field is moreintensive at the contact perimeter and along the polarizationdirection and therefore result in a state of high stress. Fur-ther, the ratio of elastic to piezoelastic maximum principaltensile stress (s1

E/s1P) was found to be 1.23.

When piezoelectric ceramics are subjected to strongalternating or direct current fields opposing the directionof poling, depolarization can occur thereby reducing thepiezoelectric effect. The point at which depolarizationoccurs depends on the grade of material, the duration of

FIG. 5. Contour plots showing the distribution of the maximum prin-

cipal tensile stress in PZT-4 underneath the spherical indenter of size

(f) = 9.52 mm for different indentation test combinations. (a)

Unpoled. (b) Poled/conducting indenter. (c) Poled/insulating indenter.



application of the electric field, and temperature. Hence,understanding the electric field distribution due toindentation becomes important. Breakdown occurswhen the electric field exceeds the dielectric strength ofpiezoelectric ceramic. Electric-field strengths of 5 � 105

to 106 V/m are sufficient to depolarize PZT.16 Forthe case of indentation of poled PZT with the conduc-ting indenter, the electric field is more concentratedwithin contact region [Fig. 6(b)]. In contrast, it is moresensitive at contact perimeter and along the directionof polarization for the case of poled PZT indented withan insulating indenter [Fig. 6(a)]. Hence, the contactregion and contact perimeter are the critical regionswhere depoling may occur and dielectric breakdowntakes place.

V. DISCUSSION

A. Microscopic observations

In a piezoelectric material the electrical field caneither aid or decrease the indentation strength of the

materials unlike in non-piezoelectric materials. Asshown by Giannakopoulos and Suresh19 and Chen andDing,22 the mechanical stress and electrical displace-ment has a

ffiffir

psingularity in an indentation of poled

piezoelectric material. This singularity leads to micro-damage and dissipates energy. Fracture experiments ofZhang7 show an increase in fracture toughness caused bylarger dissipation of energy around cracks, which is ab-sent in the case of unpoled (nonpiezoelectric) materials.This observation suggests that an increase in the inden-tation strength of a PZT block is plausible because of theinteraction of the electrical and mechanical deformationfields as observed in the present study. Hence, the prob-ability of indentation-induced cracking is less with theindentation of poled PZT with conducting indenter.Results of both the experiments and finite element simu-lations suggest that the indentation strength of thePZT depends on the condition that material is in as wellas the type of indenter used. Possible micromechanisticreasons for this were investigated by recourse to detailedfractography.

FIG. 6. Resultant electric field distribution for spherical indentation of poled PZT-4 with indenter size (f) = 9.52 mm for different indentation

test combinations.

TABLE III. Summary of the finite element analysis results for indentation with 9.52 mm diameter indenter.

Condition of the

PZT

Electric nature of the

indenter

Maximum principle tensile stress,

s1 (MPa)

Maximum normalized stress = s1

(pa2)/PMax resultant electric field,

E (V/m)

Unpoled Conducting 108.92 0.112 . . .

Insulating 108.92 0.112 . . .

Poled Conducting 88.39 0.0898 6.10

Insulating 155.35 0.1585 14.32

Note: Because of the affine nature of the elastic stress fields, only the relative variations are of significance.

TABLE IV. Indentation strengths of broken pieces of PZT for different test combinations.

Indenter diameter, D (mm) Condition of the PZT Electric nature of the indenter Maximum load, Pmax (N) Indentation strength, sf (MPa)

9.52 Unpoled Conducting 560.0 � 29.47 330.0 � 21.00

Poled Conducting 411.7 � 39.65 214.0 � 20.61

Poled Insulating 442.7 � 34.32 294.6 � 22.84

3.00 Unpoled Conducting 588.3 � 35.67 697.9 � 42.31

Poled Conducting 540.0 � 60.00 614.9 � 68.32



Figure 7(a) is a scanning electron micrograph showingthe indentation impression made in the unpoled PZT.Here, the sample was unloaded prior to fracture. Classi-cal radial cracks emanating from the edge of the impres-sion can be seen. Magnified view of the region adjacentto the contact radius shown in Fig. 7(b) indicates severalcircumferential cracks at the boundary. Notice a majorcrack among these, which eventually leads to fracture.Figure 7(c) is an image obtained from the region belowthe indenter in a specimen that was fractured duringindentation. It shows that fracture is the result of a pre-dominant cone crack emanating from the edge of thespherical indenter impression [Fig. 7(c)]. All these indi-cate classical Hertzian kind of failure; i.e., the maximumprincipal stress becomes tensile at the contact perimeteralong the radial direction, thereby initiating a crack [seeFig. 5(a)], which is typical in a brittle ceramic subjected

to spherical indentation. The finite element simulations[Fig. 5(a)] are consistent with this.

The fracture behavior was completely different in thecase of poled samples. Whereas the unpoled samplesbroke into two or three pieces, the poled samples invari-ably shattered into several small pieces. Figure 8shows the indentation impression where the sample wasunloaded just before fracture. The features are unlikethose observed in poled case, with the absence of peri-odically arranged radial cracks. Instead, two large cracksat the edge of the impression are seen. Note these cracksare not tangential to the contact periphery. Further, thecrack-opening width appears to increase with increasingdistance away from the contact periphery. These twocracks are connected by a crack along the periphery, butonly one side. All these features indicate that the fractureinitiation site is beneath the indenter, and the cracksgrow upward toward and away from the deformedregion. This is in qualitative agreement with our full-field finite element simulations, which show that themaximum principal stresses are tensile beneath theindenter.

B. Indentation strengths

The observation that fracture initiates in the subsur-face region in the poled PZT also explains the experi-mentally observed strength differences between poledand unpoled samples. In the latter, cracking initiates onthe surface, and it is well known that surface cracks haveconsiderably higher stress concentration (vis-a-vis thoseinside the body) and hence are easy to drive. In contrast,the region of maximum tensile stress in the poled case isalso in the hemispherical expanding cavity that is con-strained by the surrounding material, and hence crackgrowth is difficult.

FIG. 7. (a) Optical micrograph of the indentation impression made in

an unpoled PZT-4 sample. (b) Higher-magnification view of the con-

tact perimeter showing circumferential cracking. (c) SEM micrograph

obtained from the region beneath the indenter after the sample has

fractured.

FIG. 8. Optical micrograph of an indentation impression obtained

from a poled PZT-4 sample. This image was obtained by stopping

the indentation test just before fracture.



In between the strengths of the poled PZT measuredwith insulating and conducting indenters, we find the for-mer to be�20% higher than the latter. This is because thelocation and magnitudes of the maximum principal stressdiffer in between these two cases. For the conductingindenter, they are far deeper beneath the indenter [seeFigs. 5(b) and 5(c)], with a far smaller magnitude (seeTable IV). Both of these factors make the load requiredfor breaking the specimen higher, as higher load isrequired for making a strength-limiting flaw that is locateddeep inside to attain criticality. Also, since resultant stressinduced is lower (because of the drainage of the electricalcharge through the conducting indenter), one has to applyhigher loads for the critical stress to be reached.

As mentioned previously, the indentation strengthsmeasured were approximately the same whether thedirection of the indentation and the direction of polariza-tion are parallel or antiparallel, when the indentationdirection opposed the direction of poling. This result isreinforced by the stress contours (not shown here) fromour finite element simulations, which indicate that themagnitude of the maximum principal stress attainedremains invariant with change in the direction of inden-tation with respect to the direction of polarization. Whenthe direction of polarization is reversed, the piezoelectricconstants change sign while the rest of the material con-stants such as Young’s modulus and dielectric strengthremain invariant. The magnitude of these constants aresame as the positive poled case, though the individualcomponents of induced electric field are equal and oppo-site in direction of individual components of inducedelectric field in the positive poled PZT case; the resultantelectric field induced is same in both the cases. Conse-quently, the maximum magnitude attained of the maxi-mum principal tensile stress and its distribution withinthe piezoelectric half-space remains invariant, and there-fore the indentation strength remains the same as in thepositive poled case.

While indentation size effect was observed (smallerindenter giving rise to higher strength) in both poledand unpoled PZT samples, the relative difference instrengths are much smaller when the conducting indenterdiameter is reduced from 9.52 to 3 mm (see Table II).While the difference between unpoled and poled sam-ples for the bigger indenter is �27%, it is only 4% forthe smaller indenter and within the experimental scatter.This is because with the smaller indenter, the subsurfacecracks that lead to fracture of poled specimens are closeenough to the surface itself, and hence the differencebetween stress concentrations of surface and subsurfacecracks are only marginal.

It is possible that residual stresses, which are inducedduring poling, could affect the strength measurements.However, polarized Raman spectroscopic studies of PZTsamples subjected to mechanical load by Deluca et al.23

suggest that residual stresses are caused by electrodeatoms only at the contact points and not intrinsic to thematerial itself per se. Since the stress and strain gradientexist only locally underneath the indenter during inden-tation, residual stresses causing the observed differencesin indentation strengths can be ruled out.We close the discussion with some interesting obser-

vations made on the strength measurements made onbroken pieces. When indentations were performed, thefractured pieces were often large enough for conductingfurther strength tests. In such cases, we measured theirindentation strength by locating the indent at locationsthat were visibly free of damage. A considerable numberof tests were performed on both poled and unpoled sam-ples, and the results are summarized in Fig. 9 andTable III. As can be seen from these, the trends instrengths are exactly opposite to those seen in the freshsamples, i.e., those tested for the first time. While at firstsight these appear perplexing, it is possible to rationalizethem by thinking in terms of the different natures ofstrength limiting flaws in poled and unpoled cases. Asdiscussed previously, surface flaws are strength limitingin the case of unpoled samples and the area sampled bystress is the contact perimeter; i.e., surface defects pres-ent far away from the indenter remain unaffected. There-fore, fractured samples are tested; these defects aresampled and those statistics need not necessarily bedifferent from those in the fresh samples. Therefore,indentation strengths of unpoled sampled remain un-changed (within the experimental scatter) with repetitionof tests on broken pieces.The strength-limiting flaws in the poled specimens are

subsurface, which require relatively large strength to

FIG. 9. Histogram showing the variation of the indentation strength

for different indentation test combinations in fresh and repeat tests

(see the text for details). The indenter diameter, f, for both the cases

is 9.52 mm. Here, UP refers to unpoled, P to poled, CI to conducting

indenter, II to insulating indenter.



initiate. Further, since these are “volume defects” thespace sampled is much larger. It is also likely that therelatively large stress applied as well as the stress wavesgenerated during shattering (recall our observation thatunpoled samples broke into two/three pieces in a “grace-ful” manner, whereas the poled pieces shattered vio-lently), activates microcracks through the specimenbody. Such microcracks weaken the ceramic bodyconsiderably, and hence when the visibly undamagedpieces are tested again, they require much lower loadsto fracture.

When the indenter is reduced, from 9.52 to 3 mm, thestress fields get much smaller and hence the “micro-cracked zone size” is likely to be confined to a smallervolume. As a result, the strength reduction is relativelysmall (from 722.5 to 614.9 MPa or �15%) vis-a-vis thatseen with the larger diameter indenter (403.4 to 214 MPaor 47%) for the poled PZT tested with a conductingindenter. In the unpoled case, the strength differencebetween fresh and repeat samples is insignificant, sincethe surface cracked areas (upon first loading) are likelyto be small and do not interact. Therefore, the strengthsof the repeated tested samples are same as those of thefresh ones.

VI. SUMMARY

The experiments and complementary finite elementsimulations conducted in this study show that the inden-tation strength of piezoelectric ceramics depends on thepoling condition as well as the type of indenter. This isbecause the electromechanical coupling in the poled pie-zoelectric material alters the state of stress distributionunderneath the indenter, which in turn leads to samplingof different types of strength-limiting defects. While thestrength-limiting flaws in the unpoled ceramic are typi-cal surface flaws, the volume defects are responsible forthe fracture initiation in poled samples, as maximumstresses move to the region beneath the indenter. As aresult of this, poled samples exhibit much higherstrengths, especially when indented with a conductingindenter. Fractography as well as repeat experimentson broken samples confirm this change in defect–stressinteraction with poling.

REFERENCES

1. R.M. McMeeking: A J-integral for the analysis of electrically

induced mechanical stress at cracks in elastic dielectrics. Int. J.Eng. Sci. 28, 605 (1990).

2. Y.E. Pak: Crack extension force in a piezoelectric material.

Trans. ASME J. Appl. Mech. 57, 647 (1990).

3. Y. Shindo, E. Ozawa, and J.P. Nowacki: Singular stress and

electric fields of a cracked piezoelectric strip. Appl. Electromag.Mater. 1, 77 (1990).

4. H.A. Sosa: Plane problems in piezoelectric media with defects.

Int. J. Solids Struct. 28, 491 (1991).

5. Z. Suo, C.M. Kuo, D.M. Barnett, and J.R. Willis: Fracture me-

chanics for piezoelectric ceramics. J. Mech. Phys. Solids 40, 739(1992).

6. Z. Suo: Models for breakdown-resistant dielectric and ferroelec-

tric ceramics. J. Mech. Phys. Solids 41, 1155 (1993).

7. T.Y. Zhang: Effects of static electric field on the fracture behav-

ior of piezoelectric ceramics. Acta Mech. Sin. 18, 537 (2002).

8. T.Y. Zhang and C.F. Gao: Fracture behaviors in piezoelectric

solids. Theor. Appl. Fract. Mech. 41, 339 (2004).

9. M.S. Zarnik, D. Belavic, and S. Macek: Evaluation of the consti-

tutive material parameters for the numerical modeling of struc-

tures with lead–zirconate–titanate thick films. Sens. Actuators, A136, 618 (2007).

10. A. Makagon, M. Kachanov, S.V. Kalinin, and E. Karapetian:

Indentation of spherical and conical punches into piezoelectric

half-space with frictional sliding: Applications to scanning-probe

microscopy.Phys. Rev. B 76, 064115 (2007).

11. F. Yang: Analysis of the axisymmetric indentation of a semi-

infinite piezoelectric material: The evaluation of the contact stiff-

ness and the effective piezoelectric constant. J. Appl. Phys. 103,074115 (2008).

12. U. Ramamurty and M.C. Kumaran: Mechanical property extrac-

tion through conical indentation of a closed-cell aluminum foam.

Acta Mater. 52, 181 (2004).

13. S. Jana, R. Bhowmick, Y. Kawamura, K. Chattopadhyay, and

U. Ramamurty: Deformation morphology underneath the Vickers

indent in a Zr-based bulk metallic glass. Intermetallics 12, 1097(2004).

14. S. Ramachandra, P. Sudheer Kumar, and U. Ramamurty: Impact

energy absorption in an Al foam at low velocities. Scr. Mater. 49,741 (2003).

15. S.J. Matysiak: Axisymmetric problem of punch pressing into a

piezoelectric half-space. Bull. Polish Acad. Techn. Sci. 33, 25(1985).

16. H.A. Sosa and M.A. Castro: On concentration load at boundary of

a piezoelectric half-plane. J. Mech. Phys. Solids 42, 1105 (1994).

17. H. Fan, K.Y. Sze, and W. Yang: Two dimensional contact on a

piezoelectric half-space. Int. J. Solids Struct. 33, 1305 (1996).

18. Wei-qiu Chen: On piezoelectric contact problem for a smooth

punch. Int. J. Solids Struct. 37, 2331 (2000).

19. A.E. Giannakopoulos and S. Suresh: Theory of indentation of

piezoelectric materials. Acta Mater. 47, 2153 (1999).

20. U. Ramamurty, S. Sridhar, A.E. Giannakopulous, and S. Suresh:Experimental study of spherical indentation on piezoelectric

materials. Acta Mater. 47, 2417 (1999).

21. S. Sridhar, A.E. Giannakopoulos, S. Suresh, and U. Ramamurty:

Electric response during indentation of piezoelectric materials: A

new method for materials characterization. J. Appl. Phys. 85, 380(1999).

22 W. Chen and H. Ding: Indentation of a transversely isotropic

piezoelectric half-space by a rigid sphere. Acta Mech. Solid. Sin.12, 114 (1999).

23. M. Deluca, C. Galassi, and G. Pezzotti: Residual stresses in

PZT investigated by polarized Raman piezospectroscopy.

Ferroelectrics Lett. 32, 31 (2005).



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