measurement and determination of dynamic biaxial flexural strength of thin ceramic substrates under...

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International Journal of Mechanical Sciences 47 (2005) 1212–1223

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Measurement and determination of dynamic biaxialflexural strength of thin ceramic substrates under high

stress-rate loading

Ming Chenga, Weinong Chenb,�

aMechanical Engineering Department, 817 Sherbrooke Street West, Montreal, Que., Canada H3A 2K6bSchools of Aero/Astro and Materials Engineering, Purdue University, West Lafayette, IN 47907-2023, USA

Received 25 May 2004; received in revised form 23 March 2005; accepted 6 April 2005

Available online 4 June 2005

Abstract

Quasi-static and dynamic piston-on-3-ball experiments have been performed on an 8-mol% yttriastabilized zirconia (8YSZ) ceramic material and its doped versions to investigate the effects of the dopantson the biaxial flexural strength of 8YSZ thin substrates. To facilitate the extraction of information fromlimited number of dynamic experimental results, a new strength model with constant stress-rate as anindependent variable was developed based on Tuler and Butcher’s concept of cumulative damage. Byemploying an overall least-squares curve fitting technique, the experimental data were fitted to this model.Results revealed that the alumina (Al2O3) dopant with amount less than 3mol% and a 3-mol% yttriastabilized zirconia (3YSZ) dopant with amount less than 30wt% do not change the biaxial flexural strengthof 8YSZ thin substrates significantly under both quasi-static and dynamic loading conditions. However, allthe materials exhibited clear loading rate strengthening effects.r 2005 Elsevier Ltd. All rights reserved.

Keywords: Ceramic; Zirconia; Biaxial flexural strength; Loading rate; Stress rate

see front matter r 2005 Elsevier Ltd. All rights reserved.

ijmecsci.2005.04.004

ding author. Tel.: +1 765 494 1788; fax: +1 765 494 0307.

ress: [email protected] (W. Chen).

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M. Cheng, W. Chen / International Journal of Mechanical Sciences 47 (2005) 1212–1223 1213

1. Introduction

Measurement of the strength of a series of nominally identical ceramic specimens typicallyproduces considerable scatter in the results. This scattering nature of experimental data associatedwith the failure of ceramic materials has been a challenge in the experimental design and datainterpretations. Large numbers of identical specimens and experiments are desired in order to gaina high confidence level on the data from a statistical point of view. To obtain a statisticaldistribution at certain loading rate requires at least thirty experiments [1]. On the other hand, suchnumbers are frequently limited in experiments due to lack of sufficient resources such as time andspecimens. Furthermore, if the loading-rate is taken into consideration to obtain an insight intothe dynamic strength behavior of a ceramic material at a range of loading rates, a huge number ofspecimens are required. It was also difficult to repeat high loading-rate experiments exactly with asame loading-rate for many times as compared to the quasi-static experiments. In many practicalexperiments, in particular, during the early developing stages of new materials, large numbers ofidentical specimens are not available to satisfy the statistical requirement.However, if the physics of the failure behavior is at least partially understood, the experimental

results will not be viewed as purely statistical, and the required number of identical specimens andtests may be reduced. In the case of a recent study evaluating loading rates and dopants effects onthe failure strength of an 8-mol% yttria stabilized zirconia (8YSZ) ceramic, the number ofspecimens for each composition is limited for mechanical tests at room temperature during thescreening process of searching promising compositions for improved strength. Many specimensare used for high temperature tests to investigate their physical properties.To assist the material composition screening process, meaningful information must be drawn

from the limited amount of data. The key is that the failure processes of the ceramic material withvarious types and amounts of dopants are not totally statistically random. Physics-based materialmodels have been available to describe various aspects of the failure processes of brittle materialsfor decades (e.g., Tuler and Butcher [2]). It is therefore feasible that we adopt a relevant model forthe specific material of interest and utilize the physics-based model to derive meaningfulinformation from limited numbers of identical specimens and experiments for property evaluationduring material development. Using this approach, we introduce a strength model accounting forthe effects of loading rate for ceramic materials. With such a model, we employ an overall least-squares curve fitting technique, which incorporates all available data at different loading ratesincluding quasi-static loading, to estimate model parameters. This approach can largely elicitinformation of mechanical strength from a paucity of experiments. This capability is especiallysignificant in a case in which the purpose of the experimental investigation is to compare therelative dynamic strength made of different candidate material compositions. In the developmentof new materials, by adding small amount additions to a baseline material, the model parametersprovide a criterion for determining whether or not the addition significantly changes the dynamicstrength of the baseline material.In this paper, experiments designed to determine the relative dynamic biaxial flexural strength

of thin 8YSZ substrates with various types and amounts of dopants are briefly described. Amaterial model that describes both the quasi-static and dynamic strength behavior based on adamage accumulation mechanism under constant stress-rate loading is then introduced. Thematerial constants are determined using overall least-squares curve fitting from the limited

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M. Cheng, W. Chen / International Journal of Mechanical Sciences 47 (2005) 1212–12231214

amount of experimental data. It is shown that, with the material model, material selectiondecisions can be made based on the limited amount of available experimental data.

2. Biaxial flexural experiments

A high oxygen ion conductivity over wide ranges of temperature and oxygen pressures in 8YSZhas led to its use as a solid oxide electrolyte in a variety of electrochemical applications such ashigh temperature solid-oxide fuel cells (SOFCs) [3] and molecular filters to generate pure oxygenfrom oxygen bearing gases such as carbon dioxide, water vapor, and air [4]. The solid-oxideelectrolytes are typically made by a tape-cast process. After sintering, the products are usually inthe form of thin sheets with a thickness of much less than 2mm. Recent applications of the solid-oxide electrolyte involve space explorations, which inevitably expose it to loading conditions ofmultiaxial and dynamic bending during launching and landing. To investigate the dynamic biaxialflexural strength of thin ceramic substrates, a novel dynamic piston-on-3-ball experimentaltechnique was developed [5]. This new experimental technique provides a method to obtain thebiaxial flexural strength at desired loading rates.In our recent research on developing improved 8YSZ material, certain amounts of alumina

(Al2O3) and 3-mol% yttria stabilized zirconia (3YSZ) were doped into 8YSZ. Experiments wereconducted to examine the effects of the dopants on the biaxial flexural strength of 8YSZ underboth quasi-static and dynamic piston-on-3-ball loading conditions. An ASTM standardizedmethod [6] was used for the quasi-static experiments, while a newly developed dynamic piston-on-3-ball method [5] was used for the dynamic experiments.

2.1. Specimen preparation

The specimens were made from TZ-8YSZ powder (TOSOH USA, Inc., Atlanta, GA). Thepowder was mixed with dopants and was then processed into a slurry with dispersant, binder, andplasticizer. The slurry was then tape-cast. Then, the specimens were laser-cut out of green sheetsand sintered at 1450 �C for 3 h. The surface roughness of as-fired specimens is between 20 and30mm as observed with a Zeiss IM 35 inverted microscope. The specimen geometry is the onerecommended by ASTM F 394-78 [6]: 32mm in diameter and 0.75mm in thickness. An X-raydiffraction (XRD) analysis performed on the materials revealed that only the cubic phase waspresent in all of these specimens [7].

2.2. Quasi-static experiments

In the piston-on-3-ball experimental method, a thin ceramic substrate is placed on three ballssitting 120� apart on a 25.4-mm diameter circle. A piston pushes on the center of the circle fromthe other side of the ceramic sheet, thus producing a biaxial flexural loading condition. The quasi-static experiments were performed using a hydraulically driven material testing system (MTS 810)with a piston speed of less than 1:27mm=s at room temperature.Static strength is the main concern in this project. It is the basic characterizing property for the

experimental studies of the effects of high temperatures or high loading rates. Therefore, thirty

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nominally identical specimens were tested under quasi-static loading for each materialcomposition as recommended by ASTM F 394-78 [6]. The experimental results indicate thatthe biaxial flexural strength data are scattered, which is consistent with the characteristics ofbrittle failures. The distribution of these strength data were fitted to a Weibull cumulativeprobability distribution function using the method of maximum likelihood [8],

PF ¼ 1� exp �ass0

� �m� �, (1)

where s is the biaxial flexural strength, s0 is the mean of the biaxial flexural stress data at fracture,m is the Weibull parameter, and a is a scale factor. The fitting results from a Matlab 6 function(weibfit.m) are listed in Table 1, which show that the Weibull parameter, m, exhibits somevariations. However, the fact that their 95% confidence intervals overlap each other indicates thatthe variations of the Weibull parameter for these dopants are not significant. A similarphenomenon is observed from the strength data, as shown in Table 1.

2.3. Dynamic experiments

In order to delineate the effects of stress rates on dynamic strength, it is necessary to maintainthe same experimental conditions, except for the loading rates. The test sections of both the quasi-static and dynamic experimental facilities are thus designed to be identical—a standard piston-on-3-ball setup. The dynamic version of the piston-on-3-ball method, shown schematically in Fig. 1,has been developed. A detailed description and validation of this technique have been givenelsewhere [5]. Here, we only briefly describe the principle of this technique.The principle and test procedure for the dynamic piston-on-3-ball experiment are similar to that

of a split Hopkinson pressure bar (SHPB) test [9]. An impulse load is applied on the center of athin ceramic substrate specimen by an instrumented hammer through the incident bar. Part of theloading pulse is then transmitted through the specimen to the transmission bar (Fig. 1). The time-resolved impulse forces in the hammer and the transmission bar are recorded simultaneously.Because of the wave propagation effects, the force signal recorded at the transmission barpossesses a phase delay to the force signal from the hammer. The specimen is fractured when theimpulse force applied by the hammer reaches a high enough amplitude. The impulse peak andthe loading rate can be determined from the recorded signals. Once the load peak is determined,the biaxial flexural strength at the center of the specimen surface in tension can be calculated using

Table 1

Weibull parameters fitted from static experimental results and biaxial flexural strengths

Material composition m a s0

Pure 8YSZ 7.44 (5.24, 9.65)a 0.62 (0.40, 0.84) 310.8

1mol% Al2O3-doped 8YSZ 8.36 (4.51, 12.21) 0.61 (0.31, 0.92) 322.3

3mol% Al2O3-doped 8YSZ 8.69 (2.13, 15.25) 0.62 (0.14, 1.10) 333.0

20wt% 3YSZ-doped 8YSZ 10.04 (4.48, 15.60) 0.61 (0.29, 0.92) 317.4

30wt% 3YSZ-doped 8YSZ 8.81 (2.69, 14.92) 0.62 (0.24, 1.00) 342.6

aInside the brackets is the 95% confidence interval.

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Transmission bar Quartz disk

Instrumented hammer

Incident bar

Specimen

Supports

ChargeAmplifier

Charge Amplifier

A/D Converter

Fig. 1. A schematic of the dynamic piston-on-3-ball setup.


Kirstein and Woolley’s [10] equation in the case where the specimen is in a dynamic equilibriumstate [11].

sf ¼ �3

4pP

d2ð1þ nÞ ln

r2

r0

� �2

þ1� n2

r2

r0

� �2

� ð1þ nÞ 1þ lnr1

r0

� �2" #

� ð1� nÞr1

r0

� �2 !

,

(2)

where P is the load peak causing fracture, n is Poisson’s ratio of the specimen material, r0 is theradius of the specimen, r1 is the radius of the support circle, r2 is the radius of the loading area,and d is the specimen thickness.Typical force traces involving a 32-mm-diameter, 0.72-mm-thick, 1mol% Al2O3-doped 8YSZ

ceramic specimen are presented in Fig. 2. This figure shows that the loading is of a constant stress-rate (equivalent to constant force-rate) after the load exceeds 60N. When the load is below 60N,the stress is well within the elastic range without causing damage. It is very difficult to achieve aconstant stress rate over the entire duration of the loading. Our attention is therefore focused onthe critical stages of the experiment where high stress, damage, and fractures are involved. Thespecimen was in the dynamic force equilibrium state, since the loading rate in the hammerð1:1293� 105 N=sÞ was nearly the same as in the quartz ð1:1382� 105 N=sÞ just before fracture.Cheng et al. [5] have presented the details of experimental procedure and theoretically derivedcriteria to judge the validity of experimental data by determining if the specimen is in a dynamicforce equilibrium state before fracture.Unlike quasi-static case, it is very difficult to obtain a group of experimental results with exactly

identical loading rates since the loading rate can only be controlled within a certain range.Therefore, a regular statistical method to process data with the same loading rates is infeasiblesince a huge number of experiments would be required. Under such conditions, a strength model

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0 0.001 0.002 0.003

Time(s)

-40

0

40

80

120

160

Forc

e (N

)

HammerQuartz

Fig. 2. Typical force records with a dynamic piston-on-3-ball experiment.


accounting for the effects of loading rate is desired to extract strength behavior information fromlimited number of experimental data by employing an overall least-squares curve fittingtechnique.

3. Strength model under constant stress-rate loading

The behavior of brittle materials such as ceramics under high-speed loading has been thesubject of many studies. Many experiments revealed that rapidly loaded structures could bearstresses that considerably exceeded the critical levels under static loading conditions (Grady [12],Bourne et al. [13], Clifton [14]). The most recent review of the field of dynamic failure mechanics isthat of Rosakis and Ravichandran [15]. Most modeling efforts on dynamic failure behaviors ofbrittle materials have been based on fracture mechanics and/or damage mechanics. Fracturemechanics is well developed to describe the crack propagation and dynamic fracture ofbrittle materials. However, it can only be applied when an initial crack is well defined or assumed,with its shape, size, location, and orientation determined. The material bulk is considered tobe a continuum other than the crack. On the other hand, damage mechanics describes theeffects of micro-cracking on the mechanical properties (such as elastic stiffness degradation,induced anisotropy, and anelastic strains) of the brittle material as micro-cracking develops.Distributed crack models are inevitably included in the dynamic material models based ondamage mechanics [16].Freund [17] formulated a dynamic material model to describe the time-dependent strength of

brittle materials under high strain-rate loading by applying a Weibull’s crack density model andan effective elastic stiffness model proposed by Delameter et al. [18]. He stated that themacroscopic stress in a brittle specimen subjected to high strain-rate loading can continue toincrease as the macroscopic deformation proceeds during the incubation time, which is the delaytime between the initial crack propagation and the final failure of the structure. During this

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incubation period, the critical level of stress needed to produce crack growth can be reached atmany other defects in the specimen. Crack growth can thus be initiated at many flaws if theloading is applied rapidly. As the cracks in the specimen continue to grow, the effective elasticstiffness of the cracked specimen continues to degrade. Finally, the stress in the specimen fails toincrease further. The stress at this moment was defined by Freund as the impact strength at acertain constant strain-rate. Later on, many researchers modified Freund’s constant strain-ratemodel by applying different effective elastic stiffness models. For example, Chen and Ravichadran[19] modified this model using an effective elastic stiffness model developed by Kemeny and Cook[20]. Chen [21] modified Freund’s model using another effective elastic stiffness model given byBudiansky and O’Connell [22].Most existing constitutive models for high-speed loading are constant strain-rate models.

However, the dynamic piston-on-3-ball experimental technique provides only constant stress-ratedata. In order to avoid the introduction of material properties into data interpretation, it isnecessary to employ high stress-rate models. Denoual et al. [23] developed a model forfragmentation of brittle materials under constant stress-rate loading using a probabilisticapproach. This model covers only part of loading-rate range, and the starting point of this rangeis unknown. Bouzid et al. [24] proposed a damage model to describe the failure of a glass underconstant stress-rate impact loading. Their model shows that the degree of damage is a function ofloading rate and applied stress, but provides no explicit relationship between loading rates and thedynamic strength of brittle materials. It is then desired to formulate explicit relationships betweendynamic strength and stress loading-rate to fit the dynamic experimental data. In addition,fracture mechanics points out that the strength of a brittle material is dependent on the loadingmodes and the loading rates, as well as the initial crack sizes and orientations [17]. This physicalbackground indicates that the dynamic strength of a brittle material is related to its static strengthunder static loading conditions. When loading rate approaches zero, quasi-static strength shouldbe recovered from a dynamic model.Tuler and Butcher [2] proposed a general failure criterion based on the concept of damage

accumulation. The spall stress s is dependent on the stress pulse duration in the formZ ts

0

½sðtÞ � s0l dt ¼ Constant, (3)

where s0 is a threshold stress, ts is the time to failure from sðtÞ ¼ s0, and l is a material constant.This model indicates that a longer ts corresponds to a lower spall stress, ss ¼ sðtsÞ. They found agood correlation between the spall stress values estimated by Eq. (3) and those obtainedexperimentally for a spalling layer of aluminum specimens. Freund [17] recast Tuler and Butcher’smodel in the formZ ts

0

sðtÞsw

� 1

� �bdt ¼ C, (4)

where sðtÞ is a representative stress (such as the remote tensile stress in the case of uniaxial loadingand/or the central stress on the tensile surface in the case of biaxial loading, e.g., piston-on-3-ballloading), sw is the stress threshold for the beginning of damage accumulation, ts is the timerequired for the stress to start from sw and reach its maximum level; b and C are experimentally

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determined material constants. Freund [17] proved that this model is consistent with his constantstrain-rate loading model.Since Tuler and Butcher’s general failure criterion is based on damage accumulation, which

introduces integration of stress history to time, as shown in Eq. (4). Therefore, a relation can becreated between the dynamic strength and constant stress-rate with an emphasis on the stress-rateeffects. Two idealized assumptions have been introduced in the development of our new model.First, the threshold stress, sw, is assumed to be approximately the same as the strength of the samematerial under static loading. The dynamic strength of a brittle material increases from its staticstrength with increasing strain rate as commonly observed from experiments (Grady [12], Bourneet al. [13], Clifton [14]). Second, the dynamic strength depends only on the loading history afterthe stress load exceeds the static strength. In other words, the crack size distribution in thematerial bulk does not change if the applied load is below its static strength. Loading modes haveno effects on the dynamic strength if the stress load is less than the static strength. Thisassumption is consistent with the ‘‘weakest link’’ concept in fracture mechanics [8], whichindicates that the crack propagation determines the strength of a brittle material. A crack starts topropagate when the stress intensity factor at this crack reaches its critical value.Therefore, as pointed out previously, the initial time can be set at the point when stress load

passes the static strength. The loading stress history after passing sw can then be expressed as

sðtÞ ¼ ð _st þ swÞHðtÞ, (5)

where HðtÞ is the Heaviside function and _s is the applied constant stress-rate.Substituting Eq. (5) into Eq. (4), we obtain

_sb

sbw

tbþ1s

bþ 1¼ C. (6)

The stress, sðtÞ, increases linearly from sw at t ¼ 0 to the dynamic strength, ss, at t ¼ ts in thisconstant stress-rate strength model, i.e.,

ss ¼ _sts þ sw. (7)

By eliminating ts in Eqs. (5) and (6), we obtain

ss ¼ sw 1þ_s_sr

� �1=1þb" #

, (8)

where the reference stress-rate, _sr, is defined as

_sr ¼sw

ð1þ bÞC. (9)

The model for dynamic strength at high stress-rate loading expressed by Eq. (8) recovers thestrength at the quasi-static state as the loading rate approaches zero. Thus, this model is capableof describing the ceramic strength over a wide range of loading rates from quasi-static to dynamic.

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4. Discussions

Ten or more specimens were tested for each material composition except the pure 8YSZbaseline material since a limited number of specimens were available. The experimental dynamicbiaxial flexural strengths with corresponding loading rates are listed in Table 2.The dynamic experimental data are fitted to Eq. (8) using an overall nonlinear least-squares

method. The threshold stress for each material composition is chosen as the corresponding meanstrength at the quasi-static state, see Table 1. Fig. 3(a) gives a typical fitting curve of 20wt%3YSZ-doped 8YSZ for a stress-rate range from 0 to 1600GPa/s. The curve starts from the meanquasi-static strength, which is due to the assumption on which the model was based. The dynamicexperimental data with various stress-rate scatter around the fitting curve, and the trend that thestrength increases as the stress-rate increases can be recognized from both experimental data andfitting curve. This curve provides a description of the mean strength variation with stress-rate.All of the five fitting curves corresponding to different chemical compositions studied in this

research are shown in Fig. 3(b). In comparison with the scatter associated with the experimental

Table 2

Experimental data of dynamic biaxial flexural strength (MPa) / stress-rate (GPa/s) and corresponding model

parameters by curve fitting

Material 8YSZ 1mol% Al2O3 3mol% Al2O3 20wt% 3YSZ 30wt% 3YSZ

composition -doped 8YSZ -doped 8YSZ -doped 8YSZ -doped 8YSZ

1 534.3/495.5 372.6/828.7 466.2/678.4 539.7/1308.1 424.4/1238.9

2 518.1/1446.9 364.1/831.8 584.0/908.2 369.2/578.4 447.6/1010.6

3 360.9/404.2 501.5/524.5 519.9/1016.9 481.6/1312.1 447.5/404.8

4 531.3/500.4 425.8/722.0 516.7/1088.0 471.9/1238.2 339.9/435.9

5 217.5/494.8 398.9/358.3 553.6/490.2 492.0/933.4 490.3/578.3

6 374.3/402.6 536.4/1450.3 440.1/829.8 570.5/1382.6

7 719.3/1043.5 378.8/405.6 456.8/1092.9 537.9/684.5

8 592.3/836.8 548.5/510.2 489.1/813.8 407.1/792.5

9 459.0/1207.0 238.5/500.5 393.4/1016.3 532.9/843.7

10 529.4/931.6 471.6/980.7 443.2/515.8 378.3/365.5

11 516.4/1068.1

12 376.8/595.0

13 581.8/678.9

14 503.7/1350.2

15 430.2/1093.6

16 367.3/836.3

17 465.1/1059.4

18 369.2/659.1

Reference stress-rate

_sr 2574 2589 2530 3121 3709

Parameter

b 0.490 0.414 0.459 0.454 0.470

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0 400 800 1200 1600Stress rate (GPa/s)

0

200

400

600B

iaxi

al fl

exur

al s

tren

gth

(MPa

)

0 400 800 1200 1600Stress rate (GPa/s)

0

200

400

600

Bia

xial

flex

ural

str

engt

h (M

Pa)

8YSZ1 mol% Al2O3-doped 8YSZ3 mol% Al2O3-doped 8YSZ20 wt% 3YSZ-doped 8YSZ30 wt% 3YSZ-doped 8YSZ

(a) (b)

Fig. 3. Dynamic experimental results. (a) Experimental data of 20wt% 3YSZ-doped 8YSZ with fitted curves, (b) fitted

curves of 5 different compositions.


results, the variations within these curves are relatively small, which indicates that the effects ofthe dopants on the dynamic strengths of 8YSZ material are not significant. This is consistent withthe quasi-static strength results on these compositions, as shown in Table 1.Microstructure studies of these material compositions under a scanning electronic microscope

(SEM) show that the variation of grain size of these compositions is insignificant [25]. Forexample, the average grain size of pure 8YSZ is 2:1mm. The average grain size of 1-mol% Al2O3

doped 8YSZ is 3:0mm, slightly larger than that of 8YSZ, while the average grain size of 3-mol%Al2O3 doped 8YSZ is 1:8mm, slightly smaller than that of 8YSZ. In general, the grain size of8YSZ with Al2O3 dopants amount up to 3-mol% changes insufficiently. Similarly, the averagegrain size of 3YSZ doped 8YSZ keeps in a same level with 3YSZ amount up to 30-wt%. Sincethere is a strong relationship between the grain size and strength of ceramic materials [26], nosignificant effects of those dopants on the material compositions are then expected. Thiscorroborates our experimental results from another point of view. The results presented inFig. 3(a) and (b) thus indicate that the limited amounts of experimental data do not affect thenature of our conclusion to the dynamic strength of the material compositions.

5. Conclusions

A new material model for dynamic strength under constant high stress-rate loading for brittlematerials has been developed based on the concept of cumulative damage. The model describesstrength for brittle materials under both quasi-static and dynamic loading conditions in a unifiedmanner. The parameters in this model were experimentally identified using an overallleast-squares curve fitting technique for all the data at different loading stress-rates. With the

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employment of this model, the required number of identical specimens and experiments weresignificantly reduced without affecting the nature of the results.Ceramic material 8YSZ and four of its doped compositions with alumina and 3YSZ were tested

using a standard piston-on-3-ball method under quasi-static loading and a newly developeddynamic piston-on-3-ball method under high stress-rate loading, for the purpose to investigate theeffects of those dopants on the biaxial flexural strength of 8YSZ baseline material. Theexperimental results show that the effects of those dopants are insignificant.

Acknowledgements

This research was supported by NASA through a Grant (NAG 8-1469) to The University ofArizona.

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