cohesive zone modeling of grain boundary microcracking...

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Computational Materials Science 43 (2008) 440–449

Cohesive zone modeling of grain boundary microcracking inducedby thermal anisotropy in titanium diboride ceramics

M. Pezzotta a, Z.L. Zhang a,*, M. Jensen b, T. Grande b, M.-A. Einarsrud b

a Department of Structural Engineering, Norwegian University of Science and Technology (NTNU), Trondheim, Norwayb Department of Material Technology, Norwegian University of Science and Technology (NTNU), Trondheim, Norway

Received 26 February 2007; received in revised form 16 October 2007; accepted 10 December 2007Available online 11 February 2008

Abstract

This paper addresses the residual stresses and their effect on microcracking in polycrystalline ceramic materials. Residual stresses atmicrostructural level in titanium diboride ceramics, as a result of thermal expansion anisotropy, were analyzed by finite element methodusing Clarke’s model. Damage mechanics based cohesive zone model was applied to study grain boundary microcracking, propagationand arrest. Quantitative relations between temperature variation, grain boundary energy, grain size, final microcrack length as well asmicrocracking temperature are established.� 2007 Elsevier B.V. All rights reserved.

PACS: 62.20.Mt; 81.05.Je; 65.40.De; 61.72.Mm

Keywords: Cohesive zone modeling; Thermal anisotropy; Residual stresses; Titanium diboride; Microcracking

1. Introduction

Grain level thermal expansion anisotropy in polycrystal-line ceramics will induce residual stresses during the cool-ing from sintering temperature. The residual stresses exerta strong influence on the mechanical integrity of materials.Depending on the grain size and grain boundary energy,microcracking occurs and the strength and fracture tough-ness will be reduced. Several numerical and experimentalstudies on predicting and measuring the residual stressesin polycrystalline ceramics have been reported in the liter-ature [1–4]. Most of the papers focused on polycrystallinealumina, zirconia/alumina composites, silicon carbide-rein-forced alumina, silicon carbide/silicon nitride composites[5]. The residual stresses in another technological impor-tant material – titanium diboride (TiB2), are less under-stood. TiB2 is a ceramic material with high strength,hardness and melting point, and good wear resistance. It

0927-0256/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.commatsci.2007.12.011

* Corresponding author. Tel.: +47 73592530.E-mail address: [email protected] (Z.L. Zhang).

is an attractive candidate for the cathode material in thealuminum electrolysis process, because of its good wettingability with liquid aluminum and good electrical conductiv-ity and chemical inertness at high temperature [6–8]. How-ever, microcracking as a result of residual stresses andmechanical property degradation due to liquid aluminumpenetration at grain boundary, are some of the potentiallimiting factors for wide industrial application as cathodematerial [9]. This paper studies the residual stresses in sin-gle phase TiB2 due to thermal expansion anisotropy andtheir effect on microcracking initiation and arrest.

In general there are two methods to analyze the residualstresses in ceramic materials: one is micromechanical modelbased deterministic method [10–15] and another is realmicrostructure based statistical method [3,4,16–18]. Theformer method will be used in this study. A representativefinite element model based on the micromechanical modelby Clarke [10] is constructed and a damage mechanics basedcohesive zone model approach is applied to simulate thesubsequent microcracking. Clarke’s model consists of fourgrains surrounded by the thermally isotropic material

mailto:[email protected]

matrix

x

y

O

4 grains ensemble

Fig. 1. Clarke’s model: four thermal anisotropic grains surrounded by athermal isotropic matrix material.

M. Pezzotta et al. / Computational Materials Science 43 (2008) 440–449 441

(matrix). Fig. 1 schematically shows the model. Clarke’smodel has recently been applied to study the strengtheningeffect of nano-particles in ceramic composites [13]. Fig. 2shows the typical microstructure of hot-pressed TiB2 pro-duced by the authors and the circles indicate the four grainconjunctions. In the analysis, both the grains and the matrixare assumed to possess elastic isotropy. The matrix repre-sents the global average behavior of the material with amean thermal expansion coefficient, while the four grainsare thermally anisotropic. The chosen model system is bothstress and defect free at the beginning of the cooling process.Because each grain attempts to strain differently than thematrix material and the neighboring grains, a residual stressfield will be generated when the sample cools down. Themodel has been chosen such that a tensile stress field occursin the center of the model. Depending on the grain bound-ary energy, grain boundary microcracking initiates in thecenter when the residual stress field is sufficiently large.

2. Modeling procedures and material data

2.1. Material parameters

In the present analysis, the data of the elastic and ther-mal properties of TiB2 from Munro [7] are used. Both thefour grains and surrounding matrix materials are assumed

Fig. 2. Microstructure of TiB2 hot pressed without additives. Circles inthe figure show the possible four grain junctions.

to possess the same elastic isotropy. The Poisson ratio (t) isset to the value of 0.108, independent of the temperature.The elastic modulus (E) is a decreasing function of temper-ature, Fig. 3.

The in-plane thermal expansion coefficient is in generaldependent on the crystallographic orientation of the grain.In this study, both the maximum and minimum values ofthe thermal expansion coefficients (a) have been chosenfor the grains and the mean value has been assigned tothe matrix material. Fig. 4 displays the thermal expansioncoefficients used in the analyses.

The grain size strongly influences the microcrackingbehavior. Three grain sizes have been considered: 10, 20and 50 lm. Unless otherwise mentioned, a typical coolingtemperature range (DT) from T = 1500 �C to the roomtemperature T = 20 �C has been applied.

2.2. Finite element model

The micromechanical model by Clarke has been mod-eled by the finite element method. As depicted in Fig. 1,the plane stress Clarke’s model is made of four squaregrains, surrounded by the matrix material which is boththermally and elastically isotropic. The matrix representsthe average material behavior. The four grains arethermally anisotropic. A texture free microstructure isconsidered and a worst case is examined: the direction of±45� from the x-axis represents the maximum/minimumexpansion directions.

Only a tensile mode has been considered in this study.Because of both the geometry and material symmetrychosen for the model, one quarter of the model has beenmodeled. In particular, since the microcracking is assumedto occur along the grain boundary, the attention will be puton the stress and strain distributions along the x-axis.Fig. 5 shows the finite element model used for the analyses.A square region in the middle of the model is specified as

Fig. 3. Temperature dependent Young’s modulus for TiB2 [7].

Fig. 5. Finite element model for the residual stress and microcrackinganalyses: entire model with the grain of 50 lm indicated by a black thickline.

Fig. 4. Thermal expansion coefficients used in the study [7].

traction

separation

t0

Gc, cohesive energy

< : linear elastic deformation

< < c: damage

= c=fracture; crack initiation

δ

δ

δ σ

δ

δc

δ

δ

δ0

δ0

δ0

Fig. 6. Schematic plot of the cohesive zone elements and traction–separation law used in the analysis.

442 M. Pezzotta et al. / Computational Materials Science 43 (2008) 440–449

the grain (Fig. 5). The region consists of only square ele-ments. The same model but with different grain sizes hasbeen used. The model contains about 18,000 elementsand a similar number of nodes. The used mesh modelproved, after a sensitivity analysis, to be suitable todescribe the system. Refining the mesh at the grain-matrixboundary did not change significantly the results concern-ing stress distribution and crack size. The only differencebetween the grains and matrix materials concerns the ther-mal properties. Two types of analyses have been carriedout. In the first analysis no cracking is permitted alongthe grain boundaries. The purpose of this type of analysisis to investigate the residual stress fields. In the second typeof analysis, a cohesive zone model is applied which allows

the prediction of microcracking. ABAQUS has been usedfor both analyses.

2.3. Cohesive zone model

The microcracking along the grain boundary is modeledby the cohesive zone model in ABAQUS. Cohesive ele-ments were used by Nguyen et al. to describe the crackingbehavior in solid oxide fuel cell’s materials and a compar-ison between discrete and continuum modeling capacitywas performed in [15]. The cohesive model is characterizedby a traction–separation law [19]. Fig. 6 schematicallyshows the cohesive zone element and the traction–separa-tion relation. The law characterizing these elementsdescribes a damage initiation criterion and the damageevolution. The cohesive element behavior is assumed tobe linear-elastic initially until a critical traction/displace-ment has been reached. Beyond the critical stress, an irre-versible damage process will begin. The damage of thematerial is described by the rate of degradation of thematerial stiffness after the above mentioned critical trac-tion/displacement has been reached. The basic parametersnecessary to describe the traction–separation law are twoamong the critical traction (t0), the critical separation (dc)and the cohesive, or fracture energy (Gc) [19]. When thecohesive zone model is used to model the grain boundaryseparation, the cohesive energy is equivalent to the grainboundary energy density. It should be noted that symmetryconditions have been utilized in the finite element analysis


and the reported Gc in the following is half of the totalgrain boundary separation energy (density). This studyfocuses on brittle fracture in ceramic materials and thedamage evolution model used here is a simplification ofthe model presented by Tvergaard and Hutchinson [19].In the cohesive zone modeling, the tractions will decreaseonce a critical traction value is reached, until they are zero.When the traction reaches zero, material separation, i.e.,microcracking, will occur. Because the residual stress fieldis a self-balanced stress field, the residual stress field willbe re-distributed due to the initiation and propagation ofa microcrack. As mentioned in Tvergaard and Hutchinson[19], the exact form of the traction–separation law may notbe critical for microcracking prediction. Microcracking isdetermined by the cohesive energy and critical traction val-ues. The critical traction is usually not known and in thepresent analysis, a critical traction of 1000 MPa has beenused. The effect of the critical traction’s value on the simu-lation results will be discussed later. The elastic stiffness ofthe cohesive zone has been assumed to the same as the bulkmaterial 565 GPa, independent of the temperature. A verysmall thickness (10�8 mm) has been used for the cohesiveelements.

Fig. 7. (a) Opening stresses at different temperatures for grain size 50 lm,(b) normalized opening stress distributions for different temperatures butsame grain size 50 lm and (c) normalized opening stress distributions forthree grain sizes at room temperature.

3. Results

3.1. Residual stress field in TiB2

An analytical study on the residual stress distributionhas been carried out by Clarke [10]. In the following, thenumerical results from the current analysis are presentedand compared with the ones by Clarke. Fig. 7a shows theresidual stress distribution of the opening stress along thex-axis at three different temperatures (1415 �C, 734 �Cand 20 �C) for the case with grain size 50 lm. As it canbe expected, the maximum tensile stress occurs in the centerof the model where it is prone to microcracking. The stressdecreases with the x, i.e. while moving from the center ofthe reference system toward the positive x direction, andbecomes compressive at a distance about 0.55 times ofthe grain size. A compressive stress peak occurs at theinterface with the matrix material. The compressive stressin the matrix material will decrease and approach zero atthe outer free boundary. It is interesting to observe thatthe maximum opening stress in the middle of the modelfor the TiB2 material could be as large as 3000 MPa. Themaximum compressive stress is about half of the tensilepeak stress. It should be noted that microcracking willnot only depend on the peak tensile stress value. Mostimportantly, it depends on the total energy available inthe material system.

The opening stresses in Fig. 7a have been normalized bya reference stress r* in Fig. 7b. The reference stress r* isdefined as [10,12,20]:

r� ¼ �EðT Þe22ðDT Þ ¼ EðT ÞDaDT ; ð1Þ

where E is the Young’s modulus; Da is amax�amin and a arethe thermal expansion coefficients; DT corresponds toTinit�Troom = (Tinit�20) �C. The Young’s modulus values

Fig. 8. Opening stress r22 distribution along the x-axis in the grain fordifferent steps of the cooling process, l = 50 lm, (a) Gc = 0.33 N/mm and(b) Gc = 0.025 N/mm.


used in the normalization are 514, 542 and 565 GPa for tem-peratures 1415 �C, 734 �C, and 20 �C, respectively. It can beseen that the opening stresses at different temperatures canbe well normalized by using the reference stress (Fig. 7b).The reference stress is related to the use of the Eshelby tech-nique in the analysis of the deformation of a grain in a poly-crystalline structure subjected to thermal stresses.

Attempts have also been made to normalize the openingstresses for different grain sizes at room temperature(Fig. 7c). In Fig. 7c the opening stress fields at T = 20 �Cfor the three grain sizes have been normalized by a refer-ence stress value of r* = 365.9 MPa, calculated with thecorresponding values of Young’s modulus and strain.

It can be seen that the residual stresses distribution dur-ing the cooling process can be normalized by temperatureand grain size. An important conclusion which can bedrawn from Fig. 7c is that the residual stress field is depen-dent on the grain geometry but independent of the grainsize. This is because that in the residual stress analysis nolength scale is involved. In the following it will be shownthat the grain size plays a crucial role in microcracking.

3.2. Microcracking

In the second type of analysis, the cohesive zone modeldescribed in section II, has been applied to study the resid-ual-stress-driven microcracking and its intrinsic relation-ships with the microstructure parameters. Three grainsizes have been analyzed and for each grain size severalgrain boundary energy values have been used to studythe microcrack length. The analysis is quasi-static and nodynamic effect related to the microcrack propagation, hasbeen considered.

3.2.1. Re-distribution of the residual stresses

Fig. 8 shows the opening stress distributions for the casewith grain size l = 50 lm and grain boundary separationenergy density Gc = 0.33 and 0.025 N/mm, respectively.In the following, the expressions ‘‘grain boundary energy”or ‘‘cohesive energy” refer to grain boundary separationenergy density. It can be found from Fig. 8a that for thecase with Gc = 0.33 N/mm, the critical traction has beenreached when the temperature cooled down to about1030 �C. Further cooling induces damage at the grainboundary and irreversible deformation will occur. Themaximum opening stress is controlled by the critical trac-tion value specified. In order to balance the damageoccurred in the tensile part, the residual stress will be re-distributed and the magnitude of the compressive stresswill be increased.

For the case with a lower grain boundary energyGc = 0.025 N/mm, Fig. 8b, significant crack growth willoccur at room temperature. The stress will be redistributedand both the peak tensile stress and zero stress point willmove along the positive x direction.

Fig. 9a and b shows the stress distribution contours forthe component r22 in the grain area at different stages of

the cooling process. Fig. 10 shows the stress temperaturehistory in the center of the model (x = 0,y = 0) for thecohesive energy Gc = 0.025 N/mm. The stress is first line-arly increasing with temperature and then, after reachingthe specified critical traction for the damage criterion, itdecreases. For the case analyzed, it seems that at about1030 �C the maximum traction’s value has been reached.After that point, the stress decreases with the temperature.The shape resembles the plot of the used damage criterionwhich represents the traction–separation law. At about181 �C the opening stress decreases to zero and accordingto the definition, material separation and microcrackingoccurs at this material point. It should be noted that micro-cracking depends on grain size and grain boundary energy.

3.2.2. Microcracking and microcrack length

Experimental studies [21,22] have shown that micro-cracking due to residual stress will occur once the grainsize has reached a critical value. With the damage based

Fig. 9. Stress distribution contours of r22 in the grain, for grain size 50 lmand Gc = 0.025 N/mm, (a) at T = 882 �C; (b) at T = 20 �C.

Fig. 10. Opening stress r22 in x = 0 during the cooling process, grain size50 lm, Gc = 0.025 N/mm.


cohesive zone model approach, the critical grain boundaryenergy for microcracking, or the critical grain size at whichmicrocracking occurs can be predicted. For the model withgrain size 50 lm, Fig. 11 shows the opening stress distribu-tion along the x-axis at room temperature for different val-ues of grain boundary energy. At room temperature, thedamage due to residual stress for the case with Gc = 1 N/mm is not large enough to cause microcracking. Micro-cracking will just occur for the case with grain boundaryenergy about 0.03 N/mm and significant amount of micro-crack propagation will occur for the case with grain bound-ary energy 0.02 N/mm.

It can be clearly seen that for the same grain size thelower the grain boundary energy, the easier the microcrackwill initiate and the longer the final microcrack length. It isinteresting to note that even the final distribution of theresidual stress is disturbed by the critical grain boundaryenergy; the peak value of the compressive stress is ratherinsensitive to the grain boundary energy.

Fig. 12 plots the predicted final microcrack length vs.the critical grain boundary energy for the three grain sizes.It is interesting to note that the microcrack length has adecreasing but linear relation with the grain boundaryenergy and the grain size has a strong influence on the val-ues of the critical grain boundary energy for microcrackingGc. For a material with larger grain size, the crack willoccur easily (i.e. for smaller values of Gc) and higher grainboundary energy is needed to avoid microcracking.

One important aspect in manufacturing ceramics is toavoid microcracking. The above analysis indicates thatresidual stress induced microcracking can be avoided byeither reducing grain size or improving grain boundaryenergy. For the case with grain size 50 lm, a grain bound-ary energy value larger than about 0.03 N/mm is necessaryto avoid microcracking in TiB2. This grain boundaryenergy is equal to the reported maximum fracture tough-ness for TiB2, 6.2 MPa

ffiffiffiffi

mp

[7].

Fig. 11. Final stress distribution along the x-axis in the grain, atT = 20 �C, for different values of the cohesive energy (N/mm), l = 50 lm.

Fig. 12. Microcrack length for different grain size microstructures anddifferent cohesive energies.


Fig. 13 shows the calculated critical grain boundaryenergy vs. grain size. For a given grain size, if the grainboundary energy is larger than the one shown in the figure,no microcracking will occur. On the other hand, for a fixedgrain boundary energy value, the critical grain size foravoiding microcracking can be calculated.The critical grainboundary energy shown in Fig. 13 has been fitted by thefollowing equation:

Gc ¼ 0:0006lþ 0:00014 ð2Þ

where Gc has the unit N/mm and grain size l has a unit oflm.

These data could be important for material design:known the grain boundary energy of the microstructure,the critical grain size can be estimated. The link amongthe critical grain size, the temperature variation and othermaterial and process parameters may be useful. Exposureto liquid aluminum at high temperatures, for example,

Fig. 13. Grain size vs. critical grain boundary energy for avoidingcracking.

can cause the degradation of the boundary energy. Thistheme will be addressed in a companion paper.The presentrelation has been compared with the models available inthe literature [22–26] in Fig. 14. In general a linear relationbetween the critical grain size (or critical particle size) andthe surface fracture energy is expected:

lc ¼ mcf

E � ðDa � DT Þ2ð3Þ

where m is a coefficient, cf the surface fracture energy whichcorresponds to Gc in this study. In Fig. 14, the followingvalues for TiB2 have been used: E = E(T = 20 �C)=565 G-Pa, Da = amax�amin = 2.8 � 10�6 K and DT = Tfin�Tinit =�1480 �C. The literature models are based on simplifiedanalytical solutions. A comparison of the literature modelswith experimental results, for various materials, has beencarried out by Rice and Pohanka in [22] together with adiscussion of the way to estimate the parameters. For a gi-ven grain boundary energy, the first model listed by Riceand Pohanka yields the lowest critical grain size: the coef-ficient m is about 10.7. The one by Cleveland and Bradt[23] lies in the middle as well as the present study, with aproportionality coefficient of about 15. The largest valueof the predicted critical grain size is given by the third mod-el listed in [22] with m � 20. The exact value of the coeffi-cient m depends on the grain structure of the materialand the model assumptions.

The first model in [22] refers to Davidge and Green’swork [24]. This 3D model describes the cracking in com-posites made of a glassy matrix with embedded ceramicspherical particles. Four glasses differing by the thermalexpansion coefficients have been experimentally studied.In the absence of applied stresses, when the particle has athermal expansion coefficient higher than that of thematrix, the presence of flaws near a particle and the fulfill-ment of an energy criterion are necessary for cracking tooccur. The first model represents the critical particle size

Fig. 14. Critical grain (or particle) size prediction by different models.

Fig. 15. Crack-size evolution during the change of temperature fordifferent cohesive energy values (in N/mm), grain size l = 50 lm.

Fig. 16. Opening stress r22 distribution at T = 20 �C, for two differentinitial temperatures of the cooling process, l = 50 lm, Gc = 0.025 N/mm.


derived from energy balance. The second model listed in[22] concerns JFP Clarke’s study [25] which focuses onintergranular cracking as a result of the growth of an exist-ing grain boundary pore for two adjacent hexagonalgrains. The energy for crack growth along the grain bound-ary comes from two sources, strain relaxation around agrowing crack and relief of residual strains in the twoneighboring grains. The Griffith energy balance methodtogether with two conditions, one about the initial cracksize and the other about the crack growth length, has beenapplied to derive a relation among grain size and otherparameters. The third model in [22] is from the analysisof the crack propagation at interface between two dissimi-lar materials by Mulville et al. [26]. A fracture criterion interms of strain energy release rate has been developed todetermine the fracture resistance of interfacial cracks.Experiments on specimens of epoxy bonded to aluminumwere also done to compare with the analytical results.Cleveland and Bradt’s [23] relation is derived from anenergy criterion for the case of pseudobrookite structure,with high level of thermal expansion anisotropy. Assumingdodecahedral grains, an expression for the total energy isgiven. This is used to obtain, from an approximated energybalance, the critical grain size as a function of the abovementioned parameters.

It can be observed that all the considered models con-cern slightly different situation than the one in the presentstudy but still are indicative for the understanding of therole of the involved factors in the relation ‘critical grainsize-critical grain boundary energy’.

3.2.3. Microcracking temperature

Microcracking usually starts at a higher temperaturethan room temperature. Microcracking temperature is animportant process parameter and has been experimentallystudied in the literature. Several empirical works have beencarried out [27,28]. With the damage-mechanics-basedcohesive model approach, the microcracking temperaturefor a given grain size and grain boundary energy can bepredicted.

Fig. 15 displays the evolution of microcracking withtemperature for different values of grain boundary energy.The crack size is remaining zero until a critical temperatureat which the crack starts. The crack will increase its sizewhile the temperature further decreases. Obviously micro-cracking starts at a higher temperature for the cases withlower values of grain boundary energy.

3.2.4. Effect of cooling temperature range

The residual stress field and consequently the micro-cracking behavior are proportional to the temperature var-iation the material is subjected to. A lower startingtemperature of the cooling process will result in lowerstress, lower microcracking temperature and shorter finalmicrocrack length.

Fig. 16 shows the final stress distributions for two caseswith starting temperatures 1000 �C and 1500 �C, respec-

tively. For the case with starting temperature Tinit =1000 �C, at room temperature the damage process has justbeen initiated and the residual stress field is not sufficient tocause microcracking. However, for the case with startingtemperature Tinit = 1500 �C, a microcrack has not only ini-tiated in the center but also propagated for 7.5 lm uponreaching the room temperature.

4. Discussion and concluding remarks

Residual stress due to thermal anisotropy plays animportant role in mechanical properties of polycrystallineceramics. The residual stress field and associated micro-cracking in TiB2 ceramics has been studied for the first timeusing finite element method. Three grain sizes, 10, 20 and50 lm, have been considered. The Clarke model has been

Fig. 17. Comparison of the predicted critical grain boundary toughnesswith the statistical macroscopic toughness distribution compiled byMunro.


used to represent microstructure features and cohesive zonemodel is applied to model the material microcracking pro-cess. It should be noted that two dimensional plane stressmodel has been used and no dynamic effect was consideredin modeling the microcracking and propagation. It hasbeen demonstrated that microstructural level residual stressdistribution is solely determined by the microstructure andsintering process parameters. The calculation of residualstress does not involve a length scale and the residual stressdistribution can be normalized by both the grain size andcooling temperature. The calculated peak tensile residualstress can be as high as 3000 MPa for TiB2 while the com-pressive peak stress is approximately half of the tensilepeak. On the other hand, microcracking due to thermalresidual stress is strongly dependent on the grain size. Eventhough the distribution is the same, material with largegrain size has more available elastic energy and is proneto microcracking.

With the damage-mechanics-based cohesive zonemodel, a quantitative relation between the grain bound-ary energy, grain size and microcrack length has beenestablished. The critical grain boundary energy has beenshown to be linearly proportional to the grain size – lar-ger grain size needs higher grain boundary energy toavoid residual-stress-induced microcracking. True criticalgrain boundary energy value for a material is difficultto measure and almost no data are available for TiB2.The critical grain boundary energy value predicted by thismodel can be related to the macroscopic fracture tough-ness. However, the macroscopic fracture of TiB2 dependson many factors, residual stress, sintering additives, frac-ture modes, et al. It is out of the scope of the presentstudy to perform well defined experimental study to verifythe predictions. Nevertheless the predicted critical grainboundary energy agrees with the experimental observa-tions of macroscopic fracture reported in [7]: residualstresses and consequent grain boundary cracking contrib-ute partly to the weakening and macroscopic cracking ofthe material. The following comparison is concerned withthe understanding of the ‘‘grain-boundary-cracking” con-tribution to the macroscopic fracture toughness. The cal-culated minimum grain boundary energy for avoidingmicrocracking are 0.0302, 0.01225 and 0.0061 N/mm forthe cases with grain sizes 50, 20 and 10 lm, respectively.For TiB2 material, the maximum measured macroscopicfracture toughness has been reported to be 6.2 MPa

ffiffiffiffi

mp

[7] which corresponds to Gc = 0.03 N/mm. The predictedcritical grain boundary energy has been converted to crit-ical stress intensity factor and is compared with the statis-tical relation between grain size and macroscopic fracturetoughness by Munro [7] in Fig. 17. If the maximum frac-ture toughness is assumed to be 6.2 MPa

ffiffiffiffi

mp

, the presentanalysis indicates that even no microcracking occurs, littleor no additional external load can be carried out for thematerial with grain size 50 lm. According to the analysis,maximum fracture toughness can only occur at smallergrain size. Munro observed the grain size which results

in the maximum fracture toughness is about 8–10 lm.It should be noted that a large scatter exists in the datacompiled by Munro since they refer to materials with dif-ferent chemical impurity content and subjected to differ-ent test conditions, and no fracture modes (mechanisms)were distinguished in the statistics. It can also beobserved that two trends are present in the statisticalfracture toughness – grain size relation. When the grainsize is less than about 8 lm the fracture toughnessincreases with the increase of grain size, while the trendbecomes opposite when the grain size is larger than about8 lm.

The residual stress analysis shows that about 55% of thegrain size close to the four grain junction center is undertensile stress. The final microcrack length is dependent onthe grain size and grain boundary energy. The analysis with10 lm grain size shows that if the grain boundary energy issmall enough, a microcrack length approaching half of itsgrain size (5 lm) can be reached at room temperature.

In this study, the interface has no thickness and largecohesive zone interface stiffness (5.6 � 1013 N/mm3) is pre-ferred. However, large stiffness may induce numericalproblems such as spurious oscillations of the tractions inan element [29]. The shape of the traction–separation curvehas a very weak influence on the fracture/damage process.The cohesive energy and the critical traction are the funda-mental parameters describing the microcracking process.Increasing cohesive energy will delay the damage processfor a given grain size. The exact value of the critical stressis not known. A value of 1000 MPa for the critical tractionhas been chosen for the cohesive traction–separation law.Other values have also been used to study the microcrack-ing behavior. A lower value of the critical stress for separa-tion results in a slightly shorter crack length. Further workshould be spent on designing experimental methods todetermine the critical stress.


Acknowledgements

This study is one part of the ongoing research Project(2003–2008) – ThermoTech (Application of Thermodynam-

ics to materials technology) at the Norwegian University ofScience and Technology (NTNU), Trondheim.

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cohesive zone modeling of grain boundary microcracking...

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