Computational Materials Science 80 (2013) 43–50

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

A two-scale weakest link model based on a micromechanical approach

0927-0256/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.commatsci.2013.04.018

⇑ Corresponding author. Tel.: +49 0 721 608 48852; fax: +49 0 721 608 44187.E-mail addresses: thomas.boehlke@kit.edu (T. Böhlke), yamen.othmani@kit.edu

(Y. Othmani).

Thomas Böhlke ⇑, Yamen OthmaniChair for Continuum Mechanics, Institute of Engineering Mechanics, Karlsruhe Institute of Technology (KIT), Kaiserstrabe 10, 76131 Karlsruhe, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 7 January 2013Received in revised form 15 April 2013Accepted 15 April 2013Available online 18 May 2013

Keywords:Weakest link approachHashin–Shtrikman boundsSecond-order estimatesFailure probability

For brittle materials in engineering, it is well known that the defect structure governs the macroscopicstrength. However, there is a class of materials [34] where the failure events occur on the micro-structurelevel with characteristic length smaller than the characteristic length of the defect. In this case, a meanfield approach in which the stress fluctuations are estimated by a second-order moment approach incombination with a quantification of failure probabilities may give useful insight to the level of failure.Weibull’s type failure probabilities are introduced in this work without reference to a defectmicrostructure.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Silicon nitride ceramics are prime structural materials for sev-eral challenging assignments and applications such as automobileindustry, bearings and cutting tools, among others. This is due totheir outstanding high stiffness, high-temperature strength and,especially, their high fracture toughness. However, the structuraluse of this material is restrained because of the limitation of itsreliability and the occurrence of damage phenomenon under se-vere working conditions. To mention only a few examples of de-fects: Failure of rollers, made of silicon nitride, was reportedwhen materials with a very high-deformation resistance wererolled [33]. Surface ring cracks are found on the surface of siliconnitride’s balls, used for rolling contact. These kinds of defects de-grade the rolling contact’s performance, considerably, through sev-eral failure processes such as crack initiation, crack propagationand spall sheet separation [48]. Another example of degradationprocess is the creep rupture that often results in silicon nitride athigh temperature from subcritical crack growth mechanism [38].Silicon nitride, like the majority of ceramics, shows brittle fracturephenomenon that originates from the stress concentration inducedby defects [12]. Jayatilaka and Trustrum [21] showed that the dis-tribution of strength data of a brittle materials is controlled by thedistribution of flaws. The fracture strength of brittle materials de-pends substantially on the properties of the critical defect such assize, shape, orientation and the strain energy release rate [34]. Theminimum size of the critical defect depends in return on the mate-

rial microstructure including the kind of defect itself and the localstress field [11], especially the size of the grains as a microstruc-tural characteristic that influences the overall strength of ceramics[34]. Many approaches, including fracture mechanics, could beadopted to predict the failure of ceramics. Nevertheless, thestrengths of such materials are statistical in nature and the damageprogression is stochastic. Therefore, the application of statisticalmethods to evaluate the failure of ceramics is needed. The applica-tion of probabilistic approach to fracture phenomena is nowadaysan appropriate and established method for the statistical analysisof the strength variability of ceramic materials [29].

Pioneering contributions to the probabilistic failure approachwere made for brittle materials. Weibull [49] was the first whoproposed a statistical theory for brittle fracture. His analysis isbased on the weakest link model which has long been used to de-scribe the strength of brittle materials. Slatcher and Evandt [44],Landes and Shatfer [31] and Bruckner and Munz [7], among others,applied the weakest link model to fracture toughness problems.Jeulin [22] used a multiscale weakest link model to study the frac-ture statistics in materials with multiscale heterogeneous textures.Lamon [28] examined and compared three weakest link based ap-proaches, which are the Weibull approach [49], the Batdorf ap-proach [2] and a multiaxial elemental strength model [30]. Theauthor studied the limitation of these theories on the basis of cera-mic reliability prediction, discussed their legitimacy for treatingdelayed fracture and provided an alternative solution.

Despite some criticisms, Weibull’s model is nowadays the basisof mechanical dimensioning and design of ceramic components.Jayatilaka and Trustrum [21] developed a general expression forthe failure probability of brittle material by using the propertiesof flaw size distribution and the stress necessary to fracture an

44 T. Böhlke, Y. Othmani / Computational Materials Science 80 (2013) 43–50

inclined crack. They proved that the flaw population in a brittlematerial, with a monotonically decreasing density of flaw sizes, fol-lows a Weibull distribution of strength data. This has been ex-tended by Danzer [11], Danzer et al. [14] and Danzer et al. [13]to an inhomogeneous flaw population. Cannillo and Carter [9] useda microstructural-based finite element code, in conjunction withWeibull’s theory, to compute the local stresses and microcracksformation in brittle materials. Danzer et al. [15] gave an overviewon cases where Weibull’s model deviates from the fracture statis-tics of ceramics. As for the application of Weibull’s approach to sil-icon nitride, there exist many contributions in the literature suchas the work by Shinozuka and Lenoe [43]. There, a probabilisticmodel, combined with the finite element method, was used todetermine the spatial distribution of silicon nitride’s strength.Bergman [4] used the weakest link method to determine the Wei-bull modulus for silicon nitride. However, for all of these models,probabilistic analysis has been applied to the macroscale level,the microstructural aspects of silicon nitride have not been takeninto consideration.

The probabilistic failure prediction can be enhanced by investi-gating the local fields, as required by the theories of damage nucle-ation [5]. In the context of field statistics, it is well known that themean value or the first-order moment of the local stress and strainis a limited statistical information, since it does not consider theheterogeneity of the local fields [36]. Therefore, to have richer sta-tistical data such as the covariance, the consideration of the secondstatistical moments permits a better knowledge of the estimates oflocal fields. Kreher [24], Kreher and Molinari [25], Buryachenkoand Kreher [8] and Bobeth and Diener [6], among others, usedsecond-order moments of the local fields to determine the stressfluctuation and accordingly the macroscopic toughness andstrength in linear random composites and polycrystals.

In the present work, a micromechanically based weakest link ap-proach is developed in conjunction with a stochastic analysis of thelocal stress fields in sintered silicon nitride. The main contributionof this work is the employment of second-order moments of stressin a probabilistic and mechanical fracture approach. The extendedweakest link approach accounts for volume defects together withdamage occurring across the interface between the phases. Theparameters used in the present study are roughly estimated param-eters that are intended to mimic qualitatively the behavior of realmaterials. We consider a specific type of micro-structures withlength scales smaller than the defects. Therefore, we do not focuson the macroscopic behavior of silicon nitride. The present modelmay be applied to its mesoscopic scale and to other categories ofceramics, such as zinc oxide. As reported in Lu et al. [34], such mate-rials are characterized by an elevated degree of porosity. In additionto the weakened mechanical properties of these ceramics, experi-ments on fracture strength showed that the size of specimens hasalmost no influence on their mean strength. The authors attributedthe latter behavior to the high density of porosity and especially tothe effects of pore to grain-size coupling.

The paper is outlined as follows. The material modeling and thehomogenization model of silicon nitride are presented in Section 2.The description of the statistical approach is presented in Section 3.The parametric study and the numerical results are presented inSection 4. Section 5 is devoted to discussion and conclusions.

Notation 1. A direct tensor notation is preferred throughout thetext. If tensor components are used, then Latin indices are used andthe Einstein’s summation convention is applied. Vectors and 2nd-order tensors are denoted by lowercase and uppercase bold letters,e.g., a 2 V and A 2 Lin, respectively. The composition of two 2nd-order or two 4th-order tensors is formulated by AB and AB. Alinear mapping of 2nd-order tensors by a 4th-order tensor iswritten as A ¼ C½B�. The scalar product and the dyadic product are

denoted, e.g., by A�B and A � B, respectively. We define(AhB)[C] = ACB "A, B, C 2 Lin and ða� bÞ � ðCsa� btÞ ¼ ða� aÞ�ðC½b� b�Þ 8a;b 2 V . The tensor I is the identity on vectors. Theidentity on symmetric 2nd-order tensors is denoted by IS. Thebrackets h�i, e.g., hei, indicate ensemble averaging which for ergodicmedia can be identified with volume averages in the infinitevolume limit. A superimposed bar, e.g. �e, indicates that thequantity refers to the macroscale.

2. Homogenization of linear elastic properties

2.1. Elastic properties of b-silicon nitride and glassy phase

The microstructure of silicon nitride consists of two phases:rod-like b-Si3N4 prismatic grains and the intergranular glassyphase, formed by the sintering additives. The prismatic grains, withvarious lengths and aspect ratios, are randomly oriented and lo-cated. The volume fraction of the glassy phase is about 12% [53].

At the micro-level, we assume a geometrically and physicallylinear elastic behavior of silicon nitride, i.e., r ¼ C½e�with the stresstensor r, the infinitesimal strain tensor e and the stiffness tensor C.Since the grain boundaries are assumed to be perfectly bondedwithin the probabilistic fracture mechanical approach (see, e.g.,[39]), also the overall behavior of silicon nitride is linear elasticand Hooke’s law on the macro scale is given by: �r ¼ C½�e�. A super-imposed bar indicates effective quantities.

b-Si3N4 grains (S) show locally a transversely isotropic behavior.The effective elastic properties of silicon nitride is isotropic formost of the material variants. This macroscopic isotropy is due tothe statistical isotropy of the microstructure, which consists of anisotropic axes’ orientation distribution and, approximately, a bimo-dal size distribution of the b-Si3N4 grains [52]. In the present work,this local anisotropy is neglected for the sake of simplicity. Thetransversely elastic stiffness tensor CS, determined by Vogelgesanget al. [47], are used for an estimation of the quasi effective elasticbehavior of the b-Si3N4 grains. As an isotropic approximation, theprojection of the transversely anisotropic stiffness tensor on thespace of isotropic fourth order tensors is used. Such an approachimplies 3KS ¼ CS � P1 and 2GS ¼ CS � P2=5 where KS and GS repre-sent the bulk and the shear modulus, respectively. The volumetricand deviatoric isotropic projectors are determined by P1 ¼ 1=3I � Iand P2 ¼ IS � P1. As a result, the Young modulus and the Poissonratio are given by ES = 377 GPa and mS = 0.41, respectively. Theglassy phase (G) is assumed to be isotropic and linear elastic withEG = 133 GPa and mG = 0.29 [19]. Hence for both phases the effectivestiffness tensor is given by

C ¼ E1� 2m

P1 þE

1þ mP2; ð1Þ

Note that the temperature dependence of the elastic properties isnot considered in the present work.

2.2. Second-order bounds of the effective elastic properties

Due to the fact that the elementary bounds predict a ratherlarge range of admissible effective elastic properties in case of apronounced phase contrast of the local elastic behavior, higher-or-der bounds have been developed [16,50,27,51,41]. In contrast tothe first-order bounds, higher-order bounds do not only take intoaccount volume fraction information of the microstructure but alsocorrelation functions of second and higher order.

In this work, Hashin–Shtrikman bounds [50,51] will be used asthe main homogenization technique to determine the upper andlower bounds for the effective elastic properties of silicon nitride.The Hashin–Shtrikman (HS) bounds as second-order bounds take

T. Böhlke, Y. Othmani / Computational Materials Science 80 (2013) 43–50 45

into account two-point correlation functions and are based on theassumption of piecewise constant stress polarizations in the singlephases. For statistical homogeneous and isotropic microstructureswith no long-range order, the HS bounds can be formulated solelyin terms of volume fraction information. For microstructures withanisotropic two-point correlation functions of ellipsoidal symme-try additionally a second-order tensor enters describing the mor-phological anisotropy [50,51,42]. In this case the HS bounds havethe following form

CHS ¼ hCAi; A ¼ ðIS þ P0dCÞ�1hðIS þ P0dCÞ�1i�1 ð2Þ

with the strain localization tensor A, the polarization tensor P0 anddC ¼ C� C0. The effective compliance tensor is given by an analo-gous expression

SHS ¼ hSBi; B ¼ ðIS þQ0dSÞ�1hðIS þQ0dSÞ�1i�1 ð3Þ

with Q0 ¼ C0 � C0P0C0; dS ¼ S� S0 and S0 ¼ C�10 .

The stiffness tensor of the (homogeneous) comparison materialC0 has to be chosen as to satisfy the following inequalities for allmaterial points in the microstructure

C� C0 6 0; C� C0 P 0: ð4Þ

Both inequalities have to be interpreted in the sense of bilinearforms. The minimum comparison stiffness satisfying Eq. (4)1 resultsin combination with (2) in an upper bound whereas the maximumcomparison stiffness satisfying Eq. (4)2 gives in combination with(2) a lower bound of the strain energy density. In the following,the stiffness tensors corresponding to the upper and lower boundare denoted by CHSþ and CHS�, respectively.

Willis [50] has shown that for the special case of constant polar-izations and ellipsoidal two-point statistics, the polarization tensorP0 is given by

P0ðC0;AÞ ¼1

4pffiffiffiffiffiffiffiffiffiffiffiffidet Ap

Zknk¼1ðn � AnÞ�3=2

HðC0;nÞdn ð5Þ

with H ¼ ISðN�ðn� nÞÞIS;N ¼ K�1 and K ¼ C0sn� nt. Here dn de-notes the surface element on the two-dimensional unit sphere.The eigenvalues of the symmetric 2nd-order tensor A correspondto the inverse of the aspect ratios of the average inclusion. For anisotropic comparison medium with stiffness C0 ¼ c0

1P1 þ c02P2 and

isotropic two-point statistics, i.e. A = I, the tensors P0 and Q0 are gi-ven by (see, e.g., [16])

P0 ¼ p1P1 þ p2P2; p1 ¼1

c01 þ 2c0

2

; p2 ¼2

5c02

c01 þ 3c0

2

c01 þ 2c0

2

ð6Þ

and

Q0 ¼ q1P1 þ q2P2; q1 ¼2c0

1c02

c01 þ 2c0

2

; q2 ¼c0

2

53c0

1 þ 4c02

c01 þ 2c0

2

; ð7Þ

respectively.Without loss of generality, for the determination of the HS

bounds the comparison material can be chosen to be isotropic[37]. Due to the fact that the silicon nitride microstructure is iso-tropic (resulting in isotropic effective properties). In the presentwork, isotropic two-point statistics are used to specify P0 and Q0

which are then given by (6) and (7), respectively.

3. Probabilistic failure model

3.1. Weakest-link theory

According to the weakest link theory, a material fails when, atany point of the volume, a critical stress state is reached. For ahomogeneous volume V subjected to a homogeneous uniaxialstress state r, the failure probability is given by

PF ¼ 1� exp �cvðrÞ VV0

� �; ð8Þ

where V0 is a normalization parameter. The exponential term in Eq.(8) is the probability of finding zero defects in V i.e. the survivalprobability. The function cv(r) is often modeled by

cvðrÞ ¼ rr0

� �m

; ð9Þ

where r0 is the Weibull [49] scale parameter. The exponent m is theWeibull shape parameter. The latter is a measure for the scatter ofstrength data. The combination between Eqs. (8) and (9) providesthe failure probability

PF ¼ 1� exp � VV0

rr0

� �m� �: ð10Þ

When r = r0 and V = V0, the failure probability is equal to 63.21%. Incase of a heterogeneous material, the stress field varies stochasti-cally with position. Therefore, Eq. (10) cannot be directly appliedto micro-heterogeneous materials. Applying the weakest link the-ory to inhomogeneous materials with heterogeneous stress fields,the failure probability in the bulk material is determined by an inte-gration over the total volume V, taking into account the positiondependent stress field r = r(x)

PvF ¼ 1� exp � 1

V0

ZV

cvðr; xÞdV� �

: ð11Þ

The scalar valued tensor function cv(r,x) depends, in general, on themulti-axial stress state r. Applying (11) to silicon nitride, the two-phase structure can be taken into account by setting

cvðr; xÞ ¼ IGðxÞcvG rðxÞð Þ þ ISðxÞcv

S ðrðxÞÞ; ð12Þ

where IG and IS are the indicator functions of the glassy phase andsilicon nitride, respectively. Silicon nitride phase boundaries arealso subjected to damage nucleation that depend on the tractiont = rn. The vector n is the unit normal vector of the interface. Thefailure probability across the interface depends on r and n byt = rn. The failure probability, only due to interface loading, canbe defined analogously to Pv

F by

PsF ¼ 1� exp � 1

A0

ZA

csðt; xÞdA� �

; ð13Þ

where A0 is a normalization parameter. A denotes the interface be-tween the phases. The probability of survival of the whole materialis defined as the product between the probabilities of survival of thebulk material and the interface. It follows that the overall failureprobability of silicon nitride PF can be expressed by

PF ¼ 1� 1� PvF

� �1� Ps

F

� �¼ 1� exp � 1

V0

ZV

cvðr; xÞdV � 1A0

ZA

csðt; xÞdA� �

: ð14Þ

The direct integral computation in (14) is impossible for real micro-geometries. For this reason, we introduce the volume stress andinterface traction probability densities f v

r ðrÞ and f srðr;nÞ, respec-

tively. Furthermore, the probability density of surface orientationf snðnÞ is introduced. The latter densities are defined by

dVVðrÞ ¼ f v

r ðrÞdr ð15Þ

and

dAAðrÞ ¼ f s

rðr;nÞ ¼ f vr ðrÞf s

nðnÞdrdn: ð16Þ

In the last equation, statistical independence between r and n onthe interface has been assumed for simplicity. Then, by makinguse of (15) and (16), Eq. (13) is written as follows

Table 1Non-dimensional parameters used for the sensitivity analysis.

mint rint/EG rG/EG (rS/EG)

5; 10; 15; 20 0.0025; 0.01; 0.015 0.01; 0.1; 0.2 0.005; 0.01; 0.02

46 T. Böhlke, Y. Othmani / Computational Materials Science 80 (2013) 43–50

PF ¼1�exp � VV0

Zr

cv ðrÞf vr ðrÞdr� A

A0

Zr

Zn

csðt;xÞf srðrÞf s

nðnÞdrdn� �

:

ð17Þ

Note that, the evaluation of the last equation requires the integra-tion over the stress and the normal vectors n spaces. The computa-tion of cv(r) is based on the maximum positive eigenvalue kmax

a ofthe stress field in each phase. cv has the functional formcv ¼ cv

G þ cvS with cv

G ¼ kmaxG =rv

G0

� �mG and cvS ¼ kmax

S =rvS0

� �mS . rva0 and

ma are Weibull’s scale and shape parameters of each phase, respec-tively. Whereas cs(t,x) requires the determination of the normal andshear tractions across the interface tr = r � (n � n) and

ts ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin � r2n� ðn � rnÞ2

q, respectively. For cs we assume an additive

split cs ¼ csr þ cs

s, with csr ¼ ðtr=tr0Þmr and cs

s ¼ ðts=ts0Þms . Theparameters tr0 and mr are respectively Weibull’s scale and shapeparameters of the interface corresponding to the normal separation.Those corresponding to the tangential separation are denoted by ts0

and ms, respectively. For sake of simplicity, we assume that mr =ms = mint and tr0 = ts0 = rint. Note that, since Hashin–Shtrikmanbounds are used as the main homogenization technique, two failureprobabilities PþF ¼ PFðCHSþÞ and P�F ¼ PFðCHS�Þ, corresponding to theupper and the lower bound, respectively, are computed. For sake ofsimplicity, the failure probability of silicon nitride is computed byPF ¼ 1=2 PþF þ P�F

� �.

3.2. Second-order moment and covariance of the stress field

In the following, we aim to estimate fv(r) by a mean field esti-mation, taking into account the first and second-order moment ofstress. The effective behavior of a linear elastic material is charac-terized by the stress potential W , defined by

W ¼ 12

�S � ð�r� �rÞ; ð18Þ

where �r is the macroscopic stress. When the material is composedof multiple homogeneous phases, having each the elastic compli-ance Sa, the second order-moment of the stress field over the phasea is given by (see, e.g., [20])

hr� ria ¼2ca

@W@Sa

; ð19Þ

ca denoting the volume fraction of phase a. A detailed proof of (19)can be found in Kreher [24]. The effective second-order moment ofstress h r � ria is employed to measure the local fluctuation of thestress field together with the covariance tensor of the stress Kr

a , de-fined by

Kra ¼ hr� ria � hria � hria; ð20Þ

which quantifies the fluctuations of the local stress fields in thephases with respect to their average. In order to compute hr � riaand accordingly Kr

a , one has to compute the partial derivativeappearing in Eq. (19). For this purpose, a numerical differentiationis performed. It is based on a finite difference two point formula, gi-ven by

@ �S

@Sa’

�SðSa þ dSaÞ � �SðSa � dSaÞ2dSa

: ð21Þ

According to (2), the small variation of the effective elastic compli-ance in Eq. (21), implies a corresponding variation of the polariza-tion tensor P0. Consequently, the numerical evaluation of thelatter tensor is performed based on a Gaussian quadrature [1].

3.3. The multivariate Gaussian distributions of phases stresses

Multivariate random variables, that are normally distributed,are characterized by their mean value and their covariance matrix

[26]. A normally distributed stress with mean value hria andcovariance tensor Kr

a , can be represented in phase a by the proba-bility density function

f ðraÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð2pÞ6 det Kra

� �q� exp �1

2ðra � hriaÞ � Kr

a

� ��1½ra � hria�� �

; ð22Þ

where det Kra

� �is the generalized determinant of Kr

a [26].For a given macroscopic stress with its corresponding local

stress covariance Kra and average stress h ria, 106 trials have been

carried out for the set of the local stress field. The Marsaglia polarmethod algorithm [46] is used to generate the set of standard nor-mally distributed random components of the local stress ra. Themethod consists in choosing random points (u,v) in the squareu2 < 1, v2 < 1 with the condition t = u2 + v2 < 1. The obtained stan-dard normally distributed quantities are

a ¼ u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 ln t

t

r; b ¼ v

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 ln t

t

r: ð23Þ

The components of the stress tensor s0a, built on the basis of (23), are

normally distributed with stress covariance IS and zero mean. In or-der to make these components follow a multivariate normal distri-bution with stress covariance Kr

a and mean h ria, the followingtransformation is performed

ra ¼ Ta s0a

� �þ hria; ð24Þ

where Ta is the Cholesky factor of Kra and satisfies T2

a ¼ Kra [3]. The

normally distributed phase stresses ra are used for the computationof the terms cv and cs in (17).

4. Numerical results

The values of Weibull’s parameters of the interface and the bulkmaterial, are not precisely known for silicon nitride. In order tohave an estimation of these parameters, a sensitivity analysis isperformed. The study consists in varying four non-dimensionalparameters and determine the appropriate ones that minimizetwo error measures, determined by means of experimental data.The non-dimensional parameters, used in the analysis, are the fol-lowings, Weibull’s shape parameter of the interface (mint), Wei-bull’s scale parameters of the interface, the glassy phase and thegrains, divided by the glassy phase Young modulus, respectively,(rint/EG), (rG/EG) and (rS/EG). The parameters used in the numericalstudy are listed in Table 1. The glassy phase exists in form of glasspockets that fill the space and form inter-granular films betweenthe grains [17]. Thus, Weibull’s shape parameter of the glassyphase mG is approximated by the one of glass (see, e.g.,[54]). Forb-Si3N4 grains, there is a lack of experimental data with respectto Weibull’s shape parameter. Since the investigated silicon nitridematerial is made with 88% of b-Si3N4 grains and only a qualitativestudy is provided, we approximate Weibull’s shape parameter mS

of the b-Si3N4 grains by an overall value typical for silicon nitride.The values of the latter parameters are mS = 15 [15] and mG = 7[32]. The loadings considered in this study are the uniaxial tension(u.t.) and pure shear (s.). Taking into account the combination ofthe four non-dimensional parameters, 108 simulations are consid-ered for each kind of loadings. The analysis is based on the minimi-

Table 2Non-dimensional parameters corresponding to minimum 1.

rG/EG rS/EG rint/EG mint Dm/mexp (%) Dr/rexp (%)

0.01 0.005 0.01 10 3.61 17.110.01 0.005 0.01 15 3.44 17.070.01 0.005 0.01 20 3.52 17.090.01 0.005 0.015 5 0.64 16.730.01 0.005 0.015 10 3.90 17.170.01 0.005 0.015 15 3.45 17.080.01 0.005 0.015 20 3.61 17.11

Table 3Non-dimensional parameters corresponding to minimum 2.

rG/EG rS/EG rint/EG mint Dm/mexp (%) Dr/rexp (%)

0.1 0.005 0.01 10 6.61 17.440.1 0.005 0.01 15 6.65 17.440.1 0.005 0.01 20 6.61 17.440.1 0.005 0.015 5 3.56 17.080.1 0.005 0.015 10 6.62 17.440.1 0.005 0.015 15 6.62 17.440.1 0.005 0.015 20 6.64 17.44

Table 4Non-dimensional parameters corresponding to minimum 3.

rG/EG rS/EG rint/EG mint Dm/mexp (%) Dr/rexp (%)

0.2 0.005 0.01 10 6.59 17.440.2 0.005 0.01 15 6.64 17.440.2 0.005 0.01 20 6.64 17.440.2 0.005 0.015 5 3.56 17.080.2 0.005 0.015 10 6.64 17.440.2 0.005 0.015 15 6.65 17.440.2 0.005 0.015 20 6.63 17.44

10

15

par

amet

er [G

Pa]

σ0:(u.t.)

σ0:(s.)

σGσ

T. Böhlke, Y. Othmani / Computational Materials Science 80 (2013) 43–50 47

zation of two error measures. The first one is defined by Dr/rexp,where Dr = r0 � rexp. The parameters r0 and rexp are the Weibullscale parameters of silicon nitride, computed at each simulationand the one provided by experiments, respectively. Likewise, thesecond error measure is Dm/mexp, where Dm = m0 �mexp. Here,the parameters m0 and mexp are the Weibull shape parameters ofsilicon nitride, computed at each simulation and provided byexperiments, respectively. For GR12 sintered silicon nitride,rexp = 0.7 GPa and mexp = 16 [23].

On the macroscopic scale, the parameters r0 and m0 are com-puted, by considering the macroscopic estimation of failure proba-bility, expressed by

PvsF ¼ 1� exp � V

V0

�kmax

r0

� �m0 !

; ð25Þ

where �kmax is the maximum positive eigenvalue of the macroscopicstress field. Note that, the estimation of r0 and m0 on the basis of(25) is an approximation of silicon nitride Weibull’s parameters.For one dimensional model, the estimation of Weibull’s macro-scopic parameters, on the basis of a macroscopic failure probability,is made with an error that lies between 0% and 30% compared to afailure probability that considers microscopic fields.

Fig. 1a illustrates the sum of the error measures Dr/rexp andDm/mexp for all the non-dimensional parameters combinations.Weibull’s scale and shape parameters, corresponding to thenon-dimensional variables, are shown in Fig. 1b for all the 108combinations as in Fig. 1a. The errors are approximately the samefor uniaxial tension and shear. Three absolute minimums areobserved in Fig. 1a. The minima and the ranges of the non-dimensional parameters are given in Tables 2–4 with thecorresponding errors measures. In the latter tables, each minimumcorresponds to a different value of rG/EG, i.e. the number ofminimums is determined by the number of the values given tothe latter non-dimensional parameter. One can observe also thatthe parameter rS/EG = 0.005, is commonly repeated in the three ta-bles. As for rint/EG, this parameter takes the two values 0.01 and0.015. Finally, the optimization of the errors Dr/rexp + D m/mexp

permit the adequate choice of the non-dimensional parameters,among the three tables. The optimal errors are Dr/rexp = 16.73%and Dm/mexp = 0.64%. Therefore, rG/EG = 0.01, rS/EG = 0.005,rint/EG = 0.015 and mint = 5 is the satisfactory choice of the latterparameters.

0 10 20 30 40 50 60 70 800

5

Simulation’s number [−]

Wei

bull’s

sca

le

(a)

Sσint

0 10 20 30 40 50 60 70 800

5

10

15

20

Simulation’s number [−]

Wei

bull’s

sha

pe p

aram

eter

[−]

(b)

m0:(u.t.)m0:(s.)mGmSmint

Fig. 2. Relation between microscopic and macroscopic Weibull’s parameters. (a)Graphical representation of the local and macroscopic Weibull’s scale parameters.(b) Graphical representation of the local and macroscopic Weibull’s shapeparameters.

0 20 40 60 80 100 1200

50

100

150

200

250

Simulation’s number [−]

Δσ/σ

exp+Δ

m/m

exp [%

]

(a)

(u.t.)(s.)

0 20 40 60 80 100 1200

10

20

30

Wei

bull’s

sca

le p

aram

eter

[GPa

]

Simulation’s number [−]

(b)0 20 40 60 80 100 120

5

10

15

20

min

t [−]

σGσSσintmint

Fig. 1. Estimation of the appropriate non-dimensional parameters. (a) Illustrationof Dr/rexp + Dm/mexp for all the 108 simulations. (b) Graphical representation ofWeibul’s parameters.

Fig. 2a and b illustrate the variation of the macroscopic andmicroscopic Weibull’s scale and shape parameters with the differ-

0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

Macroscopic stress [GPa]

P F

(a)

PFG

PFS

PFSG

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

Macroscopic stress [GPa]

P F

(b)

PFστ(u.t.)

PFστ(s.)

PFσ(u.t.)

PFσ(s.)

PFτ (u.t.)

PFτ (s.)

Fig. 3. Influence of the local Weibull scale parameters on the failure probabilities: (a) In the bulk material (u.t.; rint/EG = 0.0025; rG/EG = 0.01; rS/EG = 0.005). (b) Across theinterface (rint/EG = 0.015; rG/EG = 0.01; rS/EG = 0.005).

0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

Macroscopic stress [GPa]

P F

(a)

PFSG

PFστ

PFSGστ

0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

Macroscopic stress [GPa]

P F

(b)

PFSG

PFστ

PFSGστ

Fig. 4. Influence of the local Weibull scale parameters on PSGrsF : (a) PSGrs

F is controlled by PrsF (u.t.; rint/EG = 0.0025; rG/EG = 0.01; rS/EG = 0.005). (b) PSGrs

F is controlled by PSGF

(u.t.; rint/EG = 0.015; rG/EG = 0.01; rS/EG = 0.005).

48 T. Böhlke, Y. Othmani / Computational Materials Science 80 (2013) 43–50

ent combinations of the non-dimensional parameters, for uniaxialtension and shear. For the clarity of the illustration, the figuresshow only the results corresponding to the first 72 simulations.The same conclusions can be drawn from the rest of the simula-tions. As depicted in the latter figures, Weibull’s macroscopicparameters are approximately the same for both cases of loadings.Thus, the macroscopic parameters are independent from the ap-plied loading. Fig. 2a shows that Weibull’s scale parameter of sili-con nitride is always determined by the minimum between thethree local scale parameters rS, rG and rint. This is a result of theweakest link theory. Therefore, Weibull’s scale parameter of siliconnitride is given by the weakest micro-structural element amongthe grains, the glassy phase and the interface. By comparing thethree Weibull’s scale parameters obtained from last optimizationstudy, the optimum corresponds to rS/EG = 0.005 and in otherwords, Weibull’s scale parameter of silicon nitride is given byrS = 0.665 GPa which is very close to rexp. In this case, the grainsare the weakest micro-structural element. The local Weibull’s scaleparameters controls also the choice of the macroscopic shapeparameter. As illustrated in Fig. 2b, the Weibull’s shape parameterof silicon nitride is always determined by the microscopic shapeparameter of the phase with the smallest scale parameter. Conse-quently, Weibull’s shape parameter of silicon nitride is given by mS.

To illustrate the influence of the microscale parameters on themacroscopic ones, the followings examples are investigated.Fig. 3a shows the failure probabilities of: the bulk material,b-Si3N4 grains and the glassy phase, PSG

F ; PSF and PG

F , respectively.Considering only the bulk material, b-Si3N4 grains are the weakestmicro-structural element and the failure probability PSG

F is equal to

PSF . Fig. 3b depicts the failure probabilities corresponding to the

interface, for both shear and uniaxial tension. In case of shear,the failure probabilities corresponding to the normal and totangential directions Pr

F and PsF , respectively, are both equal to

the failure probability of the interface PrsF . In case of uniaxial

tension, PsF is less pronounced compared to Pr

F which is equal toPrs

F . This is attributed to the predominance, across the interface,of normal traction compared to shear, in case of uniaxial tension.Despite the adoption of the same Weibull’s scale parameter forboth normal and tangential directions, the nature of the appliedloading influences the competition between Pr

F and PsF . However,

the failure probability of the interface PrsF is approximately the

same for both shear and uniaxial tension.As previously mentioned, the failure probability of silicon ni-

tride PSGrsF is controlled by the weakest micro-structural element

among the different phases. This is illustrated in Fig. 4a and bwhich depicts the failure probabilities of: silicon nitride PSGrs

F , thebulk material PSG

F and the interface PrsF . In a first case, Weibull’s

scale parameter of the interface is the smallest. Accordingly, theinterface is the weakest micro-structural element and PSGrs

F ¼ PrsF ,

as shown in Fig. 4a. In a second case, the bulk material is the weak-est micro-structural element and PSGrs

F ¼ PSGF , as shown in Fig. 4b.

Besides the sensitivity analysis, the effect of the grains sizes onthe failure probability of silicon nitride have been investigated.This is illustrated in Fig. 5a and b. If the weakest micro-structuralelement is the interface, the failure probability of silicon nitridePSGrs

F is equal to PrsF . In this case, Fig. 5a shows that PSGrs

F decreaseswith the grains size and vice versa. Indeed, the surface of the inter-face is inversely proportional to the radius a of spherical inclusions.

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

Macroscopic stress [GPa]

P FSGστ

(a)

a=0.5μma=1.0μma=1.5μm

0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

Macroscopic stress [GPa]

P FSGστ

(b)

a=0.5μma=1.0μma=1.5μm

Fig. 5. Influence of the grains sizes on PSGrsF . (a) Case of PSGrs

F ¼ PrsF : decrease of PSGrs

F with increasing grains’ size (u.t.; rint/EG = 0.0025; rG/EG = 0.01; rS/EG = 0.005). (b) Case ofPSGrs

F ¼ PSGF : PSGrs

F is insensitive to grains’ size (u.t.; rint/EG = 0.015; rG/EG = 0.01; rS/EG = 0.005).

T. Böhlke, Y. Othmani / Computational Materials Science 80 (2013) 43–50 49

Then, by decreasing the radius a, the size of the interface will in-crease and accordingly a growing failure probability. Now, if thebulk material is weakest micro-structural element, Fig. 5b showsthat the size of the grains has no influence on PSGrs

F . In the presentwork, the model is based on the weakest link approach. The failureprobability of the bulk material (the glassy phase and the grains)depends on the volume of the specimen under investigation.Therefore, considering a fixed volume of the specimen, a constantvolume fraction of the grains and varying their sizes by varyingtheir numbers, does not influence the failure probability in thebulk material. Whereas the failure probability of the interface issize dependent. When the bulk material fails for the applied com-bination of materials parameters, then the whole material fails andits failure probability is size independent. Therefore the indepen-dence on the number of grains is compatible to the applied Wei-bull’s approach.

5. Discussion and conclusions

The weakest link model has been extended to account, not onlyfor the volume defects, but simultaneously for the damage acrossthe interface between silicon nitride’s phases. The model brings asignificant improvement over the classical weakest link approach.It is based on a multivariate approach and is used in conjunctionwith the second-order estimates of the local fields. The second-or-der moment and covariance of the local fields have been formu-lated and computed numerically, based on a finite difference twopoint formula. The required effective properties have been com-puted based on Hashin–Shtrikman bounds. The failure probabilityof silicon nitride has been computed by integration of the localstress and traction fields over the phase volumes and interfaces.An exhaustive parametric study has been carried out, for the casesof shear and uniaxial tension loadings, on the basis of the variationof four non-dimensional parameters and the definition of two errormeasures. The minimization of the two error measures allowed thedetermination of the range of the appropriate parameters. An esti-mation of the parameters is given. Weibull’s scale and shapeparameters of the interface have been estimated as rint = 1.995 GPaand mint = 5, respectively. Weibull’s scale parameter of b-Si3N4

grains is estimated as rS = 0.665 GPa and is already in the strengthrange that lies between 0.651 and 0.905 GPa as reported by Danzeret al. [15]. As for Weibull’s scale parameter of the glassy phase, it isestimated as rG = 1.33 GPa. The values of the scale parameters ofthe interface and the glassy phase represent only rough approxi-mations that may probably overestimate the material and theinterface parameters. However, these values are of the same order

of magnitude of strength parameters existing in the literature. In-deed, as reported in Mecholsky [35], the predicted strength of Sil-ica glass is about 9 GPa. Wippler and Böhlke [53] used a value of10 GPa for the strength of the glassy phase. The authors consideredthat the latter value is reasonable since the volume of the glassypockets are very small which implies a small failure probabilityin this phase. Also, according to Colonna and Kailer [10], basedon ab initio simulations, the cohesive strength across the interfacebetween two b-Si3N4 grains is about 19.3 GPa. The latter parame-ters represent an approximation of silicon nitride Weibull’s param-eters. Therefore, the recourse to ab initio simulations forestimating these parameters is a necessary tool for subsequentsimulations.

The parametric analysis has shown that the failure in silicon ni-tride is always controlled by the phase which constitute the weak-est micro-structural element. In fact, it was shown that Weibull’sscale parameter of silicon nitride fits with the smallest among Wei-bull’s microscopic scale parameters of the phases and accordinglyWeibull’s shape parameter. Hence, by comparing Weibull’s scaleparameters estimated by the parametric study, it comes out thatthe failure probability of silicon nitrite is controlled by the failureprobability of b-Si3N4 grains which constitute the weakest micro-structural element. Such conclusion is not usually valid for real sil-icon nitride. Indeed, the fracture of the latter material depends onseveral constitutive and micro-structural parameters. Even basedon the experimental observations there is no unique conclusionconcerning the weakest micro-structural element for silicon ni-tride. Indeed, Guo et al. [18] showed that the fracture behavior ofsilicon nitride ceramics with Lu2O3 additives depends substantiallyon the amount of Lu2O3, in addition to grains’ morphologies. Aspointed out by the author, for silicon nitride with 1.68 wt.% ofLu2O3, a predominantly inter-granular crack path is observed.The grain boundary phase remains connected and acts as elasticbridging over the crack faces. This observation implies that theglassy phase is the first to fail, i.e., the glassy phase is the weakestmicro-structural element. However, by increasing the amount ofLu2O3 up to 6.52 wt.%, the crack is deflected around the largergrains and the elongated grains are cutted. In this case, b-Si3N4

grains are the weakest micro-structural elements. Park et al. [40]investigated the influence of the amount of Ytterbium oxide(Yb2O3) as a sintering aid on the mechanical properties of siliconnitride. For less than 8 wt.% of Yb2O3 added, only small homoge-neous grains were formed. By increasing the amount of Ytterbiumoxide up to 8 wt.%, larger elongated grains are observed and thefracture toughness of silicon nitride increases in return. However,if the amount of Yb2O3 exceeds 8 wt.%, the cohesive energy be-tween the grains and the glassy phase increases extensively and

50 T. Böhlke, Y. Othmani / Computational Materials Science 80 (2013) 43–50

the fracture toughness decreases in return. Thus the fracture andcohesive strengths of the local phases depend on the amount ofYb2O3 added. The determination of the weakest micro-structuralelement in silicon nitride is nontrivial for silicon nitride withoutthe knowledge of Weibull’s scale parameters of the phases.

The influence of the grain’s size on the failure probability of sil-icon nitride has been also investigated. The study has shown thatgrains’ size effect is to be taken into consideration, only if the inter-face is the weakest micro-structural element. In this case, the fail-ure probability of silicon nitride decreases with grain’s size. Tanet al. [45] have studied the influence of particle size effect on theoverall mechanical behavior of a composite material reinforcedby spherical inclusions with nonlinear interface decohesion andsubjected to uniaxial tension. The macroscopic stress–strain curveof the composite shows a hardening behavior of the material incase of small inclusions and a softening behavior in case of largeinclusions. It has been shown that in case of nonlinear debondingand in terms of mechanical performance, the composite is muchmore stiffer, when it is reinforced by smaller particles, than whenit is reinforced by larger particles. Considering a RVE, from the lat-ter composite, with a constant volume V. The fixed volume fractionof the inclusions is denoted by f. The surface of the interface be-tween the inclusions and the matrix is S = (3fV)/R. R is the radiusof the inclusions. R and S are inversely proportional which meansthat by decreasing the radius of the inclusions, the total surfaceof interface will increase. Consequently, the probability of failureacross will increase in return. This is similar to increasing the fail-ure probability as a result of increasing the volume of defects. Con-sidering Weibull’s parameters estimated on the basis of thesensitivity analysis, b-Si3N4 grains are the weakest micro-struc-tural element and therefore the grain size has no effect on the fail-ure probability of silicon nitride.

Acknowledgement

The research leading to these results has received funding fromthe European Union’s Seventh Framework Programme FP7/2007–2013 under Grant Agreement No. 263476.

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