International Journal of Plasticity 23 (2007) 665–689

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Fracture of bicrystal metal/ceramic interfaces:A study via the mechanism-based strain

gradient crystal plasticity theory

A. Siddiqa, *

, S. Schmaudera

, Y. Huangb

a Institut fu¨ r Materialpru ¨ fung, Werkstoffkunde und Festigkeitslehre (IMWF), University of Stuttgart, Germanyb Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, USA

Received 4 March 2006Available online 21 November 2006

Abstract

Two continuum mechanical models of crystal plasticity theory namely, conventional crystal plas-ticity theory and mechanism-based crystal plasticity theory, are used to perform a comparative studyof stresses that are reached at and ahead of the crack tip of a bicrystal niobium/alumina specimen.Finite element analyses are done for a stationary crack tip and growing cracks using a cohesive mod-elling approach. Using mechanism-based strain gradient crystal plasticity theory the stresses reachedahead of the crack tip are found to be two times larger than the stresses obtained from conventionalcrystal plasticity theory. Results also show that strain gradient effects strongly depend on the intrin-sic material length to the size of plastic zone ratio ( l /R0). It is found that the larger the ( l /R0) ratio,the higher the stresses reached using mechanism-based strain gradient crystal plasticity theory. Aninsight into the role of cohesive strength and work of adhesion in macroscopic fracture is also pre-sented which can be used by experimentalists to design better bimaterials by varying cohesive

strength and work of adhesion.Ó 2006 Elsevier Ltd. All rights reserved.

Keywords: Metal/ceramic interface; Mechanism-based strain gradient crystal plasticity; Cohesive model; Macro/micro fracture analysis

0749-6419/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijplas.2006.08.007

*

Corresponding author. Tel.: +49 711 685 62579; fax: +49 711 685 62635.E-mail address: [email protected] (A. Siddiq).

International Journal of Plasticity 23 (2007) 665–689www.elsevier.com/locate/ijplas

mailto:[email protected]

mailto:[email protected]



1. Introduction

Metal/ceramic interfaces play a vital role in modern materials technology, as evident bytheir use in a variety of applications. For example, high-strength materials such as metal– matrix composites consist of internal interfaces between ceramic (e.g. SiC or Al 2O3) par-ticles or laments within a metallic host. Because of their high melting points, extremehardness, and favourable corrosion resistance, ceramic coatings are often used to protectan underlying metallic component in environments subject to high temperatures and wearrates (i.e., cutting tools). In microelectronics packaging, interfaces between metallic (Cuand/or Al) interconnects and SiO 2, carbide/nitride (TiCN) or oxide (Al 2O3) ceramicsare commonplace, and impact the performance and longevity of solid state devices.Despite their widespread use, a basic understanding of these interfaces has been elusive.For example, given a particular metal/ceramic interface, it is not yet possible to accuratelypredict such fundamental properties as its fracture energy. In most of the cases, improve-ments in interface properties proceed via a costly and time consuming trial-and-error pro-cess in which numerous materials are evaluated until suitable performance is obtained.Computational methods provide a wide range of possibilities to study the fracture behav-iour of such metal/ceramic interfaces. Scientists and engineers have been studying differentmetal/ceramic interfaces for a long time, in order to create a better understanding of thefactors that inuence the fracture behaviour of the metal/ceramic interfaces, such as plas-ticity induced around the crack tip and interface strength.

In order to study and understand the adhesion at a fundamental quantum-mechanicallevel (Methfessel et al., 1992; Finnis et al., 1995; Baitrev et al., 1999; Zhang and Smith,

2000) ab initio calculations have been performed. Methfessel et al. (1992) performed ab ini-tio calculations to compute the surface energies of 4d transition metals (Y, Zr, Nb, Mo,Tc, Ru, Rh, Pd, Ag). It was found that surface energies change with changing orientationof single crystalline materials, for example in the case of niobium, surface (100) is found tohave highest surface energy as compared to surfaces (110) and (111).

Finnis et al. (1995) performed ab initio calculations of metal/ceramic interfaces. Thesecalculations not only predicted the electronic structure of the interface but also the idealwork of adhesion. The calculations were based on niobium(111)/ a -alumina(0001) inter-faces. The computed value of ideal work of adhesion of a niobium monolayer to the oxidesurface amounts to 9.8 J/m 2 . Baitrev et al. (1999) also performed the ab initio calculations

for niobium/alumina interfaces. The work of adhesion for O- and Al-terminated nio-bium(111)/ a -alumina(0001) interfaces was reported to be in the range of 9.8–10.8 J/m 2.It was found that after relaxing the atoms, alumina is always found to oxygen termi-nated. It was also found that oxygen terminated alumina at the interface has higher workof adhesion ( $ 9.8 J/m 2) as compared to aluminium terminated alumina ( $ 2.8 J/m 2).Zhang and Smith (2000) examined the relative stability of both stoichiometric and non-stoichiometric niobium(111)/ a -alumina(0001) interfaces. The work of adhesion was com-puted using two different techniques local-density approximation (LDA) (for details seeZhang and Smith, 2000 ) and generalized-gradient approximation (GGA) (for details seeZhang and Smith, 2000 ). The work of adhesion computed was consistent with the resultsof Baitrev et al. (1999).

The mechanical properties of single crystalline sapphire/niobium/sapphire joints wereinvestigated ( Soyez, 1996; Soyez et al., 1998) for different orientation relationshipsbetween metal and ceramic. The orientation of single crystalline materials was found to

666 A. Siddiq et al. / International Journal of Plasticity 23 (2007) 665–689



have signicant effect on the energies of the Nb/Al 2O3 interface fracture. The inuence of the chemical composition of regions at or close to the interface on the strength of a metal/ceramic interface was studied ( Fischmeister, 1987 ). It was found that the strength of ametal/ceramic interface depends upon the density of the metal–oxide bonds. Ni/Al 2O3

and Cu/Al 2O3 joints were studied ( Nicholas and Lee, 1989; Nicholas et al., 1980; Kurkjianand Kingery, 1956 ), and it was found that higher strength values can result from addingreactive alloying elements, such as, chromium and titanium. This addition of alloying ele-ments raises the work of adhesion.

Various research groups ( Evans et al., 1990; Evans and Dalgleish, 1992; McMahon andVitek, 1979; Jokl et al., 1980; Hirth and Rice, 1980; Rice et al., 1990 ) postulated that thefracture energy of interfaces depends not only on the work of adhesion W adhesion , but alsoon the amount of plasticity that occurs in the metal during the fracture process, on theinterfacial atness, and on their interrelation. Fuller et al. (1980) and Chiao and Clarke(2003) observed dislocation emission from crack tips, and also proposed that brittlenessor ductility of a crack tip can be characterized by its response to mechanical loading.Fischmeister et al., 1988 found that the fracture energy of niobium/sapphire and copper/sapphire interfaces change dramatically with the relative crystallographic orientation of thetwo constituents. Beltz and Rice (1992), Beltz and Wang (1992), Korn (1993) and Kornet al. (2002) observed that the amount of plasticity initiated in the metal during fractureof such interfaces is strongly inuenced by both, the interface chemistry and the orienta-tions of slip systems relative to the crack surface. Kysar (2000) studied experimentallythe copper/sapphire interface and the directional dependence of the crack for various ori-entations. Jameel et al. (2003) studied the brittleness and ductility of cracks along the inter-

face of copper bicrystals. The brittleness and ductility of the crack tip was explained in termsof angles that various slip systems make with the crack anks. They found out that the rel-ative angles of slip planes play a critical role in the process of deformation at the crack tip.

Continuum mechanical studies have been done by many researchers to study andunderstand the fracture behaviour of metal/ceramic interfaces. Armstrong (1966) andPrice and Kelly (1964) proposed that the distinction between the intrinsic brittlenessand ductility can be considered as a competition between cleavage failure and plastic shearat the crack tip.

McHugh et al. (1989) and Varias et al. (1990) studied aluminium/rigid and copper/rigidmaterial interfaces using rate dependent crystal plasticity theory ( Asaro, 1979, 1983a,b ). A

hybrid decohesion model was proposed to describe the crack propagation along the inter-face. These results show that the interface strength changes the stress–strain eld around acrack tip. It was also found that increasing the interface strength increases the J -resistanceof the bimaterial system.

Clegg et al. (1988) and Beltz and Rice (1992) used rate dependent crystal plasticity the-ory, whereas Asaro (1983a,b) studied the crack tip behaviour of a copper/sapphire inter-face. Beltz and Rice (1992) and Beltz and Wang (1992) interpreted the experimental resultsof Beltz and Wang (1992) based on dislocation nucleation arguments. It was shown thatbrittle failure occurs when the energy required for dislocation nucleation is larger than theenergy required for cleavage, while failure is ductile if the energy required for dislocationnucleation is less than the energy required for cleavage.

Wang and Anderson (1991) performed a series of interfacial fracture experiments onspecially oriented copper symmetric tilt bicrystals and reported a directional dependenceof crack growth; that is the interfacial crack oriented in one crystallographic direction

A. Siddiq et al. / International Journal of Plasticity 23 (2007) 665–689 667



propagates much more readily than the crack on the same interface but oriented to prop-agate in the opposite crystallographic direction. Wang (1998) proposed a micromechanicalmodel for interface cracking. Three different systems were studied, i.e., gold/sapphire, alu-minium/sapphire and nickel/sapphire. The model proposed in their work, predicted thecrack tip behaviour, i.e., ductility or brittleness of the crack tip by including the local modemixity at the crack tip in the model ( Beltz and Wang, 1992 ).

In addition to the dislocation nucleation analyses, there have been other approaches topredict relative brittleness and ductility. Kysar (2001a,b) and Kysar and Briant (2002)studied the directional dependence of crack growth at copper/sapphire interfaces. Finiteelement analyses of copper/sapphire bicrystal speciement were performed using crystalplasticity theory. It was shown that relative ductility/brittleness can be explained on thebasis of the normal opening stress.

Nakatani et al. (2003) analyzed the small-scale yielding around a stationary crack alonga ductile single crystal/rigid material interface. Both continuum slip and discrete disloca-tion plasticity theory were used with either two or three slip systems. Results obtainedusing both theories were found to be consistent. The discrete dislocation simulations pro-vide a valuable window into the actual micromechanics of how near-tip plastic deforma-tion develops, and what the actual underlying dislocation distribution is that correspondsto the macroscopic elastic–plastic deformation pattern. However limitations on how manydislocations can be handled and how many computer runs can be made, make it difficult todetermine the near-tip stress and deformation elds that occur on a larger scale near such acrack tip. This information can be obtained from the continuum slip crystal plasticity solu-tions, which gave full-eld results for the elastic–plastic continuum slip crystal plasticity

solutions. It was also shown that both, discrete dislocation simulations and continuum slipcrystal plasticity theory accurately predict the crack tip opening behaviour, which isimportant in understanding the type of failure anticipated.

Kohnle et al. (2000, 2002) analyzed polycrystalline niobium/sapphire interfaces. Theinuence of the plastic properties of the metal part on the interface strength and on theenergy release rate is examined. It was shown that low yield stress results in high plasticdeformation in the metal part and consequently higher energy is stored before the criticalstress value for fracture is reached.

Crystal orientation effects on the crack initiation energies of the bicrystal niobium/alu-mina specimen have been studied ( Siddiq and Schmauder, 2005b, in press ). The results

show that by changing the orientation of the single crystalline material, one can changethe amount of plasticity induced due to the activation of different slips systems, also,the level of stresses induced at and near the crack tip can be altered. The reason of vari-ation in fracture energies for various orientations were explained on the basis of inducedstresses and plasticity around the crack tip.

Interface fracture of a metal/ceramic bicrystal specimen using a cohesive modellingapproach and conventional crystal plasticity theory was studied by Siddiq and Schmauder(2006). The effect of different cohesive law parameters, such as, cohesive strength and workof adhesion on the macroscopic fracture energy was studied. An insight on the role of thecohesive strength and work of adhesion on fracture energy was presented. It was foundthat the effect of cohesive strength on fracture energies is more profound as comparedto the work of adhesion. A comparison between experimental and simulation resultswas presented and a theoretical interlink among cohesive strength, work of adhesion, yieldstress and macroscopic fracture energy was presented.




Fracture analysis of a bimaterial interface using conventional theory of mechanism-based strain gradient plasticity was performed by Qu et al. (2004). It was shown that byusing the conventional theory of mechanism-based strain gradient plasticity theory, thetensile stress at a distance of 0.1 l m to the interface crack tip reaches 13.3 r y, where r y

is the yield stress of niobium, which when compared to classical plasticity theory (3.6 r y)is nearly 4 times higher (Qu et al., 2004) which is due to the strong strain gradient effectsnear the interface.

The aim of the presented work is to perform a comparative study of the tensile stressesat the interface crack tip using mechanism-based strain gradient crystal plasticity theoryHan et al. (2005a,b) and conventional crystal plasticity theory ( Hill and Rice, 1972 ).The study will be carried out, both for a stationary crack tip and for growing cracks usinga cohesive modelling approach ( Scheider and Brocks, 2003 ).

2. Conventional crystal plasticity theory

Crystal plasticity theory is based on the assumption that plastic deformation is thesum of the crystalline slip in all activated slip systems. Schmidt (1931) pointed out thatplastic slip occurs when the resolved shear stress onto a crystallographic plane in thedirection of slip reaches a critical value. The precise theory was formulated by Hilland Rice (1972).

The resolved shear stress s (a ), on the a th slip system is computed from the stress tensorr ij as

sðaÞ

¼ r ij lðaÞij ð1Þ

with l ðaÞij being the Schmidt factor given by

l ðaÞij ¼

12

sðaÞi nðaÞ

j þ sðaÞ j nðaÞ

i ð2Þ

where s(a ) is the vector tangent to slip system a and n(a ) is the vector normal to slip systema .

While the plastic strain rate is given by

_e p ij ¼

Xl ðaÞ

ij _cðaÞ ð3Þ

where _cðaÞ is the shear strain rate in slip system a .The stress rate is then dened as

_r ij ¼ C ijkl _eekl ¼ C ijkl ð_ekl À _e p

kl Þ ð4Þ

A viscoplastic constitutive model is used in this study in the form of a power law expres-sion as

_cðaÞ ¼ _cðaÞ0 sgnðsðaÞÞ

sðaÞ

g ðaÞ

m

ð5Þ

where g (a )

is the current strength of each slip system a , c0 is the reference (initial) shearstrain, while n is the strain rate sensitivity parameter (a larger value of n ensures the rateindependent case, such as n = 50). As n ! 1 , the plastic constitutive formulation formallybecomes rate-independent.




The rate of change of the current strength ( g (a )) of each slip system a is given by

_ g ðaÞ ¼

X

N

b¼1hab _cðaÞ ð6Þ

where the matrix hab contains the hardening moduli for each slip system. haa is known asself hardening (hardening of a slip system during deformation) while hab (a 6¼ b) is knownas latent hardening (hardening of a slip system due to the hardening in another slipsystem).

Different hardening models have been developed since the evolution of crystal plasticitytheory (Taylor, 1938; Nakada and Keh, 1966; Peirce et al., 1982; Bassani and Wu, 1991 ).For the present study Bassani and Wu (1991) has been used to simulate the deformationbehaviour of single crystalline niobium. Bassani and Wu (1991) model is based upon theanalytical characterization of the hardening moduli at any of the three stages during defor-

mation. Their expression for self and latent hardening depends on the shear strains ca

of allslip systems

haa ¼ ðh0 À hsÞsech2 ðh0 À hsÞca

s s À s0 þ hs G ðcb ; b 6¼ aÞ

hab ¼ qh aa ða 6¼ bÞ ð7Þ

G ðcb ; b 6¼ aÞ ¼1 þ Xb6¼a

f ab tanh ðcb =c0Þ

where, h0 is the initial hardening modulus, s0 the initial yield stress, s s the saturation stress,

ca

the total shear strain in slip system a , hs the hardening modulus during stage I deforma-tion, f ab the interaction strength between slip system a and b , and cb the total shear strainin slip system b while sech is the secant hyperbolic function.

The hardening moduli are described with an initial hardening ( h0) which saturates afterreaching the resolved shear stress ( s s). After the diminishing of the hyperbolic secant term,the saturation hardening ( hs) term species that each slip system has a nite hardeningrate. The function G deals implicitly with cross-hardening that occurs between slip systemsduring stage II hardening.

Other crystal plasticity models based on dislocations mechanics are also available ( Cui-tino and Ortiz, 1992; Kameda and Zikry, 1998; Ohashi and Asakawa, 2001; Harder, 1999;

Acharya, 2001; Arsenlis and Parks, 2002 ). These models explicitly account for the quan-tication of dislocations density evolution and the kinetics of dislocation interaction.These types of models involve a large number of parameters and coefficients that mustbe derived from experiments. Cuitino and Ortiz (1992) proposed a viscoplastic dislocationmechanics based crystal plasticity model using statistical mechanics. This model requiresin total 12 parameters to be identied for each family of slip system from experiments.Kameda and Zikry (1998) proposed a model by including the mobile and immobile dislo-cation densities as internal variables in the constitutive formulations. The relation for theevolution of mobile and immobile dislocation densities that correspond in an averagesense to the generation, interaction, trapping and recovery of dislocations. This modelrequires the identication of seven parameters for each family of slip system from uniaxialtension/compression experiments. Ohashi (2001) proposed a model in which the evolutionof plastic slip was dened using models of dislocation movement for fee crystals. The meanfree path in this model is described as a function of the densities of statistically stored as




well as geometrically necessary dislocations. This model is scale dependent where the edgeand screw components of the geometrically necessary dislocations are obtained from thestrain gradients. This model requires eleven parameters to be identied for each familyof slip system from experiments. Harder (1999) proposed a crystallographic model forpure fee metals in the low temperature range. Rate dependent constitutive equations arebased on either isotropic as well as kinematic hardening, whereby the mutual interactionsof dislocation processes on the different slip systems are taken into account. This modelrequires the identication of eleven parameters and was applied to fee metals in the past(Harder, 1999 ). Arsenlis and Parks (2002) and Acharya (2001) , proposed a continuumconstitutive model based on internal variables characterizing crystallographic dislocationdensities. The evolution of plastic strain is given by Orowan’s relation as a function of dis-location ux and average dislocation velocity. A large number of parameters are to beidentied in this model, approx. 6–18, depending on the dislocation interactions. Theabove discussed dislocations based models leads to the conclusion that these modelsexplicitly account for such things as the quantication of dislocations density and kineticsof dislocation interactions. On the other hand, these types of crystal plasticity modelsinvolve many parameters and coefficients that have to be derived from experiments. Also,limitations on how many dislocations can be handled and how many computer runs can bemade, make it difficult to determine the near-tip stress and deformation elds present on alarger scale near a crack tip ( Nakatani et al., 2003 ).

Present work is performed using the crystal plasticity theory (Hill and Rice, 1972) alongwith the two hardening laws by Peirce et al. (1982) and Bassani and Wu (1991) , respec-tively, has been implemented numerically within the UMAT written by Huang (1991) .

The numerical implementation of conventional crystal plasticity theory has been discussedin detail in Huang (1991) and Kysar (2001a,b) . This UMAT subroutine has been used withABAQUS for the present work.

The crystal plasticity model has been implemented for nite deformations and rota-tions. The Jaumann stress rate is used in the formulation, which is the corotational stressrate on axes that rotate with the crystal lattice.

The orientations of the slip systems are updated during every call to the UMAT sub-routine and stresses are updated using the updated orientations of the slip systems.

3. Mechanism-based strain gradient crystal plasticity theory

Steep strain gradients in the vicinity of the crack tip produce locally high levels of strainhardening due to a high density of geometrically necessary dislocations which ultimatelyend up with higher levels of stresses around the crack tip. Various plasticity models havebeen proposed ( Huang et al., 2000a,b, 2002, 2004 ; Shi et al., 2000; Guo et al., 2001; Hwanget al., 2002; Wallin and Ristinmaa, 2005 ) to study the effect of plasticity length scale atmicron and submicro scales. The mostly used approach to study the size effects throughstrain gradients is adding of a variable (tensorial) involving the gradients of plastic strain.There are some models ( Acharya and Bassani, 2000 ) where strain gradients are incorpo-rated in the hardening moduli only, this way the classical mathematical framework of standard plasticity is preserved.

Strain gradient crystal plasticity formulations have been proposed by many groups(Smyshlyaev and Fleck, 1996; Shu and Fleck, 1998; Acharya and Bassani, 2000; Regueiroet al., 2002; Shizawa and Zbib, 1999; Menzel and Steinmann, 2000; Gurtin, 2002; Yemov




and Van der Giessen, 2005; Han et al., 2005a,b; Cheong et al., 2005; Ohashi, 2005 ). Thesecontinuum models take into account for the different role of geometrically necessary dis-locations and statistically stored dislocations. Acharya and Bassani (2000) described thedependence of the work hardening moduli on strain gradients with the help of dislocationdensity tensor. Regueiro et al. (2002) derived a relation between dislocation density tensorand back stresses. Shizawa and Zbib (1999), Menzel and Steinmann (2000) and Gurtin(2002) also used the dislocation density tensor with kinematic hardening which incorpo-rated second order strain gradients. Yemov and Van der Giessen (2005), Han et al.(2005a,b) and Cheong et al. (2005) incorporated the geometrically necessary dislocationdensities with in the slip system resistance. Ohashi (2005) suggested a model where the geo-metrically necessary dislocations densities are incorporated in the strain hardening moduliwhile like other strain gradient crystal plasticity theories geometrically necessary disloca-tion densities are obtained from spatial gradients of the plastic shear strain on slip systems.Brinckmann et al. (2006) proposed a rst-order strain gradient model based on dislocationdensity. The results are compared with the mechanism-based strain gradient plasticity the-ory of Huang et al. (2004) and are found to be consistent. Al-Rub and Voyiadjis (2006)proposed a physically motivated mathematical model of strain gradient plasticity. The dis-location processes which include generation, motion, immobilization, recovery, andannihilation are included in statistically stored dislocations while geometrically necessarydislocations are dened as a function of the gradient of plastic strain (shear)gradient.

Strain gradient theories are commonly classied as higher- and lower-order theories,higher-order theories include higher-order stress which increase the order of governing

balance equations and also additional boundary conditions are needed (for example,Shu and Fleck, 1998; Shu et al., 1999; Gurtin, 2002; Al-Rub and Voyiadjis, 2005 ). Inlower-order theories neither additional higher-order stress quantities are involved noradditional boundary conditions ( Acharya and Bassani, 2000; Meissonnier et al., 2001;Yemov and Van der Giessen, 2005; Han et al., 2005a,b; Cheong et al., 2005; Ohashi,2005). Therefore, the lower-order strain gradient theories preserve the classical structureof conventional crystal plasticity (as discussed in Section 2) and also it does not requirehigher-order stress or additional boundary conditions, on the other hand, these lower-order strain gradient theories cannot capture, for example, the formation of boundary lay-ers in case of hindered plasticity ( Shu and Fleck, 1999; Gurtin, 2002 ). The numerical

implementation of higher-order theories with additional boundary conditions andhigher-order governing balance equations have been discussed in detail in Shu et al.(1999) and Al-Rub and Voyiadjis (2005) .

The mechanism-based strain gradient theory (MSG) which is being used here isdeveloped in Han et al. (2005a,b) . It is a generalization of the conventional crystal plas-ticity theory ( Hill, 1967; Rice, 1971; Hill and Rice, 1972) and as discussed above is alower-order theory. The mechanism-based strain gradient crystal plasticity theoryreduces to conventional crystal plasticity theory when strain gradients vanish or whenthe scale of the non-uniform deformation is larger than the intrinsic length scale l . Thiswill be shown in the following. The intrinsic length scale is given by (Han et al.,2005a,b):

l ¼a 2l 2b

ð g 0Þ2 ð8Þ




where a is an empirical coefficient ranging from 0.3 to 0.5, l is the shear modulus, b themagnitude of Burgers vector while g 0 denotes a reference slip resistance which is takento be l /100 (Han et al., 2005a ).

In mechanism-based strain gradient crystal plasticity theory ( Han et al., 2005a,b ), theeffective slip resistance g aT is the resultant of the slip resistance due to strain hardeningwhich is caused by statistically stored dislocations g aS for slip system a and the slip resis-tance due to the effective density of geometrically necessary dislocations g aG .

g aT ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffið g aS Þ2 þ ð g aG Þ

2q ð9Þ

The slip resistance due to strain hardening which is caused by statistically stored disloca-tions g aS is computed using equation, given as

_ g ðaÞs ¼

Xb

hab _cðbÞ

The intrinsic length scale l enters into the effective slip resistance g aT through the slip resis-tance contribution due to the effective density of geometrically necessary dislocations ga

G .

gaG ¼

o ca

o x

which is given by Han et al. (2005a)

gaG ¼ ma Â Xb

sab r cb Â mb ð10Þ

where ma is the slip plane normal while sab = sa Æsb with sa is the slip direction and · de-notes the vector product.

The slip resistance due to the effective density of geometrically necessary dislocations g aG is given by

g aG ¼ ffiffiffiffiffiffiffil gaG p ð11Þ

The modied effective slip resistance is, therefore, given by

g aT ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffið g aS Þ

2 þ l gaG

q ð12Þ

Using the above equation of effective slip resistance, the plastic slip rate is then modied to

_ca ¼ _ca0sgnðsðaÞÞ

sðaÞ

g ðaÞT

( )m

ð13Þ

In what follows next, the above discussed relations for mechanism-based strain gradientcrystal plasticity theory have been used to study strain gradient effect on the stressesaround an interface crack tip along with the J -integral values. As the mechanism-basedstrain gradient crystal plasticity theory is a lower-order theory, therefore, it neither in-volves any higher-order stress nor additional boundary conditions beyond those in con-

ventional crystal plasticity theory. In other words, all governing equations are identicalto those in conventional crystal plasticity theory except that the incremental plasticitymoduli (see Eq. (12)) are changed to account for the strain gradient effects ( Niordsonand Hutchinson, 2003 ).




The UMAT subroutine consisting of conventional crystal plasticity theory Huang(1991) has been modied (the implementation of conventional crystal plasticity theoryin the nite element code, ABAQUS, has been discussed in detail in Huang, 1991 andKysar, 2001a,b ) and above discussed relations of mechanism-based crystal plasticity the-ory (Eqs. (11)–(16)) have been implemented in the UMAT subroutine in order to study theeffect of strain gradient theory on the stresses around the crack tip and fracture energies.The implementation is straightforward, only few modications are made in the conven-tional crystal plasticity subroutine ( Huang, 1991 ). The intrinsic length scale l has beencomputed using Eq. (11), l ¼ a2l 2b

ð g 0Þ2 . The effective density of geometrically necessary dislo-cations ga

G is computed using Eq. (13), gaG ¼ ma Â Pb s

ab r cb Â mb . The slip resistancedue to the effective density of geometrically necessary dislocations g aG is computed usingEq. (14), g aG ¼ ffiffiffiffiffiffiffil ga

G p . Finally the modied effective slip resistance is obtained using Eq.

(15), g aT ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffið g aS Þ

2 þ l gaG

q . This modied effective slip resistance is then used in the visco-

plastic law _ca ¼ _ca0sgnðsðaÞÞ sðaÞ

g ðaÞT

malready implemented in Huang (1991) .

4. The cohesive model

Cohesive models have found increasing interest to simulate fracture in metallic,polymeric, and ceramic materials and their composites ( Barenblatt, 1959; Dugdale,1960; Xu and Needleman, 1994; Tvergaard and Hutchinson, 1992; Camacho and Ortiz,1996; Geubelle and Baylor, 1998; Scheider and Brocks, 2003 ). Using cohesive models,the fracture can be simulated for any structure with or without a crack. The idea forthe cohesive model is based on the consideration that the fracture mechanics analysispresupposes the existence of an innitely sharp crack leading to singular crack tipelds. However, in real materials neither the sharpness of the crack nor the stress levelsnear the crack tip region can be innite. Barenblatt (1959) and Dugdale (1960) werethe rst ones to propose the concept of cohesive model to overcome thesedifficulties.

Barenblatt (1959) , who investigated the fracture of brittle materials, dened the tractionalong the crack path as a function of the crack tip distance along the crack front. The typ-ical traction-crack tip distance is shown in Fig. 1(a).

Dugdale (1960) introduced the nite stress at the crack tip to be the yield stress, which isnot true always as the crack-opening stress can be much higher than the yield stress. Thetypical curve is shown in Fig. 1(b).

Xu and Needleman (1994) was the one of the rst to use polynomial and exponentialtypes of traction–separation equations to simulate the particle debonding in metal matri-ces. The exponential t is used for normal traction while trigonometric t for shear trac-tion. The traction–separation curves are plotted in Fig. 1(c). Xu and Needleman (1994)further used the above models to study the void nucleation at the interface of particleand matrix metal. Tvergaard and Hutchinson (1992) used a trapezoidal shape of the trac-tion–separation model to calculated the crack growth resistance in elasto-plastic materials.

The typical curve is shown in Fig. 1(d). Tvergaard (1990) also extended the Needleman(1987) model of pure normal separation for mixed mode loading. This is one of the mostpopular cohesive laws used by many authors (e.g., Chaboche et al., 1997 ). The typicalcurve is shown in Fig. 1(d).




Camacho and Ortiz (1996) employed a linear cohesive fracture mode to propagate mul-tiple cracks along arbitrary paths during impact damage in brittle materials. This modelpredicts failure by both shear and normal separation in tension and by shear separationin compression. The typical curve is plotted in Fig. 1(f).

Fig. 1. Various cohesive laws proposed by different authors: (a) Barenblatt (1959) ; (b) Dugdale (1960) ; (c) Xu andNeedleman (1994) ; (d) Tvergaard and Hutchinson (1992) ; (e) Scheider and Brocks (2003) ; (f) Camacho and Ortiz(1996); (g) Geubelle and Baylor (1998) .




Geubelle and Baylor (1998) utilized a bilinear cohesive model to simulate spontane-ous initiation and propagation of cracks in thin composite plates subjected tolow-velocity impact. The traction separation curves for this model are shown inFig. 1(g).

A cohesive law proposed by Scheider and Brocks (2003) has been used for the case of agrowing crack. Cohesive models have found increasing interest to simulate fracture inmetallic, polymeric, and ceramic materials and their composites. Using cohesive models,the fracture can be simulated for any structure with or without an initial crack. The ideafor a cohesive model is based on the consideration that the fracture mechanics analysispresupposes the existence of an innitely sharp crack leading to the singular crack tipelds. However, in real materials neither the sharpness of the crack nor the stress levelsnear the crack tip region can be generally innite.

In the cohesive law proposed by Scheider and Brocks (2003) the initial stiffness of the cohesive element can be varied by changing d1, as shown in Fig. 2. If d1 is increasedthe slope of the curve before r c is reached, decreases and similarly the slope of thecurve increases if d1 is decreased. Also a region can be dened, where the traction inthe cohesive element is kept constant. This has been achieved by using two additionalparameters d1 and d2 (Fig. 2), leading to the following formulation for the functionr (d):

r ðdÞ ¼ r c

2 dd1 À d

d1 3

d 6 d1

1 d1 6 d 6 d2

2 dÀd2

dcÀd2

3

À3 dÀdc

dcÀd2

2

þ1 d

26 d 6 d

c

8>>><>>>:

ð14Þ

Both normal and shear behaviour are described by the same relation as given in Eq. (14).In order to take into account the mode mixity effects an additional normal and shear cou-pling function is introduced (see Eq. (15)), this function reduces the maximum traction astangential separation increases.

g ðdÞ ¼ 2dd0

3

À 3dd0

2

þ 1 ð15Þ

With normal and shear coupling term (Eq. (15)), the mixed mode traction separation law

is then given byr m mod eðdÞ ¼ r ðdÞ g ðdÞ ð16Þ

Fig. 2. Cohesive law proposed by Scheider and Brocks (2003) .




where r m mod e(d) is the traction with mixed mode coupling term. The surface plot of Eq.(16) is shown for normal traction in Fig. 3. It can been seen as tangential separation dT

increases the traction value decreases for the same value of normal separation dn .

5. Finite element model

The nite element model is based on the experiments performed in Korn (1993) andKorn et al. (2002) , in which the inuence of orientation and impurities on the fracturebehaviour of Nb/sapphire interfaces were studied using notched bending tests. The spec-imens to perform four-point bending test experiments in Korn (1993) and Korn et al.(2002) were prepared by diffusion bonding single crystals of niobium and alumina in aultra high vacuum furnace for different interface orientations. It was found that forundoped bicrystals bonded at 1400 ° C, the computed interfacial fracture energy rangedfrom 77 to 2100 J/m 2 depending on the interface planes of Nb and sapphire. For fractureevaluation, interfacially notched bending test specimens of dimensions 2 · 4 · 36 mm3

were prepared. The specimen is shown in Fig. 4. The notch length is 0.4 mm. The specimenis loaded to the fracture load F c in a four-point bending tests device at a crosshead speed of

Fig. 3. Mixed mode traction–separation law ( Scheider and Brocks, 2003 ).

Nb single crystal

Polycrystalline Nb sheet

Alumina Shank

60 οm

Alumina single crystal

Niobium Alumina

Crack Tip

Alumina Shank

Nb single crystal

Poly crystalline Nb sheet

Alumina Shank

F F

60 m60 μm

Alumina single crystal

Niobium Alumina

Crack Tip

Alumina Shank

Fig. 4. Four-point bending test specimen.




96.8 l m/min. The load F and the crosshead deection at the load points v are simulta-neously recorded.

The nite element model is constructed based on the information provided in Korn(1993) and Korn et al. (2002) . As shown in Fig. 4, for the case of stationary crack tipthe nite element model of niobium single crystal is bonded to the alumina using bondoption available in ABAQUS while for the case of growing crack the bond option atthe interface was removed with a layer of cohesive elements of zero thickness. The cohesivelaw used for this study is proposed by Scheider and Brocks (2003). Finite element modelsof alumina shanks and polycrystalline niobium sheet was always joined with the corre-sponding niobium single crystal nite element model and alumina nite element modelusing tie constraint available in ABAQUS. The nite element mesh consisted of 39,508plane strain four-noded quadrilateral elements, as shown in Fig. 5. As the mechanism-based strain gradient crystal plasticity theory is a lower-order theory, therefore linear ele-ments can be used because strain gradient never appears directly as an independent vari-able. While the use of linear elements for the higher-order theory causes problems due tothe fact that in higher-order theory not only strain, but also strain gradient play as inde-pendent variables, therefore if linear displacement element are used for higher-order the-ory then the strain is constant and strain gradient always vanishes. In other words,mechanism-based strain gradient crystal plasticity theory is a lower-order theory whichdoes not involve the strain gradient directly. Only the gradient of plastic strain (not totalstrain) is evaluated via the interpolation from the Gaussian integration points ( Huanget al., 2004). Throughout this work implicit integration scheme is used.

Fig. 5. Finite Element Mesh of four-point bending test bicrystal specimen.




For all simulations, both outer alumina shanks (ceramic) and the alumina single crystalat the middle of the specimen were treated as purely elastic with a Young’s modulus of 390 GPa and a Poisson’s ratio of 0.27. The polycrystalline niobium sheet is always mod-elled with an elastic–plastic constitutive law. Young’s modulus and Poisson’s ratio for thepolycrystalline niobium sheet were the same for all simulations ( E = 104.9 GPa,v = 0.397). These elastic and plastic data are adjusted to alumina and niobium, respec-tively, given in Kohnle et al. (2000, 2002). The plastic behaviour of the stress–strain curveof the polycrystalline niobium sheet is approximated by a Ramberg–Osgood relation(Kohnle et al., 2000 ), which is described in a one-dimensional case by the followingequation:

E ce ¼ r þ arr 0

nÀ1

r ð17Þ

Here, n denotes the hardening exponent, a the yield offset and r0

the yield stress. Thismaterial law is nonlinear from the beginning, but for commonly used hardening exponents(n P 5) the divergence from linearity is only slight for stresses below r 0. The chosen plas-ticity theory is the deformation plasticity theory (for details see Kohnle et al., 2000 ) andreferences therein, which describes not a plastic material behaviour, but a nonlinear elasticmaterial. This means, that no unloading criterion exists. This choice of the theory to modelpolycrystalline niobium sheet has no effect on the current analyses as during all the simu-lations it is found that niobium polycrystalline sheet is always in the elastic region. Theparameters of the above equation are adjusted to the niobium stress–strain curves in(Kohnle et al., 2000, 2002). The parameter used are n = 6, r 0 = 180 MPa and a = 0.3.

As mentioned before single crystal niobium is modelled using the hardening law of Bassaniand Wu (1991, 1993). The hardening parameters for each slip system are derived in Siddiqand Schmauder (2005a) . The hardening parameters used for each family of slip systems,i.e., (110)[111] and (112)[111], are given in Table 1 . Slip systems (123)[111] were notconsidered during the simulation, this is based on the experimental ndings of Bowenet al. (1967) and Ma and Raabe (2004) , his experimental results showed that (123)[111]slip systems are only activated when substitutional impurities are present in the single crys-talline niobium.

For the case of mechanism-based strain gradient crystal plasticity, the values of thecoefficients a , and b used in dening the intrinsic material length parameter l (Eq.

(11)) are taken from Qu et al. (2004) with the magnitudes of Burgers vectorb = 0.25 nm, coefficient a = 0.5. These parameters lead to the value of the intrinsic mate-rial length for angle crystalline niobium to be, l = 5.29 l m while for the case of conven-tional crystal plasticity theory the intrinsic material length ( l ) was taken to be zero. Allthe simulations have been performed for the niobium(l1 0)[00 1] ialumina(11À20)[0001]-interface.

Table 1Hardening parameters for Bassani & Wu hardening law

s0 (MPa) s s (MPa) h 0 (MPa) h s (MPa) c0 coI f ab f ab I q q1

(110)[111] 13.7 16.4 292.262 1.4 0.25 0.25 10.0 9.9 0.2891 0.2315(1 12)[1 11] 13.07 16.344 49.03325 39.2266 0.1 0.1 0.34 0.3 0.01 0.011




6. Results and discussion

6.1. Stationary crack tip

The distribution of normal stress along the bicrystal niobium/alumina interface aheadof the crack tip is plotted in Fig. 6 for the case of stationary crack tip. The plot shows theresults of both, conventional crystal plasticity and mechanism-based strain gradient crystalplasticity theory.

Both conventional crystal plasticity theory and mechanism-based strain gradient crystalplasticity theory give the same stress distribution at a distance of more than 2.28 l m fromthe crack tip. For the distance less than 2.28 l m from the crack tip, mechanism-basedstrain gradient crystal plasticity theory gives higher normal stress than conventional crys-tal plasticity. This is due to the strain gradient effect at the metal/ceramic interface. If wecompare the ratio of the normal stress at the crack tip to yield stress of single crystallineniobium ( r /r y) then for conventional crystal plasticity theory this ratio is 5.0953 while formechanism-based strain gradient crystal plasticity theory this ratio is increased to 5.7075(with r y = 50.63 MPa for niobium (110)[001]).

This suggests that a strain gradient does increase the normal stress near the crack tipsignicantly, but the difference apart from the crack tip is of the factor larger than 2. Thisdifference (factor larger than 2) in stress can be explained on the basis of results obtainedin Wei and Hutchinson (1998) where the effect of J 2 strain gradient plasticity theory on thefracture energy of the bimaterial interface is discussed in Wei and Hutchinson (1998) .Fig. 7 shows the ratio of fracture energy to work of adhesion ( J c/W adh ) as a function of

cohesive strength to yield stress ratio ( r c/r y) for various values of l /R0 for a homogeneousmaterial, where R0 is the size of the plastic zone. It can be seen that at a given ( J c/W adh )value, (r c/r y) increases as l /R0 increases and that at a given r c/r y ratio J c/W adh increases asl /R0 decreases. These results show that l /R 0 has a strong inuence on the level of stress thatwill be achieved around the crack tip due to strain gradient effects, i.e., the higher the l /R0

value the higher will be the stresses around the crack tip.If we now look into the current niobium/alumina bicrystal interface the value of intrin-

sic material length is l = 5.29 l m as computed above. Also, the size of the plastic zone at

0

50

100

150

200

250

300350

0 4 8 12 16 20

Distance fromthe crack tip ( μm)

N o r m a

l s t r e s s a

l o n g

t h e c r a c k

f r o n

t ( M P a

) MSG – Crystal Plasticity

Conventional Crystal Plasticity

Niobium(110)[001]|alumina(11-20)[0001]

2.28 μm

0

50

100

150

200

250

300350

MSG – Crystal Plasticity


Niobium(110)[001]|alumina(11-20)[0001]

2.28 m

Fig. 6. Normal stress distribution along the crack front for the case of conventional crystal plasticity and MSG – crystal plasticity.




this stress level is found to be 32.5 l m. The size of plastic zone is an averaged size aroundthe crack tip which is obtained by taking the mean of all the distances around the crack tipuntil which the plastic deformation occurs. The size of plastic zone is also obtained byusing the relation proposed by Tvergaard and Hutchinson (1993) (for details see referencestherein) which also gives a similar value for the current niobium/alumina bicrystal inter-face R0 $ 32 l m. This gives a value of l /R0 = 0.1628. This is the region where ( r c/r y) is lesssensitive to the l /R0 value due to the reason that the scale of plastic deformation is larger

than the intrinsic material length l and that is why mechanism-based strain gradient crystalplasticity theory has only a small inuence on the ( r c/r y) ratio for the case of niobium/alu-mina bicrystal interfaces for all interface orientations. It should also be noted that if amaximum stress criterion is used as fracture criterion rather than the cohesive model,where a critical stress value at a distance ahead of the crack tip is prescribed then the dif-ference in maximum stress obtained from both conventional and mechanism-based crystalplasticity theory will be sensitive to the distance prescribed ahead of the crack tip. For thecase of niobium/alumina interface the region is found to be in the range of 0.0–1.9 l m.

The fracture energies computed for the case of mechanism-based strain gradient crystalplasticity theory and conventional crystal plasticity theory were found to be identical, as

shown in Fig. 8. This is due to the limited region (0.1–2 l m) where stresses were foundto be higher for the case of mechanism-based strain gradient crystal plasticity theory.It is also found that for niobium (110)[001] orientation (1 À12)[À111], (À211)[111]

and (211)[À111] systems play a major role in the total plastic slip and eventually in theplastic strain (see Fig. 9). The reason is the highest Schmidt factor for these(1À12)[À111], (À21 1)[111] and (21 1)[À111] systems, i.e. $À 0.4714, À0.2357 andÀ0.408 while other slip systems have smaller Schmidt factor $À 0.2 or even less. Fig. 10shows the relative magnitudes of the slip activity, where the slip activity at a radius of 15 l m (it is a reasonable distance from the crack tip where inuence of all the activatedslip system was seen during the simulations; distances other than 15 l m give similarresults, i.e. same slip systems contributing in total accumulated slip but only the magnitudeof plastic strain changes) from the crack tip is plotted for each of the activated slip systemsrelative to the polar angle u ; 0° coincides with the prolongation of the crack and 180 ° coin-cides with the single crystal niobium crack ank. It can be easily identied which slip sys-

0

1

2

3

4

56

7

8

9

10

0 5 10 15σc /σy

J C / W

a d h

l / R0 = 0.1

.2 .3 .5 1.0

Fig. 7. The inuence of strain gradient plasticity on ( J c/W adh ) for homogeneous material ( Wei and Hutchinson,

1998, for details see text).




tem is contributing the highest amount of plastic slip in different regions around the cracktip. Fig. 10 shows that the maximum contribution of plastic slip comes from the slip sys-tem (1 À12)[À111] for the polar angle u range of 0–60° while the contribution of slip sys-tems (À211)[111] and (211)[À111] is maximum for the range of 60–120 ° .

6.2. Quasistatically growing crack

All nite element analyses of crack growth have been performed for the case when nio-bium (110)[001] is bonded with alumina (11 À20)[00 01] using crystal plasticity theory

Fig. 9. Plastic strain on various slip systems at the tip of a stationary crack for orientation II.

0

0.5

1

1.5

2

2.5

3

3.5

4

0 5 10 15 20 25 30

Loading point displacement ( μm)

J - I n t e g r a

l ( J / m 2 )


MSG - Crystal Plasticity

0

0.5

1

1.5

2

2.5

3

3.5

4

0 5 10 15 20 25 30

Loading point displacement ( m)

J - I n t e g r a

l ( J / m 2 )



Fig. 8. J -integral vs loading point displacement for conventional crystal plasticity and MSG-crystal plasticity.




where single crystalline niobium is always modelled using the Bassani & Wu hardeninglaw. The experimental fracture energy ( Korn et al., 2002 ) for this orientation was foundto be 370 J/m 2 for a bonding temperature of 1400 ° C.

Parametric studies have been performed to study the effect of cohesive law parameters,such as cohesive strength and cohesive energy (also known as work of adhesion). This isbeing done for calibrating the cohesive law parameters, in order to reach the experimen-tally found fracture energy of this metal/ceramic system. Both conventional crystal plas-ticity theory and mechanism-based strain gradient crystal plasticity theory will be usedfor the analyses.

The effect of cohesive strength on the fracture energy of bicrystal nio-bium(110)[001]ialumina(11 À20)[0001] has been studied by keeping the work of adhe-sion to be 4.0 J/m 2. Fracture energy for various values of cohesive strength/yield stressratios is plotted in Fig. 11. These results depict that as the cohesive strength increases,the fracture energy also increases gradually until r c/r y @ 4, after this value of r c/r y theslope changes and the fracture energy becomes more sensitive to cohesive strength, thisis due to the higher plastic dissipation as cohesive strength increases. Tile results also show

( 1 -1 2 )[ -1 1 1 ]

( 2 1 1 )[ -1 1 1 ]

( -2 1 1 )[ 1 1 1 ]

Total Accumulated Slip

λ Nb

( 1 -1 2 )[ -1 1 1 ]

( 2 1 1 )[ -1 1 1 ]

( -2 1 1 )[ 1 1 1 ]

Total Accumulated Slip

λ Nbϕ Nb

30 60 90 120 150 1800

0.005

0.015

0.02

0.025

Angle (degrees)

0

0.01

P l a s

t i c s

l i p ( s t r a i n )

Fig. 10. Plastic strain on slip systems at 15 l m radius from the tip of a stationary crack for orientation II.

0

50

100

150

200

250

300

350

400

0 2 4 6 8σ c /σ y

J c ( J / m

2 )

Convenional Crystal Plasticity


Experimental J c = 370 J/m 2 [53](Korn et al. 2002)

Wadh = 4 J/m 2

Fig. 11. Fracture energy obtained from conventional and mechanism-based strain gradient crystal plasticitytheory for different cohesive strength/yield stress ratios.




that the r c/r y ratio required for mechanism-based crystal plasticity theory is 5.7 while ther c/ r y value obtained using conventional crystal plasticity theory is 5.1 for the same mac-roscopic fracture energy value of 370 J/m 2.

From the same set of analyses, cohesive strength ratio ( r (MSG) / r (CCP) ) is plotted as func-tion of (r /r y)(MSG) in Fig. 12 for the work of adhesion value of 4 J/m 2, where CT (MSG)

denotes the cohesive strength obtained from mechanism-based strain gradient crystal plas-ticity theory while r (CCP) is the cohesive strength obtained from conventional crystal plas-ticity theory (cohesive strength, in general, is the peak stress level attained by the cohesiveelements during deformation at which damage initiates). It shows that as ( r /r y)(MSG)

increases, the value of ( r (MSG) /r y(CCP) ) also increases and then starts to saturate after a(r / r y)(MSG) value of 3.5. This is due to the decrease in l /R0 ratio as shown in Fig. 13, whichdecreases as the cohesive strength increases due to the higher plastic dissipation and largerplastic zone.

A similar study has been done by keeping the cohesive strength constant and by varyingthe work of adhesion from 1 J/m 2 to 9.8 J/m 2, this change of work of adhesion does notshow any inuence on the size of the plastic zone and the l /R0 ratio is always the samewhile the macroscopic fracture energy increases linearly with increasing work of adhesionproviding the cohesive strength to be constant.

11.021.041.061.08

1.1

1.121.14

σ/σ y(MSG )

σ ( M S G ) / σ ( C

C P )

Wadh = 4.0 J/m 2

11.021.041.061.08

1.1

1.121.14

Wadh = 4.0 J/m 2

2 42 3 5 6

Fig. 12. Ratio of the cohesive strength obtained using MSG-crystal plasticity theory and conventional crystalplasticity theory ( r (MSG) /r y(CCP) ) for different ratios of stress normal to the crack tip to the yield stress obtainedfrom MSG-crystal plasticity theory ( r /r

y)

(MSG).

σ/σ y(MSG )

0

10

20

30

40

50

60

2 3 4 5

l / R 0

6

W adh = 4.0 J/m 2

0

10

20

30

40

50

60

2 3 4 5

l / R 0

6

W adh = 4.0 J/m 2

Fig. 13. Ratio of the intrinsic material length to size of plastic zone ( l /R0) for different ratios of stress normal tothe crack tip to the yield stress obtained from MSG-crystal plasticity theory ( r /r y)(MSG) .




Table 2 shows the nal results of the cohesive strength, corresponding values of work of adhesion, i.e. 1.0, 4.0, and 9.8 J/m 2, size of plastic zone obtained from conventional andmechanism-based strain gradient crystal plasticity theory for a macroscopic fractureenergy of 370 J/m 2 (Korn et al., 2002 ). It was found that as the l /R0 ratio decreases the

(r (MSG) /r y(CCP) ) value also decreases as is discussed above and also shown in Figs. 12and 13.

7. Conclusion

A comparative study of the tensile stresses at the interface crack tip was presented usingmechanism-based strain gradient crystal plasticity theory and conventional crystal plastic-ity theory. Fracture analyses were performed for a stationary crack tip and for a growingcrack using a cohesive modelling approach.

It is found that by using mechanism-based strain gradient theory we can predict higherstresses at the crack tip as compared to the conventional crystal plasticity theory. Also, ourresults show that strain gradient effects strongly depend upon the intrinsic material lengthto size of the plastic zone ratio ( l /R 0). We have found that as the l /R0 ratio increases thenormal stresses reached using mechanism-based strain gradient crystal plasticity theoryare much larger than the stresses obtained from conventional crystal plasticity theory.We also presented an insight about the role of cohesive strength and work of adhesionon fracture energies. This information will help experimentalists design better materials.

Acknowledgements

The presented work is funded by the Deutsche Forschungsgemeinschaft within theGraduiertenkolleg ‘‘Internal interfaces in Crystalline Materials’’, which is gratefullyacknowledged. Thanks are also due to Prof. M. Finnis, Dr. R.M. Cannon and Dr. I. Sche-ider for useful discussion.

References

Acharya, A., 2001. A model of crystal plasticity based on the theory of continuously distributed dislocations. J.Mech. Phys. Solids 49, 761–784.

Acharya, A., Bassani, J.L., 2000. Lattice incompatibility and a gradient theory of crystal plasticity. J. Mech. Phys.

Sol. 48, 1565–1595.Al-Rub, R.K., Voyiadjis, G.Z., 2005. A direct nite element implementation of the gradient-dependent theory.

Int. J. Numer. Meth. Eng. 63, 603–629.Al-Rub, R.K., Voyiadjis, G.Z., 2006. A physically based gradient plasticity theory. Int. J. Plast. 22, 654–684.

Table 2Comparison of parameters obtained from conventional and mechanism-based strain gradient crystal plasticitytheory

Conventional crystal plasticity MSG-crystal plasticity R0 (l m) l /R 0 r c(CCP) /r c(MSG)

r c W adh r c W adh

283 1.0 319.6 1.0 27.4 0.193 1.13258 4.0 289 4.0 32.5 0.163 1.12225 9.8 248 9.8 34.8 0.152 1.102




Armstrong, R.W., 1966. Cleavage crack propagation within crystals by the Griffith mechanism versus adislocation mechanism. Mater. Sci. Eng. 1, 251.

Arsenlis, A., Parks, D.M., 2002. Modeling the evolution of crystallographic dislocation density in crystalplasticity. J. Mech. Phys. Solids 50, 1979–2009.

Asaro, R.J., 1979. Geometrical effects in the inhomogeneous deformation of ductile single crystals. Acta Metall.27, 445–453.

Asaro, R.J., 1983a. Micromechanics of crystals and polycrystals. Advances in Applied Mechanics 23, 1, Ed. J.W.Hutchinson.

Asaro, R.J., 1983b. Crystal plasticity. J. Appl. Mech. 50, 921–934.Baitrev, I.G., Alavi, A., Finnis, M.W., 1999. First-principle calculations of the ideal cleavage energy of bulk

niobium(111)/alpha-alumina(00 01) interfaces. Phys. Rev. Lett. 82, 1510–1513.Barenblatt, G.I., 1959. The formation of equilibrium cracks during brittle fracture. General ideas and hypothesis.

Axisymmetrical cracks. PMM. 23, 434–444.Bassani, J.L., Wu, T.-Y., 1991. Latent hardening in single crystals II. Analytical characterization and predictions.

Philos. Trans. Roy. Soc. London A 435, 1–19.Bassani, J.L., Wu, T.-Y., 1993. Latent hardening in single crystals I. Theory and experiments. Math. Phys. Sci.

435, 21–41.Beltz, G.E., Rice, J.R., 1992. Dislocation nucleation at metal/ceramic interfaces. Acta Metall. Mater. 40, S321– S331.

Beltz, G.E., Wang, J.S., 1992. Crack direction effects along copper/sapphire interfaces. Acta Metall. Mater. 40,1675–1683.

Bowen, D.K., Christian, J.W., Taylor, G., 1967. Deformation properties of niobium single crystals. Can. J. Phys.45, 903–938.

Brinckmann, S., Siegmund, T., Huang, Y., 2006. A dislocation density based strain gradient model. Int. J. Plast.22, 1784–1797.

Camacho, G.T., Ortiz, M., 1996. Computational modelling of impact damage in brittle materials. Int. J. SolidsStruct. 33, 2899–2938.

Chaboche, J., Girard, R., Levasseaur, P., 1997. On the interface debonding models. Int. J. Damage Mech. 6, 220–

231.Cheong, K.S., Busso, E.P., Arsenlis, A., 2005. A study of microstructural length scale effects on the behaviour of FCC polycrystals using strain gradient concepts. Int. J. Plast. 21, 1797–1814.

Chiao, Y.H., Clarke, D.R., 2003. Direct observation of dislocation emission from crack tips in silicon at hightemperatures. Acta Metall. 37, 203–219.

Clegg, W.J., Horsfall, I., Mason, J.F., Edwards, L., 1988. The tensile deformation and fracture of Al-‘‘Shaffil’’metal–matrix composites. Acta Metall. 36, 2151–2159.

Cuitino, A.M., Ortiz, M., 1992. Computational modelling of single crystals. Modell. Simul. Mater. Sci. Eng. 1,225–263.

Dugdale, D.S., 1960. Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100–104.Evans, A.G., Dalgleish, B.J., 1992. The fracture resistance of metal ceramic interfaces. Acta Metall. Mater. 40,

S295–S306.Evans, A.G., Ru¨hle, M., Dalgleish, B.J., Charalambides, P.G., 1990. The fracture energy of bimaterial interfaces.

In: Metal–Ceramic Interfaces. Pergamon Press, Oxford, p. p. 340.Finnis, M.W., Kruse, C., Scho ¨nberger, U., 1995. Ab initio calculations of metal/ceramic interfaces: what have we

learned, what can we learn? Nanostruct. Mater. 6, 145–155.Fischmeister, H.F., 1987. In: Pask, J.A., Evans, A.G. (Eds.), Ceramic Microstructures ‘86: Role of Interfaces.

Plenum Press, New York.Fischmeister, H.F., Mader, W., Gibbesch, B., Elssner, G., 1988. Joining of ceramic, glass and metal. Mater. Res.

Soc. Symp. Proc. 122, 529.Fuller, E.R., Lawn, B.R., Thomson, R.M., 1980. Atomic modelling of chemical interactions at crack tips. Acta

Metall. 28, 1407–1414.Geubelle, P.H., Baylor, J., 1998. The impact-induced delamination of laminated composites: a 2D simulation.

Composites Part B 29, 589–602.Guo, Y., Huang, Y., Gao, H., Zhuang, Z., Hwang, K.C., 2001. Taylor-based nonlocal theory of plasticity:

numerical studies of the micro-indentation experiments and crack tip elds. Int. J. Solids Struct. 38, 7447– 7460.




Gurtin, M.E., 2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessarydislocations. J. Mech. Phys. Solids 50, 5–32.

Han, C.S., Gao, H., Huang, Y., Nix, W.D., 2005a. Mechanism-based strain gradient crystal plasticity – I. Theory.J. Mech. Phys. Solids 53, 1188–1203.

Han, C.S., Gao, H., Huang, Y., Nix, W.D., 2005b. Mechanism-based strain gradient crystal plasticity – II.Analysis. J. Mech. Phys. Solids 53, 1204–1222.

Harder, J., 1999. A crystallographic model for the study of local deformation processes in polycrystals. Int. J.Plast. 15, 605–624.

Hill, R., 1967. Generalized constitutive relations for incremental deformation of metal crystals by multislip. J.Mech. Phys. Solids 15, 79–95.

Hill, R., Rice, J.R., 1972. Constitutive analysis of elastic–plastic crystals at arbitrary strain. J. Mech. Phys. Solids20, 401–413.

Hirth, J.P., Rice, J.R., 1980. On thermodynamics of adsorption at interfaces as it inuences decohesion. Metall.Trans. A 11, 1501–1511.

Huang, Y., 1991. A user-material subroutine incorporating single crystal plasticity in the ABAQUS FiniteElement Program. Mech. Report 178, Division of Applied Sciences, Harvard University, Cambridge, MA.

Huang, Y., Gao, H., Nix, W.D., Hutchinson, J.W., 2000a. Mechanism-based strain gradient plasticity – II. Anal.J. Mech. Phys. Solids 48, 99–128.Huang, Y., Xue, Z., Gao, H., Nix, W.D., Xia, Z.C., 2000b. A study of microindentation hardness tests by

mechanism-based strain gradient plasticity. J. Mater. Res. 15, 1786–1796.Huang, Y., Qu, S., Hwang, K.C., Li, M., Gao, H., 2004. A conventional theory of mechanism based strain

gradient plasticity. Int. J. Plast. 20, 753–782.Hwang, K.C., Jiang, H., Huang, Y., Gao, H., Hu, N., 2002. A nite deformation theory of strain gradient

plasticity. J. Mech. Phys. Solids 50, 81–99.Jameel, M.A., Peralta, P., Laird, C., 2003. Initiation and propagation of stage-I cracks in copper single crystals

under load control. Mater. Sci. Eng. A 342, 279–286.Jokl, M.L., Vitek, V., McMahon, C.J., 1980. A microscopy theory of brittle fracture in deformable solids. Acta

Metall. Mater. 28, 1479–1488.

Kameda, T., Zikry, M.A., 1998. Three dimensional dislocation-based crystalline constitutive formulation forordered intermetallics. Scr. Mater. 38, 631–636.Kohnle, C., Mintchev, O., Brunner, D., Schmauder, S., 2000. Fracture of metal/ceramic interfaces. Comput.

Mater. Sci. 19, 261–266.Kohnle, C., Mintchev, O., Schmauder, S., 2002. Elastic and plastic fracture energies of metal/ceramic joints.

Comput. Mater. Sci. 25, 211–212.Korn, D., 1993. Bruchenergie und plastische Verformung von grenzachendotierten Metall-Keramic-Verbunden.

Dissertation, University of Stuttgart.Korn, D., Elssner, G., Cannon, R.M., Ru ¨hle, M., 2002. Fracture properties of interfacially doped Nb–Al 2O3

bicrystals: I, fracture characteristics. Acta Mater. 50, 3881–3901.Kurkjian, C.R., Kingery, W.D., 1956. Surface tension at elevated temperatures, III. Effect of Cr, In, Sn, and Ti on

liquid Ni surface tension and interfacial energy with alumina. J. Am. Ceram. Soc. 60, 961–963.Kysar, J.W., 2000. Directional dependence of fracture in copper/sapphire bicrystal. Acta Mater. 48, 3509–3524.Kysar, J.W., 2001a. Continuum simulations of directional dependence of crack growth along a copper/sapphire

bicrystal interface: Part I, Experiments and crystal plasticity background. J. Mech. Phys. Solids 49, 1099– 1128.

Kysar, J.W., 2001b. Continuum simulations of directional dependence of crack growth along a copper/sapphirebicrystal interface: Part II, Crack tip stress and deformation analysis. J. Mech. Phys. Solids 49, 1129–115i3.

Kysar, J.W., Briant, C.L., 2002. Crack tip elds in ductile single crystals. Acta Mater. 50, 2367–2380.Ma, A., Raabe, D., 2004. MPI, Du ¨sseldorf (unpublished data).McHugh, P.E., Varias, A.G., Asaro, R.J., Shih, C.F., 1989. Computational modelling of microstructures. Future

Generat. Comput. Syst. 5, 295–318.McMahon, C.J., Vitek, V., 1979. The effects of segregated impurities on intergranular fracture energy. Acta

Metall. Mater. 27, 507–513.Meissonnier, F.T., Busso, E.P., O’Dowd, N.P., 2001. Finite element implementation of a generalised non-local

rate-dependent crystallographic formulation for nite strains. Int. J. Plast. 17, 601–640.Menzel, A., Steinmann, P., 2000. On the continuum formulation of higher gradient plasticity for single and

polycrystals. J. Mech. Phys. Solids 48, 1777–1796.




Methfessel, M., Hennig, D., Scheffler, M., 1992. Trends of the surface relaxations, surface energies, and workfunctions of the 4d transition metals. Phys. Rev. B 46, 4816–4829.

Nakada, Y., Keh, A.S., 1966. Latent hardening in iron single crystals. Acta Metall. 14, 961.Nakatani, A., Drugan, W.J., Giessen, E.V., Needleman, A., 2003. Crack tip elds at the ductile single crystal-rigid

material interface. Int. J. Fract. 122, 131–159.Needleman, A., 1987. A continuum model for void nucleation by inclusion debonding. ASME J. Appl. Mech. 54,

525–531.Nicholas, M.G., Lee, R.J., 1989. Joining dissimilar materials. Metals Mater. 1, 348–360.Nicholas, M.G., Valentine, T.M., Waite, M.J., 1980. The wetting of alumina by copper alloyed with titanium and

other elements. J. Mater. Sci. 15, 2167–2206.Niordson, C.F., Hutchinson, J.W., 2003. On lower order strain gradient plasticity theories. Eur. J. Mech. A/

Solids 22, 771–778.Ohashi, T., 2005. Crystal plasticity analysis of dislocation emission from micro voids. Int. J. Plast. 21, 2071–2088.Ohashi, T., Asakawa, K., 2001. Evaluation of dislocation storage by means of crystal plasticity analysis, Mater.

Res. Soc. Symp. Proc. 683E, BBS5.4.1.Peirce, D., Asaro, R.J., Needleman, A., 1982. An analysis of nonuniform and localized deformation in crystalline

solids. Acta Metall. 30, 1087–1119.Price, R.J., Kelly, A., 1964. Deformation of age-hardened aluminium alloy crystals – II. Fract. Acta Metall. 12,979–992.

Qu, S., Huang, Y., Jiang, H., Liu, P.D., Wu, C., Hwang, K.C., 2004. Fracture analysis in the conventional theoryof mechanism-based strain gradient (CMSG) plasticity. Int. J. Fract. 129, 199–220.

Regueiro, R.A., Bammann, D.J., Marin, E.B., Garikipati, K., 2002. A nonlocal phenomenological anisotropicnite deformation plasticity model account for dislocation defects. J. Eng. Mater. Tech. 124, 380–387.

Rice, J.R., 1971. Inelastic constitutive relations for solids: an internal-variable theory and its application to metalplasticity. J. Mech. Phys. Solids 19, 433–455.

Rice, J.R., Suo, Z., Wang, J.S., 1990. Mechanics and thermodynamics of brittle interfacial failure in bimaterialsystems. In: Riihle, M., Evans, M.A.G., Ashby, M.F., Hirth, J.P. (Eds.), Metal–Ceramic Interfaces. PergamonPress, Oxford, p. 269.

Scheider, I., Brocks, W., 2003. Simulation of cup-cone fracture using the cohesive model. Eng. Fract. Mech. 70,1943–1961.Schmidt, E., 1931. U ber die Schubverfestigung von Einkristallen bei plastischer Deformation. Z. Phys. 40, 54–60.Shi, M.X., Huang, Y., Gao, H., Hwang, K.C., 2000. Non-existence of separable crack tip eld in mechanism-

based strain gradient plasticity. Int. J. Solids Struct. 37, 59995–66010.Shizawa, K., Zbib, H.M., 1999. A thermodynamical theory of gradient elastoplasticity with dislocation density

tensor. I: Fundamentals. Int. J. Plast. 15, 899–938.Shu, J.Y., Fleck, N.A., 1998. The prediction of a size effect in micro-indentation. Int. J. Solids Struct. 35, 1363–

1383.Shu, J.Y., Fleck, N.A., 1999. Strain gradient crystal plasticity: size-dependent deformation of bicrystals. J. Mech.

Phys. Solids 47, 292–324.Shu, J.Y., King, W.E., Fleck, N.A., 1999. Finite elements for materials with strain gradients effects. Int. J. Numer.

Mech. Eng. 44, 373–391.Siddiq, A., Schmauder, S., 2005a. Simulation of hardening in high purity niobium single crystals during

deformation. Steel grips. J. Steel Relat. Mater. 3, 281–286.Siddiq, A., Schmauder, S., 2005b. Crystal orientation effects on the fracture energies of a bimaterial interface.

Abstract no. 2.74, Proc. McMAT 2005 Mechanics and Materials Conference (CD-ROM), Louisiana, USA.Siddiq, A., Schmauder, S., 2006. Interface fracture analyses of a bicrystal niobium/alumina specimen using

cohesive modelling approach. Modell. Simul. Mater. Sci. Eng. 14, 1015–1030.Siddiq, A., Schmauder, S., in press. Modelling of crystal plasticity effects on the crack initiation energies of a

bicrystal interface. Comput. Assist. Mech. Eng. Sci. 13.Smyshlyaev, V.P., Fleck, N.A., 1996. The role of strain gradients in the grain size effect for polycrystals. J. Mech.

Phys. Solids 44, 465–495.Soyez, G., 1996. Plastische Verformung und Rißbildung in druckbeanspruchten niobium/alumina Verbunden,

Dissertation, University of Stuttgart.Soyez, G., Elssner, G., Ru¨hle, M., Raj, R., 1998. Constrained yielding in niobium single crystals bonded to

sapphire. Acta Mater. 46, 3571–3581.Taylor, G.I., 1938. Plastic strain in metals. J. Inst. Metals 62, 307–325.




Tvergaard, V., 1990. Effects of bre debonding in a whisker-reinforced metal. Mater. Sci. Eng. A 125, 203–213.Tvergaard, V., Hutchinson, J.W., 1992. The relation between crack growth resistance and fracture process

parameters in elastic–plastic solids. J. Mech. Phys. Solids 40, 1377–1397.Tvergaard, V., Hutchinson, J.W., 1993. The inuence of plasticity on mixed mode interface toughness. J. Mech.

Phys. Solids 41, 1119–1135.Varias, A.G., O’Dowd, N.P., Asaro, R.J., Shih, C.F., 1990. Failure of bimaterial interfaces. Mater. Sci. Eng. A

126, 65–93.Wallin, M., Ristinmaa, M., 2005. Deformation gradient based kinematic hardening model. Int. J. Plast. 21, 2025–

2050.Wang, J.S., 1998. A micromechanical model for interface crack extension in metal/ceramic bimaterial systems.

Acta Mater. 46, 4973–4984.Wang, J.S., Anderson, P.M., 1991. Fracture behaviour of embrittled FCC metal bicrystals. Acta Metall. Mater.

39, 779–792.Wei, Y., Hutchinson, J.W., 1998. Models of interface separation accompanied by plastic dissipation at multiple

scales. Int. J. Fract. 95, 1–17.Xu, X.P., Needleman, A., 1994. Numerical simulation of fast crack growth in brittle solids. J. Mech. Phys. Solids

42, 1397–1434.Yemov, S., Van der Giessen, E., 2005. Size effects in single crystal thin lms: nonlocal crystal plasticitysimulations. Eur. J. Mech. A Solids 24, 183–193.

Zhang, W., Smith, J.R., 2000. Stoichiometry and adhesion of Nb/Al 2O3. Phys. Rev. B 61, 16883–16889.


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International Journal of Plasticity 23 (2007) 665–689