Computational Modeling of Dual-Phase Ceramics with Finsler-Geometric Phase Field Mechanics

John D.Clayton

1US ARL,Impact Physics Branch,Aberdeen Proving Ground,MD,21005-5066,USA.

Abstract:A theory invoking concepts from differential geometry of generalized Finsler space in conjunction with diffuse interface modeling is described and implemented in finite element (FE) simulations of dual-phase polycrystalline ceramic microstructures.Order parameters accounting for fracture and other structural transformations,notably partial dislocation slip,twinning,or phase changes,are dimensionless entries of an internal state vector of generalized pseudo-Finsler space.Ceramics investigated in computations are a boron carbide-titanium diboride (B4C-TiB2) composite and a diamond-silicon carbide (C-SiC) composite.Deformation mechanisms-in addition to elasticity and cleavage fracture in grains of any phase-include restricted dislocation glide (TiB2 phase),deformation twinning (B4C and β-SiC phases),and stress-induced amorphization (B4C phase).The metric tensor of generalized Finsler space is scaled conformally according to dilatation induced by cavitation or other fracture modes and densification induced by phase changes.Simulations of pure shear consider various morphologies and lattice orientations.Effects of microstructure on overall strength of each composite are reported.In B4C-TiB2,minor improvements in shear strength and ductility are observed with an increase in the second phase from 10 to 18% by volume,suggesting that residual stresses or larger-scale crack inhibition may be responsible for toughness gains reported experimentally.In diamond-SiC,a composite consisting of diamond crystals encapsulated in a nano-crystalline SiC matrix shows improved strength and ductility relative to a two-phase composite with isolated bulk SiC grains.

Keywords:Micromechanics,ceramic composites,Finsler space,differential geometry,phase field,finite elements.

1 Introduction

Polycrystalline ceramics are solid materials that generally are mechanically stiff yet prone to brittle fracture.Heterogeneous composite ceramics consisting of grains of different chemical compositions or phases have been engineered to provide increased strength,ductility,and toughness relative to monolithic ceramics.

The current work focuses on two such ceramic composites,each consisting of two distinct phases.One such dual-phase material contains mostly crystalline boron carbide (B4C,trigonal structure)interspersed with secondary grains of crystalline titanium diboride(TiB2,hexagonal structure),wherein grain sizes of both phases are comparable.Versions of this material system are discussed in sigl et al.[Sigl and Kleebe (1995);White and Dickey (2011)].Single-phase boron carbide is prone to brittle failure by cleavage,and at high stresses,may undergo a phase change to an amorphous solid as well as deform by deformation twinning [Chen,McCauley and Hemker (2003);Clayton (2012);An and Goddard (2015)].Titanium diboride may impart higher fracture resistance by cleavage crack deflection or bridging,and some ductililty may be afforded by dislocation motion that has been suggested from shock experiments[Vanderwalker and Croft(1988)].Compressive residual stresses induced in B4C from TiB2by thermal processing may also improve performance.

The second dual-phase material studied is comprised of diamond crystals (C,cubic structure)either interspersed with or encapsulated by silicon carbide micro-or nano-crystals(β-SiC,cubic structure).Though the hardest of natural materials,diamond is prone to cleavage fracture,and incorporation of a softer SiC phase appears to improve overall toughness [Zhao,Qian,Daemen et al.(2004)].As with the other composite,possible reasons include crack deflection at grain and phase boundaries as well as compressive residual stresses in grains of the stiffer phase.Twinning is also prevalent inβ-SiC as a result of its low stacking fault energy[Van Torne(1966);Ning and Ye(1990)].

This paper invokes,in FE simulations,a theory combining aspects of phase field mechanics and differential geometry on generalized Finsler spaces.Perhaps the most comprehensive description of this model,including versions accounting for multiple inelastic deformation mechanisms,finite strain kinematics,and nonlinear elasticity,is given in Clayton et al.[Clayton and Knap (2018a)].A pseudo-Finsler metric tensor is assigned that depends on a vector of internal state variables,whose entries when rendered dimensionless upon division by a regularization length,are interpreted as order parameters in the context of phase field theory.Dependence of the metric on internal state permits changes in lengths,areas,and volumes of material in conjunction with microstructure evolution.For example,voids may lead to dilatation,or structure collapse in B4C may lead to densification[Clayton(2012);An and Goddard (2015)].Incorporation of this metric tensor in the theory leads to a rescaled energy density and augmented balance laws that enable additional physics(e.g.,pressure-shear coupling)without prescription of ad-hoc kinetic equations.The phase field-related components of the model are based on theory in Clayton et al.[Clayton and Knap (2011,2014,2016)],while the Finsler-geometric concepts entering the theory follow from Clayton [Clayton (2017b,c,a,2018a)].Potential advantages of the present computational implementation over other methods[Tomar,Zhai and Zhou(2004);Clayton(2005,2009);Bammann and Solanki(2010)]for modeling inelasticity and fracture include the following:intrinsic regularization by gradient-dependent energy,little if any calibration of parameters,and preservation of mesh topology,e.g.,no element deletion or separation along element boundaries.

A recent study by Clayton et al.[Clayton,Leavy and Knap (2019)] invoked a simpler version of the present theory towards study of the same two ceramic composites.That work invoked pure phase field mechanics[Clayton and Knap(2016)]without incorporation of the pseudo-Finsler metric.Furthermore,uniaxial compression loading was simulated in Clayton et al.[Clayton,Leavy and Knap (2019)],while pure shear (i.e.,biaxial tension-compression) is simulated in the present study.Shear strength is of high interest as it is thought to be an indicator of resistance to ballistic penetration in armor ceramics[Bourne(2008);Clayton(2015,2016)].

Remaining sections of this work are organized as follows.The continuum theory and aspects of its computer implementation are reviewed in §2.Polycrystalline simulations,with results from shear loading,on B4C-TiB2are given in §3.Parallel treatment of diamond-SiC is contained in§4,followed by conclusions.

2 Finsler-geometric phase field theory

The continuum theory invoked in subsequent simulations is tersely summarized here.The present approach extends that of Clayton et al.[Clayton,Leavy and Knap(2019)]to include a generalized pseudo-Finsler metric tensor [Clayton (2017c)],rather than the Euclidean(and more specifically,rectangular Cartesian) metric invoked in Clayton et al.[Clayton,Leavy and Knap (2019)].The present theory implements linearized kinematics [Clayton and Knap (2015)] and quasi-static balance laws following from a variational approach[Clayton(2017b)].Geometrically nonlinear and dynamic models are discussed elsewhere[Clayton(2017a);Clayton and Knap(2018a)].

2.1 Internal state

A vectorDof internal state variables coincides with a director vector of Finsler geometry,though here this state vector need not be of unit length.LetXwith Cartesian componentsXKdenote the reference position vector of a material particle.In a preferred coordinate system,

Entries ofDnormalized by regularization lengthlare order parametersξandη.Parameterξ ∈[0,1]depicts fracture:

Other structure changes are captured byη ∈[0,1]:

For ceramics considered herein,η ＞0 can track shearing by slip or twinning dislocations and monitor localized slip in amorphous bands.It also can represent density variations due to phase changes.

Metric tensorGdepends potentially on coordinatesXas well as internal stateD.Homogeneity is not required in the present generalized pseudo-Finsler theory [Bejancu(1990)],in contrast to strictly Finsler theories wherein only the direction ofDis important.Though more general representations are admissible [Clayton (2017c)],the current model invokes Weyl-type isotropic scaling of the Cartesian metric 1,i.e.,a conformal transformation[Clayton(2017b);Clayton and Knap(2018a)]:

Explicit dependence onXis not necessary for rectangular coordinates.Constantsmandkquantify volume changes associated with respective fracture/cavitation and phase changes.The determinant isG=detG=exp(mξ2+kη2).

2.2 Kinematics

Displacement isu=u(X).Let?(·)denote the covariant derivative with respect toX,here a partial coordinate derivative.Displacement gradient is decomposed as

Elastic distortion isβE(X).The state-dependent contribution from changes in microstructure is[Clayton and Knap(2018a)]

The first term on the right accounts for dilatation from cavitation or cracks,the second for compression or extension normal to a material plane from phase changes,the third for inelastic shearing.VectorMdenotes the plane of structural transformations,slip,twinning,and/or localized shear,where the orthogonal shear direction isS.Total strain is?,and elastic strain is?E:

Polynomial interpolation functions in(6)are[Clayton,Leavy and Knap(2019)]

Damage deformation is idealized as isotropic withUniaxial strain normal to the basal plane commensurate with the phase change in B4C [Clayton (2012,2017c)] is quantifeid byTwinning on habit planeMis a simple shear whose maximum magnitude isγ0.The same functions account for preferred single slip of full or partial dislocations alongSwith slip plane normalM,whereγ0is a saturation limit[Clayton(2018b)].

2.3 Energy functional and equilibrium equations

A total energy functional for a heterogeneous body ? with boundary?? is

Strain energy per unit volume isW,phase energy isf,and their sum isψ=W+f.In a reference state of pseudo-Finsler space,a volume element of the body is scaled as dV=,where dV0is the element in Cartesian space.Properties may vary withX,e.g.,among grains in a polycrystal.PositionXis dropped from the explicit list of arguments in subsequent derivations,where it is understood that properties are homogeneous within each sub-volume of a grain[Clayton and Knap(2016)].

Internal state contributions,including gradient effects,are embedded in functionf:

Denote the mechanical traction vector byt,the conjugate surface force toηbyr,and the conjugate surface force toξbys.Stationary points of Ψ =are solutions to the following variational equation at fixedX:

Euler-Lagrange equations resulting from this principle and application of Rund’s divergence theorem[Rund(1975)]are the static balance of linear momentum

and the static equilibrium equations for internal state:

Right sides of(13)and(14)vanish identically for a Riemannian metric(m=k=0)and in such cases coincide with phase field mechanics[Clayton and Knap(2016);Clayton,Leavy and Knap(2019)].StressPis symmetric:

Natural boundary conditions on?? with outward unit normalnare

Strain energy density is linear elastic with damage-dependent moduliλ,μ,K:

Gradient-independent energy functions in(10)are,with H(·)a Heaviside function,

The first term inf0is a double-well with minima atη= (0,η0),a local maximum atη=η0/2,and a cut-offη0＞0.Quadratic forms are the remainder off0andg0.Constants entering(20)areA,η0,?A,andB.Let Γ denote twin boundary or stacking fault energy,Υ fracture surface energy.Material constants are related by

2.4 Numerical methods

The FE method is used to solve 3D problems in an incremental fashion.Boundary conditions are updated at each load increment.Numerical algorithms apply conjugate gradient energy minimization[Clayton and Knap(2016)].Solution fieldsu(X),η(X),ξ(X)minimize Ψ subject to boundary constraints,where the second of(9)is used.In this context,V0is the volume of the FE domain in its initial undeformed state in Euclidean space,prior to structural transformations.Force residuals computed numerically from Ψ must be smaller than a prescribed tolerance;(13) and (14) are not evaluated explicitly.Irreversibility is ensured by constraintsδξ(X)≥0 forξ(X)≥ξTandδη(X)≥0 forη(X)≥ηT,whereξT=ηT=0.9.According to the present variational model based on energy minimization similar to Bourdin et al.[Bourdin,Francfort and Marigo (2000)],strain rate does not influence the results,which are effectively time-independent.Inertia and stress waves are excluded,as are kinetics of internal state that can be addressed using methods described elsewhere[Clayton and Knap(2018a)].

3 Boron carbide-titanium diboride polycrystals

Properties and FE renderings of B4C-TiB2are presented,followed by numerical results for pure shear loading.

3.1 Material representation

Material properties and FE models are discussed in brief,noting that detailed justification and sources of property data are available in Clayton et al.[Clayton,Leavy and Knap(2019)].Parameters entering models of B4C and TiB2are summarized in Tab.1.

First consider B4C.Twinning occurs on0001}and fracture on{0001}.Amorphous shear bands can form inside twins.Twinning shear isγ0,and twinning surface energy is Γ.Lattice collapse normal to(0001)producesxη,andxξmeasures expansion from cavitation[An and Goddard (2015)].The crystal-to-glass energy barrier is.Fracture energy is Υ and cleavage anisotropy is.

Now consider TiB2.Fracture occurs on{0001},with Υ the surface energy[Du and Chen(2018)].Order parameterηaddresses stacking fault energy and line energy of (partial)dislocations viaAand,respectively.Dilatation from fractures is represented byxξ ＞0.Denote the Burgers vector of a partial dislocation byb,cumulative dislocation density byρD.The lowest energy barrier corresponds to0001}slip [Du and Chen (2018)].Stacking fault energy Γ is inferred from Vanderwalker et al.[Vanderwalker and Croft(1988)].Letdenote the maximum distance a(partial)dislocation travels prior to arrest.LetρSdenote the saturation dislocation density.Stored line energy is[Clayton(2011)],with full Burgers vectorb0.The phase field representation of single partial slip in TiB2thus obeys

Residual stresses may affect toughness [White and Dickey (2011);Scharf and Rubink(2018)].These may be addressed in future work along with other defects.

Domains consist of 50 crystal polyhedra comprising a cube ? of edge lengthL= 10μm.Average grain size isL/501/3≈2.5μm.Numerals 1 and 2 pertain to B4C and TiB2,respectively.Morphologies with volume fractionsV2of 10% and 18% are simulated.These are shown in Fig.1.Volume fractions are physically comparable to those of real microstructures [Scharf and Rubink (2018)].Two random orientation distributions(S,M,N)are assigned to grain lattices of each morphology,such that four simulations are reported in total.

Table 1:Material constants for B4C and TiB2[Clayton,Leavy and Knap(2019)]

Figure 1:FE meshes;(B4C,TiB2)crystals are(teal,yellow)and V2 is the volume fraction of TiB2:(a)microstructure 1,V2 =0.10(b)microstructure 2,V2 =0.18

Displacement-controlled boundary conditions on part of?? correspond to pure shear[Clayton and Knap (2018b)],consisting of tension and compression applied in equal magnitudes along perpendicularY- andZ-directions.Letdenote the load parameter.Essential boundary conditions are applied to four of the six faces of the initially cube-shaped polycrystalline aggregate:

Along the remaining two faces of the domain (X= 0,X=L),traction-free conditions allow the body to expand or contract freely in theX-direction.Regarding conjugate forces to internal state,free boundary conditionsr=s= 0 on all of??.In the current implementation of these boundary conditions,average shear strainis increased incrementally in steps of 5×10-4.Conjugate gradient energy minimization[Clayton and Knap(2016)]is used to seek an equilibrium configuration of the body at each increment.

3.2 Computational results

Microstructures for grains of lattice orientation set 1 are shown in Fig.2 at an applied shear strain of= 0.01 = 1%.Order parameterηdenotes twinning,shear localization,and amorphization in the B4C phase and basal slip in the TiB2phase.A quadratic scale is used to visualize low transformation magnitudes (0η0.25).Elements withξ ＞0.8 are visually removed to show fractures.As explained following (18) and (19),such elements possess very low tangent stiffness,withμ →ζμ0asξ →1 in shear for example,where hereinζ= 0.01.These elements are never fully removed from the calculation;original mesh topology is retained.

Fracture paths are quite similar for microstructures 1 and 2 in respective Fig.2(a) and Fig.2(b).Cleavage cracks tend to traverse the more brittle B4C grains.Transformation behaviors(η ＞0)are enhanced in microstructure 2 relative to microstructure 1,notably in grains of the TiB2phase which have a lower stacking fault energy(0.12 versus 0.54 J/m2)and thus a lower resistance to slip than the twinning threshold of B4C.In both of these cases,and in others not shown in Fig.2 for orientation set 2,inelasticity mechanisms of slip,twinning,and amorphization are relatively scarce(η ?1 in most grains),in agreement with experimental observations from samples of comparable sizes[Clayton and Knap(2018a);Scharf and Rubink(2018)].

Volume-averaged maximum shear stress and order parameters are labeled,,and,respectively.Definitions of the latter two are obvious,while the stress averaged over the reference domain is

Values are shown versus applied shear strainin Fig.3.Average stresses in Fig.3(a) demonstrate modest effects of lattice orientation and volume fractions,recalling microstructure 1vs.2 in Fig.1 with 10% and 18% TiB2,respectively.Stiffer responses are obtained for the same lattice orientation set when the fraction of TiB2is increased.

Figure 2:B4C-TiB2,lattice 1,shear strain=1%:(a)V2 =0.10(b)V2 =0.18

Figure 3:Averaged results for B4C-TiB2 versus applied shear strain :(a)shear stress (b)fracture(c)twinning,phase change,or slip

This results from the higher elastic shear modulus of TiB2as well this phase’s higher fracture energy,greater ductility due to slip,and lower tendency to cleave(Tab.1).Average damagein Fig.3(b) evolves similarly for all cases except microstructure 1 with lattice 1.Interestingly,this is the structurally weakest case in terms of average shear stress,yet it demonstrates the lowest accumulation rate of average micro-crack density.The same trend has been observed in simulations of pure B4C [Clayton and Knap (2018a,b)]:overall stiffness reduction and failure are most closely correlated with the presence of a few dominant cracks and are inversely correlated with diffuse crack density.Average structural changes are reported in Fig.3(c).Microstructure 2 demonstrates more inelasticity than microstructure 1 since the former contains more TiB2that slips more easily than shear-induced inelastic modes(twinning and localization)in B4C.

Tab.2 lists average maximum shear stress and values of applied shear strain and averaged order parameters at the same peak loading increment.Peak stressPCincreases from 1%to 10%when the volume fraction of TiB2is increased from 10%to 18%.TransformationsηCincrease withV2for reasons outlined in the context of Fig.3(c).Trends are less obvious for ductility?Cand peak damageξC.Different trends are evident in these variables forlattice 1 and lattice 2.Lattice orientation affects overall ductility and damage via the action of resolved driving forces for inelasticity and fracture that depend on orientations of basal planes for slip,twinning,and cleavage.Differences due to lattice orientation can be comparable in magnitude to differences due to volume fraction of the second phase.

Table 2:Peak shear stress (PC),associated strain (?C),and averaged order parameters(ξC,ηC)for B4C-TiB2

Experiments suggest fracture toughness of polycrystalline B4C may be improved by 10-100%and flexure strength by 10-30%via addition of 13-23%TiB2[Scharf and Rubink(2018)].The current simulations show more modest improvements with increasing TiB2.Residual stresses omitted in the simulations may partially explain the discrepancy.Since the FE domain may not be large enough to capture long-crack blunting by TiB2,it may be enlarged to include more crystals in the future.

4 Diamond-silicon carbide polycrystals

Properties and FE representations for diamond (C)-β-SiC are reported,followed by simulation results for pure shear loading.

4.1 Material representation

Model features for each crystal type are summarized briefly here;details and supporting references can be found in Clayton et al.[Clayton,Leavy and Knap (2019)].Properties are given in Tab.3.First consider diamond.Cleavage occurs on{111},with surface energy Υ.Dilatation from crack opening is quantified byxξ ＞0,with the same value used for all ceramics of present interest.Silicon carbide can fracture and twin.Cleavage planes are{110}.Twinning in the zinc blende structure occurs on111}withγ0=[Van Torne (1966)].Stacking fault energy Γ is notably low [Ning and Ye(1990)].Regularization lengthlis universal among all ceramics addressed.As in §3,residual stresses and other microstructure heterogeneities(initial pores,tertiary phases,etc.)are deferred to future work.

Two very different morphologies are modeled,as shown in Fig.4.Both domains are cubes of the same dimensions of§3,containing 50 polyhedral grains of average sizeL/501/3≈2.5μm.Denote byV1andV2volume fractions of diamond and SiC phases,respectively.Microstructure 1 features a layer of uniform thickness (200 nm) of nano-crystalline SiC fully surrounding each diamond crystal,contributingV2=20%.Microstructure 2 containsfive bulk SiC grains contributingV2= 10%.Two possible random lattice orientation sets give(S,M,N)for the 50 grains.

Table 3:Material constants for C and SiC[Clayton,Leavy and Knap(2019)]

Figure 4:FE meshes;(C,SiC)crystals are(blue,green)and V2 is the volume fraction of SiC:(a)microstructure 1,V2 =0.20(b)microstructure 2,V2 =0.10

A uniform orientation for nanocrystalline SiC layer is used in microstructure 1:boundary layer grains are too small to be resolved by the FE mesh.A{110}potential fracture plane is oriented at 45°to the direction of compression-tension,providing a maximum resolved stress for shear fracture.If loading is normal to[010](tension)and[001](compression),the active twinning system for(S,M)is(111),andNis prescribed as(101)for orientation set 1 andfor set 2.

Boundary conditions for pure shear are prescribed nearly identically to those discussed in§3.For each of the two orientation sets with microstructure 1,one simulation is performed with boundary conditions of (23) and≥0,and a second with the sign ofreversed.In the forthcoming discussion of results of each of these second simulations,refers to the magnitude of (negative) applied shear strain for ease of comparison.Twinning in the nanocrystalline layer is promoted in the former case but precluded in the latter since the driving force,i.e.,resolved shear stress for twinning,becomes negative.For microstructure 2,? ≥0 only.The total number of simulations reported in§4 is six.

4.2 Finite element simulations and analysis of results

Shown in Fig.5 are contours ofηthat show twinning in the SiC phase.Results correspond to lattice orientation set 1,where the habit plane and twinning direction are oriented to promote twinning in the nano-crystalline grain boundary(GB)layer in Fig.5(a).Quadratic scales enable visualization of low to moderate,i.e.,incomplete,twinning(0η0.25)that occurs in this GB layer and in a few preferentially oriented bulk grains of SiC in Fig.5(b).Physically,incomplete twinning corresponds to planar stacking faults typical inβ-SiC.Diamond,which comprises the majority of each microstructure,does not undergo twinning or slip,soη= 0 in all diamond crystals.Fracture behaviors are notably different for the two cases shown,where cleavage cracks in the diamond grains are evident at different locations in each microstructure.A dominant cleavage crack cutting diagonally across multiple diamond crystals through entire domain is noteworthy in microstructure 2,while smaller cracks arrested or deflected at phase boundaries are evident in microstructure 1.Intergranular fractures are more common in the latter microstructure.

Volume-averaged shear stressis defined as in (24),and volume-averaged order parametersandare calculated similarly.Values are shown versus applied shear strainin Fig.6.Stresses for all six simulations of diamond-SiC are reported in Fig.6(a).Recall that for two of the simulations with microstructure 1,the twin system in the GB phase is oriented with respect to the loading mode such that twinning is precluded.Lattice orientations of diamond grains are maintained identically with their counterparts in the legend immediately above.Initial elastic stiffness is larger in microstructure 2,which has a lower volume fraction than microstructure 1(V2=0.10vs.0.20)of the elastically stiffer diamond phase.However,maximum stresses are often lower for microstructure 2 and always occur at a lower applied strain for microstructure 1.When twinning is suppressed in microstructure 1,effective stiffness increases since local deformation in the GB phase remains elastic until fracture.No trends are discernible for behaviors in the strain softening regime,which differ substantially among microstructures and orientations.The same can be inferred with regard to average damagein Fig.6(b),and correlations with average stress are not all obvious.The rate of damage accumulation does decrease,however,at strains beyond those corresponding to maximum average stress.Transformation behavior quantified in Fig.6(c) is logically explained.Microstructure 1 demonstrates significant activity due to the preferential orientation of its twinning system in the nano-crystalline GB layers.Minor transformation behavior is reflected byfor microstructure 2 as a result of twinning in a few bulk grains,e.g.,see Fig.5(b).When twinning is suppressed in microstructure 1,→0 since the diamond phase is elastic-brittle.

Figure 5:C-SiC,lattice 1,shear strain=0.9%:(a)V2 =0.20,(b)V2 =0.10

Figure 6:Averaged results for diamond-SiC versus applied shear strain:(a)shear stress(b)fracture (c)twinning

Table 4:Peak shear stress (PC),associated strain (?C),and averaged order parameters(ξC,ηC)for C-SiC composites

Tabulated results for peak shear stress,corresponding ductility or peak strain,and order parameters are found in Tab.4.Peak shear stressPCdecreases by up to 10% when the GB phase is able to twin in microstructure 1,while ductility is about the same or slightly improved with twinning.Compared to microstructure 1 which contains half as much of the weaker SiC phase by volume(10%versus 20%),microstructure 2 demonstrates lower peak shear stress than three of the four simulations with microstructure 1.Ductilityis lowered by 15 to 30% in microstructure 2 compared to microstructure 1.Transformation measured byηCis much larger in microstructure 1(when twinning is enabled)than 2 since it contains a larger percentage of SiC preferentially oriented for twinning.Peak damageξCis lower in microstructure 2 than 1,suggesting dominant cracks in the former and diffuse damage in the latter,more durable aggregates.The present findings agree qualitatively with experiments[Zhao,Qian,Daemen et al.(2004)]:fracture toughness of polycrystalline diamond is improved by intergranular nano-crystalline SiC layers.

As was also the case for B4C-TiB2reported in§3,trends in results for diamond-SiC depend on lattice orientation.In the present material,fracture is strongly affected by orientation of octahedral cleavage planes in the diamond phase,and ductility by orientations of twinning habit planes in the SiC phase.As an exception to other trends implied already,for lattice 2 with slip/twins active,PCis lower in microstructure 1 than microstructure 2.Differences in strength between microstructures due to phase volume fractions and interface morphology depend strongly on enabling of twinning activity or lack thereof.This suggests important coupling exists among mechanisms of twinning,interfacial failure,and cleavage fracture.

5 Conclusions

A constitutive theory incorporating ideas from Finsler differential geometry and phase field mechanics has been implemented to model deformation and failure of dual-phase ceramics under pure shear loading.Simulations of polycrystalline B4C-TiB2and diamond-SiC composites demonstrate different twinning,amorphization,slip,and fracture behaviors of these ceramics depending on crystal type,lattice orientation,and phase morphology.Addition of TiB2to B4C provides an increase in overall shear strength and ductility,where the former phase is more likely to shear inelastically and possesses a higher fracture energy,thereby enabling blunting of cleavage cracks initiated in the more brittle B4C phase.The SiC phase,when incorporated as nano-crystalline layers along grain boundaries,apparently obstructs paths of cleavage cracks in brittle diamond crystals.For improved failure resistance,encapsulation of diamond in very fine grains of the softer second phase is recommended over addition of SiC grains of commensurate size of the diamond.For quasi-static deformations of aggregates of 50 bulk crystals with grain sizes on the order of severalμm,maximum shear strengths are on the order of 1.5 GPa for B4C-TiB2and 2.9 GPa for diamond-SiC.Respective applied shear strains at maximum load are on the order of 0.9%and 0.6%for the two material systems.

Acknowledgement:Prof.Jincheng Du of the University of North Texas is thanked for providing DFT findings for fracture and slip in TiB2.Prof.Tom Scharf of the University of North Texas is thanked for providing characterization of B4C-TiB2.Dr.Shawn Coleman of ARL is thanked for providing literature on diamond-SiC.Prior collaborative efforts on software development and execution at ARL with Dr.Jarek Knap and Brian Leavy that preface and support the current work are also appreciated.