Integr grate ted C Computa tatio tional Ma l Materia ials ls Science & Sc & Engineering ( (IC ICMSE SE ) ) Appr pproache hes t to Problems wi with h Evolving D Domai mains Somnat nath G h Ghosh Depar De artments of C Civ ivil il & Me Mechanic ical l Engin gineerin ing Johns H Hopkins U Univers rsity Baltimore, Maryland USA Worksho shop o on C Compu puta tati tional M Methods ds for P r Prob roblems w with Evolv lvin ing Do Domain ins an and Dis Discontin inuit itie ies AHPCRC, S AHP , Stanfor ord U Univers rsity, C y, CA Decem ecember er 4 4-5, 5, 2013 2013
Integ egrated ated M Materi terials als S Scien ence & e & Engine neering ( (ICMSE) SE) P Paradi digm ICMSE SE philoso sophy phy “ entails i integra ration on o of i inform ormation on a acros oss le lengt gth an and t tim ime s scale ales for all all rele levan ant m mat ateria ials ls p phenomena an a and enable ables concurrent an analy alysis o of man anuf ufac acturin ing, d design ign, an and mat aterial ials w wit ithin in a a holis listic ic s system ” ” J. Allison, D. Backman, and L. Christodoulou, "Integrated Computational Materials Engineering: A new paradigm for the global materials profession," JOM, pp. 25-27, 2006.
Two C Cas ase S Studies i in t the I e ICMSE E Parad radig igm 1. 1. Multi ti-scale le m model f l for ductile ile failu ilure in hete terogeneous m meta tallic mate terials 2. Im . Image-ba based m modelin ling of fatig igue ue f failu ilure in m metall llic ic allo loys
Ductile F e Failure o ure of H Hetero erogen eneous us Metall allic ic M Mater erial ials • Ductile failure in heterogeneous materials typically initiates with particle cracking or interfacial debonding. • Voids grow near nucleated regions with deformation, and subsequently coalesce with neighboring voids to result in localized matrix failure. Automotive Engine Block Evolving Ductile Failure in Aluminum Microstructure • Evolution of matrix failure causes stress and strain redistribution in the microstructure that leads to ductile fracture at other sites. • Eventually, the phenomena leads to catastrophic failure of the microstructure. Microstructure: Cast Aluminum Alloy with Si Particulates and Intermetallics Stress-Strain plot showing ductility
Fatigue i Fat in Aerosp space E Engine Mat aterials s Dwell fatigue 9 times during hold (2 min) 9 times during hold (2 min) Stress Stress 20 times 20 times during cycle during cycle 1 sec 1 sec 1 sec 1 sec time time 50 times during one cycle 50 times during one cycle Stress Stress time time Regular fatigue Effect of microstructure important in predicting fatigue life: e.g. nucleation at location of extreme values of grain morphology, • Crack initiation site is sub-surface orientation and misorientation, micro-texturing . • Initiation location depends on local microstructure • Initiation area is faceted with limited evidence of plasticity • Away from initiation site, crack growth is ‘normal’, i.e. striations • 2 min. dwell can lead to 2-10 x reduction in fatigue life
Structur cture-Materi erial Inte teracti tion Chal hallenges • Mo Modelin ling at at the macros roscopic s scales cannot ot prov rovide a accura rate estim imates o of d duc uctilit ility an and fat atigu igue lif life • Lac acks ap appropriat iate lo local ge al geometric ic an and thermo-mechan anical al in information o of the in incip ipie ient d dam amage age s sit ites • Mo Modelin ling at at the micros ostru ructura ral s scales is comput utat atio ional ally ly i intractabl ble • Need a approp ropri riate mult ulti-scale le t techniq ique ues in in spat atia ial an l and temporal l domai ains that at w will ill up uphold ld t the e effic icie iency o of sim imula ulations, w while ile not compro romising t the require red r resol olution ons
Case ase St Study dy 1 Adapt daptive Multi Spat Spatial-Sca cale le Mo Modelin ling o g of Ductile Fr Frac acture i in Heterogeneous M Metal allic Ma Materia ials ls S. Ghosh, “Micromechanical Analysis and Multi-Scale Modeling Using the Voronoi Cell Finite • Element Method”, CRC Press/Taylor & Francis, 2011, 729 pages. S. Ghosh and D. Paquet, “Adaptive Multi-Level Model for Multi-Scale Analysis of Ductile Fracture • in Heterogeneous Aluminum Alloys”, Mechanics of Materials, (in press), 2013. S. Ghosh, J. Bai and D. Paquet, Jour. Mech. Physics Solids , Vol. 57, 2009. • C. Hu, J. Bai and S. Ghosh, Modeling and Simulation in Materials Science and Engineering , Vol. 15, • pp. S377-S392, 2007
Two-Wa Way C y Couple led Adap Adaptive Conc oncurrent t Multi ti-Scal Scale Mode del Physics-Based Reduced Homogenization Theory- Ordered Models Based Swing Region for Error Level-0 Analysis Level-1 RVE Homogenization T O Micromechanical Analysis in P Level-2 B Critical Regions O D T O T Localization W O N M U P Discretization Error Modeling Error Increase DOF e.g. by h-p-adaptation Introduce multiple-level hierarchy
Fra ramew ework rk f for C or Conc oncurren ent M Multi-Sc Scal aling 1. 1. Mult Multi-Sc Scal ale C Characterizat atio ion: : Morpholo logy gy-bas based D Domai ain P Partitio ionin ing g an and RV RVE Id Identifi fication High Resolu olutio ion Dom omain Recon construct uctio ion Step 1. Wavelet interpolation A of low res. images 48 µm Step 2. Correlation-based enhancement from limited high res. images = + ' ' ' g g g ( ', ') ( ', ') ( ', ') I x y I x y I x y hrsm wvlt diff Recur cursiv ive R Refin inement based d on on Mor orph phol olog ogica cal l 2304 µm Cha haract cteristic c Fun unct ctions
Frame mework for C Concur urrent M Mult ulti-Scal cale s 2. 2. Micromechan anic ical al Analy alysis is: Voron onoi oi Cell ll FEM f M for Duc Ductile ile Fractur ure VCFEM Optical micrograph S. Ghosh, “Micromechanical Analysis and Multi-Scale Modeling Using the Voronoi Cell Finite Element Method”, CRC Press/Taylor & Francis, 2011, 729 pages.
Vor oron onoi C Cell FE FEM M For Formulation for or ix Crac acking Partic icle le a and Matrix VCFEM for VCFEM for VCFEM for VCFEM for VCFEM for VCFEM for Matrix Cracking Matrix Cracking undamaged particle undamaged particle damaged particle damaged particle Stress function: Φ = Φ + Φ Φ = Φ +Φ m m m c / c c c cr / +Φ / m cr poly rec poly rec rec ( ) ( ) ∏ ∫ ∫ ∆ ∆ = − ∆ ∆ Ω − ∆ Ω σ u B σ σ ε σ , , : d d Assumed Stress-Hybrid FEM Ω Ω + + m c m c e ( ) ( ) ∫ ∫ + + ∆ ⋅ ⋅∆ ∂Ω − + ∆ ⋅∆ Γ e σ σ n u t t u d d ∂Ω Γ e tm ( ) ∫ − + ∆ − − ∆ ⋅ ⋅∆ ∂Ω m m c c c σ σ σ σ n u ' d ∂Ω c ( ) ∫ ′′ − + ∆ ⋅ ⋅∆ ∂Ω σ σ c c cr n u d ∂Ω For local softening in stress-strain cr ( ) ∫ ∫ + ∆ ∆ Ω + ∆ Ω A σ ε σ ε s s s s response: Higher-order displacement , : d d Ω Ω s s interpolated regions is embedded in ( ) ∫ − + ∆ ⋅ ⋅∆ ∂Ω σ σ s s n u d the stress-interpolated VCFEM ∂Ω s domain C. Hu and S. Ghosh, IJNME , 2008.
Microstruc uctur ural P Partic icle le a and Matrix ix Cra racking Matrix Cracking Nucleation: Particle Cracking Nucleation: Gurson-Tvergaard-Needleman Weibull distribution based (GTN) type Models crack initiation criterion 2 ( ) 3 q q p Φ = + − − + * *2 2 2 cosh 1 q f q f σ 1 σ 3 2 0 0 ( ) = + = ε ε + − ε p p p (1 ) df df df A d f d nucleation growth kk ≤ f for f f c = * − * f f f ( ) + − > u c f f f for f f − c c c f f F c Non on-lo loca cal void oid grow owth h rate 1 ( ) ( ) ( ) ∫ = − f x f x w x x dV ( ) local x W v
Frame mework for C Concur urrent M Mult ulti-Scal cales 3. Macros 3. roscopic Mo Modelin ling: : Hom Homogenized C Con ontinuum M Mod odel f for or Plas lastic icit ity an and Dam Damage age E Evolu lution wit ith H Heteroge geneit itie ies Anisotropic yield surface in the GTN model Σ 2 Σ 3 ( ) 2 eq hyd φ = + − − = 2 cosh 1 0 Q f Q f ( ) ( ) 1 1 2 2 Y W Y W f p f p ( )( ) ( ) ( ( )( ) ( ) 2 ) 2 2 Σ 2 = Σ − Σ + Σ − Σ + Σ − Σ + Σ 2 F W G W H W C W eq p yy zz p zz xx p xx yy p xy F, G, H and C : Anisotropic YS parameters calibrated from homogenization of micromechanics in principal material-damage coordinates ( ) ( ) Q = − + p = Σ = Σ + Σ + Σ 1 f f e A e e 1 ; Q kk − 1 hyd xx yy zz (1 ) f inclusion 1 ( ) ( ) ( ) ∫ = − x x x x f f w dV ( ) local W x v
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