Relating Properties of Metals to Microstructure and Processing through Grain-Scale Modeling

Transcription

1 Relating Properties of Metals to Microstructure and Processing through Grain-Scale Modeling David Littlewood Jing Lu Antoinette Maniatty Mechanical, Aerospace, and Nuclear Engineering Rensselaer Polytechnic Institute Tech Valley Engineering Symposium 2007 slide 1/17

2 Outline Motivation Methodology Multiscale approach Constitutive model Finite element formulation Implementation Simulation results Properties: simulation of a bulk forming process Behavior: fatigue and plastic deformation Future work Tech Valley Engineering Symposium 2007 slide 2/17

3 Motivation Properties (anisotropy, moduli, yield strength) determined at grain scale Goal: Improved polycrystalline material models Relationship between processing and properties Material response under loading conditions Approach: Multiscale model, focus on microstructure Tech Valley Engineering Symposium 2007 slide 3/17

4 Multiscale Nature of the Problem [MISIOLEK ET AL.] Tech Valley Engineering Symposium 2007 slide 4/17

5 Methodology: Multiscale Approach Component Scale Homogeneous Displacement b.c. Von Mises plasticity Grain Scale Heterogeneous Periodic b.c. Crystal plasticity Tech Valley Engineering Symposium 2007 slide 5/17

6 Methodology Constitutive Model Deformation gradient Green strain tensor e E = 1 2 F = e F p F ( e F T e F I ) Hyperelastic potential Second Piola-Kirchoff stress Anisotropic elasticity ψ = 1 2 e E: C: e E S = C: e E C ijkn = C jikn = C ijnk = C knij Tech Valley Engineering Symposium 2007 slide 6/17

7 Methodology Slip Model Plastic velocity gradient pˆl = 12 α=1 γα P α Schmid tensor Slip rate Hardness evolution P α = s α m α γ α τ = γ α τ α 1 1 m o g α g α ġ α = G o ( go g α g s g o ) β Hαβ γ β Tech Valley Engineering Symposium 2007 slide 7/17

8 Methodology Governing equations: Finite Element Formulation ( ) σ ij + p δ ij = 0, 1,j 3 σ ii p = 0 Weak forms (total Lagrangian): ) (σ ij h + p h δ ij ψ α,k F 1 Kj JdΩ o } Ωo {{} f iα int(ū, p) 1 Ω 2o ˆt i ψ α da da o dγ o }{{} f ext iα ( ) 1 Ωo K 3 σh ii p h ψ ρ JdΩ o }{{} hρ(ū, p) = 0 = 0 Tech Valley Engineering Symposium 2007 slide 8/17

9 Methodology Linearized equations: Finite Element Formulation K r iαjβ ū jβ + G r iαϕ p ϕ = f ext iα f int iα (ū r, p r ) H r ρjβ ū jβ + M r ρϕ p ϕ = 0 h ρ (ū r, p r ) Discontinuous pressure field allows for a p solution on the element level: ( p ϕ = M r 1 ϕρ hρ (ū r, p r ) + H r ρjβ ū jβ) Tech Valley Engineering Symposium 2007 slide 9/17

10 Implementation PETSc [ANL] FemLib [Cornell] RPI Crystal Plasticity Model RPI Quasistatic FEM Driver MPICH [ANL] Tech Valley Engineering Symposium 2007 slide 10/17

11 Implementation Running on large-scale systems SCOREC RedHat Linux cluster (Borg) 50 nodes, 4-way Opteron, 6 GB RAM & 11 nodes, 2-way Opteron, 2 GB RAM Cornell Theory Center Windows cluster 170 nodes, dual Xeon 3.6 GHz processors, 4 GB RAM RPI SUR Blue Gene 1024 nodes, dual 700 MHz PowerPC 440 processors, 1 GB RAM Performance analysis [Scott Owens] Tech Valley Engineering Symposium 2007 slide 11/17

12 Simulation of Bulk Forming Process Macro-scale model and grain-scale model MACRO SCALE [Misiolek et al.] GRAIN SCALE Tech Valley Engineering Symposium 2007 slide 12/17

13 Results: Material Processing UNDEFORMED MODEL Tech Valley Engineering Symposium 2007 slide 13/17

14 Results: Material Processing 8% STRAIN Tech Valley Engineering Symposium 2007 slide 13/17

15 Results: Material Processing 16% STRAIN Tech Valley Engineering Symposium 2007 slide 13/17

16 Results: Material Processing 25% STRAIN Tech Valley Engineering Symposium 2007 slide 13/17

17 Polycrystal with Embedded Particle ORIENTATION SET #1 ORIENTATION SET #2 Tech Valley Engineering Symposium 2007 slide 14/17

18 Results: Material Response TOTAL ACCUMULATED SLIP, TIME = 2.0 SEC. Tech Valley Engineering Symposium 2007 slide 15/17

19 Results: Material Response TOTAL ACCUMULATED SLIP, TIME = 4.0 SEC. Tech Valley Engineering Symposium 2007 slide 15/17

20 Results: Material Response TOTAL ACCUMULATED SLIP, TIME = 6.0 SEC. Tech Valley Engineering Symposium 2007 slide 15/17

21 Future Work Investigate misorientation distribution Focus on formation of subgrains Comparison to experimental data Project results into high-cycle regime Move between different time scales More realistic grain structures Increase number of grains Tech Valley Engineering Symposium 2007 slide 16/17

22 Questions? Relating Properties of Metals to Microstructure and Processing through Grain-Scale Modeling David Littlewood Jing Lu Antoinette Maniatty Tech Valley Engineering Symposium 2007 slide 17/17