A Method for Combining Experimentation and Molecular Dynamics Simulation to Improve Cohesive Zone Models for Metallic Mi

1. A Method for Combining Experimentation and Molecular Dynamics Simulation to Improve Cohesive Zone Models for Metallic Microstructures Jacob Hochhalter1,2 Miguel Aguilo1 Prof. Anthony Ingraffea1 Dr. Edward Glaessgen2 Prof. Wilkins Aquino1 1School of Civil and Envr. Engineering Cornell University; Ithaca, NY 2Durability, Damage Tolerance & Reliability Branch NASA Langley Research Center; Hampton, VA

2. Outline • Multiscale Fracture Simulation • Molecular Dynamics (MD) Simulation • Extracting Cohesive Zone Model (CZM) Parameters • The Scale Issue • Experimental Measurements • Full-Field, High-Resolution Strain Mapping • Inverse Problem • Reconstructing CZM Parameters • The Uniqueness Issue • Combine 2, 3, and 4 to Help Reconcile the ‘Issues’ 2

3. 5.7 mm Row of bolt holes Structural Component Multiscale Fracture Simulation RD ND 250 m RD Structural Detail Microstructural Scale 3

4. Multiscale Fracture Simulation: The Scale Issue Must assert the crack growth mechanisms and calibrate critical values of metrics that simulate those mechanisms Observed Crack Path Plasticity 1 1Maniatty et al., IJNME 60 (2004) Fracture 2 Cohesive Zone Model (CZM) 4 2Park et al., JMPS 57 (2009)

5. Combine Two Approaches to Determine CZM Parameters yy ESCM 0.002 0.001 40 nm 100 m str 0 VIC • The Scale Issue: • Computing atomic motion becomes intractable • 1 m cube of aluminum => 15 billion atoms • < 1 s (with Accelerated Dynamics) • Highly idealized material (purity, structure) • The Scale Issue: • Observing atomic motion is currently not possible • 1 nm displacement resolution • ~ 1 s observation snapshots • Fabrication of a pure crystal is difficult/impossible ? • What comparisons can be made - qualitatively, quantitatively? • What damage processes can we begin to reconcile? • What inherent differences between modeling and testing must be overcome? 5

6. MD Simulation: Extracting MD-Based Cohesive Zone Model (CZM) Parameters 1.9 nm 2.9 nm 1.9 nm 40 nm 21 CZVE Along 40nm GB Interface p  p 0 System size: MD domain: ~40 nm (360,000 atoms) FEM region: ~1 m (15,000 d.o.f.) Nanometer-Scale CZVE: 1250 atoms Response from Concurrent Multiscale Typical Values at Microscale: p = ~500 [MPa] 0 = ~1 [m] Tractions & Displacements Crack Tip Constitutive Response 6 1Yamakov et al., JMPS 54 (2006) 1899

7. Orientation 2 Orientation 1 E-beam weld heat 11.1 4.95 -1.2 polycrystalline -7.35 -13.5 200 nm Experimental Procedure Vacuum Arc Melter / Crystal Puller Single Crystal Aluminum Development of Bi-crystals 100 m Load Stage 10-50 nm gold spheres m VIC Displacement Field SEM Displacement resolution < 1 nm 7

8. Inverse Problem: Reconstructing CZM Parameters from Experiment The spatial distribution of material properties can be reconstructed from the measured system response to an applied load using the finite element method and optimization algorithms. Reconstruct Material Properties e.g. Young’s Modulus, Poisson Ratio, CZM Parameters, etc. Known Geometry, Boundary Conditions, etc. Measured System Response e.g. displacements 8

9. Inverse Problem: Flowchart 40 nm 100 m Measure displacement field, ue Define potential-based CZM Initialize MD-based CZM No Take current values as solution Yes Reconcile the distinct CZMs 9

10. Gradient-Based Optimization: Computing the Gradient - The cost is 2m+1 times the cost of solving one forward problem - Additional perturbation parameter that has no physical meaning + Easy Finite Difference Too Costly for Typical Microstructures - The cost is m+1 times the cost of solving one forward problem + No perturbation parameter Direct Method - The math is tedious + The cost is 2 times the cost of solving one forward problem + No perturbation parameter Adjoint Technique 10

11. Inverse Problem: Mathematical Statement Where is the solution to a variational boundary value problem, e.g. equilibrium of elastic isotropic material Consider the following optimization problem 11

12. Inverse Problem: • Reconstructing Material Model Parameters Geometry & Boundary Conditions Shear Modulus Target Solution Adjoint Technique Approximation 12

13. Combine to Help Reconcile the Scale and Uniqueness ‘Issues’ Stacking Fault Partial Dislocation 50 nm Deformation Twinning Partial Dislocation 10 nm ~ Ductile Brittle • MD-Based CZMs Supply a Physical Basis to the Uniqueness Issue • Inverse Methods Supply an Approach to Reconcile MD with Experiment Park et al., JMPS 57 (2009) 891 13 Yamakov et al., JMPS 54 (2006) 1899

14. Summary • Methods for extracting cohesive zone models (CZM) from molecular dynamics (MD) simulations have been developed and provide qualitatively accurate models. • A VIC experimental method that uses an SEM to measure displacements at sub-nanometer resolution provides surface data during crack growth. • An inverse method has been developed that requires only 2 forward runs and is independent of the number of spatially varying search parameters. • These 3 methods are being combined to provide improved techniques for determining CZM parameters and, in turn, provide a method to reconcile experiment with MD simulation of crack growth. 14