Materials Process Design and Control Laboratory MOLECULAR DYNAMICS APPROACH FOR INVESTIGATION OF GRAIN BOUNDARY RESPONSE WITH APPLICATIONS TO CONTINUUM SIMULATION OF FAILURE IN NANO-CRYSTALLINE MATERIALS Baskar Ganapathysubramanian, Veeraraghavan Sundararaghavan and Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY URL:

Materials Process Design and Control Laboratory ACKNOWLEDGEMENTS FUNDING SOURCES:  Air Force Research Laboratory  Air Force Office of Scientific Research  National Science Foundation (NSF)  ALCOA  Army Research Office COMPUTING SUPPORT:  Cornell Theory Center (CTC)

Materials Process Design and Control Laboratory OVERVIEW –Motivation –Problem definition –Molecular dynamics simulation –Cohesive model: ISV method –Conclusions –Scope for further work

Materials Process Design and Control Laboratory MOTIVATION –Design polycrystalline materials with tailored properties –Accurate design of processes to obtain tailored properties in crystals –Simulation of response/failure of crystalline materials –Simulation of grain boundary failure –Enhanced models of grain boundary (GB) separation and sliding mechanisms based on quasi-static MD simulations in a bicrystal Meshing GB Properties Molecular dynamics Control loads FEM models Intra- granular: Crystal plasticity models Grain boundary (Cohesive zone models)

Materials Process Design and Control Laboratory Cohesive zone models are used to describe the grain boundary response and allow for natural initiation of intergranular cracks. The tool can be used for metallic polycrystal systems including nanostructured materials. Reducing the characteristic length scale of the grains, closes the gap between meso-scale simulations and atomistic simulations.  Allows calibration of constitutive models using MD simulations MOTIVATION Kumar, Ritchie, Gao (llnl/ucb) Sethna, Cornell

Materials Process Design and Control Laboratory Modelling grain boundary response: 1) Simple isotropic hardening rule (Anand- Staroselsky (1998), Fu et al (2004)): no fracture with slip inside grains 2)Use of cohesive zones: Espinosa, Ingraffea, Anand, Mcdowell, Ortiz ect 3)Zavattieri & Espinosa (2001): simple cohesive law with rate dependence accounting for different Tmax, weibull distribution to account for uncertainty in the form of misorientations, no plastic effects 4)McDowell (2004): simple cohesive law with rate independent form, No plastic effects 5)Anand(2004): Cohesive law based on state variable with hardening of grain boundary, reversible law with elasto plastic decomposition of displacement jump 6)Bower(2004) : Arrhenius law to calculate strain rates based on activation energy of grain boundary LITERATURE

Materials Process Design and Control Laboratory GRAIN BOUNDARY MODEL Grain boundary x+x+ x-x- a a a 1.Integration is carried out over the centerline of the element. 2.The displacement jumps are interpolated using shape functions of the centerline element 4 noded cohesive element Finite Element Method

Materials Process Design and Control Laboratory EAM potentials for quantitative studies Choosing geometries Goal: Find general law for strength of grain boundaries, depending on geometry, temperature and strain rate Limitations: computionally intensive, large domain/long time, need a compromise ATOMISTIC MODELING OF GRAIN BOUNDARY BEHAVIOR Pull grain apart with constrained atoms – measure stress at each step Motivation: feed cohesive laws to FEM simulations Bicrystal arrangement

Materials Process Design and Control Laboratory ATOMISTIC MODELING OF GRAIN BOUNDARY BEHAVIOR METHODOLOGY Initialize atoms Set boundary conditions Randomize velocities, set temperature Thermalize for 5 ps (NVE with velocity rescaling) Set to NVT canonical Pull boundary atoms at prescribed rate Set force on Boundary atoms to zero Calculate average stress of mobile atom FCC Cu Bi-crystal with a 45º misorientation atoms Strain rate of 1 A/ps EAM potential

Materials Process Design and Control Laboratory ATOMISTIC MODELING OF GRAIN BOUNDARY BEHAVIOR L x B x H: 40 x 20 x 40 atomic planes Normal stress-displacement response. Dominant peak with associated peak stress. Peak stress at 5.8 Å Compared with Spearot et. al (Mech. Mat. 36 (2004)) Smaller domain size Peak stress 5.6 Å

Materials Process Design and Control Laboratory COHESIVE ELEMENT FORMULATION -1 GB Constitutive law (Anand’s Model) Far field displacement (A) Normal Stress (GPa) 13 parameter l 2 norm fit MD results Some continuum scale parameters: Interface friction = Interface normal stiffness = GPa/nm Exponent of 0.5 for initial hardening

Materials Process Design and Control Laboratory INTRAGRANULAR SLIP MODEL Crystallographic slip and re- orientation of crystals are assumed to be the primary mechanisms of plastic deformation Evolution of various material configurations for a single crystal as needed in the integration of the constitutive problem. Evolution of plastic deformation gradient The elastic deformation gradient is given by Incorporates thermal effects on shearing rates and slip system hardening (Ashby; Kocks; Anand)

Materials Process Design and Control Laboratory Pure grain boundary sliding separation SIMULATION OF GB FAILURE IN BICRYSTALS RESPONSE IN SHEAR(PURE SLIDING)

Materials Process Design and Control Laboratory SIMULATION OF GB FAILURE IN BICRYSTALS RESPONSE IN TENSION(PURE OPENING)

Materials Process Design and Control Laboratory SIMULATION OF GB FAILURE IN BICRYSTALS RESPONSE IN TENSION(MIXED MODE)

Materials Process Design and Control Laboratory SIMULATION OF GB FAILURE IN BICRYSTALS RESPONSE IN TENSION Response of crystal is still in elastic regime GB produces a plastic response at low strains Length of simulation tailored to MD simulation Boundary conditions: Pulled along y Compressed in x

Materials Process Design and Control Laboratory TEMPERATURE COMPENSATED STRAIN RATE Equivalence of the effects of change in strain rates and in temperature upon the stress strain relation in metals Intended for investigation of behavior of steels at very high deformation rates. Obtained by tests at moderate strain rates at low temperatures Use the equivalence relation to extract behavior at low strain rates by simulating high- strain rate deformation at higher temperatures. Isothermal deformation Zener and Hollomon: J. Applied Physics (1943) Z is the Zener-Hollomon parameter έ is the strain rate R is the universal gas constant Q is the activation energy Q shown to be the equal to the self-diffusion for pure metals ( Kuper et. al Physical Review 96 (1954) )

Materials Process Design and Control Laboratory TEMPERATURE COMPENSATED STRAIN RATE Q for copper is 213 KJ/mole R: J/mole/K Look at realistic pulling rate 0.1 mm/s Ratio of strain rates Å/ps at 300 K equivalent to 10 Å/ps at 317 K 0.1 mm/s at 300 K equivalent to 100 m/s at 381 K Temp compensated pulling rate of 0.1 mm/s Pulling rate of 100 m/s

Materials Process Design and Control Laboratory LAWS FOR DIFFERENT LOADING REGIMES Strain rate dependence Temperature dependence Deformation mode dependence? Orientation dependence Size effects Triple points and other grain junctions? Material dependence PARAMETRIC STUDIES Effect of temperature variation on peak stress and magnitude of deformation at peak stress Effect of strain rate variation on peak stress Vary mis-orientation angle. How does deformation and failure proceed?

Materials Process Design and Control Laboratory Copper bi-crystal Tension test at different temperatures T = 300 K, 400 K, 500 K Slope constant with temperature Displacement associated with peak stress decreases Peak stress decreases TEMPERATURE EFFECTS LAWS FOR DIFFERENT LOADING REGIMES

Materials Process Design and Control Laboratory MISORIENTATION EFFECTS Misorientation has effect on peak stress Displacement associated with peak stress not sensitive Slope remains constant At higher temperatures thermalization leads to diffusion of the grain boundary. LAWS FOR DIFFERENT LOADING REGIMES

Materials Process Design and Control Laboratory Aluminium bi-crystal Tension test at different strain rates v = 10 A/ps, 1 A/ps, 0.1 A/ps Displacement associated with peak stress increases Peak stress increases with strain rate STRAIN RATE EFFECTS LAWS FOR DIFFERENT LOADING REGIMES

Materials Process Design and Control Laboratory LAWS FOR DIFFERENT LOADING REGIMES SIZE EFFECTS The simulation domain size affects the magnitude of the peak stress and the displacement associated with it. Can we extract the asymptotic limit? Finite size scaling A(L) = A o + c/L n Yield stress decreases with increasing size Ductility increases with size

Materials Process Design and Control Laboratory Espinosa (2001): Rate and temperature dependent law: Bower (2004): Misorientation dependence: Weibull distribution (Espinosa 2002) ? COHESIVE ELEMENT FORMULATION -2 Conrad and Narayan (1999):

Materials Process Design and Control Laboratory TRIPLE JUNCTION LAWS FOR DIFFERENT LOADING REGIMES Trijunction (T: 30, 60) Peak stress lower, displacement lesser

Materials Process Design and Control Laboratory SEVERAL OPEN ISSUES Open issues: Should triple points cohesive zones have special constitutive models (using MD)? Can the response be generalized considering large complexity since a space of misorientations of 3 different grains need to be explored? Triple point element quad junction element Complexity: Orientation space x Grain junctions x Deformation mode dependence x Temperature dependence x Strain rate dependence x Material type = Large data set that needs to be explored (Statistical learning?)

Materials Process Design and Control Laboratory SCHEMATIC OF A GB DATABASE Strain rate dependence Temperature dependence Orientation dependence Deformation mode dependence Triple points and other grain junctions Material Multiphase Continuum cohesive law trained using gradient optimization State variable evolution, traction separation law Sethna et al

Materials Process Design and Control Laboratory Extend to complex interfaces Look at other failure/deformation mechanisms Simulation of larger length scales A database of grain boundary properties Design of processes to tailor properties SCOPE FOR FUTURE WORK

Materials Process Design and Control Laboratory INFORMATION RELEVANT PUBLICATIONS Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY URL: Prof. Nicholas Zabaras CONTACT INFORMATION V. Sundararaghavan and N. Zabaras, "A dynamic material library for the representation of single phase polyhedral microstructures", Acta Materialia, Vol. 52/14, pp , 2004 S. Ganapathysubramanian and N. Zabaras, "Modeling the thermoelastic- viscoplastic response of polycrystals using a continuum representation over the orientation space", International Journal of Plasticity, Vol. 21/1 pp , 2005