1. Application of Asymptotic Expansion Homogenization to Atomic Scale N Chandra and S Namilae Department of Mechanical Engineering FAMU-FSU College of Engineering Florida State University

2. Why link atoms and continuum ? “Nanotechnology”  Atomic details (structural and Material) have profound influence on properties -Thermomechanical -Physical, electrical, magnetic However computational problems -100 nm cube of Si ~ billion atoms  Macroscopic phenomenon effected by atomic scale details Fracture  Crack tip Plasticity  Dislocation Grain boundaries “Materials by design” Creep /SP 

3. Problems in Atomic scale domain Grain boundaries play a important role in the strengthening and deformation of metallic materials. Some problems involving grain boundaries : Grain Boundary Structure Grain boundary Energy Grain Boundary Sliding Effect of Impurity atoms

4. Equilibrium Grain Boundary Structures [110]  3 and [110]  11 are low energy boundaries, [001]  5 and [110]  9 are high energy boundaries [110]  3 (1,1,1)[001]  5(2,1,0) [110]  9(2,21)[110]  11(1,1,3) GB

5. Experimental Results 1 1 Proceeding Symposium on grain boundary structure and related phenomenon, 1986 p789 Grain Boundary Energy Computation Calculation GBE =  (E atoms in GB configuration) – N  E eq(of single atom)

6. Grain Boundary Sliding Simulation Generation of crystal for simulation of sliding. Free boundary conditions in X and Y directions, periodic boundary condition in Z direction. X’ Y’ A state of shear stress is applied T = 450K Simulation cell contains about 14000 to 15000 atoms Grain boundaries studied:  3(1 1 1),  9(2 2 1),  11 ( 1 1 3 ),  17 (3 3 4 ),  43 (5 5 6 ) and  51 (5 5 1)

7. Sliding Results Fig.6 Extent of sliding and Grain boundary energy Vs misorientation angle Grain boundary sliding is more in the boundary, which has higher grain boundary energy Monzen et al 1 observed a similar variation of energy and tendency to slide by measuring nanometer scale sliding in copper Monzen, R; Futakuchi, M; Suzuki, T Scr. Met. Mater., 32, No. 8, pp. 1277, (1995) Monzen, R; Sumi, Y Phil. Mag. A, 70, No. 5, 805, (1994) Monzen, R; Sumi, Y; Kitagawa, K; Mori, T Acta Met. Mater. 38, No. 12, 2553 (1990) 1 Reversing the direction of sliding changes the magnitude of sliding

8. Mg Segregation in Al Grain Boundaries Position 1 Position 2 E GB For Pure Al =0.65 x 10 -2 (eV/A 2 ) Segregation of Mg atoms to particular locations in grain boundary is based on size effect and hydrostatic pressure Variation of grain boundary energy in presence of Mg atom

9. Hydrostatic Stress and Segregation Energy Grain boundary energy and segregation are influenced by changes in coordination of atoms at grain boundary Simulation results also indicate that there is an increase In grain boundary sliding when Mg atoms are present Effect of Mg on sliding Distribution of atoms around impurity atom in  9 STGB

10. Problems in macroscopic domain influenced by atomic scale MD provides useful insights into phenomenon like grain boundary sliding Problems in real materials have thousands of grains in different orientations Multiscale continuum atomic methods required A possible approach is to use Asymptotic Expansion Homogenization theory with strong math basis, as a tool to link the atomic scale to predict the macroscopic behavior

11. Sinclair (1975) Hoagland et.al (1976) Mullins (1982) Gumbusch et.al. (1991) Tadmor et.al. (1996), Shenoy et.al. (1999) Flexible Border Technique Finite element Atomistic method FE-At method Quasicontinuum method Continuum-Atomics linking

12. Rafii Tabor (1998) Broughton et.al. (2000) Lidorkis et. al. (2001) Friesecke and James Three scale model Coarse grained molecular dynamics Handshaking methods -CLS Multiscale scheme Continuum-Atomics linking Other efforts: CZM based, description of continuum in atomic Regions, lipid membranes etc

13. Homogenization methods for Heterogeneous Materials Heterogeneous Materials e.g. composites, porous materials Two natural scales, scale of second phase (micro) and scale of overall structure (macro) Computationally expensive to model the whole structure including fibers etc Asymptotic Expansion Homogenization (AEH) Schematic of macro and micro scales

14. AEH idea Overall problem decoupled into Micro Y scale problem and Macro X scale problem

15. AEH literature Functional analysis Bensoussan et.al. (1978), Sanchez Palencia (1980) Elasticity well established Kikuchi et. al. (1990) Adaptive mesh refinement Hollister et. al. (1991) Biomechanics Application Ghosh et. al. (1996),(2001) AEH combined with VCFEM Buannic et. al. (2000) Beam theory with AEH Inelastic Problems Fish et.al. (2000) Plasticity Chung et.al. (2001) Viscoplasticty Transport Problems in Porous media

16. Formulation Let the material consist of two scales, (1) a micro Y scale described by atoms interacting through a potential and (2)a macro X scale described by continuum constitutive relations. Periodic Y scale can consist of inhomogeneities like dislocations impurity atoms etc Y scale is Scales related through  Field equations for overall material given by

17. Contd The functions u (i) (x,y) are Y periodic in variable y. and are independent of the scaling parameter . The basic concept in AEH is to expand the primary variables as an asymptotic series. Using the expansion for displacement u From the definition of the scaling parameter, for any g(x,y)

18. Hierarchical Equations Strain can be expanded in an asymptotic expansion Substituting in equilibrium equation, constitutive equation and separating the coefficients of the powers of  three hierarchical equations are obtained as shown below. Micro equation Macro equation

19. Microscale Equation Using the following transformation Micro equation can be solved as In Variational form c  corrector term in macro scale due to microscale perturbations. series of vectors

20. Microscale Equation The Y scale here is composed of atoms interacting through an interatomic potential. If we consider a finite element mesh refined to atomic scale in the Y region then, would denote the atomic level stiffness matrix W is the total strain energy density of the Y scale and q dente the displacements of individual atoms. Micro equation can be solved as at atomic level (6xN) B T C loc  q  Atomic displacements C loc  Local elastic constants determined from MD

21. Macroscale equation Given by apply the mean operator on this equation, by virtue of Y-periodicity of u (2) equation reduces to C H is the homogenized elasticity matrix for the overall region given by (A) Equation (A) solved by FEM with appropriate BC gives solution corrected for atomic scale effects

22. Local Elastic Constants Based on Kluge et al J. of App. Phy. (1990) Knowing local strain and local stress in a small region V of MD Simulation local elastic constants system of N interacting atoms in a parallelepiped whose edges are described by vectors a, b and c with H=(a,b,c) Constant strain application H=H o to H=H o +  H o (Parinello –Rahman Variable cell MD)

23. Local Elastic constants Local stress in small area defined as W  volume, r ij  distance between i th and j th atoms, U  interatomic potential function q  unit step function d  Dirac delta function R ij  center of mass of particles i and j This Method has been applied to grain boundaries using EAM and pair potentials

24. Computational Procedure Create an atomic model of microscopic Y scale Use molecular dynamics to obtain the material properties at various defects such as GB, dislocations etc. Form the  matrix and homogenized material properties Make an FEM model of the overall (X scale) macroscopic structure and solve for it using the homogenized equations and atomic scale properties

25. Summary Incorporating atomic-scale effects in determining the material behavior is important in a number of engineering applications. Grain boundaries structure and deformation characteristics can be studied at atomic scale. Using Molecular Dynamics it has been shown that extent of grain boundary sliding is related to grain boundary energy The formulation for AEH to link atomic to macro scales has been proposed with detailed derivation and implementation schemes. Work is underway to implement the computational methodology.