1. Peipei Li - Civil Engineering pl474@cornell.edu Shule Hou - Civil Engineering sh983@cornell.edu Jiaqi Qu - Civil Engineering jq57@cornell.edu Coupled Atomistic and Discrete Dislocation method (CADD)

2. Topics  Background  What is CADD  Model of CADD  1D Model  1D Model Example  Implementation  How to run the code  Results

3. Background Some phenomena (dislocation nucleation, cross-slip, crack formation and growth) involving plastic deformation and fracture of ductile materials are intrinsically atomistic. Atomistic studies are usually unable to address large- scale deformation except with supercomputers. So multi-scale methods are introduced in which certain key regions are modeled atomistically while most of the domain is treated with an approximate continuum model(such as FEM) and able to reduce computational cost.

4. What is CADD Coupled atomistic and discrete dislocation method(CADD) CADD is one of the multi-scale methods. CADD minimizes the number of atoms and replaces atomic degrees of freedom by continuum DOFS describing the continuum elastic displacements and the dislocation lines with little or no loss of accuracy.

5. Model of CADD Ⅰ : contain all the singularities and discontinuities (Discrete dislocation) Ⅱ : smooth, continuous and ideally suited to FE Ⅱ : smooth, continuous and ideally suited to FE (Linear elastic body bvp) (Linear elastic body bvp) Ⅲ : atomistic region

6. Pad: Passing of dislocations Ensure that real atoms at and near the interface are properly coordinated Mitigate the effect of the free surface that would be created on the atomistic region during the cutting process Model of CADD

7. 1D Model The total potential energy of CADD: Where is the energy functional for chain of atoms, is the total continuum energy. Where k 1 is the stiffness for first-neighbour interaction, k 2 is the stiffness for second –neighbour interactions.

8. 1D Model The total potential energy of CADD: Where is the energy functional for chain of atoms, is the total continuum energy. Where k c is the effective stiffness for the element. For a proper value for k c in a state of uniform deformation,

9. 1D Model Example A chain of 101 atoms, The displacement of atom 0 is fixed, A force f =1 applied to atom 100, K 1 =1,K 2 =1,K c =6, Interface I = 50, Considering inhomogeneous deformation, apply additional force of magnitude f = 0.1 to atoms/nodes I- 2, I-1, I. The distance a between atoms is constant, the value is 1.

10. 1D Model Example Using MATLAB to solve this problem, [K]{d}={ f } K a : Stiffness of atoms part K c : Stiffness of continuum part

11. 1D Model Example W A Curtin and Ronald E Miller Atomistic/continuum coupling in computational materials science Our MATLAB solution

12. We get the code package from http://qcmethod. org/ http://qcmethod. org/ (This website serves as a clearinghouse for multi-scale method-related information.) Unzipped the package Download the terminal ( Cygwin under windows) Implementation

13. How to run the code Commands: % cd ~/QC/GB-example % Make QCCOMPILER=gnu (gFortran compiler) After compile, we'll get executable—gb. Use commands % cd ~/QC/GB-example/Shear %../ gb gb_shear.out Run gb, we’ll get outputs. Finally, we need some tools to visualize the outputs. Here we used Tecplot to get the plots and even videos.

14. Example This example builds an Al bi-crystal consisting of two face-centered cubic (fcc) crystals separated by a (111) twin plane. The twin plan has a step, the height of which is equal to three (111) interplanar spacings. The bi-crystal is subjected to an increasing uniform shear which causes the twin boundary to migrate in the direction perpendicular to the twin plane.

15. Code: FEM part The example presented here uses three-node linear elements with one Gauss point at the centroid of each element. The iso-parametric formulation is used. A utility routine that can be used by the user_mesh routine to generate regular or symmetric meshes. Eg. Set SymmetricMesh=.true, We get the finite mesh for the continuum region as: The element, local node numbering and shape functions

16. Results Final mesh Final mesh in atom shape Video

17. Thank you !