peipei li - civil engineering pl474@cornell.edu shule hou - civil engineering sh983@cornell.edu...

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)

Topics Background What is CADD Model of CADD

1D Model 1D Model Example

Implementation How to run the code Results

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.

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.

Model of CADD • Ⅰ: contain all the singularities and discontinuities

(Discrete dislocation)

• Ⅱ: smooth, continuous and ideally suited to FE

(Linear elastic body bvp)

• Ⅲ: atomistic region

• 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

1D Model• The total potential energy of CADD:

• Where is the energy functional for chain of atoms,

is the total continuum energy.

• Where k1 is the stiffness for first-neighbour interaction, k2 is the stiffness for second –neighbour interactions.

1D Model• The total potential energy of CADD:

• Where is the energy functional for chain of atoms,

is the total continuum energy.

• Where kc is the effective stiffness for the element.• For a proper value for kc in a state of uniform

deformation,

1D Model Example• A chain of 101 atoms,• The displacement of

atom 0 is fixed,• A force f =1 applied to

atom 100,• K1=1,K2=1,Kc=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.

1D Model Example• Using MATLAB to solve this problem,

[K]{d}={ f }

Ka: Stiffness of atoms part Kc: Stiffness of continuum part

1D Model Example

W A Curtin and Ronald E Miller

Atomistic/continuum coupling

in computational materials science

46 47 48 49 50 51 52 53 540.1

0.15

0.2

0.25

Str

ain

Atom/Node Number

Point Force at Interface: FE solution

Our MATLAB solution

We get the code package from

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

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.in>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.

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.

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

Results • Final mesh Final mesh in atom shape

• Video

Thank you !