Post on 31-Dec-2015
description
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 !