PSI - Issue 37

Larisa Stepanova et al. / Procedia Structural Integrity 37 (2022) 900–907 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

902

3

e M

mixity parameter

2. Molecular dynamics simulations In the computer experiments the copper plate with the central crack was chosen to make comparisons between the atomistic stress field and the classical Williams series expansion: 2 ( ) /2 1 , 1 ( ) ( ), k m k ij k m ij m k r, a r f     − = =− =   (1) with index m associated to the fracture mode; m k a amplitude coefficients related to the geometric configuration, load and mode; ( ) , ( ) k m ij f  angular functions depending on stress component and mode. Analytical expressions for angular eigenfunctions ( ) , ( ) k m ij f  are available (Karihaloo and Xiao (2001)): ( ) ( ) ( ) ( ) 1,11 ( ) 1,22 ( ) 1,12 ( ) 2 / 2 ( 1) cos( / 2 1) ( / 2 1) cos( / 2 3) , 2 ( ) 2 / 2 ( 1) cos( / 2 1) ( / 2 1) cos( / 2 3) , 2 ( ) / 2 ( 1) sin( / 2 1) ( / 2 1) sin( / 2 3) , 2 k k k k k k k f k k k k k f k k k k k f k k k k            = + + − − − − −     = − − − − + − −     = − + − − + − −   (2) ( ) ( ) ( ) ( ) 2,11 ( ) 2,22 ( ) 2,12 ( ) 2 / 2 ( 1) sin( / 2 1) ( / 2 1) sin( / 2 3) , 2 ( ) 2 / 2 ( 1) sin( / 2 1) ( / 2 1) sin( / 2 3) , 2 ( ) / 2 ( 1) cos( / 2 1) ( / 2 1) cos( / 2 3) . 2 k k k k k k k f k k k k k f k k k k k f k k k k            = − + − − − − − −     = − − + − − + − −     = − − − − + − −   (3) The displacement fields around the crack tip can be described as ( ) ( ) 2 / 2 ( ) , 1 ( , ) / , m k m k k i k m i m k u r a G r g   = = = =− =  

where the following notations are adopted ( ) ( ) ( ) ( ) ( ) 1,1 ( ) 1,2 ( ) ( ) k k k k g k g k       = + + − = − − −

/ 2 ( 1) cos / 2 ( / 2) cos( / 2 2) , / 2 ( 1) sin / 2 ( / 2)sin( / 2 2) , k k k k   − − + − / 2 ( 1) sin / 2 ( / 2)sin( / 2 2) , / 2 ( 1) cos / 2 ( / 2) cos( / 2 2) . k k k k k k k     + − + − ) ( ) ) ( )

(

( ) 2,1 ( ) 2,2 k k

( ) ( )

g g

 

k = − + − = − + − −

(

 

m k a are the unknown mode I parameters. The SIFs can be computed from the coefficients as

The coefficients

1 2 a is related to T-stress as

1 2 4 . a

1 1 2

2 1 2

 = −

II K a = −

I K a =





and

.

The goal of this study is to determine the

1

o

m k a in the multi-point series expansion (1) for the central crack in a plate under Mode I and

higher order coefficients

Mixed Mode loadings. All computations have been performed in the open flexible simulation tool LAMMPS. The implementation was realized with the Embedded-Atom Method potential (EAM) for the computation of material properties. The total number of atoms were varying from 300000 to 800000. The cracks are created by removing atoms of crystalline structures. The simulation was carried out at a temperature of 10 0 K. The typical plate has the dimensions 723x723x14.46 Angstroms. The most experiments are implemented for a plate with 720 thousand atoms. The crack is obtained by removing 180 atoms in the center. The crack length is 72.3 Angstroms and the crack thickness is 7.23