PSI - Issue 39

Larisa Stepanova et al. / Procedia Structural Integrity 39 (2022) 748–760 Author name / Structural Integrity Procedia 00 (2019) 000–000

750

3

continuum and atomistic approaches it is still debated whether a continuum - based concept should be applied to a discrete system and whether continuum fracture mechanics parameters could be computed from MD simulations. Thus, one can conclude that MD simulations need to be performed to confirm the conclusions. This must be done with the most severe scrutiny lest errors at comparing the results of continuum and atomistic approaches. The key purpose of this study is to obtain stress intensity factors, T-stress and higher order coefficients of the Williams approximation by MD method and to compare their values with the corresponding values obtained by the conventional linear elastic fracture mechanics for different values of the mixity parameter.

Nomenclature ij σ

stress tensor components around the crack tip displacement vector components around the crack tip polar coordinates of the system with its origin at the crack tip coefficients of the terms of the Williams series expansion

i u

, r θ

m

k a

, I II K K , ( ) k m ij f θ , ( ) k m i g θ T ( ) ( )

mode-I and mode II stress intensity factors

T-stresses

circumferential functions included in stress distribution related to the geometric configuration, load and mode circumferential functions included in displacement distribution related to the geometric configuration, load and mode

m

index associated to the fracture mode

shear modulus mixity parameter

G

e M

2. Molecular dynamics modelling In the computational 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 σ θ θ ∞ − = =−∞ = ∑ ∑ (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 θ angular functions depending on stress component and mode. Analytical expressions for angular eigenfunctions ( ) , ( ) k m ij f θ are available (Karihaloo and Xiao (2001), Hello et al. (2012), Hello (2018)): ( ) ( ) ( ) ( ) 1,11 ( ) 1,22 ( ) 1,12 ( ) ( / 2) 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) , k k k k k k f k k k k k f k k k k k f k k k k k θ θ θ θ θ θ θ θ θ   = + + − − − − −     = − − − − + − −     = − + − − + − −   (2)

(

)

( ( ) ( / 2) 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) . k k k k k k f k k k k k f k k k k k f k k k k k θ θ θ θ θ θ θ θ θ   = − + − − − − − −     = − − + − − + − −     = − − − − + − −   ) ( ) ( ) 2,11 ( ) 2,22 ( ) 2,12

(3)

The displacement fields around the crack tip can be described as