PSI - Issue 32

L.V. Stepanova et al. / Procedia Structural Integrity 32 (2021) 261–272 Author name / Structural Integrity Procedia 00 (2019) 000–000

269 9

( (

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

( ) k

( ) ( ) ( ) ( )

g g g g

k k = + + − = − − − k = − + − = − + − − (

θ κ θ κ

1,1

( ) k

1,2

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

θ κ

(

θ κ

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

and

.

The goal of this study is to determine the

I K a =

II K a = −

σ = −

π

π

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. Eqs. (1) can be written in the matrix form CA Σ = ,

(4) where Σ is the vector consisting of the numerical data estimated from molecular dynamics modeling, C is a rectangular matrix of order 2 m n × , A is the vector consisting of unknown mode I and mode II fracture parameters. The values of fracture parameters are estimated by minimizing the objective function: ( ) ( ) ( ) . T J A A A σ σ = − Σ − Σ The closed form solution for the unknown vector of parameters can be found as ( ) 1 T T A C C C − = Σ , (5)

Fig. 13.Various contours surrounding the crack tip with different numbers of atoms.