PSI - Issue 40

L.V. Stepanova et al. / Procedia Structural Integrity 40 (2022) 392–405 Stepanova L.V., Belova O.N. / Structural Integrity Procedia 00 (2022) 000 – 000

400

9

( (

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

( ) k

( ) ( ) ( ) ( )

g g g g

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

   

1,1

( ) k

1,2

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

 

(

 

m k a are the parameters which should be found. The SIFs can be computed from the coefficients

The coefficients

1 2 a is related to T-stress as

1 2 4 . a

1 1 2

2 1 2

as

and

.

The aim of this study is to determine

I K a =

II K a = −

 = −





1

o

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

the higher order coefficients

and Mixed Mode loadings. The theoretical solution (1) can be written in the matrix form CA  = ,

(4) where  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. Two methods for computing the unknown coefficients were utilized. The first one is based on the minimization of the objective function in the framework of which the values of fracture parameters are estimated by minimizing the objective function: ( ) ( ) ( ) . T J A A A   = − − The second approach is based on the closed form solution for the unknown vector of parameters when the closed form solution for the linear system (4) can be found as ( ) 1 T T A C C C − =  (5)

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