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

399

8

e M =

0.5

12  at 15000 (left)and 20000 (right) steps of simulation. Mixed mode loading,

Fig. 12. Distributions of

.

1.2. Theoretical solution for the crack-tip stress and displacement fields in isotropic linear elastic media The present study is an attempt to determine the higher order coefficients in Williams’ series expansion in the classical specimen for linear fracture mechanics – a plate with central crack under Mode I and Mixed Mode loadings using molecular dynamics method. For a crack subjected to both tensile and shearing stresses the two-dimensional stress field in the neighborhood of the crack tip may be approximated by the following functions in polar coordinates using Williams ’ approach: ( ) 2 / 2 1 ( ) , 1 ( , ) m k m k k ij k m ij m k r a r f    = = − = =− =   (1) where the index m is associated to the fracture mode; coefficients 1 k a are related to symmetric mode and 2 k a are corresponded to antisymmetric mode, the circumferential functions ( ) , ( ) k m ij f  depend on stress components and mode. Analytical expressions for angular eigenfunctions ( ) , ( ) k m ij f  are available (Karihaloo and Xiao (2001), 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            = + + − − − − −     = − − − − + − −     = − + − − + − −   (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            = − + − − − − − −     = − − + − − + − −     = − − − − + − −   ) ( ) ( ) 2,11 ( ) 2,22 ( ) 2,12

(3)

The displacement field in the vicinity of 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