PSI - Issue 32

8

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

268

e M =

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

0.5

.

Fig. 12. Distributions of

1.2. Analytical description of the crack-tip stress and displacement fields The present study is aimed at determination of the higher order coefficients in Williams’ series expansion in the classical specimen for linear fracture mechanics – a plate with double edge notches using molecular dynamics method. The Williams solution uses the mathematical form of an infinite series to describe the crack tip stresses according to which asymptotic expressions for the stress field in a plane medium with a traction-free crack submitted to mode I and mode II loading are presented as: ( ) 2 / 2 1 ( ) , 1 ( , ) m k m k k ij k m ij m k r a r f σ θ θ = =∞ − = =−∞ = ∑ ∑ (1) with index m associated to the fracture mode; coefficients m k a related to the geometric configuration, load and mode, angular functions ( ) , ( ) k m ij f θ depending on stress components and mode. Analytical expressions for circumferential eigenfunctions ( ) , ( ) k m ij f θ 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 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 ( ) ( ) 2 / 2 ( ) , 1 ( , ) / , m k m k k i k m i m k u r a G r g θ θ = =∞ = =−∞ = ∑ ∑

where the following notations are adopted