PSI - Issue 50

L.V. Stepanova et al. / Procedia Structural Integrity 50 (2023) 275–283 Author name / Structural Integrity Procedia 00 (2021) 000 – 000

279

5

The values of the elastic tensor components and values of Poisson ratio both for monocrystalline copper and aluminum are in good compliance with the experimental information for these macroscopic mechanical quantities. 3. Asymptotic solution of classical fracture mechanics of brittle materials The current study is endeavored at determining the scaled coefficients in the Max Williams series expansion considering higher order terms using molecular dynamics method and comparison of the coefficients with the Williams series expansion known from the analytical solution of classical fracture mechanics. The Williams solution uses the mathematical form of an infinite series to characterize the notch and crack tip stresses according to which asymptotic expressions for the stresses in a plane medium with a central traction-free crack subjected to mixed mode I/II loadings are presented as:   2 /2 1 ( ) , 1 ( , ) m k m k k ij k m ij m k r a r f            (5) where index m is related to the failure mode; amplitude coefficients m k a are associated with the geometric configuration, load and mode, circumferential functions ( ) , ( ) k m ij f  depend on stress tensor elements and loading mode. Analytical formulae for circumferential eigenfunctions ( ) , ( ) k m ij f  are widely accessible and given in (Karihaloo and Xiao (2001)):       ( ) 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                                              (6)





 ( ) ( /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

(7)

The displacement vector components in the proximity the notch and crack tip can be described via the Max Williams power series expansion as     2 /2 ( ) , 1 ( , ) / , m k m k k i k m i m k u r a G r g          (8)

where in above equations the following notations are adopted         ( ) 1,1 ( ) 1,2 ( ) /2 ( 1) cos /2 ( /2)cos( /2 2) , ( ) /2 ( 1) sin /2 ( /2)sin( /2 2) , k k k k g k k k g k k k                             ( ) 2,1 ( ) 2,2 ( ) /2 ( 1) sin /2 ( /2)sin( /2 2) , ( ) /2 ( 1) cos /2 ( /2)cos( /2 2) . k k k k g k k k g k k k                    

m k a are the unknown mode I and mode II parameters and should be found theoretically,

The coefficients

numerically or experimentally. The SIFs are expressed from the factors m k a as

1 1 2

2 1 2

I K a 

II K a 





and

.

1 2 a is called T-stress as

1 2 4 . a

 

Eqs. (6) are valid for pure crack opening loads whereas Eqs. (7) are valid for

1

o