PSI - Issue 50

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

277

3

Nomenclature ij 

stress tensor components in the proximity of the notch and crack tip the distance from the notch or crack tip and the polar angle scaled multipliers of the terms of the Max Williams power series expansion for mixed mode I+II loading stress intensity factors for pure tensile loadings and for pure shear loadings circumferential functions included in stress distributions associated with the geometry of the cracked or notched specimen and types of loading angular functions included in displacement distributions associated with the geometry of the cracked or notched specimen and types of loading

, r 

m

k a

, I II K K , ( ) k m ij f  ( )

( ) , ( ) k m i g 

E

Young’s modulus Poisson’s ration shear modulus



G

( ) U 

the potential energy of the system

 i 

the strain tensor

the strain tensor components in Voigt notation (or Voigt form)

the elastic matrix elements in Voigt notation

ij C

2. Molecular dynamics simulations 2.1. Mechanical properties of monocrystalline fcc copper and aluminum

First and foremost, the aim of our study is to determine the elastic properties of copper and aluminum single crystals by the molecular dynamics method effectuated in the open code LAMMPS. This is indispensable in order to be confident that the potential we have taken correctly models the properties of the material we are studying. The tensor of elastic modules can be determined using the potential energy of atoms ( ) U  , which depends on the strain tensor  and can be decomposed into a Taylor series:

2

1 2

U

U





6

6

1 , 1       i  i i j    

( ) U U 

(0)  

,

(1)

i j  

i 

i

j

where the elastic tensor components can be expressed as

2

1

U



,

C

(2)



ij

V

 

 

0

i

j

where ij C is the elastic matrix elements, 0 V is the volume of the atomistic system. By modeling of various types of loadings, the values of potential energy and strains were obtained. According to the data computed, a curve of the dependence of energy on strains was constructed and an approximating curve was carried out. The coefficient standing in the second degree of the polynomial approximating curve was substituted into the formula (2). Thus, we obtained the tensor of elastic modules (in GPa):