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

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numerical method for determination of SIFs from atomistic simulations is proposed. Using atomic coordinates to describe the displacement field around a crack tip, this method projects observed displacements onto the set of continuous displacement fields defined by the Williams series expansion, with the expansion coefficients determining the SIFs. In (Machova et al. (2017)) the results of 3D molecular dynamics simulations are presented. The stress fields on the atomistic level in 3D bcc iron crystals with edge cracks are obtained. Atomistic results were compared with continuum mechanics predictions. The results of 3 D atomistic simulations are in agreement with the Beltz-Rice continuum predictions (Beltz and Machova (2004), Beltz and Machaova (2007)). The overarching objective of the paper (Roy and Roy (2019)) is to investigate the validity of application of continuum-based linear elastic fracture mechanics (LEFM) methodology, which is often employed by researchers to model fracture processes at the “discrete” atomic scale. A detailed methodology for computing atomistic J -integral was presented, and its critical value was used as a metric for fracture toughness prediction for a covalently bonded material. The methodology was verified by applying it to a single graphene sheet with zig-zag morphology using the bond-order based ReaxFF potential in the LAMMPS MD software. Due to its inherent ability to model bond breakage, ReaxFF potential was used to successfully model crack propagation in a graphene sheet. The critical J integral value obtained from the methodology discussed in this paper were found to be in good agreement with the value available in literature. Thus, predictions obtained using the atomistic J-integral are compared with LEFM predictions for the case of a single (zig-zag) graphene sheet with a center-crack under tensile loading at room temperature, and show significant deviation from LEFM for crack lengths below a certain threshold. After this short glimpse one can conclude that the research activity currently carried out on determination of continuum fracture mechanics parameters using computational modelling through molecular dynamics method has not yet led to univocal and conclusive results and causes the need for further extensive computational experiments aimed at carefully funding the parameters of the fracture mechanics. The main purpose of this study is to compute the parameters of classical fracture mechanics, namely, stress intensity factors, T-stress and higher-order coefficients of the Williams series expansion using the method of molecular dynamics and to compare the angular distributions of the stress tensor components obtained using computer modelling with the known analytical distributions of conventional fracture mechanics.

Nomenclature ij 

stress tensor components around the crack tip

, r 

polar coordinates of the system with its origin at the crack tip coefficients of the terms of the Williams series expansion

m

k a

, I II K K , ( ) k m ij f  , ( ) k m i g  ( ) ( )

mode-I stress intensity factors

angular functions included in stress distribution related to the geometric configuration, load and mode angular functions included in displacement distribution related to the geometric configuration, load and mode

m G

index associated to the fracture mode

shear modulus mixity parameter

e M

1.1. Atomistic modelling of a central crack All atomistic simulations are performed by means of molecular dynamics method with a classical molecular dynamics software Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) using an embedded atom model EAM potential a copper crystal (Cu_u3.eam). FCC copper is considered. The plate with a central crack under tensile loading and mixed mode loading has been modeled. For the full MD model simulations start with the static minimization of the whole system that is followed by the dynamics equilibration to achieve a stress – free state and temperature of 10 K. NPT ensemble is used for the dynamics equilibration. Next, displacement-controlled