﻿ MD simulation of crack propagati
 MD simulation of crack propagation in brittle crystals 118094 - Introduction to Computational Physics - Fall 2012/13 Liron Ben-Bashat Bergman
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Theoretical Background
Fracture of brittle crystals is important because it signals catastrophic failure which immensely reduces the reliability of brittle crystals, which are nowadays a main building block in many high tech industries, from MEMS and NEMS to bio-inspired devices and solar cells. It is complex because of its multi-scale nature and because crack initiation and propagation is difficult to predict. Cracks in brittle crystals are atomically sharp and propagate by breaking bonds by a complicated sequence. This process is greatly influenced by the anisotropy of the atomic arrangement at the crack tip. These characteristics prevent from continuum mechanics to describe atomic scale phenomena.

Fig. 1 Ductile fracture of Al vs. brittle fracture in mild steel

### Linear elastic fracture mechanics

Griffith [1] (1921) was the first to establish the relationship between material strength and a flaw in the material. He dealt with crack initiation only and didn’t consider the dynamics involved with the fracture process. The energy associated with crack initiation in the system is consisted of the external mechanical work, , elastic strain energy, , and  free surface energy, :

(1)

According to Griffith a crack will initiate when the following apply:

(2)                    ,
namely, when the driving force for crack initiation (the mechanical energy) equilibrates with the internal surface energy per unit area, the material resistance for initiation. Irwin [2], generalized Griffith’s approach by including a new and general strain energy release rate quantity (SERR), , defined as the change in mechanical energy per unit area:
(3)
The SERR is the mechanical driving force for crack initiation.
The material resistance for crack initiation includes all the dissipation mechanisms:

(4)

where the lower bound for   is , twice the free surface energy, appropriate for ideal brittle materials. Crack initiation will take place when:

(5)

The thermodynamics energy balance states that fracture initiates when the quasi-static strain energy release rate (SERR) equals the free surface energy of the cleaved planes. Freund [3] suggested that crack propagation occurs when the dynamic energy released per unit area of crack advance, G(V), equals the dynamic material property that resists crack propagation, . Freund used Navier elastodynamics equation for boundary value problem (BVP) of uniformly distributed compressive stresses on the crack surfaces of half infinite body and Rice J-Integral for the calculations of the SERR for crack propagating at constant speed. He defined a similar criterion to Irwin's except that he used a crack moving at a temporal velocity, where G(V) is the temporal total energy release rate:

(6)

The criterion may be approximated as a linear function of the crack velocity:

(7)

where
V is crack velocity, G(V) is net energy available to propagate the crack, which is the multiplications of the quasi-static SERR, , and a universal dynamic function, g(V).  is Rayleigh free surface wave speed, considered as a crack speed limit in solids under pure Mode I. The linear approximation of g(V) provides a sufficient and simple expression which is commonly used by the community of researchers in the field. The above relationship implies that faster cracks involve larger driving forces. Dynamic fracture experiments and numerical observations show that crack speed never attains . The maximal speed a crack can attain is 70%–90% of . This behavior is attributed to the onset of energy dissipation mechanisms such as thermal phonon emission.

Silicon crystal

Silicon crystal has a diamond like structure and is considered an ideally brittle material below ~700 degrees celsius. Silicon tends  to cleave along low energy cleavage systems. It has two low energy cleavage planes: {111} and {110} while the later has higher energies.

Fig. 2 Diamond structure unit cell

.
References:
[1] A.A. Griffith. Phenomena of rupture and flow in solids. Philosophical Transactions
of the Royal Society of London A221 163-198 (1921).
[2] Irwin, G. R. Fracture. In handbook of physics, Springer-Verlag, Berlin 6 551 (1958).
[3] L. B. Freund. Dynamic fracture mechanics. Cambridge university press (1990).