Validity of linear elasticity in the crack-tip region of ideal brittle solids

Abstract

It is a well known that, according to classical elasticity, the stress in the crack-tip region is singular, which has led to a debate over the validity of linear elasticity in this region. In this work, comparisons of finite and small strain theories have been made in the crack-tip region of a brittle crystal to comment on the validity of linear elasticity in the crack tip region. We find that linear elasticity is capable of accurately defining the state of stress very close ($\sim$1 nm) to a static crack tip.

Appendix

The elasticity tensor $c_{ijkl}$ linearly relates the stress tensor $\sigma _{ij}$ to the strain tensor $\epsilon _{ij}$ as $\sigma _{ij} = c_{ijkl} \epsilon _{ij}$. Symmetries of $\mathbf {c}$ and $\epsilon$, and the requirement for the strain energy to be positive definite, reduce the fourth order tensor $c_{ijkl}$ to a $6\times 6$ matrix $C_{ij}$. For cubic cystal symmetry $C_{ij}$ has three independent elements and takes the form

$$\begin{aligned} \left( \begin{array}{cccccc} C_{11} &{} C_{12} &{} C_{12} &{} 0 &{} 0 &{} 0\\ C_{12} &{} C_{11} &{} C_{12} &{} 0 &{} 0 &{} 0\\ C_{12} &{} C_{12} &{} C_{11} &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} C_{44} &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} C_{44} &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} C_{44}\end{array} \right) \end{aligned}$$

where $C_{11}=$ 151.35 GPa, $C_{12}=$ 76.409 GPa and $C_{44}=$ 56.422 GPa in the atomic crystal frame of reference (Kermode et al. 2008). This elasticity tensor has to be rotated from the crystal frame to the sample frame ($x=[11\bar{2}]$, $y=[111]$, $z=[1\bar{1}0]$), corresponding to cleavage along the lowest energy $(111)$ cleavage plane.