Assumptions:
- Cross terms in strain tensor are defined as equal $\varepsilon_{xy} = \varepsilon_{yx}$.
- pure mode I crack.
- Far from crack tip, material is purely elastic and we are way below yield stress => $\varepsilon_{ij} \propto \sigma_{ij}$.
- Cross term $\sigma_{yx} $in linear elastic fracture mechanics (LEFM) contains factor $\sin{\frac{\theta}{2}}$.
Now, cross term is not symmetric respect to $\theta$ (contains sin). The material however should be symmetric and thus $\varepsilon_{xy} = -\varepsilon_{yx}$. What am I missing?
What is the difference between positive and negative shear stress?
LEFM formulas can be found e.g. in page 6 in: