A peierls criterion for the onset of deformation twinning at a crack tip
E. B. Tadmor and S. Hai
Journal of the Mechanics and Physics of Solids, 51, 765-793 (2003).

ABSTRACT
A criterion for the onset of deformation twinning (DT) is derived within
the Peierls framework for dislocation emission from a crack tip due to
Rice (1992).  The critical stress intensity factor (SIF) is obtained
for nucleation of a two-layer microtwin, which is taken to be a precursor
to DT. The nucleation of the microtwin is controlled by the <i>unstable
twinning energy</i> &gamma;<sub>ut</sub>, a new material parameter identified
in the analysis. &gamma;<sub>ut</sub> plays the same role for DT as
&gamma;<sub>us</sub>, the unstable stacking energy introduced by Rice,
plays for dislocation emission. The competition between dislocation emission
and DT at the crack tip is quantified by the <i>twinning tendency</i>
<i>T</i> defined as the ratio of the critical SIF's for dislocation nucleation
and microtwin formation.  DT is predicted when <i>T</i>&gt;1 and dislocation
emission when <i>T</i>&lt;1. For the case where the external loading is
proportional
to a single load parameter, <i>T</i> is proportional to
sqrt(&gamma;<sub>us</sub>/&gamma;<sub>ut</sub>).
The predictions of the criterion are compared with atomistic simulations
for aluminum of Hai and Tadmor (2003) for a number of different crack
configurations
and loading modes. The criterion is found to be qualitatively exact for all
cases, predicting the correct deformation mode and activated slip system.
Quantitatively, the accuracy of the predicted nucleation loads varies from
5% to 56%. The sources of error are known and may be reduced by appropriate
extensions to the model.