The origin and development of cracks in earth materials is a subject of
general interest and wide application: from seismologists studying
earthquakes to engineers studying the strength of materials. Therefore
many laboratory experiments have been carried out to investigate the
response of rocks to an applied differential stress, often using
acoustic emissions (AE) to track intermittent crack growth inside the
rock specimen prior to system-sized sample failure. Under a constant
applied stress in double-torsion tensile tests with a guide groove and a
single dominant crack, independent observations of the stress intensity
factor and crack growth velocity imply the size of the largest (sub-)
critical crack grows with time t according to an inverse power-law.
A similar law holds for the mean crack length in a
population of micro-cracks growing under compressional stress with no
pre-defined fault plane, often treated using mean field models that
ignore localisation of damage clearly seen in the experiments. Here I
present a new hypothesis for the origin of this formula by combining
expressions for crack population growth and localisation in a single
model. The model is tested and some of its parameters inferred from
analysis of the AE rate and the spatial clustering of its source
locations from laboratory data.