Scaling Relations for the Lengths and Widths of Fractures

Fault-fracture patterns have been studied in slabs of clay during extensional deformations. Fractures nucleate and grow on many scales. A new scaling relation is proposed for the length $l$ of a fracture as a function of the area $l\sim A^{\beta }$, with the same exponent $\beta =0.68\mathrm{}\pm 0.03$ for many deformation types. A consequence of this scaling relation is that the width of a fracture scales with the length as $w\sim l^{(1-\beta )/\beta }$. A spring network model is shown to reproduce the pattern, both visually and statistically, with the same scaling exponents.