Fracture Network Mathematics

2D elastostatic displacement solutions for the Yoffe Mode I and Rice Mode II crack models are reviewed. These solutions are used to introduce the elastostatic displacement solution for a 2D Mode I/II multi-crack configuration in terms of meromorphic differential forms on a hyperelliptic curve. The complex dimension of the vector space of mermorphic forms is demonstrated to be g+1, where g is the genus of the hyperelliptic curve, using the Riemann-Roch theorem. Limitations of the 2D multi-crack model to modeling 3D fracture networks are identified, and a mathematical description of 3D fracture network dynamics preceding an earthquake based on singular spectrum analysis of crack phase fields is conjectured.