While focusing heavily on LEFM, the text addresses conditions where significant yielding occurs at the crack tip. J-Integral

In a large industrial plant, a critical component, a high-pressure pipeline, failed catastrophically, resulting in significant damage and downtime. The pipeline was made of a high-strength steel alloy, with a wall thickness of 2 inches and an outside diameter of 12 inches. It was designed to operate at pressures up to 1000 psi.

Griffith’s energy balance: crack grows if released elastic energy ≥ surface energy.

By integrating the Paris Law, engineers can predict the remaining operational lifespan of a structure containing a known defect: