### Engineering Motivation

This problem considers the linear elastic response of a center crack in a planar "test specimen" to an imposed tensile loading. From symmetry of the geometry and loading, we consider just one quarter of the domain: a half-crack of length *μ*_{1} in a slab of half-thickness unity and of half-length (hence aspect ratio) *μ*_{2}. Our intermediate output is the compliance — displacement-stress product over the loaded boundary — which is equivalent to the total strain energy (per unit depth); our "final" output — the square root of the derivative of the compliance with respect to the crack length — is the Stress Intensity Factor (SIF). The SIF is crucial in the analysis of cracks and failure.

### Physical Principles

From the physical point of view, this Worked Problem emphasizes the scaling of the Stress Intensity Factor with geometric parameters; the connections between the total strain energy (or compliance), the Energy Release Rate (ERR), and the Stress Intensity Factor (SIF); and the application of the Energy Release Rate and Stress Intensity Factor to the prediction of (*i*) brittle failure by Griffith's criterion, and (*ii*) fatigue-induced crack growth by Paris's law.