Damage tolerant strategies for managing the lives of structural components, such as aircraft engines, require a probabilistic description of the results of inspections, particularly the probability of flaw detection and of false alarms. These quantities are controlled by the distributions of signal responses from nominally identical flaws and of noise signals in the absence of flaws. Traditional approaches to determine such distributions are empirical, but the incorporation of insight from physical models would have many advantages. These advantages are discussed and one particular aspect of the approach, the use elastodynamic theories to quantifying the effect of material microstructure on signal and noise distributions is discussed in detail. Emphasis is placed on the interrelationship of contributions from mechanics, materials science and statistics. Included is a discussion of the basic physical assumptions, the key model results and the experiments that have been conducted to verify those model predictions. The paper describes new applications of elastodynamics that rest heavily on a foundation built by a number of pioneers in the field, one of them was Professor Achenbach.