# Advanced Structural Analysis/Part I - Theory/Failure Modes/Fatigue/Crack Initiation/The Similarity Principle

Wöhler curves relate directly to specific test specimens and environments. Test specimens for assessment of the fatigue properties of materials are often small unnotched peaces of material with smooth surfaces. Thus, the corresponding Wöhler curve represents something that is very dissimilar, and therefore not immediately pertinent to many structural elements in real engineering applications.

One important concept in expanding the applicability domain of fatigue test data is the Similarity Principle which implies that high cycle fatigue at a local surface is governed by the local stress field. Consequently, according to the similarity principle, the Wöhler curve of an unnotched material can be directly applied to a notched specimen of the same material, if the analysis is based on the local stress field of the considered regions. For instance.

${\displaystyle \sigma _{fNotched}={\frac {\sigma _{f}}{K_{t}}}}$

Where:

${\displaystyle \sigma _{fNotched}=}$ the nominal fatigue limit of a notched specimen

${\displaystyle \sigma _{f}=}$ the fatigue limit of the unnotched reference specimen

${\displaystyle K_{t}=}$ the stress concentration factor of the notch

It should be noted that experiments indicate that the Similarity Principle is conservative.

Clearly, detailed finite element analysis may yield the local stress level directly, and thus eliminate the need to assess any stress concentration factors.