The analysis of scaled cracked components

Abstract

A recent arrival in the open literature is the scaling theory finite similitude bringing into existence a countably infinite number of new similitude rules. The application of the theory to fracture mechanics has revealed that the first-order similitude rule is applicable in the sense that information gleaned from two fractured scaled components can be combined to capture the response of a third. This paper examines an unexplored aspect of the finite similitude theory, in that, it is shown how it can be re-configured for the purposes of scaling analysis. This is achieved by retaining the first derivative (with respect to scale) in the first-order rule and on involving an additional transport equation. The approach assumes the first-order rule to be exact, which is confirmed here, but in addition, has the advantage of automatically accounting for any defect-size related size effects that are present. It is a top-down scheme with analysis performed under the constraint of an imposed rule. To instigate the methodology, fracture mechanics for the first time is defined on a new scaling space, where all physical quantities are deemed dependent on a single dimensional scaling parameter β. This approach is demonstrated to facilitate the imposition of similitude rules and has led to the discovery of new relationships that connect fatigue responses at two sizes. Although analysis here is limited to linear elastic fracture mechanics (LEFM), it is demonstrated through proof, analytical, and experimentally validated numerical models, that the approach provides an exact scaling theory for fatigue.