What is a straight line? Does this matter in mechanical analysis?

It is an assumption that engineers make every day: we draw a straight line to define our product geometry, and then we analyze the product performance - but do we ever challenge the reality of the geometry and how that would influence the performance? Small and smooth variations to a product geometry are generally anticipated in the design drawing tolerance definitions, and for a simple product or component these can be taken into account in analysis by assuming the maximum or minimum dimensions. For a more complex component, containing multiple load paths, it is not necessarily the case that choosing to analyze the maximum or minimum dimension cases would identify the strongest or weakest component within the tolerance range. This becomes further complicated when considering non-smooth geometry. How is lack of smoothness defined and recognized, and what kinds of influence does it have on the performance of the component? For example, component inspection might identify a small geometrical defect triggering a concession analysis of the stress raiser feature. On a more systematic level, should we not have a more systematic approach? Non-smooth geometry can be represented in mathematics by fractals and wavelets. In this paper I will explore non-smooth surface effects on the surface of an axi-symmetric uni-axial test specimen, considering how the size of the non- smooth surface features disturb the nominal stress state, and the development of localized plasticity. The analysis will also include fatigue cycling.