‘Explicit’ and ‘Implicit’ Non-local Continuum Descriptions: Plate with Circular Hole

Abstract

Classical theory of elasticity fails to reflect the true behaviour of solids with internal material organization when internal and external length scales are of comparable orders. This drawback leads to emergence of non-classical continuum theories which are offered to be distinguished as ‘implicit’/‘weak’ and ‘explicit’/‘strong’ non-local models according to different interpretations of the incorporation of characteristic length scales. As an extension of recent works of authors, the presented chapter is focused on the correspondence between ‘implicit’ type Cosserat (micropolar) and ‘explicit’ type Eringen’s two-phase local/non-local models, in terms of characteristic quantities. To this end, an example problem of practical importance; a plate with a circular hole, is studied by employing standard displacement based finite element method. The non-locality of Eringen’s model is optimized regarding stress concentration factors reported for infinite Cosserat plates. The analysis of Eringen’s model is repeated by adopting both Euclidean and geodetical distance during incorporation of long range interactions. According to the results, stress fields of explicitly and implicitly non-local models seem to be in a very good agreement considering plates with large scale ratios, as the missing neighbour relations appeared at domain boundaries of Eringen’s model are not effective at the vicinity of the hole. Yet, obtaining different ‘explicit’ material parameters for each sample problem reveals that it is unlikely to have a unified relationship between characteristic quantities of Cosserat’s and Eringen’s models.