‘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 nonclassical 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 model is optimized regarding stress concentration factors reported for infinite Cosserat plates. The analysis of Eringen 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 model are not effective at the vicinity of the hole. Yet, obtaining different 'explicit' material parameters for each sample problem reveals

Stress distribution around an elliptic hole in a plate with ‘implicit’ and ‘explicit’ non-local models

“…The nonconventional (enlarged) symmetric model of the present study is validated by comparing displacement/stress fields of the model in Fig. 2(b), with the corresponding full model in [56], and identical results are obtained.…”

“…Considering the herein presented example with an elliptic hole of different aspect ratios, which can be considered as a generalisation of the problem treated in an authors' previous article [56], the formulation of Eringen's model is customised to include the geodetical distance into the kernel function. Accordingly, for any two points in a domain that the diffusion process cannot follow a straight line, geodetical path will be calculated by following the steps described below.…”

“…For mesh sensitivity analysis the results of Model 1 are compared with [56] considering same material parameters. Although, the number of elements in the present study is increased by 25% with respect to the ones in [56], the stress concentration factor at point A, (i.e. SCF = (σ xx ) A /σ 0 ) only changed by 1.4%.…”

“…Possessing additional kinematic and dynamic descriptors significantly enriches the description of the non-local theories, while in order to better understand their capabilities, limitations and similarities, many comparative studies have been conducted [51][52][53][54][55][56]. With a similar motivation, the focus here is on plates having either -easier to handlecircular hole or -more generic-elliptic hole.…”

“…With a similar motivation, the focus here is on plates having either -easier to handlecircular hole or -more generic-elliptic hole. Literature survey shows that; besides a recent paper of the authors [56], the considered problem has been investigated only on the basis of micropolar theory looking for numerical or exact solutions [15,[57][58][59][60][61][62][63].…”