Unified finite element methodology for gradient elasticity

Abstract

In this paper a unified finite element methodology based on gradient-elasticity is proposed for both two-and three-dimensional problems, along with some considerations about the best integration rules to be used and a comprehensive convergence study. From the convergence study it has emerged that for both two and three-dimensional problems, the implemented elements show a convergence rate virtually equal to the corresponding theoretical values.Recommendations on optimal element size are also provided. Furthermore, the ability of the proposed methodology to remove singularities in statics has been demonstrated through a couple of examples, in both two and three dimensions.

Gradient elasticity: a transformative stress analysis tool to design notched components against uniaxial/multiaxial high‐cycle fatigue

“…(2) to be calculated as two uncoupled systems of second-order partial differential equations. This results in a straightforward and effective C 0 finite element implementation [12,13].…”

“…In this paper, the theory developed by Aifantis and co-workers in the early 1990s [4,10,11] and recently implemented in a unified finite element framework [12,13] is considered. This approach consists in enriching the constitutive relations with the Laplacian of the strains as follows:…”

“…Independently of the specific geometry, under-integrated bi-quadratic quadrilateral elements were used. The mesh in the vicinity of the stress concentrators was gradually refined according to the recommendations on optimal element size given in Refs [12,13]. In particular, in the highly stressed regions the average mesh size was equal to about 0.025 mm, with the minimum size approaching 0.002 mm (Fig.…”

“…In the finite element methodology discussed in Refs [12,13] and used for the numerical simulations in the present investigation, the second step was considered in terms of stresses as follows:…”

Gradient-enriched finite element methodology for axisymmetric problems

“…Here gradient elasticity theories come into play; in fact, as extensively discussed and proven in the literature (see, for instance, [10,15,17,19,20,42,46]), they are able to overcome the deficiencies of classical elasticity in accurately determining the stress fields in specific problems, especially those for which it is necessary to account for microand nanoscale effects in the description of the overall macroscopic response. In particular, gradient-enriched theories are able to remove singularities from (micro-)stress and (micro-)strain fields in the neighbourhood of sharp crack tips and, in general, they have a smoothing effect on the stresses in the presence of stress concentrators, such as notches and holes.…”