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ABSTRACT
Basic concepts and numerical integration issues in the 2D boundary element implementation of strain gradient elasticity problems
Ney Augusto Dumont
Pontifical Catholic University of Rio de Janeiro, Brazil 22451-900 Rio de Janeiro, Brazil
* Corresponding Author: Ney Augusto Dumont,
The International Conference on Computational & Experimental Engineering and Sciences 2021, 23(1), 2-2. https://doi.org/10.32604/icces.2021.08187
Abstract
The mathematical modeling of microdevices, in which structure and
microstructure have approximately the same scale of magnitude, as well as of
macrostructures of markedly granular or crystal nature (microcomposites),
demands a nonlocal approach for strains and stresses. The present proposition is
based on a simplified strain gradient theory laid down by Aifantis, which has also
been applied mainly by Beskos and collaborators in the context of the boundary
element method. This paper is an extension of a presentation made during the
ICCES 2014 Conference in Crete, Greece, now relying on machine-precision
evaluation of all singular and hypersingular integrals required in a 2D
implementation. The singular fundamental solution needed in a boundary element
formulation is rederived from the developments made by Polyzos and co-workers
and conceptually assessed. The basic formulation is presented for 2D and 3D
problems, but the numerical issues and the displayed examples only concern the
2D case. An important contribution is the evidence that normal displacement
gradients along the boundary must be approximated independently from
displacements, which seems to better agree with Mindlin's pioneer proposition on
the subject of strain gradient elasticity and results that no additional treatment of
“corner nodes” is actually required. The terms related to hypersingular integrals
are newly assessed in order to cope with the strict requirements of a variationallybased formulation particularly when applied to curved boundaries. The
implementation of an accurate numerical quadrature scheme for improper integrals
(due to the occurrence of Bessel functions) over a curved boundary is also
addressed. The implementation is done for in principle arbitrarily high order 2D
elements, with results assessed both conceptually (fulfilment of spectral properties)
and in terms of numerical convergence for some academic applications.
Keywords
Cite This Article
Dumont, N. A. (2021). Basic concepts and numerical integration issues in the 2D boundary element implementation of strain gradient elasticity problems.
The International Conference on Computational & Experimental Engineering and Sciences, 23(1), 2–2.