Convergence and performance of the h- and p-extensions with mixed finite element C0-continuity formulations, for tension and buckling of a gradient elastic beam

Abstract

Mixed formulations with C 0 -continuity basis functions are employed for the solution of some types of one-dimensional fourth-and sixth-order equations, resulting from axial tension and buckling of gradient elastic beams, respectively. A basic characteristic of gradient elasticity type equations is the appearance of boundary layers in the higher-order derivatives of the displacements (e.g., in the stress fields). This is due to the small parameters (related to the size of the microstructure) entering the governing equations. The proposed mixed formulations are based on generalizations of the well-known Ciarlet-Raviart mixed method, where the new main variables are related to second-order (or fourth order, for the buckling problem) derivatives of the displacement field. The continuous and discrete Babuška-Brezzi inf-sup conditions are established. The mixed formulations are numerically tested for both the uniform h-and p-extensions. With regard to the axial tension problem, the standard quasi-optimal rates of convergence are numerically verified in all cases (i.e., algebraic rate of convergence for the h-extension and exponential rate for the p-extension). On the other hand, the h-extension observed convergence rates of the critical (buckling) load for the second model problem are slightly higher than the theoretical ones found in the literature (especially for polynomial order p = 1). The respective observed rates of convergence of the buckling load for the p-extension are still exponential.

Plane-strain crack problems in microstructured solids governed by dipolar gradient elasticity

“…In addition, efficient numerical techniques (see e.g. Shu et al, 1999;Amanatidou and Aravas, 2002;Tsepoura et al, 2002;Tsamasphyros et al, 2007) have been developed to deal with problems analyzed by the ToupinMindlin theory.…”

Problems of the Flamant–Boussinesq and Kelvin Type in Dipolar Gradient Elasticity

“…Cook and Weitsman [11], and Eshel and Rosenfeld [12,13]). More recently, this approach and related extensions for microstructured materials have been employed to analyze various problems involving, among other areas, wave propagation (Georgiadis et al [3], Vardoulakis and Georgiadis [14]), fracture (Georgiadis [8], Grentzelou and Georgiadis [9], Shi et al [15]), mechanics of defects (Lazar and Maugin [16]), finite elasticity (Fosdick and Royer-Carfagni [17]), plasticity (Fleck et al [18], Vardoulakis and Sulem [19], Begley and Hutchinson [20], Huang et al [21]), and numerical techniques (Shu et al [22], Amanatidou and Aravas [23], Tsepoura et al [24], Giannakopoulos et al [25], Tsamasphyros et al [26]). Based on the existing results, it is concluded that the Mindlin theory does extend the range of applicability of the 'continuum' concept in an effort to bridge the gap between classical continuum theories and atomic-lattice theories.…”

Energy theorems and the J-integral in dipolar gradient elasticity

“…In particular, recent work by Gao et al (1999) and Huang et al (2000Huang et al ( , 2004 introduced a mechanismbased strain gradient theory of plasticity that is established from the Taylor dislocation model. In addition, efficient numerical techniques (see, e.g., Shu et al, 1999;Amanatidou and Aravas, 2002;Engel et al, 2002;Tsepoura et al, 2002;Giannakopoulos et al, submitted for publication;Tsamasphyros et al, 2005) have been developed to deal with some problems analyzed with the Toupin-Mindlin approach. Based on the existing results, it is concluded that the Toupin-Mindlin theory does extend the range of applicability of the Ôcontin-uumÕ concept in an effort to bridge the gap between classical continuum theories and atomic-lattice theories.…”