Beams, Plates and Shells in the Special Theory of Gradient Elasticity (GRADELA)

Bending deformations of various linear and surface structures, such as beams, plates and shells are studied within the context of a special theory of gradient elasticity, known as Aifantis’ GRADELA model. Since its original publication in 1992, various versions of strain gradient elasticity have been proposed often resorting to the pioneering contributions of generalized continuum mechanics of the 1960’s (e.g., Toupin, Mindlin, Eringen) which, however, contained too many phenomenological coefficients and were mainly used in relation to wave propagation studies. Thus, a number of problems related to the elimination of stress and strain singularities along with the interpretation of size effects, which could not be addressed with classical elasticity, remained unresolved. Emerging applications in micro/nanotechnology (MEMS/NEMS, nanotubes, graphene, cells, biomembranes) have revived the interest of the mechanics community in gradient elasticity models and their use in micro/nano mechanical problems. A recent comprehensive review with a large number of references can be found in a recent article by Askes and Aifantis (Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results, Int. J. Solids Struct. 48, 1962-1990, 2011). The goal of the present article is to provide a brief account on the application of the GRADELA model to deformation problems of bending and buckling for statics and dynamics. Some of these results have been obtained in a more complex form in previous publications of Lazopoulos. Thus, one of the main contributions of the article is the presentation of simpler results that can be of direct use for comparisons with related experiments.