A review of continuum mechanics models for size-dependent analysis of beams and plates

Abstract

This paper presents a comprehensive review on the development of higher-order continuum models for capturing size effects in small-scale structures. The review mainly focus on the size-dependent beam, plate and shell models developed based on the nonlocal elasticity theory, modified couple stress theory and strain gradient theory due to their common use in predicting the global behaviour of small-scale structures. In each higher-order continuum theory, various size-dependent models based on the classical theory, first-order shear deformation theory and higher-order shear deformation theory were reviewed and discussed. In addition, the development of finite element solutions for size-dependent analysis of beams and plates was also highlighted. Finally a summary and recommendations for future research are presented. It is hoped that this review paper will provide current knowledge on the development of higher-order continuum models and inspire further applications of these models in predicting the behaviour of micro-and nano-structures.

Bifurcation analysis of the rotating axially compressed nano‐rod with imperfections

“…with being the normalized solution (26), (2) being given by (28), and̃being given through = 3̃+ , see (38). Function , appearing in the bifurcation Equation (39), is considered as a function of and a bifurcation parameter Δ 1 , since Δ 2 = ′ ( 01 )Δ 1 , see (25).…”

“…Unlike the atomic modelling of small‐scale structures, continuum approach requires less computational effort and, moreover, in many cases of buckling and free vibrations of such structures as well as for the wave propagation analytical solutions are also available. The application of non‐local theory in the static and dynamic stability problems of different types of rods is quite extensive, see the review articles . and book…”

“…The application of non-local theory in the static and dynamic stability problems of different types of rods is quite extensive, see the review articles. [20,38] and book. [17] There are two approaches in constructing the governing equations for rod's equilibrium.…”

Existence and integral representation of solutions for plane deformations of a micropolar elastic solid with surface elasticity

“…The incorporation of surface elasticity into continuum-based models of deformation has become widespread in the recent literature primarily due to the desire to accommodate size effects in elastic solids (see, for example, [29,31] and the references contained therein). The starting point in the development of such models is in selecting an appropriate theory of surface elasticity.…”

8‐node hexahedral unsymmetric element with rotation degrees of freedom for modified couple stress elasticity

“…Many experimental observations have shown that the micro/nano structures, which have been widely used in modern engineering applications, may experience size‐dependent mechanical behaviors. Various non‐classical higher‐order continuum theories, such as the strain gradient theory (SGT) and the couple stress theory (CST), have been developed for describing such size‐dependent phenomena . However, as these higher‐order theories are more complex than the classical one, the analytical or semi‐analytical solutions are available only for restricted problems with simple geometries and certain boundary conditions.…”