Analytical length scale calibration of nonlocal continuum from a microstructured buckling model

Abstract

International audienceThis paper deals with the buckling of a column which ismodeled by some finite rigid segments and elastic rotational springs and relating its solution to continuum nonlocal elasticity. This problem, which can be referred to Hencky's chain, can serve as a basic model to rigorously investigate the effect of the microstructure on the buckling behaviour of a simple equivalent continuum structural model. The buckling problem of the pinned-pinned discretized column is analytically investigated by introducing a Lagrange multiplier. Such a buckling problem is mathematically treated as an iterative eigenvalue problem. It is shown that the buckling load of this finite degree-of-freedom system is exactly obtained by a recursive formula involving Chebyschev polynomials. Euler's buckling load is asymptotically obtained at larger scales. However, at smaller scales, the buckling model highlights some scale effect that can be only captured by nonlocal elasticity for the equivalent continuum. We show that Eringen's nonlocal continuum is well suited to capture this scale effect. The small scale coefficient of the equivalent nonlocal continuum is then identified from the specific microstructure features, namely the length of each cell. It is shown that the small length scale coefficient valid for this buckling problem is very close to the one already identified from a comparison with the Born-Karman model of lattice dynamics using dispersive wave properties

Discrete and non-local elastica

“…It is worth mentioning that the spatial chaotic behavior of discrete chains has been also observed for non-conservative systems by Kocsis and Károlyi [12]. More recently, Wang et al [13] or Challamel et al [14,15] showed the possibility of linking the linearized Hencky system to non-local mechanics. Non-local mechanics is understood herein as an Eringen's type non-local differential model [16] where an implicit differential equation is assumed between the stress and the strain (or the bending moment and the curvature at the beam scale).…”

“…The analytical solution for the fundamental buckling load issued of the linearization process, has been calculated by Wang [4,5] -see also [11] or more recently by Challamel et al [14,15]. The reasoning is repeated here for completeness.…”

Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory

“…Based on the Eringen's nonlocal model, Reddy (2007) studied the bending, buckling and vibration of different linear beam models. Challamel et al (2014) presented an analytical method to calibrate the nonlocal parameter for buckling analysis of micro-structures. However, it was reported that the capability of nonlocal elasticity theory provided to study the size-dependent effects on the mechanical properties of small-scaled structures may exist some limited problems (Eltaher, Hamed, Sadoun, & Mansour, 2014;Li, Yao, Chen, & Li, 2015a;Lim, Zhang, & Reddy, 2015;Ma, Gao, & Reddy, 2008).…”

Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory

“…The basic property of the nonlocal functions can be seen in Lim et al [52]. Some studies [66][67][68][69][70][71] have been presented to identify and incorporate the nonlocal parameter e 0 a for different small-scaled structures. It is worth mentioning that, different materials produce different values of the material characteristic parameter l, some studies [53,72,73] have studied the material characteristic parameter l for different materials.…”