Buckling may be the key factor in the morphogenesis of natural and biological entities (Darcy Thomson, 1959); and it is certainly a structural limit state that determines the design of many engineering structures. Buckling can occur over a large spectrum of length scales from nanotechnology investigations to the civil engineering structural design scale. Yet, a stability theory applicable to nano, micro and macro structures remains a challenge. More specifically, such a theory must be able to capture specific length scale effects at nano, micro and macro scales, in order to be applicable in both classical fields of structural engineering (timber engineering, reinforced concrete engineering, steel structures or steel‐concrete composite structures) and innovative micro‐ and nano‐engineering of structures.
We investigate the elastic and inelastic stability of small‐scale or large‐scale structures with modern mechanics concepts, such as nonlocal and non‐smooth mechanics. Such nonlocal mechanics models permit the consideration of scale transition in stability analysis, and highlight the interest of gradient and integral‐based nonlocal mechanics for engineering design. Specifically, we focus on the predictive understanding of the static and dynamic behavior of microstructured solids at small and large scales, with the help of gradient elasticity and nonlocal theories. The first theoretical results obtained for the buckling or vibrations of nonlocal columns are very recent (e.g. buckling of Eringen's column or gradient elasticity columns in 2003). The structural paradigm of Galileo's cantilever beam has been reconsidered with both nonlocal elasticity and plasticity models. Specifically, scale effects and localization propagation have been analysed for this structural problem; while nonlocality was related to the material or structural microstructure dependence.
Based on a two‐scale nonlocal model, we recently obtained a unified theory for the buckling of gradient and nonlocal elasticity beam models. Specifically, we found that this engineering model can fit well molecular dynamics data for nanoscale applications. We also noticed that composite beams with interlayer slip, sandwich beams or nanobeams are ruled by some similar nonlocal elasticity models. In other words, the stability of macroscale beam models (at the civil engineering scale) and nanoscale beam models exhibit similar scale effects, even if the source of physical nonlocality depends on the application considered. A further application of nonlocal macroscopic model is the accurate modeling of soil‐structure interaction. In this quest, we found that the Reissner soil‐structure interaction model can be classified as a two length‐scale nonlocal elasticity model. In similar developments, we identified higher‐order beam shear models as gradient elasticity Timoshenko models.
The introduction of nonlocal concepts in elastic and inelastic stability analysis of structures opens new perspectives for innovative structural design, that range from very small scale structures for e.g. nanotechnology applications, to microstructured members composed of ecomaterials or nature‐ based materials, and large scale civil engineering structures, such as repetitive frame structures. Further developments of nonlocal plasticity and damage concepts will enhance current predictive capabilities at the structural scale to account for e.g. material softening behavior, related to microcracking that may well redefine Galileo's cantilever problem.
1. Challamel N. and Wang C.M., The small length scale effect
for a non‐local cantilever beam: a paradox solved,
Nanotechnology, 19, 345703, 2008.
2. Challamel N, Lanos C, Casandjian C, Plastic failure of
nonlocal beams, Phys. Rev. E, 78, 026604, 2008.
3. Zhang Y.Y., Wang C.M. and Challamel N., Bending,
buckling and vibration of hybrid nonlocal beams, J. Eng.
Mech., ASCE, 136, 5, 562‐574, 2010.
4. Challamel N., Andrade A., Camotim D. and Milisavlevich
B.M., On the flexural‐torsional buckling of cantilever
strip beam‐columns with linearly varying depth, J. Eng.
Mech., ASCE, 136, 6, 787‐800, 2010.
5. Challamel N., Bernard F. and Casandjian C., Out‐of‐plane
behaviour of partially composite or sandwich beams by
exact and Finite Element Methods, Thin‐Walled Structures,
48, 561‐580, 2010.
6. Challamel N., A variationally‐based nonlocal damage model
to predict diffuse microcracking evolution, Int. J. Mech. Sc.,
52, 1783‐1800, 2010.
7. Challamel N., Higher‐order shear beam theories and
enriched continuum, Mech. Res. Comm., 38, 388‐392, 2011.
8. Challamel N., Mechab I., Elmeiche N. and Krour B., Buckling
of composite nonlocal or gradient connected beams, Int. J.
Structural Stability Dynamics, 11, 6, 1015‐1033, 2011.
9. Challamel N. and Girhammar U.A., Boundary layer effect in
composite beams with interlayer slip, J. Aerospace
Engineering, ASCE, 24, 2, 199‐209, 2011.
10. Elishakoff I., Pentaras D., Dujat K., Versaci C., Muscolino G.,
Storch J., Bucas S., Challamel N., Natsuki T., Zhang Y.Y.,
Wang C.M., Ghyselinck G., Carbon nanotubes and
nanosensors: vibrations, buckling and ballistic impact, Wiley
- ISTE, 2012.