Nonlocal Elasticity Models in Nanoscale Structures

Nonlocal elasticity models extend classical continuum theory by incorporating interactions at a distance, thus capturing size-dependent mechanical responses of structures whose dimensions approach the nanoscale. Traditional local theories assume stress at a point depends solely on strains at that point, whereas nonlocal formulations introduce spatial kernels or higher-order strain gradients to reflect long-range interatomic forces. Integral and differential representations of nonlocal elasticity, including two-phase local/nonlocal schemes, have been developed to resolve paradoxes in beam bending, vibration and buckling analyses. Coupling with surface elasticity theories accounts for the dominant role of surface energy in nanostructures. Advanced models further integrate piezoelectricity, flexoelectricity and viscoelastic damping to predict resonant frequencies and stability thresholds of beams, plates and shells. Across diverse materials—carbon nanotubes, functionally graded nanoplates and polymeric nanoshells—nonlocal theories enable accurate design of ultrasensitive mass sensors, energy harvesters and fluid-coupled devices. By reconciling atomistic insights with continuum approximations, these frameworks underpin reliable performance predictions and inform the optimisation of next-generation nanoscale systems.

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Nonlocal Elasticity Models in Nanoscale Structures publication trend

The graph below shows the total number of articles in nonlocal elasticity models in nanoscale structures across all publications each year (not limited to Nature Index journals).

Technical terms

Nonlocal elasticity: A continuum approach in which stress at a point depends on strains over a surrounding region rather than solely on local strains.

Strain gradient theory: A model that augments classical elasticity by incorporating derivatives of strain to capture size-dependent stiffening or softening.

Surface energy: The additional energy associated with the atoms at the surface of a nanoscale structure due to altered bonding environments.

Functionally graded material: A material whose composition or microstructure varies spatially to achieve gradual changes in properties.

Nanoplate: A plate-like element whose thickness is measured in nanometres, exhibiting dominant surface and small-scale effects.

References

1. Size effect in ultrasensitive micro- and nanomechanical mass sensors. Mechanical Systems and Signal Processing (2023).
2. Study on the effect of viscosity and fluid flow on buckling behavior of nanoplate with surface energy. Results in Engineering (2023).
3. Exact solutions for the static bending of Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model. AIP Advances (2016).
4. A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells. Applied Mathematics and Mechanics (2019).
5. On the Dynamics of a Visco–Piezo–Flexoelectric Nanobeam. Symmetry (2020).