Dynamics of Stress-Driven Two-Phase Elastic Beams

The dynamic behaviour of micro- and nano-beams is investigated by the nonlocal continuum mechanics, a computationally convenient approach with respect to atomistic strategies. Specifically, size effects are modelled by expressing elastic curvatures in terms of the integral mixture of stress-driven local and nonlocal phases, which leads to well-posed structural problems. Relevant nonlocal equations of the motion of slender beams are formulated and integrated by an analytical approach. The presented strategy is applied to simple case-problems of nanotechnological interest. Validation of the proposed nonlocal methodology is provided by comparing natural frequencies with the ones obtained by the classical strain gradient model of elasticity. The obtained outcomes can be useful for the design and optimisation of micro- and nano-electro-mechanical systems (M/NEMS).

Modelling flexural wave propagation by the nonlocal strain gradient elasticity with fractional derivatives

The flexural wave propagation in a microbeam is studied based upon the nonlocal strain gradient model with the spatial and time fractional order differentials in the present work. To capture the dispersive behaviour induced by the inherent nanoscale heterogeneity, the stress gradient elasticity and the strain gradient elasticity are often used to model the mechanical behaviour. The present model incorporates the two models and introduces the fractional order derivatives which can be understood as a generalization of integral order nonlocal strain gradient model. The Laplacian operator in the constitutive equation is replaced with the symmetric Caputo fractional differential in the present model. To illustrate the flexibility of the present model, the flexural wave propagation in a microbeam is studied. The fractional order in the present model as a new material parameter can be adjusted appropriately to describe the dispersive properties of the flexural waves. The numerical results based on the new nonlocal strain gradient elasticity with fractional order derivatives are provided for both Euler–Bernoulli beam and Timoshenko beam. The comparisons with the integer order nonlocal strain gradient model and the molecular dynamic simulation are performed to validate the flexibility of the fractional order nonlocal strain gradient model.

A one-dimensional model for elasto-capillary necking

We derive a nonlinear one-dimensional (1d) strain gradient model predicting the necking of soft elastic cylinders driven by surface tension, starting from three-dimensional (3d) finite-strain elasticity. It is asymptotically correct: the microscopic displacement is identified by an energy method. The 1d model can predict the bifurcations occurring in the solutions of the 3d elasticity problem when the surface tension is increased, leading to a localization phenomenon akin to phase separation. Comparisons with finite-element simulations reveal that the 1d model resolves the interface separating two phases accurately, including well into the localized regime, and that it has a vastly larger domain of validity than 1d models proposed so far.