A high-order FEM formulation for free and forced vibration analysis of a nonlocal nonlinear graded Timoshenko nanobeam based on the weak form quadrature element method

The purpose of this paper is to provide a high-order finite element method (FEM) formulation of nonlocal nonlinear nonlocal graded Timoshenko based on the weak form quadrature element method (WQEM). This formulation offers the advantages and flexibility of the FEM without its limiting low-order accuracy. The nanobeam theory accounts for the von Kármán geometric nonlinearity in addition to Eringen’s nonlocal constitutive models. For the sake of generality, a nonlinear foundation is included in the formulation. The proposed formulation generates high-order derivative terms that cannot be accounted for using regular first- or second-order interpolation functions. Hamilton’s principle is used to derive the variational statement which is discretized using WQEM. The results of a WQEM free vibration study are assessed using data obtained from a similar problem solved by the differential quadrature method (DQM). The study shows that WQEM can offer the same accuracy as DQM with a reduced computational cost. Currently the literature describes a small number of high-order numerical forced vibration problems, the majority of which are limited to DQM. To obtain forced vibration solutions using WQEM, the authors propose two different methods to obtain frequency response curves. The obtained results indicate that the frequency response curves generated by either method closely match their DQM counterparts obtained from the literature, and this is despite the low mesh density used for the WQEM systems

A gradient flow theory of plasticity for granular materials

A flow theory of plasticity for pressure-sensitive, dilatant materials incorporating second order gradients into the flow-rule and yield condition is suggested. The appropriate extra boundary conditions are obtained with the aid of the principle of virtual work. The implications of the theory into shear-band analysis are examined. The determination of the shear-band thickness and the persistence of ellipticity in the governing equations are discussed

On the microscopic origin of the plastic spin

Considering the configuration of a single slip and employing a scale invariance argument, it is possible to deduce a set of microscopic plastic flow relations having a direct counterpart in the macroscopic formulation of plasticity and viscoplasticity. In particular, a microscopic form of the plastic spin and its macroscopic counterpart for the case of anisotropy induced by kinematic hardening are obtained in terms of elementary physical arguments. Moreover the evolution equation for the back-stress is rigorously derived. Parameters which were assumed to be constant and/or independent from each other in a macroscopic development, are now found to be interrelated and dependent on the accumulated plastic strain. These findings are used for the analysis of the simple shear problem, especially in evaluating the development of the axial stress normal to the shear plane. A preliminary qualitative comparison with available data from fixed-end torsion experiments is discussed

Gradient Elasticity Formulations for Micro/Nanoshells

Many unexpected applications were found due to superior thermochemomechanical and optoelectromagnetic material properties noted at the nanoscale. Nanoscale structures, such as nanobeams, nanoplates, and nanoshells, were used in many MEMS and NEMS applications. Therefore, understanding the static and dynamic behaviour of them is important for reliable design of micro- and nanodevices. Experimental studies are generally difficult at the nanoscale due to resolution limitation of available nanoprobes. Molecular dynamics (MD) experiments were therefore normally employed to understand nanoscale behavior. Unfortunately, MD studies are limited to small number of atoms and short time intervals. Continuum models were then proposed as an alternative solution method. For mechanical behaviour modelling of nanostructures, the classical continuum mechanics models are not adequate because these models only contain bulk material properties and cannot capture inhomogeneously evolving microstructures and related size effects. To simulate nanostructures, a number of continuum theories have been used to predict the influence of nanoscale effects, such as couple stress and Cosserat theories, nonlocal elasticity, and gradient elasticity. The gradient theory is an extension of classical theory to include additional higher-order spatial derivatives of strain and/or stress, as well as (internal) acceleration. It has been shown to be a powerful alternative tool for dealing with nanostructures without resorting to expensive MD computations.GSRT under grantsThales Intermonu and ERC-13, as well as a HiCi grant from KAU

A simple approach to solve boundary-value problems in gradient elasticity

We outline a procedure for obtaining solutions of certain boundary value problems of a recently proposed theory of gradient elasticity in terms of solutions of classical elasticity. The method is applied to illustrate, among other things, how the gradient theory can remove the strain singularity from some typical examples of the classical theory

On the role of microstructure in the behavior of soils: Effects of higher order gradients and internal inertia

The role of microstructure in the behavior of soils is outlined by considering two typical examples: fluid flow and shear banding. In particular, the modifications resulting on Darcy\u27s law and flow rule of soil plasticity due to the development of high strain gradients and internal inertia are discussed. This is done by resorting to complete balance laws for the internal variables. In the case of Darcy\u27s law, the internal variable is an interaction stress between the porous solid and the interpenetrating fluid phase, which is required to obey an internal momentum balance equation. In the case of flow rule, the internal variable is the so-called back stress which is also required to obey a complete balance law containing both a rate and flux term. © 1994