Reducible and irreducible forms of stabilised gradient elasticity in dynamics

The continualisation of discrete particle models has been a popular tool to formulate higher-order gradient elasticity models. However, a straightforward continualisation leads to unstable continuum models. Pade approximations can be used to stabilise ´ the model, but the resulting formulation depends on the particular equation that is transformed with the Pade approximation. In this contribution, we study two different stabilised ´ gradient elasticity models; one is an irreducible form with displacement degrees of freedom only, and the other is a reducible form where the primary unknowns are not only displacements but also the Cauchy stresses — this turns out to be Eringen’s theory of gradient elasticity. Although they are derived from the same discrete model, there are significant differences in variationally consistent boundary conditions and resulting finite element implementations, with implications for the capability (or otherwise) to suppress crack tip singularitie

A new multi-scale dispersive gradient elasticity modelwith micro-inertia: Formulation and C0-finiteelement implementation

Motivated by nano-scale experimental evidence on the dispersion characteristics of materials with a lattice structure, a new multi-scale gradient elasticity model is developed. In the framework of gradient elasticity, the simultaneous presence of acceleration and strain gradients has been denoted as dynamic consistency. This model represents an extension of an earlier dynamically consistent model with an additional micro-inertia contribution to improve the dispersion behaviour. The model can therefore be seen as an enhanced dynamic extension of the Aifantis' 1992 strain-gradient theory for statics obtained by including two acceleration gradients in addition to the strain gradient. Compared with the previous dynamically consistent model, the additional micro-inertia term is found to improve the prediction of wave dispersion significantly and, more importantly, requires no extra computational cost. The fourth-order equations are rewritten in two sets of symmetric second-order equations so that C0-continuity is sufficient in the finite element implementation. Two sets of unknowns are identified as the microstructural and macrostructural displacements, thus highlighting the multi-scale nature of the present formulation. The associated energy functionals and variationally consistent boundary conditions are presented, after which the finite element equations are derived. Considerable improvements over previous gradient models are observed as confirmed by two numerical examples

Mass matrices for elastic continua with micro-inertia

In this paper, the finite element discretization of non-classical continuum models with micro-inertia is analysed. The focus is on micro-inertia extensions of the one-dimensional rod model, the beam bending theories of Euler-Bernoulli and Rayleigh, and the two-dimensional membrane model. The performance of a variety of mass matrices is assessed by comparing the natural frequencies and their modes with those of the associated discrete systems, and it is demonstrated that the use of higher-order mass matrices reduces errors and improves convergence rates. Furthermore, finite element sizes larger than the corresponding physical length scale are shown to be sufficient to capture the natural frequencies, thus facilitating numerical models that are not only reliable but also computationally efficient.The authors acknowledge support from MCIN/ AEI/10.13039/501100011033 under Grants numbers PGC2018-098218-B-I00 and PRE2019-088002. FEDER: A way to make Europe. ESF invests in your future

Gradient elasticity: a new tool for the multiaxial high-cycle fatigue assessment of notched components

In this paper, the accuracy of gradient elasticity in estimating the fatigue strength of engineering components, characterised by the presence of stress risers and subjected to multiaxial high-cycle fatigue loadings, is assessed. In particular, a new approach, based on the combination of the Ru-Aifantis theory of gradient elasticity and the Theory of Critical Distances (TCD), is proposed for the fatigue assessment of notched metallic components. The proposed methodology represents an important step forward respect to the state of the art, allowing an accurate fatigue assessment of engineering components, by post-processing the relevant gradient-enriched stresses directly on the surface of the component, with evident advantages from a practical point of view

A classification of higher-order strain gradient models - linear analysis

The use of higher-order strain-gradient models in mechanics is studied. First, existing second-gradient models from the literature are investigated analytically. In general, two classes of second-order strain-gradient models can be distinguished: one class of models has a direct link with the underlying microstructure, but reveals instability for deformation patterns of a relatively short wave length, while the other class of models does not have a direct link with the microstructure, but stability is unconditionally guaranteed. To combine the advantageous properties of the two classes of second-gradient models, a new, fourth-order strain-gradient model, which is unconditionally stable, is derived from a discrete microstructure. The fourth-gradient model and the second-gradient models are compared under static and dynamic loading conditions. A numerical approach is followed, whereby the element-free Galerkin method is used. For the second-gradient model that has been derived from the microstructure, it is found that the model becomes unstable for a limited number of wave lengths, while in dynamics, instabilities are encountered for all shorter wave lengths. Contrarily, the second-gradient model without a direct link to the microstructure behaves in a stable manner, although physically unrealistic results are obtained in dynamics. The fourth-gradient model, with a microstructural basis, gives stable and realistic results in statics as well as in dynamics

Mono-scale and multi-scale formulations of gradient-enriched dynamic piezomagnetics

In this contribution, we combine and extend earlier work on static piezomagnetics and dynamic gradient elasticity to develop novel dynamic piezomagnetic continuum models. The governing equations can be formulated as a mono-scale model or as multi-scale models. The latter include full coupling between micro and macro-scale displacements and micro and macro-scale magnetic potentials, which allows to denote these as “multi-scale multi-physics” models. The field equations and boundary conditions are given together with the underlying energy functionals. An analysis of coupled dispersive waves is carried out to illustrate the behaviour of the models and their ability to simulate dispersive piezomagnetic waves

Gradient-enriched finite element methodology for axisymmetric problems

Due to the axisymmetric nature of many engineering problems, bi-dimensional axisymmetric finite elements play an important role in the numerical analysis of structures, as well as advanced technology micro/nano-components and devices (nano-tubes, nano-wires, micro-/nano-pillars, micro-electrodes). In this paper, a straightforward (Formula presented.)-continuous gradient-enriched finite element methodology is proposed for the solution of axisymmetric geometries, including both axisymmetric and non-axisymmetric loads. Considerations about the best integration rules and an exhaustive convergence study are also provided along with guidances on optimal element size. Moreover, by applying the present methodology to cylindrical bars characterised by a circumferential sharp crack, the ability of the present methodology to remove singularities from the stress field has been shown under axial, bending, and torsional loading conditions. Some preliminary results, obtained by applying the proposed methodology to notched cylindrical bars, are also presented, highlighting the accuracy of the methodology in the static and fatigue assessment of notched components, for both brittle and ductile materials. Finally, the proposed methodology has been applied to model the unit cell of the anode of Li-ion batteries showing the ability of the methodology to account for size effects

Microstructural length scale parameters to model the high-cycle fatigue behaviour of notched plain concrete

The present paper investigates the importance and relevance of using microstructural length scale parameters in estimating the high-cycle fatigue strength of notched plain concrete. In particular, the accuracy and reliability of the Theory of Critical Distances and Gradient Elasticity are checked against a number of experimental results generated by testing, under cyclic bending, square section beams of plain concrete containing stress concentrators of different sharpness. The common feature of these two modelling approaches is that the required effective stress is calculated by using a length scale which depends on the microstructural material morphology. The performed validation exercise demonstrates that microstructural length scale parameters are successful in modelling the behaviour of notched plain concrete in the high-cycle fatigue regime

Energy absorption in lattice structures in dynamics: Nonlinear FE simulations

An experimental study of the stress–strain behaviour of titanium alloy (Ti6Al4V) lattice structures across a range of loading rates has been reported in a previous paper [1]. The present work develops simple numerical models of re-entrant and diamond lattice structures, for the first time, to accurately reproduce quasi-static and Hopkinson Pressure Bar (HPB) test results presented in the previous paper. Following the development of lattice models using implicit and explicit non-linear finite element (FE) codes, the numerical models are first validated against the experimental results and then utilised to explore further the phenomena associated with impact, the failure modes and strain-rate sensitivity of these materials. We have found that experimental results can be captured with good accuracy by using relatively simple numerical models with beam elements. Numerical HPB simulations demonstrate that intrinsic strain rate dependence of Ti6Al4V is not sufficient to explain the emergent rate dependence of the re-entrant cube samples. There is also evidence that, whilst re-entrant cube specimens made up of multiple layers of unit cells are load rate sensitive, the mechanical properties of individual lattice structure cell layers are relatively insensitive to load rate. These results imply that a rate-independent load-deflection model of the unit cell layers could be used in a simple multi degree of freedom (MDoF) model to represent the impact behaviour of a multi-layer specimen and capture the microscopic rate dependence