SUMMARYIn gradient elasticity strain gradient terms appear in the expression of virtual work, leading to the need for C 1 continuous interpolation in finite element discretizations of the displacement field only. Employing such interpolation is generally avoided in favour of the alternative methods that interpolate other quantities as well as displacement, due to the scarcity of C 1 finite elements and their perceived computational cost. In this context, the lack of three-dimensional C 1 elements is of particular concern. In this paper we present a new C 1 hexahedral element which, to the best of our knowledge, is the first three-dimensional C 1 element ever constructed. It is shown to pass the single element and patch tests, and to give excellent rates of convergence in benchmark boundary value problems of gradient elasticity. It is further shown that C 1 elements are not necessarily more computationally expensive than alternative approaches, and it is argued that they may be more efficient in providing good-quality solutions.

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Multi-spheres and Superquadrics are popular approaches for addressing particle shape effect in the Discrete Element Method (DEM). This study focuses on the mechanical characteristics of cubical particles, modelled by the two methods (using EDEM and LIGGGHTS), through conducting a series of numerical case studies at both single particle and bulk levels. In the first part of the study, several testing scenarios, which clarify the impact, interlocking, sliding and tilting characteristics of the particle, are discussed and the respective simulations are carried out. The results emphasize the importance of surface bumpiness and edge sharpness in the single-particle behaviour and are used for informing the bulk response. Further, role of the two shape descriptors on bulk response is evaluated in angle of repose, Jenike shear and silo flow simulations. The results of these tests are assessed both at the micro, directly through DEM outputs, and at the meso-and macro-scales, using a coarse graining technique. It is seen that the properties of edge and surface in superquadric and multi-sphere particles considerably influence the heap profile in the angle of repose test. However, in a Jenike direct shear, the shape complexity only significantly affects the shear strength, porosity and mode of motion when the packing is dense. Additionally, in silo discharge, the effect of shape features is even less on the flow pattern and mass flow rate but is found to have a significant influence on the stress distribution.

Chirality is, generally speaking, the property of an object that can be classified as left-or right-handed. Though it plays an important role in many branches of science, chirality is encountered less often in continuum mechanics, so most classical material models do not account for it. In the context of elasticity, for example, classical elasticity is not chiral, leading different authors to use Cosserat elasticity to allow modelling of chiral behaviour. Gradient elasticity can also model chiral behaviour, however this has received much less attention than its Cosserat counterpart. This paper shows how in the case of isotropic linear gradient elasticity a single additional parameter can be introduced that describes chiral behaviour. This additional parameter, directly linked to threedimensional deformation, can be either negative or positive, with its sign indicating a discrimination between the two opposite directions of torsion. Two simple examples are presented to show the practical effects of the chiral behaviour.