Sliding and rolling dissipation in Cosserat plasticity

Based on micromechanical considerations at the level of grain contacts and taking into account the way in which kinematic and static quantities are introduced at grain surface and grain centre, we identify appropriate measures related to energy dissipation due to rolling and sliding between grains, within both a discrete and a Cosserat continuum description. This allows us, within the framework of Cosserat plasticity, to identify appropriate invariants and formulate simple forms of the respective yield surfaces. The resulting model is shown to be a multiple-yield-surface generalisation of the model by Mühlhaus and Vardoulakis (Géotechnique 37(3):271-283, 1987). By introducing separate and clearly identified rolling- and sliding-resistance parameters, the model allows for separate activation of the respective dissipative mechanisms. © 2011 Springer-Verlag

Chirality in isotropic linear gradient elasticity

AbstractChirality 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 three-dimensional 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

Evolution of deformation and breakage in sand studied using X-ray tomography

International audienceParticle breakage of a granular material can cause significant changes in its microstructure, which will govern its macroscopic behaviour; this explains why the mechanisms leading to particle breakage have been a common subject within several fields, including geomechanics. In this paper, X-ray computed micro-tomography is used, to obtain three-dimensional images of entire specimens of sand, during high-confinement triaxial compression tests. The acquired images are processed and measurements are made on breakage, local variations of porosity, volumetric strain, maximum shear strain and grading. The evolution and spatial distribution of quantified breakage and the resulting particle size distribution for the whole specimen and for specific areas are presented here for the first time and are further related to the localised shear and volumetric strains. Before peak stress is reached, compaction is the governing mechanism leading to breakage; neither compressive strains nor breakage are significantly localised and the total amount of breakage is rather low. Post peak, in areas where strains localise and breakage is present, a dilative volumetric behaviour is observed locally, as opposed to the overall compaction of the specimen. Some specimens exhibited a compaction around the shear band at the end of the test, but there was no additional breakage at that point. From the grading analysis, it is found that mainly the grains with diameter close to the mean diameter of the specimen are the ones that break, whereas the biggest grains that are present in the specimen remain intact

Continua with microstructure: second gradient theory.

Second-gradient theories represent a frequently used subset of theories of continua with microstructure. This paper presents an extended overview of second-gradient theories, starting from a simple one-dimensional example, proceeding with a thorough description of gradient elasticity and additionally briefly describing some other theories of this kind. A series of characteristic examples is presented to demonstrate the main aspects and applications of second-gradient theories. Finally, the complications in the finite-element implementation of second-gradient theories are presented, along with a review of the finite elements that have been developed for this purpose

Polynomial C1 shape functions on the triangle

We derive generic formulae for all possible C1 continuous polynomial interpolations for triangular elements,by considering individual shape functions, without the need to prescribe the type of the degreesof freedom in advance. We then consider the possible ways in which these shape functions can be combinedto form finite elements with given properties. The simplest case of fifth-order polynomial functionsis presented in detail, showing how two existing elements can be obtained, as well as two new elements,one of which shows good numerical behaviour in numerical tests

Numerical solution of crack problems in gradient elasticity

Gradient elasticity is a constitutive framework that takes into account the microstructure of an elastic material. It considers that, in addition to the strains, second-order derivatives of the displacement also affect the energy stored in the medium. Three different yet equivalent forms of gradient elasticity can be found in the literature, reflecting the different ways in which the second-order derivatives can be grouped to form other physically meaningful quantities. This paper presents a general discretisation of gradient elasticity that can be applied to all three forms, based on the finite-element displacement formulation. The presence of higher order terms requires C 1-continuous interpolation, and some appropriate two- and three-dimensional elements are presented. Numerical results for the displacement, stress and strain fields around cracks are shown and compared with available solutions, demonstrating the robustness and accuracy of the numerical scheme and investigating the effect of microstructure in the context of fracture mechanics

Computational challenges in using strain-gradient theories in three dimensions

Strain-gradient models are useful for modelling scale-dependent phenomena such as boundary-layer formation and localisation of deformation. Most problems involving straingradient models in the literature are studied in two dimensions, since this allows for easier theoretical and numerical treatment. In order to determine the applicability of strain-gradient models to real-world problems it is however necessary to consider these models in threedimensional boundary value problems. Additionally, some types of material behaviour (e.g. behaviour in torsion) are only observable in three dimensions. We therefore consider here the computational challenges arising when implementing straingradient models in three dimensions, in the common case where the finite element method is used. Using both theoretical arguments and practical examples, we present issues related to the computational cost of the available finite elements and the different possible plasticity models and integration algorithms. Moreover, we consider in more detail the modes of localisation of deformation in three-dimensional problems, showing the dependence of the results on the way localisation is triggere

A framework for formulating and implementing non-associative plasticity models for shell buckling computations

In modelling the behavior of thick-walled metal shells under compressive loads, the use of J2 flow theory can lead to unrealistic buckling estimates, while alternative ‘corner’ models, despite offering good predictions, have not been widely adopted for structural computations due to their complexity. The present work develops a new and efficient plasticity model for predicting the structural response of compressed metal shells. It combines the simplicity of the Von Mises yield surface, with a non-associative flow rule, mimicking the effect of a yield surface corner. This allows for tracing the equilibrium path of the loaded shell and identifying consistently structural instability, employing a single constitutive model. A robust backward-Euler integration scheme, suitable for both three-dimensional (solid) and shell elements is developed, along with the corresponding consistent algorithmic moduli for nonlinear isotropic hardening materials, accounting rigorously for the nonlinear dependence of plastic straining on the direction of strain increments. The model is implemented in ABAQUS as a user material subroutine. Simulations of thick-walled metal cylinders under compression predict structural instability in good agreement with experimental data. © 2022 Elsevier Lt

A three dimensional C1 finite element for gradient elasticity

In gradient elasticity strain gradient terms appear in the expression of virtual work, leading to the need for C1 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 C1 finite elements and their perceived computational cost. In this context, the lack of three-dimensional C1 elements is of particular concern. In this paper we present a new C1 hexahedral element which, to the best of our knowledge, is the first three-dimensional C1 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 C1 elements are not necessarily more computationally expensive than alternative approaches, and it is argued that they may be more efficient in providing good-quality solution