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

A gradient approach to localization of deformation. I. Hyperelastic materials

By utilizing methods recently developed in the theory of fluid interfaces, we provide a new framework for considering the localization of deformation and illustrate it for the case of hyperelastic materials. The approach overcomes one of the major shortcomings in constitutive equations for solids admitting localization of deformation at finite strains, i.e. their inability to provide physically acceptable solutions to boundary value problems in the post-localization range due to loss of ellipticity of the governing equations. Specifically, strain-induced localized deformation patterns are accounted for by adding a second deformation gradient-dependent term to the expression for the strain energy density. The modified strain energy function leads to equilibrium equations which remain always elliptic. Explicit solutions of these equations can be found for certain classes of deformations. They suggest not only the direction but also the width of the deformation bands providing for the first time a predictive unifying method for the study of pre- and post-localization behavior. The results derived here are a three-dimensional extension of certain one-dimensional findings reported earlier by the second author for the problem of simple shear.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42674/1/10659_2004_Article_BF00040814.pd

On higher order gradient continuum theories in 1-D nonlinear elasticity. Derivation from and comparison to the corresponding discrete models

Higher order gradient continuum theories have often been proposed as models for solids that exhibit localization of deformation (in the form of shear bands) at sufficiently high levels of strain. These models incorporate a length scale for the localized deformation zone and are either postulated or justified from micromechanical considerations. Of interest here is the consistent derivation of such models from a given microstructure and the subsequent comparison of the solution to a boundary value problem using both the exact microscopic model and the corresponding approximate higher order gradient macroscopic model.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42682/1/10659_2004_Article_BF00043251.pd