A three-dimensional finite element for gradient elasticity based on a mixed-type formulation

Zybell, Lutz; Mühlich, Uwe; Kuna, Meinhard; Zhang, Z.L.

doi:10.1016/j.commatsci.2011.02.026

Cited by 36 publications

(43 citation statements)

References 17 publications

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“…Since we use a constant interpolation for λ , the matrix N λ is a 4 × 4 unit matrix. A three‐dimensional version of QU34L4, named BR153L9, has also been developed …”

Section: Revisiting Existing Mixed Elementsmentioning

confidence: 99%

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A novel efficient mixed formulation for strain‐gradient models

Papanicolopulos

Gulib

Marinelli

2018

Numerical Meth Engineering

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Summary Various finite elements based on mixed formulations have been proposed for the solution of boundary value problems involving strain‐gradient models. The relevant literature, however, does not provide details on some important theoretical aspects of these elements. In this work, we first present the existing elements within a novel, single mathematical framework, identifying some theoretical issues common to all of them that affect their robustness and numerical efficiency. We then proceed to develop a new family of mixed elements that addresses these issues while being simpler and computationally cheaper. The behavior of the new elements is further demonstrated through two numerical examples.

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“…Since we use a constant interpolation for λ , the matrix N λ is a 4 × 4 unit matrix. A three‐dimensional version of QU34L4, named BR153L9, has also been developed …”

Section: Revisiting Existing Mixed Elementsmentioning

confidence: 99%

“…To overcome the restrictions in element choice imposed by the C 1 requirement, alternative element formulations have been proposed, especially using mixed formulations based on Lagrange multipliers or penalty methods . Elements based on mixed formulations have also been developed for couple stress theories, where similar issues exist.…”

Section: Introductionmentioning

confidence: 99%

A novel efficient mixed formulation for strain‐gradient models

Papanicolopulos

Gulib

Marinelli

2018

Numerical Meth Engineering

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“…Based on the idea of Ru et al, in which the fourth‐order gradient elasticity problem of Aifantis is split into two second order problems, a C 0 ‐continuous corresponding finite element formulation is possible . While the latter is restricted to the special formulation of Aifantis, in Shu et al and Zybell a mixed approach is investigated, which enables C 0 ‐continuous discretizations in the framework of the more general Mindlin‐Toupin gradient elasticity theory.…”

Section: Introductionmentioning

confidence: 99%

Three‐field mixed finite element formulations for gradient elasticity at finite strains

et al. 2019

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Gradient elasticity formulations have the advantage of avoiding geometry‐induced singularities and corresponding mesh dependent finite element solution as apparent in classical elasticity formulations. Moreover, through the gradient enrichment the modeling of a scale‐dependent constitutive behavior becomes possible. In order to remain C0 continuity, three‐field mixed formulations can be used. Since so far in the literature these only appear in the small strain framework, in this contribution formulations within the general finite strain hyperelastic setting are investigated. In addition to that, an investigation of the inf sup condition is conducted and unveils a lack of existence of a stable solution with respect to the L2‐H1‐setting of the continuous formulation independent of the constitutive model. To investigate this further, various discretizations are analyzed and tested in numerical experiments. For several approximation spaces, which at first glance seem to be natural choices, further stability issues are uncovered. For some discretizations however, numerical experiments in the finite strain setting show convergence to the correct solution despite the stability issues of the continuous formulation. This gives motivation for further investigation of this circumstance in future research. Supplementary numerical results unveil the ability to avoid singularities, which would appear with classical elasticity formulations.

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“…[8] and [9] for the small strain framework). Alongside the essential boundary condition u =ū on Γ D on the higher order Dirichlet boundary displacement gradients are prescribed in normal direction with ∇u · n =h.…”

Section: Introductionmentioning

confidence: 99%

A New C0‐Continuous FE‐Formulation for Finite Gradient Elasticity

Riesselmann

Ketteler

Schedensack

et al. 2019

Proc Appl Math and Mech

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Three field mixed C 0 -continuous gradient elasticity finite element formulations contain both displacement and displacement gradients as solution variables. In this contribution a new formulation is proposed, in which the displacements are not part of the problem anymore, but only displacement gradients, leading to a reduced number of variables to be discretized. In numerical examples the proposed formulations are compared to finite strain adaptations of existing mixed three field approaches for small strains and unveil favorable computing time. For the 3D case a further reduction of variables through a penalty formulation is investigated.

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A three-dimensional finite element for gradient elasticity based on a mixed-type formulation

Cited by 36 publications

References 17 publications

A novel efficient mixed formulation for strain‐gradient models

A novel efficient mixed formulation for strain‐gradient models

Three‐field mixed finite element formulations for gradient elasticity at finite strains

A New C0‐Continuous FE‐Formulation for Finite Gradient Elasticity

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