Generalization of strain-gradient theory to finite elastic deformation for isotropic materials

Beheshti, Alireza

doi:10.1007/s00161-016-0542-x

Cited by 11 publications

(9 citation statements)

References 25 publications

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“…The frame indifference and isotropy of W nloc was proven in Triantafyllidis and Aifantis . The constitutive parameter κ ≥ 0 of the gradient enrichment can be linked to a microstructural length‐parameter with

l = \sqrt{κ false/ μ}

. For the following numerical examples the nonlocal length‐parameter is set to l = 0.1mm.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…[22] The constitutive parameter ≥ 0 of the gradient enrichment can be linked to a microstructural length-parameter with = √ ∕ . [3,21] For the following numerical examples the nonlocal length-parameter is set to = 0.1 mm.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Investigations with respect to evaluation of constitutive gradient elasticity constants are undertaken in Peerlings and Fleck . While most of the gradient elasticity models are restricted to the regime of small strains, investigations of gradient elastic constitutive models for large deformations can be found in Beheshti and Triantafyllidis et al…”

Section: Introductionmentioning

confidence: 99%

“…Investigations with respect to evaluation of constitutive gradient elasticity constants are undertaken in Peerlings and Fleck. [20] While most of the gradient elasticity models are restricted to the regime of small strains, investigations of gradient elastic constitutive models for large deformations can be found in Beheshti [21] and Triantafyllidis et al [22] Despite its before mentioned advantages over classical formulations, gradient elasticity formulations have not found a significant employment in general practical applications yet. One reason for this is a difficult numerical implementation due to their higher continuity requirements for corresponding finite element formulations.…”

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confidence: 99%

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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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l = \sqrt{κ false/ μ}

. For the following numerical examples the nonlocal length‐parameter is set to l = 0.1mm.…”

Section: Numerical Examplesmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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

et al. 2019

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“…The theoretical foundation for the small strain gradient-enriched continuum theory dates back to the works of [1] and [2] and the more recent adaptation of [3]. A historical overview is given in [4] and an investigation of finite strain models can be found in [5] and [6]. When finding suitable numerical solution schemes such as e.g.…”

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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Finite element analysis of plane strain solids in strain-gradient elasticity

Beheshti

2017

Acta Mech

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Generalization of strain-gradient theory to finite elastic deformation for isotropic materials

Cited by 11 publications

References 25 publications

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

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

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

Finite element analysis of plane strain solids in strain-gradient elasticity

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