Two Finite-Element Discretizations for Gradient Elasticity

Zervos, A.; Papanicolopulos, Stefanos‐Aldo; Vardoulakis, I.

doi:10.1061/(asce)0733-9399(2009)135:3(203)

Cited by 89 publications

(78 citation statements)

References 23 publications

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“…A plane-strain, thick hollow cylinder subjected to external pressure is considered [19,45]. Only a quarter of the cylinder is analysed due to symmetry (shaded region in Figure 2).…”

Section: Errors and Convergence Ratesmentioning

confidence: 99%

“…This is because higher order terms appear in the weak form, thus requiring the derivatives of displacements to be continuous -C 1 -continuity requirement. In principle, the problem can be solved by Hermitian finite elements [18,19], mixed methods [20], meshless methods [21], penalty methods [22,23], Langrange multipliers [24] and subdivision surfaces [25]. However, all these methods have their drawbacks in terms of efficiency or implementational convenience.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Convergence analysis of Laplacian-based gradient elasticity in an isogeometric framework

Kolo

Askes

Borst

2017

Finite Elements in Analysis and Design

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“…A plane-strain, thick hollow cylinder subjected to external pressure is considered [19,45]. Only a quarter of the cylinder is analysed due to symmetry (shaded region in Figure 2).…”

Section: Errors and Convergence Ratesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Convergence analysis of Laplacian-based gradient elasticity in an isogeometric framework

Kolo

Askes

Borst

2017

Finite Elements in Analysis and Design

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“…(14). The most desirable solution would be, obviously, the possibility to use the same integration rule used in the first step of the Ru-Aifantis theory.…”

Section: Numerical Integrationmentioning

confidence: 99%

“…The first one comprehends approaches that leave the continuum mechanics equations intact, by using Meshless methods [3][4][5][6][7][8][9][10][11], Penalty methods [12][13][14], Hermitian finite elements [15][16][17][18], next nearest neighbour interaction (instead of the simpler nearest neighbour interaction used in the standard finite element software) [19], etc. The second one includes approaches that transform the governing equations, in order to obtain less demanding continuity requirements; among these is the Ru-Aifantis theorem [20] which splits the original fourth-order p.d.e.…”

Section: Introductionmentioning

confidence: 99%

Unified finite element methodology for gradient elasticity

Bagni

Askes

2015

Computers & Structures

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In this paper a unified finite element methodology based on gradient-elasticity is proposed for both two-and three-dimensional problems, along with some considerations about the best integration rules to be used and a comprehensive convergence study. From the convergence study it has emerged that for both two and three-dimensional problems, the implemented elements show a convergence rate virtually equal to the corresponding theoretical values.Recommendations on optimal element size are also provided. Furthermore, the ability of the proposed methodology to remove singularities in statics has been demonstrated through a couple of examples, in both two and three dimensions.

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“…done by Askes et al [2] who used the Element Free Galerkin method. Zervos et al [8] applied two C 1 finite elements, the Bell triangle and the Bogner-Fox-Schmitt element to linear gradient elasticity and compared the results to implicit methods. In our contribution, we compare the performance of the Argyris, the Clough-Tocher (HCT) and the isoparametric BognerFox-Schmidt (BFS) element with the C 1 Natural Element Method (C 1 NEM) [6] in nonlinear gradient elasticity.…”

Section: Motivationmentioning

confidence: 99%

𝒞¹ continuous discretization of gradient elasticity

Fischer

Mergheim

Steinmann

2009

Proc Appl Math and Mech

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For the modeling of size effects, gradient continua can be applied. In this contribution, strain gradients are used in a nonlinear hyperelastic material model. The resulting partial differential equations are solved numerically in terms of a Galerkin scheme requiring C 1 continuity of the shape functions. The performance of three C 1 continuous finite elements and the C 1 Natural Element method is compared. motivationThe modeling of materials with size effects is an issue of high interest. Size effects can be observed in various types of materials such as concrete, metals and composite materials, an overview on the topic is found in [1]. Within the classical Boltzmann theory these effects can not be captured. To overcome this problem, extended models like gradient elasticity can be used. This theory which was established by Mindlin [4] and Toupin [7] incorporates the strain gradient into the definition of the free energy. The introduction of strain gradients and an additional material parameter, associated with the material microstructural length, allows for the modeling of size effects which was proven by Aifantis [1]. The crux in handling gradient elasticity is that a C 1 continuous approximation of the deformation is required. This increases the complexity of the numerical methods compared to the usual C 0 continuous interpolation. Several alternatives like implicit methods and mixed element formulations have been developed to avoid the requirement of C 1 continuity, e.g. [3,5]. However since these alternatives come along with different numerical problems, it is worth studying C 1 methods and evaluating their numerical behavior. This was e.g. done by Askes et al. [2] who used the Element Free Galerkin method. Zervos et al.[8] applied two C 1 finite elements, the Bell triangle and the Bogner-Fox-Schmitt element to linear gradient elasticity and compared the results to implicit methods. In our contribution, we compare the performance of the Argyris, the Clough-Tocher (HCT) and the isoparametric BognerFox-Schmidt (BFS) element with the C 1 Natural Element Method (C 1 NEM) [6] in nonlinear gradient elasticity. basic equationsIn gradient elasticity the free energy density depends not only on the deformation gradient but additionally on its gradient. By introducing the Piola stresses P := ∂W/∂F and the double stress Q := ∂W/∂G balance of momentum then results asFirst and second order boundary conditions are prescribed as t P = t P , ϕ = ϕ and t Q = t Q , ∇ N X ϕ =φ N using the abbreviations numerical methodsThe Argyris element with 21 degrees of freedom and the HCT macro element with 12 resulting degrees of freedom are chosen to represent the class of subparametric finite elements. The degrees of freedom of these elements are the normal derivatives at the center of the edges as well as the displacements and the derivatives up to the second order at the corner nodes. Furthermore, the isoparametric quadrilateral BFS element is selected whose degrees of freedom are the displacement, the first derivatives and the mix...

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Two Finite-Element Discretizations for Gradient Elasticity

Cited by 89 publications

References 23 publications

Convergence analysis of Laplacian-based gradient elasticity in an isogeometric framework

Convergence analysis of Laplacian-based gradient elasticity in an isogeometric framework

Unified finite element methodology for gradient elasticity

𝒞¹ continuous discretization of gradient elasticity

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Two Finite-Element Discretizations for Gradient Elasticity

Cited by 89 publications

References 23 publications

Convergence analysis of Laplacian-based gradient elasticity in an isogeometric framework

Convergence analysis of Laplacian-based gradient elasticity in an isogeometric framework

Unified finite element methodology for gradient elasticity

𝒞1 continuous discretization of gradient elasticity

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𝒞¹ continuous discretization of gradient elasticity