Finite-Element Analysis of Polyhedra under Point and Line Forces in Second-Strain Gradient Elasticity

Reiher, Jörg Christian; Giorgio, Ivan; Bertram, Albrecht

doi:10.1061/(asce)em.1943-7889.0001184

Cited by 78 publications

(63 citation statements)

References 50 publications

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“…In this case, only 7 independent parameters remain (a 1 · · · a 7 ) [2,5] W = a 1 (ε ··I) 2 + a 2 ε ··ε + a 3 (κ ··I) · (κ ··I) + a 4 I ··κ · κ ··I + a 5 (I ··κ) · (I ··κ) + a 6 κ ··· κ + a 7 κ ··· κ, (4) where I is the identity tensor. We can further reduce the model by assuming centro-symmetric isotropic materials.…”

Section: Strain Gradient Elasticitymentioning

confidence: 99%

“…Another model which we will consider here is the one introduced by Reiher et al [4]. Another model which we will consider here is the one introduced by Reiher et al [4].…”

Section: Strain Gradient Elasticitymentioning

confidence: 99%

“…It has been shown earlier that the first strain gradient theory should suffice to regularize the stress fields under these kinds of boundary conditions [4]. In strain gradient elasticity, higher-order gradients of the displacement field are deployed to overcome shortcomings of the classical models.…”

Section: Introductionmentioning

confidence: 99%

“…However, in practice, due to the complexities of measuring the new parameters, various simplified versions of the theory are being exploited, among them we name Altan et al [3] and Reiher et al [4]. In fact, these theories are approximations of physical systems neglecting the underlying microstructure.…”

mentioning

confidence: 99%

“…In fact, these theories are approximations of physical systems neglecting the underlying microstructure. However, in practice, due to the complexities of measuring the new parameters, various simplified versions of the theory are being exploited, among them we name Altan et al [3] and Reiher et al [4]. a continuum with an elastic energy determined as a function of the gradients of its macroscopic displacement, can only approximate the behavior of a physical system sufficiently as long as the microstructure has a much smaller length-scale than the macrostructure [1].…”

mentioning

confidence: 99%

See 4 more Smart Citations

A comparison between the performance of different Mindlin's theory based strain gradient models in local singularities

Makvandi

Juhre

2019

Proc Appl Math and Mech

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The classical continuum mechanics theories are originally supposed to determine deformations in ranges from millimeter to meter, the so-called macroscopic scale. In fact, these theories are approximations of physical systems neglecting the underlying microstructure. For instance, a Cauchy continuum, i.e. a continuum with an elastic energy determined as a function of the gradients of its macroscopic displacement, can only approximate the behavior of a physical system sufficiently as long as the microstructure has a much smaller length-scale than the macrostructure [1]. Although these models were exploited in studies for large and small scales, experiments have shown that the classical models are not able to properly cover the smaller scales; in particular, problems in micron-and nano-dimensions are frequently observed. Size-effects, which cannot be captured exploiting these theories, seem to be the source of this issue. On top of that, the appearance of local singularities at the crack tips (or more broadly, in the presence of point and line loads) is one of the known limitations of the classical continuum mechanics theories. Generalizing these models by introducing additional kinematic terms to consider the underlying microstructure effects at macroscopic levels is one way of overcoming the already mentioned problems. In this contribution, we will focus on the Mindlin's theory of elasticity with microstructures [2] and its different forms. Therein, it is shown that for the first strain gradient theory, five additional parameters must be introduced. However, in practice, due to the complexities of measuring the new parameters, various simplified versions of the theory are being exploited, among them we name Altan et al. [3] and Reiher et al. [4]. Our aim here is to compare the performance of these simplified theories in removing the local singularities of the conventional continuum mechanics theory.

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Section: Strain Gradient Elasticitymentioning

confidence: 99%

“…Another model which we will consider here is the one introduced by Reiher et al [4]. Another model which we will consider here is the one introduced by Reiher et al [4].…”

Section: Strain Gradient Elasticitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

A comparison between the performance of different Mindlin's theory based strain gradient models in local singularities

Makvandi

Juhre

2019

Proc Appl Math and Mech

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A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations

Turco

Dell’Isola²,

Misra

2019

Num Anal Meth Geomechanics

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Summary We formulate a discrete Lagrangian model for a set of interacting grains, which is purely elastic. The considered degrees of freedom for each grain include placement of barycenter and rotation. Further, we limit the study to the case of planar systems. A representative grain radius is introduced to express the deformation energy to be associated to relative displacements and rotations of interacting grains. We distinguish inter‐grains elongation/compression energy from inter‐grains shear and rotations energies, and we consider an exact finite kinematics in which grain rotations are independent of grain displacements. The equilibrium configurations of the grain assembly are calculated by minimization of deformation energy for selected imposed displacements and rotations at the boundaries. Behaviours of grain assemblies arranged in regular patterns, without and with defects, and similar mechanical properties are simulated. The values of shear, rotation, and compression elastic moduli are varied to investigate the shapes and thicknesses of the layers where deformation energy, relative displacement, and rotations are concentrated. It is found that these concentration bands are close to the boundaries and in correspondence of grain voids. The obtained results question the possibility of introducing a first gradient continuum models for granular media and justify the development of both numerical and theoretical methods for including frictional, plasticity, and damage phenomena in the proposed model.

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Trefftz collocation method for two‐dimensional strain gradient elasticity

Solyaev

Lurie

2020

Numerical Meth Engineering

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Indirect Trefftz method is proposed for solving two‐dimensional boundary value problems of the strain gradient elasticity theory (SGET). A system of trial functions satisfying the fourth‐order equilibrium equations of SGET are developed based on the generalized Papkovich‐Neuber potentials. The classical part of the displacement solution is represented through the T‐complete system of functions satisfying the Laplace equation. The gradient part of the solution is represented through the system of heuristic functions satisfying the Helmholtz equation. The least squares collocation method is used to enforce the boundary conditions. Numerical examples are presented for the square domain under non‐uniform tensile and bending loads. It is shown, that the advantage of the presented method is that it allows to directly control the accuracy of the fulfillment of all nonstandard boundary conditions, that are prescribed in SGET on the surfaces and edges of the body.

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Finite-Element Analysis of Polyhedra under Point and Line Forces in Second-Strain Gradient Elasticity

Cited by 78 publications

References 50 publications

A comparison between the performance of different Mindlin's theory based strain gradient models in local singularities

A comparison between the performance of different Mindlin's theory based strain gradient models in local singularities

A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations

Trefftz collocation method for two‐dimensional strain gradient elasticity

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