A geometrically nonlinear Euler–Bernoulli beam
model within strain gradient elasticity with isogeometric analysis and lattice
structure applications

Tran, Loc V.; Niiranen, Jarkko

doi:10.2140/memocs.2020.8.345

Cited by 32 publications

(14 citation statements)

References 65 publications

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“…In fact, the process of micro-macro identification has been fully developed only when the variational postulations of Mechanics have been recovered [43][44][45][46][47]. Galileo did not conjecture the right linear dependence of the contact force intensity on the distance from neutral axis in beam theory: clearly, it is extremely useful, if one wants to develop generalized beam theories, to understand how the progenitor theory has been formulated [14,[48][49][50][51][52][53][54][55][56][57][58].…”

Section: The Recovery Of Ancient Hellenistic Mechanics: Middle Ages Mechaniciansmentioning

confidence: 99%

The Study of the Genesis of Novel Mathematical and Mechanical Theories Provides an Inspiration for Future Original Research

Spagnuolo

Dell’Isola

Cazzani

2021

Advanced Structured Materials

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In this introductory chapter we present the motivations that prompted the authors and editors to work on this volume. The fil rouge followed in the discussion presented below is based on the following consideration: the study of the genesis of mathematical and mechanical theories does not have a merely philological purpose, but can influence and even inspire the development of new original ideas. We present some examples that clarify our thesis: the development of the model of planetary motion and a historical-critical study of the development of Continuum Mechanics. Clearly understanding the main errors and misunderstandings that other than pure research logics have introduced in the scientific discussion is the only way to learn a serious and rigorous approach to Science. An idea that has guided us in the development of this chapter is that fragmentation of culture brings to inability to deal with complexity. The only way to study and, above all, understand the complexity of our world is through a unified vision of knowledge.

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Section: The Recovery Of Ancient Hellenistic Mechanics: Middle Ages Mechaniciansmentioning

confidence: 99%

The Study of the Genesis of Novel Mathematical and Mechanical Theories Provides an Inspiration for Future Original Research

Spagnuolo

Dell’Isola

Cazzani

2021

Advanced Structured Materials

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“…Further efforts at identification of higher-gradient constants are found in the works related to micro-architectural thin structures. For instance, to model the small-scale bending experiments of [38] and others, it has been shown that a single non-classical parameter is enough, in addition to the classical Young's modulus, for capturing size effects present in the bending of planar lattice metamaterial beam structures described by 1D generalized beam bending model for both linear [33,35] and geometrically nonlinear [61] analyses. On the other hand, for modeling a 2D model discussed in [66] more that one non-classical parameters are needed although there is no clarity on the which of these are most relevant.…”

Section: Introductionmentioning

confidence: 99%

Granular micromechanics‐based identification of isotropic strain gradient parameters for elastic geometrically nonlinear deformations

et al. 2021

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Although the primacy and utility of higher-gradient theories are being increasingly accepted, values of second gradient elastic parameters are not widely available due to lack of generally applicable methodologies. In this paper, we present such values for a second-gradient continuum. These values are obtained in the framework of finite deformations using granular micromechanics assumptions for materials that have granular textures at some 'microscopic' scale. The presented approach utilizes so-called Piola's ansatz for discrete-continuum identification. As a fundamental quantity of this approach, an objective relative displacement between grain-pairs is obtained and deformation energy of grain-pair is defined in terms of this measure. Expressions for elastic constants of a macroscopically linear second gradient continuum are obtained in terms of the microscale grain-pair parameters. Finally, the main result is that the same coefficients, both in the 2D and in the 3D cases, have been identified in terms of Young's modulus, of Poisson's ratio and of a microstructural length.

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“…Besides, different numerical solution methods such as higher-order finite element methods [24][25][26][27][28][29], and isogeometric analysis [30][31][32][33] have been developed. Second, the generalized continuum theories can be usefully employed to consider the microarchitecture-dependent mechanics of lattice or cellular structures [34][35][36][37][38][39][40]. Indeed, by using these higher-order theories, the internal microarchitecture can be homogenized via generalized constitutive relations, possibly together with dimension reduction models, which remarkably decreases both modeling and computation costs.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…A literature survey discloses a rather limited number of studies on the mechanics of cellular materials under the higher-order continuum theories [34][35][36][37][38][39][40]. Since solids with different microarchitectures under various loading conditions and deformation states might not be appropriately modeled by using a unified generalized continuum theory, different models such as the Cosserat theory [41], micropolar theory [35,42,43] and higher gradient theories [34,[37][38][39][40]44,45,[46][47][48] have been developed. Closely connected to the present study, micropolar plate models were adopted in [42,43] to present a 2D equivalent singlelayer for lattice core sandwich plates and panels, and the resulting problem was solved by linear Lagrange interpolation finite elements along with the selective reduced integration approach.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Nonlinear Finite Element Analysis of Free and Forced Vibrations of Cellular Plates Having Triangularly Prismatic Metamaterial Cores: A Strain Gradient Plate Model Approach

Torabi¹,

Niiranen²

2021

Preprint

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The main objective of this paper is to develop a theoretically and numerically reliable and efficient methodology based on combining a finite element method and a strain gradient shear deformation plate model accounting for the nonlinear free and forced vibrations of cellular plates having equitriangularly prismatic metamaterial cores. The proposed model based on the nonlinear finite element strain gradient elasticity is developed for the first time to provide a computationally efficient framework for the simulation of the underlying nonlinear dynamics of cellular plates with advanced microarchitectures. The corresponding governing equations follow Mindlin’s SG elasticity theory including the micro-inertia effect applied to the first-order shear deformation plate theory along with the nonlinear von Kármán kinematics. Standard and higher-order computational homogenization methods determine the classical and strain gradient material constants, respectively. A higher-order \({C}^{1}\)-continuous 6-node finite element is adopted for the discretization of the governing variational formulation with respect to the spatial domain, and an arc-length continuation technique along with time periodic discretization is implemented to solve the resulting nonlinear time-dependent problem. Through a set of comparative studies with 3D full-field models as references, the accuracy and efficacy of the proposed dimension reduction methodology are demonstrated for a diverse range of problem parameters for analyzing the large-amplitude dynamic structural response.

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A geometrically nonlinear Euler–Bernoulli beam model within strain gradient elasticity with isogeometric analysis and lattice structure applications

Cited by 32 publications

References 65 publications

The Study of the Genesis of Novel Mathematical and Mechanical Theories Provides an Inspiration for Future Original Research

The Study of the Genesis of Novel Mathematical and Mechanical Theories Provides an Inspiration for Future Original Research

Granular micromechanics‐based identification of isotropic strain gradient parameters for elastic geometrically nonlinear deformations

Nonlinear Finite Element Analysis of Free and Forced Vibrations of Cellular Plates Having Triangularly Prismatic Metamaterial Cores: A Strain Gradient Plate Model Approach

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