On the formulation of elastic and electroelastic gradient beam theories

Lurie, S. A.; Solyaev, Yury

doi:10.1007/s00161-019-00781-3

Cited by 21 publications

(6 citation statements)

References 69 publications

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“…Figure 12 shows a stroboscopic image of the motion of the duoskelion structure. 7 As for the previously reported cases, Figures 11 and 12 show that, after some small initial vibrations, there is a sudden buckling and the slope of the displacement increases rapidly. Looking at Figure 12, it is possible to notice that the initial vibrations are mainly due to an extensional deformation mode and that, following buckling, the first (bending) mode is activated, at least initially.…”

Section: Influence Of the Loading Ratesupporting

confidence: 72%

“…Recent research in the field of metamaterials [1][2][3] has stimulated the rediscovery and improvement of previously developed models of structural elements-such as beams [4][5][6][7][8][9][10][11][12], plates [13][14][15][16], shells [17][18][19][20][21], and solids [22][23][24]-which have been adapted to study the exotic mechanical behaviors exhibited by metamaterials.…”

Section: Introductionmentioning

confidence: 99%

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In-plane dynamic buckling of duoskelion beam-like structures: discrete modeling and numerical results

Turco

Barchiesi

Dell’Isola

2021

Mathematics and Mechanics of Solids

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In this contribution, a previously introduced discrete model for studying the statics of duoskelion beam-like structures is extended to dynamics. The results of numerical simulations performed using such an extended model are reported to discuss the in-plane dynamic buckling of duoskelion structures under different loading and kinematic boundary conditions. The core instrument of the analysis is a discrete beam element, which, in addition to flexure, also accounts for extension and shearing deformations. Working in the setting of dynamics, inertial contributions are taken into account as well. A stepwise time integration scheme is employed to reconstruct the complete trajectory of the system, namely before and after buckling. It is concluded that the duoskelion structure exhibits exotic features compared with classical beam-like structures modeled at macro-scale by Euler–Bernoulli’s model.

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Section: Influence Of the Loading Ratesupporting

confidence: 72%

Section: Introductionmentioning

confidence: 99%

In-plane dynamic buckling of duoskelion beam-like structures: discrete modeling and numerical results

Turco

Barchiesi

Dell’Isola

2021

Mathematics and Mechanics of Solids

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“…It is notable that if we use the Variant I of the surface boundary conditions (29) and neglect the edge boundary conditions (33), then the SGET solution will coincide with the classical one. 61 This classical polynomial solution is directly included into the system of trial functions that follows from the Papkovich-Neuber representation (12), (17), such that the Trefftz solution will coincide with the analytical one. Nevertheless, for the full correct statement of the problem with the edge boundary conditions, the SGET solution deviates from the classical one.…”

Section: Square Domain Under Bendingmentioning

confidence: 99%

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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“…The resulting equivalent elastic modulus  b is 60.39 GPa. For this choice of material and geometric properties, the analytical solution 61 of the displacements for a simply supported beam with uniform load in the absence of a piezoelectric effect, the displacement at mid-span is 2.156 × 10 −4 m. The elastic stiffness of the beam will be increased by the piezoelectric effect with the effective elastic modulus 62 given by…”

Section: Validation Using a Piezoelectric Beammentioning

confidence: 99%

“…The resulting equivalent elastic modulus

{\mathcal{E}}_b

is 60.39 GPa. For this choice of material and geometric properties, the analytical solution 61 of the displacements for a simply supported beam with uniform load in the absence of a piezoelectric effect, the displacement at mid‐span is

2.156\times 1{0}^{-4}

m. The elastic stiffness of the beam will be increased by the piezoelectric effect with the effective elastic modulus 62 given by

{\hat{\mathcal{E}}}_b={\mathcal{E}}_b+\frac{{\left[{e}_b^{311}\right]}^2}{\kappa_b^{33}}.

The equivalent piezoelectric coefficient can also be computed using stress relaxations, as

{e}_b^{311}=-16.53

\mathrm{C}/{\mathrm{m}}^2

.…”

Section: Numerical Examplesmentioning

confidence: 99%

Vibration analysis of piezoelectric Kirchhoff–Love shells based on Catmull–Clark subdivision surfaces

Liu

McBride

Saxena

et al. 2022

Numerical Meth Engineering

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An isogeometric Galerkin approach for analysing the free vibrations of piezoelectric shells is presented. The shell kinematics is specialized to infinitesimal deformations and follow the Kirchhoff–Love hypothesis. Both the geometry and physical fields are discretized using Catmull–Clark subdivision bases. This provides the required C1$$ {C}^1 $$‐continuous discretization for the Kirchhoff–Love theory. The crystalline structure of piezoelectric materials is described using an anisotropic constitutive relation. Hamilton's variational principle is applied to the dynamic analysis to derive the weak form of the governing equations. The coupled eigenvalue problem is formulated by considering the problem of harmonic vibration in the absence of external load. The formulation for the purely elastic case is verified using a spherical thin shell benchmark. Thereafter, the piezoelectric shell formulation is verified using a one dimensional piezoelectric beam. The piezoelectric effect and vibration modes of a transverse isotropic curved plate are analyzed and evaluated for the Scordelis–Lo roof problem. Finally, the eigenvalue analysis of a CAD model of a piezoelectric speaker shell structure showcases the ability of the proposed method to handle complex geometries.

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On the formulation of elastic and electroelastic gradient beam theories

Cited by 21 publications

References 69 publications

In-plane dynamic buckling of duoskelion beam-like structures: discrete modeling and numerical results

In-plane dynamic buckling of duoskelion beam-like structures: discrete modeling and numerical results

Trefftz collocation method for two‐dimensional strain gradient elasticity

Vibration analysis of piezoelectric Kirchhoff–Love shells based on Catmull–Clark subdivision surfaces

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