High-Contrast Multi-layered Plates. Statics, Dynamics and Buckling

Boutin, Claude

doi:10.1007/978-3-031-24141-3_4

Cited by 3 publications

(1 citation statement)

References 18 publications

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“…where η = x 2 h 1 + h 2 (10) is the dimensionless transverse variable; A 1 , A 2 and A 3 are arbitrary real constants; and the parameters α 1 and α 2 are defined by…”

Section: Dispersion Relationmentioning

confidence: 99%

Degenerated Boundary Layers and Long-Wave Low-Frequency Motion in High-Contrast Elastic Laminates

Aghalovyan,

Ghulghazaryan,

Kaplunov

et al. 2023

Mathematics

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The effect of high contrast on the multiscale behaviour of elastic laminates is studied. Mathematical modelling in this area is of significant interest for a variety of modern applications, including but not limited to advanced sandwich structures and photovoltaic panels. As an example, the antiplane shear of a symmetric, three-layered plate is considered. The problem parameters expressing relative thickness, stiffness and density are assumed to be independent. The high contrast may generally support extra length and time scales corresponding to degenerated boundary layers and propagating long-wave low-frequency vibration modes. The main focus is on the relation between these two phenomena. The developed multiparametric approach demonstrates that those do not always appear simultaneously. The associated explicit estimates on contrast parameters are established. In addition, the recent asymptotic extension of the classical Saint-Venant’s principle is adapted for calculating the contribution of the degenerate boundary layer or long-wave low-frequency propagation mode. The peculiarity of the limiting absorption principle in application to layered media is also addressed.

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“…where η = x 2 h 1 + h 2 (10) is the dimensionless transverse variable; A 1 , A 2 and A 3 are arbitrary real constants; and the parameters α 1 and α 2 are defined by…”

Section: Dispersion Relationmentioning

confidence: 99%

Degenerated Boundary Layers and Long-Wave Low-Frequency Motion in High-Contrast Elastic Laminates

Aghalovyan,

Ghulghazaryan,

Kaplunov

et al. 2023

Mathematics

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Asymptotic long-wave model for a high-contrast two-layered elastic plate

Mikhasev

2024

Mathematics and Mechanics of Solids

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The paper is concerned with the derivation of asymptotically consistent equations governing the long-wave flexural response of a two-layered rectangular plate with high-contrast elastic properties. In the general case, the plate is under dynamic and variable surface, volume, and edge forces. Performing the asymptotic integration of the three-dimensional (3D) elasticity equations in the transverse direction and satisfying boundary conditions on the faces and interface, we derived the sequence of two-dimensional (2D) differential equations with respect to required functions in the first two approximations. The eight independent restraints for the generalized displacements and stress resultants are considered to formulate the 16 independent variants of boundary conditions. One of the main results of the paper is the Timoshenko–Reissner type equation capturing the effect of the softer layer and taking into account the in-plane deformation induced by the edge forces. Comparative calculations of natural frequencies were carried out based on our and alternative models.

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A historical overview on static and dynamic analyses of sandwich or partially composite beams and plates

Challamel,

Atashipour,

Girhammar

et al. 2024

Mathematics and Mechanics of Solids

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This paper gives a brief overview of the static and the dynamic analyses of sandwich beams and sandwich plates composed of two elastic layers connected by a soft core. We show the strict correspondence between the sandwich beam, sandwich plate, and the analogous composite beam and composite plate problem with partial interaction. The beam problem is governed by a sixth-order differential equation in space for the deflection, whereas the plate problem is governed by a tri-Laplacian equation for the deflection, as first shown by Hoff in 1950 for symmetrical three-layer component (sandwich plate or composite plate theory with partial interaction). It is shown herein that the tri-Laplacian partial differential equation is also valid for general unsymmetrical sandwich plate or partially composite plate. In the limit cases, i.e., for non-composite action and for full-composite action, the beam model reduces into the Euler–Bernoulli beam model and the plate model into the Kirchhoff–Love model. Boundary layer phenomena may be predominant in the asymptotic stiff problem. The nonlocal bending moment–curvature of the sandwich beam (or composite beam with partial interaction) is analysed, and extended to a nonlocal Marcus bending moment–curvature for the composite plate theory. The consistency of the tri-Laplacian plate theory is commented, regarding the bi-Laplacian Reissner sandwich plate theory, which neglects the bending stiffnesses of each face layer. The paper finally presents results for the static bending and vibrations of sandwich beams and plates on simply supported boundary conditions, with a discussion on asymptotic limit cases.

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High-Contrast Multi-layered Plates. Statics, Dynamics and Buckling

Cited by 3 publications

References 18 publications

Degenerated Boundary Layers and Long-Wave Low-Frequency Motion in High-Contrast Elastic Laminates

Degenerated Boundary Layers and Long-Wave Low-Frequency Motion in High-Contrast Elastic Laminates

Asymptotic long-wave model for a high-contrast two-layered elastic plate

A historical overview on static and dynamic analyses of sandwich or partially composite beams and plates

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