Near-field transient waves in anisotropic elastic plates for two and three dimensional problems

An exact three-dimensional solution is presented for the deformation and stress distribution in an elliptical plate under uniform normal loading of the lateral surfaces, and clamped along its edge. The plate is assumed to be of constant, moderate thickness and composed of anisotropic elastic material which is inhomogeneous in the through-thickness direction but symmetric about the mid-plane. The only material symmetry assumed is that of reflectional symmetry in planes parallel to the mid-plane. A transfer matrix method is used which, without making any further assumptions, gives the exact solution at each point in the plate in terms of the stress and displacement at the mid-plane. The two-dimensional differential equations governing these mid-plane values are found to be the same as those for an equivalent homogeneous plate whose constant elastic moduli are determined by appropriate through-thickness weighted averages of the inhomogeneous moduli. The solution of the two-dimensional problem is known for such a plate when subject to the specified surface and edge conditions, and yields a closed form analytical solution that satisfies all the governing equations and surface conditions of the full three-dimensional elasticity problem, with edge displacement conditions satisfied on the mid-plane. The important special case of an anisotropic laminated plate is given by assuming piecewise constant properties through the thickness.

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From near-field to far-field: Radiative coupling of particle plasmon resonances in three-dimensional geometries

2011 Conference on Lasers and Electro-Optics Europe and 12th European Quantum Electronics Conference (CLEO EUROPE/EQEC) ◽

10.1109/cleoe.2011.5943224 ◽

2011 ◽

Author(s):

Richard Taubert ◽

Ralf Ameling ◽

Thomas Weiss ◽

Andre Christ ◽

Harald Giessen

Keyword(s):

Near Field ◽

Three Dimensional ◽

Far Field ◽

Radiative Coupling ◽

Plasmon Resonances ◽

Particle Plasmon

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On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

Journal of Elasticity ◽

10.1007/s10659-021-09819-7 ◽

2021 ◽

Author(s):

Olivier Ozenda ◽

Epifanio G. Virga

Keyword(s):

Free Energy ◽

Dimension Reduction ◽

Three Dimensional ◽

Energy Functional ◽

The Other ◽

Variational Model ◽

Two Dimensional ◽

Elastic Plates ◽

Kinematic Constraint ◽

Free Energy Functional

AbstractThe Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard $\varGamma $ Γ -convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate’s mid surface (alongside the expected measures of bending). The incompatibility with standard $\varGamma $ Γ -convergence also appears to be removed in the cases where contact with that method and ours can be made.

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Imaging of bi-anisotropic periodic structures from electromagnetic near-field data

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2020-0114 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Dinh-Liem Nguyen ◽

Trung Truong

Keyword(s):

Inverse Scattering ◽

Maxwell Equations ◽

Field Data ◽

Periodic Structures ◽

Near Field ◽

Scattering Problem ◽

Three Dimensional ◽

Inverse Scattering Problem ◽

Factorization Method ◽

Unique Determination

AbstractThis paper is concerned with the inverse scattering problem for the three-dimensional Maxwell equations in bi-anisotropic periodic structures. The inverse scattering problem aims to determine the shape of bi-anisotropic periodic scatterers from electromagnetic near-field data at a fixed frequency. The factorization method is studied as an analytical and numerical tool for solving the inverse problem. We provide a rigorous justification of the factorization method which results in the unique determination and a fast imaging algorithm for the periodic scatterer. Numerical examples for imaging three-dimensional periodic structures are presented to examine the efficiency of the method.

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A Fast Three-Dimensional Near-Field Imaging Algorithm for MIMO Cross-Array

2020 13th UK-Europe-China Workshop on Millimetre-Waves and Terahertz Technologies (UCMMT) ◽

10.1109/ucmmt49983.2020.9296053 ◽

2020 ◽

Author(s):

Feifan Wang ◽

Qi Yang ◽

Bin Deng ◽

Ye Zhang ◽

Hongqiang Wang

Keyword(s):

Near Field ◽

Three Dimensional ◽

Imaging Algorithm ◽

Near Field Imaging ◽

Field Imaging

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Mathematical Theory of Transversally Isotropic Shells of Arbitrary Thickness at Static Load

Materials Science Forum ◽

10.4028/www.scientific.net/msf.968.496 ◽

2019 ◽

Vol 968 ◽

pp. 496-510

Author(s):

Anatoly Grigorievich Zelensky

Keyword(s):

Mathematical Theory ◽

Three Dimensional ◽

Nonlinear Problems ◽

Theory Of Elasticity ◽

Elastic Plates ◽

Dimensional Problem ◽

Boundary Problems ◽

Complex Boundary ◽

Plates And Shells ◽

Basic Equations

Classical and non-classical refined theories of plates and shells, based on various hypotheses [1-7], for a wide class of boundary problems, can not describe with sufficient accuracy the SSS of plates and shells. These are boundary problems in which the plates and shells undergo local and burst loads, have openings, sharp changes in mechanical and geometric parameters (MGP). The problem also applies to such elements of constructions that have a considerable thickness or large gradient of SSS variations. The above theories in such cases yield results that can differ significantly from those obtained in a three-dimensional formulation. According to the logic in such theories, the accuracy of solving boundary problems is limited by accepted hypotheses and it is impossible to improve the accuracy in principle. SSS components are usually depicted in the form of a small number of members. The systems of differential equations (DE) obtained here have basically a low order. On the other hand, the solution of boundary value problems for non-thin elastic plates and shells in a three-dimensional formulation [8] is associated with great mathematical difficulties. Only in limited cases, the three-dimensional problem of the theory of elasticity for plates and shells provides an opportunity to find an analytical solution. The complexity of the solution in the exact three-dimensional formulation is greatly enhanced if complex boundary conditions or physically nonlinear problems are considered. Theories in which hypotheses are not used, and SSS components are depicted in the form of infinite series in transverse coordinates, will be called mathematical. The approximation of the SSS component can be adopted in the form of various lines [9-16], and the construction of a three-dimensional problem to two-dimensional can be accomplished by various methods: projective [9, 14, 16], variational [12, 13, 15, 17]. The effectiveness and accuracy of one or another variant of mathematical theory (MT) depends on the complex methodology for obtaining the basic equations.

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Transient waves in inhomogeneous isotropic elastic plates

Acta Mechanica ◽

10.1007/bf01177947 ◽

1984 ◽

Vol 53 (3-4) ◽

pp. 141-161 ◽

Cited By ~ 5

Author(s):

H. Cohen ◽

R. S. D. Thomas

Keyword(s):

Elastic Plates ◽

Transient Waves

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Three-Dimensional Finite-Difference Time-Domain Method Modeling of Nanowire Optical Probe

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.602-605.3359 ◽

2014 ◽

Vol 602-605 ◽

pp. 3359-3362

Author(s):

Chun Li Zhu ◽

Jing Li

Keyword(s):

Finite Difference ◽

Time Domain ◽

Finite Difference Time Domain ◽

Incident Light ◽

Near Field ◽

Three Dimensional ◽

Polarization Direction ◽

Near Field Imaging ◽

Difference Time ◽

Field Imaging

In this paper, output near fields of nanowires with different optical and structure configurations are calculated by using the three-dimensional finite-difference time-domain (3D FDTD) method. Then a nanowire with suitable near field distribution is chosen as the probe for scanning dielectric and metal nanogratings. Scanning results show that the resolution in near-field imaging of dielectric nanogratings can be as low as 80nm, and the imaging results are greatly influenced by the polarization direction of the incident light. Compared with dielectric nanogratings, metal nanogratings have significantly enhanced resolutions when the arrangement of gratings is perpendicular to the polarization direction of the incident light due to the enhancement effect of the localized surface plasmons (SPs). Results presented here could offer valuable references for practical applications in near-field imaging with nanowires as optical probes.

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Three-Dimensional Green’s Functions in Anisotropic Elastic Bimaterials With Imperfect Interfaces

Journal of Applied Mechanics ◽

10.1115/1.1546243 ◽

2003 ◽

Vol 70 (2) ◽

pp. 180-190 ◽

Cited By ~ 50

Author(s):

E. Pan

Keyword(s):

Green's Functions ◽

Green’S Functions ◽

Three Dimensional ◽

Imperfect Interface ◽

Boundary Integral ◽

Stress Fields ◽

Superposition Method ◽

Interface Conditions ◽

Complementary Part ◽

Anisotropic Elastic

In this paper, three-dimensional Green’s functions in anisotropic elastic bimaterials with imperfect interface conditions are derived based on the extended Stroh formalism and the Mindlin’s superposition method. Four different interface models are considered: perfect-bond, smooth-bond, dislocation-like, and force-like. While the first one is for a perfect interface, other three models are for imperfect ones. By introducing certain modified eigenmatrices, it is shown that the bimaterial Green’s functions for the three imperfect interface conditions have mathematically similar concise expressions as those for the perfect-bond interface. That is, the physical-domain bimaterial Green’s functions can be obtained as a sum of a homogeneous full-space Green’s function in an explicit form and a complementary part in terms of simple line-integrals over [0,π] suitable for standard numerical integration. Furthermore, the corresponding two-dimensional bimaterial Green’s functions have been also derived analytically for the three imperfect interface conditions. Based on the bimaterial Green’s functions, the effects of different interface conditions on the displacement and stress fields are discussed. It is shown that only the complementary part of the solution contributes to the difference of the displacement and stress fields due to different interface conditions. Numerical examples are given for the Green’s functions in the bimaterials made of two anisotropic half-spaces. It is observed that different interface conditions can produce substantially different results for some Green’s stress components in the vicinity of the interface, which should be of great interest to the design of interface. Finally, we remark that these bimaterial Green’s functions can be implemented into the boundary integral formulation for the analysis of layered structures where imperfect bond may exist.

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