scholarly journals Elastodynamic behavior of mechanical cloaks designed by direct lattice transformations

Wave Motion ◽  
2020 ◽  
Vol 92 ◽  
pp. 102419 ◽  
Author(s):  
Muamer Kadic ◽  
Martin Wegener ◽  
André Nicolet ◽  
Frédéric Zolla ◽  
Sébastien Guenneau ◽  
...  
Keyword(s):  
Direct Lattice

Some simple theorems concerning crystal lattice sums of electric potential, electric field, and electric field gradient in crystals

1967 ◽  
Vol 20 (12) ◽  
pp. 2551 ◽  
Author(s):  
CK Coogan
Keyword(s):  
Electric Field ◽  
Field Gradient ◽  
Electric Potential ◽  
Higher Symmetry ◽  
Lattice Sums ◽  
Symmetry Properties ◽  
Lattice Sum ◽  
Direct Lattice ◽  
Electric Potential Field ◽  
Conver Gence

The conditions under which direct lattice sums of electric potential, field, and field gradient converge are discussed. The analogous conditions under which differences in these lattice sums, for two points in the crystal, converge are also outlined. These conditions are applied to direct lattice sum calculations in crystals in which the ideal lattice is distorted close to a defect of some kind. The conver- gence conditions are then applied to the case of determining the direct lattice sums in crystals in which higher symmetry properties can be invoked, which leads to a knowledge by inspection of the lattice sum at one point in the unit cell.

Electrostatic lattice energy in ionic crystals: optimization of the convergence of Ewald series

1978 ◽  
Vol 34 (6) ◽  
pp. 974-979 ◽  
Author(s):  
M. Catti
Keyword(s):  
Lattice Energy ◽  
Rates Of Convergence ◽  
Unit Cell ◽  
Group Symmetry ◽  
Cube Root ◽  
Cell Dimensions ◽  
Madelung Energy ◽  
Direct Lattice ◽  
Unit Cell Dimensions ◽  
Theoretical Results

Ewald's method is reconsidered to express the dependence of Madelung energy on the ionic charges explicitly, also taking into account the space-group symmetry of the structure. Upper bounds for the residues of the two partial series have been calculated by integral approximation; that relative to the direct-lattice series is shown to depend on the cube root of the unit-cell volume. The optimum value of the parameter A, which equalizes the rates of convergence of the two sums and minimizes the total number of terms, has been determined numerically for a given termination error and for a range of unit-cell dimensions. Theoretical results are tested by calculations on some specific crystal structures.

scholarly journals Mechanical cloak design by direct lattice transformation

2015 ◽  
Vol 112 (16) ◽  
pp. 4930-4934 ◽  
Author(s):  
Tiemo Bückmann ◽  
Muamer Kadic ◽  
Robert Schittny ◽  
Martin Wegener
Keyword(s):  
Lattice Theory ◽  
Solid Mechanics ◽  
Elastic Solid ◽  
Lattice Points ◽  
Coordinate Transformations ◽  
Constituent Material ◽  
Elasticity Tensors ◽  
Complex Boundary ◽  
Direct Lattice ◽  
Theory And Experiment

Spatial coordinate transformations have helped simplifying mathematical issues and solving complex boundary-value problems in physics for decades already. More recently, material-parameter transformations have also become an intuitive and powerful engineering tool for designing inhomogeneous and anisotropic material distributions that perform wanted functions, e.g., invisibility cloaking. A necessary mathematical prerequisite for this approach to work is that the underlying equations are form invariant with respect to general coordinate transformations. Unfortunately, this condition is not fulfilled in elastic–solid mechanics for materials that can be described by ordinary elasticity tensors. Here, we introduce a different and simpler approach. We directly transform the lattice points of a 2D discrete lattice composed of a single constituent material, while keeping the properties of the elements connecting the lattice points the same. After showing that the approach works in various areas, we focus on elastic–solid mechanics. As a demanding example, we cloak a void in an effective elastic material with respect to static uniaxial compression. Corresponding numerical calculations and experiments on polymer structures made by 3D printing are presented. The cloaking quality is quantified by comparing the average relative SD of the strain vectors outside of the cloaked void with respect to the homogeneous reference lattice. Theory and experiment agree and exhibit very good cloaking performance.

scholarly journals Energy-filtered electron diffuse scattering of ferroelectrics PMN and PMN-xPT

2014 ◽  
Vol 70 (a1) ◽  
pp. C623-C623
Author(s):  
Jerome Pacaud ◽  
Wajdi Saidi ◽  
Ray Withers ◽  
Brahim Dkhil ◽  
Jian-Min Zuo
Keyword(s):  
Diffuse Scattering ◽  
Phonon Scattering ◽  
Relaxor Ferroelectrics ◽  
Charge Ordering ◽  
Diffuse Intensity ◽  
Sufficient Energy ◽  
Direct Lattice ◽  
Imaging Plates ◽  
Dielectric And Piezoelectric Properties

PbMb1/3Nb2/3O3 (PMN) and its solid solution (1-x)PbMb1/3Nb2/3O3-(x)PbTiO3 (PMN-xPT) are relaxor ferroelectrics which have attracted attention in the last few decades because of their very interesting dielectric and piezoelectric properties and have since be two of the most extensively studied. All the previous studies emphasized the role of the local structural fluctuations leading to local changes in symmetries [1] due to displacements of ions in the unit-cell. We studied PMN and PMN-xPT by electron diffuse scattering using an in-column energy filter and Imaging-Plates as detector. We found evidences for streaks of intensity along the [110]* direction as previously found in PbZn1/3Nb2/3O3 (PZN) with neutron diffraction [2]. Moreover, weak diffuse scattering sheets can be observed along (111)* reciprocal planes showing the existence of correlations along the [111] directions of the direct lattice. Figure 1 shows a diffraction pattern taken along [02-1] zone axis presenting both diffuse features. This can be related to the displacement of Pb ions along the diagonals of the cube found by simulation [3] but greatly complexify the analysis of the shape of the diffuse intensity. Compared to the neutron, electron diffraction has the advantage of two dimensional recording of diffuse scattering and eventually sensitivity to charge ordering but quantitative analysis is limited due to the complication of multiple scattering and the lack of sufficient energy resolution for the study of inelastic phonon scattering.

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