Lattice Elements Research Articles

Low-velocity impact and compressive behavior for shifted-tri-axial composite panels

This paper presents the experimental behavior of low-energy impact and quasi-static compression test of shifted-tri-axial structural wood-fiber-based composite panels made from laminated paper. The experimental results were analyzed based on design parameters and configurations of panels for the further design and optimization. The results showed that the face stiffness and strength was a significant factor to improve both impact performance and compressive performance. The panels made with additional carbon fiber fabric composite faces had higher energy absorption compared with the same panels made without it. The core configuration also affected the impact behavior of the panels, the foam filled core integrated with the shifted-tri-axial rib structure improved the impact load and absorbed more energy than the same panels without the foam. Further, the structure and size of the element in the core influenced the impact performance and energy absorption. The location for both compression and impact at the triangular lattice element center of the ribs had higher absorbed energy than the location at the hexagonal lattice element center of the ribs. A 3D contour surface map of maximum energy absorption was made based on the experimental data, the contour shows localized energy absorption based on the impact location on the core, the small triangular lattice element of the core had highest maximum energy absorption of panels. For both the quasi-static compression tests and the low-velocity impact tests, the panels with the same core configuration had similar compressive load–displacement trends during the early contact phase. However, the peak load was higher in compression than the peak load for the low-velocity impact for panels with the same configuration.

The [formula omitted]-invariant and catenary data of a matroid

The catenary data of a matroid M of rank r on n elements is the vector (ν(M;a0,a1,…,ar)), indexed by compositions (a0,a1,…,ar), where a0≥0, ai>0 for i≥1, and a0+a1+⋯+ar=n, with the coordinate ν(M;a0,a1,…,ar) equal to the number of maximal chains or flags (X0,X1,…,Xr) of flats or closed sets such that Xi has rank i, |X0|=a0, and |Xi−Xi−1|=ai. We show that the catenary data of M contains the same information about M as its G-invariant, which was defined by H. Derksen (2009) [9]. The Tutte polynomial is a specialization of the G-invariant. We show that many known results for the Tutte polynomial have analogs for the G-invariant. In particular, we show that for many matroid constructions, the G-invariant of the construction can be calculated from the G-invariants of the constituents and that the G-invariant of a matroid can be calculated from its size, the isomorphism class of the lattice of cyclic flats with lattice elements labeled by the rank and size of the underlying set. We also show that the number of flats and cyclic flats of a given rank and size can be derived from the G-invariant, that the G-invariant of M is reconstructible from the deck of G-invariants of restrictions of M to its copoints, and that, apart from free extensions and coextensions, one can detect whether a matroid is a free product from its G-invariant.

Transition from energy dissipation to crack openings during continuum–discontinuum fracture of concrete

Fracture in quasi-brittle materials like concrete is mainly governed by two processes: energy dissipation and discontinuous displacement fields or discrete cracks. In this study we discuss the simultaneous development of both types of fracture. Energy dissipation and crack openings are analyzed during the fracture development by acoustic emission and digital image correlation (DIC) respectively. DIC results are processed using nonlinear damage modelling and lattice element modelling to reveal the microcracking zone. It is observed that overall failure mechanism is governed by three processes (1) high energy dissipation phase during which energy release rate increases (2) continuum–discontinuum transitional phase where energy dissipation rate attains its maximum value, and (3) discontinuum phase where energy dissipation rate drops and crack openings start increasing.

Digital Morphing Wing: Active Wing Shaping Concept Using Composite Lattice-Based Cellular Structures.

We describe an approach for the discrete and reversible assembly of tunable and actively deformable structures using modular building block parts for robotic applications. The primary technical challenge addressed by this work is the use of this method to design and fabricate low density, highly compliant robotic structures with spatially tuned stiffness. This approach offers a number of potential advantages over more conventional methods for constructing compliant robots. The discrete assembly reduces manufacturing complexity, as relatively simple parts can be batch-produced and joined to make complex structures. Global mechanical properties can be tuned based on sub-part ordering and geometry, because local stiffness and density can be independently set to a wide range of values and varied spatially. The structure's intrinsic modularity can significantly simplify analysis and simulation. Simple analytical models for the behavior of each building block type can be calibrated with empirical testing and synthesized into a highly accurate and computationally efficient model of the full compliant system. As a case study, we describe a modular and reversibly assembled wing that performs continuous span-wise twist deformation. It exhibits high performance aerodynamic characteristics, is lightweight and simple to fabricate and repair. The wing is constructed from discrete lattice elements, wherein the geometric and mechanical attributes of the building blocks determine the global mechanical properties of the wing. We describe the mechanical design and structural performance of the digital morphing wing, including their relationship to wind tunnel tests that suggest the ability to increase roll efficiency compared to a conventional rigid aileron system. We focus here on describing the approach to design, modeling, and construction as a generalizable approach for robotics that require very lightweight, tunable, and actively deformable structures.

Lattice Element Models and Their Peculiarities

This paper presents the lattice element models, as a class of discrete models, in which the structural solid is represented as an assembly of one-dimensional elements. This idea allows one to provide robust models for propagation of discontinuities, multiple cracks interaction or cracks coalescence. Many procedures for computation of lattice element parameters for representing linear elastic continuum have been developed, with the most often used ones discussed herein. Special attention is dedicated to presenting the ability of this kind of models to consider material disorder, heterogeneities and multi-phase materials, which makes lattice models attractive for meso- or micro-scale simulations of failure phenomena in quasi-brittle materials, such as concrete or rocks. Common difficulties encountered in material failure and a way of dealing with them in the lattice models framework are explained in detail. Namely, the size of the localized fracture process zone around the propagating crack plays a key role in failure mechanism, which is observed in various models of linear elastic fracture mechanics, multi-scale theories, homogenization techniques, finite element models, molecular dynamics. An efficient way of dealing with this kind of phenomena is by introducing the embedded strong discontinuity into lattice elements, resulting with mesh-independent computations of failure response. Moreover, mechanical lattice can be coupled with mass transfer problems, such as moisture, heat or chloride ions transfer which affect the material durability. Any close interaction with a fluid can lead to additional time dependent degradation. For illustration, the lattice approach to porous media coupling is given here as well. Thus, the lattice element models can serve for efficient simulations of material failure mechanisms, even when considering multi-physics coupling. The main peculiarities of such an approach have been presented and discussed in this work.

On the Zariski topology over an $L$-module $M$

Let $L$ be a multiplicative lattice and $M$ be an $L$-module. In this study, we present a topology said to be the Zariski topology over $\sigma (M),$ the collection of all prime elements of an $L$-module $M.$ We research some results on the Zariski topology over $\sigma (M).$ We show that the topology is a $T_{0}$-space and a $T_{1}$-space under some conditions. Some properties and results are studied for the topology over $\sigma (L)$, the collection of all prime elements of a multiplicative lattice $L.$

Lower bound on the number of meet-irreducible elements in extremal lattices

Extremal lattices are lattices maximal in size with respect to the number n of their join-irreducible elements with bounded Vapnik-Chervonekis dimension k. It is natural, however, to estimate the size of a lattice also with respect to the number of its meet-irreducible elements. Although this number may differ for nonequivalent (n, k + 1)-extremal lattices, we show that each (n, k + 1)-extremal lattice has k disjoint chains of meet-irreducible elements, each of length n − k + 2.

Studying the Krull dimension of finite lattices under the prism of matrices

The Krull dimension of a finite lattice (X;6) is equal to the height of the poset of join prime elements of X minus 1. To every partially ordered set we assign an order-matrix, and we use these ordermatrices to characterize the join prime elements of finite lattices. In addition, we present a reduction algorithm for the computation of the height of a finite poset. The algorithm is based on the concept of the incidence matrix. Our main objective, ultimately, is to use these processes to calculate the Krull dimension of any given finite lattice.

On the question of predicting the service life of lattice steel structural elements

Present paper deals with the evaluation of fatigue strength at crack initiation stage in the double angle elements of lattice steel structures. The current construction norms for steel structures do not include check for fatigue strength at crack initiation stage, and the calculation of elements endurance limit does not account actual stress-strain state in stress concentration zones, material deformation characteristics in these areas, and strength properties of steel. Experimental study of the stress-strain state kinetics in the areas of gusset plate breakage of double angle welded joints was carried out by strain gauge method. Results were obtained regarding the stress state stabilization under cyclic loading and stress intensity factor. During cyclic tests, maximum load level varied depending on the yield stress. The formation of fatigue cracks was detected in the zone of gusset plate breakage. The curve at fatigue cracks initiation stage, by deformational fracture criteria at 'stiff' loading, was described by the equations presented in the 'Standards of strength calculation of equipment and pipelines of nuclear power plants'. Calculation analysis of results for number of cycles before fatigue crack initiation has been made and good agreement with test results has been obtained.

Deterministic composite nanophotonic lattices in large area for broadband applications.

Exotic manipulation of the flow of photons in nanoengineered materials with an aperiodic distribution of nanostructures plays a key role in efficiency-enhanced broadband photonic and plasmonic technologies for spectrally tailorable integrated biosensing, nanostructured thin film solarcells, white light emitting diodes, novel plasmonic ensembles etc. Through a generic deterministic nanotechnological route here we show subwavelength-scale silicon (Si) nanostructures on nanoimprinted glass substrate in large area (4 cm2) with advanced functional features of aperiodic composite nanophotonic lattices. These nanophotonic aperiodic lattices have easily tailorable supercell tiles with well-defined and discrete lattice basis elements and they show rich Fourier spectra. The presented nanophotonic lattices are designed functionally akin to two-dimensional aperiodic composite lattices with unconventional flexibility- comprising periodic photonic crystals and/or in-plane photonic quasicrystals as pattern design subsystems. The fabricated composite lattice-structured Si nanostructures are comparatively analyzed with a range of nanophotonic structures with conventional lattice geometries of periodic, disordered random as well as in-plane quasicrystalline photonic lattices with comparable lattice parameters. As a proof of concept of compatibility with advanced bottom-up liquid phase crystallized (LPC) Si thin film fabrication, the experimental structural analysis is further extended to double-side-textured deterministic aperiodic lattice-structured 10 μm thick large area LPC Si film on nanoimprinted substrates.

Microscale Testing and Modelling of Cement Paste as Basis for Multi-Scale Modelling.

This work aims to provide a method for numerically and experimentally investigating the fracture mechanism of cement paste at the microscale. For this purpose, a new procedure was proposed to prepare micro cement paste cubes (100 × 100 × 100 µm3) and beams with a square cross section of 400 × 400 µm2. By loading the cubes to failure with a Berkovich indenter, the global mechanical properties of cement paste were obtained with the aid of a nano-indenter. Simultaneously the 3D images of cement paste with a resolution of 2 µm3/voxel were generated by applying X-ray microcomputed tomography to a micro beam. After image segmentation, a cubic volume with the same size as the experimental tested specimen was extracted from the segmented images and used as input in the lattice model to simulate the fracture process of this heterogeneous microstructure under indenter loading. The input parameters for lattice elements are local mechanical properties of different phases. These properties were calibrated from experimental measured load displacement diagrams and failure modes in which the same boundary condition as in simulation were applied. Finally, the modified lattice model was applied to predict the global performance of this microcube under uniaxial tension. The simulated Young’s modulus agrees well with the experimental data. With the method presented in this paper the framework for fitting and validation of the modelling at microscale was created, which forms a basis for multi-scale analysis of concrete.

From lattices to Hv-matrices

Abstract In this paper we study the concept of sets of elements, related to results of an associative binary operation. We discuss this issue in the context of matrices and lattices. First of all, we define hyperoperations similar to those used when constructing hyperstructures from quasi-ordered semigroups. This then enables us to show that if entries of matrices are elements of lattices, these considerations provide a natural link between matrices, some basic concepts of the hyperstructure theory including Hv-rings and Hv-matrices and also one recent construction of hyperstructures.

The size effect in quasi-brittle materials: Experimental and numerical analysis

The size effect in structures is responsible for the materials apparent properties variations as function of size. Different methodologies are proposed in the literature to address this phenomenon; however, there is still no consensus on how to specifically deal with this. For instance, if the material investigated were a quasi-brittle material, it would present a fissures development in different scales during the damaging process. In the present work, an experimental study applying uniaxial tension over expanded polystyrene specimens of different sizes is proposed. The acoustic emission events occurring during the tests were also recorded. A tailored version of a lattice discrete element method was used to simulate the tests. This numerical approach take into account several phenomena related with the damage process in quasi-brittle materials, such as the localization effect, the fractal dimension nature of the region over which the damage evolves; the collaborative effect between cluster of fissures and the avalanche effect during the damage process. It is important to mention that these characteristics are associated with the correct simulation of the acoustic emission registry. The obtained experimental and simulated results were in great agreement and clearly showed a size effect, despite the narrow range of dimension explored. The size effect evaluated during the simulations, in terms of the dissipated energy, is shown to be in agreement to the known fractal theory proposed by Alberto Carpinteri and coworkers. Moreover, results in terms of acoustic emission are preliminarily explored to determine the correlation between the acoustic emission events and the fracture mode that governs the source of these events. Finally, some conclusions related to the size effect captured during the tests and the possibilities for simulations of the fracturing process in quasi-brittle materials provided by the numerical method are point out.

A GCD and LCM-like inequality for multiplicative lattices

Let $A_1,\ldots,A_n$ $(n\ge 2)$ be elements of an commutative multiplicative lattice. Let $G(k)$ (resp., $L(k)$) denote the product of all the joins (resp., meets) of $k$ of the elements. Then we show that $$L(n)G(2)G(4)\cdots G(2\lfloor n/2 \rfloor ) \leq G(1)G(3)\cdots G(2\lceil n/2 \rceil -1).$$ In particular this holds for the lattice of ideals of a commutative ring. We also consider the relationship between $$G(n)L(2)L(4)\cdots L(2\lfloor n/2 \rfloor ) \text{ and } L(1)L(3)\cdots L(2\lceil n/2 \rceil -1)$$and show that any inequality relationships are possible.

Propagation properties of Gaussian Schell-model array beams in non-Kolmogorov turbulence

A stochastic beam generated by a recently introduced Gaussian Schell-model array (GSMA) source (2015 Opt. Lett. 40 5662) is investigated. We derive the analytical propagation formulae for the spectral density and the propagation factor in non-Kolmogorov turbulence by utilizing the extended Huygens–Fresnel principle and second-order moments of the Wigner distribution function. Numerical results show that the lattice patterns of GSMA beams, which keep propagation-invariant in free space, are destroyed by the turbulence at sufficiently large distances. The GSMA beams have significant advantage over the Gaussian Schell-model beam in the robustness of the destructive effect of non-Kolmogorov turbulence, especially for the GSMA beam with more lattice elements and bigger relative separation distance of each lattice element. The effects of beam parameters and non-Kolmogorov turbulence on the propagation factor are analyzed in detail.

Diffusivity and derivatives for interstitial solutes: activation energy, volume, and elastodiffusion tensors

Computational atomic-scale methods continue to provide new information about geometry, energetics and transition states for interstitial elements in crystalline lattices. This data can be used to determine the diffusivity of interstitials by finding steady-state solutions to the master equation. In addition, atomic-scale computations can provide not just the site energy, but also the stress in the cell due to the introduction of the defect to compute the elastic dipole. We derive a general expression for the fully anistropic diffusivity tensor from site and transition state energies, and three derivatives of the diffusivity: the elastodiffusion tensor (derivative of diffusivity with respect to strain), the activation barrier tensor (logarithmic derivative of diffusivity with respect to inverse temperature) and activation volume tensor (logarithmic derivative of diffusivity with respect to pressure). Computation of these quantities takes advantage of crystalline symmetry, and we provide an open-source implementation of the algorithm. We provide analytic results for octahedral–tetrahedral networks in face-centred cubic, body-centred cubic, hexagonal closed-packed lattices, and conclude with numerical results for C in Fe.

Rotating system for four-dimensional transverse rms-emittance measurements

Knowledge of the transverse four-dimensional beam rms-parameters is essential for applications that involve lattice elements that couple the two transverse degrees of freedom (planes). Of special interest is the removal of inter-plane correlations to reduce the projected emittances. A dedicated ROtating System for Emittance measurements (ROSE) has been proposed, developed, and successfully commissioned to fully determine the four-dimensional beam matrix. This device has been used at the High Charge injector (HLI) at GSI using a beam line which is composed of a skew quadrupole triplet, a normal quadrupole doublet, and ROSE. Mathematical algorithms, measurements, and results for ion beams of 83Kr13+ at 1.4 MeV/u are reported in this paper.

“Truly concurrent” and nondeterministic semantics of discrete-time Petri nets

In the paper, a "truly concurrent" and nondeterministic semantics is defined in terms of branching processes of discrete-time Petri nets (DTPNs). These nets may involve infinite numbers of transitions and places, infinite number of tokens in places, and (maximal) steps of concurrent transitions, which allows us to consider this class of DTPNs to be the most powerful class of Petri nets. It is proved that the unfolding (maximal branching process) of the DTPN is the greatest element of a complete lattice constructed on branching processes of DTPNs with step semantics. Moreover, it is shown that this result is true also in the case of maximal transition steps if additional restrictions are imposed on the structure and behavior of the DTPN.

Ferromagnetic resonance and resonance modes in kagome lattices: From an open to a closed kagome structure

We present ferromagnetic resonance (FMR) investigations of 20 nm thick permalloy (${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$) elements (width $W=200$ nm, length $L=470$ nm, period $a=500$ nm) arranged in open and closed artificial kagome lattices. The measurements were done at 9.4 and 34 GHz to ensure a saturated or near-saturated magnetic state of the kagome structures. The FMR data are analyzed in the framework of an analytical macrospin model which grasps the essential features of the bulk and edge modes at these microwave frequencies and is in agreement with the results of micromagnetic simulations. Polar plots of the resonance fields versus the field angle made by the direction of the magnetic field with respect to the main symmetry directions of the kagome lattice are compared with the results of the analytical model. The measured FMR spectra with a sixfold rotational symmetry qualitatively reproduce the structure expected from the theory. Magnetic dipolar interactions between the elements of the kagome lattices result in the mixing of edge and bulklike excitations at 9.4 GHz and in a systematic deviation from the model, especially for the closed kagome lattice.

Supersonic flow past a flat lattice of cylindrical rods

Two-dimensional supersonic laminar ideal gas flows past a regular flat lattice of identical circular cylinders lying in a plane perpendicular to the free-stream velocity are numerically simulated. The flows are computed by applying a multiblock numerical technique with local boundary-fitted curvilinear grids that have finite regions overlapping the global rectangular grid covering the entire computational domain. Viscous boundary layers are resolved on the local grids by applying the Navier–Stokes equations, while the aerodynamic interference of shock wave structures occurring between the lattice elements is described by the Euler equations. In the overlapping grid regions, the functions are interpolated to the grid interfaces. The regimes of supersonic lattice flow are classified. The parameter ranges in which the steady flow around the lattice is not unique are detected, and the mechanisms of hysteresis phenomena are examined.