Repeating Unit Cell Research Articles

Lieb-Schultz-Mattis type theorems for Majorana models with discrete symmetries

We prove two Lieb-Schultz-Mattis type theorems that apply to any translationally invariant and local fermionic $d$-dimensional lattice Hamiltonian for which fermion-number conservation is broken down to the conservation of fermion parity. We show that when the internal symmetry group $G^{\,}_{f}$ is realized locally (in a repeat unit cell of the lattice) by a nontrivial projective representation, then the ground state cannot be simultaneously nondegenerate, symmetric (with respect to lattice translations and $G^{\,}_{f}$), and gapped. We also show that when the repeat unit cell hosts an odd number of Majorana degrees of freedom and the cardinality of the lattice is even, then the ground state cannot be simultaneously nondegenerate, gapped, and translation symmetric.

Evaluation of energy absorption capabilities and mechanical properties in FDM printed PLA TPMS structures

Additive manufacturing is an advanced technology used to fabricate complex geometries with unique properties like cellular structures which accommodate repeated unit cells located in the x, y and z direction. These structures can be used as infill patterns due to their self-supporting structure. Among the cellular structures, Triply Periodic Minimal Surface (TPMS) structures such as Gyroid, Diamond and Schwarz Primitive (SchwarzP) structures can be tailored to produce complex structures for various applications like tissue engineering scaffolds and replace the conventional polymeric foams. TPMS structures are designed and manufactured by using the Fused Deposition Modelling (FDM) technique using Poly-Lactic Acid (PLA) as material. Among TPMS structures, Gyroid is having a unique property like structurally symmetric which design was modified to enhance the mechanical properties. The modified Gyroid or deformed Gyroid undergone a quasi-static compression test and compare the results with Diamond and SchwarzP structures. Porosity and permeability coefficients are evaluated and an optical microscope is used to verify the fabricated components. As well as, Failure patterns of the structures were evaluated and energy absorption capabilities determined. The main objective of this paper is to evaluate the impact of design and porosity on the mechanical and morphological properties of TPMS structures. In conclusion, the deformed Gyroid has more energy absorption capability up to the 11.6% strain than other TPMS structures. After 11.6% of strain, SchwarzP structure dominates.

A new progressive damage model for predicting the tensile behavior of the three-dimensional woven carbon/carbon composites using micromechanics method

A new progressive damage model for the three-dimensional (3 D) woven carbon/carbon (C/C) composites is developed at fiber-matrix level using the micromechanics method. A woven architecture based Representative Volume Element (RVE) model composed of yarns, matrix and yarn/matrix interface is constructed, in which the manufacturing void defects are accounted for. The fiber-matrix concentric cylinder model is employed as a repeating unit cell to represent the yarn, and the matrix micro strain field is computed analytically by the micromechanics method. The maximum stain criteria is utilized for fiber longitudinal breakage, and the Von-Mises criterion is applied for the damage initiation of matrix in both intra-yarns and inter-yarns. The damaged fiber and matrix are modeled by the stiffness degradation method combined with exponential damage evolution equations. The zero thickness cohesive elements governed by bilinear traction-separation constitutive are adopted for yarn/matrix interfacial debonding behavior. The micro progressive damage and failure behavior of the 3 D woven C/C composites subjected to tension is implemented through a developed user-defined material subroutine in commercial software ABAQUS. The predicted stress-strain response is in a good agreement with experimental results. In addition, the effect of manufacturing void defects is also examined by the developed model.

Framework for Predicting Failure in Polymeric Unidirectional Composites through Combined Experimental and Computational Mesoscale Modeling Techniques

As composites continue to be increasingly used, finite element material models that homogenize the composite response become the only logical choice as not only modeling the entire composite microstructure is computationally expensive but obtaining the entire suite of experimental data to characterize deformation and failure may not be possible. The focus of this paper is the development of a modeling framework where plasticity, damage, and failure-related experimental data are obtained for each composite constituent. Mesoscale finite elements models consisting of multiple repeating unit cells are then generated and used to represent a typical carbon fiber/epoxy resin unidirectional composite to generate the complete principal direction stress-strain curves. These models are subjected to various uniaxial states of stress and compared with experimental data. They demonstrate a reasonable match and provide the basic framework to completely define the composite homogenized material model that can be used as a vehicle for failure predictions.

Vibro-acoustic modeling of aircraft structures using Finite Element- informed Statistical Energy Analysis

This paper addresses the problem of predicting the structure born and airborne sound transmission in aircraft using Statistical Energy Analysis (SEA). Often analytical formulations are used to approximate the SEA parameters. In the present prediction method, a finite element (FE)-informed SEA approach is employed. To compute the coupling coefficient, the structure is represented with a repetition of unit cell and an FE model of the unit cell is assigned to evaluate the direct field dynamic stiffness matrix of the SEA subsystems at the connections. An efficient strategy is employed to determine the equivalent material properties of the FE model. Thus, a two-dimensional unit cells of different constructions such as composite, sandwich, visco-elastic laminate and ribbed section sections can be used. To evaluate the equivalent properties of multilayers structures, each layer is assumed as thick laminate with orthotropic orientation. Moreover, rotational inertia and transversal shearing, membrane and bending deformations are accounted for. First order shear deformation theory is employed. The developed approach handles symmetrical layouts of unlimited number of transversal compressible or incompressible layers. The accuracy of this modeling approach is confirmed through comparison to alternate validated theoretical approaches. Representative examples of spacecraft structural response and interior noise predictions for typical load cases are shown and the use of SEA models as a tool for guiding construction of complex structures to meet acoustic performance targets and optimize designs are presented. Conclusions about the application and advantages of this approach is presented.

Overall dynamic properties of locally resonant viscoelastic layered media based on consistent field integration for oblique anti-plane shear waves

In this work, the analysis of oblique anti-plane shear waves propagation and scattering in low frequency resonant micro-structured layered media with viscoelastic constituent layers is presented. The band structure of the infinitely periodic systems and scattering off a finite thickness slab of such media are determined using the transfer matrix method. A consistent dynamic field homogenization approach is applied, in which the micro-scale field equations are integrated and the overall macro-scale quantities are defined to be compatible with these integral forms. A reduced set of constitutive tensors is presented for general asymmetric repeating unit cells (RUC), utilizing the proposed homogenized macro-scale quantities combined with Onsager’s principle and presumed material form of elastodynamic reciprocity. This set can be further restricted by studying the form of the dispersion equation leading to a unique constitutive tensor. It is shown that for an asymmetric RUC, the full constitutive tensor, including off-diagonal elastic moduli and Willis coupling terms are required in order to match the scattering and band structure of the micro-structured media, but all the off-diagonal parameters vanish for a symmetric RUC. Therefore, it is possible to create an equivalent homogenized representation with a uniquely determined diagonal constitutive tensor for a symmetric RUC, though, as is the case also with asymmetric RUCs, all non-zero components will be wave-vector dependent. Numerical examples are presented to demonstrate the application and consistency of the proposed method. All the diagonal terms are converging to their appropriate Voigt or Reuss averages at the long-wavelength limit for all wave directions. The wave-vector dependent nature of the off-diagonal coupling constants can still be observed even in this limit. The conditions for lossy (passive) or lossless (fully elastic) systems are presented and are shown to impose weak requirements on overall constitutive tensors.

Stiffness Prediction of Triaxial Braided Composites Accounting for Manufacturing Parameters

Textile composites are applied to aerospace and automotive industries because they have good fatigue characteristics, and higher specific stiffness and strength compared to metallic materials. In addition, textile composites have better out-of-plane performances such as the resistance of impact and inter-laminar delamination, in addition to greater structural flexibility than the traditional unidirectional composite materials. In this paper, we studied a stiffness prediction method for the effective stiffness of triaxial braided composites accounting for manufacturing parameters such as fiber volume fraction, number of filaments in a yarn, braiding angle, and gap between fiber yarns. The repeating unit cell (RUC) of the triaxial braided composites and the geometric model of yarn architecture were defined in consideration of the gap between adjacent yarns. The gap size between yarns was measured from the photographic image of the cross-section of the triaxial braided composite specimens. The fiber yarn discretization method and the stress averaging method were applied to predict the effective mechanical properties of triaxial braided composites. The predicted properties by the proposed model have good agreement with experimental results.

Assessing Tetrahedral Lattice Parameters for Engineering Applications Through Finite Element Analysis.

Minimizing weight while maintaining strength in components is a continuous struggle within manufacturing industries, especially in aerospace. This study explores how controlling the dimensions of the geometric parameters of a lattice yields ideal mechanical properties for aerospace-related applications. A previously developed Bubble-mesh based computational method was used to generate a novel type of tetrahedral lattice that allows for the manipulation of three geometric parameters: cell size/density, strut diameter, and strut intersection rounding. Topology optimization and lattice generation within components are typical methods used to decrease weight while maintaining strength. Although these are robust optimization methods, each have their faults. Highly topology-optimized components may fail under unexpected loads, and lattice generation within commercial software is often limited in its ability to create ideal lattices with controlled geometric parameters, resulting in lattices with repeating unit cells. In this study, we used finite element methods (FEM)-based compression tests on latticed cubes with various parameter combinations to determine the best balance of lattice parameters. The results showed that strut diameter and strut intersection rounding were the best parameters to control to maintain strength and reduce weight. This understanding of the lattice structures was then applied to two aerospace components: a jet engine bracket and an airplane bearing bracket. By applying tetrahedral lattices with specified strut diameters and strut intersection rounding, the weight of the jet engine bracket was reduced by 51.8%, and the airplane bearing bracket was reduced by 20.5%.

Effects of general imperfect interface/interphase on the in-plane conductivity of thermal composites

The effects of general imperfect interphase/interface models on effective and local thermal conductive responses of fiber reinforced-composites are investigated. A physical interphase model is firstly introduced and then extended to the weakly conducting (WC) and highly conducting (HC) interfaces by satisfying the discontinuity conditions of temperature and normal heat flux, respectively. To simulate different geometric arrangements of fibers, hexagonal and square repeating unit cells (RUCs) are used in this paper. Regarding simulation technique, the locally-exact homogenization theory (LEHT) is extended by considering the aforementioned interphase/interface models to re-produce the localized temperature and heat flux distributions in the unit cells and then to predict the macroscopic thermal conductivities. The present method does not require mesh discretization and pre-processing, but directly obtains the accurate analytical expression of internal domains after solving the thermal differential equations. The unknown coefficients are obtained through applying imperfect interphase/interface conditions and weak-form periodic boundary conditions. Finally, the effective Fourier's formula is established to obtain the overall thermal conductivities of composites. The LEHT is advantageous in fast convergence, numerical stability and computational efficiency, thus providing a good numerical tool in studying the influence of interface parameters on composites. The generated numerical results are in good agreement with the analytical or numerical simulations in the literature. On this basis, the effects of interfacial geometric and physical parameters on the macroscopic coefficients and local thermal responses of the composites are tested. Finally, the thickness of a physical interphase is varied to test its applicability in generating equivalent effect with WC interface and HC interface. Generally, the imperfect interfacial effects play important roles in the thermal conductive response of composite materials. In the meantime, the proposed method is of great significance for the study of thermal conductions of composite microstructures.

A finite beam element multifunctional fabric modeling framework demonstrated with SMA knitted actuators

Multifunctional textiles have gained recent attention due their intrinsic properties that provide actuator, energy dampening, or sensory capabilities within small form factor textiles without external attachments. Such technologies are specifically appealing for medical and aerospace wearables, where active compression, haptic feedback, or the tracking of bodily functions are important tasks that are ideally conducted in a minimally-intrusive fashion. Current design capabilities of multifunctional textile are limited as present predictive tools lack accuracy and universality. This paper presents a finite beam element modeling tool for shape memory alloy (SMA) knitted architectures. The temperature-dependent variation of material properties within the SMA knitted loop affects the macroscopic force-extension behavior of SMA knitted architectures leading to an actuated and a relaxed knitted architecture response. This difference is exploited as the active property in SMA knitted architectures. The modeling architecture defines interfaces between sub-models organized in modules, specifically the material constitutive module, repetitive unit cell module, manufacturing module, contact module, and a boundary condition module. The SMA knitted architecture is modeled utilizing a 1D SMA constitutive model, quarter loop knit unit cell, a differential geometry-based manufacturing model, while assuming 3D Coulomb friction conditions. Kinematically-suitable boundary conditions are applied and the simulation predictions are compared quantitatively to macroscopic tensile experimental results, as well as qualitatively to microscopic x-ray diffraction phase analysis. The verification against experimental data supports the ability of the modeling tool to accurately predict the SMA knitted architecture thermo-mechanical performance with mean force-extension errors of less than 5%. The modeling tool provides the basis to understand, design, and optimize the lightweight, large force and deformation SMA knitted actuator textiles for novel applications. Additionally, the multifunctional textile modeling tool is implemented based on highly interchangeable sub-models to create synergies and propel the modeling of any multifunctional textile.

Exploring the property space of periodic cellular structures based on crystal networks

The properties of periodic cellular structures strongly depend on the regular spatial arrangement of their constituent base materials and can be controlled by changing the topology and geometry of the repeating unit cell. Recent advances in three-dimensional (3D) fabrication technologies more and more expand the limits of fabricable real-world architected materials and strengthen the need of novel microstructural topologies for applications across all length scales and fields in both fundamental science and engineering practice. Here, we systematically explore, interpret, and analyze publicly available crystallographic network topologies from a structural point of view and provide a ready-to-use unit cell catalog with more than 17,000 unique entries in total. We show that molecular crystal networks with atoms connected by chemical bonds can be interpreted as cellular structures with nodes connected by mechanical bars. By this, we identify new structures with extremal properties as well as known structures such as the octet-truss or the Kelvin cell and show how crystallographic symmetries are related to the mechanical properties of the structures. Our work provides inspiration for the discovery of novel cellular structures and paves the way for computational methods to explore and design microstructures with unprecedented properties, bridging the gap between microscopic crystal chemistry and macroscopic structural engineering.

Influence of SLM parameters on the compressive behaviour of lattice structures in 17-4PH stainless steel

Additive manufacture (AM) technologies allow innovative structural design, including complex lattice structures. Selective laser melting (SLM) is an AM process that enables the manufacture of space filling lattice structures. Lattice structures are topologically ordered, three-dimensional open-celled structures composed of one or more repeating unit cells. From a mechanical designer viewpoint, a great advantage offered by cellular materials is high strength accompanied by a relatively low mass. Due to their complex structure, fine geometry and the absence of supports structures, the setting of the best parameters to print lattice structures is difficult and could negative influence their mechanical response. This study investigated how different parameters settings influence the compression behaviour of lattice structures in stainless steel 17-4PH (AISI-630) printed by SLM. A comparison between compressive response of structures printed with different parameters is presented and considerations about their behaviour exposed.

Modeling and characterization of viscoelastic origami structures using a temperature variation-based model

The use of origami cores in sandwich structures is attracting increasing attention in many engineering applications due to origami cores’ good compromise between weight and strength and their ability to dissipate energy. Modeling is one of the best tools for estimating design parameters for high performance. The primary outcome of this investigation is a general platform for modeling polymeric origami cores that takes into account the viscoelastic properties of the base material at elevated temperatures and periodic boundary conditions. The model considers a single origami unit cell as a representative unit element, which consists of repeating elements of the origami structure. Periodic boundary conditions were applied in the shared edges of the repeated unit cells. Furthermore, the work shows the results of multi-unit cell origami panel using the viscoelastic model.

Mechanical behavior and failure of glass/carbon fiber hybrid composites: Multiscale computational predictions validated by experiments

A comprehensive, novel and computationally low cost multiscale model based on finite element analysis is proposed which includes repeating unit cell micromechanics, material nonlinearity, and progressive damage analysis to predict the mechanical behavior and failure of glass/carbon fiber hybrid composites subjected to various loading conditions. The computational micromechanics is used to predict the homogenized properties of the composite from the mechanical properties of its constituents (fiber, and matrix), together with the volume fraction and spatial distribution of the fibers within it. Hashin’s failure criteria at the meso-scale is implemented to determine failure of lamina while nonlinearity of epoxy matrix is introduced to model through J2 deformation theory of plasticity. It is shown that the introduced multiscale model can be used for 3D macroscale structural analysis through a user defined material (UMAT) subroutine developed at the finite-element software ABAQUS/Implicit. The results of 3-point bending and tensile tests accompanied by acoustic emission, and in-plane shear tests with digital image correlation analysis, are used to validate the proposed multiscale model. It is shown that damage progress and strain distribution under various loading conditions can be predicted successfully, thus a reasonable correlation between the model and collected experimental data is achieved.

Interpenetrating lattices with enhanced mechanical functionality

Metamaterials derive their unusual properties from their architected structure, which generally consists of a repeating unit cell designed to perform a particular function. However, existing metamaterials are, with few exceptions, physically continuous throughout their volume, and thus cannot take advantage of multi-body behavior or contact interactions. Here we introduce the concept of multi-body interpenetrating lattices, where two or more lattices interlace through the same volume without any direct connection to each other. This new design freedom allows us to create architected interpenetrating structures where energy transfer is controlled by surface interactions. As a result, multifunctional or composite-like responses can be achieved even with only a single print material. While the geometry defining interpenetrating lattices has been studied since the days of Euclid, additive manufacturing allows us to turn these mathematical concepts into physical objects with programmable interface-dominated properties. In this first study on interpenetrating lattices, we reveal remarkable mechanical properties including improved toughness, multi-stable/negative stiffness behavior, and electromechanical coupling.

Finite element model of laminate construction element with multi-phase microstructure

Abstract The article describes a method of creating a mesoscale finite element model of a fabric reinforced laminate that replicates the smallest repetitive fragment of its microstructure – RUC (Repetitive Unit Cell). The model takes into account the influence of the number and orientation of layers, the weave of the reinforcement fabric as well as manufacturing technology on the strength and stiffness of the laminate. The constants of the finite elements forming RUC (equivalent cross-sectional parameters, limit values of forces ensuring layer integrity) are determined experimentally by performing uncomplicated tests of specimens of a particular laminate. A special preprocessor was developed to generate the finite element model of the construction element from laminate, which automatically creates the so-called batch file defining the model. The usefulness of the preprocessor was checked by simulating a three-point bending test of a laminate door beam of a passenger car. The obtained calculation results were verified experimentally.

Ultimate Strength Prediction of Fibrous Composites only upon Independently Measured Constituent Properties

This study presents an accurate and easy-to-use micromechanical model to predict the ultimate strength of unidirectional polymer composites under an arbitrary load condition only upon independently measured constituent properties. In this model, the micromechanical method based on generalized model of cells (GMC), which effectively predict the nonlinear deformation of unidirectional composites, is used to analyze the repeating unit cells of composites. At the same time, a unified plastic theory (i.e. modified Ramaswamy-Stouffer model) is incorporated into the GMC's analytical framework to describe viscoplastic behaviors of matrix phase. Additionally, because of the stress concentration effect which causes the difference between the matrix in-situ and original strength behaviors, a stress concentration factor is introduced in order to utilize the measured constituent properties directly. The prediction results with SCF match better with the experimental results than the prediction results without SCF. In addition, the prediction results show that the presence of thermal residual stress and material plastic effects generally has important influence on the strength prediction of a composite.

Impact of damage on the propagation of Rayleigh waves in lattice materials

We analyze in this contribution the phase velocities of Rayleigh waves in periodic beam-lattices materials. The effective mechanical properties for the virgin and damaged structures are evaluated. The damaged lattice is modeled by removing beams within full networks made of repetitive unit cells. An evaluation of the phase velocities for the longitudinal and transverse versus the amount of damage is done for different relative densities evaluated versus the percentage of damaged beams for the square and triangular network. The effective mechanical properties of the overall network are evaluated as a function of the increasing damage based on a numerical procedure. Computations show that the square lattice has higher phase velocities in comparison with the triangular lattice. This work sets the basis of a methodology for evaluating the state of damage in network materials based on the changes in the wave propagation velocities.

Elastic response of hollow truss lattice micro-architectures

Elastic meta-materials, which we will treat herein as synonymous with elastic micro-architected materials, derive their unique combination of properties from the fine scale geometry of their solid and void regions. Recent advances in additive manufacturing have made the fabrication of such micro-architected materials feasible, and have driven interest in their properties. We consider the class of elastic meta-materials whose micro-architectures are composed of a network of straight struts connected at nodes arranged within a repeating unit cell; the octet truss is one well-known example of such truss lattices. Heretofore, the majority of work characterizing such truss lattices has considered solid struts and has been based on rod theory, a simplification of continuum elasticity which is accurate in the limit of low relative densities. In this paper, we use full continuum theory, solved with finite element analysis on adaptively refined meshes, to characterize the elastic performance of seven truss lattices with cubic geometric symmetry, and whose struts and nodes are hollow; we consider a wide range of strut wall thicknesses and truss relative densities. Varying the wall thickness provides an additional dimension of control, so that stiffness and anisotropy can, to an extent, be decoupled from the relative density of a given truss lattice. We show that a truss’ stiffness can be increased at fixed relative density, with bend-dominated truss lattices showing significantly greater improvement than stretch-dominated designs. Increases in stiffness by a factor of greater than five are observed, and increases by a factor of 100 or more are obtainable. Bend-dominated structures with hollow struts can be found that are stiffer than stretch-dominated ones at nearly all relative densities, contradicting a common rule of thumb. For some trusses, including the octet truss, choosing the wall thickness appropriately allows one to obtain isotropic response over a large range of volume fractions. We use topology optimization to find maximally stiff multiscale structures comprised of micro-architected materials and find that all the trusses we consider perform similarly, with anisotropic trusses resulting in slightly stiffer structures for a given load.

Quasiperiodic Dielectric Gratings for Multiband Fiber-To-Chip Couplers

Dielectric gratings that couple optical fibers and planar waveguide circuits are key for optical-to-electronic (electronic-to-optical) signal conversion, but their applicability to platforms that require broader bandwidths and higher capacity is limited by their single-wavelength response. Herein, we present the design of a quasi-periodic grating coupler with multiband fiber-to-waveguide (waveguide-to-fiber) coupling response, where the grating consists of a periodic repetition of unit cells made of alternating silicon and air grooves according to the Fibonacci sequence. Through finite-difference time-domain (FDTD) calculations, we show that this new device could be used for coupling multi-wavelength fiber modes in a single grating structure. The results were obtained for fibers operating in the wavelength range from 1000 nm to 2000 nm, but the concept can be readily extended to other frequency ranges. Moreover, the allowed modes in the grating are almost insensitive to fiber misalignments and small fabrication errors for high Fibonacci steps, which is useful when alignment of optical components is impractical. It is hoped that properly designed gratings overlapping multiple modes may lead to ultra-broadband fiber-waveguide couplers that can cope with the growing demand for higher capacity and bandwidth in optical communications.