Periodic Microstructure Research Articles

On Attractors of Ginzburg–Landau Equations in Domain with Locally Periodic Microstructure: Subcritical, Critical, and Supercritical Cases

On Attractors of Ginzburg–Landau Equations in Domain with Locally Periodic Microstructure: Subcritical, Critical, and Supercritical Cases

Side-polished photonic crystal fiber sensor with ultra-high figure of merit based on Bloch-like surface wave resonance

A Bloch surface wave (BSW) resonance configuration is introduced for biosensing with an ultra-high figure of merit (FOM). The BSW excitation is realized through the evanescent field of the core-guided fundamental mode of a side-polished photonic crystal fiber (PCF). By taking advantage of the air hole periodic microstructure of the PCF cladding, the BSW platform can be achieved with only a single high refractive index dielectric layer on its flat surface. The dielectric layer deposited on the polished surface of the PCF modifies the local effective refractive index, allowing direct manipulation of the BSWs, whereby the resonance wavelength of the surface wave can be adjusted by choosing the material and thickness of this layer. Here, we numerically investigate Bloch-like surface wave (BLSW) resonance conditions around telecom wavelengths for silicon, titanium dioxide, copper monoxide, and aluminum oxide termination layers. The BLSW excitation platform materials have low loss, which results in higher surface field enhancements and narrower resonances, which are advantageous properties for the sensors. The obtained results open new avenues for the application of optical surface waves in biosensing with high FOM. Furthermore, these results show a much higher figure of merit (FOM) than traditional approaches, allowing for increased sensitivity and accuracy.

Bidirectional periodic pore structure Si-C-N multiphase ceramic with high thermostability and excellent microwave absorption properties over a wide temperature range

High-temperature electromagnetic wave (EMW) absorption materials are urgently needed for high thrust-to-weight ratio aero-engine. A novel EMW absorbing Si-C-N ceramic at elevated temperatures was fabricated by chemical vapor infiltration (CVI) Si3N4 and SiC on 2D plain carbon fiber preform forming periodic framing and then oxidation to remove carbon fiber. The three-level gradient periodic microstructure was in-situ constructed in the Si-C-N ceramics to meet the impedance matching characteristics and effectively attenuate as much as EMWs. Compared with traditional EMW absorption materials, the Si-C-N ceramics with bidirectional periodic pore had a lower density of 1.2 g/cm3 and better frequency dispersion behavior which could obtain an effective absorption bandwidth (EAB) covering the whole X-band at 2.45 mm and the lowest reflection coefficient (RCmin) of − 40 dB at 3.5 mm. The optimal Si-C-N ceramics (S2) show excellent EMW absorption performance over a wide temperature range. The Si-C-N ceramic had a promising applications at high temperatures.

Inverse design of mechanical springs with tailored nonlinear elastic response utilizing internal contact

This work employs a contact-aware topology optimization approach for the design of nonlinear springs with a broad range of prescribed load–displacement responses in both tension and compression. By leveraging the third medium contact approach to model internal contact, this method enables the utilization of collision between parts to achieve the desired load–displacement response while minimizing material consumption. The effectiveness of the proposed computational design approach is demonstrated using axially loaded springs as a benchmark, but the proposed method is also applicable to more general cases, including the design of periodic nonlinear material microstructures and metamaterials.

Process applicability and hole wall microstructure evolution mechanism of femtosecond laser spiral drilling

In this work, a 1.5 mm thick GH3044 nickel-based superalloy is processed by femtosecond laser to obtain a micro-hole for the fuel nozzle of an aerospace engine with a large thrust-to-weight ratio. To address the issue of sidewall adhesion during micro-hole machining, a spiral drilling method implemented with a rotating wedge scanning module is used. Combined with a two-step drilling strategy, it achieves flexible removal of adhesion while ensuring machining efficiency and further improving the quality of drilling. The effect of different drilling strategies on the sidewall roughness of micro-holes is investigated. After process optimization, the roughness of the the micro-hole sidewalls is reduced to below Rq 0.4 μm, which is approximately 8% compared to the traditional one-step drilling strategy. Based on the EBSD analysis results, it is observed that the microstructure of the micro-hole sidewalls remains largely unchanged regardless of the drilling strategy employed. The analysis of the micro-hole sidewalls shows the presence of periodic microstructures. When aiming for low roughness, the geometric characteristics of these periodic microstructures can be adjusted by manipulating the laser's single pulse energy. These experimental findings present a new perspective for optimizing the femtosecond laser drilling process and provide theoretical support for future enhancements in the actual fuel injection effect.

Efficient thermomechanically coupled FE‐FFT‐based multiscale simulation of polycrystals

AbstractIn general, the overall macroscopic material behavior of any structural component is directly dependent on its underlying microstructure. For metal components, the associated microstructure is given in terms of a polycrystal. To enable the simulation of the related microstructural and overall elasto‐viscoplastic material behavior, a two‐scale simulation approach can be used. In this context, we use a FE‐FFT‐based two‐scale method, which is an efficient alternative to the classical FE2 method for the simulation of periodic microstructures. In addition, we consider a thermomechanically coupled framework to account for both thermal and mechanical loads. Finally, we incorporate a model order reduction technique based on a coarsely discretized microstructure to develop an efficient two‐scale simulation technique. As a demonstration of the feasibility of the proposed simulation framework, a numerical example will be investigated.

FFT-based multiscale scheme for homogenisation of heterogeneous plates including damage and failure

This paper introduces a new approach that combines FFT-based multiscale homogenisation with the Lippman–Schwinger equation to efficiently and accurately analyse plate structures with periodic micro-structures. For the first time, the plate periodic Lippman–Schwinger equation is derived to address the auxiliary cell problem. The proposed method comprises two key components: (i) a 3D plate formulation using a novel finite prism approach, and (ii) solving the Lippman–Schwinger equations by means of Green’s functions. Solutions for both the classical and first-order plate theories are derived and implemented for linear and nonlinear problems, completed with an open-source code provided.The efficiency and accuracy of the proposed method are assessed through several case studies, including complex woven composites. It is demonstrated that proposed method achieves a comparable level of accuracy to 3D solid problems while significantly reducing the required computing effort (i.e., milliseconds in computing time as opposed to hours required for 3D solid models). Moreover, nonlinear progressive damage problem involving an actual plate woven composite model is investigated. The results obtained show good agreement with experimental measurements for the progressive damage in plain woven composite, further highlighting the effectiveness of the proposed method.

Multiscale topological design of coated structures with layer-wise bi-material lattice infill for minimum dynamic compliance

In this paper, a novel multiscale topological design strategy for coated structures infilled with layer-wise bi-material lattice microstructures is presented, where the objective is to minimize the dynamic compliance of the macrostructure under time-harmonic excitation frequency. The optimization of bi-material microstructures is concurrently performed with the topological design of the macrostructure to fully consider the interaction of structural evolution between the two scales. At the macroscale, the material distribution for the macrostructure is optimized using the velocity field-based level set method, which inherits the implicit geometrical representation, and smooth structural boundaries can be obtained. Due to the signed distance property, the thickness of the outer coating is able to be well-maintained during the iteration. As for the microscale, the density method is chosen to conduct topology optimization of lattice microstructures. The lattice infill for coated structures is designed to be layer-wise by dividing it uniformly (or near-uniformly) into different layers, and each layer is composed of periodic and uniform bi-material microstructures, which can enlarge the design space and generate various microscale structural patterns to better resist the vibration response. The sensitivity analysis for the dynamic compliance with respect to the design variables at different scales is carefully performed and the influences of the number of divided layers, excitation frequency, coating thickness, damping coefficients, element meshes, static constraint, and division direction for the lattice infill on the optimized macroscale and microscale structures are studied by several numerical examples. Moreover, a body-fitted mesh for a Computer-Aided Design (CAD) model is constructed and the finite element analysis results further verify the effectiveness of the proposed approach.

Advanced Fabrication of Mechanical Metamaterials Based on Micro/Nanoscale Technology

In recent years, mechanical metamaterials have shown good mechanical properties such as ultralightness, programmable stiffness, and tunable Poisson's ratio through clever structural design of microstructures. Micro/nanotechnology has been used to ensure mechanical metamaterials’ fabrication quality and performance for use as structural substrates in multifunctional applications. Mechanical metamaterials are intentionally engineered with periodic microstructures in diverse shapes, allowing for the creation of complex structural substrates. However, this sophisticated design poses significant challenges in the fabrication process, particularly at the micro/nanoscale level, where current technology hinders mechanical metamaterials’ performance improvement and function realization. This review explores the fabrication processes of mechanical metamaterials using micro/nanotechnology and showcase recent progress and future trends. Particular attention is given to recent development, challenges, and future trends in advanced fabrication technologies for mechanical metamaterials. This survey aims to explain why mechanical metamaterials have significant benefits and are well‐suited as structural substrates for multifunctional applications, what advanced fabrication technologies at the micro/nanoscale have been introduced to prepare mechanical metamaterials and how to overcome manufacturing technology bottlenecks of mechanical metamaterials in terms of synthesis, materials, and design method.

High frequency multi-field continualization scheme for layered magneto-electro-elastic materials

Magneto-electro-elastic (MEE) heterogeneous materials with periodic microstructure are investigated, with special focus on the particular case of a layered MEE material. By exploiting the transfer matrix method in combination with the Floquet–Bloch boundary conditions, the frequency dispersion spectrum can be derived by solving an eigenproblem, involving a 12 × 12 transfer matrix with palindromic characteristic polynomial. In particular, by assuming the cubic symmetry of layers, the problems involving longitudinal and transverse elastic waves are uncoupled. This circumstance has the advantage of simplifying the treatment and the size of the subproblems to be solved, corresponding to a 4 × 4 transfer matrix for the longitudinal case, and a 8 × 8 transfer matrix of the transverse problem, both characterized by palindromic characteristic polynomials. Afterward, a dynamic multi-field continualization approach is proposed to investigate wave propagation in MEE layered periodic material by matching the Z-transform of the vector collecting the nodal fields to the two-sided Laplace transform of the same vector at the macroscale. The continualization technique allows identifying multi-field integral-type non-local continua as well as multi-field generalized gradient-type (higher-order) non-local continua. Finally, to verify the accuracy of the proposed approach, the dispersion curves derived from the continualization technique are compared with the curves provided by the Floquet–Bloch theory.

An eigenstrain-based micromechanical model for homogenization of elastic multiphase/multilayer composites

This paper presents a novel micromechanical procedure for the linear elastic homogenization of composites with periodic microstructures. The procedure is developed for composites with arbitrary number of phases and geometric shapes of the inhomogeneities, in contrast with most existing homogenization approaches. Also, no restriction is made in relation to the mismatch between the properties of the phases and volume fractions of the inhomogeneities. The proposed procedure is based on the Eshelby equivalent inclusion approach and extends a model originally derived for evaluating the effective elastic moduli of periodic two-phase composites. The procedure represents the fluctuating elastic fields within each multiphase repeating unit cell (RUC) using Fourier series, resulting in Lippmann-Schwinger integral equations governing the unknown eigenstrain fields of the inclusions. Unlike traditional iterative algorithms used in Fast Fourier Transform (FFT)-based approaches, the procedure solves the integral equations straightforwardly from a scheme of partition of the domain of each inclusion. The efficiency of the proposal procedure is demonstrated through applications to composites with different arrays of coated fibers and constituent materials.

A normalization method of measured elastic properties of glass NCF composites with respect to fiber volume fraction based on periodic microstructure micromechanics

International standards have requested the normalization of fiber-dominated properties in composites with respect to fiber volume fraction. However, the normalization of matrix-dominated properties has not yet been requested because there is no clear model. This partial normalization might deteriorate accuracy in design and analysis. This study developed a systematic approach for normalizing both fiber-dominated and matrix-dominated properties of composites, based on periodic microstructure micromechanics (PMM). To evaluate the accuracy of PMM, this study started from comparing the elastic properties calculated by PMM to those calculated by FEA. Then, the eight measured elastic properties were obtained by testing three different types of glass NCFs/epoxy laminates. Since PMM needs the elastic properties of raw materials, the three unknown properties among them were determined by using the back-calculation procedure developed in this study. After all, the PMM gave us highly accurate properties of the laminates, resulting in a maximum error of less than 1% when comparing the calculated elastic properties with the measured ones. Lastly, the orthotropic elastic properties of the laminates were normalized with respect to fiber volume fraction. This study makes a significant step forward in composite structures by providing a systematic approach for normalizing full elastic properties in composites.

Reduced Order Modeling of Dynamic Mechanical Metamaterials for Analysis of Infinite and Finite Systems

Abstract Dynamic mechanical metamaterials (MMs) are artificial media composed of periodic micro-structures, designed to manipulate wave propagation. Modeling and designing MMs can be computationally demanding due to the broad design space spanned by the geometric and material parameters. This work aims to develop a generalized reduced order modeling approach for determining MM dynamics in low frequency ranges with accuracy and speed, using a limited number of parameters and small matrices. The MM unit cells are treated as assemblies of structural elements with discrete degrees-of-freedom, whose effective stiffness and inertia are determined by optimizing energy criteria based on continuum results derived from a small number of eigen-study simulations. This proposed approach offers a parameterized and discretized representation of MM systems, which leads to fast and accurate computation of eigen-study results for periodic arrays, as well as dynamic responses in time domain for finite-sized arrays. The high computational efficiency and physical accuracy of this method will help streamline the modeling process and aid in design discovery and optimization, especially in combination with machine learning and data-driven techniques.

Homogenisation of Nonlinear Heterogeneous Thin Plate When the Plate Thickness and In-Plane Heterogeneities are of the Same Order of Magnitude

Summary In this work, we propose a new two-scale finite-strain thin plate theory for highly heterogeneous plates described by a repetitive periodic microstructure. For this type of theory, two scales exist, the macroscopic one is linked to the entire plate and the microscopic one is linked to the size of the heterogeneity. We consider the case when the plate thickness is comparable to in-plane heterogeneities. We assume that the nonlinear macroscopic part of the model is of Kirchhoff–Love type. We obtain the nonlinear homogenised model by performing simultaneously both the homogenisation and the reduction of the initial three-dimensional plate problem to a two-dimensional one. Since nonlinear equations are difficult to solve, we linearise this homogenised Kirchhoff–Love plate theory. Finally, we discuss the treatment of edge effects in the vicinity of the lateral boundary of the plate.

Enhancement of photothermal effect using a hierarchical plasmonic structure

The photothermal effect exhibited by plasmonic structures has garnered considerable attention because of its versatile application prospects. However, the conversion efficiency of broadband light sources and large area fabrication still pose challenges. Herein, a hierarchical plasmonic structure was proposed as a photothermal substrate, which integrated a two-dimensional periodic microstructure with a disordered nanostructure. The microstructure and nanostructure were obtained through litho-etching and metal-assisted chemical etching technologies, respectively. Titanium nitride was deposited on the hierarchical structure as the mediator between the irradiation and generation of heat. The metallic nanostructure on the hierarchical structure played a crucial role in generating the thermoplasmonic effect. Moreover, the type (vertical and tapered shapes) of the microstructure and morphology (porous texture and deep trench) of the nanostructure were also altered to investigate the photothermal effect. The addition of deeper nanotrenches on the tapered microstructure exhibited a synergistic effect on the photothermal efficiency. The hierarchical plasmonic structures were experimentally demonstrated to exhibit a 3.6 times greater photothermal efficiency (change in temperature) compared to a titanium nitride film. These engineered hierarchical structures have remarkable potential for use as non-classical photothermal devices.

Machine learning aided multiscale magnetostatics

Computational material modeling using advanced numerical techniques speeds up the design process and reduces the costs of developing new engineering products. In the field of multiscale modeling, huge computation efforts are expected for modeling heterogeneous materials while trying to reach high accuracy levels. In this work, a machine learning approach, namely the convolutional neural network (CNN), is developed as a solution providing a high level of accuracy while being computationally efficient. The input for the CNN model consists of two/three-dimensional images of artificial periodic and biphasic microstructures in the form of nonoverlapping and overlapping, mono- and polydisperse circular/spherical inclusion systems, which are generated by a random sequential inhibition process. These correspond to Statistical Volume Elements (SVE). Considering linear magnetostatics at the microscale, the output is the apparent permeability of the SVE. Training and testing data for the apparent properties is produced with finite element method-based two-scale asymptotic homogenization. The model efficiency is revealed by employing some representative examples in two and three-dimensional settings. In this regard, the performance of the CNN model is assessed with the applied computational homogenization method relating to the accuracy and computational efficiency. The results with the CNN model show high accuracy in predicting the homogenized permeability and a significant decrease in computation time.

A comparative study of the fatigue behavior of Ti‐6Al‐4V circular lattices using numerical and experimental methods

AbstractLattice materials are a subset of cellular solids with a periodic microstructure and are of interest in a number of applications because of their potential weight‐saving properties. In this study, circular lattices are machined out of plates made of Ti‐6Al‐4V and fatigue tested to generate an S–N curve. These results are compared to a FEM with fatigue data based on MIL‐HDBK‐5H. A good agreement with the FEM was found, suggesting that the fatigue life of lattices can be modeled without specifically detailed fatigue data. The model accurately predicted the point of failure in most of the lattice specimens and was within 4% of the scatter band. Fractographic analysis showed significant shear features in the fracture surfaces, indicating a rapid shear fracture is likely the dominant mode of failure. Finally, the relationship between equivalent stress and relative density was determined using a variety of optimization schemes.

A novel homogenization method for periodic piezoelectric composites via diffused material interface

In this article, we propose a novel homogenization method for piezoelectric composites with periodic microstructures. In the proposed homogenization method, one can find the analytical expressions for the material properties of the composites in the representative volume element (RVE) via diffused material interface method. The availability of analytical expressions for the material properties in the entire domain of the RVE enables one to determine the effective homogenized material properties with the standard finite element method. As the proposed homogenization method regularizes the discontinuity in material properties across the interfaces, there is no need to explicitly track the material interfaces while implementing the finite element method for determining the effective material properties. Hence, one can implement the proposed homogenized method for any piezoelectric composites with complicated material interfaces and determine the effective material properties. In this study, we have considered the piezoelectric composites to be comprised of periodic microstructures where the RVE constitutes a matrix with an inclusion of a specific shape. We have carried out a study on the effect of the shape of inclusion viz. square-shaped, I-shaped, T-shaped, and plus-shaped, and the size of inclusion on the effective piezoelectric material properties. We have also studied the influence of shape and size of inclusion on the electro-mechanical response of a homogenized piezoelectric continuum. To implement the proposed homogenization method, we have used Gridap, an open-source finite element toolbox in Julia that provides very compact codes freely available to all the researchers and makes a third-party verification of the proposed method straightforward.

On optimization of heterogeneous materials for enhanced resistance to bulk fracture

We propose a novel approach to optimize the design of heterogeneous materials, with the goal of enhancing their effective fracture toughness under mode-I loading. The method employs a Gaussian processes-based Bayesian optimization framework to determine the optimal shapes and locations of stiff elliptical inclusions within a periodic microstructure in two dimensions. To model crack propagation, the phase-field fracture method with an efficient interior-point monolithic solver and adaptive mesh refinement, is used. To account for the high sensitivity of fracture properties to initial crack location with respect to heterogeneities, we consider multiple cases of initial crack and optimize the material for the worst-case scenario. We also impose a minimum clearance constraint between the inclusions to ensure design feasibility. Numerical experiments demonstrate that the method significantly improves the fracture toughness of the material compared to the homogeneous case.

Soft elastic composites: Microstructure evolution, instabilities and relaxed response by domain formation

This work provides a review of several complementary analytical methods for characterizing the macroscopic response of ‘soft’ elastic composites subjected to large deformations. For this purpose, we first recall the variational definitions of the homogenized response for hyperelastic composites with periodic and random microstructures. We distinguish between the principal solution for the ‘mesoscopic’ stored-energy function, which is based on one-cell-periodic solutions for the periodic microstructures and on the macroscopically uniform solution for the random microstructures, and the ‘relaxed’ or ‘macroscopic’ homogenized stored-energy function, which is obtained after the onset of ‘microscopic’ (periodic on multiple cells) or ‘macroscopic’ (long wavelength) instabilities. The various techniques include variational bounds of the Voigt, and generalized Reuss and Hashin–Shtrikman types, variational estimates based on the use of suitably optimized ‘linear comparison composites,’ and bounds based on the relaxation of the principal solution for the mesoscopic energy by means of a generalized Maxwell procedure consisting in the quasiconvexification of the energy and leading to domain formation. Although the methods are completely general, for simplicity, we review here illustrative results for neo-Hookean laminates and fiber-reinforced elastomers subjected to plane strain loading conditions. Among the main results, it was found that the predicted formation of twins or lamellar domains and associated soft modes of deformation for the laminates is consistent with experimental observations for thermoplastic elastomers with highly ordered lamellar microstructures. In addition, it was found that the predictions for the macroscopic instabilities by means of the relaxation of the mesoscopic energy are generally different (more conservative) and more consistent with the variational definitions of the macroscopic homogenized energy than those based on loss of ellipticity of the incremental elasticity homogenization problem.