Unit Cell Of Microstructure Research Articles

A Pragmatic Design of Elastic Metamaterial with Extreme Anisotropic Stiffness

Traditional elastic metamaterials encounter difficulties when trying to independently adjust the operating frequency in two perpendicular directions. In this research, we present a practical active elastic metamaterial with the capability to achieve extreme stiffness anisotropy. Our approach involves incorporating piezoelectric patches, coupled with adjustable conductance, into the microstructure unit cell to control the effective elastic stiffness in both primary directions. We conduct simulations to manipulate wave propagation within such metamaterials and conduct experimental investigations to validate the design. The obtained results closely align with mathematical analysis and numerical predictions, demonstrating the effectiveness of our proposed metamaterial. This novel approach offers a versatile and innovative design methodology for elastic metamaterials.

Vibration Bandgap Analysis of Periodic Composite Microplates using Isogeometric Approach

For periodic composite microplates, the microstructure dependent effect has great influence on the elastic wave vibration band gaps. In this paper, an effective isogeometric analysis method based on the modified couple stress theory (MCST) is proposed to calculate the vibration band gaps in periodic composite microplates. The MCST is employed to capture the microstructure effect of vibration band gaps in periodic composite microplates. The high order continuity requirement of MCST can be easily satisfied by applying the NURBS based isogeometric analysis approach. The elasto-dynamics and discrete equations of isogeometric analysis are obtained under the Mindlin kinematics assumptions. The present approach is validated by comparison with the analytical solution. And the influence of microstructure effect, volume fraction and unit cell length on the band gaps are investigated in details.

Optimization and design on multi-layers of dielectric metasurface as broadband terahertz quarter wave plate

Planar metasurface-based quarter wave plates (QWPs) have significant advantages over conventional devices in terms of compactness, flexibility, and simplicity of manufacture; however, they offer a relatively narrow operational bandwidth. A broadband terahertz achromatic QWP is realized in the 0.68–1.48 THz spectral region in this work, which consists of several separate metasurface layers of dielectric elliptic pillars stacked together with various rotation angles. Meanwhile, an improved simulated annealing method is proposed, which introduces the evolutionary strategies to optimize the distinct fundamental microstructural unit cells, and the running speed is greatly increased. Furthermore, the proposed multi-layers metasurface may pave the way for arbitrary polarization control of incident waves and be ideally suited for application by virtue of subwavelength thickness in the other frequency bands as well.

A multi-scale framework to predict damage initiation at martensite/ferrite interface

Martensite/ferrite (M/F) interface damage largely controls failure of dual-phase (DP) steels. In order to predict the failure and assess the ductility of DP steels, accurate models for the M/F interfacial zones are needed. Several M/F interface models have been proposed in the literature, which however do not incorporate the underlying microphysics. It has been recently suggested that (lath) martensite substructure boundary sliding dominates the M/F interface damage initiation and therefore should be taken into account. Considering the computationally infeasibility of direct numerical simulations of statistically representative DP steel microstructures, while explicitly resolving the interface microstructures and the sliding activity, a novel multi-scale approach is developed in this work. Two scales are considered: the DP steel mesostructure consisting of multiple lath martensite islands embedded in a ferrite matrix, and the microscopic M/F interfacial zone unit cell resolving the martensite substructure. Based on the emerging microscopic damage initiation pattern, an effective indicator for the M/F interface damage initiation is determined from the interface microstructural unit cell response, along with the effective sliding in this unit cell. Relating these two effective quantities for different interface microstructural configurations leads to an effective mesoscale model relating the interface damage indicator to the sliding activity of the martensite island in terms of the mesoscopic kinematics. This microphysics-based M/F interface damage indicator model, which could not be envisioned a-priori, is fully identified from a set of interfacial unit cell simulations, thus enabling the efficient prediction of interface damage initiation at the mesoscale. The capability of the developed effective model to predict the mesoscopic M/F interface damage initiation is demonstrated on an example of a realistic DP steel mesostructure.

Large-Scale Synthesis of Metal Additively-Manufactured Microstructures Using Markov Random Fields

A data-driven framework is developed and examined for creating spatially-varying crystallographic textures over component-scale Computer-Aided Design (CAD) models. Here, a set of three orthogonal 2D micrographs of an Additively-Manufactured (AM) specimen are first obtained experimentally through Electron Backscatter Diffraction (EBSD) and subsequently converted to a 3D representative unit cell using the Markov Random Field (MRF) technique. Features such as grain size, crystallographic orientation, and grain boundary misorientation distributions are used to validate the reconstructed 3D microstructure against input experimental EBSD images. The variations of microstructural features during a powder-based additive manufacturing process are subsequently modeled by merging patches from the 3D snapshot of AM microstructural unit cell in a part-scale geometry using a tensor-based optimization process. The optimization algorithm repeatedly pastes microstructural elements from the reconstructed MRF unit cell onto the geometrical CAD domain until it is entirely covered. Here, through a simple Graphical User Interface (GUI), the user specifies a tensor field over the volumetric CAD model, defining the local control over grain-scale, anisotropy, and crystal growth orientation. This new approach provides a workflow for reconstructing global maps of AM microstructures in real-time by embedding site-specific images based on known AM microstructural patterns seen in experimental characterization techniques. The numerical results are helpful specifically for the visualization of process–microstructure relationships in metal additive manufacturing techniques.

Numerical simulation for deformation of laminates combining the novel shell element with the decoupled two-scale viscoelastic analysis of FRP

This study proposes a numerical simulation for the deformation of laminates incorporating a combination of the novel shell element, whose thickness is allowed to change in a macro-scale model, with a decoupled two-scale viscoelastic analysis of FRP. The dependence of resin properties on the degree of cure (DOC) is considered in the simulation. Because the shell element of interest is enriched with degrees of freedom (DOFs) to represent the transverse deformations, it is capable of evaluating and controlling the thickness change during curing process of laminates. Since the additional DOFs are introduced to each element independently, they are condensed out at the element level in assembling the global finite element (FE) equation. Besides the force, the displacement can also be imposed on the outermost DOFs to control the change in thickness. Thus, numerical simulations where the plate thickness is controlled according to the requirements for molded products can be realized without introducing solid-shell-type formulations. The macroscopic mechanical behavior of FRP can be represented by the orthotropic version of the model employed for resin whose DOC-dependent macroscopic viscoelastic properties (the macroscopic coefficient of thermal expansions (CTEs) and coefficient of cure shrinkages (CCSs)) are identified from the relaxation curves obtained by results of numerical material tests (NMTs) conducted on the periodic microstructure (unit cell). The usefulness of the proposed approach was clarified by the results of the numerical verifications.

Design of metasurface beam splitter based on polarization characteristics of incident wave

A transmissive metasurface polarization beam splitter (MPBS) with three-layer dielectric substrate was designed to meet the application requirements of polarization beam splitting devices. The microstructure unit cell was composed of an elliptical patch and a square resonant ring, and the phase and amplitude responses to the x- and y-polarized incident waves were independent. Here 16 transmissive unit cells designed with a 90° phase difference and more than 0.8 transmission coefficient could be used to realize different functions by pre-designing distribution of “00”, “01”, “10”, “11” encoding particles in x- and y-direction. We experimentally demonstrated a 2-bit encoding MPBS that separated x- and y-polarization incident waves into two different directions at 15 GHz respectively using specially designed encoding unit cells. The work may alleviate the shortage of advanced multifunctional devices and provide important application prospect in imaging, MIMO system, etc.

A Novel Twofold Symmetry Architected Metamaterials with High Compressibility and Negative Poisson's Ratio

This study presents the compression response of additively manufactured novel soft porous structures with architected microstructure. Six porous additively manufactured architected periodic structures with two‐fold and four‐fold symmetry were considered. The effect of pore shape and fold symmetry of microstructure on the non‐linear response of a square array of architected pores in a soft polymeric matrix is experimentally investigated. The digital image correlation (DIC) is used for investigating the evolution of strains and deformation during uniaxial tensile tests and compression tests of porous structures. Compression induced instability lead to negative Poisson's ratio, and compaction of porous structures, which is found to depend not only on the shape of the architecture but also the fold symmetry exists in the microstructure's unit cell. Unique architectures with multiple buckling modes and shape transformation are also observed. Two‐fold symmetry structures are found to buckle at lower strains compared to the four‐fold symmetric structure at the same porosity level and produced high compaction and negative Poisson's ratio. The results showed that in addition to pore shape, the fold symmetry could be used effectively to design a new class of soft, active, and reconfigurable devices over a wide range of length scales with desired characteristics.

Continuum representation of nonlinear three-dimensional periodic truss networks by on-the-fly homogenization

Thanks to scale-bridging fabrication techniques, truss-based metamaterials have gained both popularity and complexity, ultimately resulting in structural networks whose description based on classical discrete numerical calculations becomes intractable. We here present a framework for the efficient and accurate simulation of large periodic three-dimensional (3D) truss networks undergoing nonlinear deformation (accounting for linear elastic beams undergoing finite rotations). Although the focus is on elastic beams, the method is sufficiently general to extend to inelastic material behavior. Our approach is based on a continuum representation of the truss (and its numerical implementation via finite elements) whose constitutive behavior is obtained from on-the-fly periodic homogenization at the microstructural unit cell level. We pursue a semi-analytical strategy (previously reported only in two dimensions) which admits the analytical calculation of consistent tangents for convergent implicit solution schemes; the extension to 3D – through the addition of torsional deformation modes and the handling of 3D rotations – results in a powerful tool for the prediction of the complex mechanical response of large structural networks. We validate the small-strain response by comparison to analytical solutions, followed by finite-strain benchmarks that compare simulation results to those of fully-resolved discrete calculations. The homogenization of beam unit cells results in a regularized macroscale model with an intrinsic length scale, which manifests especially when modeling bifurcations or localization. We finally apply our approach to macroscopic boundary value problems involving complex-shaped truss metamaterials (with truss unit cells near the body's boundary mapped onto a conformal surface), which reveal only an insignificant effect of boundary layers on the overall mechanical response, again supporting the applicability of our homogenization approach.

Mathematical Model for Analyzing Heat Transfer Characteristics of Ablative Thermal Insulating Material

A mathematical model based on minimal thermal resistance and equal law of specific equivalent thermal conductivity is developed to discuss the heat transfer characteristics of ablative thermal insulating material from the mesoscopic scale. Based on the statistical results of mesoscopic parameters, the microstructure unit cell model was established to analyze the influence rule of mesoscopic parameterization which includes the size, distribution, and positional relation of microsphere and fiber. The results show that the equivalent thermal conductivity decreases with the density, size, distribution area, and distance of microsphere and the space distance and volume fraction of fiber decreasing. Besides, the equivalent thermal conductivity will become larger when more quality of heat transfers along the fiber direction. Exploring the relationship between the macroscopic heat transfer process and the microstructure is meaningful for exploring the heat transfer behavior of thermal insulating material and improvement of the processing technology.

Two-scale concurrent topology optimization of lattice structures with connectable microstructures

Two-scale concurrent design of lattice material microstructures and their macroscale distributions can effectively enlarge the design space, and thus achieve lightweight structures with desirable mechanical and multi-physics performances. Most of the existing inverse homogenization-based design methods do not take into consideration the connectivity issues of the microstructures. In practice, this may hinder the manufacturing and application of the optimized two-scale designs. To handle this issue, the present paper proposes a designable connective region method to obtain connectable microstructures in the context of two-scale concurrent structural topology optimization, considering structures composed of lattice materials with repetitive unit cells and prescribed porosity. On the microscale, the microstructures topologies are represented by the density model, and the effective material properties are computed with the homogenization method. On the macroscale, the distribution of different lattice materials is described with the discrete material optimization method, which can effectively reduce the amounts of macroscale elements with intermediate densities. The connectivity between any two types of the microstructures is naturally ensured by introducing pre-defined connective regions in the microstructural unit cells and keeping these regions sharing the same topology. This method can be conveniently implemented through microscale design variable linking and requires no evaluation of extra connectivity constraints and the corresponding sensitivities. It is exemplified by two-scale concurrent structural topology optimization for compliance minimization problems in two- and three-dimensional design domains. Numerical results show that this method is able to generate connectable lattice structures, which exhibits improved stiffness as compared with their uniform-lattice counterparts.

A design framework for gradually stiffer mechanical metamaterial induced by negative Poisson's ratio property

Gradually stiffer (GS) mechanical metamaterial is a novel kind of adaptive structures under nonlinear varying loads for cushioning and vibration damping in engineering. These man-made metamaterials in applications are usually tailored by trial-and-error. This study develops a design framework that integrates the topology optimization, parametric design and compression experiment for the GS mechanical metamaterials induced by the negative Poisson's ratio (NPR) property. Firstly, a parametric level set method is incorporated with the numerical homogenization method to topologically optimize a series of microstructural unit cells with different NPRs. Then, the nonlinear static analysis is respectively applied to different unit cells to check their GS behavior. Secondly, typical microstructures with GS property are selected and simplified to perform the subsequent parametric design to analyze the expected property. Thirdly, the obtained mechanical metamaterials are fabricated by using the additive manufacturing (AM). GS properties of the obtained designs are analyzed and verified by both the simulation and compression experiments. Results show that we could obtain the GS mechanical metamaterial based on NPR properties for their similar characteristics. This study demonstrates that the proposed design framework is effective to tailor innovative layouts of the GS mechanical metamaterials.

Topology optimization of magnetic composite microstructures for electropermanent magnet

This paper presents topology optimization for the design of magnetic composite applicable to electropermanent magnet. Here, the magnetic composite is built from a periodic microstructure consisting of air, iron and permanent magnet (PM) materials. The combination of non-magnetic, soft and hard magnetic materials in a microscopic scale enables to produce its own persistent magnetic field like PM material, and also enables the control of the magnetic field by an external current like iron material. This work aims to find the optimal microstructure unit cell layout of the electropermanent magnet, and estimate its cross-property bounds. Here, the cross-property bounds connect the effective magnetic permeability and residual flux density, which are calculated using the asymptotic homogenization method. The design objectives (i.e. desired effective properties) are theoretically studied with consideration of application to electromechanical devices. Then, an multi-objective optimization problem to achieve desired effective properties is formulated and solved with a multi-material gradient-based topology optimization formulation. As a result, the optimal composite unit cell layouts that constitute the Pareto fronts are successfully obtained. From the Pareto fronts, cross-property bounds of the electropermanent magnet are numerically constructed and discussed.

Decoupled two-scale viscoelastic analysis of FRP in consideration of dependence of resin properties on degree of cure

In consideration of the dependence of resin properties on the degree of cure (DOC), a method of decoupled two-scale analysis of unidirectional fiber reinforced plastics (FRP) is proposed to simulate the curing process subjected to the temperature history that may cause residual stress and deformation at the macro-scale. Within the framework of computational homogenization, a series of numerical material tests (NMTs) are conducted on the periodic microstructure (unit cell), which is composed of fibers and resin, for various cure states to characterize the macroscopic material behavior and its dependency on DOC. The resin’s material behavior at the micro-scale is assumed to be viscoelastic based on the generalized Maxwell model whose properties depend on DOC, and be accompanied with thermal expansion/contraction and cure shrinkage. On the assumption that the macroscopic mechanical behavior of FRP can be represented by the orthotropic version of the model employed for the resin, its DOC-dependent macroscopic viscoelastic properties as well as the macroscopic coefficient of thermal expansions (CTEs) and coefficient of cure shrinkages (CCSs) are identified from the relaxation curves obtained by the NMT results. In addition to this key ingredient, the noteworthy contribution in this study is the introduction of a new configuration of the rheology elements to formulate the macroscopic CTEs and CCSs in both the equilibrium and non-equilibrium parts of the generalized Maxwell model. After some verification analyses for these DOC-dependent material properties, we present a macroscopic analysis followed by a localization analysis for estimating the macro- and microscopic residual deformations and stresses of a laminate subject to curing to exemplify the successful decoupling for two-scale computational homogenization.

Extending the Ramberg–Osgood Relationship to Account for Metal Porosity

This article proposes an extended Ramberg–Osgood (R–O) relationship accounting for the effect of porosity on porous metal behavior under tension. Initially, microstructural unit cells satisfying continuity boundary conditions were employed to account for pore intensity and its effect on elastic-plastic behavior. Results obtained from micromechanical simulations coupled with regression analyses were utilized to express elastic-plastic behavior as a function of porosity. Therefore, mathematical relationships were successfully developed to extend the R–O model to account for porosity. Finally, for validation and assessment of the developed relation, analytical and macromechanical finite element (FE) results were compared with those of testing. Comparisons at low-porosity range, i.e., < 10 pct, proved to have excellent agreement. It is concluded that multiscale FE analyses conducted successively on the micro- and macro-scales efficiently delineated the effect of porosity on mechanical behavior. Moreover, these analyses enabled extending the R–O relationship for accurate modeling of porous metals with low-range porosity.

Topology optimization for concurrent design of layer-wise graded lattice materials and structures

Mechanical properties of hierarchical lattice structures depend not only on their overall shapes and topologies, but also on their microstructural configurations. This paper proposes a new method for concurrent topology optimization of structures composed of layer-wise graded lattice microstructures. Both macroscale design variables representing the distribution of different lattice materials and microscale design variables defining the topologies of the microstructural unit cells are to be simultaneously optimized. This formulation thus integrates the microstructure design into the structural design, instead of pursuing a grey macroscale design and then interpreting the intermediate densities into certain microstructures. The proposed method also enlarges the design space by allowing for graded microstructures. Two new design constraints, namely the structural coverage constraint and the average porosity constraint, are introduced into the proposed optimization formulation to reduce the complexity of the constraints in the layer-wise graded design. The macroscale and microscale designs are linked by using the Asymptotic homogenization method to compute the effective elastic properties of the microstructured materials. Numerical examples show validity of the proposed method. It is also found that layer-wise graded lattice structures outperform those with uniform lattice microstructures in terms of structural stiffness. Finite element simulations of constructed models of the optimized designs suggest that graded lattice structures exhibit higher buckling resistance and ultimate load bearing capacity than single-scale solid material structures or uniform lattice structures under the same material usage.

Topology optimization of material microstructures using energy-based homogenization method under specified initial material layout

Due to multiple local optimums in the society of periodic material microstructure deign using topology optimization, it is widely recognized that the initial microstructure configuration can significantly influence the optimal design. This study investigates the effect of different initial material layouts of microstructure unit cell upon the optimal design, and proposes a microstructure initialization scheme to alleviate the dependency of the optimal design on its initial design. The effective elastic properties of microstructures are evaluated by the efficient energy-based homogenization method. Topology optimization models are formulated to seek the best microstructures with the extreme properties or desired properties under the prescribed volume constraints. Several typical numerical examples are presented to demonstrate the effectiveness of the proposed scheme. Numerical results show that the proposed scheme can reduce the sensitivity of the optimal design upon optimization parameters and be beneficial to achieve a stable and rapid convergence.

Impact of micro-scale residual stress on in-situ tensile testing of ductile cast iron: Digital volume correlation vs. model with fully resolved microstructure vs. periodic unit cell

The understanding of the mechanisms controlling deformation of ductile iron at the micro-scale and their coupling to the manufacturing conditions is still far from complete. In this respect, recent synchrotron-based studies have demonstrated that the thermal contraction mismatch between the graphite particles and the matrix during solid-state cooling leads to a complex residual stress state in the microstructure. To investigate its impact on the room-temperature tensile deformation, a computational-experimental analysis extendable to other similar composite materials is presented in this paper. First, a miniaturized specimen is loaded and imaged in-situ with X-ray tomography. Then, the microscale displacement is reconstructed using digital volume correlation (DVC) and used to prescribe the boundary conditions in a finite element model of the full microstructure between two cross-sections. The model predictions at both the macroscale – tensile force and lateral contraction – and the microscale – strain field – are compared to the corresponding experimental and DVC-based data for several choices of the initial stress state, particles’ mechanical behavior and strength of the particles-matrix interface. It is proved that the micro-scale residual stress and a low interface strength are the key to explain the early stages of the tensile deformation of ductile iron. Finally, it is shown that a simple unit cell model of the microstructure would lead to significantly different results, thus demonstrating the superior accuracy and robustness of the present approach.

Compatibility in microstructural optimization for additive manufacturing

Microstructures with spatially-varying properties such as trabecular bone are widely seen in nature. These functionally graded materials possess smoothly changing microstructural topologies that enable excellent micro and macroscale performance. The fabrication of such microstructural materials is now enabled by additive manufacturing (AM). A challenging aspect in the computational design of such materials is ensuring compatibility between adjacent microstructures. Existing works address this problem by ensuring geometric connectivity between adjacent microstructural unit cells. In this paper, we aim to find the optimal connectivity between topology optimized microstructures. Recognizing the fact that the optimality of connectivity can be evaluated by the resulting physical properties of the assemblies, we propose to consider the assembly of adjacent cells together with the optimization of individual cells. In particular, our method simultaneously optimizes the physical properties of the individual cells as well as those of neighbouring pairs, to ensure material connectivity and smoothly varying physical properties. We demonstrate the application of our method in the design of functionally graded materials for implant design (including an implant prototype made by AM), and in the multiscale optimization of structures.

Quantifying numerically forecast size effects in the free vibration of heterogeneous beams

This paper reports on the influence that a periodic microstructure has on the unconstrained flexural vibration of geometrically similar but differently sized heterogeneous beam samples. A numerical investigation was conducted by finite element analysis (FEA) incorporating the detailed heterogeneity to identify and quantify any effect of beam size on the transverse modal frequencies when the microstructural scale is comparable to the overall size. Finite element models of the macroscopic beam samples were created by firstly specifying microstructural scale unit cells containing a single void or inclusion using ANSYS Mechanical APDL and then repeatedly regenerating these as required. Four beam sizes consisting of one, two, three or four layers of unit cells were created while the length to depth aspect ratio was kept constant for all sizes. Void or inclusion volume fraction was also altered while keeping the homogenised mass and stiffness properties of each beam fixed. The influence of the beam boundary texture on the results was also investigated. The ANSYS results were compared to the analytical solution for a conventional Timoshenko beam and a nonlocal Timoshenko beam. Using the nonlocal Timoshenko analysis, the Eringen small length scale coefficients were estimated but found to be size dependent. Numerical predictions obtained from a novel control volume based finite element (CVFEM) procedure incorporating micropolar constitutive behaviour were therefore matched to the ANSYS results and thereby used to identify the two additional constitutive parameters featuring in planar micropolar elasticity theory, namely the characteristic length in bending and coupling number.