Mechanically tunable phononic band gaps in three-dimensional periodic elastomeric structures

Three-dimensional periodic structures have many applications in acoustics and their properties are strongly related to structural details. Here we demonstrate through simulations the ability to tune the phononic band gaps of 3D periodic elastomeric structures using deformation. The elastomeric nature of the material makes the transformation of the band gaps a reversible and repeatable process, providing avenues for the design of tunable 3D phononic crystals such as sonic switches.     

增强六手臂缺失支柱手性拉胀超材料力学性能理论研究

前期研究工作中, 基于有限元分析, 作者发展了一种在大变形范围内具有可调恒定负泊松比的新型增强六手臂缺失支柱手性拉胀超材料. 为了揭示微观结构?力学性能关系, 并进一步指导超材料目标参数设计, 本文在小变形框架下基于能量法建立了表征该拉胀材料等效泊松比和弹性模量的理论模型. 增强六手臂缺失支柱手性拉胀材料由“Z”型手臂元件组成. “Z”型手臂可以被假设为两端简支的欧拉?伯努利梁. 因此, 本文首先推导了两端受集中力和力偶的任意形状欧拉?伯努利梁的应变能. 然后, 考虑平衡条件和变形协调条件进一步给出了材料等效泊松比和弹性模量的理论表达式. 研究表明只有“Z”型梁的内外手臂比为2:1时, 理论表达式才有简洁的形式. 为了更好地利用所推导的理论表达, 基于理论推导, 本文开发了MATLAT图形用户界面 (GUI). 在GUI中输入可描述该超材料几何形状的独立几何参数, 即可直接获取其等效泊松比和弹性模量. 最后, 基于理论结果, 系统讨论了超材料微结构几何参数对其等效力学性能的影响, 并将理论解与有限元计算结果进行了对比. 结果表明, 可以通过调控微结构几何参数获取大范围的目标力学性能.      

周期性两相层状带隙材料优化模型

研究并建立了一种在给定频段具有带隙性质的周期性两相层状材料的优化设计模型。首先基于层状材料波传播问题的解析解,得到了波数余弦函数与层状材料微结构参数间的解析表达式。进而分析了波数余弦函数与衰减系数的关系,提出了以波数余弦函数的平方在给定频段的积分为弹性波带隙特性的描述指标,以最大化该指标实现在给定频段使弹性波衰减系数最大化的思想,建立了设计在给定频段具有最优带隙性质的周期性两相层状材料优化提法和求解方法。最后,以几个典型的设计算例为对象,得到了给定微结构尺度约束下在特定频段具有最优带隙性质的材料微结构参数,讨论了材料微结构尺寸对最优材料结构参数的影响,以及最优结构参数对材料带隙性质的鲁棒性,验证了本文优化模型的有效性。     

Extension/Compression-Controlled Complete Band Gaps in 2D Chiral Square-Lattice-Like Structures

Achieving tunable band gaps in a structure by external stimuli is of great importance in acoustic applications. This paper aims to investigate the tunability of band gaps in square-lattice-like elastic periodic structures that are usually not featured with notable band gaps.Endowed with chirality, the periodic structures here are able to undergo imperfection-insensitive large deformation under extension or compression. The influences of geometric parameters on band gaps are discussed via the nonlinear finite element method. It is shown that the band gaps in such structures with curved beams can be very rich and, more importantly, can be efficiently and robustly tuned by applying appropriate mechanical loadings without inducing buckling. As expected, geometry plays a more significant role than material nonlinearity does in the evolution of band gaps. The dynamic tunability of band gaps through mechanical loading is further studied. Results show that closing, opening, and shifting of band gaps can be realized by exerting real-time global extension or compression on the structure. The proposed periodic structure with well-designed chiral symmetry can be useful in the design of particular acoustic devices.     

Rupture mechanisms in combined tension and shear—Micromechanics

A micromechanics model based on the theoretical framework of plastic localization into a band introduced by Rice is developed. The model consists of a planar band with a square array of equally sized cells, with a spherical void located in the centre of each cell. The periodic arrangement of the cells allows the study of a single unit cell for which fully periodic boundary conditions are applied. The micromechanics model is applied to analyze failure by ductile rupture in experiments on double notched tube specimens subjected to combined tension and torsion carried out by the present authors. The stress state is characterized in terms of the stress triaxiality and the Lode parameter. Two rupture mechanisms can be identified, void coalescence by internal necking at high triaxiality and void coalescence by internal shearing at low triaxiality. For the internal necking mechanism, failure is assumed to occur when the deformation localizes into a planar band and is closely associated with extensive void growth until impingement of voids. For the internal shearing mechanism, a simple criterion based on the attainment of a critical value of shear deformation is utilized. The two failure criteria capture the transition between the two rupture mechanisms successfully and are in good agreement with the experimental result.     

Bilinear Bäcklund transformation,Lax pair and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics/fluid mechanics/plasma physics

In this paper, outcomes of the study on the Bäcklund transformation, Lax pair, and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics, fluid mechanics, and plasma physics are presented. Via the Hirota bilinear method, a bilinear Bäcklund transformation is obtained, based on which a Lax pair is constructed. Via the symbolic computation, mixed rogue–solitary and rogue–periodic wave solutions are derived. Interactions between the rogue waves and solitary waves, and interactions between the rogue waves and periodic waves, are studied. It is found that (1) the one rogue wave appears between the two solitary waves and then merges with the two solitary waves; (2) the interaction between the one rogue wave and one periodic wave is periodic; and (3) the periodic lump waves with the amplitudes invariant are depicted. Furthermore, effects of the noise perturbations on the obtained solutions will be investigated.

     

Investigation of flexural wave band gaps in a locally resonant metamaterial with plate-like resonators

This paper presents an investigation of flexural wave band gaps in locally resonant metamaterials (LRMs). An LRM is a periodic structure consisting of repeated unit cells containing a local resonator. Due to the local resonance occurring in the unit cell, the LRM induces a band gap (a frequency band in which no waves propagate). Discrete-like or beam-like resonators have generally been used to realise LRMs in previous research. By extending the beam-like resonator configuration, this paper studies LRMs with a plate-like resonator to exploit its advantages with respect to large design freedom. In order to understand flexural wave band gaps in an LRM with plate-like resonators, parametric studies are conducted with the development of a finite element model. Further, the influences of the plate-like resonator design parameters on flexural wave band gaps are investigated. Based on the parametric studies, the rules governing band gap properties are determined. Finally, tailoring flexural wave band gaps by adjusting the parameters is discussed.     

Nonlinear elastic wave propagation in a phononic material with periodic solid–solid contact interface

Phononic materials enable enhanced dynamic properties, and offer the ability to engineer the material response. In this work we study the wave propagation in such a structure when introduced with nonlinearity. Our system is comprised of pre-compressed material with periodic solid–solid contacts, which exhibit a quadratic nonlinearity for small displacements. We suggest a new approach to modeling this system, where we discretize the unit cell in order to derive an approximate analytical solution using a perturbation method, which we are then able to easily validate numerically. With these methods, we study the band structure in the system and the second harmonic generation originating from the nonlinearity. We qualitatively analyze the second harmonic response of the system in terms of the single-crack response with linear band structure considerations. Significant band structure manipulation by changing system parameters is demonstrated, including possible in-situ tuning. The system also exhibits effective frequency doubling, i.e. the transmitted wave is primarily comprised of the second harmonic wave, for a certain range of frequencies. We demonstrate very high robustness to disorder in the system, in terms of band structure and second harmonic generation. These results have possible applications as frequency-converting devices, tunable engineered materials, and in non-destructive evaluation.     

Topology design and optimization of nonlinear periodic materials

This paper explores optimal topologies yielding large band gap shifts in one- and two-dimensional nonlinear periodic materials. The presence of a nonlinearity in a periodic material system results in amplitude-dependent dispersion behavior, leading to novel wave-based devices such as tunable filters, resonators, and waveguides. The performance of these devices over a broad frequency range requires large, tunable band gaps, motivating the present study. Consideration of a one-dimensional bilayer system composed of alternating linear and nonlinear layers shows that optimal designs consist of thin, compliant nonlinear layers. This is at first surprising considering the source of the shift originates from only the nonlinear layer; however, thin layers lead to localized stresses that activate the nonlinear character of the system. This trend persists in two-dimensional materials where optimization studies are performed on plane-stress models discretized using bilinear Lagrange elements. A fast algorithm is introduced for computing the dispersion shifts, enabling efficient parametric analyses of two-dimensional inclusion systems. Analogous to the one-dimensional system, it is shown that thin ligaments of nonlinear material lead to large dispersion shifts and group velocity variations. Optimal topologies of the two-dimensional system are also explored using genetic algorithms aimed at producing large increases in complete band gap width and shift, or group velocity variation, without presupposing the topology. The optimal topologies that result resemble the two-dimensional inclusion systems, but with small corner features that tend to enhance the production of dispersion shift further. Finally, the study concludes with a discussion on Bloch wave modes and their important role in the production of amplitude-dependent dispersion behavior. The results of the study provide insight and guidance on selecting topologies and materials which can yield large amplitude-dependent band gap shifts and group velocity variations, thus enabling sensitive nonlinear devices.     

低频弹性波超材料的若干进展

超材料是一类新兴的具有超常物理性质的人造周期/拟周期材料, 能够改变电磁波、声波以及弹性波等在介质中的传播特性. 因在航天、国防以及民用科学等方面的巨大应用潜力, 超材料自被提出后便受到极大的关注并引发研究热潮. 弹性波超材料是超材料的一种, 能够基于弹性波与超材料结构的相互耦合作用实现对弹性波的操控. 带隙是评估弹性波超材料实现弹性波操控的重要工具, 其性质与超材料的材料参数、晶格常数以及局域振子的固有频率相关. 受制于超材料的承载能力、外观尺寸以及局域振子结构等因素, 利用传统超材料开启低频(约100 Hz)弹性波带隙依然存在较大困难. 文章首先简要介绍超材料开启弹性波带隙的基本原理, 然后从低频弹性波超材料基本结构与低频带隙实现方法、低频带隙优化与调控策略、低频带隙潜在应用等三个方面详细总结低频弹性波超材料的研究工作. 其中, 低频带隙超材料的基本结构主要包括布拉格散射型超材料、传统局域共振型超材料以及准零刚度局域共振超材料. 文章通过总结低频弹性波超材料的研究进展, 分析了目前研究中的不足并对未来低频弹性波的研究方向进行了展望.      

The effect of porosity distribution on ductile failure

The effect of a nonuniform distribution of porosity on flow localization and failure in a porous material is analyzed numerically. The void density distribution and properties used to characterize the material behavior were obtained from measurements on partially consolidated and sintered iron powder. The calculations were carried out using an elastic viscoplastic constitutive relation for porous plastic solids. Local material failure is incorporated into the model through the dependence of the flow potential on void volume fraction. The region modelled is a small portion of a larger body, subject to various triaxial stress conditions. Both plane strain and axisymmetric deformations are considered with imposed periodic boundary conditions. Interactions between regions with higher void fractions promote plastic flow localization into a band. Local failure occurs by void growth and coalescence within the band. The results suggest a failure criterion based on a critical void volume fraction that is only weakly dependent on stress history. The critical void fraction does. however, depend on the initial void distribution and material hardening characteristics.     

Surface effect on band structure of magneto-elastic phononic crystal nanoplates subject to magnetic and stress loadings

This paper presents a theoretical model for the size-dependent band structure of magneto-elastic phononic crystal(PC) nanoplates according to the Kirchhoff plate theory and Gurtin-Murdoch theory, in which the surface effect and magneto-elastic coupling are considered. By introducing the nonlinear coupling constitutive relation of magnetostrictive materials, Terfenol-D/epoxy PC nanoplates are carried out as an example to investigate the dependence of the band structure on the surface effect, magn...     

A class of auxetic three-dimensional lattices

We propose a class of auxetic three-dimensional lattice structures. The elastic microstructure can be designed to have an omnidirectional Poisson's ratio arbitrarily close to the stability limit of −1. The cubic behaviour of the periodic system has been fully characterized; the minimum and maximum Poisson's ratio and the associated principal directions are given as a function of the microstructural parameters.The initial microstructure is then modified into a body-centred cubic system that can achieve Poisson's ratio lower than −1 and that can also behave as an isotropic three-dimensional auxetic structure.     

Bifurcations and exact solutions of an asymptotic rotation-Camassa–Holm equation

This paper investigates the rotation-Camassa–Holm equation, which appears in long-crested shallow-water waves propagating in the equatorial ocean regions with the Coriolis effect due to the earth’s rotation. The rotation-Camassa–Holm equation contains the famous Camassa–Holm equation and is a special case of the generalized Camassa–Holm equation. By using the approach of dynamical systems and singular traveling wave theory to its traveling wave system, in different parameter conditions of the five-parameter space, the bifurcations of phase portraits are studied. Some exact explicit parametric representations of the smooth solitary wave solutions, periodic wave solutions, peakons and anti-peakons, periodic peakons as well as compacton solutions are obtained.

     

舰艇新型宏观负泊松比效应蜂窝舷侧防护结构

提出一种具有宏观负泊松比效应的新型蜂窝舷侧防护结构,通过对负泊松比效应蜂窝胞元特殊结构构型设计,实现中等弹速下良好抗爆抗冲击性能。利用有限元动力学分析软件,研究鱼雷或导弹水下对舷侧防护结构的撞击侵入和穿透过程,对比研究了不同蜂窝构型、材料、胞元尺寸和胞壁厚度对舷侧结构抗冲击性能的影响。结果表明,蜂窝防护结构具有良好的抗冲击性能,负泊松比蜂窝构型较正泊松比蜂窝构型抗冲击性能更优。     

The SEA Approach to Vibration Localization in a Nearly Periodic Structure

Abstract

The phenomenon of vibration localization occurring in a nearly periodic structure was investigated through a statistical energy analysis (SEA) approach. The phenomenon has been examined mostly through a wave propagation approach, where a localization factor was often employed to evaluate the strength of vibration localization. The wave propagation approach properly predicted the factor close to Monte Carlo calculations in nearly periodic structures for both weak and strong couplings. In this analytical study, the localization factor was derived from the SEA approach for a nearly periodic structure monocoupled with a weak coupling. The SEA approach sequentially breaks the structure into two-oscillator blocked substructures and proposes a way of determining the vibration localization factor with equations of energy balance. This article shows that the SEA approach is quite appropriate for calculating the vibration localization factor compared to the wave propagation approach.     

3D Strain Field Evolution and Failure Mechanisms in Anisotropic Paperboard

Background

Experimental analyses of the 3D strain field evolution during loading allows for better understanding of deformation and failure mechanisms at the meso- and microscale in different materials. In order to understand the auxetic behaviour and delamination process in paperboard materials during tensile deformation, it is essential to study the out-of-plane component of the strain tensor that is, in contrast to previous 2D studies, only achievable in 3D.

Objective

The main objective of this study is to obtain a better understanding of the influence of different out-of-plane structures and in-plane material directions on the deformation and failure mechanisms at the meso- and microscale in paperboard samples.

Methods

X-ray tomography imaging during in-situ uniaxial tensile testing and Digital Volume Correlation analysis was performed to investigate the 3D strain field evolution and microscale mechanical behaviour in two different types of commercial paperboards and in two material directions. The evolution of sample properties such as the spatial variation in sample thickness, solid fraction and fibre orientation distribution were also obtained from the images. A comprehensive analysis of the full strain tensor in paperboards is lacking in previous research, and the influence of material directions and out-of-plane structures on 3D strain field patterns as well as the spatial and temporal quantification of the auxetic behaviour in paperboard are novel contributions.

Results

The results show that volumetric and deviatoric strain, dominated by the out-of-plane normal strain component of the strain tensor, localize in the out-of-plane centre already in the initial linear stress-strain regime. In-plane strain field patterns differ between samples loaded in the Machine Direction (MD) and Cross Direction (CD); in MD, strain localizes in a more well-defined zone close to the notches and the failure occurs abruptly at peak load, resulting in angular fracture paths extending through the stiffer surface planes of the samples. In CD, strain localizes in more horizontal and continuous bands between the notches and at peak load, fractures are not clearly visible at the surfaces of CD-tested samples that appear to fail internally through more well-distributed delamination.

Conclusions

In-plane strain localization preceded a local increase of sample thickness, i.e. the initiation of the delamination process, and at peak load, a dramatic increase in average sample thickening occurred. Different in-plane material directions affected the angles and continuity of the in-plane strain patterns as well as the sample and fracture properties at failure, while the out-of-plane structure affected how the strain fields distributed within the samples.

     

Phononic layered composites for stress-wave attenuation

The aim is to design a layered metamaterial with high attenuation coefficient and high in-plane stiffness-to-density ratio using homogenization to calculate and optimize the dynamic effective stiffness and mass density of layered periodic composites (phononic layers) over a broad frequency band. This is achieved by: (1) minimizing the frequency range of the first pass band, (2) maximizing the frequency range of the stop band, and (3) creating local resonance over the second pass band. To verify the theoretical calculation, laboratory samples were fabricated and their attenuation coefficient were measured and compared with the theoretical results. It is observed that over 4–20 kHz frequency range the attenuation per unit length in the optimally designed composite can exceed 500 dB/m; which increases with increasing frequency. A dynamic Ashby chart, depicting attenuation coefficient vs. in-plane stiffness-to-density ratio, is presented for various engineering materials and is compared with the fabricated metamaterial to show the significance of our design. This method can be used in variety of applications for stress wave management, e.g., in addition to match the impedance of the resulting composite to that of its surrounding medium to minimize (or essentially eliminate) stress wave reflection.     

Optical properties of two-dimensional polymer photonic crystals after deformation-induced pattern transformations

The photonic band structure and optical transmittance of two-dimensional periodic elastomeric photonic crystals are studied computationally to understand the effects of large strains on optical properties of the structures. The large compressive deformation patterns of the two-dimensional periodic structure studied by Mullin and coworkers [Mullin, T., Deschanel, S., Bertoldi, K., Boyce, M.C., 2007. Pattern transformation triggered by deformation. Physical Review Letters 99(8), 084301] are first reproduced using hyperelastic material models for the elastomer SU-8. Finite element analysis is then used to solve Maxwell's equations to obtain light transmittance through both the undeformed and deformed structures; simultaneously the wave equation resulting from the appropriate two-dimensional form of Maxwell's equations is solved as an eigenvalue problem to obtain the band structure. The deformation-induced shift in transmission spectrum valleys for different bands is calculated, and the changes in the width of these reflectance peaks are also obtained. The band structure calculation shows that there are no complete photonic band gaps as expected for the low dielectric contrast system. However, the effect of the observed reversible, symmetry-breaking deformation pattern is to uncouple many of the photonic bands in all three high symmetry directions, i.e. Γ–X, X–M, and Γ–M. New non-degenerate deformation-induced optical modes appear in both the real space transmittance spectra and the band structure with lower reflectance values. Analyses of the deformation pattern, the optical mode shapes, and the photonic band structure reveal that localized regions of large rotation are responsible for the significant changes in optical transmittance. The results have practical importance for the design of strain-tunable optomechanical materials for sensing and actuation.     

Parametric design of the band structure for lattice materials

Lattice materials are often investigated to determine how small parameter variations in the periodic microstructrure can influence the elastic wave propagation. A general hierarchical scheme, based on asymptotic perturbation techniques, is outlined to analytically assess the parametric sensitivity of the material band structure to a generic multi-parametric perturbation (direct problem). Modeling refinements, parameters updates, microstructural damages and manufacturing irregularities can be treated indifferently and simultaneously. According to a converse strategy, based on the inversion of the sensitivity problem, a hierarchical scheme is sketched to identify the parameter combinations which realize a design band structure (inverse problem). The direct and inverse problem are applied to the sensitivity analysis and band structure design of the anti-tetrachiral lattice material. Despite the high spectral density and the high-dimensional parameter space, the multi-parameter perturbation technique demonstrates its suitability in, first, analytically—although asymptotically—describe the material spectrum and, second, designing the material microstructure to obtain the desired spectral components. The inverse problem solution is discussed in terms of existence, uniqueness, asymptotic consistency and physical admissibility.