Periodic Plate Structures Research Articles

Bandgap and Wave Propagation Properties of 2D Curved and Chiral Hybrid Star‐Shaped Metamaterials

Designing artificial microstructural metamaterials that fulfill the single‐phase vibration dampening and lightweight twin goals is challenging and critical. 2D curved and chiral hybrid star‐shaped metamaterials (CCHSM) are proposed to perform vibration reduction filtering in the midfrequency region by combining the properties of chiral structure and concave structure. The generating process of the bandgap of in‐plane elastic waves, the properties of directional attenuation, and the flow direction of energy in the dispersion curve are examined using the modal shape, dispersion surface, and group velocity. In a specific frequency range, the equivalent medium parameters satisfy the conditions of negative refraction transverse and longitudinal waves, so the structure also has double‐negative characteristics. The filtering performance is validated using the CCHSM periodic plate structure and a finite‐element simulation of transmission loss. In addition, the correlation between wave attenuation characteristics and structural parameters is explored. The radius of the circular arc and the deflection angle of the connecting rod have a significant impact on the frequency range of the structural bandgap. The results show that the created simple‐configured, single‐phase lightweight acoustic metamaterials exhibit strong local resonance properties and double‐negative features.

Numerical Implementation of Variational Asymptotic Homogenization Method for Periodic Plate Structures

The variational asymptotic homogenization (VAM) theory is extended to access freely to commercial finite element (FE) software to deal with periodic plate structures. In this work, the finite element format for periodic plate structures based on the variational asymptotic homogenization is developed, ensuring the commercial finite element software can be utilized to obtain the effective plate stiffness. A standard numerical framework and an integration algorithm are proposed for unifying the dimensional reduction analysis and the homogenization analysis in a formalized manner. As for model validation, the periodic plates composed of unit cells with three-dimension (3D) heterogeneous geometry are simulated by various elements and modeling techniques using the commercial FE software rather than programming in-house code. Compared to the results provided in the existing literature, the proposed approach shows excellent performance in terms of computational efficiency and time without compromising the VAM accuracy. It is preferable to enhance the application of the variational asymptotic homogenization theory for the more sophisticated heterogeneous plate structures.

Dynamic modeling and characteristic analysis of the periodically coupled plate structure based on the dynamic stiffness method

Abstract In this paper, the dynamic model of periodically coupled plate structure is formulated by the dynamic stiffness method (DSM). Geometrical construction is assumed to vary periodically in the whole plate structure. According to the coupling condition, the whole periodic plate structure is divided into appropriate individual rectangular plates firstly. To the individual plate, the displacement fields both of in-plane and out-of-plane are derived from the governing equation with a strong-form methodology. The dynamic stiffness matrix is developed by the Projection method. Once the dynamic stiffness matrixes of the individual plates are obtained, the global dynamic stiffness matrix of the whole periodic structure can be assembled in the similar manner with the finite element method (FEM). Several examples are presented to validate the convergence and accuracy of the present formulation. From the harmonic responses, the band gap behaviors can be observed obviously. Furthermore, a parametric analysis is implemented to study the effects of geometrical distributions and plate dimensions on the vibration characteristics.

Band gap property analysis of periodic plate structures under general boundary conditions using spectral-dynamic stiffness method

Vibration band gap properties of periodic rectangular plate structures with general boundary conditions are studied using the spectral-dynamic stiffness method (S-DSM). Material and dimensional properties are assumed to vary periodically in the whole plate structure. In the S-DSM, the general solution governing each unit plate is firstly derived with the aid of spectral method and then the dynamic stiffness matrix is established by substituting the general solution into corresponding boundary equations. Finally, the global spectral dynamic matrix for the whole periodic plate structure is assembled in a similar way to the finite element method (FEM). The harmonic responses are calculated to illustrate the band gap properties of the periodic structures. By comparing the results obtained by the FEM, it can be verified that the S-DSM is of superior accuracy and convergence rate. A parametric study is also conducted to investigate the effects of the material properties, geometrical parameters and structural damping on the vibration band gap behaviors of the periodic structures.

Shear stiffness prediction of Reissner–Mindlin plates with periodic microstructures

ABSTRACTIn this article, an intuitionistic interpretation of the new numerical implementation of asymptotic homogenization for effective bending stiffness of in-plane periodic plate structures is presented. Based on this interpretation, a two-step method of effective shear stiffness prediction for their Reissner–Mindlin model is developed. An equivalent displacement field of linear curvature is applied to a unit cell in the first step; in the second step, a new unit cell problem is constructed and solved, and the effective shear stiffness is obtained through energy equivalence. This method can be easily implemented in commercial software and several examples are given to demonstrate its validity.

Two-scale topology design optimization of stiffened or porous plate subject to out-of-plane buckling constraint

This paper studies maximum out-of-plane buckling load design of thin bending plates for a given amount of material. Two kinds of plates are considered. One is made of periodic homogeneous porous material. Another is uniformly stiffened solid plate. The plate material, thickness, design domain of its middle plane and boundary conditions are given. The pattern of prescribed in-plane external load or displacements along the part of boundaries, which move freely, is given. Both plate topology and micro-structural topology of porous material or stiffener layout are concurrently optimized. The artificial element material densities in both macro and micro-scale are chosen as design variables. The volume preserving nonlinear density filter is applied to obtain the black-white optimum topology and comparison of its different sensitivities is made to show the reason for oscillation during optimization process in Appendix. The new numerical implementation of asymptotic homogenization method (NIAH, Cheng (Acta Mech Sinica 29(4): 550---556, 2013) and Cai (Int J Solids Struct 51(1), 284---292, 2014) is applied to homogenization of periodic plate structures and analytic sensitivity analysis of effective stiffness with respect to the topological design variables in both macro-scale and micro-scale. On basis of that, this paper implements the sensitivity analysis of out-of-plane buckling load by using commercial FEA software and enables the application of gradient-based search algorithm in optimization. Several numerical implementation details are discussed. Three numerical examples are given to show the validity of this method.

Analysis of flexural wave bandgaps in periodic plate structures using differential quadrature element method

By employing the first order shear deformation plate theory and the Bloch–Floquet theorem, the dispersion equation of flexural wave in the periodic composite plate structure with piezoelectric patches is derived and solved by the use of the differential quadrature element method. Moreover, wave modes for the dispersion curves of the considered periodic plate are compared with those of a homogeneous plate, from which the reason of the frequency band gap is revealed. Then, a comprehensive parametric study is conducted to highlight the influences of the physical parameters and the geometrical parameters on the frequency band gaps. The results show that the method is efficient and accurate and the bandwidth can be enlarged by changing the physical and geometrical parameters. The special band gap property of periodic plate structure has many potential applications in wave/vibrations attenuation areas for mechanical, aerospace and civil engineering structures.

Novel numerical implementation of asymptotic homogenization method for periodic plate structures

The present paper develops and implements finite element formulation for the asymptotic homogenization theory for periodic composite plate and shell structures, earlier developed in Kalamkarov (1987) and Kalamkarov (1992), and thus adopts this analytical method for the analysis of periodic inhomogeneous plates and shells with more complicated periodic microstructures. It provides a benchmark test platform for evaluating various methods such as representative volume approaches to calculate effective properties. Furthermore, the new numerical implementation (Cheng et al., 2013) of asymptotic homogenization method of 2D and 3D materials with periodic microstructure is shown to be directly applicable to predict effective properties of periodic plates without any complicated mathematical derivation. The new numerical implementation is based on the rigorous mathematical foundation of the asymptotic homogenization method, and also simplicity similar to the representative volume method. It can be applied easily using commercial software as a black box. Different kinds of elements and modeling techniques available in commercial software can be used to discretize the unit cell. Several numerical examples are given to demonstrate the validity of the proposed methods.

Study on vibration characteristics in periodic plate structures using the spectral element method

Periodic structures have great scientific potential because of their superior structural dynamic properties. The aim of this study is to analyze the dynamic behavior of periodic plate structures using the spectral element method (SEM). The spectral equations of the plate elements with two parallel sides that are simply supported are established. Then, the spectral dynamic stiffness matrix of the whole periodic plate structure is assembled. The frequency responses are obtained by the calculation of the spectral equations to illustrate the characteristics of the band gaps. From the results, it is seen that the SEM can be effectively applied to study the vibration properties of the periodic structures. Compared with the results calculated by the finite element method, it can be observed that more accurate results in high-frequency ranges can be achieved by the SEM. These results indicate that the band gap characteristics depend on both the material properties and unit cell numbers. Furthermore, the effects of the structure damping and type on the frequency responses are investigated.

Higher order vibrations of thin periodic plates

A problem of higher order vibrations of thin periodic plates related to a plate periodicity is discussed. The applied model, based on the Kirchhoff plate theory assumptions and additional hypothesis of tolerance averaging [Woźniak Cz, Wierzbicki E. Averaging techniques in thermomechanics of composite solids. Tolerance averaging versus homogenization. Poland, Częstochowa: Częstochowa University of Technology; 2000], makes it possible to take into account the effect of the periodicity cell size on the overall plate behaviour. Assuming more functions describing oscillations of the periodicity cell, we can investigate within the model not only fundamental vibrations but also higher order vibrations related to a periodic plate structure. These functions have to be properly chosen approximations of solutions to eigenvalue problem for natural vibrations of separated cell with periodic boundary conditions, [Jędrysiak J. On vibrations of thin plates with one-dimensional periodic structure. Int J Eng Sci 2000;38:2023-2043]. Using the known two-dimensional plate models—the orthotropic plate model and finite element method as well as the discrete model, certain justifications of the proposed model are made.

Waveguiding and frequency selection of Lamb waves in a plate with a periodic stubbed surface

In this paper, we numerically and experimentally study the waveguiding of Lamb modes in a thin plate with a periodic stubbed surface and propose a frequency-selection method based on the found complete band gaps of Lamb waves in the periodic structure. In the numerical simulations, we employ finite-element method to analyze the waveguiding effect of a line defect created in the periodic plate structure; and on the experimental side, we utilize a pulsed laser to generate broadband elastic-wave energy and a laser interferometer to receive the wave signals inside the line-defect waveguide. In the experiment, well-confined acoustic energy in the acoustic band gaps is observed. Furthermore, a polyline sharply bent waveguide is designed and used for the frequency selection of Lamb waves. Measurements show that acoustic energy with frequencies in the band gaps can be separated out and guided by the bent waveguiding route. The characteristics of deaf bands found in the experiment are discussed as well.

A study on sound radiation from isotropic plates stiffened by unsymmetrical beams

The determination of sound pressure radiated from periodic plate structures is fundamental in the estimation of noise level in aircraft fuselages or ship hull structures. As a robust approach to this problem, here a very general and comprehensive analytical model is developed for predicting the sound radiated by a vibrating plate stiffened by periodically spaced orthogonal unsymmetrical beams subjected to a sinusoidally time varying point load. The plate is assumed to be infinite in extent, and the beams are considered to exert both line force and moment reactions on it. Using this theoretical model, the sound pressure levels on axis in a semi-infinited fluid (water) bounded by the plate were calculated using three numerical tools such as the Gauss-Jordan method, the LU decomposition method and the IMSL numberial package. Especially, the variation in the sound pressure levels and their modes were investigated according to the change in frequency, bay spacing and bay distance.