Symplectic analysis for regulating wave propagation in a one-dimensional nonlinear graded metamaterial

An analytical method,called the symplectic mathematical method,is proposed to study the wave propagation in a spring-mass chain with gradient arranged local resonators and nonlinear ground springs.Combined with the linearized perturbation approach,the symplectic transform matrix for a unit cell of the weakly nonlinear graded metamaterial is derived,which only relies on the state vector.The results of the dispersion relation obtained with the symplectic mathematical method agree well with those achieved by the Bloch theory.It is shown that wider and lower frequency bandgaps are formed when the hardening nonlinearity and incident wave intensity increase.Subsequently,the displacement response and transmission performance of nonlinear graded metamaterials with finite length are studied.The dual tunable effects of nonlinearity and gradation on the wave propagation are explored under different excitation frequencies.For small excitation frequencies,the gradient parameter plays a dominant role compared with the nonlinearity.The reason is that the gradient tuning aims at the gradient arrangement of local resonators,which is limited by the critical value of the local resonator mass.In contrast,for larger excitation frequencies,the hardening nonlinearity is dominant and will contribute to the formation of a new bandgap.

TRANSIENT THERMAL RESPONSE IN THICK ORTHOTROPIC HOLLOW CYLINDERS WITH FINITE LENGTH:HIGH ORDER SHELL THEORY

The transient thermal response of a thick orthotropic hollow cylinder with finite length is studied by a high order shell theory. The radial and axial displacements are assumed to have quadratic and cubic variations through the thickness, respectively. It is important that the radial stress is approximated by a cubic expansion satisfying the boundary conditions at the inner and outer surfaces, and the corresponding strain should be least-squares compatible with the strain derived from the strain-displacement relation. The equations of motion are derived from the integration of the equilibrium equations of stresses, which are solved by precise integration method （PIM）. Numerical results are.obtained, and compared with FE simulations and dynamic thermo-elasticity solutions, which indicates that the high order shell theory is capable of predicting the transient thermal response of an orthotropic （or isotropic） thick hollow cylinder efficiently, and for the detonation tube of actual pulse detonation engines （PDE） heated continuously, the thermal stresses will become too large to be neglected, which are not like those in the one time experiments with very short time.

In-Plane Dynamic Crushing Behaviors of a Vertex-Based Hierarchical Auxetic Honeycomb

Auxetic metamaterials,which exhibit the negative Poisson’s ratio(NPR)effect,have found wide applications in many engineering fields.However,their high porosity inevitably weakens their bearing capacity and impact resistance.To improve the energy absorption efficiency of auxetic honeycombs,a novel vertex-based hierarchical star-shaped honeycomb(VSH)is designed by replacing each vertex in the classical star-shaped honeycomb(SSH)with a newly added self-similar sub-cell.An analytical model is built to investigate the Young’s modulus of VSH,which shows good agreement with experimental results and numerical simulations.The in-plane dynamic crushing behaviors of VSH at three different crushing velocities are investigated,and empirical formulas for the densification strain and plateau stress are deduced.Numerical results reveal more stable deformation modes for VSH,attributed to the addition of self-similar star-shaped sub-cells.Moreover,compared with SSH under the same relative densities,VSH exhibits better specific energy absorption and higher plateau stresses.Therefore,VSH is verified to be a better candidate for energy absorption while maintaining the auxetic effect.This study is expected to provide a new design strategy for auxetic honeycombs.

Tailored energy absorption for a novel auxetic honeycomb structure under large deformation

In comparison to conventional hexagonal honeycomb structures,auxetic metamaterials with re-entrant configurations have exhibited superior mechanical properties in terms of energy absorption.To further enhance the energy absorption capacity of these materials,a novel re-entrant honeycomb configuration,named novel auxetic re-entrant honeycomb(NARH),is developed by incorporating“<>”-shaped cell walls into the conventional auxetic re-entrant honeycomb(ARH).Two analytical models for the plateau stress are formulated to consider the plastic deformation of NARH during quasi-static compression and the dynamic impact using the linear momentum theorem.Quasi-static compression tests on 3D printed NARH honeycomb specimens and finite element simulations are performed to verify the effectiveness of the theoretical models.NARH exhibits higher plateau stresses compared with ARH during compression,which can be attributed to the presence of more plastic hinges formed in NARH.These hinges,the embedded parts with inclined cell walls,not only improve stability by forming stable triangles during compression but also enhance the energy absorption capacity.A parametric study is conducted to analyze the effect of impact velocity,thickness,and incline angle of cell walls on crashworthiness.Numerical simulations demonstrate higher sensitivity of the mechanical properties to impact velocity and cell wall thickness.Adding ribs to the“<>”-shaped cell walls in NARH further reduces the initial peak force during dynamic crushing while maintaining high energy absorption.The research provides valuable guidelines for the design of energy absorption metamaterials.

热-电-弹压电/石墨烯复合纳米板在粘弹性地基上的非线性受迫振动

本文研究了非对称压电/石墨烯复合纳米板在粘弹性地基上、热环境及外部简谐激励作用下的动力学行为.根据Hamilton原理和Kirchhoff板理论,建立复合纳米板的动力学方程,并通过Galerkin方法对简支边界下的动力学方程进行求解,获得了一个包含有二次非线性项的Duffing-Helmholtz方程.从解析和数值方面分别分析了非局部弹性参数、温度、粘弹性基体、电压以及纳米板的几何尺寸对系统非线性与线性频率比和非线性振动行为的影响.此外,本文揭示了系统的非对称性和二次非线性项之间的对应关系.本研究的动力学结果将为压电-石墨烯复合纳米谐振器力学性能的实验表征提供理论依据.

SYMPLECTIC ANALYSIS FOR WAVE PROPAGATION OF HIERARCHICAL HONEYCOMB STRUCTURES

Wave propagation in two-dimensional hierarchical honeycomb structures with two- order hierarchy is investigated by using the symplectic algorithm. By applying the variational prin- ciple to the dual variables, the wave propagation problem is transformed into a two-dimensional symplectie eigenvalue problem. The band gaps and spatial filtering phenomena are examined to find the stop bands and directional stop bands. Special attention is directed to the effects of the relative density and the length ratio on the band gaps and phase constant surfaces. This work provides new opportunities for designing hierarchical honeycomb structures in sound insulation applications.

Dynamic Crushing Strength Analysis of Auxetic Honeycombs

The in-plane dynamic crushing behavior of re-entrant honeycomb is analyzed and compared with the conventional hexagon topology.Detailed deformation modes along two orthogonal directions are examined,where a parametric study of the effect of impact velocity and cell wall aspect ratio is performed.An analytical formula of the dynamic crushing strength is then deduced based on the periodic collapse mechanism of cell structures.Comparisons with the finite element results validate the effectiveness of the proposed analytical method.Numerical results also reveal higher plateau stress of re-entrant honeycomb over conventional hexagon topology,implying better energy absorption properties.The underlying physical understanding of the results is emphasized,where the auxetic effect（negative Poisson＇s ratio） induced in the re-entrant topology is believed to be responsible for this superior impact resistance.

A PRECISE METHOD FOR SOLVING WAVE PROPAGATION IN HOLLOW SANDWICH CYLINDERS WITH PRISMATIC CORES

Wave propagation in infinitely long hollow sandwich cylinders with prismatic cores is analyzed by the extended Wittriek-Williams （W-W） algorithm and the precise integration method （PIM）. The effective elastic constants of prismatic cellular materials are obtained by the homogenization method. By applying the variational principle and introducing the dual variables the canonical equations of Hamiltonian system are constructed. Thereafter, the wave propagation problem is converted to an eigenvalue problem. In numerical examples, the effects of the prismatic cellular topology, the relative density, and the boundary conditions on dispersion relations, respectively, are investigated.