Analysis On The Elastic Wave Dispersions And Micro-Structural Design Of Cellular Materials

Cellular materials are two-dimensional periodic lattice materials,which have excellent sound absorption,vibration isolation,vibration damping,elastic waveguide properties,etc.These emerging materials have been widely used in the areas of aerospace,aviation,marine,light industry,construction and acoustic functional devices.Meanwhile,much attention has been paid on the wave propagation characteristics of cellular materials in recent two decades.There are two main ways: the equivalent continuum method and discretization lattice dynamic method.The former method takes the cellular topologies and their geometrical parameters into consideration,while the latter one does not.Based on these two modelling approaches,the propagation problems are focused here for the honeycomb sandwich structures and the two-dimensional infinite periodic lattices.The main contents and conclusions are summarized as follows:(1)An improved homogenization model taking account of the shear effect for the cell sheets has been developed on the basis of the equivalent strain energy principle for predicting the in-plane elastic constants of sandwich cores.Adopting the continuum theory of small deformation,the macro-strain of unit cell is firstly obtained by the strain of cell sheets modelled as Timoshenko beams.Thereafter,the strain energy density of the unit cell is expressed as a function of the macro-strain,and the second order partial derivatives of this strain energy density yield the effective elastic constants.By comparing the structural responses and the first five natural frequencies,the proposed Timoshenko effective model is validated with the Euler effective model(taking the shear factor in the Timoshenko effective model as zero)and the complete model.The results also indicate that the shear effect has a significant effect on the in-plane equivalent properties of cellular materials,especially for the cases with large slenderness ratio(or relative density).The development of the homogenization method provides valuable theoretical basis for the wave propagation problem of lightweight sandwich structures in the subsequent work.(2)The combination of the extended Wittrick-Williams algorithm and the precise integration method,which is very easy to deal with the boundary value problems,is extended to analyze the propagation of elastic wave in infinitely long hollow sandwich cylinders with prismatic cores.As the frequency of the wave stays in a low frequency range,the prismatic core is equalized to an anisotropic and homogenous medium with the homogenization method developed in Section(1).Subsequently,the Lagrangian function is reconstructed in the cylindrical coordinate system according to the principle of virtual displacement.And then,taking Legendre transformation,the dual variables are introduced to construct the state space formalism.Finally,in this formalism,the extended Wittrick-Williams algorithm under the piece-wise constant approximation,and in combination with the precise integration method,is implemented in deducing the dispersion relation.Numerical results reveal the effectiveness of this new method by comparing with the polynomial approach.Meanwhile,potential applications of the method are discussed by the parameter studies,including topological configurations(square,diamond,triangle-6 and triangle-8),relative density and the boundary conditions,on the dispersion relations.(3)Star-shaped honeycombs are analyzed in terms of the equivalent mechanical behaviors and wave propagation properties(band gaps).By applying the Castigliano's second theorem,the effective Young's modulus and Poisson's ratio are derived by an analytical method used in structural mechanics.And then,on the basis of the Bloch's theorem,the wave propagation characteristics of an infinite periodic system are simplified and reduced to the analysis of unit representative cell.By applying the finite element method,the dispersion characteristics are analyzed by the dynamic matrix in conjunction with the extended Wittrick-Williams algorithm.Special attentions are paid to the effects of the geometrical parameters on the effective constants and band gaps.Meanwhile,the relationship between the effective Poisson's ratio and the band gaps is investigated by numerical simulations.The results demonstrate that the negative Poisson's ratio provides an enhanced effective Young's modulus of the considered honeycombs.In addition,the desirable band gaps are also observed with Poisson's ratio in the negative values.It can be conducted that the Poisson's ratio can be considered as an important factor for the optimal design of cellular structures.(4)The concept of variable cross-section is firstly implemented into the design of two dimensional cellular structures in order to tailor the band structures.Adopting the numerical method developed in Section(3),the effects of the internal angle,the slenderness ratio,and the material distribution on the band gaps and directionality of hexagonal and re-entrant lattices with cell walls of non-uniform thickness are analyzed.The results show that the band gaps are affected significantly by those three parameters.In particular,the cellular materials with cell walls of non-uniform thickness have more and wider band gaps comparing with the ones arranged by traditional uniform cross-section cell walls.As to the wave directional behavior,the effect associated with the internal angle is found to be more prominent than that of the other two factors.Thus,in conclusion,our results offer the possibility of designing novel wave filters by modifying lattice configurations.(5)Two types of square crystal lattices are adopted to assess the reliability of the band gaps,obtained by limiting the variation of wave vectors along the contour of the irreducible Brillouin zone(IBZ).One type,named System ?,is composed of primitive cells that have the same degree of symmetry as the square crystal lattices,i.e.,four-folded reflective and rotational symmetries,such as the square or re-entrant square lattices.The other type,named System ?,is arranged by the unit cells having solely four-folded rotational symmetry,e.g.,square zigzag or tetrachiral lattices.Numerical studies on the first eight frequencies,including the extreme values and their locations,are conducted by applying the reduced finite element method.Corresponding results demonstrate that:(i)For System ?,it is enough to highlight the presence and determine the stop gaps by covering wave vector along boundaries of the IBZ;(ii)For System ?,just as the chiral configurations,several extreme frequencies including minima and maxima deviate from boundaries of the IBZ.Thus,it can be concluded that the symmetry of the primitive cells should be fully considered when one investigates the band gap characteristics of lattices.