Elastic wave dispersion in microstructured membranes

The effect of microstructural properties on the wave dispersion in linear elastic membranes is addressed in this paper. A periodic spring-mass lattice at the lower level of observation is continualized and a gradient-enriched membrane model is obtained to account for the characteristic microstructural length scale of the material. In the first part of this study, analytical investigations show that the proposed model is able to correctly capture the physical phenomena of wave dispersion in microstructured membrane which is overlooked by classical continuum theories. In the second part, a finite-element discretization of microstructured membrane is formulated by introducing the pertinent inertia and stiffness terms. Importantly, the proposed modifications do not increase the size of the problem compared wiith classical elasticity. Numerical simulations confirm that the vibrational properties are affected by the microstructural characteristics of the material, particularly in the high-frequency regime.

[1]  M. Born,et al.  Wave Propagation in Periodic Structures , 1946, Nature.

[2]  P. Rosenau,et al.  Dynamics of dense lattices. , 1987, Physical review. B, Condensed matter.

[3]  Umesh A. Korde,et al.  Membrane vibration experiments: An historical review and recent results , 2006 .

[4]  Ching S. Chang,et al.  Second-gradient constitutive theory for granular material with random packing structure , 1995 .

[5]  J. Wattis.  Quasi-continuum approximations to lattice equations arising from the discrete nonlinear telegraph equation , 2000 .

[6]  David R. Owen,et al.  an Introduction to Finite Element Method , 1979 .

[7]  Fusao Oka,et al.  Dispersion and wave propagation in discrete and continuous models for granular materials , 1996 .

[8]  A. G. Every,et al.  Determination of the dispersive elastic constants of the cubic crystals Ge, Si, GaAs, and InSb , 2008 .

[9]  J. Reddy.  An introduction to the finite element method , 1989 .

[10]  R. Langley.  THE RESPONSE OF TWO-DIMENSIONAL PERIODIC STRUCTURES TO IMPULSIVE POINT LOADING , 1997 .

[11]  J. Awrejcewicz,et al.  Continuous models for 2D discrete media valid for higher-frequency domain , 2008 .

[12]  Jacob Fish,et al.  Non‐local dispersive model for wave propagation in heterogeneous media: one‐dimensional case , 2002 .

[13]  A. Louisa,et al.  コロイド混合体における有効力 空乏引力から集積斥力へ | 文献情報 | J-GLOBAL 科学技術総合リンクセンター , 2002 .

[14]  H. Askes,et al.  Finite element modelling of wave dispersion with dynamically consistent gradient elasticity , 2009 .

[15]  A R Bishop,et al.  Continuum approach to discreteness. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  R. de Borst,et al.  Enhanced continua and discrete lattices for modelling granular assemblies , 2005, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[17]  E. Aifantis,et al.  Finite element analysis with staggered gradient elasticity , 2008 .

[18]  A. Tyas,et al.  Asymptotic equivalence of homogenisation procedures and fine-tuning of continuum theories , 2008 .

[19]  J. Fish,et al.  A Dispersive Model for Wave Propagation in Periodic Heterogeneous Media Based on Homogenization With Multiple Spatial and Temporal Scales , 2001 .

[20]  D. Beskos,et al.  Wave dispersion in gradient elastic solids and structures: A unified treatment , 2009 .

[21]  Wieslaw Beres,et al.  Book Review: Finite Element Analysis with Error Estimators , 2006 .

[22]  V. Erofeyev.  Wave processes in solids with microstructure , 2003 .

[23]  H. Askes,et al.  One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure: Part 2: Static and dynamic response , 2002 .

[24]  Andrei V. Metrikine,et al.  One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure: Part 1: Generic formulation , 2002 .

[25]  A. Metrikine.  On causality of the gradient elasticity models , 2006 .

[26]  Andrei V. Metrikine,et al.  Comparison of wave propagation characteristics of the Cosserat continuum model and corresponding discrete lattice models , 2001 .

[27]  R. Taylor.  The Finite Element Method, the Basis , 2000 .

[28]  J. Awrejcewicz,et al.  Continuous models for 1D discrete media valid for higher-frequency domain , 2005 .

[29]  Philip Rosenau,et al.  Dynamics of nonlinear mass-spring chains near the continuum limit , 1986 .