Overall Dynamic Constitutive Relations of Micro-structured Elastic Composites

A method for homogenization of a heterogeneous (finite or periodic) elastic composite is presented. It allows direct, consistent, and accurate evaluation of the averaged overall frequency-dependent dynamic material constitutive relations. It is shown that when the spatial variation of the field variables is restricted by a Bloch-form (Floquet-form) periodicity, then these relations together with the overall conservation and kinematical equations accurately yield the displacement or stress modeshapes and, necessarily, the dispersion relations. It also gives as a matter of course point-wise solution of the elasto-dynamic field equations, to any desired degree of accuracy. The resulting overall dynamic constitutive relations however, are general and need not be restricted by the Bloch-form periodicity. The formulation is based on micro-mechanical modeling of a representative unit cell of the composite proposed by Nemat-Nasser and coworkers; see, e.g., [1] and [2].Comment: 23 pages, 6 figures, submitted to JMP

Bounds on Effective Dynamic Properties of Elastic Composites

We present general, computable, improvable, and rigorous bounds for the total energy of a finite heterogeneous volume element or a periodically distributed unit cell of an elastic composite of any known distribution of inhomogeneities of any geometry and elasticity, undergoing a harmonic motion at a fixed frequency or supporting a single-frequency Bloch-form elastic wave of a given wave-vector. These bounds are rigorously valid for \emph{any consistent boundary conditions} that produce in the finite sample or in the unit cell, either a common average strain or a common average momentum. No other restrictions are imposed. We do not assume statistical homogeneity or isotropy. Our approach is based on the Hashin-Shtrikman (1962) bounds in elastostatics, which have been shown to provide strict bounds for the overall elastic moduli commonly defined (or actually measured) using uniform boundary tractions and/or linear boundary displacements; i.e., boundary data corresponding to the overall uniform stress and/or uniform strain conditions. Here we present strict bounds for the dynamic frequency-dependent constitutive parameters of the composite and give explicit expressions for a direct calculation of these bounds

Micromechanically Based Constitutive Relations for Polycrystalline Solids

A basic method to estimate the overall mechanical response of solids which contain periodically distributed defects is presented. The method estimates the shape and growth pattern of voids periodically distributed over the grain boundaries in a viscous matrix. The relaxed moduli are obtained for a polycrytalline solid that undergoes relaxation by grain boundary sliding which accounts for the interaction effects. The overall inelastic nonlinear response at elevated temperatures in terms of a model which considers nonlinear power law creep within the grains, and linear viscous flow in the grain boundaries is discussed

Two-dimensional Phononic Crystals with Acoustic-Band Negative Refraction

A two-dimensional phononic crystal (PC) can exhibit longitudinal-mode negative energy refraction on its lowest (acoustical) frequency pass band. The effective elastodynamic properties of a typical PC are calculated and it is observed that the components of the effective density tensor can achieve negative values at certain low frequencies on the acoustical branches for the longitudinal-mode pass-band, and that negative refraction may be accompanied by either positive or negative effective density. Furthermore, such a PC has a high anisotropy ratio at certain low frequencies, offering potential for application to acoustic cloaking where effective material anisotropy is essential.Comment: in Proceedings of the ASME 2016 Conference on Smart Materials, Adaptive Structures and Intelligent Systems, Stowe, VT (2016

Torsional Instability of Cantilevered Bars Subjected to Nonconservative Loading

Cantilever bar torsional instability under nonconservative compression loadin

On the stability of equilibrium of continuous systems Technical report no. 65-1

Stability of equilibrium of linear elastic continuum - Galerkin metho

On the Stability Equilibrium of Continuous Systems

Sufficiency theorem for stability of linearly viscoelastic solid subjected to partial follower surface traction

Refraction Characteristics of Phononic Crystals

The refraction properties of phononic crystals are revealed by examining the anti-plane shear waves in doubly periodic elastic composites with unit cells containing rectangular and/or elliptical inclusions. The band-structure, group velocity, and energy-flux vector are calculated using a powerful variational method which accurately and efficiently yields all the field quantities over multiple frequency pass-bands. Equifrequency contours and energy-flux vectors are calculated as functions of the wave-vector. By superimposing the energy-flux vectors on equifrequency contours in the plane of the wave-vector components, and supplementing this with a three-dimensional graph of the corresponding frequency surface,a wealth of information is extracted essentially at a glance. This way it is shown that a composite with even a simple square unit cell containing a central circular inclusion can display negative or positive energy and phase-velocity refractions, or simply performs a harmonic vibration (standing wave), depending on the frequency and the wave-vector. Moreover that the same composite when interfaced with a suitable homogeneous solid can display: 1. negative refraction with negative phase-velocity refraction; 2. negative refraction with positive phase-velocity refraction; 3. positive refraction with negative phase-velocity refraction; 4. positive refraction with positive phase-velocity refraction; or even 5. complete reflection with no energy transmission, depending on the frequency, and direction and the wave length of the plane-wave which is incident from the homogeneous solid to the interface. By comparing our results with those obtained using the Rayleigh quotient and, for the layered case, with the exact solutions, the remarkable accuracy and the convergence rate of the present solution method are demonstrated. MatLab codes with comments will be provided