Ewaryst Wierzbicki

On the dynamic behavior of honeycomb based composite solids

SummaryThe aim of this contribution is twofold. First, a dispersive model of periodic composite solids made of an isotropic matrix reinforced by a hexagonal system of slender fibres or by a honeycomb-like slender skeleton is formulated. Second, this model is applied to the analysis of vibration and wave propagation problems in the above honeycomb based composites. Contrary to the known homogenized models the main feature of the proposed model is that it describes the effect of cell size on the overall dynamic behavior of a composite solid. It is shown that on the macro-level the response of honeycomb based composites is isotropic. It is also proved that there exist dispersive dilatational-type and shear-type waves, which can propagate in these composites. Simple formulae for lower and higher free vibration frequencies are derived, and the existence of certain restrictons imposed on the physically allowable wave propagation speeds is shown.

A macroscopic model of the diffusion and heat transfer processes in a periodically micro-stratified solid layer

SummaryA macroscopic mathematical model for the analysis of the diffusion and heat transfer processes in a bounded periodically micro-stratified thick solid layer will be derived. Contrary to the known asymptotic homogenization approach the proposed modelling method retains the microstructure length in a macroscopic description of the process. It will be shown that this effect makes it possible to formulate correctly boundary conditions on the boundaries which intersect the stratification and is responsible for certain transient and boundary layer phenomena.

A MACROSCOPIC MODEL FOR THE HEAT PROPAGATION IN THE MICROPERIODIC COMPOSITE SOLIDS

This article deals with the problem of heat propagation in periodic composites described by the Cattaneo-type constitutive law. The simplest model for the macroscopic behavior of these composites can be obtained using results of the well-known asymptotic homogenization theory. However, this model is not able to describe the phenomena related to higher order motions and higher propagation speeds that play an important role in wave propagation analysis. The aim of this contribution is to propose a more general macroscopic model for the heat propagation in periodic composites that can be used for the analysis of the aforementioned phenomena and is based on the tolerance averaging of equations with functional coefficients.

On the modelling of transient micro-motions and near-boundary phenomena in a stratified elastic layer

An averaged (macroscopic) model for the analysis of overall behavior of periodically micro-stratified linear-elastic thick bounded layer is derived. The main property of the obtained model equations is that their coefficients depend on the period of stratification contrary to the known homogenized model equations. It is shown that this model property is responsible for the description of certain transient micro-motions and near-boundary phenomena in a bounded laminated layer.

On Dynamics of Thin Plates With a Periodic Structure

The subject of investigations is elastodynamics of periodic plates like that shown in Fig. l. So far, two general approaches to the modelling of these plates leading to the effective 2D-models (i.e. represented by equations with constant coefficients) have been proposed. The first is based on the asymptotic analysis and multi-scale expansions, cf. [4], and yields what are called homogenized plate equations which have the same form as equations of homogeneous plates, [5,9]. Hence, the main attention in this approach is concentrated on calculations of constant coefficients in a homogenized equation, which are referred to as the effective plate stiffness (cf. [5,9] and [ 10] for an overview of papers on this subject). The second approach applies the tolerance averaging technique as a tool of modelling, [ 14]. We can mention here a series of papers [ 1-3,7,8,11-13] related to dynamics and stability of elastic plates and beams with a periodic structure. The tolerance averaging approach, in contrast to homogenization, makes it possible to describe the local dynamic plate behaviour (e.g. to determine higher order vibration frequencies caused by the plate periodic structure) and takes into account the rotational inertia effect on the plate dynamics. However, until now the effective 2D-models for dynamics of periodic plates, based on the tolerance averaging technique, have been restricted to situations in which the period length is sufficiently large when compared to the plate thickness.