Czesław Woźniak

A nonstandard method of modelling of thermoelastic periodic composites

Abstract The purpose of the paper is to derive from the equations of the nonlinear thermoelasticity a certain class of homogenized models of periodic composite materials. The main feature of the proposed approach is a possibility of modelling local stresses and heat fluxes with various approximations. The approach is based on the methods of nonstandard analysis. The linearized equations and an example of application to laminated materials are discussed.

Micromorphic effects in a modelling of periodic multilayered elastic composites

Abstract The paper deals with the problems of modelling of multilayered periodic composites. The proposed models take into account certain micromorphic effects resulting from the periodic structure of the body. The governing equations describing motions and stresses of the nonhomogeneous elastic materials with microperiodic structure in the nonlinear as well as linear cases have been derived. As an example the time-harmonic vibrations of laminated medium consisting of alternating layers of two homogeneous, isotropic, linear-elastic materials have been discussed.

ON THE MODELING OF THE HYPERBOLIC HEAT TRANSFER PROBLEMS IN PERIODIC LATTICE-TYPE CONDUCTORS

The aim of this contribution is to propose, compare, and apply two kinds of simplified mathematical models for the analysis of hyperbolic problems describing heat transfer in dense periodic lattice-type conductors of an arbitrary form. The considerations are based on the Cattaneo-type constitutive heat transfer law. We begin with the formulation of a discrete model represented by a system of ordinary differential equations that have a finite difference form with respect to the spatial coordinates. By using some smoothness operations we derive continuum models from the aforementioned finite difference formulation. The general results are illustrated and compared in the example of a temperature wave propagating in a certain special lattice-type periodic conductor.

Boundary-Layer Phenomena in a Laminated Rigid Heat Conductor

In the contribution it is shown that a temperature distribution in a laminated rigid conductor can suffer certain near-boundary and near-initial fluctuations. These fluctuations are caused by periodic heterogeneity of a conductor. The aim of the research is to investigate the above fluctuations. To this end a certain approximate mathematical model for the heat transfer analysis in a laminated conductor is proposed. The main feature of the model is a decomposition of the heat conduction problem into an independent boundary-layer problem for temperature fluctuations and a problem of finding the averaged temperature. This fact makes it possible to satisfy boundary and initial conditions for temperature not only by its averaged part but also by temperature fluctuations. For selected problems the obtained results are compared with those derived from the homogenized model as well as from the exact Fourier heat transfer equation. It is shown that in contrast to the known homogenized model the proposed model describes the initial and material boundary layer effect on the heat transfer in a laminated rigid conductor.

Analytical Mechanics of Elastic Media

Using the term “analytical mechanics” we usually mean the specific kind of presentation of mechanics of the finite systems of particles; the concept of constraints and that of the generalized coordinates plays an important role in this presentation. In this paper we are to deal with mechanics of uncountable systems of particles constituting material continua. However, our way of presentation is different from that used in the classical approach to continuum mechanics. We start from the dynamics of an uncountable system of homogenously deformable elastic particles and we impose certain restrictions on the motion or on the internal forces. Such system has much more general structure then the well known classical elastic continuum being characterized not only by its material properties but also by the restrictions imposed on the motion and the system of internal forces. It can be observed that all known “technical” or “approximate” theories of elasticity (theories of elastic plates, shells, rods as well as different finite element approaches, formulations based on the Ritz or Galerkin methods, etc.) can be directly derived form the analytical mechanics of elastic media presented in this contribution. At the same time, mechanics of such “structured” media is also of the more general nature than the mechanics of constrained continuum [1,2], where only restrictions for the deformations are taken into account.