Macroscopic Modelling of Vibration and Stability Problems in Prestressed Microperiodic Elastic Solids

Abstract

The aim of this contribution is to propose and apply a certain averaged (macroscopic) mathematical model for the analysis of vibration and stability problems of prestressed microperiodic elastic solids. In contrast to the known homogenized model [2], [3] the proposed model describes the effect of the period length on the macroscopic solid behaviour. Formulation of the problemThe object of considerations is a description of dynamic behaviour of a microperiodic elastic composite solid subjected to initial stresses. This problem is not new. We can mention here approaches based on the known equation of the elasticity theory, [1] or on the homogenized model equations [2], [3]. However, the first approach can be applied only to some special periodic structures (like laminates) and the second approach neglects the effect of the period length on the overall solid behaviour. The main aim of this contribution is to formulate a new macroscopic model of the prestressed periodic solid describing the aforementioned effect. We also derive a 2D-model of a thin prestressed plate with the thickness of an order of the period length. Moreover, we apply the above 2D-model to a certain vibration and stability problem for a thin uniperiodic plate. The proposed approach is based on the tolerance averaging technique of PDEs with periodic coefficients and is a certain generalization of results given in [4]. In order to make this note selfconsistent we outline the basic concepts and assumptions of the applied modelling technique.Let ∆(x) ≡ ∆ + x be a periodic cell with a center at point x in the physical space such that ∆(x) ⊂ Ω where Ω is a region occupied by a solid in this space. The solid under consideration is assumed to be microperiodic; it means that the diameter of ∆ is negligibly small when compared to the smallest characteristic length dimension of Ω. The tolerance averaging technique is based on the well known definition of the averaging operation hf i(x) for an arbitrary integrable function f.The basic concept is that of a slowly varying function of an argument x. It is a function F satisfying the following tolerance averaging approximationwhich has to hold for every integrable function f , here ' is a certain tolerance relation, [4]. Let u(x, t), x ∈ Ω, be a displacement field at time t from the reference configuration of the periodic solid. Define w(x, t) ≡ hui(x, t). The first assumption of the modelling technique applied to the linear elasticity theory equations is that w(·, t) is a slowly varying function together with all derivatives. Hence in the decomposition of displacement field u = w + r into the averaged w and the residual r part we obtain hri ' 0. It follows that r can be interpreted as a fluctuation displacement field caused by the periodic nonhomogeneous structure of the solid. The second modelling assumption is that the fluctuation displacement field r can be approximated by means of the formula, summation over A = 1, ..., N holds:where h A (·) are the known a priori periodic shape functions, such that hh A i =...

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