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On the propagation of elastic waves in a multiperiodically reinforced medium

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Abstract

By a multiperiodically reinforced medium (multiperiodic composite) we mean a composite in which the matrix material is reinforced by two or more families of periodically spaced fibres. Moreover, at least along one direction the periods corresponding to different families are different. An example of this composite is shown in Fig. 1, where along the x 1-axis we deal with two different periods \(l_1^{1}, l_2^{1}\). The aim of the contribution is twofold. First, we propose a macroscopic (averaged) model of a multiperiodic composite, describing the effect of period lengths on the overall dynamic behaviour of the medium, in contrast to the known homogenized models. Second, we apply this model to the analysis of elastic waves propagating across a composite reinforced by two pairs of families of parallel periodically spaced fibres with different periods along certain direction.

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Abbreviations

0x 1 x 2 :

The Cartesian orthogonal co-ordinate system

\(\mathbb{R}^{2}\) :

0x 1 x 2-planes

x=(x 1, x 2):

A point on a plane 0x 1 x 2

∂ k  = ∂/ ∂x k :

Partial differentiation with respect to the x k-coordinate, k = 1,2

\(\nabla =(\partial _{1},\partial _{2})\) :

Differentiation operator

t :

Time co-ordinate

\(\dot{F}=\partial F/ \partial t\) :

Time differentiation

\({\bf K} \cdot {\bf L}, {\bf K}:{\bf L},{\bf K} \otimes {\bf L}\) :

Scalar product (e.g. K. ij L jk .), double scalar product (e.g. K. ij L ij .) tensor product (e.g. K. ij L kl .), respectively

•:

Full contraction of the fourth-order tensors

DF :

Runs over \(\{F, \partial_{1} F, \partial_{2} F, \dot{F}\}\) if F = F(x, t)

Dom F :

Domain of F

α, β:

Run over {1, 2}, unless otherwise stated summation convention does not hold

a, b :

Run over {1, 2}, unless otherwise stated summation convention does not hold

A, B :

Run over \(\{1,{\ldots},N\}\), summation convention holds

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Authors and Affiliations

  1. Department of Structural Mechanics, Łódź University of Technology, al. Politechniki 6, 93-590, Łódź, Poland

    Jarosław Jędrysiak

  2. Institute of Mathematics and Computer Sciences, Technological University of Częstochowa, ul. Dąbrowskiego 73, 42-200, Częstochowa, Poland

    Czesław Woźniak

Authors
  1. Jarosław Jędrysiak
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  2. Czesław Woźniak
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Correspondence to Jarosław Jędrysiak.

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Jędrysiak, J., Woźniak, C. On the propagation of elastic waves in a multiperiodically reinforced medium. Meccanica 41, 553–569 (2006). https://doi.org/10.1007/s11012-006-9003-0

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  • DOI: https://doi.org/10.1007/s11012-006-9003-0

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