The influence of random microstructure on wave propagation through heterogeneous media
 First Online:
 Received:
 Accepted:
DOI: 10.1007/s1070401601702
 Cite this article as:
 Song, Y., Gitman, I.M., Parnell, W.J. et al. Int J Fract (2017) 204: 115. doi:10.1007/s1070401601702
 786 Downloads
Abstract
In this paper the influence of mechanical and geometrical properties, both deterministic and stochastic in nature, of a heterogeneous periodic composite material on wave propagation has been analysed in terms of the occurrence of stopbands. Numerical analyses have been used to identify those parameters that have the most significant effect on the wave filtering properties of the medium. A striking conclusion is that randomness in geometrical properties has a much larger effect than randomness in mechanical properties.
Keywords
Stopband Wave filter Composite material Laminate Randomness Wave propagation1 Introduction
In a material with disorder there is no longer the clear notion of a stopband as in the periodic case (Sheng 1995). The literature discusses strong and weak disorder, most commonly in the positional disorder of e.g. inclusions in a matrix medium. The transition from weak to strong disorder results in the loss of the bandgap structure present in periodic media. The effective wavenumber becomes complex at all frequencies and so for an infinite medium the theory predicts broadband attenuation, although this depends on the relative magnitude of the imaginary and real parts of the effective wavenumber. Furthermore, in reality the priority is to understand wave propagation through media of finite extent and so what is perhaps most important is the magnitude of a transmission coefficient across the medium, measuring the amount of energy that has passed through the system. The aim of this work is therefore to understand the influence of nonperiodic internal structure of materials on timeharmonic elastic wave propagation, and specifically how this affects the presence or otherwise of stopbands.
The mass density and Young’s modulus of each phase comprising the unit cell, their volume fraction and the size of the unit cell relative to the medium itself are all parameters that influence the stopband phenomenon. However a thorough study of their influence on the properties of stopbands does not appear to exist in the literature. Furthermore, when these parameters are subject to various degrees of random perturbation it is not clear how this will affect the stopband properties. Thus, in this paper randomness in both mechanical and geometric properties will be studied; the analyses will be carried out numerically and the wave filter effects will be compared with those of the undisturbed, periodic medium.
2 Setup of the numerical experiment
Figure 1 illustrates the configuration analysed throughout: numerical simulations, using the Newmark constant average acceleration time integration method, are conducted on a finite bar of total length L. This bar comprises four different regions \(V_0 , V_1 ,V_2 \) and \(V_3 \) with the last three being split up into subdomains \(V_j^l \) and \(V_j^r \) which are located on the left and right of the domain \(V_0 \), respectively. In order to ensure that no waves are reflected back into the domain of interest, we have chosen a bar length with sufficiently large zones beyond the actual domain of interest. In order to slow down wave propagation significantly, two zones of impedancematched layers have been taken on either side of the central zone, leading to \(L/L_0 =2.2\) (see below for full details). The medium in \(V_1 \) is chosen to have properties that correspond to the harmonic mean of the Young’s modulus and arithmetic mean of the density of the material that occupies \(V_0 \) (this will be discussed in more detail below).
The source of longitudinal elastic waves is located at the centre of region \(V_1^l \) and the receiver is placed at the centre of the region \(V_1^r \). Regions \(V_2 \) and \(V_3 \) are socalled Perfectly Matched Layers (PMLs) and are impedance matched to \(V_1 \). PMLs are used here as an alternative to absorbing boundary conditions. PMLs slow the wave down, ensuring that no reflections can be generated which would travel back into the domain of interest over the timescale of the simulation. In order to ensure equal impedance across regions \(V_1 , V_2 \) and \(V_3 \), we set \(\sqrt{\rho _1 E_1 }=\sqrt{\rho _2 E_2 }=\sqrt{\rho _3 E_3 }\). The density and Young’s modulus contrasts in these domains are taken as follows: \(\rho _2 =10\rho _1 , \rho _3 =50\rho _1 \) and \(E_2 =0.1E_1 , E_3 =0.02E_1 \). This implies that the wave speeds \(c_1 , c_2 \) and \(c_3 \) in the outer subdomains are related by the expressions \(c_2 =\sqrt{E_2 /\rho _2 }=0.1c_1 \) and \(c_3 =\sqrt{E_3 /\rho _3 }=0.02c_1 \), noting that the wave speeds in the PMLs are very small as required.
The transmission coefficient can be defined as \(T(f)=\frac{A(f)}{B(f)}\), with amplitudes A(f) and B(f) being obtained after Fourier transform of a received displacement, following a continuous sine wave passing through homogeneous (resulting in B(f)) and heterogeneous (resulting in A(f)) specimens. The sine wave starts at \(t=0\) with angular frequency \(\omega \), amplitude \({{\mathcal {F}}}\) and associated forcing \(F=\mathcal{F}cos\left( {\omega t} \right) \) at the source point and, as usual, frequency \(f=\frac{\omega }{2\pi }.\)
In a finite domain simulated numerically, it is expected that there may always be a very small amount of energy transmitted; thus a stopband criterion is adopted according to which a frequency resides in a stopband when \(T\le 0.05\). In all tests an angular frequency ranging from \(\omega =10^{5}\,\hbox {rad}/\hbox {s}\) to \(4.5\times 10^{6}\,\hbox {rad}/\hbox {s}\), in intervals of \(5\times 10^{4}\,\hbox {rad}/\hbox {s}\) is considered.
3 Influence of mechanical and geometrical properties of a periodic composite
Contrast in Young’s moduli: Vary the contrast parameter \(\beta _E \) whilst keeping \(\beta _\rho =1\) and \(\ell /L_0 =0.1\) Four different contrasts have been analysed: \(\beta _E =0.05\), 0.1, 0.25, 0.5,
Contrasts in mass densities: Vary the density contrast parameter \(\beta _\rho \) whilst keeping \(\beta _E =1\) and \(\ell /L_0 =0.1\). Four different contrasts have been analysed: \(\beta _\rho =0.05\), 0.1, 0.25, 0.5,
Variation in unit cell lengths: Take \(\ell =0.002\hbox { m}, \ell =0.004\,\hbox { m}, \ell =0.01\,\hbox { m}\) and \(\ell =0.02\,\hbox { m}\), while keeping constant \(\beta _E =0.25,\, \beta _\rho =0.1\) and \(L_0 =0.1\,\hbox {m}\).
Note that transmission coefficients are presented here as functions of normalised frequencies. The normalisation has been performed with respect to the characteristic time scale \(t_c =L_o /c\) (with averaged microstructural properties used in order to compute c) via \(\bar{\bar{f}} =f*t_c \).
Increasing the contrast in Young’s moduli (decreasing \(\beta _E )\), leads to a bandgap at lower frequency and the transmission coefficient in the passband drops slightly. Low frequency bandgap widths are relatively insensitive to changes in \(\beta _\rho \) however (Fig. 3left);
Increasing the contrast in density leads to a significant increase in the width of the first stopband and the transmission coefficient associated with the second passband also decreases (Fig. 3centre);
Increasing the unit cell length whilst keeping \(L_0 \) fixed gives rise to a stopband at lower frequency (Fig. 3right).
4 Influence of randomness on the bandgap structure of composites
So far the discussion has focussed on materials with heterogeneous but strictly periodic structure. In this section, the influence of randomness in the mechanical and geometrical parameters will be studied.
Randomness in mechanical and geometrical parameters: associated random properties
Case  \(C_v \left( {E_a } \right) \)  \(C_v \left( {E_b } \right) \)  \(C_v \left( {\rho _a } \right) \)  \(C_v \left( {\rho _b } \right) \)  \(C_v \left( {l_a } \right) \)  \(C_v \left( {l_b } \right) \) 

Periodic  0  0  0  0  0  0 
Random Young’s moduli  0.05  0.05  0  0  0  0 
0.1  0.1  0  0  0  0  
0.2  0.2  0  0  0  0  
Random densities  0  0  0.05  0.05  0  0 
0  0  0.1  0.1  0  0  
0  0  0.2  0.2  0  0  
Random geometry  0  0  0  0  0.05  0.05 
0  0  0  0  0.1  0.1  
0  0  0  0  0.2  0.2 
In Fig. 4 the average transmission coefficients as functions of frequency are plotted for the cases of randomness introduced in Young’s moduli (Fig. 4left), densities (Fig. 4centre) and geometry (Fig. 4right).
This can be understood as follows. From Fig. 3right it is clear that the position of the first passband scales directly with the value of the unit cell length, and higher passbands appear at certain intervals along the frequency axis. However, when this is translated into corresponding wave lengths \(\lambda \) according to \(\lambda =c/f\) (taking the averaged material properties to compute c), it becomes clear that the higher passbands are associated with smaller wave lengths; these smaller wave lengths eventually become smaller than the length of the unit cell. Thus, a randomised unit cell length has very little influence on the position and extent of the first passband, but it affects the subsequent passsbands.
5 Conclusions
In this study, the influence of both heterogeneous mechanical and geometrical properties on wave propagation has been tested, in particular their effects on stopbands. Randomness in the mechanical properties does not appear to affect bandgap structure significantly. On the other hand, randomness in the geometrical properties, even in the form of moderate perturbations, can lead to a significant reduction of the transmission coefficient in the second passband, and, eventually, with sufficient randomness, this second passband can be transformed into a stopband. This difference can be ascribed to the fact that in this study the source of heterogeneity is predominantly a geometrical distribution of material phases configured in series.
Acknowledgements
We gratefully acknowledge the Leverhulme Trust for financial support under grant number F/00 120/CC. W.J. Parnell is also grateful to the Engineering and Physical Sciences Research Council for his research fellowship (EP/L018039/1).
Funding information
Funder Name  Grant Number  Funding Note 

Leverhulme Trust (GB) 
 
Engineering and Physical Sciences Research Council 

Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.