Microstructure-dependent Band Gaps for Elastic Wave Propagation in a Periodic Microbeam Structure


A new model for producing band gaps for flexural elastic wave propagation in a periodic microbeam structure is developed using an extended transfer matrix method and a non-classical Bernoulli–Euler beam model that incorporates the strain gradient, couple stress and velocity gradient effects. The band gaps predicted by the new model depend on the three microstructure-dependent material parameters of each constituent material, the beam thickness, the unit cell length and the volume fraction. A parametric study is conducted to quantitatively illustrate these factors. The numerical results reveal that the first band gap frequency range increases with the increases of the three microstructure-dependent material parameters, respectively. In addition, the band gap size predicted by the current model is always larger than that predicted by the classical model, and the difference is large for very thin beams. Furthermore, both the unit cell length and volume fraction have significant effects on the band gap.

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a :

Unit cell length of a two-phase periodic composite beam

\({a}_{\mathrm{1}}\) :

Length of material I in the unit cell

\({a}_{\mathrm{2}}\) :

Length of material II in the unit cell

b :

Width of the beam

f :

Wave frequency

h :

Thickness of the beam

i :

Imaginary unit

k :

Wave number

l :

Material length scale parameter

\(l_{m}\) :

Couple stress coefficient

\(l_{s}\) :

Strain gradient coefficient

\(l_{v}\) :

Velocity gradient coefficient

\(m_{{0}}, m_{\mathrm{2}}\) :

Mass inertia

q :

Bloch wave number in the x-direction

w :

Deflection on the beam centroidal axis

A :

Cross-sectional area of the beam

\(C_{\mathrm{0}}, C_{\mathrm{2}}, D, S\) :

Convenient material parameters

E :

Young’s modulus

I :

Second moment of cross-sectional area of the beam

\({{\varvec{I}}}\) :

6 by 6 identity matrix

\({{\varvec{T}}}\) :

Transfer matrix

\(V_{f}^{\mathrm{(I)}}\) :

Volume fraction of material I

W :

Coefficient for a harmonic propagating wave

\({\omega }\) :

Angular frequency of the wave

\({\mu }\) :

Shear modulus

\({\rho }\) :

Mass density of the beam material

\({\nu }\) :

Poisson’s ratio

\(^{\mathrm{(I)}}\) :

Item for material I

\(^{\mathrm{(II)}}\) :

Item for material II


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The work reported here is funded by the National Natural Science Foundation of China [grant numbers 12002086, 11872149 and 11472079] and the Fundamental Research Funds for the Central Universities [grant number 2242020R10027]. These supports are gratefully acknowledged.

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Correspondence to Gongye Zhang or Changwen Mi.

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Zhang, G., Zheng, C., Qiu, X. et al. Microstructure-dependent Band Gaps for Elastic Wave Propagation in a Periodic Microbeam Structure. Acta Mech. Solida Sin. (2021). https://doi.org/10.1007/s10338-021-00217-z

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  • Band gaps
  • Wave propagation
  • Bernoulli–Euler beam
  • Strain gradient
  • Couple stress
  • Velocity gradient