Effective material properties for shear-horizontal acoustic waves in fiber composites

The effective dynamic properties of composites made of elastic cylindrical fibers randomly distributed in another elastic solid can be evaluated with plane shear-horizontal acoustic waves. In this paper, it is shown that the effective mass density and the effective shear stiffness are complex valued and frequency dependent. Simple formulas are derived for these effective quantities. The low-frequency limit of these formulas is found to be in agreement with physical expectations. The derivation is based on the multiple-scattering approach of Waterman and Truell, where each cylinder of finite cross section is replaced with an equivalent line scatterer. Numerical results are presented for the effective mass density and effective shear stiffness for various values of frequency, cylinder concentration, and elastic properties of the cylinders and matrix.

Figure 1

Schematic diagram of SH waves propagating inside and outside a layer of thickness $2h$. The propagation vectors are in the $y_3$ direction. The particle displacements are in the $y_2$ direction.

Figure 2

Real and imaginary parts of the effective mass density $\rho ∕{\rho }_0$ $(={\tilde{\rho }}^{\prime }+i{\tilde{\rho }}^{″})$ for empty cylindrical cavities in an elastic solid versus the dimensionless frequency $\tilde{\omega }$ for three concentrations $\varphi$.

Figure 3

Real and imaginary parts of the effective shear stiffness $M∕{\mu }_0$ $(={\tilde{M}}^{\prime }-i{\tilde{M}}^{″})$ for empty cylindrical cavities in an elastic solid versus the dimensionless frequency $\tilde{\omega }$ for three concentrations $\varphi$.

Figure 4

Real and imaginary parts of the effective mass density $\rho ∕{\rho }_0$ $(={\tilde{\rho }}^{\prime }+i{\tilde{\rho }}^{″})$ for elastic circular cylinders in elastic solids versus the dimensionless frequency $\tilde{\omega }$ for various stiffness ratios $\mu$. $\varphi =15.71\%$ and $c_0=c_1$.

Figure 5

Real and imaginary parts of the effective shear stiffness $M∕{\mu }_0$ $(={\tilde{M}}^{\prime }-i{\tilde{M}}^{″})$ for elastic circular cylinders in elastic solids versus the dimensionless frequency $\tilde{\omega }$ for various stiffness ratios $\mu$. $\varphi =15.71\%$ and $c_0=c_1$.

Figure 6

Modulus of the reflection coefficient versus the frequency for a semi-infinite sample of a boron-epoxy composite. The radius and concentration of the boron fibers are $50\mu \mathrm{m}$ and 9.42%, respectively. The reflection coefficient is $R=-Q$, where $Q$ is given by (14, 17). In (17), $\Theta =\rho k_0∕{\rho }_{0}K$. Solid line: $\rho$ is given by (18). Dashed line: $\rho$ is given by the mixture law (26). ${\rho }_0=1261\mathrm{kg}∕{\mathrm{m}}^3$, ${\rho }_1=2682\mathrm{kg}∕{\mathrm{m}}^3$, ${\mu }_0=1.8\mathrm{GPa}$, and ${\mu }_1=161\mathrm{GPa}$.