Multiple Wave Scattering and Calculated Effective Stiffness and Wave Properties in Unidirectional Fiber-Reinforced Composites

Abstract

Analytic methods of elastic wave scattering in fiber-reinforced composite materials are investigated in this study to calculate the effective static stiffness (axial shear modulus, m) and wave properties (axially shear wave speed, B and attenuation, Y) in composites. For simplicity only out-of-plane shear waves are modeled propagating in a plane transverse to the fiber axis. Statistical averaging of a spatially random distribution of fibers is performed and a simultaneous system of linear equations are obtained from which the effective global wave numbers are numerically calculated. The wave numbers, K=Re(K)+iIm(K), are complex numbers where the real parts are used to compute the effective axial shear static stiffness and wave speed; the imaginary parts are used to compute the effective axial shear wave attenuation in composites.

Three major parts of this study are presented. The first part is the discussion of multiple scattering phenomena in a successive-events scattering approach. The successive-events scattering approach is proven to be mathematically exact by comparing the results obtained by the many-bodies-single-event approach. Scattering cross-section is computed and comparison of the first five scattering orders is made. Furthermore, the ubiquitous quasi-crystalline approximation theorem is given a justifiable foundation in the fiber-matrix composite context. The second part is to calculate m, B and Y for fiber-reinforced composites with interfacial layers between fibers and matrix. The material properties of the layers are assumed to be either linearly or exponentially distributed between the fibers and matrix. A concise formula is obtained where parameters can be computed using a computationally easy-to-program determinant of a square matrix. The numerical computations show, among other things, that the smoother (more divisional layers), or thinner, the interfacial region the less damped are the composite materials. Additionally composites with exponential order distribution of the interfacial region are more damped than the linear distribution ones. The third part is to calculate m, B and Y for fiber-reinforced composites with interfacial cracks. The procedures and computational techniques are similar to those in the second part except that the singularity near the crack tip needs the Chebychev function as a series expansion to be adopted in the computation.

Both the interfacial layers and interfacial crack cases are analyzed in the low frequency range. The analytic results show that waves in both cases are attenuated and non-dispersive in the low frequency range. The composites with interfacial layers are transversely isotropic, while composites with interfacial cracks are generally transversely anisotropic.