Blur Effect in a Multiple Particle Inverse Problem for Fiber-Reinforced Composites

Abstract

The Prony method of scattering data analysis is extended to an inverse problem for a fiber-reinforced composite. Unidirectional fibers of shear moduli $\mu _k$ ($k=1,2, \ldots , n$) are embedded in the host of shear modulus $\mu$. We consider antiplane strain of the fibrous composite when a section perpendicular to the axis of fibers is the unit disk which contains n non-overlapping inclusions. The contact between the components is supposed to be perfect. The main attention is paid to rigid inclusions when $\mu _k \gg \mu$. Let the longitudinal displacement u be given on the unit circle. Other components of displacement vanish in the unit disk in the antiplane statement. The considered problem is written in terms of complex potentials and solved by a method of functional equations. In particular, the out-of-plane traction proportional to the normal derivative $\frac{\partial u}{\partial \textbf{n}}$ is found on the unit circle. This yields a constructive method to the symbolic approximation of the Dirichlet-to-Neumann operator for an arbitrary multiply connected circular domain. The method is applied to the inverse problem for non-overlapping equal disks whose centers $a_k$ ($k=1,2,\ldots ,n$) have to be determined. Let the displacement u and the traction $\mu \frac{\partial u}{\partial \textbf{n}}$ be given on the outer unit circle. We construct explicitly a polynomial $P_n(z)$ whose complex roots coincide with the centers of inclusions $a_k$. This result can be considered as a solution to the special Prony problem. The considered examples demonstrate the effect of blurring for large n when disks in the near-boundary vicinity are properly determined. The location of the deeper disks is blurry and can be determined by the same equation $P_n (z) = 0$ but solved with higher accuracy.