Torsional rigidity of a bar with multiple fibers

Abstract

The classical problem of torsion is newly considered with the complex fast multipole method used to determine the torsional rigidity of a bar with multiple fibers. New analytical formulas are given for the rigidity in the case of circular contours of the fibers and the bar. It is shown that the method ensures the results which, up to three significant digits, agree well with the solutions obtained by series expansions. For a fixed concentration of a great many (up to 540) thin fibers whose shear modulus is significantly (30 times) greater than the shear modulus of the matrix, the torsional rigidity weakly depends on the diameter and the distance between the fibers. The torsional rigidity G becomes 2.5 times larger as the fiber concentration c increases from 0 to 0.16 for a very small concentration interval (0 ≤ c ≤ 0.03), where the dependence G(c) is linear. The inverse quantity 1/G (torsional compliance) varies linearly in a much wider range of concentrations (0 ≤ c ≤ 0.16).