Helical inclusion in an elastic matrix

Abstract

An elastic space containing an elastic helical rod subjected to both axial and radial extension as well as torsion is considered. Due to translation-rotation helical symmetry, the resulting elastic fields in the matrix can be expressed in terms of a two-dimensional helix-associated coordinate system. In this problem, a ‘helical elastic foundation’ as a generalization of the Winkler foundation is determined by means of which the interacting force and moment at the rod/matrix interface can be expressed in terms of the rod displacement. The matrix is assumed to be linear elastic while the geometric nonlinearity of the helical rod is taken into account. Using superposition of fundamental solutions for a homogeneous elastic space (in the absence of the rod), and the constitutive and equilibrium equations for the rod, the internal forces and moments in the rod as well as the displacement and elastic fields in the matrix are obtained. Along with the general results, two asymptotic solutions are presented. The first, corresponding to a small curvature but not too small pitch, allows an analytical integration of the rod–matrix interaction over the rod cross-section boundary. The second corresponds to an almost straight helical rod: the helix becomes a straight line, but in the limit the main normal to its axis describes a screw surface as in the case of a ‘genuine’ helix. In this case, the helical elastic foundation has a closed-form parametric expression which is valid for a rather large range of the helix parameters. The foundation stiffness is found as a function of the helix pitch and the rod radius; the problem thus is reduced to a system of finite, nonlinear equations.