Homogenization of a micro-periodic helix

Abstract

Equations of motion governing the dynamics of helix are studied in the situation when the microstructure is micro-periodic. Using the asymptotic homogenization method, we derive these equations in the case of waves much longer than the length scale of a periodic unit cell and for any finite number of phases in the cell. The procedure of constructing a formal asymptotic expansion solution is derived. Generally, the constitutive coefficients are harmonic averages, while the mass density and polar moment of inertia are arithmetic averages. These results are illustrated numerically on the case of a two-phase helix.

Acknowledgement

We have benefited from comments of an anonymous reviewer. The work reported herein has been supported by the Canada Research Chairs program and the NSERC.

Notes

†Recall that in the SI system: lb = 4.44822 N; lb/ft² = 47.88 N/m²; ft lb = 1.3558 m N.