Thermomechanics of helical strands and helical-fiber-reinforced rods
Helical strands, or helically wound cables, are made of layers of individual wires wrapped around a common central axis. They are seen in ropes and power transmission cables. Similar structures are also present in biological tissues in the form of helical fiber reinforced composites. Regardless of the distinct applications, the helical wrapping in all such structures introduces mirror asymmetry, i.e. chirality, resulting in effective properties not present in the base material. One most prominent effect of the presence of helices is the coupling between tension and torsion, which is widely modeled and studied in the literature. However, complex issues arise when there is bending. First, the effective bending stiffness is difficult to estimate and existing analytical models require careful validation. We conduct a full-fledged finite element analysis of the bending of a single layered helical strand, with internal friction and pretension. The effects of the pretension level, bending amplitude, and the friction coefficient on the effective bending stiffness are elucidated. Second, in the no-slip regime, the existing Euler-Bernoulli framework for helical strands is extended. A Timoshenko rod model is established for helical strands, with a 6 by 6 stiffness matrix governed by five independent elastic moduli. The model is capable of capturing the bending-shearing coupling in helical strands due to the underlying chirality, and correctly predicting the cross section forces and moments under several boundary value problems when compared with finite element results, whereas the existing Euler-Bernoulli model wrongly predicts particular forces or moments to be zero. The bending-shearing coupling is also demonstrated by the non-planar bending of helical strands under a single transverse force, or a single bending moment. The equations of vibration are then derived, with the eigenfrequencies and mode shapes identified. Due to chirality, the longitudinal and torsional modes are coupled, as are the bending-shearing modes in the two principal directions of the cross section. The Timoshenko rod model for helical strands is further extended by considering thermal expansion. The finite element analysis demonstrates a coupling between thermal expansion and torsion, due to the structural chirality, which is also incorporated into the final form of the thermomechanical constitutive relation of helical strands. With the thermomechanical constitutive relation, the thermoelastic waves in a helical strand are solved. With Fourier-type heat conduction, the thermelastic wave solutions are governed by four non-dimensional parameters: two thermoelastic coupling constants (in the longitudinal and torsional directions), a chirality parameter, and the Fourier number. The longitudinal and the torsional waves are dispersive and damped, and are dependent on the temperature. The adiabatic-isothermal transition of the wave propagation is dictated by the Fourier number. With Maxwell-Cattaneo-type heat conduction, the heat propagation follows a hyperbolic differential equation, with the heat wave celerity depending on the thermal relaxation $\tau$. The full thermoelastic wave solutions for helical strands are governed by a sixth-order algebraic equation. An additional non-dimensional parameter comes into play; it characterizes the speed of heat propagation compared with that of mechanical perturbations. The solutions show distinct behaviors for fast heat propagation v.s. slow heat propagation.
adiabatic-isothermal transition, telegraph equation, chirality, bending stiffness, stick-slip, contact, pretension, Timoshenko rod, finite element analysis, thermomechanical constitutive relation, vibration, eigenfrequency, bending-shearing coupling, thermal-torsional coupling, thermoelastic wave, discontinuity, phase velocity, group velocity, spectral finite element, Maxwell-Cattaneo heat conduction, dispersion relation, thermal relaxation time
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign