  External loads typically have a indirect influence on phonon curves, i.e.,
they influence the phonon curves by changing the state about which
linearization is performed. In this paper, we show that in nanotubes, the axial
load has a direct first-order influence on the long-wavelength behavior of the
transverse acoustic (TA) mode. In particular, when the tube is force-free the
TA mode frequencies vary quadratically with wave number and have curvature
(second derivative) proportional to the square-root of the nanotube's bending
stiffness. When the tube has non-zero external force, the TA mode frequencies
vary linearly with wave number and have slope proportional to the square-root
of the axial force. Therefore, the TA phonon curves -- and associated transport
properties -- are not material properties but rather can be directly tuned by
external loads. In addition, we show that the out-of-plane shear deformation
does {\em not} contribute to this mode and the unusual properties of the TA
mode are exclusively due to bending. Our calculations consist of 3 parts:
first, we use a linear chain of atoms as an illustrative example that can be
solved in close-form; second, we use our recently-developed symmetry-adapted
phonon analysis method to present direct numerical evidence; and finally, we
present a simple mechanical model that captures the essential physics of the
geometric nonlinearity in slender nanotubes that couples the axial load
directly to the phonon curves. We also compute the Density of States and show
the significant effect of the external load.
