## 2 Theoretical Model  
Spiral interfaces of a rolled-up microtube with multiple windings (Figure 1a,b) are modelled by cylindrical interfaces of a multilayer tube that consists of coaxial shells (multishell) as shown schematically in Figure 1c. We use the elastic continuum model because it works well for the considered dimensions of the structures. It was shown previously that such a model gives results in good agreement with the lattice dynamics and MD simulations [34, 39, 40, 41, 42]. Moreover, we are primarily interested in the acoustic phonon confinement energies near the Brillouin zone (BZ) center. These phonon modes are primarily responsible for heat transfer and can be directly studied with Brillouin spectroscopy. The elastic continuum model is particularly accurate for the BZ center phonons. We consider theoretically infinitely long structures because, practically, the length of micro- and nano-tubes is much larger than their diameter. The effect of the finite length on the phonon energies is negligible in our case. Figure 1: Microtube with multiple windings fabricated using roll-up technology, a scheme (**a**) and a STEM image of a radial superlattice from a rolled-up 20 nm InGaAs/10 nm Ti/46 nm Au layer (**b**) (From Ref. [58] © IOP Publishing. Reproduced with permission. All rights reserved). Cross-section in the plane orthogonal to the \(X\)-axis of a multilayer tube consisting of coaxial shells (multishell) (**c**). The tube core and the outer medium are vacuum. The axis of the structure is selected as the \(X\)-axis. The polar coordinates in the \(Y\)Z-plane are (\(r\),\(\varphi\)). The picture corresponds to a multishell with a periodic alternation of two materials. The displacement vector \(\mathbf{u}_{m}\) in each layer (\(m=1,...,\)\(N\)), treated as an elastic continuum, obeys the equations of elastodynamics [64]:  
\[\mu_{m}\
abla^{2}\mathbf{u}_{m}+(\lambda_{m}+\mu_{m})\
abla\
abla\mathbf{\cdot u }_{m}=\rho_{m}\frac{\partial^{2}\mathbf{u}_{m}}{\partial t^{2}}. \tag{1}\]  
At every intershell boundary \(r_{m}\), (\(m=1,\)\(\ldots,\)\(N\)-1) the following six boundary conditions represent continuity of the stress tensor components:  
\[ \begin{split}\sigma_{rm}&=\sigma_{rm+1};\\ \sigma_{rm}&=\sigma_{rm+1};\\ \sigma_{r\phi m}&=\sigma_{r\phi m+1},\end{split}  \tag{2}\]  
and of the displacement vector components:  
\[ \begin{split} u_{m}&=u_{m+1};\\ u_{m}&=u_{m+1};\\ u_{\phi m}&=u_{\phi m+1}.\end{split}  \tag{3}\]  
At the internal boundary \(r_{0}\), the following three boundary conditions represent vanishing of the stress tensor components:  
\[ \begin{split} 0&=\sigma_{r\
u};\\ 0&=\sigma_{r1};\\ 0&=\sigma_{r\rho 1}.\end{split}  \tag{4}\]  
Similarly, at the external boundary \(r_{N}\), the following three boundary conditions represent vanishing of the stress tensor components:  
\[ \begin{split}\sigma_{rN}&=0;\\ \sigma_{rN}&=0;\\ \sigma_{r\rho N}&=0.\end{split}  \tag{5}\]  
In total, there are 6\(N\) boundary conditions. The boundary problem described by the Equation (1) and the boundary conditions (2) to (5) for an arbitrary number of the shell pairs satisfies the correspondence principle with respect to the case of a two-shell composite coaxial tube [65,66]. The solutions to the equations of elastodynamics are sought in the form of a combination of dilatational waves and shear (equivvoluminal) waves:  
\[\mathbf{u}_{m}=\Delta\Phi_{m}+\
abla\mathbf{\times H}_{m}. \tag{6}\]  
Here the scalar potential of the dilatational motion and the vector potential of the shear motion along the axis of the structure (which has only two independent components) satisfy the equations:  
\[ \begin{split}\
u_{1m}^{2}\
abla^{2}\Phi_{m}&=\ddot {\Phi}_{m},\\ \
u_{2m}^{2}\
abla^{2}\mathbf{H}_{m}&=\ddot{\mathbf{ H}}_{m}.\end{split}  \tag{7}\]  
The velocities \(\
u_{1m}\) and \(\
u_{2m}\) correspond to dilatational and shear waves in the material of the \(m\)-th layer. The solutions to the wave equations are sought as plane waves traveling along the axis of the structure.