whose performance depends on the geometry of the in- ner hollow cavity of nanotubes. The current work has been inspired by the discovery of the self-healing properties of carbon nanostructures under electron-beam irradiation. 28–32 Banhart et al. 28 found that carbon onions, assemblies of concentric spher- ical carbon shells, shrink toward smaller onion struc- tures when irradiated. This shrinkage is attributed to the knock-on collision of carbon atoms from constituent shells; the resulting vacancies are healed by annealing reconstruction 33 that shrinks the system with reduced intershell spacings. As a result, extreme pressure arises in the core of spherical carbon shells. Using a similar irra- diation technique, Sun et al. 31 synthesized MWNTs with reduced interwall spacing and revealed high pressures of several tens GPa within the hollow cavity. It is thus ex- pected that self-contraction of only outermost walls of a MWNT (possibly realized by finely-focused beam) will provide the setting for radial corrugation of the interior part of the MWNT. It should be borne in mind that in the carbon onion ex- periments, diamond nucleation occurs at the center of the onion as the induced pressure is too high for the original sp 2 bonds to persist. 28 To make the current work con- vincing, therefore, it is important to examine whether the critical pressure p$_{c}$ for the corrugation is low enough to prevent the breaking of sp 2 bonds within the inner walls. We will revisit this issue in Section III B. Figs. 1(b) and (c)). As a consequence, various corruga- tion patterns arise in the cross section of the N -walled nanotube. Only a few examples of the corrugation pat- terns are shown in Fig. 1(d) together with the associated mode index n representing the wave number in the cir- cumferential direction. It is noteworthy that the mechanical 38,39 and structural 40 consequences of irradiation in few-walled nanotubes have already been explored by atomistic simulations. Xu et al. numerically reproduced the irradiation-induced high pressure within a double-walled nanotube $^{38}$; they artificially removed carbon atoms from the outer wall and observed the subsequent self-healing process that causes an extreme pressure acting on the inner wall. A similar healing process is assumed to oc- cur when the outer walls of a many-walled nanotube are eroded locally, as shown in Fig. 1(b). It should be noted that in the existing experiment, it is possible to remove carbon atoms from an MWNT with monolayer precision. 41 This fact supports the validity of our hy- pothesis that a part of the carbon atoms within only a desired number of outermost walls is removed locally. The stable cross-sectional shape of the embedded tube is obtained by minimizing its mechanical energy U per unit axial length, 21 When treating many-walled nanotubes, atomistic sim- ulations are realistic and accurate but they demand huge computational cost in general. Thus we used a simplified model based on the continuum approximation. 18,34–36 First, we map an MWNT onto concentric cylindrical shells endowed with 37 the in-plane stiffness C = 345 nN/nm, the flexural rigidity D = 0 . 238 nN · nm, and Pois- son’s ratio ν = 0 . 149, in which the inter-shell separation is defined by 0.3415 nm. Figures 1(a)-(c) illustrate the shrinking process considered in this paper. We assume an ( M + N )-walled nanotube under irradiation that kicks off a portion of carbon atoms located within M ( ≫ N ) outer walls surrounding N inner walls. Then, self-contraction of the M walls results in high pressure that exerts on the N walls that they encapsulate. For the sake of analyti- cal arguments, we simplify the M walls by a continuum elastic medium and assume the outermost boundary of the N -walled tube to be in contact with the inner sur- face of the medium throughout the shrinking process (see B. Continuum model of radially shrinking MWNT C. Energy formulation A. Self-healing nature of graphite sheets II. METHODOLOGY 2 (a) (b) (c) (d) n=4 n=9 n=5 FIG. 1: (color online) Illustration of radial shrinking hypoth- esis of MWNTs. (a) An ( M + N )-walled nanotube is sub- jected to electron-beam irradiation. (b) Irradiation with fine resolution kicks off a portion of carbon atoms located within M ( ≫ N ) outermost walls. (c) The eroded M walls are healed and radially contracting, as a result of which extreme pressure is exerted on the N innermost walls. (d) Typical corrugation modes of the pressed N -walled nanotubes are shown with the mode index n . U = U$_{D}$ + U$_{I}$ + U$_{M}$ + Ω . (1)