# Embedding Effect on the Mechanical Stability of Pressurised Carbon Nanotubes

Authors: Motohiro Sato; Hisao Taira; Tetsuro Ikeda; Hiroyuki Shima
Journal: Journal of Nanomaterials
 (2013)

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## Abstract

We elaborate on the cross-sectional deformation of carbon nanotubes embedded into a self-contracting host medium. The continuum elastic approach is used to formulate the mechanical energy of both the embedded nanotubes and the self-contracting outer medium with finite thickness. Our formula allows us to evaluate the critical radial pressure applied on the interface between the embedded nanotube and the outer contracting medium as well as the deformation mode that arises immediately above the critical pressure.
An interesting mechanical implication of the embedding effect, that is, the power-law dependence of the critical pressure on the elastic
modulus of the medium, is deduced by the theoretical approach established.


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## Body

## 1. Introduction

The salient structural feature of a carbon nanotube is its self-repairing behaviour that arises under high-energy beam irradiation [1, 2]. When the kinetic energy transferred from the incident beam to the constituent carbon atoms is sufficiently large, the atoms are pushed away from the original equilibrium positions, leaving vacancies in the host hexagonal lattice [3, 4]. In typical solids, such irradiation-induced vacancies survive without curing as time passes. However, this is not the case in carbon nanotubes; the removal of carbon atoms from the purely hexagonal lattice leads to a local reconstruction that acts to maintain its coherent network structure with cylindrical geometry. For this reason, carbon nanotubes are often referred to as self-repairing (or self-healing) materials [5–7].This self-repairing nature provides beneficial effects for manipulating the carbon nanotube morphology, especially when combined with heat treatment. For instance, fine-tuning of electron beam irradiation makes it possible to synthesize multiwall carbon nanotubes (MWNTs) with reduced interwall spacings [8]; the spontaneous shrinkage in the radial direction is a result of the knock-on collision of carbon atoms followed by annealing reconstruction of the vacancies. This experimental finding implies that when the outermost carbon walls of an MWNT are eroded selectively by irradiation, the self-contraction of the outermost walls exerts high pressure on the encapsulated, undamaged innermost walls [9–11]. Application of high pressure may then trigger a novel class of cross-sectional transformations of the inner walls [12], similar to the case of pristine (irradiation-free) MWNTs under hydrostatic pressure [13, 14]. Another class of radial contraction has been observed in MWNTs synthesized in the presence of nitrogen [15]. The yielded nanotubes showed polygon-shaped cross-sections rather than ordinary circular ones, a phenomenon that is partly attributed to the interwall thermal contraction upon cooling, as verified numerically by molecular dynamics simulations [16].From a nanoengineering perspective, the tunability of cross-sectional geometry may be useful for developing nanofluidic [17–19] or nanoelectrochemical devices [20] based on carbon nanotubes, because both utilize the hollow cavity within the innermost tube. A very interesting issue from an academic viewpoint is the effect of the core tube deformation on the physicochemical properties of intercalated molecules confined in the hollow cavity. It is indeed known that various types of intercalated molecules can fill the innermost hollow cavities of nanotubes [18] and exhibit intriguing behaviours that are distinct from their macroscopic counterparts [21–23]. These distinct behaviours originate from the similarity between the intermolecular spacings and the linear dimension of the nanoscale compartment. Therefore, cross-sectional deformation that breaks cylindrical symmetry will provide a clue to improving the performance of nanotube-based devices.In this paper, we establish the continuum elastic approach that describes the cross-sectional deformation of carbon nanotubes surrounded by a self-contracting host medium. The mechanical energy of both the pressurized nanotubes and the contracting medium with finite thickness are formulated using thin-shell theory. The obtained formula allows us to evaluate the critical radial pressure applied on the interface between the inner nanotube and the outer medium. Our numerical calculations have unveiled a power-law dependence of the critical pressure on the elastic modulus of the medium that is independent of the medium thickness.


## 2. Methodology

### 2.1.  Relevant Energy Components

Figure1 illustrates the self-contraction process of MWNTs subjected to high-energetic beam irradiation. Under irradiation, the induced beam kicks off a portion of carbon atoms at outer walls, causing vacancies followed by spontaneous shrinkage as marked by color in Figure 1(b). Even during the outer-walls self-contraction, the inner walls remain undamaged by irradiation, and thus they tend to keep their initial tube radii. As a result, the contracting outerwalls exert a high pressure, designated by p, on the inner undamaged walls (see Figure 1). A possible consequence of the radiation-induced high-pressure application is a circumferentially wrinkling structure, called radial corrugation [13], in the inner undamaged walls embedded in the eroded region, as examined later in an approximation based on the thin-shell theory.Schematics of the pressurized carbon nanotubes. (a) A pristine MWNT prior to high-energy beam irradiation. (b) An irradiated MWNT whose outermost carbon walls contract radially because of the self-repairing property of the eroded region (indicated in orange).
(a)
MWNT before irradiation
(b)
MWNT after irradiationThe stable cross-sectional shape of the embedded tube is obtained by minimizing its mechanical energyU per unit axial length [13] as
(1)U=UD+UI+Um+Ω.
The first term UD=∑i=1N‍UD(i) with the definition
(2)UD(i)=ri2(C1-ν2∫02π‍ℐi2dθ+D∫02π‍ℬi2dθ)
represents the deformation energy of the embedded nanotubes. ℐi and ℬi are, respectively, in-plane and bending-induced strains of the ith wall, and θ is the circumferential coordinate. For (2), we supposed that each ith wall had a radius ri prior to cross-sectional deformation and that the deformation caused a displacement xi=(ui,vi) of a volume element of the ith wall at (ri,θ) in the polar coordinate representation. The two strain terms ℐi and ℬi are described in terms of the displacement components by [14]
(3)ℐi=ui+∂θviri+12(∂θui-viri)2,ℬi=-∂θ2ui-∂θviri2,
where ∂θ=∂/∂θ.For quantitative discussions, the elastic coefficientsC and D need to be carefully determined. In conventional thin-shell theory for macroscopic objects, C and D are related to the Young’s modulus E of the wall and its thickness h as
(4)C=Eh,D=Eh312(1-ν2)formacroscopicshells.
However, for carbon nanotubes, the macroscopic relations for C and D noted earlier fail because there is no unique way of defining the thickness of the graphene wall [24]. Thus, the values of C and D should be evaluated ab initio from direct measurements or computations of carbon sheets, without reference to the macroscopic relations. In actual calculations, we substitute C=345 nN/nm,  D=0.238 nN·nm for carbon nanotubes, and ν=0.149 from prior work [25] based on the density functional theory.The second term,UI=∑i,j=i±1‍UI(i,j), in (1) accounts for the van der Waals (vdW) interaction energy, which determines the equilibrium distance between adjacent concentric walls. Thus far, several continuum models for the vdW interactions have been proposed. Expressions for the wall-wall interaction proposed in [26] were based on the surface integration of the vdW force and its derivative over the cylindrical walls, while disregarding the vectorial nature of the force. The significance of the vectorial nature of the force was addressed in [27], where analytical expressions for the wall-wall interaction were obtained by considering only the component of the vdW force normal to the wall. In accordance with the result of [27], we define the interaction energy by
(5)UI(i,j)=cij(ri+rj)4∫02π‍(ui-uj)2dθ,
where the coefficients cij are derived through a harmonic approximation of the interwall force [28] associated with the vdW intermolecular potential
(6)V(r)=4α[(βr)12-(βr)6]
with the definitions of α=2.39meV and β=0.3415nm the same as those in [29]. Equation (5) takes into account correctly the normal-to-wall component of vdW forces, and it is valid for infinitesimal deformation, which we address in the present work.The final termΩ in (1) is the negative of the work done by p during cross-sectional deformation; it can be written as [13]
(7)Ω=p∫02π‍(rNuN+uN2+vN2-vN∂θuN+uN∂θvN2)dθ.
Note that all three terms are functions of ui(p,θ) and vi(p,θ) of the ith wall under p.The remaining term,Um, in (1) is the elastic energy of the eroded medium surrounding the inner part of the nanotubes. To derive it, we assume that the medium has finite thickness of rm-rN, where rm is the outmost radius of the cylinder-shaped surrounding medium and rN is the outmost tube radius of the embedded nanotube. In addition, the medium is assumed to be homogeneous and isotropic with Young’s modulus Em and Poisson’s ratio νm. The validity of the latter assumption depends on the following two effects of irradiation on the mechanical stiffness of MWNTs. Irradiation reduces the axial stiffness because it creates vacancies [30, 31], and it simultaneously enhances the radial stiffness, owing to the production of covalent bonding between adjacent walls [32]. The possible value of Em ranges from 100 GPa (for amorphous carbon) [33, 34] to much less. An explicit form of Um is presented in the next section.


### 2.2. Corrugation Mode Analysis

Our objectives are to determine (i) the optimal displacementsui and vi that minimize U under a given value of p and (ii) the critical pressure pc above which the circular cross section of the embedded nanotube is elastically deformed into a noncircular one. These are accomplished by decomposing the displacement ui(p,θ) just after buckling as follows:
(8)ui(p,θ)=ui(0)(p)+δui(θ).
In (8), the term ui(0)(p)∝p describes a uniform radial contraction at p<pc, at which the cross section remains circular and thus the displacement is independent of θ. The other term δui(θ) describes a deformed (noncircular) cross-section observed immediately above pc. Note that the superscript (0) attached to ui(0)(p) differentiates it from the θ-dependent displacement ui(p,θ) observed at p>pc. As to vi(p,θ), we can write
(9)vi(p,θ)=δvi(θ),
because no circumferential displacement arises at p<pc (i.e., vi(0)(p)≡0).Applying the variation method toU with respect to ui and vi, we obtain a system of 2N linear differential equations with regard to δui(θ) and δvi(θ). To derive the 2N linear differential equations, quadratic or cubic terms in δui and δvi are omitted since we consider elastic deformation with sufficiently small displacements. In addition, the terms consisting only of ui(0) and p are also omitted; the sum of such terms should be equal to zero since ui(0) represents an equilibrium circular cross-section under p. In fact, the function form of ui(0)(p) is determined by the fact that the sum of those terms equals zero. The differential equations can be solved using the Fourier series expansions
(10)δui(θ)=∑n=1∞‍δμ-i(n)cosnθ,δvi(θ)=∑n=1∞‍δν-i(n)sinnθ,
wherein we took into account that δui, δvi, and their derivatives are periodic in θ. Substituting the expansions into the differential equations results in the matrix equation Mu=0; the vector u consists of δμ-i(n) and δν-i(n) with all possible i and n, and the matrix M involves one variable p and other material parameters. It should be noted that, due to the orthogonality of cosnθ and sinnθ, the matrix M can be expressed by a block diagonal matrix of the form
(11)M=Mn=1⊕Mn=2⊕⋯.
Here, Mn=k for arbitrary integer k is a 2N×2N submatrix that satisfies Mn=kun=k=0, where un=k is a 2N-column vector composed of δμ-i(k) and δν-i(k). As a result, the secular equation det(M)=0 is rewritten by
(12)det(Mn=1)det(Mn=2)⋯=0.
Solving (12) with respect to p, we obtain a sequence of discrete values of p. Among these values, the minimum one serves as the critical pressure pc that is associated with a specific integer k. From the definition, the pc associated with a specific k allows only δμ-i(n=k) and δν-i(n=k) be finite, but it requires δμ-i(n≠k)≡0 and δν-i(n≠k)≡0. Immediately above pc, therefore, the cross section of embedded nanotubes becomes radially deformed as described by
(13)ui(θ)=ui(0)(pc)+δμ-i(k)coskθ,vi(θ)=δν-i(k)sinkθ,
where the value of k is uniquely determined by the one-to-one relation between k and pc.


## 2.1.  Relevant Energy Components

Figure1 illustrates the self-contraction process of MWNTs subjected to high-energetic beam irradiation. Under irradiation, the induced beam kicks off a portion of carbon atoms at outer walls, causing vacancies followed by spontaneous shrinkage as marked by color in Figure 1(b). Even during the outer-walls self-contraction, the inner walls remain undamaged by irradiation, and thus they tend to keep their initial tube radii. As a result, the contracting outerwalls exert a high pressure, designated by p, on the inner undamaged walls (see Figure 1). A possible consequence of the radiation-induced high-pressure application is a circumferentially wrinkling structure, called radial corrugation [13], in the inner undamaged walls embedded in the eroded region, as examined later in an approximation based on the thin-shell theory.Schematics of the pressurized carbon nanotubes. (a) A pristine MWNT prior to high-energy beam irradiation. (b) An irradiated MWNT whose outermost carbon walls contract radially because of the self-repairing property of the eroded region (indicated in orange).
(a)
MWNT before irradiation
(b)
MWNT after irradiationThe stable cross-sectional shape of the embedded tube is obtained by minimizing its mechanical energyU per unit axial length [13] as
(1)U=UD+UI+Um+Ω.
The first term UD=∑i=1N‍UD(i) with the definition
(2)UD(i)=ri2(C1-ν2∫02π‍ℐi2dθ+D∫02π‍ℬi2dθ)
represents the deformation energy of the embedded nanotubes. ℐi and ℬi are, respectively, in-plane and bending-induced strains of the ith wall, and θ is the circumferential coordinate. For (2), we supposed that each ith wall had a radius ri prior to cross-sectional deformation and that the deformation caused a displacement xi=(ui,vi) of a volume element of the ith wall at (ri,θ) in the polar coordinate representation. The two strain terms ℐi and ℬi are described in terms of the displacement components by [14]
(3)ℐi=ui+∂θviri+12(∂θui-viri)2,ℬi=-∂θ2ui-∂θviri2,
where ∂θ=∂/∂θ.For quantitative discussions, the elastic coefficientsC and D need to be carefully determined. In conventional thin-shell theory for macroscopic objects, C and D are related to the Young’s modulus E of the wall and its thickness h as
(4)C=Eh,D=Eh312(1-ν2)formacroscopicshells.
However, for carbon nanotubes, the macroscopic relations for C and D noted earlier fail because there is no unique way of defining the thickness of the graphene wall [24]. Thus, the values of C and D should be evaluated ab initio from direct measurements or computations of carbon sheets, without reference to the macroscopic relations. In actual calculations, we substitute C=345 nN/nm,  D=0.238 nN·nm for carbon nanotubes, and ν=0.149 from prior work [25] based on the density functional theory.The second term,UI=∑i,j=i±1‍UI(i,j), in (1) accounts for the van der Waals (vdW) interaction energy, which determines the equilibrium distance between adjacent concentric walls. Thus far, several continuum models for the vdW interactions have been proposed. Expressions for the wall-wall interaction proposed in [26] were based on the surface integration of the vdW force and its derivative over the cylindrical walls, while disregarding the vectorial nature of the force. The significance of the vectorial nature of the force was addressed in [27], where analytical expressions for the wall-wall interaction were obtained by considering only the component of the vdW force normal to the wall. In accordance with the result of [27], we define the interaction energy by
(5)UI(i,j)=cij(ri+rj)4∫02π‍(ui-uj)2dθ,
where the coefficients cij are derived through a harmonic approximation of the interwall force [28] associated with the vdW intermolecular potential
(6)V(r)=4α[(βr)12-(βr)6]
with the definitions of α=2.39meV and β=0.3415nm the same as those in [29]. Equation (5) takes into account correctly the normal-to-wall component of vdW forces, and it is valid for infinitesimal deformation, which we address in the present work.The final termΩ in (1) is the negative of the work done by p during cross-sectional deformation; it can be written as [13]
(7)Ω=p∫02π‍(rNuN+uN2+vN2-vN∂θuN+uN∂θvN2)dθ.
Note that all three terms are functions of ui(p,θ) and vi(p,θ) of the ith wall under p.The remaining term,Um, in (1) is the elastic energy of the eroded medium surrounding the inner part of the nanotubes. To derive it, we assume that the medium has finite thickness of rm-rN, where rm is the outmost radius of the cylinder-shaped surrounding medium and rN is the outmost tube radius of the embedded nanotube. In addition, the medium is assumed to be homogeneous and isotropic with Young’s modulus Em and Poisson’s ratio νm. The validity of the latter assumption depends on the following two effects of irradiation on the mechanical stiffness of MWNTs. Irradiation reduces the axial stiffness because it creates vacancies [30, 31], and it simultaneously enhances the radial stiffness, owing to the production of covalent bonding between adjacent walls [32]. The possible value of Em ranges from 100 GPa (for amorphous carbon) [33, 34] to much less. An explicit form of Um is presented in the next section.


## 2.2. Corrugation Mode Analysis

Our objectives are to determine (i) the optimal displacementsui and vi that minimize U under a given value of p and (ii) the critical pressure pc above which the circular cross section of the embedded nanotube is elastically deformed into a noncircular one. These are accomplished by decomposing the displacement ui(p,θ) just after buckling as follows:
(8)ui(p,θ)=ui(0)(p)+δui(θ).
In (8), the term ui(0)(p)∝p describes a uniform radial contraction at p<pc, at which the cross section remains circular and thus the displacement is independent of θ. The other term δui(θ) describes a deformed (noncircular) cross-section observed immediately above pc. Note that the superscript (0) attached to ui(0)(p) differentiates it from the θ-dependent displacement ui(p,θ) observed at p>pc. As to vi(p,θ), we can write
(9)vi(p,θ)=δvi(θ),
because no circumferential displacement arises at p<pc (i.e., vi(0)(p)≡0).Applying the variation method toU with respect to ui and vi, we obtain a system of 2N linear differential equations with regard to δui(θ) and δvi(θ). To derive the 2N linear differential equations, quadratic or cubic terms in δui and δvi are omitted since we consider elastic deformation with sufficiently small displacements. In addition, the terms consisting only of ui(0) and p are also omitted; the sum of such terms should be equal to zero since ui(0) represents an equilibrium circular cross-section under p. In fact, the function form of ui(0)(p) is determined by the fact that the sum of those terms equals zero. The differential equations can be solved using the Fourier series expansions
(10)δui(θ)=∑n=1∞‍δμ-i(n)cosnθ,δvi(θ)=∑n=1∞‍δν-i(n)sinnθ,
wherein we took into account that δui, δvi, and their derivatives are periodic in θ. Substituting the expansions into the differential equations results in the matrix equation Mu=0; the vector u consists of δμ-i(n) and δν-i(n) with all possible i and n, and the matrix M involves one variable p and other material parameters. It should be noted that, due to the orthogonality of cosnθ and sinnθ, the matrix M can be expressed by a block diagonal matrix of the form
(11)M=Mn=1⊕Mn=2⊕⋯.
Here, Mn=k for arbitrary integer k is a 2N×2N submatrix that satisfies Mn=kun=k=0, where un=k is a 2N-column vector composed of δμ-i(k) and δν-i(k). As a result, the secular equation det(M)=0 is rewritten by
(12)det(Mn=1)det(Mn=2)⋯=0.
Solving (12) with respect to p, we obtain a sequence of discrete values of p. Among these values, the minimum one serves as the critical pressure pc that is associated with a specific integer k. From the definition, the pc associated with a specific k allows only δμ-i(n=k) and δν-i(n=k) be finite, but it requires δμ-i(n≠k)≡0 and δν-i(n≠k)≡0. Immediately above pc, therefore, the cross section of embedded nanotubes becomes radially deformed as described by
(13)ui(θ)=ui(0)(pc)+δμ-i(k)coskθ,vi(θ)=δν-i(k)sinkθ,
where the value of k is uniquely determined by the one-to-one relation between k and pc.


## 3. Mechanical Energy of the Embedding Medium

### 3.1. Total Energy Cost

In polar coordinates, the radial and circumferential components of normal stress in the medium are denoted byσr and σθ, respectively, and the shear stress is denoted by τrθ; all three quantities are functions of r and θ. Then, Um is determined by σr and τrθ at r=rN as
(14)Um=Um(0)+ΔUm(n),(15)Um(0)=rN2∫02π‍σr(0)|r=rNuN(0)dθ,(16)ΔUm(n)=rN2∫02π‍(σr(n)|r=rNδuN+τrθ(n)|r=rNδvN)dθ,
where δuN and δvN describe the corrugation amplitudes of the outermost wall of the embedded nanotube; see (10). The superscripts (0) and (n) indicate the quantities corresponding to uniform contraction and radial corrugation, respectively. In other words, Um(0) represents the energy required for uniform radial contraction of the surrounding medium keeping in contact with the embedded nanotube, and ΔUm represents the energy required for radial corrugation with mode index n.


### 3.2. Stress-Strain-Displacement Relation

The mechanics of an elastic medium is governed by the stress functionϕ, which satisfies the so-called compatibility equation [35]
(17)(∂r2+r-1∂r+r-2∂θ)2ϕ(r,θ)=0,
where ∂r=∂/∂r and ∂θ=∂/∂θ. Once ϕ is obtained, we can deduce the stress components as follows:
(18)σr=(r-1∂r+r-2∂θ2)ϕ,σθ=∂r2ϕ,τrθ=∂r(r-1∂θ)ϕ.
By definition, the strain components εr, εθ, and γrθ are given by the matrix form
(19)[εrεθγrθ]=G[σrσθτrθ]=[∂r0r-1r-1∂θr-1∂θ∂r-r-1][uv],(20)G=1Em[1-νm2-νm(1+νm)0-νm(1+νm)1-νm20002(1+νm)],
where u=u(r,θ) and v=v(r,θ) are, respectively, the radial and circumferential displacements of a volume element in the elastic medium.The general solution of (17) is given by
(21)ϕ(r,θ)=∑n=0∞‍ϕn(r,θ)=∑n=0∞‍[fn(r)cosnθ+gn(r)sinnθ].
The zeroth component ϕ0 represents a uniform contraction of the circular cross section, thus corresponding to the energy Um(0) that we have introduced in (15). The first one, ϕ1, implies a rigid-body translation, which is irrelevant to our consideration. Other components, ϕn for n≥2, describe radial corrugations with mode index n, thus providing the energy ΔUm(n) given by (16). In the following, we set [36]
(22)f0=a0logr+c0r2,fn=anr-n+bnr2-n+cnr2+n+dnrn,(n≥2)
in order to obtain physically relevant solutions of σr, σθ, and τrθ that decay with increasing r. Without loss of generality, we set gn≡0 in (21) according to our assumption of cosine-type radial displacement δui(θ); see (10). We emphasize that the hypothesized solution forms of f0 and fn≥2 differ from those in our earlier work [12]; in the present study, we introduce the positive power terms of r in (22) to take into account correctly the boundary condition at r=rm, whereas these terms were omitted in [12].


### 3.3. Energy Cost under Uniform Contraction

We now evaluate the explicit form of the energyUm(0). It results from the uniform contraction of the medium described by the r-dependent medium displacement u(0)(r). The elastic nature of the embedding medium assures that
(23)σr(0)(r)=κ(r)u(0)(r)
with the r-dependent stiffness coefficient κ(r). In addition, complete contact at the interface of the medium and the embedded nanotube implies
(24)u(0)(rN)=uN(0).
Hence, the previous expression of Um(0), given in (15), can be rewritten as
(25)Um(0)=κ02∫02π‍{uN(0)}2rNdθwithκ0≡κ(rN).
The remaining task is, therefore, to represent κ0 in terms of already-known material parameters such as Em,νm,rm, and rN.To accomplish this task, we consider a specific solution of (17) that has the form ϕ=ϕ0, and then we substitute it back into (18) to obtain
(26)σr(0)=a0r-2+2c0,σθ(0)=-a0r-2+2c0,
and τrθ(0)=0. The coefficients a0 and c0 are determined by imposing the boundary conditions of σr(0) at r=rN and r=rm as follows:
(27)a0rN-2+2c0=p,a0rm-2+2c0=0.
Eliminating a0 and c0 from (27), we have
(28)σr(0)(r)=rN2prm2-rN2(rm2r2-1),σθ(0)(r)=-rN2prm2-rN2(rm2r2+1).
The obtained σr(0)(r) and σθ(0)(r) lead to the displacements
(29)u(0)(r)=-1+νmEm·rN2prm2-rN2[(1-2νm)r+rm2r]
and v(0)(r)=0 at the medium region (rN<r<rm), as confirmed by (19). By comparing (23), (28), and (29), we conclude that
(30)κ0=-Em(1+νm)·(rm2rN2-1)·[(1-2νm)rN+rm2rN]-1.


### 3.4. Energy Cost under Radial Corrugation

Next, we consider the energyΔUm(n)(n≥2) that corresponds to the radial corrugation of the nth order. A similar procedure to the case of n=0 yields
(31)σr(n)(r,θ)=={-n(n+1)anr-2-(n-1)(n+2)bn}r-ncosnθ==+{-n(n-1)dnr-2-(n+1)(n-2)cn}rncosnθ,σθ(n)(r,θ)=={n(n+1)anr-2+(n-1)(n-2)bn}r-ncosnθ==+{n(n-1)dnr-2+(n+1)(n+2)cn}rncosnθ,τrθ(n)(r,θ)=={n(n+1)anr-2+n(n-1)bn}r-nsinnθ==-{n(n-1)dnr-2+n(n+1)cn}rnsinnθ,
leading to the results
(32)u(n)(r,θ)=(1+νm)Em(nanr-2+ξ+bn)r1-ncosnθ=-(1+νm)Em(ndnr-2+ξ-cn)r1+ncosnθ,(33)v(n)(r,θ)=(1+νm)Em(nanr-2+η-bn)r1-nsinnθ=+(1+νm)Em(ndnr-2+η+cn)r1+nsinnθ,
with ξ±=n±2(1-2νm) and η±=n±4(1-νm). Of the four coefficients an, bn, cn, and dn involved in the previous equations, two of them are eliminated by considering the boundary conditions σr(n)(rN,θ)≡0 and τrθ(n)(rN,θ)≡0.Owing to the complete contact condition, we have(34)u(n)(rN,θ)=δμ-N(n)cosnθ,v(n)(rN,θ)=δν-N(n)sinnθ.
Besides, the elastic nature of the medium at the contact interface implies the relations
(35)σr(n)|r=rN=[κ1δμ-N(n)+κ2δν-N(n)]cosnθ,τrθ(n)|r=rN=[κ2δμ-N(n)+κ1δν-N(n)]sinnθ.
From (35), the stiffness coefficients are given by
(36)κ1=σr(n)|r=rN,δμ-N(n)=1,δν-N(n)=0cosnθ,κ2=σr(n)|r=rN,δμ-N(n)=0,δν-N(n)=1cosnθ.
Note that a set of two normalization conditions of
(37){δμ-N(n)=1,δν-N(n)=0},{δμ-N(n)=0,δν-N(n)=1},
which is indicated by (36), allows us to describe the two remaining undetermined coefficients, from among an, bn, cn, and dn, in terms of materials parameters such as Em,  νm, and rm; see the statement immediately below (33). Hence, κ1 and κ2 derived as mentioned are those expressed by already-known parameters, though the explicit forms are not shown here to save space. Eventually, we obtain ΔUm(n) by substituting the results into (16).


## 3.1. Total Energy Cost

In polar coordinates, the radial and circumferential components of normal stress in the medium are denoted byσr and σθ, respectively, and the shear stress is denoted by τrθ; all three quantities are functions of r and θ. Then, Um is determined by σr and τrθ at r=rN as
(14)Um=Um(0)+ΔUm(n),(15)Um(0)=rN2∫02π‍σr(0)|r=rNuN(0)dθ,(16)ΔUm(n)=rN2∫02π‍(σr(n)|r=rNδuN+τrθ(n)|r=rNδvN)dθ,
where δuN and δvN describe the corrugation amplitudes of the outermost wall of the embedded nanotube; see (10). The superscripts (0) and (n) indicate the quantities corresponding to uniform contraction and radial corrugation, respectively. In other words, Um(0) represents the energy required for uniform radial contraction of the surrounding medium keeping in contact with the embedded nanotube, and ΔUm represents the energy required for radial corrugation with mode index n.


## 3.2. Stress-Strain-Displacement Relation

The mechanics of an elastic medium is governed by the stress functionϕ, which satisfies the so-called compatibility equation [35]
(17)(∂r2+r-1∂r+r-2∂θ)2ϕ(r,θ)=0,
where ∂r=∂/∂r and ∂θ=∂/∂θ. Once ϕ is obtained, we can deduce the stress components as follows:
(18)σr=(r-1∂r+r-2∂θ2)ϕ,σθ=∂r2ϕ,τrθ=∂r(r-1∂θ)ϕ.
By definition, the strain components εr, εθ, and γrθ are given by the matrix form
(19)[εrεθγrθ]=G[σrσθτrθ]=[∂r0r-1r-1∂θr-1∂θ∂r-r-1][uv],(20)G=1Em[1-νm2-νm(1+νm)0-νm(1+νm)1-νm20002(1+νm)],
where u=u(r,θ) and v=v(r,θ) are, respectively, the radial and circumferential displacements of a volume element in the elastic medium.The general solution of (17) is given by
(21)ϕ(r,θ)=∑n=0∞‍ϕn(r,θ)=∑n=0∞‍[fn(r)cosnθ+gn(r)sinnθ].
The zeroth component ϕ0 represents a uniform contraction of the circular cross section, thus corresponding to the energy Um(0) that we have introduced in (15). The first one, ϕ1, implies a rigid-body translation, which is irrelevant to our consideration. Other components, ϕn for n≥2, describe radial corrugations with mode index n, thus providing the energy ΔUm(n) given by (16). In the following, we set [36]
(22)f0=a0logr+c0r2,fn=anr-n+bnr2-n+cnr2+n+dnrn,(n≥2)
in order to obtain physically relevant solutions of σr, σθ, and τrθ that decay with increasing r. Without loss of generality, we set gn≡0 in (21) according to our assumption of cosine-type radial displacement δui(θ); see (10). We emphasize that the hypothesized solution forms of f0 and fn≥2 differ from those in our earlier work [12]; in the present study, we introduce the positive power terms of r in (22) to take into account correctly the boundary condition at r=rm, whereas these terms were omitted in [12].


## 3.3. Energy Cost under Uniform Contraction

We now evaluate the explicit form of the energyUm(0). It results from the uniform contraction of the medium described by the r-dependent medium displacement u(0)(r). The elastic nature of the embedding medium assures that
(23)σr(0)(r)=κ(r)u(0)(r)
with the r-dependent stiffness coefficient κ(r). In addition, complete contact at the interface of the medium and the embedded nanotube implies
(24)u(0)(rN)=uN(0).
Hence, the previous expression of Um(0), given in (15), can be rewritten as
(25)Um(0)=κ02∫02π‍{uN(0)}2rNdθwithκ0≡κ(rN).
The remaining task is, therefore, to represent κ0 in terms of already-known material parameters such as Em,νm,rm, and rN.To accomplish this task, we consider a specific solution of (17) that has the form ϕ=ϕ0, and then we substitute it back into (18) to obtain
(26)σr(0)=a0r-2+2c0,σθ(0)=-a0r-2+2c0,
and τrθ(0)=0. The coefficients a0 and c0 are determined by imposing the boundary conditions of σr(0) at r=rN and r=rm as follows:
(27)a0rN-2+2c0=p,a0rm-2+2c0=0.
Eliminating a0 and c0 from (27), we have
(28)σr(0)(r)=rN2prm2-rN2(rm2r2-1),σθ(0)(r)=-rN2prm2-rN2(rm2r2+1).
The obtained σr(0)(r) and σθ(0)(r) lead to the displacements
(29)u(0)(r)=-1+νmEm·rN2prm2-rN2[(1-2νm)r+rm2r]
and v(0)(r)=0 at the medium region (rN<r<rm), as confirmed by (19). By comparing (23), (28), and (29), we conclude that
(30)κ0=-Em(1+νm)·(rm2rN2-1)·[(1-2νm)rN+rm2rN]-1.


## 3.4. Energy Cost under Radial Corrugation

Next, we consider the energyΔUm(n)(n≥2) that corresponds to the radial corrugation of the nth order. A similar procedure to the case of n=0 yields
(31)σr(n)(r,θ)=={-n(n+1)anr-2-(n-1)(n+2)bn}r-ncosnθ==+{-n(n-1)dnr-2-(n+1)(n-2)cn}rncosnθ,σθ(n)(r,θ)=={n(n+1)anr-2+(n-1)(n-2)bn}r-ncosnθ==+{n(n-1)dnr-2+(n+1)(n+2)cn}rncosnθ,τrθ(n)(r,θ)=={n(n+1)anr-2+n(n-1)bn}r-nsinnθ==-{n(n-1)dnr-2+n(n+1)cn}rnsinnθ,
leading to the results
(32)u(n)(r,θ)=(1+νm)Em(nanr-2+ξ+bn)r1-ncosnθ=-(1+νm)Em(ndnr-2+ξ-cn)r1+ncosnθ,(33)v(n)(r,θ)=(1+νm)Em(nanr-2+η-bn)r1-nsinnθ=+(1+νm)Em(ndnr-2+η+cn)r1+nsinnθ,
with ξ±=n±2(1-2νm) and η±=n±4(1-νm). Of the four coefficients an, bn, cn, and dn involved in the previous equations, two of them are eliminated by considering the boundary conditions σr(n)(rN,θ)≡0 and τrθ(n)(rN,θ)≡0.Owing to the complete contact condition, we have(34)u(n)(rN,θ)=δμ-N(n)cosnθ,v(n)(rN,θ)=δν-N(n)sinnθ.
Besides, the elastic nature of the medium at the contact interface implies the relations
(35)σr(n)|r=rN=[κ1δμ-N(n)+κ2δν-N(n)]cosnθ,τrθ(n)|r=rN=[κ2δμ-N(n)+κ1δν-N(n)]sinnθ.
From (35), the stiffness coefficients are given by
(36)κ1=σr(n)|r=rN,δμ-N(n)=1,δν-N(n)=0cosnθ,κ2=σr(n)|r=rN,δμ-N(n)=0,δν-N(n)=1cosnθ.
Note that a set of two normalization conditions of
(37){δμ-N(n)=1,δν-N(n)=0},{δμ-N(n)=0,δν-N(n)=1},
which is indicated by (36), allows us to describe the two remaining undetermined coefficients, from among an, bn, cn, and dn, in terms of materials parameters such as Em,  νm, and rm; see the statement immediately below (33). Hence, κ1 and κ2 derived as mentioned are those expressed by already-known parameters, though the explicit forms are not shown here to save space. Eventually, we obtain ΔUm(n) by substituting the results into (16).


## 4. Results and Discussions

Figure2 shows the critical pressure pc required for the cross-sectional deformation of embedded carbon nanotubes. Different types of curves in the plots (dashed, dashed-dotted, and solid) correspond to different values of Young’s modulus ratio Em/E, where E=1 TPa is assumed to be the nanotube’s modulus. We found that, independent of the Em/E value, the pc curves are upward convex as functions of the medium radius rm in units of rN. The growth of pc is rapid for rm/rN<2, and then, it saturates for larger rm/rN. The rapid growth in pc indicates a “hardening effect” caused by the surrounding medium, that is, an enhancement in the radial stiffness of the embedded nanotube by encapsulation. This hardening effect disappears with a further increase in rm/rN; the results imply that the surrounding medium with thickness rm>5rN no longer enhances the hardening effect and thus can be identified as a medium with infinitely large thickness (rm→∞), which is the case considered in our earlier work [12].Critical pressurepc for cross-sectional deformation of embedded carbon nanotubes. Monotonic increases with rm/rN, the ratio between the radii of the medium (rm) and the outermost tube (rN), are observed regardless of the Young’s modulus ratio Em/E between the elastic medium (Em) and the carbon nanotubes (E): Em/E=10-3 (dashed curves), Em/E=10-2 (dash-dotted curves), and Em/E=10-1 (solid curve).
(a)
(b)
(c)
(d)
(e)
(f)Figure3 provides the index of radial deformation modes observed immediately above pc. Successive transformation of deformation modes with increasing medium thickness rm was confirmed, as a result of the fact that the energy required to deform the surrounding medium needed to be responsible for determining the stable corrugation pattern of the embedded nanotube. Again, we found that the mode variation disappears for rm/rN>3, and a larger value of n arises for a higher modulus ratio Em/E. The nonmonotonic variance in the corrugation mode n observed within 1<rm/rN<2 is what can be clarified for the first time by the present work; it is applicable to the condition rm~rN, which lay beyond the scope of our earlier work [12].Mode indexn of the radial corrugation observed in the embedded nanotubes. The modulus ratio is set to Em/E=10-3, 10-2, and 10-1 from the bottom to the top.
(a)
SWNT
(b)
10-walled NTFigure4 shows the pc dependence on the modulus ratio (Em/E), where the radius ratio α≡rm/rN is fixed to be α=1.1 or 5.0. For every choice of α and N, the pc curves in this plot obey a power law represented by
(38)pc∝(EmE)βwithβ≃23.
We have confirmed that the power-law behaviour holds for any values of α and N within the ranges of 1.1<α<5 and 1<N<10, respectively. Recall here that, for α>5, pc becomes almost constant even with increasing α. These facts lead to the conjecture that the power law described by (38) is universal to all embedded nanotubes, at least under the numerical conditions that we have addressed. The power law of pc represented by (38) is an interesting manifestation of the embedding effect of nanotubes into an elastic medium.Figure 4
Double logarithm plot of the critical pressurepc as a function of the Young’s modulus ratio Em/E between the elastic medium (Em) and the nanotubes (E). The parameter α≡rm/rN represents the ratio of the medium radius rm to the outermost tube radius rN, where N=1 for single-walled nanotubes and N=10 for MWNTs. For all the parameter values chosen, the two-third power law of pc given by (38) is observed.It is worthy to note that the two-third power law ofpc has been also proposed for the buckling of macroscopic “tunnels" embedded in an infinite soft ground [36]. The radial collapse of tunnel liners, which is one of the major civil engineering disasters, was analysed to reveal a closed form of pc; several approximations led to the conclusion of a power-law of pc similar to (38), though an infinitely large medium and only the case of one cylindrical shell (i.e., N=1) are hypothesized. In this context, our results suggest that the approximation theory proposed for macroscopic tunnels [36] is valid for nanoscopic cylinders, even though they consist of more than one concentric walls.The previous discussion may pose a further question: does the power law ofpc hold true for macroscopic counterparts of “many”-walled nanotubes? In fact, macroscaled pipe-in-pipe structures (i.e., a pipe inserted inside another pipe) are known to be an efficient design solution for subsea-pipeline developments in deep water [37, 38], wherein buckling resistance to large amounts of external hydrostatic pressure is a key structural design requirement. Addressing these problems is expected to shed light on the development of multiple-cylindrical structures from multidisciplinary viewpoints and will be considered in our future work.


## 5. Conclusion

We have demonstrated a continuum elastic approach that describes the cross-sectional deformation of carbon nanotubes surrounded by a self-contracting host medium. The approach enables quantitative discussions of the critical pressure for radial corrugation and the stable corrugation mode of the nanotube surrounded by the contracting medium. Numerical calculations based on the established theory revealed the power-law dependence of the critical pressure on the elastic modulus of the medium, which is independent of the medium thickness. Further studies are expected to shed light on other mechanical properties of embedded nanotubes and to suggest applications based on their unique cross-sectional deformations.



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*Source: 767249-2013-05-19.xml*