## Abstract

The Flügge shell theory is frequently used for the analysis of carbon nanotubes (CNTs) due to the relatively accurate results it provides in spite of its theoretical simplicity. Based on the Flügge shell theory, a tubular beam theory was established by considering non-locality. In order to convert a cylindrical shell theory for a curved plate per unit width into a tubular beam theory by contour integration, the longitudinal coordinate that passes through the centre of the circular contour was defined, and, based on this, the radial coordinate was defined. In this way, a generalized beam theory (GBT) was obtained as a further refined form of the Flügge theory. This GBT coincides with the Flügge theory, if the refined form and the non-locality are ignored. After obtaining the phase-velocity curve and the group-velocity curve with respect to single- to triple-walled CNTs, the influences of multiplicity, reduction of the plate-bending stiffness and the stiffness of the surrounding matrix were investigated.

## 1. Introduction

In recent years, various characteristics of nanomaterials such as fullerenes and carbon nanotubes (CNTs) have attracted much attention. The molecular dynamics method is known to be one of the successful methods for analysing the structural properties of CNTs; however, such analysis has not been adequately performed so far due to the immense memory storage and the long calculation time required. The traditional continuum mechanics approximation is another method that can be used for the analysis of CNTs. A CNT is usually modelled as a beam, or a frame structure of truss or beam elements, and a cylindrical shell. Other continuum models have also been proposed (Li & Guo 2008).

The continuum mechanics model is classified into beam theory and shell theory. The beam theory is simple and applicable for comprehensive static and dynamic problems of the CNTs (Wang & Cai 2006).

For example, the shear-deformation behaviour is overlooked if the CNT is replaced with the hollow section of a Bernoulli–Euler beam, and the cross-sectional distortion behaviour is overlooked if the CNT is replaced with the hollow section of a Timoshenko beam. Therefore, the cylindrical shell theory is the most reliable analytical theory at this point. Among the cylindrical shellal theories, the Donnell (1976) theory is well known for its ease of use (Raichura *et al*. 2005; Li & Kardomateas 2007). However, the Donnell theory is not suitable for long and narrow CNTs since the approximation is for a cylindrical shell with a larger cross-sectional radius. On the other hand, the Flügge (1962) theory is known as a highly reliable theory that can be used for most shapes regardless of the size of their cross-sectional radius. However, the cylindrical shell theory itself is not appropriate for application to thin tubular beams since it involves a large number of variables such as displacement, strain, stress and stress resultant. Ignoring the cross-sectional distortion is not desirable from the point of view of reliability of the beam analysis. It is desirable to develop a beam theory that has reliability equal to or greater than that of the cylindrical shell theory. Based on this assumption, this research was conducted in order to establish a tubular beam theory on the basis of the Flügge theory, which is generally regarded as providing highly accurate analysis results.

The difference between the tubular beam theory and the cylindrical shell theory is whether a unitary coordinate axis is defined or not, and whether the contour integration of the governing equation is performed or not. Schardt (1989) developed the generalized beam theory (GBT) on the basis of the Flügge theory. His theory allows axial bending and cross-sectional distortion to be considered while the fourth-order ordinary differential equation corresponding to the traditional beam theory is being expressed. However, neither the circumferential normal strain nor the shear strain are taken into consideration, nor is the torsional deformation. Thus, beam theory was developed on the basis of Flügge's theory, where the circumferential normal strain, the shear deformation and the torsional deformation were taken into consideration.

In the cylindrical shell theory, the centre of the curved-plate thickness constituting the shell is the origin of the coordinate in the thickness direction, and, if the axial coordinate is taken from that point, it is defined by numerous parallel lines. By contrast, in the beam theory, the origin is the centre of the cross section, and the axial coordinate is defined by a single line from the centre. This difference influences the values associated with the cross-sectional quantities, such as the moment of inertia of the cylindrical shells. In particular, if the value of the thickness-to-radius ratio of the cross section becomes large, the reliability of the cylindrical shell theory decreases. However, by developing a beam theory that is truly based on the cylindrical shell theory, it becomes possible to establish an analytical technique that ensures the reliability of both shells and beams. This is the focus of this research, in which the newly developed beam theory was established.

In the comparison of analyses of nano-scale cylindrical shells performed with continuum approximation and the molecular dynamics method, the accuracy decreases in certain instances.

The theory of non-local elasticity (Eringen 1983) is promising as a theory capable of targeting this issue. In one analysis, although the non-locality in longitudinal direction is taken into consideration for a cylindrical shell (Zhang *et al*. 2004), the non-locality of the circumferential direction is not considered.

Based on the above discussion, this research was undertaken as follows. By using the unitary coordinate axis in a longitudinal direction that passes through the centre of the circular cross section, Flügge's equation of motion was derived, which takes the non-local elasticity based on the energy principle into consideration. At the same time, the rotational inertia was taken into consideration, which is usually not considered in the Flügge theory due to the rigorous integral in the thickness direction. Then, in order to derive the cylindrical tube theory from the cylindrical shell theory, integration was performed in a circumferential direction with the displacement function of the GBT proposed by Schardt (1989). Subsequently, the symmetric property with respect to the principal diagonal was verified with the obtained stiffness matrix. At that point in the present research, it was discovered that only the term of the van der Waals (vdW) force that governs the coupling with the neighbouring tubes disrupts the symmetric property with respect to the main diagonal. As a result of using a unitary coordinate axis in an axial direction, the obtained equation turned out to be an advanced form of the Flügge theory, and, after performing an approximation, it was found that it completely coincides with the traditional Flügge theory.

Then, by applying the strain condition to the obtained stiffness matrix, the equations indicated that the beam can be restored to a Timoshenko beam or a Schardt generalized beam.

As a numerical calculation example, the phase-velocity curve was obtained for a single to triple-walled cylindrical shell of infinite length, and various aspects, such as the multi-layered structure of the tube, the reduction coefficient of the plate bending stiffness, the stiffness of the surrounding matrix and the non-locality, were investigated. The influence of the non-locality on the group-velocity curve was also investigated.

## 2. Coordinate systems and relevant assumptions

Let us assume that a multi-walled cylindrical shell comprises *M* cylindrical shells with length *l*, which are nested concentrically in a stationary state. The cylindrical shells are numbered *m*=1, 2, …, *M* starting from the innermost one.

As shown in figure 1*a*, the axial direction of the cylindrical shell coincides with the *x*-axis taken from the centre S of the circular cross section. Instead of the circumferential coordinate, an angle *φ* is used, which is taken from the vertex of the vertical axis on the circular cross section, the radial coordinate *Z* was taken from the centre S. *t*_{m} is the thickness of the *m*th cylindrical shell and *a*_{m} is the mean radius to the centre of thickness. Here, the curvilinear surface that bisects the wall thickness of the cylindrical shell is referred to as the middle surface. Given that the radial coordinate from the middle surface (origin) is *z*_{m}, the following equation can be derived, as illustrated in figure 1*b*:(2.1)

The displacement of the middle surface (*z*_{m}=0) corresponding to the direction of each of the coordinates (*x*, *φ*, *z*_{m}) is denoted by *u*_{m}, *v*_{m}, *w*_{m} and the displacement of an arbitrary point (*z*_{m}=*z*_{m}) on the cross section is denoted by *u*_{zm}, *v*_{zm}, *w*_{zm}.

The assumptions are as follows.

The thickness

*t*_{m}is relatively smaller than the mean radius*a*_{m}of the cross section.The displacement of an arbitrary point on the normal line drawn from the middle surface is determined according to the displacement on the middle surface to which the arbitrary point belongs, and the normal line is perpendicular to the new middle surface resulting from the displacement.

The displacement is infinitesimal with respect to the shell thickness.

The normal stress

*σ*_{zz}is perpendicular to the middle surface and is relatively small with respect to other stresses, and therefore it is approximately zero.

Assumption (ii) corresponds to the shear deformation of the curved plate constituting the cylindrical shell and is assumed to be zero.

## 3. Governing equation of the Flügge theory

### (a) Displacement and strain

According to Flügge's cylindrical shell theory (Flügge 1962), based on assumption (ii), by using the displacement on the middle surface, the displacement of an arbitrary point P(*x*, *φ*, *z*_{m}) on the cross section can be expressed as(3.1)However, based on assumption (ii), ignoring the shear deformation of the shell, the cross-sectional rotation *ψ*_{m}(*x*, *φ*) of a plate is defined as(3.2)

Based on assumption (iii), the infinitesimal displacement, the displacements of the normal strain *ϵ*_{xx} in the *x*-direction, the normal strain *ϵ*_{φφ} in the *φ* direction and the shear strain γ_{xφ} in the *xφ* plane can be expressed as(3.3)

Therefore, by substituting equation (3.1) into equation (3.3), the following equation is obtained:(3.4)This equation can be decomposed into membrane strain components and the curvature components of the middle surface.

### (b) Constitutive law of macroscopic stress

*σ*_{xx} is the normal stress acting on the plane perpendicular to the longitudinal *x*-direction, *σ*_{φφ} is the normal stress acting on the plane perpendicular to the *φ*-axis, that is, the profile coordinate of the beam section wall, and *σ*_{xφ} is the shear stress within the *x*–*φ* plane. By substituting equation (3.4) into the stress–strain relation with Poisson ratio *ν*, the following results are derived:(3.5)

### (c) Relation between macroscopic and microscopic stress

According to the non-local elasticity theory, the stress at reference point ** x** of an object is dependent not only on the strain at point

**, but also on the strain at its neighbouring points**

*x***′. This idea has been confirmed by the atomic hypothesis of lattice dynamics and the observations of phonon dispersion experiments. If the influence on the reference point**

*x***from the neighbouring strain is completely ignored, the theory is reduced to the classical (local) elasticity theory. Here, the vector expression and the correspondence of the macroscopic (classic) stress and the microscopic stress are written as(3.6)The positive direction of the macroscopic (classic) stress**

*x**σ*

_{ij}and the microscopic stress

*t*

_{ij}are depicted in figure 2

*a*,

*b*, respectively.

Under the assumption that the non-local modulus function is the Green function of a certain linear differential operator , the following equation is obtained (Eringen 1983):(3.7)The appropriate two-dimensional linear operator is written as(3.8)In this equation, *a* on the right-hand side is the characteristic length of the material and the material constant can be determined as *e*_{0}≃0.39.

### (d) Definition of stress resultants of shell

Figure 3*a* illustrates the positive stress resultants of the force and figure 3*b* illustrates the positive direction of the stress resultants of the moment. These stress resultants can be defined by the cross-sectional integral of the microscopic stress *t*_{ij}. After multiplying the linear operator from the left to both sides of the definition of the stress resultant, if equation (3.7) is taken into consideration, this can be rewritten (Lu *et al*. 2007) as(3.9)

### (e) Strain energy and kinetic energy

The strain energy *U* of the multi-walled cylindrical shell consisting of *M* concentrically nested cylindrical shells is given by the following equation:(3.10)

Let *ρ* be the unit weight of the cylindrical shell consisting of a thin curved plate whose wall thickness is *t*_{m}. If the rotational inertia of the plate is taken into consideration, by using the displacement (*u*_{zm}, *v*_{zm}, *w*_{zm}), the kinetic energy can be expressed as (Seide 1975)(3.11)Here, *t*_{m}, which represents the integral interval, is the thickness of the *m*th cylindrical shell, while *t* inside the integrand is time as an independent variable.

### (f) Work of external forces

The forces acting on the cylindrical shell are the body forces {*b*_{xm}, *b*_{φm}, *b*_{rm}}^{T} acting on the coordinate axis (*x*, *φ*, *z** _{m}*) of the

*m*th cylindrical shell, the distributed surface force {

*s*

_{xm},

*s*

_{φm},

*s*

_{rm}}

^{T}and the ring-distributed force {

*S*

_{xm},

*S*

_{φm},

*S*

_{rm}}

^{T}acting along the profile at location

*x*=

*x*

_{i}(

*i*=1, 2, …) along the longitudinal axis. The surface force acting on the inside and outside surfaces of the shell is taken into consideration by using the Dirac delta function

*δ*(

*z*).

The vdW force is the distributed load acting in the radial direction of the cylindrical shell. The vdW force applied to the *m*th shell by the *m*+1st cylindrical shell is denoted as *s*_{m,m+1}. The first subscript represents the number of the shell receiving the action and the second subscript represents the number of the shell generating the action. The vdW force acting on the *m*th cylindrical shell is given by(3.12)where *c*_{m} is the interaction coefficient of the vdW force between the *m*−1st and the *m*th cylindrical shell, and *c*_{m+1} is the interaction coefficient of the vdW force between the *m*th and the *m*+1st cylindrical shell. Both *w*_{z,0} and *w*_{z,M+1} are zero since they do not exist. Also, if the cylindrical shell on the outermost shell (*m*=*M*) is covered with an elastic medium, the elastic coefficient is denoted by *c*_{M+1}.

Based on the above discussion, the work of the external force *V* can be expressed as(3.13)Here,(3.14)

### (g) The Hamilton principle

The Hamilton principle is expressed as follows:(3.15)Equations (3.10), (3.11) and (3.13) are substituted into the Hamilton principle equation (3.15) and enclosed by the displacement components (*δu*_{m}, *δv*_{m}, *δw*_{m}). In order to satisfy the equation for an arbitrary virtual displacement (*δu*_{m}, *δv*_{m}, *δw*_{m}), the following equilibrium condition must be proven for the stress resultant:(3.16)

### (h) Displacement expression for stress resultant

By substituting equation (3.5) into the stress *σ*_{kl} on the right-hand side of the definition (3.9) of the stress resultant, the following displacement expression can be obtained:(3.17)where *I*_{m} and in the above equations are defined as(3.18)The reduction coefficient *α*, which is multiplied by the plate-bending modulus *k*_{m}, is *α*=1 for a normal cylindrical shell. However, in the case of CNTs, depending on the experimental and other results, the value of the plate-bending modulus *k*_{m} needs to be lowered by the reduction coefficient *α* (Ru 2000). On the other hand, based on the thin wall assumption, if *I*_{m}→1, the constitutive equation (3.17) completely coincides with Flügge's (1962) constitutive equation. More specifically, the constant *I*_{m} provides the present theory with higher accuracy than Flügge's theory.

### (i) Equivalent shear force

The boundary condition is written as follows:(3.19)where *Q*_{vm} and *Q*_{wm} are the equivalent shear forces corresponding to the displacements *v*_{m} and *w*_{m}, respectively, and and are the external force components associated with the shear forces. Likewise, the bending moment and the external force in the axial direction are defined as follows:(3.20)(3.21)If equation (3.17) is substituted into the stress resultant , , and on the right-hand side after multiplying the operator on the left to both sides of this equivalent shear force equation (3.20), they can be expressed with the deformation components as follows:(3.22)In addition, by using equation (3.2), the cross-sectional rotation *ψ*_{m} can be replaced with the derivative of the deflection *w*_{m}.

## 4. Governing equation for beams obtained by Flügge's theory

### (a) Separation of variables for the displacement function

The displacement (*u*_{m}, *v*_{m}, *w*_{m}) corresponding to the coordinate direction of (*x*, *φ*, *z** _{m}*) can be expressed by separating the variables as follows:(4.1)Based on definition (3.4)

_{3}, which is the shear strain on the middle surface, in order to be able to ignore the shear strain , the following can be used:(4.2)

The unit warping function ^{k}*u*(*φ*) and its derivatives are set as shown in table 1 (Schardt 1989).

Under assumption (ii), in order to ignore the shear deformation, the cross-sectional rotation *ψ*_{m}(*x*, *φ*, *t*) of the curved plate constituting a cylindrical shell can be set as below, according to equation (3.2),(4.3)In this regard, the following definition is adopted:(4.4)

The distributed loads *p*_{xm}, *m*_{xm}, *p*_{φm}, *m*_{φm}, *p*_{rm} are also expanded by the unit warping function ^{k}*u*(*φ*) or its derivative function(4.5)

### (b) The Hamilton principle

The definitional equations of the equivalent shear force (3.20) and the ring-distributed force acting along the profile (3.21) and the displacement expression (3.17) of the stress resultant are substituted into the Hamilton principle function described above(4.6)The constitutive law of the stress resultant *N*_{xm}, *Q*_{vm}, *M*_{xm} has already been expressed in equation (3.17). Based on equations (4.1) and (4.3), the equation below is used as the virtual displacement(4.7)

### (c) Definition of stress resultant for beams

The stress resultant and the concentrated load on the entire cross section of a cylindrical tube are defined as follows:(4.8)where the stress resultants per unit width of a curved plate constituting a cylindrical shell are *M*_{xm}, *N*_{xm}, *Q*_{vm}, *Q*_{wm} and the distributed loads on the profile line are , , , .

With respect to the deformation pattern *k*, the stress resultant is found by contour integration on the bending moment along the midline of the curved plate constituting the cylindrical shell. The stress resultant gives the axial force (*k*=2), the bending moments around the vertical axis (*k*=3) and the horizontal axis (*k*=4) of the cross section, as well as the warping moment (*k*=5 and above) associated with the cross-sectional distortion. The stress resultant gives the torsional torque (*k*=1), the horizontal shear force (*k*=3), the vertical shear force (*k*=4) and the shear force component associated with the cross-sectional distortion (*k*=5 and above). The stress resultant gives the vertical shear force (*k*=3), that is, the shear force in the thickness direction that is applied on the curved plate constituting the cylindrical shell, the vertical shear force (*k*=4) and the shear force associated with the cross-sectional distortion (*k*=5 and above).

### (d) Constitutive law and equation of motion of beams

In the torsional deformation mode (*k*=1), based on table 1, the basis function of the in-plane displacement is , while the other is . By substituting these into the contour integration of constitutive equation (3.22)_{1} and the equilibrium equations of the stress resultant *Q*_{vm} of the Hamilton principle equation (4.6), the constitutive law and the torsional vibration equation of motion for the torque *T*_{m}(*x*, *t*) of the *m*th shell and the rotation angle *ϕ*_{m}(*x*, *t*) are given as(4.9)The rest of the equations are of the form of zero-identity. However, the expressions are rewritten as and , and the torsional resistance is simplified (Gjelsvik 1981) as follows:(4.10)

In the longitudinal expansion mode (*k*=2), the unit warping function is and . By substituting these expressions into the contour integral of the constitutive equation (3.17)_{2} and the equilibrium equation of the stress resultant *N*_{xm} in the Hamilton equation (4.6), and by subsequently dividing both sides by *a*_{m}, the constitutive law of the axial force *N*_{m}(*x*, *t*) and the elastic elongation *u*_{0m}(*x*, *t*) of the *m*th shell are obtained together with the equation of motion of the longitudinal vibration(4.11)The rest of the equations are of the form of zero-identity. However, here they are rewritten as , , .

In the bending deformation and the cross-sectional distortion modes (*k*≥3), since the unit warping function and the basis function of in-plane displacement consist of trigonometric functions, the contour integration was performed by multiplying each virtual displacement corresponding to the equilibrium condition of the cross section as obtained from the Hamilton equation (4.6) and the constitutive equations (3.17) and (3.22)_{1}. If the orthogonality condition of the functions is applied, the constitutive law and the equation of motion for beams become as follows:(4.12)(4.13)However, owing to the contour integration, the non-local elastic operator was newly rewritten as(4.14)If is eliminated from equations (4.12)_{3} and (4.13)_{1}, the displacement expression of the shear force is given as(4.15)

### (e) Displacement equation for beams and the stiffness matrix

By eliminating from the equilibrium equations of the stress resultant (4.13)_{1} and (4.13)_{4}, and by substituting the constitutive equation (4.12) into the stress resultants , , of the rest of the equilibrium equations (4.13)_{2} and (4.13)_{3}, we obtain the stiffness equation, which is the ultimate goal of this research,(4.16)Likewise, based on equation (4.4), the cross-sectional rotation ^{k}*Ψ*_{m} has been written into the first derivative of the deflection ^{k}*W*_{m}. The element stiffness matrix on the left-hand side of the equation can be defined through the following symmetrical operator components:(4.17)where the cross-sectional quantities are defined as follows, which includes definitions that will be used later:(4.18)

According to the form of the first term on the left-hand side of equation (4.16), the element stiffness matrix of the *m*th shell is symmetric with respect to the main diagonal. However, according to the second term on the left-hand side, the symmetric property with respect to the main diagonal on the global stiffness matrix of the multi-walled cylindrical shell is disturbed due to the non-local operator . In other words, if the non-locality is not considered, then , which means that the symmetric property with respect to the main diagonal of the total stiffness matrix of the multi-walled cylindrical shell can be kept intact.

If Cramer's rule is applied after obtaining the determinant of the stiffness matrix of the left-hand side of equation (4.16), the governing differential equation of the eighth order with only one single displacement magnitude, e.g. , or , can be obtained.

### (f) Correspondence with Timoshenko beam and GBT beam

Let the displacement vector of the stiffness matrix (4.16) described in §4*e* be . If the circumferential elastic strain on the middle surface of the cross section is zero (), then , and equation (4.16) becomes two simultaneous differential equations as follows:(4.19)Since *n*=1 for the bending of the rigid cross-section, by assuming that the plate bending coefficients ; *I*_{Bm} and as well as the vdW force are all zero, the equation described above yields the Timoshenko beam theory.

Next, in order to compare the theoretical equation obtained in this research and Schardt's GBT, if the shear strain on the middle surface is assumed to be zero (), then from equation (4.2). Then, by substituting this relational expression into the system of simultaneous differential equations (4.19), differentiating both sides of the upper equation by *x* once and subsequently subtracting the same sum from both sides, the governing differential equations (4.20) of fourth order are given with respect to the displacement component ,(4.20)However, each coefficient is as follows:(4.21)When the cross section is rigid (*n*=1), the warping resistance *k*_{Cm}(*n*) of a cylindrical shell for which the bending stiffness of the plate is not reduced (*α*=1) becomes(4.22)that coincides with the geometrical moment of inertia of a circular tube.

### (g) Boundary condition

The following equation describes the boundary condition that can be obtained from the Hamilton principle (4.6):(4.23)However, as given by equations (4.12), the stress resultant at the boundary is expressed by the displacement magnitude. The external force of the boundary cross section can be calculated by substituting equation (3.21) into equation (4.8). Table 2 illustrates the condition in the case where this boundary condition is applied to the actual cylindrical tube.

## 5. Phase-velocity curve

The phase-velocity curve of an *M*-walled CNT cylindrical shell of infinite length which does not receive distributed external load is obtained. For an elastic wave with wavelength *λ* and phase velocity *c* the following deformation functions are assumed:(5.1)

These functions are subsequently substituted into the equation of motion (4.16). The condition where 3*M* kinds of amplitudes , , (*m*=1, 2, …, *M*) have a non-trivial solution indicates that the determinant value of the obtained coefficient matrix becomes zero. Based on this, the phase-velocity curve is obtained.

The values that are commonly used for the CNT material are as follows (Dong & Wang 2007):

The following equation is used for the vdW interaction spring constant *c*_{m} (Girifalco & Lad 1956)(5.2)However, the *c*–*c* bond length is *d*=1.42×10^{−8} cm.

Figures 4–6 illustrate the phase-velocity curve, where the frequency *ω* (THz) is shown on the horizontal axis, and the elastic wave velocity *c* (m s^{−1}) is shown on the vertical axis. Figures 4–6 illustrate the phase-velocity curve of single-walled nanotubes (SWNTs), double-walled nanotubes (DWNTs), and triple-walled nanotubes (TWNTs), respectively. For the circumferential deformation of the cross section, four types (*n*=0, 1, 2, 3) are illustrated. In order to take into consideration the non-locality, *e*_{0}*a*=0.0554, the spring constant *c*_{mat} of the surrounding matrix is assumed to be equal to the spring constant *c*^{*} of the vdW interaction.

Depending on the size of the coefficient matrix on the eigenvalue problem, there are theoretically three wave-mode curves on the single-walled cylindrical shell, six on the double-walled and nine on the triple-walled. However, it appears that, in the second mode of the circumferential deformation mode (*n*=0) on the double-walled cylindrical shell (figure 5), there are only five dispersion curves since both the inner and the outer shells are layered on top of each other, forming a single horizontal line. Likewise, it appears that, in the second mode of the circumferential deformation mode (*n*=0) on the triple-walled cylindrical shell (figure 6), there are only seven dispersion curves since the inner, intermediate and outer shells are layered on top of each other along a single horizontal line. The first and second waves on the circumferential deformation mode (*n*=0) on the single-walled cylindrical shell (figure 4) indicate that the value is almost constant between 0 and 5 THz, while the first wave decreases significantly at approximately 5.4 THz and maintains a constant value after that. There is a cut-off frequency of the third wave that corresponds to the frequency where the first wave falls. In the circumferential deformation mode *n*=1, 2, 3, it has been confirmed that there is a cut-off frequency for all three different wave modes. The first wave of these modes becomes a dispersion curve that passes through the origin when the spring constant of the surrounding matrix is ignored (Dong & Wang 2007). However, in this case, since the spring constant is taken into consideration, the cut-off frequency is obtained. On the other hand, the cut-off frequencies of all wave modes, which correspond to the increase in the value of *n* (the number of the circumferential deformation mode), increase. This tendency is also observed in the DWNTs and TWNTs (figures 5 and 6, respectively).

The influence of the reduction coefficient *α* of the bending stiffness of the plate was found by using the triple-walled cylindrical shell. In other words, the calculation of the standard cylindrical shell theory was defined as corresponding to *α*=1, and the calculation of the CNT cylindrical shell theory was defined as *α*=1/27.5. The differences between the dispersion curves are illustrated in figure 7. In the circumferential deformation mode, where *n*=0, the second and third waves do not receive any influence from the reduction coefficient. However, the first wave indicates that the post-drop constant value where the reduction is taken into consideration (*α*=1/27.5) differs by almost 50 per cent when compared with the scenario in which the reduction is not taken into consideration (*α*=1). Furthermore, in the circumferential deformation mode *n*=1, 2, 3, the second and third waves do not receive any influence from the reduction coefficient, although the second wave indicates that the post-drop constant value in the case where the reduction is taken into consideration (*α*=1/27.5) becomes one-quarter that of the scenario in which the reduction is not taken into consideration (*α*=1).

The influence of the spring constant *c*_{mat} of the surrounding matrix was found by using the triple-walled cylindrical shell. In other words, the spring constant of the matrix that is equal to the vdW interaction spring constant is defined as *c*_{mat}=1.0*c*^{*} and the spring constant of the matrix which becomes 1/100 of the former is written as *c*_{mat}=0.01*c*^{*}. The differences between the dispersion curves are illustrated in figure 8. As shown, the cut-off frequency of the outermost shell, which is in contact with the matrix, decreases as the matrix spring constant decreases. The magnitude becomes greater as the circumferential deformation mode number *n* becomes larger. In particular, when *n*≥1, it becomes greater as the wave mode number becomes smaller. In the first wave mode, it decreases to approximately 40 per cent. However, there is no influence in the second and third modes, which is the anticipated result.

The influence of the non-locality was determined by using the triple-walled cylindrical shell. In this regard, *e*_{0}*a*=0 when the non-locality is ignored, while *e*_{0}*a*=0.0554 and *e*_{0}*a*=0.1 when it is taken into consideration. The differences between the dispersion curves are illustrated in figure 9. When the circumferential deformation mode number is *n*=0, 1, 2, there is no significant difference between these scenarios. However, when the circumferential deformation mode number is *n*≥3, the difference becomes slightly noticeable. These individual first waves of the three different cylindrical shells start from the cut-off frequency and become one curve at approximately 8 THz. When the non-locality is ignored (*e*_{0}*a*=0), the frequency after that remains constant. In contrast to this, when the non-locality is taken into consideration and *e*_{0}*a*=0.0554 the phase-velocity curve, which becomes one curve, begins to decrease its wave velocity gradually at approximately 8 THz, after which it stops approximately at 32 THz. On the other hand, when the non-locality is increased to *e*_{0}*a*=0.1, the phase-velocity curve, which becomes one curve, decreases its wave velocity faster and stops at approximately 17 THz. The second and third waves also tend to decrease the wave velocity, although not as noticeably as the first wave.

Lastly, the dispersion curve of the group velocity *c*_{g} of the single-walled cylindrical shell was calculated (figure 10) by differentiating the angular frequency *ω* with respect to the wavenumber *k*. The vertical axis represents the group velocity *c*_{g} (m s^{−1}), and the horizontal axis represents the number of waves *k* (10^{9} m^{−1}) (Wang *et al*. 2008). The circumferential deformation mode is (*n*=0, 1, 2, 3) as before, and the spring constant *c*_{mat} of the surrounding matrix is also assumed to be equal to the spring constant *c*^{*} of the vdW interaction. In the range where the wave is small, that is, from zero to the maximum group velocity, the influence of the non-locality is small. However, the difference becomes significant after that. When the non-locality is ignored, the group velocity takes a constant value at the maximum velocity after reaching it. When the non-locality is taken into consideration, the maximum group velocity decreases drastically in most cases.

## 6. Conclusion

The vibration theory of Flügge's cylindrical shell, which takes into consideration the rotational inertia, was converted into a non-local elastic beam theory by using the Hamilton principle. Since the origin of the radial coordinate of the cylindrical shell cross section was taken at the centre of the circle, this theory proved to be more accurate than the vibration theory of Flügge's cylindrical shell. By defining the stress resultant of the whole cross section for these theories, the equation of motion was derived for the stress resultant. The stiffness equation can be obtained by substituting the constitutive law of the stress resultant into the equation of motion, and the obtained element stiffness matrix represents the symmetric property with respect to the main diagonal. However, on the coupling terms with the neighbouring shells, which are attributed to the vdW force, the symmetry property was disturbed due to the non-locality. The boundary conditions used here are as described in table 2.

In addition, the validity of the obtained beam theory from the wave analysis of the multi-walled cylindrical tube was verified. This validation was specifically focused on the relation between the number of walls in the tube and the cut-off frequency, the influence of the reduction of the bending stiffness of the plate on the tube, the influence of the stiffness of the surrounding matrix and the difference between the scenarios with or without the inclusion of the non-locality. Lastly, the influence of the non-locality with respect to the group-velocity curve was also examined.

The presented beam theory that was derived from Flügge's cylindrical shell theory is simpler and requires a smaller number of state quantities than the traditional Flügge theory, while at the same time it is more accurate than the cylindrical shell theory. Hence, the present theory is now developed as a theory that can be used for tubular beams, such as multi-walled nanotubes. As the Flügge theory ignores the shear deformation of the shell wall, more accurate shell models are usually needed in cases where exact higher order dispersion relations are needed.

## Acknowledgments

T.U. gratefully acknowledges the Deutsche Akademische Austausch Dienst for providing the opportunity to study at Technische Hochschule Darmstadt. T.U. also thanks Prof. Richard Schardt for his academic support.

## Footnotes

- Received October 3, 2008.
- Accepted November 28, 2008.

- © 2009 The Royal Society