# Suppression of Brazier Effect in Multilayered Cylinders

Authors: Hiroyuki Shima; Motohiro Sato; Sung-Jin Park
Journal: Advances in Condensed Matter Physics
 (2014)

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## Abstract

When a straight hollow tube having circular cross-section is bent uniformly into an arc, the cross-section tends to ovalize or flatten due to the in-plane stresses induced by bending; this ovalization phenomenon is called the Brazier effect. The present paper is aimed at theoretical formulation of the Brazier effect observed in multilayered cylinders, in which a set of thin hollow cylinders are stacked concentrically about the common axis. The results indicate that mechanical couplings between stacked cylinders are found to yield pronounced suppression of the cross-sectional ovalization. Numerical computations have been performed to measure the degree of suppression in a quantitative manner and to explore how it is affected by the variations in the bending curvature, the number of stacked cylinders, and the interlayer coupling strength.


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## Body

## 1. Introduction

Concentrically stacked cylinders are ubiquitous in nature and technology. Primary examples in the field of condensed matter physics is a series of multiwalled nanotubes. To date, successfully synthesized was a wide variety of nanotubes made from carbon [1, 2], silver [3], gold [4], silica [5], boron nitride [6, 7], WS2 [8, 9], and TiO2 [10], most of which show potential utilities as device components owing to unique physicochemical properties and/or large surface area. In a realm of biochemistry, in addition, there are several examples such as self-assembled microtubules in living cells [11] and multilayer lipid tubes [12]. In those biotubes, the mechanical strength and morphological stability are crucial factors for their biological functions [13–15].From a practical viewpoint, coaxial multicylinder structures give a merit for designing composite materials both in macroscopic and nanoscopic scales. In civil engineering, for example, deep water pipelines [16] and fiber-reinforced polymer tubes [17] are known to be endowed with high mechanical strength by constructing them with multilayered structures. Choice of multilayered structures is also preferred in building thermal energy storage with cylindrical shape [18], wherein a concentric-tube arrangement results in significant improvement in the system performance.Another important benefit of coaxial multicylinder structures is enhanced mechanical robustness against cross-sectional deformation. Compared with a single hollow cylinder, multilayered counterparts afford resistance to cross-sectional deformation caused by external load [19, 20]. The resistance property originates mainly from the presence of internal cylinders enclosed by outer ones; the internal cylinders can push back the inward deflection of outer cylinders, thus preventing the cross-sections from severe deformation. This will hold true for bending-induced deformation in the cross-section of a hollow cylinder, commonly known as the Brazier effect [21]. Coaxial multicylinders are expected to lead feasible suppression of the Brazier effect in the systems, thus being preferred for designing composite materials that should be undeformable against external load. Despite the fundamental interest, little attempt has been made so far to elucidate the degree of suppression which multilayered structures supply.The purpose of this work is to establish the theory that quantifies the Brazier effect suppression expected to occur in multilayered cylinders. Thin-shell theory based approach is used to formulate the strain energy of multilayered cylinders under bending, the result of which allows describing the growth of ovalization with increasing the bending curvature. The dependence of the cross-sectional shape on the relevant parameters such as the bending curvature, the number of layers, and the coupling strength between adjacent layers has been also clarified.


## 2. Background: What Is the Brazier Effect?

Flattening of elastic tubes under pure bending has been a focus of much attention over many decades since Brazier first studied this problem [21]. The flattening phenomenon has been usually called the Brazier effect after his name, while other seminal works by Reissner [22] and by Seide and Weingarten [23] have played decisive roles in the progress of the issue.Figure1 illustrates the mechanism of Brazier effect. It shows an infinitesimal portion of a deformed hollow cylinder that is originally straight and has a circular cross-section. Application of pure bending to the cylinder causes the compressive and tensile stresses that act at an angle to the unrotated section. As a result, the original circular shape deforms into an oval as displayed in the bottom panel of Figure 1. This is equivalent to saying that for a certain curvature, the ovalization minimizes the mechanical energy of the cylinder.(a) Schematic of the Brazier effect. Bending application to a hollow cylinder of circular section gives rise to the stresses indicated by arrows colored in red and blue. The compressive (red arrow) and tensile (blue) stresses act at a certain angle to the unrotated section, as a result of which the original circular shape deforms into an oval or flattened shape. (b) Definition of the oblateness parameterζ.
(a)
(b)The degree of ovalization is quantified by the parameterζ, called the oblateness. Given an initially circular cross-section of radius r, the product rζ equals to the minor axis length of the oval obtained after deformation. The definition of ζ is conventionally used even for extremely deformed sections, which go beyond simple ovalization to flattening or collapse in a peanut-like shape in which ζ can be much less than unity.


## 3.  Formulation

The stable cross-sectional shape of a hollow cylinder under pure bending is evaluated using a thin shell theory. The theory states that the energy costU for a cylinder to deform within linear elastic region is written by a sum of three energy terms:
(1)U=Ucir+Uaxi+Uint⁡.
Here, Ucir is the strain energy associated with the circumferential displacement, Uaxi is that with the axial strain, and Uint⁡ is the energy caused by the interaction between cylindrical surfaces in the radial direction. All the three energy terms are functions of the curvature Γ of the deformed cylinder’s axis and the displacement of volume elements. The latter quantity, displacement, is in general represented by a spatially dependent vector. To describe it, we introduce a set of mutually perpendicular vectors u, v, and w as shown in Figure 2. Amplitudes of the vectors u, v, and w indicate the displacements of a volume element in the radial, circumferential, and axial direction, respectively.Figure 2
Diagram of the displacementsu,v,w of a volume element in a hollow cylindrical tube with radius r. The tube axis is parallel to the z axis, which is vertical to the tube’s cross-section with thickness h that lies in the x-y plane.Our objective is a series of the optimal oblateness parameters{ζi}=(i=1,…,N) for each ith layer contained in an N-layered cylinder; it is what minimizes the total energy U of the system under a given bending curvature Γ. To this aim, we first derive explicit forms of Ucir, Uaxi, and Uint⁡ for a monolayered cylinder as functions of u, v, and w (see from Section 3.1 to Section 3.4), followed by applying the variation method to U with respect to u, v, and w. The strategy for a monolayered system can be extended straightforwardly to multilayered ones, as will be demonstrated in Section 5.


### 3.1. Stress-Strain Relation

From the definition of elasticity, every elastic medium satisfies an essential relation between applied strain (εj) and the resulting stress (σj) [j=r,θ,z], which is written in terms of cylindrical coordinates {r,θ,z} by
(2)εr=1E[σr-ν(σθ+σz)],(3)εθ=1E[σθ-ν(σz+σr)],(4)εz=1E[σz-ν(σr+σθ)],
with E being Young’s modulus and ν being Poisson’s ratio of the material under discussion. As σr≡0 for a hollow cylinder, it follows from (3)-(4) that
(5)σθ=E1-ν2(εθ+νεz),σz=E1-ν2(εz+νεθ).
Suppose that an infinitesimal volume element dV=dSdz of a cylinder is displaced slightly in the j-direction (j=θ or z). The strain energy dUj caused by the deformation reads
(6)dUj=12σjεjdSdz.[j=θorz].
We thus obtain the expressions
(7)dUcir=E2(1-ν2)(εθ2+νεθεz)dSdz,dUaxi=E2(1-ν2)(εz2+νεzεθ)dSdz,
where we renamed dUθ by dUcir and dUz by dUaxi. To proceed the argument, we need to derive explicit functional forms of εθ, εz in terms of the displacements u, v, w.


### 3.2. Circumferential Strain Energy

We first considerUcir of a hollow cylinder subjected to cross-sectional deformation. Suppose a circumferential line element of length dℓ~θ lying within the annular cross-section with thickness h (see Figure 3). The tilde (g~) attached to ℓθ means that it is measured at a point distant from the centroidal circle 𝒞. We will see later that under appropriate conditions, the strain is determined by u(θ) and v(θ) that are defined just on the undeformed centroidal circle 𝒞 of the circular cross-section (see (16)).Figure 3
Displacement of a circumferential line element of the lengthdℓ~θ to that of dℓ~θ*. The centroidal circle 𝒞 of the cross-section, having the radius r before deformation, is also depicted by a dotted curve. Through the displacement described by u~ and v~, the line element is elongated slightly in the θ direction and the normal to the centroidal curve rotates with an angle β.The extensional strain of the circumferential line element, designated byε~θ, is defined by
(8)ε~θ=dℓ~θ*-dℓ~θdℓ~θ,
where dℓ~θ=r~dθ and dℓ~θ* is the length of the line element after deformation (throughout the paper, the asterisk symbolizes the quantity after deformation). The coordinates x~*, y~* of the element after deformation are given by
(9)x~*(θ)=[r~+u~(θ)]cos⁡θ-v~(θ)sinθ,y~*(θ)=[r~+u~(θ)]sinθ+v~(θ)cos⁡θ,
which imply that
(10)(dℓ~θ*)2=(dx~*)2+(dy~*)2=[r~2+2r~(u~+v~)+(u~+v~)2+(u~-v~)2](dθ)2.
Note that (8) can be rewritten as ε~θ+1=dℓ~θ*/(r~dθ). Squaring the both sides and then rearranging give
(11)ε~θ+12ε~θ2=12[1r~2(dℓ~θ*dθ)2-1].
For ε~θ≪1, the term ε~θ2 can be omitted; we substitute (10) into (11) to obtain
(12)ε~θ=u~+v~′r~+12(u~′+v~′r~)2+12(u~′-v~r~)2,
where u~′≡∂u~/∂θ and so forth.The last term in (12) is associated with the rotation of the line element due to deformation. The rotation angle β consists of two parts (see Figure 4): (i) a clockwise component ∂u~/(r~∂θ) due to the spatial variation of u~ in the circumferential direction and (ii) a counterclockwise one v~/r~ due to the circumferential displacement of the element. Combination of the two parts gives
(13)β=v~-u~′r~,
which has a positive value in the counterclockwise sense.Two different mechanisms of the rotation of a line element. Left: clockwise rotation due to the spatial variation ofu~ in the θ direction. Right: counterclockwise rotation caused by the circumferential displacement of the element.
(a)
(b)Formula (12) is valid for arbitrary large rotation β. Particularly when ε~ and β are both sufficiently small (but finite), we may neglect the second term in the right side in (12) (here we exclude the possibility that |u~| or |v~′| is of the order of r~ or larger). We further assume that normals to the undeformed centroidal circle 𝒞 remain normal and inextensional during the deformation. Under this assumption, we can write
(14)u~=u,v~=v+ρβ,
where u and v denote the displacements of a point just on 𝒞 and ρ is a radial coordinate measured from 𝒞. By substituting (14) into (12), we attain the strain-displacement relation such as
(15)ε~θ(ρ,θ)=εθ(θ)+ρκθ(θ),
with the definitions
(16)εθ=u+v′r+12(u′-vr)2,κθ=-u′′-v′r2.
Here r is the radius of the undeformed circle 𝒞.The results (15) and (16) state that the circumferential strain at points distant from 𝒞 is determined by the displacements u(θ) and v(θ) of a point just on 𝒞. The results also allow us to write the deformation energy Ucir as
(17)Ucir=Er2(1-ν2)∫0L‍dz∫-h/2h/2‍dρ∫02π‍dθ[ε~θ(ρ,θ)]2
or, more concisely,
(18)Ucir=Ehr2(1-ν2)∫0L‍dz∫02π‍dθ(εθ2+h212κθ2),
which implies that Ucir is a functional of u, v, and their derivatives as
(19)Ucir=Ucir[u,∂u∂θ,∂2u∂θ2,v,∂v∂θ].


### 3.3. Axial Strain Energy

Parallel discussions to that in Section3.2 give us the explicit form of the deformation energy Uaxi caused by the axial strain. The strain ε~z of the axial line element dℓz is defined by
(20)ε~z=dℓ~z*-dℓ~zdℓ~z.
Here dℓ~z* is the length of the line element after deformation. Using the displacement components u~, v~, and w~, the square of dℓ~z* is given by
(21)(dℓ~z*)2=(dℓ~z+∂w~∂zdz)2+(∂u~∂zdz)2+(∂v~∂zdz)2.
Note the equivalence of dℓ~z with dz in (21); then, we obtain from (20)-(21) that
(22)2ε~z+ε~z2=2w~′+(w~′)2+(u~′)2+(v~′)2,
where w~′≡∂w~/∂z and so forth. Since ε~z2≪1 and (w~′)2≃0, we have
(23)ε~z=w~′+12(u~′)2+12(v~′)2.In general,w~ differs from w that is allocated just on 𝒞. Nevertheless, w~ can be related with w and u by
(24)w~=w-ρ∂u∂z
and accordingly w~′=w′-ρu′′. Similarly, we use the approximations of u~≃u and v~≃v+ρβ, which imply that u~′≃u′, v~′≃v′. Consequently we have ε~z=εz-ρκz with the definitions of
(25)εz=w′+12(u′)2+12(v′)2,κz=u′′,
which leads us to the axial energy expression
(26)Uaxi=Ehr2(1-ν2)∫0L‍dz∫02π‍dθ(εz2+h212κz2).
It should be emphasized that the expression (26) implies the functional property of Uaxi represented by
(27)Uaxi=Uaxi[∂u∂z,∂2u∂z2,∂v∂z,∂w∂z].


### 3.4. Interaction Energy

Mechanical stability of cross-sections in multilayered cylinders owes in part to interaction between adjacent cylindrical surfaces. In typical nanotubular materials, for instance, the interaction originates from intermolecular van der Waals forces, thus sufficiently small compared with strong chemical bonding [1]; as a result, many nanotubular materials are endowed with anisotropy in mechanical stiffness, characterized by large stiffness against longitudinal stresses and flexibility to radial compression.For a linear elastic deformation, it is reasonable to represent the interaction energy betweenith and jth concentric cylinders by
(28)Uint⁡(i,j)=kri2∫0L‍dz∫02π‍dθ(ui-uj)2.
Here, k is an effective spring constant per surface area rdθdz, which serves as a measure for the interaction strength. The value of k is dependent on the material and geometric structure of the system considered; among choices, we hereafter pay attention to the weak interaction for which krN/E≤1 or lesser, by referring to the existing multiwalled nanotubes [24].


## 3.1. Stress-Strain Relation

From the definition of elasticity, every elastic medium satisfies an essential relation between applied strain (εj) and the resulting stress (σj) [j=r,θ,z], which is written in terms of cylindrical coordinates {r,θ,z} by
(2)εr=1E[σr-ν(σθ+σz)],(3)εθ=1E[σθ-ν(σz+σr)],(4)εz=1E[σz-ν(σr+σθ)],
with E being Young’s modulus and ν being Poisson’s ratio of the material under discussion. As σr≡0 for a hollow cylinder, it follows from (3)-(4) that
(5)σθ=E1-ν2(εθ+νεz),σz=E1-ν2(εz+νεθ).
Suppose that an infinitesimal volume element dV=dSdz of a cylinder is displaced slightly in the j-direction (j=θ or z). The strain energy dUj caused by the deformation reads
(6)dUj=12σjεjdSdz.[j=θorz].
We thus obtain the expressions
(7)dUcir=E2(1-ν2)(εθ2+νεθεz)dSdz,dUaxi=E2(1-ν2)(εz2+νεzεθ)dSdz,
where we renamed dUθ by dUcir and dUz by dUaxi. To proceed the argument, we need to derive explicit functional forms of εθ, εz in terms of the displacements u, v, w.


## 3.2. Circumferential Strain Energy

We first considerUcir of a hollow cylinder subjected to cross-sectional deformation. Suppose a circumferential line element of length dℓ~θ lying within the annular cross-section with thickness h (see Figure 3). The tilde (g~) attached to ℓθ means that it is measured at a point distant from the centroidal circle 𝒞. We will see later that under appropriate conditions, the strain is determined by u(θ) and v(θ) that are defined just on the undeformed centroidal circle 𝒞 of the circular cross-section (see (16)).Figure 3
Displacement of a circumferential line element of the lengthdℓ~θ to that of dℓ~θ*. The centroidal circle 𝒞 of the cross-section, having the radius r before deformation, is also depicted by a dotted curve. Through the displacement described by u~ and v~, the line element is elongated slightly in the θ direction and the normal to the centroidal curve rotates with an angle β.The extensional strain of the circumferential line element, designated byε~θ, is defined by
(8)ε~θ=dℓ~θ*-dℓ~θdℓ~θ,
where dℓ~θ=r~dθ and dℓ~θ* is the length of the line element after deformation (throughout the paper, the asterisk symbolizes the quantity after deformation). The coordinates x~*, y~* of the element after deformation are given by
(9)x~*(θ)=[r~+u~(θ)]cos⁡θ-v~(θ)sinθ,y~*(θ)=[r~+u~(θ)]sinθ+v~(θ)cos⁡θ,
which imply that
(10)(dℓ~θ*)2=(dx~*)2+(dy~*)2=[r~2+2r~(u~+v~)+(u~+v~)2+(u~-v~)2](dθ)2.
Note that (8) can be rewritten as ε~θ+1=dℓ~θ*/(r~dθ). Squaring the both sides and then rearranging give
(11)ε~θ+12ε~θ2=12[1r~2(dℓ~θ*dθ)2-1].
For ε~θ≪1, the term ε~θ2 can be omitted; we substitute (10) into (11) to obtain
(12)ε~θ=u~+v~′r~+12(u~′+v~′r~)2+12(u~′-v~r~)2,
where u~′≡∂u~/∂θ and so forth.The last term in (12) is associated with the rotation of the line element due to deformation. The rotation angle β consists of two parts (see Figure 4): (i) a clockwise component ∂u~/(r~∂θ) due to the spatial variation of u~ in the circumferential direction and (ii) a counterclockwise one v~/r~ due to the circumferential displacement of the element. Combination of the two parts gives
(13)β=v~-u~′r~,
which has a positive value in the counterclockwise sense.Two different mechanisms of the rotation of a line element. Left: clockwise rotation due to the spatial variation ofu~ in the θ direction. Right: counterclockwise rotation caused by the circumferential displacement of the element.
(a)
(b)Formula (12) is valid for arbitrary large rotation β. Particularly when ε~ and β are both sufficiently small (but finite), we may neglect the second term in the right side in (12) (here we exclude the possibility that |u~| or |v~′| is of the order of r~ or larger). We further assume that normals to the undeformed centroidal circle 𝒞 remain normal and inextensional during the deformation. Under this assumption, we can write
(14)u~=u,v~=v+ρβ,
where u and v denote the displacements of a point just on 𝒞 and ρ is a radial coordinate measured from 𝒞. By substituting (14) into (12), we attain the strain-displacement relation such as
(15)ε~θ(ρ,θ)=εθ(θ)+ρκθ(θ),
with the definitions
(16)εθ=u+v′r+12(u′-vr)2,κθ=-u′′-v′r2.
Here r is the radius of the undeformed circle 𝒞.The results (15) and (16) state that the circumferential strain at points distant from 𝒞 is determined by the displacements u(θ) and v(θ) of a point just on 𝒞. The results also allow us to write the deformation energy Ucir as
(17)Ucir=Er2(1-ν2)∫0L‍dz∫-h/2h/2‍dρ∫02π‍dθ[ε~θ(ρ,θ)]2
or, more concisely,
(18)Ucir=Ehr2(1-ν2)∫0L‍dz∫02π‍dθ(εθ2+h212κθ2),
which implies that Ucir is a functional of u, v, and their derivatives as
(19)Ucir=Ucir[u,∂u∂θ,∂2u∂θ2,v,∂v∂θ].


## 3.3. Axial Strain Energy

Parallel discussions to that in Section3.2 give us the explicit form of the deformation energy Uaxi caused by the axial strain. The strain ε~z of the axial line element dℓz is defined by
(20)ε~z=dℓ~z*-dℓ~zdℓ~z.
Here dℓ~z* is the length of the line element after deformation. Using the displacement components u~, v~, and w~, the square of dℓ~z* is given by
(21)(dℓ~z*)2=(dℓ~z+∂w~∂zdz)2+(∂u~∂zdz)2+(∂v~∂zdz)2.
Note the equivalence of dℓ~z with dz in (21); then, we obtain from (20)-(21) that
(22)2ε~z+ε~z2=2w~′+(w~′)2+(u~′)2+(v~′)2,
where w~′≡∂w~/∂z and so forth. Since ε~z2≪1 and (w~′)2≃0, we have
(23)ε~z=w~′+12(u~′)2+12(v~′)2.In general,w~ differs from w that is allocated just on 𝒞. Nevertheless, w~ can be related with w and u by
(24)w~=w-ρ∂u∂z
and accordingly w~′=w′-ρu′′. Similarly, we use the approximations of u~≃u and v~≃v+ρβ, which imply that u~′≃u′, v~′≃v′. Consequently we have ε~z=εz-ρκz with the definitions of
(25)εz=w′+12(u′)2+12(v′)2,κz=u′′,
which leads us to the axial energy expression
(26)Uaxi=Ehr2(1-ν2)∫0L‍dz∫02π‍dθ(εz2+h212κz2).
It should be emphasized that the expression (26) implies the functional property of Uaxi represented by
(27)Uaxi=Uaxi[∂u∂z,∂2u∂z2,∂v∂z,∂w∂z].


## 3.4. Interaction Energy

Mechanical stability of cross-sections in multilayered cylinders owes in part to interaction between adjacent cylindrical surfaces. In typical nanotubular materials, for instance, the interaction originates from intermolecular van der Waals forces, thus sufficiently small compared with strong chemical bonding [1]; as a result, many nanotubular materials are endowed with anisotropy in mechanical stiffness, characterized by large stiffness against longitudinal stresses and flexibility to radial compression.For a linear elastic deformation, it is reasonable to represent the interaction energy betweenith and jth concentric cylinders by
(28)Uint⁡(i,j)=kri2∫0L‍dz∫02π‍dθ(ui-uj)2.
Here, k is an effective spring constant per surface area rdθdz, which serves as a measure for the interaction strength. The value of k is dependent on the material and geometric structure of the system considered; among choices, we hereafter pay attention to the weak interaction for which krN/E≤1 or lesser, by referring to the existing multiwalled nanotubes [24].


## 4. Displacement Condition

To evaluate the optimal value ofζi for each ith cylinder, we need to relate ζi with ui, vi, and wi, through which the total strain energy U is set to be a function of ζi. In his pioneering work, Brazier hypothesized that a typical cross-section deforms according to [21]
(29)ui=riζinextcos⁡2θ,
where the parameter ζinext is a dimensionless measure of the ovalization. The suffix “inext" means that the cross-section is assumed to deform inextensionally in the θ direction, namely, when εθ in (18) is set to zero. Under this special restriction, one obtains exactly the relation [25]
(30)∂vi∂θ+ui=0
or equivalently
(31)vi=-ri2ζinextsin2θ,
where the constant of integration was set to zero since there is no net rotation of the section about the cylinder axis.The relation (30) is exact when the circumferential elongation/contraction of volume elements is forbidden εθ≡0. Even though such the circumferential deformation is permitted, the relation (30) nearly holds in a realm of linear elastic deformation. This is why we use the following expressions for displacement of volume elements contained in the ith cylinder:
(32)ui=riζicos⁡2θ,vi=-ri2ζisin2θ,(33)wi=Γ(z-L2)[(ri+ui)sinθ+vicos⁡θ],
where ζi is the oblateness whose value depends on i. The validity of the above expressions will be revisited in Appendix.


## 5. Optimal Ovalization

Discussions made in the preceding two sections allow us to formulate the total mechanical energyU as a function of the series of {ζi}:
(34)U=∑i=1N‍Ucir(i)+∑i=1N‍Uaxi(i)+∑i,j(≠i)N‍Uint⁡(i,j)=[ζi,spolynomialofthe4thdegree].
The optimal ovalization is determined by
(35)∂U∂ζi=0.(i=1,…,N).
To calculate (35), we can omit ζi-terms of higher order than the 2th degree since ζi<1 for all i. The resulting equation with respect to ζi can be reduced to the matrix representation of
(36)Mζ=γ.
Here, ζ and γ are column vectors defined by
(37)ζ=[ζ1ζ2⋮ζN-1ζN],γ=34Γ2Eh[r13r23⋮rN-13rN3],
and M is an N×N tridiagonal matrix
(38)M=[a1b10⋯⋯0b1a2b2⋮0b2a3b3⋮⋱bi-1aibi⋱⋮⋮bN-2aN-1bN-10⋯⋯0bN-1aN]
with the definitions of
(39)a1=ω1+λ1+kr12r2,ai=ωi+λi+kri2ri+1+kri3,(2≤i≤N-1)aN=ωN+λN+krN3,bi=-kriri+12,(1≤i≤N-1)ωi=3Eh34ri,λi=58Γ2Ehri3.
As a consequence, the optimal values of ζi are determined by
(40)ζ=M-1γ,
from which we derive the dependence of the optimal ζi on the curvature Γ as formally expressed by ζi=ϕ(Γ).


## 6.  Numerical Results

The strategy for evaluating the optimalζi is summarized as follows.(i)
Prepare the energy expressions ofUcir, Uaxi, and Uint⁡ given by (18), (26), and (28), respectively.(ii)
Substitute the displacement expressions ofui, vi, and wi, given by (32)-(33), into the energy expressions.(iii)
Through (i) and (ii), the total energyU has been reduced to the ζi’s polynomial of the 4th order.(iv)
Omit the 3rd and 4th order terms in theζi’s polynomial and make the partial derivatives of ∂U/∂ζi=0 to obtain the matrix equation Mζ=γ (see (36)).(v)
Solve (40), ζ=M-1γ, to reach the optimal oblateness {ζi}.In actual computations, we consider that all theN-layered cylinders with different N possess the same value of rN; namely, the radius of the outmost cylinder is fixed to be a common constant rN irrespective of N. Taking the common rN as a unit of length, we set the layer’s thickness to be h/rN=0.02. The remaining parameters to be determined are N, Γ, and k; those values are artificially tuned in order to explore the effect of their variations on the optimal cross-section.Figure5 displays the stable cross-sections under the bending curvature indicated. The curvature is normalized by multiplying the outmost cylinder’s radius rN; thus, ΓrN is dimensionless. For every three samples, we fixed N=20 and interlayer coupling strength of krN/E=0.1. It follows from the views that the ovalization gradually evolves with increasing Γ as expected. Particularly at ΓrN=0.03, the outmost layer reaches flattened shape, while the innermost layer nearly collapses into a peanut-like shape. The two bulges indicated by dashed squares imply that large excess strain is stored in the inner layers around the bulgy regions, which prevent from significant deformation of the outer layers by pushing them outward. We thus can say that the physical origin of the Brazier effect suppression in multilayered systems is the present of additional layers embedded into the hollow core of the surrounding layers.Figure 5
Variation in the cross-sectional view with increasing the curvatureΓ of the cylinder’s axis. The parameters of N=20 and krN/E=0.1 are fixed.We have found in computations that the degree of suppression monotonically increases with the number of constituent layersN. Figure 6 exhibits the dependence of ζN on N and Γ, which tell us that the larger N we apply, the smaller ζN we obtain. It is also observed that for N≥40, the suppression no longer works effectively and the optimal ζN converges to a constant.(a) OblatenessζN of the outmost cylinder in an N-layered system as a function of the curvature ΓrN. The curvature is normalized by the radius of the outmost cylinder rN. (b) Suppression in the oblateness ζN with increasing the number of constituent layers N.
(a)
(b)Added toN, the interlayer coupling quantified by k is another important factor for determining the stable cross-section. Figure 7(a) gives examples on how the variation in k contributes to the suppression of the Brazier effect. The horizontal axis indicates the interlayer spring constant normalized by multiplying rN/E, by which it is nondimensional. The sample size is fixed to be N=50, and the change in ζN with increasing k is plotted. The results show that the degree of suppression is magnified with intensifying the couplings up to krN/E~0.1, above which an increase in k gives minor contribution. The pronounced suppression at the small k region is observed universally as far as the sample is composed of modest number of layers, say N≥20. This fact is demonstrated graphically in Figure 7(b), which shows the difference, designated by δN, in the short radius rNζN of deformed cross-sections between the weak and strong coupling conditions defined as follows:
(41)δN≡(rNζN)krN/E=0.001-(rNζN)krN/E=0.1.
The plot makes clear that the cross-section endowed with strong interlayer couplings tends to retain its original circular shape when N≥20. Figure 8 displays, as examples, the robust behaviors of samples with N=30 and 50, together with collapse of those with N=1 and 10. Inward convex bulges observed in the latter two samples disappear in the former two, which is a manifestation of the coupling effect in modest-N-layered systems.(a) Oblateness suppression caused by interlayer coupling in a 50-layered system. The coupling strength is quantified by the normalized spring constantkr50/E with E being Young’s modulus. (b) Difference in the short radius rNζN of deformed cross-sections between the weak and strong coupling conditions: δN≡(rNζN)krN/E=0.001-(rNζN)krN/E=0.1.
(a)
(b)Figure 8
N-dependence of the cross-section. ΓrN=0.02 and krN/E=0.1 are fixed for all images.


## 7. Conclusion

We have proposed an approximation theory that describes the bending-induced ovalization in the circular cross-section of multilayered cylinders. Taking into account the in-plane deformation within individual layers and interlayer couplings, we have formulated the mechanical energy of multilayered cylinders, followed by variational computation to evaluate the optimal oblateness that minimizes the energy under pure bending. The oblateness was found to decrease monotonically with increasing the number of layers and/or strength in the interlayer coupling, as we demonstrated in a systematic and quantitative manner. The results will facilitate the basic understanding of the mechanics of coaxial multicylinder structures that are present universally in macroscopic and nanoscopic science.



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*Source: 923896-2014-04-28.xml*