Suppression of Brazier Effect in Multilayered Cylinders

1 Department of Environmental Sciences & Interdisciplinary Graduate School of Medicine and Engineering, University of Yamanashi, 4-4-37 Takeda, Kofu, Yamanashi 400-8510, Japan 2Division of Engineering and Policy for Sustainable Environment, Faculty of Engineering, Hokkaido University, Kita-13, Nishi-8, Sapporo, Hokkaido 060-8628, Japan 3Department of Urban and Environment Engineering, Incheon National University, 12-1 Songdo-dong, Yeonsu-gu, Incheon 406-772, Republic of Korea


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], WS 2 [8,9], and TiO 2 [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][14][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 crosssectional deformation.Compared with a single hollow cylinder, multilayered counterparts afford resistance to crosssectional 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.

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.
Figure 1 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.
The degree of ovalization is quantified by the parameter , called the oblateness.Given an initially circular cross-section of radius , the product  equals to the minor axis length of the oval obtained after deformation.The definition of  is conventionally used even for extremely deformed sections, which go beyond simple ovalization to flattening or collapse in a peanut-like shape in which  can be much less than unity.

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 cost  for a cylinder to deform within linear elastic region is written by a sum of three energy terms: Here,  cir is the strain energy associated with the circumferential displacement,  axi is that with the axial strain, and  int 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 , V, and  as shown in Figure 2. Amplitudes of the vectors , V, and  indicate the displacements of a volume element in the radial, circumferential, and axial direction, respectively.Our objective is a series of the optimal oblateness parameters {  } = ( = 1, . . ., ) for each th layer contained in an -layered cylinder; it is what minimizes the total energy  of the system under a given bending curvature Γ.To this aim, we first derive explicit forms of  cir ,  axi , and  int for a monolayered cylinder as functions of , V, and  (see from Section 3.1 to Section 3.4), followed by applying the variation method to  with respect to , V, and .The strategy for a monolayered system can be extended straightforwardly to multilayered ones, as will be demonstrated in Section 5.

Stress-Strain Relation.
From the definition of elasticity, every elastic medium satisfies an essential relation between applied strain (  ) and the resulting stress (  ) [ = , , ], which is written in terms of cylindrical coordinates {, , } by with  being Young's modulus and ] being Poisson's ratio of the material under discussion.As   ≡ 0 for a hollow cylinder, it follows from (3)-( 4) that Suppose that an infinitesimal volume element  =  of a cylinder is displaced slightly in the -direction ( =  or ).The strain energy   caused by the deformation reads We thus obtain the expressions where we renamed   by  cir and   by  axi .To proceed the argument, we need to derive explicit functional forms of   ,   in terms of the displacements , V, .

Circumferential Strain Energy.
We first consider  cir of a hollow cylinder subjected to cross-sectional deformation.Suppose a circumferential line element of length  l The centroidal circle C of the cross-section, having the radius  before deformation, is also depicted by a dotted curve.Through the displacement described by ũ and Ṽ, the line element is elongated slightly in the  direction and the normal to the centroidal curve rotates with an angle .
lying within the annular cross-section with thickness ℎ (see Figure 3).The tilde (̃) attached to ℓ  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 () and V() that are defined just on the undeformed centroidal circle C of the circular cross-section (see (16)).
The extensional strain of the circumferential line element, designated by ε , is defined by where  l = r and  l *  is the length of the line element after deformation (throughout the paper, the asterisk symbolizes the quantity after deformation).The coordinates x * , ỹ * of the element after deformation are given by which imply that Note that (8) can be rewritten as ε +1 =  l *  /(r).Squaring the both sides and then rearranging give Advances in Condensed Matter Physics For ε ≪ 1, the term ε2  can be omitted; we substitute ( 10) into (11) to obtain where ũ ≡ ũ/ and so forth.The last term in ( 12) is associated with the rotation of the line element due to deformation.The rotation angle  consists of two parts (see Figure 4): (i) a clockwise component ũ/(r) due to the spatial variation of ũ in the circumferential direction and (ii) a counterclockwise one Ṽ/r due to the circumferential displacement of the element.Combination of the two parts gives which has a positive value in the counterclockwise sense.Formula ( 12) is valid for arbitrary large rotation .Particularly when ε and  are both sufficiently small (but finite), we may neglect the second term in the right side in (12) (here we exclude the possibility that |ũ| or |Ṽ  | 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 where  and V denote the displacements of a point just on C and  is a radial coordinate measured from C. By substituting ( 14) into (12), we attain the strain-displacement relation such as with the definitions Here  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 () and V() of a point just on C. The results also allow us to write the deformation energy  cir as or, more concisely, which implies that  cir is a functional of , V, and their derivatives as

Axial Strain Energy.
Parallel discussions to that in Section 3.2 give us the explicit form of the deformation energy  axi caused by the axial strain.The strain ε of the axial line element ℓ  is defined by Here  l *  is the length of the line element after deformation.Using the displacement components ũ, Ṽ, and w, the square of  l *  is given by Note the equivalence of  l with  in (21); then, we obtain from ( 20)-( 21) that where w ≡ w/ and so forth.Since ε2  ≪ 1 and (w  ) 2 ≃ 0, we have Advances in Condensed Matter Physics 5 In general, w differs from  that is allocated just on C. Nevertheless, w can be related with  and  by w =  −    (24) and accordingly w =   −   .Similarly, we use the approximations of ũ ≃  and Ṽ ≃ V + , which imply that ũ ≃   , Ṽ ≃ V  .Consequently we have ε =   −   with the definitions of which leads us to the axial energy expression It should be emphasized that the expression (26) implies the functional property of  axi represented by 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 between th and th concentric cylinders by Here,  is an effective spring constant per surface area , which serves as a measure for the interaction strength.The value of  is dependent on the material and geometric structure of the system considered; among choices, we hereafter pay attention to the weak interaction for which   / ≤ 1 or lesser, by referring to the existing multiwalled nanotubes [24].

Displacement Condition
To evaluate the optimal value of   for each th cylinder, we need to relate   with   , V  , and   , through which the total strain energy  is set to be a function of   .In his pioneering work, Brazier hypothesized that a typical crosssection deforms according to [21] where the parameter  inext is a dimensionless measure of the ovalization.The suffix "inext" means that the cross-section is assumed to deform inextensionally in the  direction, namely, when   in ( 18) is set to zero.Under this special restriction, one obtains exactly the relation [25] V   +   = 0 (30) or equivalently 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   ≡ 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 th cylinder: where   is the oblateness whose value depends on .The validity of the above expressions will be revisited in Appendix.

Optimal Ovalization
Discussions made in the preceding two sections allow us to formulate the total mechanical energy  as a function of the series of {  }: The optimal ovalization is determined by To calculate (35), we can omit   -terms of higher order than the 2th degree since   < 1 for all .The resulting equation with respect to   can be reduced to the matrix representation of Here,  and  are column vectors defined by Advances in Condensed Matter Physics and M is an  ×  tridiagonal matrix with the definitions of As a consequence, the optimal values of   are determined by from which we derive the dependence of the optimal   on the curvature Γ as formally expressed by   = (Γ).

Numerical Results
The strategy for evaluating the optimal   is summarized as follows.
(iii) Through (i) and (ii), the total energy  has been reduced to the   's polynomial of the 4th order.
(iv) Omit the 3rd and 4th order terms in the   's polynomial and make the partial derivatives of /  = 0 to obtain the matrix equation M =  (see (36)).
In actual computations, we consider that all the -layered cylinders with different  possess the same value of   ; namely, the radius of the outmost cylinder is fixed to be a common constant   irrespective of .Taking the common   as a unit of length, we set the layer's thickness to be ℎ/  = 0.02.The remaining parameters to be determined are , Γ, and ; those values are artificially tuned in order to explore the effect of their variations on the optimal cross-section.Figure 5 displays the stable cross-sections under the bending curvature indicated.The curvature is normalized by multiplying the outmost cylinder's radius   ; thus, Γ  is dimensionless.For every three samples, we fixed  = 20 and interlayer coupling strength of   / = 0.1.It follows from the views that the ovalization gradually evolves with increasing Γ as expected.Particularly at Γ  = 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.
We have found in computations that the degree of suppression monotonically increases with the number of constituent layers . Figure 6 exhibits the dependence of   on  and Γ, which tell us that the larger  we apply, the smaller   we obtain.It is also observed that for  ≥ 40, the suppression no longer works effectively and the optimal   converges to a constant.
Added to , the interlayer coupling quantified by  is another important factor for determining the stable crosssection.Figure 7(a) gives examples on how the variation in  contributes to the suppression of the Brazier effect.The horizontal axis indicates the interlayer spring constant normalized by multiplying   /, by which it is nondimensional.The sample size is fixed to be  = 50, and the change in   with increasing  is plotted.The results show that the degree of suppression is magnified with intensifying the couplings up to   / ∼ 0.1, above which an increase in  gives minor contribution.The pronounced suppression at the small  region is observed universally as far as the sample is composed of modest number of layers, say  ≥ 20.This fact is demonstrated graphically in Figure 7(b), which shows the difference, designated by   , in the short radius     of deformed cross-sections between the weak and strong coupling conditions defined as follows: The plot makes clear that the cross-section endowed with strong interlayer couplings tends to retain its original circular shape when  ≥ 20. Figure 8 displays, as examples, the robust behaviors of samples with  = 30 and 50, together with collapse of those with  = 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--layered systems.

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.

Appendix
On the Validity of (32) In this Appendix, we examine the validity of our approximation as restated by   =     cos 2, V  = −   2   sin 2, (A.1) which describe the cross-sectional deformation.We should be reminded that the above expressions of   and V  are natural extensions of the following: which are exactly correct under the ideal condition that the circumferential strain (i.e.,   in ( 18)) must be zero.Therefore, it is nontrivial if we are allowed to make the straightforward extension to the case, where   ̸ = 0, just by replacing the coefficient from  inext to   with keeping the function forms of   and V  .We will demonstrate below that the extension should be justified under the numerical conditions we have applied.
For a single-layered cylinder (i.e.,  = 1), the fourthorder polynomial of  given by ( 35 where  ≡  1 and  ≡  1 .The spring constant  is that associated with springs illustrated in the left panel of Figure 9.

2 AdvancesFigure 1 :
Figure 1: (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 parameter .

Figure 2 :
Figure2: Diagram of the displacements , V,  of a volume element in a hollow cylindrical tube with radius .The tube axis is parallel to the  axis, which is vertical to the tube's cross-section with thickness ℎ that lies in the - plane.

Figure 3 :
Figure 3: Displacement of a circumferential line element of the length  l to that of  l * .The centroidal circle C of the cross-section, having the radius  before deformation, is also depicted by a dotted curve.Through the displacement described by ũ and Ṽ, the line element is elongated slightly in the  direction and the normal to the centroidal curve rotates with an angle .

Figure 4 :
Figure 4: Two different mechanisms of the rotation of a line element.Left: clockwise rotation due to the spatial variation of ũ in the  direction.Right: counterclockwise rotation caused by the circumferential displacement of the element.

Figure 5 :
Figure 5: Variation in the cross-sectional view with increasing the curvature Γ of the cylinder's axis.The parameters of  = 20 and   / = 0.1 are fixed.

Figure 6 :
Figure 6: (a) Oblateness   of the outmost cylinder in an -layered system as a function of the curvature Γ  .The curvature is normalized by the radius of the outmost cylinder   .(b) Suppression in the oblateness   with increasing the number of constituent layers .

Figure 9 :
Figure 9: Left: sketch of the cross-section in a monolayered cylinder assumed.The effect of the coupling parameter  is illustrated by virtual springs inside.Right: cubic curve of the -polynomial represented by /.

Figure 10 :
Figure 10: Data comparison between the approximated oblateness  and the exact one   for different values of .