Research Article Variational Principles for Buckling of ...Variational formulations were employed in a number of studies involving microtubules. In particular, small scale formulations

Research ArticleVariational Principles for Buckling of MicrotubulesModeled as Nonlocal Orthotropic Shells

Sarp Adali

Discipline of Mechanical Engineering, University of KwaZulu-Natal, Durban 4041, South Africa

Correspondence should be addressed to Sarp Adali; [email protected]

Received 15 April 2014; Revised 19 June 2014; Accepted 20 June 2014; Published 5 August 2014

Academic Editor: Chung-Min Liao

Copyright © 2014 Sarp Adali. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A variational principle for microtubules subject to a buckling load is derived by semi-inverse method.The microtubule is modeledas an orthotropic shell with the constitutive equations based on nonlocal elastic theory and the effect of filament network taken intoaccount as an elastic surrounding. Microtubules can carry large compressive forces by virtue of the mechanical coupling betweenthe microtubules and the surrounding elastic filament network.The equations governing the buckling of the microtubule are givenby a system of three partial differential equations.The problem studied in the present work involves the derivation of the variationalformulation for microtubule buckling. The Rayleigh quotient for the buckling load as well as the natural and geometric boundaryconditions of the problem is obtained from this variational formulation. It is observed that the boundary conditions are coupledas a result of nonlocal formulation. It is noted that the analytic solution of the buckling problem for microtubules is usually adifficult task. The variational formulation of the problem provides the basis for a number of approximate and numerical methodsof solutions and furthermore variational principles can provide physical insight into the problem.

1. Introduction

Understanding the buckling characteristics of microtubulesis of practical and theoretical importance since they performa number of essential functions in living cells as discussedin [1–4]. In particular, they are the stiffest components ofcytoskeleton and are instrumental in maintaining the shapeof cells [1, 5].This is basically due to the fact thatmicrotubulesare able to support relatively large compressive loads as aresult of coupling to the surrounding matrix. Since thisfunction is of importance for cellmechanics and transmissionof forces, the study of the buckling behavior of microtubulesprovides useful information on their biological functions.Consequently the buckling of microtubules has been studiedemploying increasinglymore complicated continuummodelswhich are often used to simulate their mechanical behaviorand provide an effective tool to determine their load carryingcapacity under compressive loads. The present study facili-tates this investigation of the microtubule buckling problemby providing a variational setting which is the basis of anumber of numerical and approximate solution methods.

Buckling of microtubules occurs for a number of rea-sons such as cell contraction or constrained microtubulepolymerization at the cell periphery. To better understandthis phenomenon, an effective approach is to use continuummodels to represent a microtubule. These models includeEuler-Bernoulli beambyCivalek andDemir [6], Timoshenkobeam by Shi et al. [7], and cylindrical shells by Wang etal. [8] and Gu et al. [9]. The present study provides avariational formulation of the buckling ofmicrotubules usingan orthotropic shell model to represent their mechanicalbehavior. Variational principles form the basis of a numberof computational and approximate methods of solutionsuch as finite elements, Rayleigh-Ritz and Kantorovich. Inparticular Rayleigh quotient provides a useful expression toapproximate the buckling load directly. As such the resultspresented can be used to obtain the approximate solutionsfor the buckling of microtubules as well as the variationallycorrect boundary conditions which are derived using thevariational formulation of the problem.

Continuummodeling approach has been used effectivelyin other branches of biology and medicine [10], and their

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2014, Article ID 591532, 9 pageshttp://dx.doi.org/10.1155/2014/591532

2 Computational and Mathematical Methods in Medicine

accuracy can be improved by implementing nonlocal con-stitutive relations for micro- and nanoscale phenomenoninstead of classical local ones which relate the stress at agiven point to the strain at the same point. As such localtheories are of limited accuracy at the micro- and nanoscalesince they neglect the small scale effects which can besubstantial due to the atomic scale of the phenomenon.Recent examples of microtubule models based on the localelastic theory include [11–18] where Euler-Bernoulli andhigher order shear deformable beams and cylindrical shellsrepresented the microtubules. A review of the mechanicalmodeling of microtubules was given by Hawkins et al. [19]and a perspective on cell biomechanics by Ji and Bao [20].

In the present study the formulation is based on thenonlocal theory which accounts for the small scale effects andimproves the accuracy.The nonlocal theory was developed inthe early seventies by Eringen [21, 22] and recently appliedto micro- and nanoscale structures. Nonlocal continuummodels have been used in a number of studies to investigatethe bending and vibration behavior of microtubules usingnonlocal Euler-Bernoulli [6, 23] andTimoshenko beams [24].There have been few studies on the buckling of microtubulesbased on a nonlocal theory. Nonlocal Timoshenko beammodel was employed in [25–27] and nonlocal shell modelin [1]. In a series of studies, Shen [28–30] used nonlocalshear deformable shell theory to study the buckling andpostbuckling behavior of microtubules. Nonlocal problemsalso arise in other subject areas and have been studiedusing fractional calculus in a number studies [31, 32]. Themodels based on beam or isotropic shell theories neglect thedirectional dependence of the microtubule properties. Theaccuracy of a continuum model can be improved further byemploying an orthotropic shell theory to take this directionaldependence into account as discussed in [12, 33–35].

The objective of the present study is to derive a variationalprinciple and Rayleigh quotient for the buckling load as wellas the applicable natural and geometric boundary conditionsfor a microtubule subject to a compressive load. The partic-ular model used in the study is a nonlocal orthotropic shellunder a compressive load with the effect of filament networktaken into account as an elastic surrounding. Moreover thepressure force on the microtubule exerted by the viscouscytosol is calculated using the Stokes flow theory [1]. Theinclusion of these effects in the governing equations isimportant to model the phenomenon accurately since it isknown that the microtubules can carry large compressiveforces by virtue of the mechanical coupling between themicrotubules and the surrounding elastic filament networkas observed by Brangwynne et al. [36] and Das et al. [37].Moreover, the microtubules are surrounded by the viscouscytosol in addition to the soft elastic filament network and thebuckling causes the viscous flowof the cytosol [38].These twoprocesses result in an external stress field which improves thebuckling characteristics of microtubules.

Variational formulations were employed in a numberof studies involving microtubules. In particular, small scaleformulations for the linear vibrations of microtubules werederived using the energy expression in [39, 40] and fornonlinear vibrations in [41]. Previous studies on variational

principles involving nanoscale structures includemultiwalledcarbon nanotubes. In particular, variational principles werederived for nanotubes under buckling loads [42], for nan-otubes undergoing linear vibrations [43], and for nanotubesundergoing nonlinear vibrations [44] using the nonlocalEuler-Bernoulli beam theories. Variational principles werealso derived for nanotubes undergoing transverse vibrationsusing a nonlocal Timoshenko beam model in [45] and astrain-gradient cylindrical shell model in [46]. Apart fromproviding an insight into a physical problem, the variationalformulations are often employed in the approximate andnumerical solutions of the problems, in particular, in thepresence of complicated boundary conditions [47].Moreovernatural boundary conditions can be easily derived from thevariational formulation of the problem. In the present studyvariational formulation of a problem is derived by the semi-inverse method developed by He [48–51] which was appliedto several problems of mathematical physics to obtain thevariational formulations for problems formulated in termsof differential equations [52–55]. Recently the semi-inversemethod was applied to the heat conduction equation in [56]to obtain a constrained variational principle.The equivalenceof this formulation to the one obtained by He and Lee [57]has been shown in [58, 59]. A recent application of thesemi-inverse method involves the derivation of variationalprinciples for partial differential equations modeling watertransport in porous media [60].

2. Equations Based on Nonlocal Elastic Theory

In the present study the microtubule is modeled as anorthotropic cylindrical shell of length 𝐿, radius 𝑅, andwall thickness ℎ and surrounded by a viscoelastic medium(cytoplasm). It is subject to an axial compressive load 𝑁 asshown in Figure 1.

The microtubule has the Young’s moduli 𝐸1and 𝐸

2along

the axial and circumferential directions, shearmodulus𝐺 andPoisson’s ratios 𝜇

1and 𝜇

2along the circumferential and axial

directions. Dimensionless longitudinal direction is denotedby 𝑥 = 𝑋/𝑅 and the circumferential direction by 𝜃 as inFigure 1. For an orthotropic shell, the constitutive relationsbased on nonlocal elasticity theory are given by (see [1])

𝜎𝑥−(𝑒𝑎0)2

𝑅2∇2𝜎𝑥=

𝐸1

1 − 𝜇1𝜇2

𝜀𝑥+

𝐸2𝜇2

1 − 𝜇1𝜇2

𝜀𝜃,

𝜎𝜃−(𝑒𝑎0)2

𝑅2∇2𝜎𝜃=

𝐸2

1 − 𝜇1𝜇2

𝜀𝜃+

𝐸1𝜇1

1 − 𝜇1𝜇2

𝜀𝑥,

𝜏𝑥𝜃−(𝑒𝑎0)2

𝑅2∇2𝜏𝑥𝜃= 𝐺𝜀𝑥𝜃,

(1)

where ∇2 = (𝜕2/𝜕𝑥2) + (𝜕2/𝜕𝜃2) is the Laplace operator,𝜎𝑥, 𝜎𝜃, and 𝜏

𝑥𝜃are stress components, and 𝜀

𝑥, 𝜀𝜃, and 𝜀

𝑥𝜃

are the normal and shear strains. In (1), 𝑒𝑎0is the small

scale parameter reflecting the nanoscale of the phenomenonand has to be experimentally determined. The differentialequations governing the buckling of the microtubules are

Computational and Mathematical Methods in Medicine 3

L

CytoplasmCytoplasm

NX

R

h

Cytoplasm

Microtubule

N

Microtubule

𝜃

� (x, 𝜃)

u(x, 𝜃)

w(x, 𝜃)

Figure 1: Microtubule under a compressive load and its surround-ing.

given in [1] based on the nonlocal constitutive relation (1).Thedifferential equation formulation of the problem is expressedas a system of partial differential equations given by

𝐷1(𝑢, V, 𝑤) ≡ 𝐿

1(𝑢) + 𝑀

1(V, 𝑤) = 0,

𝐷2(𝑢, V, 𝑤) ≡ 𝐿

2(V) + 𝑀

2(𝑢, 𝑤) = 0,

𝐷3(𝑢, V, 𝑤) ≡ 𝐿

3(𝑤) +𝑀

3(𝑢, V) = 0,

(2)

where 𝑢(𝑥, 𝜃), V(𝑥, 𝜃), and 𝑤(𝑥, 𝜃) are the displacement com-ponents in the axial, circumferential, and radial directions,respectively, as shown in Figure 1. The differential operators𝐿𝑖and𝑀

𝑖are defined as

𝐿1(𝑢) = 𝑘

2(1 + 𝑐

2) 𝑢𝜃𝜃+ 𝑢𝑥𝑥+𝑁

𝐾L (𝑢𝑥𝑥) , (3)

𝑀1(V, 𝑤) = (𝑘

2+ 𝜇1) V𝑥𝜃+ 𝜇1𝑤𝑥− 𝑐2𝑤𝑥𝑥𝑥

+ 𝑐2𝑘2𝑤𝑥𝜃𝜃, (4)

𝐿2(V) = 𝑘

2(1 + 3𝑐

2) V𝑥𝑥+ 𝑘1V𝜃𝜃+𝑁

𝐾L (V𝑥𝑥) , (5)

𝑀2(𝑢, 𝑤) = (𝑘

2+ 𝜇1) 𝑢𝑥𝜃+ 𝑘1𝑤𝜃− 𝑐2(3𝑘2+ 𝜇1) 𝑤𝑥𝑥𝜃, (6)

𝐿3(𝑤) = − (1 + 𝑐

2) 𝑘1𝑤 − 2𝑐

2𝑘1𝑤𝜃𝜃

− 𝑐2(𝑤𝑥𝑥𝑥𝑥

+ 𝑘1𝑤𝜃𝜃𝜃𝜃

+ (4𝑘2+ 2𝜇1) 𝑤𝑥𝑥𝜃𝜃

)

+𝜍𝑅

𝐾L (𝑤) +

𝑁

𝐾L (𝑤𝑥𝑥) +

𝑅

𝐾L (𝑃𝑌𝑌) ,

(7)

𝑀3(𝑢, V) = −𝜇

1𝑢𝑥+ 𝑐2(𝑢𝑥𝑥𝑥

− 𝑘2𝑢𝑥𝜃𝜃)

− 𝑘1V𝜃+ 𝑐2(3𝑘2+ 𝜇1) V𝑥𝑥𝜃,

(8)

where the differential operator L(⋅) is defined as L(⋅) =𝜂2∇2−1, the subscripts𝑥, 𝜃denote differentiationwith respect

to that variable, and the dimensionless small scale parameter𝜂 is given by 𝜂 = 𝑒𝑎

0/𝑅. The symbols in (3)–(8) are defined as

𝑘1=𝐸1

𝐸2

,

𝑘2=𝐺 (1 − 𝜇

1𝜇2)

𝐸1

,

𝑐2=

ℎ3

0

12𝑅2ℎ,

𝐾 =𝐸1ℎ

(1 − 𝜇1𝜇2),

(9)

where ℎ0is the effective thickness for bending. The symbol

𝜍 appearing in (7) is the elastic constraint from the filamentsnetwork and is given by 𝜍 = 2.7 𝐸

𝑐where 𝐸

𝑐is the elastic

modulus of the surrounding viscoelastic medium.The radialpressure 𝑃

𝑌𝑌exerted by the motion of cytosol in (7) is

computed from the dynamic equations of cytosol given in [1].

3. Variational Formulation

Following the semi-inverse method, we construct a varia-tional trial-functional 𝑉(𝑢, V, 𝑤) as follows:

𝑉 (𝑢, V, 𝑤) = 𝑉1(𝑢) + 𝑉

2(V) + 𝑉

3(𝑤)

+ ∫

2𝜋

0

∫

𝑙

0

𝐹 (𝑢, V, 𝑤) 𝑑𝑥 𝑑𝜃,(10)

where 𝑙 = 𝐿/𝑅, 𝑥 ∈ [0, 𝐿/𝑅] and the functionals 𝑉1(𝑢), 𝑉

2(V),

𝑉3(𝑤) are given by

𝑉1(𝑢) =

1

2∫

2𝜋

0

∫

𝑙

0

[ − 𝑘2(1 + 𝑐

2) 𝑢2

𝜃− 𝑢2

𝑥

+𝑁

𝐾(𝜂2(𝑢2

𝑥𝑥+ 𝑢2

𝑥𝜃) + 𝑢2

𝑥)] 𝑑𝑥 𝑑𝜃,

𝑉2(V) =

1

2∫

2𝜋

0

∫

𝑙

0

[ − 𝑘2(1 + 3𝑐

2) V2𝑥− 𝑘1V2𝜃

+𝑁

𝐾(𝜂2(V2𝑥𝑥+ V2𝑥𝜃) + V2𝑥)] 𝑑𝑥 𝑑𝜃,


𝑉3(𝑤)

=1

2∫

2𝜋

0

∫

𝑙

0

[− (1 + 𝑐2) 𝑘1𝑤2+ 2𝑐2𝑘1𝑤2

𝜃

− 𝑐2(𝑤2

𝑥𝑥+ 𝑘1𝑤2

𝜃𝜃)

− (4𝑘2+ 2𝜇1) 𝑤2

𝑥𝜃] 𝑑𝑥 𝑑𝜃 ⋅ ⋅ ⋅

+1

2∫

2𝜋

0

∫

𝑙

0

[−𝜍𝑅

𝐾(𝜂2(𝑤2

𝑥+ 𝑤2

𝜃) + 𝑤2) +

𝑁

𝐾

× (𝜂2(𝑤2

𝑥𝑥+ 𝑤2

𝑥𝜃) + 𝑤2

𝑥)

+2𝑅

𝐾L (𝑃𝑌𝑌) 𝑤] 𝑑𝑥 𝑑𝜃.

(11)

In (10), 𝐹(𝑢, V, 𝑤) is an unknown function to be determinedsuch that the Euler-Lagrange equations of the variationalfunctional (10) correspond to the differential equation (2).This establishes the direct relation between the variationalformulation and the governing equations in the sense thatdifferential equation (2) can be obtained from the derivedvariational principle using the Euler-Lagrange equations. Itis noted that the choice of the trial functionals defined by(10)-(11) is not unique. The review article by He [61] providesa systematic treatment on the use of semi-inverse methodfor the derivation of variational principles and the selectionof trial functionals as well as on variational methods for thesolution of linear and nonlinear problems.

It is noted that the Euler-Lagrange equations of thevariational functional 𝑉(𝑢, V, 𝑤) are

𝐿1(𝑢) +

𝛿𝐹

𝛿𝑢= 0,

𝐿2(V) +

𝛿𝐹

𝛿V= 0,

𝐿3(𝑤) +

𝛿𝐹

𝛿𝑤= 0.

(12)

Thus the functionals 𝑉1(𝑢), 𝑉

2(V), and 𝑉

3(𝑤) are the varia-

tional functionals for the differential operators 𝐿1(𝑢), 𝐿

2(V),

and 𝐿3(𝑤), respectively. In (12), the variational derivative

𝛿𝐹/𝛿𝑢 is defined as

𝛿𝐹

𝛿𝑢=𝜕𝐹

𝜕𝑢−𝜕

𝜕𝑥(𝜕𝐹

𝜕𝑢𝑥

) −𝜕

𝜕𝜃(𝜕𝐹

𝜕𝑢𝜃

) +𝜕2

𝜕𝑥2(𝜕𝐹

𝜕𝑢𝑥𝑥

)

+𝜕2

𝜕𝑥𝜕𝜃(𝜕𝐹

𝜕𝑢𝑥𝜃

) +𝜕2

𝜕𝜃2(𝜕𝐹

𝜕𝑢𝜃𝜃

)

−𝜕3

𝜕𝑥3(

𝜕𝐹

𝜕𝑢𝑥𝑥𝑥

) ⋅ ⋅ ⋅ .

(13)

Comparing (12) with (2), we observe that the followingequations have to be satisfied for Euler-Lagrange equationsof𝑉(𝑢, V, 𝑤) to represent the governing equation (2), namely,

𝛿𝐹

𝛿𝑢= 𝑀1(V, 𝑤)

= (𝑘2+ 𝜇1) V𝑥𝜃+ 𝜇1𝑤𝑥

− 𝑐2𝑤𝑥𝑥𝑥

+ 𝑐2𝑘2𝑤𝑥𝜃𝜃,

(14)

𝛿𝐹

𝛿V= 𝑀2(𝑢, 𝑤) = (𝑘

2+ 𝜇1) 𝑢𝑥𝜃+ 𝑘1𝑤𝜃

− 𝑐2(3𝑘2+ 𝜇1) 𝑤𝑥𝑥𝜃,

(15)

𝛿𝐹

𝛿𝑤= 𝑀3(𝑢, V)

= −𝜇1𝑢𝑥+ 𝑐2(𝑢𝑥𝑥𝑥

− 𝑘2𝑢𝑥𝜃𝜃)

− 𝑘1V𝜃+ 𝑐2(3𝑘2+ 𝜇1) V𝑥𝑥𝜃.

(16)

The function 𝐹(𝑢, V, 𝑤) has to be determined such that(14)–(16) are satisfied. For this purpose we first determine𝐹(𝑢, V, 𝑤) satisfying (14) to obtain

𝐹 (𝑢, V, 𝑤) = − (𝑘2+ 𝜇1) V𝑥𝑢𝜃+ 𝜇1𝑤𝑥𝑢 + 𝑐2𝑤𝑥𝑥𝑢𝑥

+ 𝑐2𝑘2𝑤𝑥𝑢𝜃𝜃+ Φ (V, 𝑤) ,

(17)

where Φ(V, 𝑤) is an unknown function of V and 𝑤. Next wecompute 𝛿𝐹/𝛿V from (17), namely,

𝛿𝐹

𝛿V= (𝑘2+ 𝜇1) 𝑢𝑥𝜃+𝛿Φ (V, 𝑤)

𝛿V(18)

which should satisfy (15). From (15) and (18) it follows that𝛿Φ (V, 𝑤)

𝛿V= 𝑘1𝑤𝜃− 𝑐2(3𝑘2+ 𝜇1) 𝑤𝑥𝑥𝜃. (19)

The expression for Φ(V, 𝑤) satisfying (19) is determined as

Φ (V, 𝑤) = −𝑘1𝑤V𝜃+ 𝑐2(3𝑘2+ 𝜇1) 𝑤𝑥𝑥V𝜃. (20)

Thus from (17) and (20), 𝐹(𝑢, V, 𝑤) is obtained as

𝐹 (𝑢, V, 𝑤) = − (𝑘2+ 𝜇1) V𝑥𝑢𝜃+ 𝜇1𝑤𝑥𝑢 + 𝑐2𝑤𝑥𝑥𝑢𝑥

+ 𝑐2𝑘2𝑤𝑥𝑢𝜃𝜃− 𝑘1𝑤V𝜃

+ 𝑐2(3𝑘2+ 𝜇1) 𝑤𝑥𝑥V𝜃.

(21)

Finally we note that 𝜕𝐹/𝜕𝑤 satisfies (16). Thus the function𝐹(𝑢, V, 𝑤) given by (21) satisfies the Euler-Lagrange equations(14)–(16) as required. Now the variational functional can beexpressed as

𝑉 (𝑢, V, 𝑤) = 𝑉1(𝑢) + 𝑉

2(V) + 𝑉

3(𝑤) + 𝑉

4(𝑢, V, 𝑤) , (22)

where𝑉4(𝑢, V, 𝑤)

= ∫

2𝜋

0

∫

𝑙

0

𝐹 (𝑢, V, 𝑤) 𝑑𝑥 𝑑𝜃

= ∫

2𝜋

0

∫

𝑙

0

(− (𝑘2+ 𝜇1) V𝑥𝑢𝜃+ 𝜇1𝑤𝑥𝑢 + 𝑐2𝑤𝑥𝑥𝑢𝑥

+ 𝑐2𝑘2𝑤𝑥𝑢𝜃𝜃− 𝑘1𝑤V𝜃

+𝑐2(3𝑘2+ 𝜇1) 𝑤𝑥𝑥V𝜃) 𝑑𝑥 𝑑𝜃

(23)

and 𝑉1(𝑢), 𝑉

2(V), and 𝑉

3(𝑤) are given by (11). To verify

the validity of the variational formulation (22), one has toshow that the Euler-Lagrange equations of 𝑉(𝑢, V, 𝑤) yieldthe governing (2). The fact that this is indeed the case can beshown easily.


4. Rayleigh Quotient

The Raleigh quotient is obtained for the buckling load 𝑁 bynoting that

𝑉1(𝑢) = −𝑅

1(𝑢) + 𝑁𝑆 (𝑢) ,

𝑉2(V) = −𝑅

2(V) + 𝑁𝑆 (V) ,

𝑉3(𝑤) = −𝑅

3(𝑤) + 𝑁𝑆 (𝑤) ,

(24)

where

𝑅1(𝑢) =

1

2∫

2𝜋

0

∫

𝑙

0

(𝑘2(1 + 𝑐

2) 𝑢2

𝜃+ 𝑢2

𝑥) 𝑑𝑥 𝑑𝜃,

𝑅2(V) =

1

2∫

2𝜋

0

∫

𝑙

0

(𝑘2(1 + 3𝑐

2) V2𝑥+ 𝑘1V2𝜃) 𝑑𝑥 𝑑𝜃,

𝑅3(𝑤)

=1

2∫

2𝜋

0

∫

𝑙

0

((1 + 𝑐2) 𝑘1𝑤2− 2𝑐2𝑘1𝑤2

𝜃

+ 𝑐2(𝑤2

𝑥𝑥+ 𝑘1𝑤2

𝜃𝜃)

+ (4𝑘2+ 2𝜇1) 𝑤2

𝑥𝜃) 𝑑𝑥 𝑑𝜃 ⋅ ⋅ ⋅

+1

2∫

2𝜋

0

∫

𝑙

0

(𝜍𝑅

𝐾(𝜂2(𝑤2

𝑥+ 𝑤2

𝜃) + 𝑤2)

−2𝑅

𝐾L (𝑃𝑌𝑌) 𝑤) 𝑑𝑥 𝑑𝜃,

𝑆 (𝑦) =1

2𝐾∫

2𝜋

0

∫

𝑙

0

(𝜂2(𝑦2

𝑥𝑥+ 𝑦2

𝑥𝜃) + 𝑦2

𝑥) 𝑑𝑥 𝑑𝜃.

(25)

Thus from (22) and (24), it follows that

𝑉 (𝑢, V, 𝑤) = − (𝑅1(𝑢) + 𝑅

2(V) + 𝑅

3(𝑢))

+ 𝑉4(𝑢, V, 𝑤)

+ 𝑁 (𝑆 (𝑢) + 𝑆 (V) + 𝑆 (𝑤))

(26)

and the Rayleigh quotient can be expressed as

𝑁 = min𝑢,V,𝑤

𝑅1(𝑢) + 𝑅

2(V) + 𝑅

3(𝑤) − 𝑉

4(𝑢, V, 𝑤)

𝑆 (𝑢) + 𝑆 (V) + 𝑆 (𝑤). (27)

5. Natural and GeometricBoundary Conditions

It is noted that the displacements are equal at the end points𝜃 = 0 and 𝜃 = 2𝜋; that is,

𝑢 (𝑥, 0) = 𝑢 (𝑥, 2𝜋) ,

V (𝑥, 0) = V (𝑥, 2𝜋) ,

𝑤 (𝑥, 0) = 𝑤 (𝑥, 2𝜋)

for 𝑥 ∈ [0, 𝑙] .

(28)

The first variations of 𝑉(𝑢, V, 𝑤) with respect to 𝛿𝑢, 𝛿V, and𝛿𝑤, denoted by 𝛿

𝑢𝑉, 𝛿V𝑉, and 𝛿𝑤𝑉, respectively, can be

obtained by integration by parts and using (28). We firstobtain the variations of 𝑉

1(𝑢) and 𝑉

4(𝑢, V, 𝑤) with respect to

𝛿𝑢 which are given by

𝛿𝑢𝑉1(𝑢)

= ∫

2𝜋

0

∫

𝑙

0

𝐿1(𝑢) 𝛿𝑢 𝑑𝑥 𝑑𝜃 + 𝐵

1(𝑢, 𝛿𝑢, 𝛿𝑢

𝑥) ,

𝛿𝑢𝑉4(𝑢, V, 𝑤)

= ∫

2𝜋

0

∫

𝑙

0

𝑀1(V, 𝑤) 𝛿𝑢 𝑑𝑥 𝑑𝜃 + 𝐵

2(𝑤, 𝛿𝑢) ,

(29)

where

𝐵1(𝑢, 𝛿𝑢, 𝛿𝑢

𝑥)

= ∫

2𝜋

0

[(−𝑢𝑥−𝑁

𝐾(𝜂2(𝑢𝑥𝑥𝑥

+ 𝑢𝑥𝜃𝜃) − 𝑢𝑥)) 𝛿𝑢

+𝑁

𝐾𝜂2𝑢𝑥𝑥𝛿𝑢𝑥]

𝑥=𝑙

𝑥=0

𝑑𝜃,

𝐵2(𝑤, 𝛿𝑢) = ∫

2𝜋

0

[𝑐2𝑤𝑥𝑥𝛿𝑢]𝑥=𝑙

𝑥=0𝑑𝜃.

(30)

Similarly

𝛿V𝑉2 (V) = ∫2𝜋

0

∫

𝑙

0

𝐿2(V) 𝛿V 𝑑𝑥 𝑑𝜃 + 𝐵

3(V, 𝛿V, 𝛿V

𝑥) ,

𝛿V𝑉4 (𝑢, V, 𝑤) = ∫2𝜋

0

∫

𝑙

0

𝑀2(𝑢, 𝑤) 𝛿V 𝑑𝑥 𝑑𝜃

+ 𝐵4(𝑢, 𝛿V) ,

(31)

where

𝐵3(V, 𝛿V, 𝛿V

𝑥)

= ∫

2𝜋

0

[(−𝑘2(1 + 3𝑐

2) V𝑥−𝑁

𝐾(𝜂2(V𝑥𝑥𝑥

+ V𝑥𝜃𝜃) − V𝑥)) 𝛿V

+𝑁

𝐾𝜂2V𝑥𝑥𝛿V𝑥]

𝑥=𝑙

𝑥=0

𝑑𝜃,

𝐵4(𝑢, 𝛿V) = ∫

2𝜋

0

[− (𝑘2+ 𝜇1) 𝑢𝜃]𝑥=𝑙

𝑥=0𝛿V 𝑑𝜃.

(32)

Finally we obtain 𝛿𝑤𝑉3(𝑤) and 𝛿

𝑤𝑉4(𝑢, V, 𝑤), namely,

𝛿𝑤𝑉3(𝑤)

= ∫

2𝜋

0

∫

𝑙

0

𝐿3(𝑤) 𝛿𝑤𝑑𝑥𝑑𝜃 + 𝐵

5(𝑤, 𝛿𝑤, 𝛿𝑤

𝑥)

+ 𝐵6(𝑤, 𝛿𝑤, 𝛿𝑤

𝑥) ,


𝛿𝑤𝑉4(𝑢, V, 𝑤)

= ∫

2𝜋

0

∫

𝑙

0

𝑀3(𝑢, V) 𝛿𝑤𝑑𝑥𝑑𝜃 + 𝐵

7(𝑢, V, 𝛿𝑤, 𝛿𝑤

𝑥)

+ 𝐵7(𝑢, V, 𝛿𝑤, 𝛿𝑤

𝑥) ,

(33)

where

𝐵5(𝑤, 𝛿𝑤, 𝛿𝑤

𝑥)

= ∫

2𝜋

0

[𝑐2(𝑤𝑥𝑥𝑥

+ (4𝑘2+ 2𝜇1) 𝑤𝑥𝜃𝜃) 𝛿𝑤

−𝑐2𝑤𝑥𝑥𝛿𝑤𝑥]𝑥=𝑙

𝑥=0𝑑𝜃,

𝐵6(𝑤, 𝛿𝑤, 𝛿𝑤

𝑥)

= ∫

2𝜋

0

[(−𝜍𝑅

𝐾𝜂2𝑤𝑥−𝑁

𝐾(𝜂2(𝑤𝑥𝑥𝑥

+ 𝑤𝑥𝜃𝜃) − 𝑤𝑥)) 𝛿𝑤

+𝑁

𝐾𝜂2𝑤𝑥𝑥𝛿𝑤𝑥]

𝑥=𝑙

𝑥=0

𝑑𝜃,

𝐵7(𝑢, V, 𝛿𝑤, 𝛿𝑤

𝑥)

= ∫

2𝜋

0

[(𝜇1𝑢 + 𝑐2(𝑘2𝑢𝜃𝜃− 𝑢𝑥𝑥) − 𝑐2(3𝑘2+ 𝜇1) V𝑥𝜃) 𝛿𝑤

+ (𝑐2(3𝑘2+ 𝜇1) V𝜃+ 𝑐2𝑢𝑥) 𝛿𝑤𝑥]𝑥=𝑙

𝑥=0𝑑𝜃.

(34)

Since the first variations of the functional 𝑉(𝑢, V, 𝑤) are zero,that is,

𝛿𝑢𝑉 (𝑢, V, 𝑤) = 𝛿V𝑉 (𝑢, V, 𝑤) = 𝛿𝑤𝑉 (𝑢, V, 𝑤) = 0 (35)

by the fundamental lemma of the calculus of variations, wehave

𝐵1(𝑢, 𝛿𝑢, 𝛿𝑢

𝑥) + 𝐵2(𝑤, 𝛿𝑢) + 𝐵

3(V, 𝛿V, 𝛿V

𝑥)

+ 𝐵4(𝑢, 𝛿V) + 𝐵

5(𝑤, 𝛿𝑤, 𝛿𝑤

𝑥)

+ 𝐵6(𝑤, 𝛿𝑤, 𝛿𝑤

𝑥) + 𝐵7(𝑢, V, 𝛿𝑤, 𝛿𝑤

𝑥) = 0

(36)

which yields the boundary conditions. We first note that (36)can be written as

5

∑

𝑖=1

𝑏𝑖= 0, (37)

where

𝑏1(𝑢, 𝑤, 𝛿𝑢, 𝛿𝑢

𝑥)

= ∫

2𝜋

0

[(−𝑢𝑥−𝑁


+ 𝑢𝑥𝜃𝜃) − 𝑢𝑥) + 𝑐2𝑤𝑥𝑥) 𝛿𝑢

+𝑁

𝐾𝜂2𝑢𝑥𝑥𝛿𝑢𝑥]

𝑥=𝑙

𝑥=0

𝑑𝜃,

𝑏2(𝑢, V, 𝑤, 𝛿V, 𝛿V

𝑥)

= ∫

2𝜋

0

[( − (𝑘2+ 𝜇1) 𝑢𝜃− 𝑘2(1 + 3𝑐

2) V𝑥

−𝑁


+ V𝑥𝜃𝜃) − V𝑥)

−𝑐2(3𝑘2+ 𝜇1) 𝑤𝑥𝜃) 𝛿V]𝑥=𝑙

𝑥=0

𝑑𝜃

+ ∫

2𝜋

0

[𝑁

𝐾𝜂2V𝑥𝑥𝛿V𝑥]

𝑥=𝑙

𝑥=0

𝑑𝜃,

𝑏3(𝑤, 𝛿𝑤)

= ∫

2𝜋

0

[(𝑐2(𝑤𝑥𝑥𝑥

+ (4𝑘2+ 2𝜇1) 𝑤𝑥𝜃𝜃) −

𝜍𝑅

𝐾𝜂2𝑤𝑥

−𝑁


+ 𝑤𝑥𝜃𝜃) − 𝑤𝑥)) 𝛿𝑤]

𝑥=𝑙

𝑥=0

𝑑𝜃,

𝑏4(𝑢, V, 𝛿𝑤)

= ∫

2𝜋

0

[(𝜇1𝑢 + 𝑐2(𝑘2𝑢𝜃𝜃− 𝑢𝑥𝑥)

−𝑐2(3𝑘2+ 𝜇1) V𝑥𝜃) 𝛿𝑤]

𝑥=𝑙

𝑥=0𝑑𝜃,

𝑏5(𝑢, V, 𝑤, 𝛿𝑤

𝑥)

= ∫

2𝜋

0

[(−𝑐2𝑤𝑥𝑥+𝑁

𝐾𝜂2𝑤𝑥𝑥+ 𝑐2(3𝑘2+ 𝜇1) V𝜃

+𝑐2𝑢𝑥) 𝛿𝑤𝑥]

𝑥=𝑙

𝑥=0

𝑑𝜃.

(38)

From (37)-(38), the natural and geometric boundary condi-tions are obtained at 𝑥 = 0 and 𝑥 = 𝑙 as

𝑢𝑥+𝑁


+ 𝑢𝑥𝜃𝜃) − 𝑢𝑥) − 𝑐2𝑤𝑥𝑥= 0 or 𝑢 = 0,

𝑢𝑥𝑥= 0 or 𝑢

𝑥= 0,

(𝑘2+ 𝜇1) 𝑢𝜃+ 𝑘2(1 + 3𝑐

2) V𝑥

+𝑁


+ V𝑥𝜃𝜃) − V𝑥)

+ 𝑐2(3𝑘2+ 𝜇1) 𝑤𝑥𝜃= 0 or V = 0,

V𝑥𝑥= 0 or V

𝑥= 0,

𝑐2(𝑤𝑥𝑥𝑥

+ (4𝑘2+ 2𝜇1) 𝑤𝑥𝜃𝜃) −

𝜍𝑅

𝐾𝜂2𝑤𝑥

−𝑁


+ 𝑤𝑥𝜃𝜃) − 𝑤𝑥) ⋅ ⋅ ⋅ + 𝜇

1𝑢

+ 𝑐2(𝑘2𝑢𝜃𝜃− 𝑢𝑥𝑥)

− 𝑐2(3𝑘2+ 𝜇1) V𝑥𝜃= 0 or 𝑤 = 0,

− 𝑐2𝑤𝑥𝑥+𝑁

𝐾𝜂2𝑤𝑥𝑥+ 𝑐2(3𝑘2+ 𝜇1) V𝜃

+ 𝑐2𝑢𝑥= 0 or 𝑤

𝑥= 0.

(39)


6. Conclusions

The variational formulation for the buckling of a microtubulewas given using a nonlocal continuum theory wherebythe microtubule was modeled as an orthotropic shell. Thecontinuum model of the microtubule takes the effects ofthe surrounding filament network and the viscous cytosolinto account as well as its orthotropic properties. Methodsof calculus of variations were employed in the derivationof the variational formulation and in particular the semi-inverse approach was used to identify suitable variationalintegrals. The buckling load was expressed in the form ofa Rayleigh quotient which confirms that small scale effectslower the buckling load as has been observed in a numberof studies [1, 26, 28]. The natural and geometric boundaryconditions were derived using the formulations developed.The variational principles presented here form the basisof several approximate and numerical methods of solutionand facilitate the implementation of complicated boundaryconditions, in particular, the natural boundary conditions.

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

Theresearch reported in this paperwas supported by researchgrants from the University of KwaZulu-Natal (UKZN) andfrom National Research Foundation (NRF) of South Africa.The author gratefully acknowledges the support provided byUKZN and NRF.

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Research Article Variational Principles for Buckling of ...Variational formulations were employed in a number of studies involving microtubules. In particular, small scale formulations