CC BY
20
УДК 539.3
Нелокальная теория упругости для описания механического поведения белковых микротрубочек
E. Ghandourah
Университет короля Абдулазиза, Джидда, 21589, Саудовская Аравия
В работе представлена теория балок Редди с нелокальными дифференциальными определяющими уравнениями Эрингена, а также уравнения движения белковых микротрубочек в терминах обобщенных смещений. Приведено аналитическое решение изгиба белковых микротрубочек для выявления влияния нелокального поведения на отклонения. Полученные теоретические результаты и численные решения могут служить основой для нелокальных теорий белковых микротрубочек или балок.
Ключевые слова: аналитические решения, белковые микротрубочки, теория балок Редди, изгиб, нелокальная эластичность
DOI 10.24412/1683-805X-2021-2-91-98
Nonlocal elasticity theory for the mechanical behavior of protein microtubules
E. Ghandourah
Department of Nuclear Engineering, Faculty of Engineering, King Abdulaziz University,
Jeddah, 21589, Saudi Arabia
In this work, Reddy beam theory is used with the nonlocal differential constitutive relations of Eringen, and the equations of motion of the protein microtubules in terms of the generalized displacements are presented. Analytical solution of bending the protein microtubules is presented to bring out the effect of the nonlocal behavior on deflections. The theoretical development as well as numerical solutions presented herein should serve as references for nonlocal theories of the protein microtubules or beams.
Keywords: analytical solutions, protein microtubules, Reddy beam theory, bending, nonlocal elasticity
1. Introduction
The nonlocal continuum mechanics adopt that the stress components at a point are a function of strain components at all points of the body in the continuum and contain information about the forces between atoms, and the internal length scale as a material parameter is introduced into the constitutive equations. On the other hand, hyperelastic constitutive relations are the base of the most classical continuum theories which adopt that the stress at a point is functions of strains at the same point. The cy-toskeleton in most eukaryotic cells composed of three types of filaments organized in networks: the
protein microtubules, the microfilaments, and the intermediate filament [1-8]. In most eukaryotic cells, the cytoskeleton comprises three types of filaments: the actins, the protein microtubules, and the intermediate filaments (Fig. 1) [9]. Experimental studies have shown that the protein microtubules are mostly bear compression but the actins and the intermediate filaments can bear only tension because of their minor cross sections [10, 11]. The microtubule filaments typically reply to mechanical excitations as different structural elements, unlike other cytoskele-tal filaments that work together as a network, such as the intermediate filaments and the actins [12]. The
© Ghandourah E., 2021
Fig. 1. The geometry of the protein microtubule [9, 12, 18] (color online)
molecular structure of microtubules is larger persistence lengths, the protein microtubules behave more like rigid bars. From this viewpoint, as one of the most rigid cytoskeletal components, the protein microtubules provide support for the cell to maintain its shape [12-14]. The rigidity of the protein micro-tubule filament results largely from its hollow cylindrical shape, composed of a-b tubulin heterodimers that form protofilaments [15]. The structure of the protein microtubule is composed of 13 parallel protofilaments oriented longitudinally [16]. Microtubules play important roles in the motility of the cell, the growth, the mitosis, and the meiosis [9], the protein microtubule act as tracks for motor proteins in order to carry cargoes across the cytoplasm [17]. The inner diameters of microtubules are 17 nm and the outer diameters are 27 nm. The length of microtubule filaments varies from tens of nanometers to hundreds of micrometers [9, 17]. Moreover, the previous experimental studies have shown that even though the persistence length of microtubule filaments is far greater than their lengths, the protein microtubules do not certainly appear straight in the cytoskeleton (Fig. 1).
It should be noted that the mechanical behavior of microtubules can be studied using couple stress model [19-22], micropolar model [23], nonlocal elasticity theory [24-36] and strain gradient models [30-35]. Recently, several HSDTs are proposed by researchers to examine static and dynamic behaviors responses of various type of structures and materials [36-42].
In the present work, we adopt the same scaling effect parameter as used by De Pablo et al. [43] in our investigation of the small-scale effect on the vibration behavior of the protein microtubules quantitatively, the nonlocal Reddy beam theory is applied to study the vibration characteristics of the protein microtubules in the surrounding cytoplasm. This study includes the influences of the nonlocal effect and shear modulus ratio on the frequency of the protein microtubules.
2. Formulation of the problem
The x-coordinate is taken along the length of the protein microtubules, z-coordinate along with the thickness (the height) of the protein microtubules, and the ^-coordinate is taken along the width of the protein microtubules. In a beam theory, all applied loads and geometry are such that the displacements (u1, u2, u3) along with the coordinates (x, y, z) are only functions of the x and z coordinates and time t. These equations for the nonlocal effects by expressing the stress resultants in terms of a nonlocal parameter. The following stress resultants are introduced for use in the coming sections:
N = J axxdA, M = J zaxxdA, P = J z3axxdA,
A r A r 2 A (1)
Q = JaxzdA, R = Jz2axzdA.
A A
The stress resultants P and R will appear only in the higher-order theories, employed the following
Ghandourah E. / OusmecKan MesoMexaHUKa 24 2 (2021) 91-98
93
displacement field [44]:
31 —w )
u1 = u(x, t) + z^(x, t) -c1z I ^H--I, (2)
I —x J
u2 = 0, u2 = w(x, t), (3)
where (u, w) are the axial and transverse displacements of the point (x, 0) on the midplane (i.e., z = 0) of the protein microtubules, ci = 4/(3h2), and h is the height of the protein microtubules. Although the displacement field in Eq. (2) was derived using the local shear stress-strain constitutive relation, it is assumed to be a valid displacement field. The nonzero strains are
du
S w — ■
ôx
7ft C72) ^ CZ3 02W 7(1 - c17 )—-- c17 —-
0x 0x2
0
= SV
-7K + 7 p,
2s xz — (1 - c—7 2) \0WW + 9I-Y + 7 2P,
ôœ
K —IT, P — -Ci
ôx
ôœ
ôx
ôx 2
(4)
(5)
where s0xx is the extensional strain and ke is the bending strain, and c2 = 4/h2. Using Eqs. (4)-(6), the
nonlocal constitutive relation for the macroscopic
stress takes the following special relations for the protein microtubules:
a xx M - 2 — Es xx, ôx
(6)
a xz — U0-^ = 2Gs xz (| = 2), (7)
ox
where E and G are Young's modulus and shear modulus, where e0 is a material constant, and a and l are the internal and external characteristic lengths, respectively.
The axial force-strain relation is given by
Ar ô2 N
N — m—— — EAs
ôx2
0,
(8)
where m = (e0a) is the nonlocal parameter, we have used the relations
A — J dA, J 7dA — 0. (9)
A A
Thus, the x axis is taken along the geometric cen-troid of the protein microtubules. The constitutive relations of the Reddy beam theory are
ô 2M
M-M—— — EI k + EJ p,
ôx2
ô2 P
P-M—- — EJ k + EK p,
ôx2
- 2Q
Q-|-f = GAy + GI p,
-x 2
—2R
R — u—-2 = GI y + GJ p.
—x2
We have M, P, Q, and R, where I denotes the second moment of area about the y-axis, and Ks denotes the shear correction factor,
(A, I, J, K) = J (1, z2, z4, z6)dA (11)
A
are the second, fourth, and sixth-order moments of the area about the y axis.
3. Governing equations in terms of displacements
Substituting for the first derivative of the axial force N, we obtain
N — EA— + M ôx
( ô3u ôf ^ ôxôt2 ôx
(12)
Substituting N from Eq. (12) into the equation of motion, we obtain
ôL.ôu ^ . ô2f
ôx IEA & )+f-m
2
ô u
ô u
vôt2 Môx 2ôt 2y
.(13)
From Eqs. (10) the nonlocal constitutive equations for the stress resultants of the Reddy beam the-
ory are
M-m
ô2M
ôx2
— EI k + EJp,
ô 2q
(14a)
Q — |-f = GAy + GI p,
-x2
- 2 P
P — u—- = EJ k + EK p,
-x2
where additional variables are used in the following equations:
I = I — c1J, J = J — c1K,
(14b)
A — A - c21, I — I - c2 J, A — A - c21. Eliminating Q from Eqs. (10), (14), we obtain
ô2 M
ô 2 P
ôx
2 ' — -c1 ôx2
ô ( — ôw
\ N— \ + mb ôx l ôx ) ôt
ô 2 w
■ m2
ô
ôxôt
2 - c1m4
( ô^
4
ô W
(15)
v ôxôt2 ôx2ôt2 ,
Substituting the above result in the first of Eqs. (14), we arrive at
.y - ô^ - ô^ ô w M = El — - c1EJ— +—-
ôx ôx ôx2
-c
ô 2p 1 ôx2
ô f — ôw
, N— I + m0 2 ôx l ôx J 0 ôt2
ô 2 w
ô3è
■m2-V - c1m4
2 ôxôt2 1 4
f ^3, -¡4
ô ffl ô w -— +--
y ôxôt2 ôx2ôt2 J
(16)
Substituting the second derivative of Q from Eqs. (10) into the second equation in Eqs. (14), we obtain:
q = GA | 1 + 1^-
ôx J ôx
-c
ô 2 P 1 ôx2
ô ^ôw , -—| N— |-< ôx l ôx
—
ôx
m.
ô 2 w 0 ôt2
ô
1 m 4-2"
1 4 ôxôt2
-cm
f «3
- c,2m6
ô3^ ô4w
4... Y
v ôxôt2 ôx2ôt2 J
(17)
Now we use M and Q from Eqs. (16) and (17) and the identity
2
1 ôx2
P
ô w ôx2
= c
EJ M - ClEK
ôx3
fôfy ôw 1
vax3 ôx4 j
(18)
To rewrite the equations of motion in terms of the generalized displacements, we obtain
GA
2,„\
ô^ + ô w ôx ôx2
ô f 7-Tôw
--1 N—
ôx l ôx
ô2 ■"S?
ô3^
ôx2
N +
ôw ôx
EJ—3 - cEK
3
ôx3
ôx3 ôx4
V
ô 2w
= m,
0 ôt2
ô3^ 2 ^ - c1m6
m
ô 4w
ôxôt
0 2 2 +c1m4 3 2
0 ôx2ôt2 1 4 ôx3ôt2
2
ô^ + ô w ôx ôxôt
(19)
ô
- c m,
f ô5^
ô 6 w
ôx3ôt2 ôx4ôt2
El
ô
ôx 2
- c1EJ
fô2è ô3w 1
7 +--T
ôx2 ôx3
^sl j. ôw 1 „ ô
- GAK + ix 1 = m22
f ô2è ô3w 1 v ôt2 ôxôt2
f ô>
m.
ô > 2 ôx2ôt2
ô5 w 1
Kôx 2ôt2 ôx3ôt2 ,
(20)
4. Analytical solutions of bending of simply supported protein microtubules
The boundary conditions of simply supported protein microtubules are
w = 0 and M = 0 at x = 0, L. (21)
The following expansions of the generalized displacements w and satisfy the boundary conditions:
w(x, t) = Z Wn sin(n%x/L)e"
n=1
(22) (23)
x, t) = cos(n%xlL)e'a«t.
n=1
For bending, we set N and all time derivatives to zero and take the distributed load be of the form:
q(x) = Z Qn sin (nrcx/L),
n=1
(24)
Qn = —J q( x)sin (nrcx/L)dx.
L 0
In particular, the coefficients Qn associated with various types of loads are given below
4q
q(x) = q0, Qn = —0, n = 1, 3,5,..., n%
q(x) = ^, Qn = ^2q°(-1)n+1, n = 1,2,3,..., L n%
q( x) = Q0§( x - x0), (25)
Qn = ^Q°sin n = 1,2,3,..., a L
q(x) = q0 sin y, Q1 = q0, Qn = 0, n = 2, 3,....
Substitution of the expansions for w, and q from Eqs. (22), (23) into the equations of motion (19), (20), we obtain
- L A„ „ + f W„ ] + X Q + X N l fî W. + c, f ^Y EJ« n
+ A,.
jjr nn _
mW — cm — O
0ry n ^1'At4 L ~n
2 nn + c1 m6~L
^ nn jjr
O n +LWn
©n = 0,
(26)
-EI
f \2 ' nn x
v L j
O — B
n n
f ,„-tt \
_ nn
O n + yW,
m2On — c1m4
2
nn
V L J
nn
O +--W
n L W
v
©2 = 0,
(27)
where
An = GA — c2GI + cf
Bn = GA — c2GI — c
vLy
v L j
EK,
EJ.
For static bending, by setting N = 0 and ©n = 0, we obtain
w( x) = x
n=1
1
Bn + EI
f \2 ' nn N
v L J
QnL4
An + cxBn (J/I) n4n4E^
sin (nnx/L)
(28)
<K x) = —X
n=1
x QnL4
Bn
An + cxBn (J/I)
n3n3 Ef
cos(nnx/L)
(29)
where An = (1 + n2n2Q), Q = EI/(GAKsL2) for any n, A,n = 1 + |(nn/L)2.
The nonlocal parameter A,n has the effect of increasing the deflection.
Fig. 3. The static deformation of simply supported microtubules under effect of uniformly distributed load for different nonlocal parameters e0a = 1.0 x 10-9 (1), 3.0 x 10-9 (2), 5.0 x 10-9 m (3) and q = 0.1 N/m (color online)
5. Results and discussions
Material and geometric values of microtubules as E = 2 x 109 N/m2, I = 105 x 10-34 m4, q = 1470 kg/m3, v = 0.3, L = 8 x 10-6 m. Figures 2-5 show the dis placement and bending moment along the length of the protein microtubules for different values of nonlocal parameters and three different boundary conditions. Two different load types are considered. It can be seen that the effect of the nonlocal parameter on the deflection of the protein microtubulesis significant. In general, the nonlocal parameter results in an increase of the transverse deflection and a bending moment of the protein microtubules under uniformly distributed load. Figure 2 shows the static deformation of simply supported microtubules under effect of the uniformly distributed load (q = 0.1, 0.3, 0.5 N/m), e0a = 0.3 x 10-9 m.
Figure 3 shows the static deformation of simply supported microtubules under effect of uniformly distributed load for different nonlocal parameters (e0a = 1.0 x 10-9, 3.0 x 10-9, 5.0 x 10-9 m) and q = 0.1 N/m.
Figure 4 shows the static deformation of simply supported microtubules under effect of the centrally concentrated load (P = 0.1, 0.3, 0.5 N), e0a = 0.3 x 10-9 m.
10 2w(x) 100
0-P--,-,-,-,-1
0.0 0.2 0.4 0.6 0.8 x!L
Fig. 2. The static deformation of simply supported micro-tubules under effect of the uniformly distributed load q = 0.1 (1), 0.3 (2), 0.5 N/m (3), e0a=0.3 x 10-9 m (color online)
Fig. 4. The static deformation of simply supported micro-tubules under effect of the centrally concentrated load P = 0.1 (1), 0.3 (2), 0.5 N (3), e0a=0.3 x 10-9 m (color online)
Fig. 5. The static deformation of simply supported microtubules under effect of centrally concentrated load for
different nonlocal parameters. e0a=2.0 x 10-9 (7), 6.0 x
10-8 (2), 8.0 x 10-7 m (3) and P = 0.1 N (color online)
Figure 5 shows the static deformation of simply supported microtubules under effect of centrally concentrated load for different nonlocal parameters (e0a = 2.0 x 10-9, 6.0 x 10-8, 8.0 x 10-7 m) and P = 0.1 N. It is also concluded that the effect of the nonlocal parame ter is more significant for concentrated load than the uniformly distributed load for simply supported protein microtubules and for all type boundary conditions. Some interesting results are also published by other authors [45] for cantilever microtubules. The results for clamped protein micro-tubules can not present in this manuscript. It interesting concluded that for simply supported of the protein microtubules [45], when the nonlocal parameter is greater than 6 nm, the deflection value becomes negative. In the present study, however, the nonlocal parameter is a significant effect on deflections and bending, and results are generally increased with this parameter.
6. Conclusions
This paper has presented an approach for obtaining accurate bending and bending moments and displacements in the protein microtubules in a computationally efficient manner using nonlocal continuum theory. Bending analysis of the protein microtubules subjected to uniformly distributed and concentrated loads is given. The equations of motion are then analytically solved for bending deflections of simply supported protein microtubules to bring out the effect of the nonlocal parameter. The inclusion of the nonlocal effect increases the magnitudes of deflections, the nonlocal effect is considerably different and more pronounced than using a shear correction coefficient. The numerical results show that the nonlocal parameter has an important effect on the static behavior of the protein microtubules. An improvement of the present formulation will be consid-
ered in the future work to consider other type of materials.
Funding
This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant No. J-177-135-1440. The authors, therefore, acknowledge with thanks DSR for technical and financial support.
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Received 13.11.2020, revised 17.12.2020, accepted 21.12.2020
Сведения об авторе
Emad Ghandourah, Dr., King Abdulaziz University, Saudi Arabia, eghandourah@kau.edu.sa
