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УДК 517.927.2
doi: 10.21685/2072-3040-2023-4-7
Investigating the buckling and vibration of a Kirchhoff rectangular nanoplate using modified couple stress theory
Majid E. Shahraki1, Jafar E. Jam2
:Ferdowsi University of Mashhad, Iran 2Malek Ashtar University of Technology, Tehran, Iran
1mjdeskandari@gmail.com
Abstract. In this paper a model is developed for the buckling and vibration analysis of a Kirchhoff rectangular nanoplate based on a modified couple stress theory. In order to consider the small scale effects, the modified couple stress theory, with one length scale parameter, is used. In modified couple stress theory, strain energy density is a function of strain tensor, curvature tensor, stress tensor and symmetric part of couple stress tensor. After obtaining the strain and kinetic energy, external work and substituting them in the Hamilton's principle, the main and auxiliary equations of the nanoplate are obtained. Then, by manipulating the boundary conditions the governing equations are solved using Navier approach for buckling and vibration of the nanoplate. The dimensionless critical force under a bi-axial surface force in x and y directions and frequencies of different modes are all obtained for various plate's dimensional ratios and material length scale to thickness ratios. The governing equations are numerically solved. The effect of material length scale, length, width and thickness of the nanoplate on the buckling ratios and frequencies are investigated and the results are presented and discussed in details.
Keywords: Modified couple stress theory, Kirchhoff nanoplate, Buckling, Vibration, Na-vier's solution
For citation: Shahraki M.E., Jam J.E. Investigating the buckling and vibration of a Kirchhoff rectangular nanoplate using modified couple stress theory. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki = University proceedings. Volga region. Physical and mathematical sciences. 2023;(4):75-89. (In Russ.). doi: 10.21685/2072-3040-2023-4-7
Исследование потери устойчивости и вибрации прямоугольной нанопластины Кирхгофа на основе модифицированной теории парных напряжений
Маджит Эскандари Шахраки1, Джафар Эскандари Джам2
1Мешхедский университет имени Фирдоуси, Иран 2Технологический университет Малека Аштара, Тегеран, Иран
1mjdeskandari@gmail.com
Аннотация. В данной статье разработана модель для анализа потери устойчивости и вибрации прямоугольной нанопластины Кирхгофа на основе модифицированной теории парных напряжений. Для учета мелкомасштабных эффектов используется модифицированная теория парных напряжений с одним параметром масштаба длины. В модифицированной теории парных напряжений плотность энергии деформации является функцией тензора деформаций, тензора кривизны, тензора напряжений и симметричной части тензора парных напряжений. После получения значений де-
© Shahraki M. E., Jam J. E., 2023. Контент доступен по лицензии Creative Commons Attribution 4.0 License / This work is licensed under a Creative Commons Attribution 4.0 License.
формации, кинетической энергии, внешней работы и подстановки их в принцип Гамильтона получаются основные и вспомогательные уравнения нанопластины. Затем, манипулируя граничными условиями, для расчета устойчивости и вибрации нанопластины используются основные уравнения, которые решаются с использованием подхода Навье. Значения безразмерной критической силы под влиянием двухкоорди-натной x и y поверхностой силы и частоты различных режимов были получены для различных соотношений размеров пластины и соотношений длины материала к толщине. Основные уравнения решаются численно. Было изучены влияние размера материала, длины, ширины и толщины нанопластины на коэффициенты и частота потери устойчивости, а результаты представлены и подробно обсуждены. Ключевые слова: Модифицированная теория парных напряжений, нанопластина Кирхгофа, потеря устойчивости, вибрация, решение Навье
Для цитирования: Shahraki M.E., Jam J.E. Investigating the buckling and vibration of a Kirchhoff rectangular nanoplate using modified couple stress theory // Известия высших учебных заведений. Поволжский регион. Физико-математические науки. 2023. № 4. С. 75-89. doi: 10.21685/2072-3040-2023-4-7
1. Introduction
Performing experiments in the atomic and molecular scales are the safest approach for the study of materials in small-scales, cause the structures are investigated in real dimensions. In order to determine the mechanical properties of nanostructures in this method, various mechanical loads are applied on nanostruc-tures using the atomic Force Microscopy (AFM) and the plate responses are measured. The difficulties of controlling the test conditions at this scale, high economic costs and time-consuming processes are some setbacks of this approach. Therefore, it is only used to validate other simple and low-cost methods.
Atomic simulation is another approach for studying small-scale structures. In this method, the behaviors of atoms and molecules are examined by considering the intermolecular and interatomic effects on their motions, which ultimately involves the total deformation of the body. In the case of large deformations and multi atomic scales the computational costs of this approach become unbearable, so it is only used for small deformation problems.
Given the limitations of the aforementioned methods for the study of small-scales, many literatures were directed toward finding more efficient solutions which are reliable and less cost and time consuming. Modelling small-scale structures using continuum mechanics is one of these solutions. There are a variety of size-dependent continuum theories that consider size effects and are suitable for this problem. Some of these theories are; micromorphic theory, microstructural theory, micropolar theory, Kurt's theory, non-local theory, modified couple stress theory and strain gradient elasticity.
In this paper a Kirchhoff nanoplate model is developed for buckling and vibration analysis of a graphene nanoplate based on the modified couple stress theory. The results are presented in figures and tables and are discussed in details.
2. Modified Couple Stress Theory
In 2002 Yang et al. [1] proposed a modified couple stress model by modifying the theory proposed by Toppin [2], Mindlin and Thursten [3], Quitter [4] and Mindlin [5] in 1964. The modified couple stress theory consists of only one material length scale parameter for projection of the size effect, whereas the classical
couple stress theory needs two material length scale parameters. In the modified theory the strain energy density in the three-dimensional vertical coordinates for a body bounded by the volume V and the area Q [6], is expressed as the follows:
U=2f V(jeij + mijXij ) h j = 1,2,3 (1)
j "2 ((j+uj,i), (2) Xj = 1 ( j + 0 ji)' (3)
Xij and £jj are the symmetric parts of the curvature and strain tensors, and ui are the displacement and the rotational vectors, respectively;
0=1 Curl u, (4)
2
Oy, the stress tensor, and m^, the deviatory part of the couple stress tensor, are defined as:
j ^kk&ij + 2№ij , (5) mi, j= 12%ij. (6)
Where X and ^ are the lame constants, 8jj is the Kronecker delta and l is
the material length scale parameter. From Equations (3) and (6) it can be seen that Xij and mij are symmetric.
3. Kirchhofes Plate Model
The displacement equations for the Mindlin's plate are defined as:
dw ( y, t) Ul (x, y, z, t )= - z-—-,
( .) dw (x, y, t)
U2 (x,y,z,t)= -z-^-,
dy
U3 (x,y,z,t)= w(x,y,t), (7)
where w is the midpoint displacement of the plate in the z-axis direction.
The strain and stress tensors, the symmetric part of the curvature tensor, and the rotational vector for the Mindlin's plate is obtained as follows:
d2 w
£xx (8)
dx
d2 w
£yy =-z TT, (9)
dy
F = F = — z
°xy °yx
d 2 w dxdy
F =F =F =F =F = 0
°zz ^xz °zx °yz °zy " :
_ dw
x _ ~dy
_ dw
y _—âx
e z = o,
d
2
w
xyy
dx dy
d 2 w dx dy
2.
xxy xyx 2
w
)2 w ï
dy2 dx2
xxz xzx xyz xzy xzz 0 :
О xx = —( + 2^)
( д2 ï d w z——
V
С d 2 ï
d w z——
— À
( Д2 ï
d w z——
/
V
dx
dx
—( + 2ц)
dy
/
С д2 ï d w z——
/
V
dy2
/
О zz = —Àz
Сд2 a2 ï d w d w
+
dx2 dy2
/
О yx = О xy = —2Ц
xy
2
d w
z
V
dxdy
/
О xz = czx =°yz =°zy = 0-
10) 11) 12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22) 23)
The variation of the strain energy is expressed as follows:
ÔU = J V (g^ 8 £^ + G^ Ô£^ + 2G8 £^ + 2GS £^ + 2Gyz S £^ ,
+mxx 8 Xxx + myy8xyy + mzz 8xzz + 2mxy 8xxy + 2mxz 8xxz + 2myz 8 Xyz )dV • (24)
For the sake of simplification, the coefficient of each variable in the above equation is named from Fi to F3 and this equation can be rewritten as shown below:
5U = J V ((w ^ + f2 5w, yy + F3 5w, ), (25)
In which
^ d" W
F =
2— - - d2
3x 2
( + 2|)z2 + |2 +d-W- (z22), (26)
J dy2 v '
F d 2 w F2 =
2 ,- -j2 1 - d w
dy2
( + 2|)z2 + |2 ](2 -|l2), (27)
J dx y '
dxdy (2 + 4|l2). (28)
F d2 w F3 =
4. Buckling Load
For a rectangular plate with length a, width b and thickness h, under the axial forces (Pxy, Py, Px), the buckling force is obtained as shown in equation (29) [7, 8]:
2 2 2 „ d w d w w , ,
Px + ^W^V ="(x'y)' (29)
where Px is the Axial force along the x axis, Py is the Axial force along the y axis, Pxy is the shear force in the xy plane, and q (x, y) is the out-of-plane force.
5. Virtual Work of External Loads
In these kind of problems the virtual work of three kinds of external forces are included in the solutions, if the middle-plane and the middle-perimeter of the plate are shown as Q and r respectively, these virtual works are [9]:
1. The virtual work done by the body forces, which is applied on the volume V= Q x (- h / 2, h /2).
2. The virtual work done by the surface tractions at the upper and lower surfaces (Q).
3. The virtual work done by the shear tractions on the lateral surfaces, S=
r x (- h / 2, h / 2).
If (fx, fy, fz) are the body forces, (cx, Cy, cz) are the body couples, (qx, qy, qz) are the forces acting on the Q plane (x ty, tZ) are the Cauchy's tractions and (Sx, Sy, Sz) are surface couples the Variations of the virtual work is expressed as:
5w=- [J q (fx5u+fy 5V+fz5w+qx5u+qy 5V+qz5w+cx50 x +
+cy50y + cz 50 z) dx dy + J r (tx 5u + ty 5V + tz 5w + sx 50 x + sy 50y + sz 50 z) (30)
Given that in this study only the external force qz was applied, virtual work becomes:
f0 J 0
the variation of kinetic energy is obtained as:
rarb
5w = J 0J 0q(x,y)5w(x,y)dxdy , (31)
ôT = J aJ2 hP(ùl8Ul + U2ÔU2 + U3ÔU3 )dAdz =
=L
3
phw ôw + ( 8фх + (y ô(Py)
(32)
where p is the density.
Finally using the Hamilton's principle, it can be said that [10]:
T
J (8T- (SU- 5w))dt = 0
(33)
where T is the kinetic energy, U is the strain energy, and W is the work of the external forces.
6. The Governing Equations of the Plate
Using Hamilton's principle, equation (33), we have:
h ( d2F1 d2f2 azF3
2т, л
дx2 by2 dxdy
и д w
+ Px—;т +
x Эх2
д 2 w
д 2w
3
+2 + = q (x, y ) + ph w - Ph- V2 w .
^ dxdy y dy2 12
(34)
7. General Equation
Considering the following constants:
A1 = ( + 2|i)/2 + |ji 2 h,
(35)
= J V
(36)
The general governing equation of the Kirchhoffs plate plate will becomce:
д 4 w
д 4 w
д 4 w
Ai —7- + Ai —- + 2 Ai —-——
1 дx4 1 дy4 1 дx 2дy2
+ Px ^w + 2 Pxy
дx 2
д 2 w дxдy
+
+P
д 2 w у э 2
= q (x, y )+ph
д w ph
д(2
12
w
д4 Л д w
дx2дt2 ду 2дt2
+
(37)
8. Solution of the Governing Equations Using Navier's Method
The Navier's solution is applicable to the rectangular plates which have simply supported boundary conditions on all edges. Since the boundary conditions are spontaneously satisfied in this method, the unknown functions of the plate's mid-plane were assumed to be double trigonometric series [7, 9] :
h
2
2
W ( x> y>t ) = Z ZWmn sin sin ßy
m=1n =1
j'rat
The force can also be calculated from the following relations:
©O ^
q=2 2««sin ax sin Py e'at,
where
m=1n=1
Qmn = \ j ! j ( x, y ) sin ^ sin ßy dx dy :
Qmn
ab! oJ 0
qQ; For sinusoidal force 16% .
, For uniform force
mnn
4Qo.
ab
; For point force in the plane center
nm _ nn /—-a =—, ß =—, i = V-1.
(38)
(39)
(40)
(41)
(42)
Simply-supported boundary conditions were also satisfied by the Navier's method according to the following equations:
x = 0 x = a
y = 0 y = b
m \ i \ ^ ■ mn . nn w (0, y ) = w (a, y ) = ZZWmn sin—x sin y = 0,
mn nn
9y (y) = 9y (ay) = ymn sin-xcosT"y = 0
ab
i iw i u\ ^^ • mn . nn w(x,0) = w(x,b) = ZZwmn sin-xsin-^y = 0,
9x (x,0) = 9x (x,b) = Xmn cos — xsin~Ty = 0
a b
(43)
(44)
9. Obtaining the General Equation Matrix of a Kirchhofes Plate
After solving the governing equations and naming the coefficient of each variable, we have:
wmn ((1a4 + 4ß4 + 24a2ß2 - Pxa2 -Pyß2 ) =
3 3
2 ph 2 2 ph 2 2
= Qmn - Phwmn ® - — wmna ® -~wmn ß ® = (( ]-W2 [^ ])[wmn ] = [Qmn ]•
(45)
(46)
In which:
N1 = 4a4 + 4p4 + 24a2p2 - Pxa2 - Pyp2, (47)
K1 = -ph-P^a2 -P^p2. (48)
1 12 12
Various materials such as epoxy, graphene, copper and so on can be considered as the plate's material. In this study, graphene is chosen as the plate's material. A single-layer graphene plate has the following properties [10]:
E = 1.06 TPa,V = 0.25 ,h = 0.34 nm, p = 2250 kg/m3 .
Also, the relationship between E, ^ and v can be expressed as: i vE E
" = (1+V)(1 -2V)' " = 2(+V) (49)
where ^ and X are the lame's coefficients and E is the Young's modulus [11]. The value of the distributed force was considered to be q = 1 N / m2.
10. Results and Discussion
Results were obtained using a computational program coded in the MATLAB software. The results have also been compared with the literature [1217] and good agreements between results were observed. The plate's dimensional parameters are chosen as follows:
a: plate's length;
b: plate's width;
h: plate's thickness;
l: material length scale parameter.
Fig. 1 shows the values of critical force for Kirchhoffs nanoplate under a biaxial surface force in x and y directions. As can be seen, this value decreases due to an increase in length scale parameter to thickness and length to thickness ratio of nanoplate. Also, for the classical theory (neglecting the effect of size parameter) the critical force reaches its lowest value, but with an increase in the size effect, the critical force values increase.
Fig. 2 shows the values of critical force for Kirchhoffs nanoplate under a biaxial surface force in x and y directions for different modes. As can be seen, this value increases due to an increase in length scale parameter to thickness ratio. It was observed that, the first mode has the lowest critical force, and it increases for the next modes.
Table 1 compares the values of critical force for different nanoplates under a bi-axial surface loading for various length scale parameter to thickness ratios. It can also be observed that, the Mindlin's nanoplate has the highest, and the Third-order nanoplate has the lowest critical force values.
Table 2 compares the values of critical force for different nanoplates under a uniaxial surface loading in x-direction for various length scale parameter to thickness ratios. It can be seen that the Mindlin's nanoplate has the highest, and the Third-order nanoplate has the lowest critical force values.
By comparing tables 1 & 2 it is found that in the Kirchhoffs nanoplate the critical force under a uniaxial load is greater than that of bi-axial load.
Fig. 3 to 6 shows the frequency of different modes of Kirchhoffs nanoplate (шц - о>12 - Ш21 - ш22). It was observed that this value decreases due to an increase in length to thickness ratio. Also, for the classical theory (neglecting the effect of size parameter) the frequency reaches its lowest value, but with an increase in the size effect, the dimensionless frequency values increase. Furthermore, it is shown that the first mode has lowest frequency and it increases for the next modes.
Fig. 1. Values of critical force for Kirchhoffs nanoplate under a bi-axial surface force in x and y directions for different material length scale to thickness and length to thickness ratios of the nanoplate (a/b = 1)
Fig. 2. Values of critical force for Kirchhoffs nanoplate for different modes under a bi-axial surface force in x and y directions for different material length scale to thickness ratios of the nanoplate (a/b = 1)
Table 3 shows that the frequency of different modes of Kirchhoffs nanoplate increases due to an increase in material length scale parameter to thickness ratio.
Table 1
Values of the critical force of buckling for different nanoplates under a bi-axial surface loading for various width to thickness ratios (l/h=, a/b=1)
o/b Kirchhoff Mindlin Third order shear N order shear
a/n plate plate deformation plate deformation plate (n = 5)
5 142.2802 233.7327 130.1058 131.5295
10 35.5701 86.0362 34.7400 34.8479
20 8.8925 23.9784 8.8394 8.8465
30 3.9522 10.8814 3.9417 3.9431
40 2.2231 6.16595 2.2198 2.2202
50 1.4228 3.9597 1.4214 1.4216
Table 2
Values of the critical force of buckling for different nanoplates under a uniaxial force in the x-direction for different material length scale to thickness ratios (l/h=, a/b=1)
a/h Kirchhoff plate Mindlin plate Third order shear deformation plate N order shear deformation plate (n = 5)
5 284.5604 467.4653 260.2116 263.0590
10 71.1401 172.0725 69.4801 69.6959
20 17.7850 47.9569 17.6789 17.6930
30 7.9045 21.7629 7.8834 7.8862
40 4.4463 12.3319 4.4396 4.4405
50 2.8456 7.9194 2.8429 2.8432
Fig. 3. Comparison of frequencies of the first mode (ran) for a Kirchhoffs nanoplate for various material length scale parameter to thickness and length to thickness ratios of the nanoplate (a/b = 1)
By comparing Tables 4 & 5 shows the frequency of different modes for various nanoplates. Based on the table the Mindlin nanoplate has the highest frequency values (except for the classical theory I = 0).
Fig. 4. Comparison of frequencies of the mode (Ю12) for a Kirchhoffs nanoplate for different material length scale parameter to thickness ratios (a/b = 1)
Fig. 5. Comparison of frequencies of the mode (0)21) for a Kirchhoffs nanoplate for different material length scale parameter to thickness ratios (a/b = 1)
Fig. 6. Comparison of frequencies of the mode (022) for a Kirchhoffs nanoplate for different material length scale parameter to thickness ratios (a/b = 1)
Table 3
Comparison of frequencies of different modes of Kirchhoffs nanoplates for different material length scale to thickness and length to width ratios (a/h = 30)
Mode l/h
0 0.5 1 2
a/b = 1
«11 13.9886 19.7828 31.2794 57.6764
«12 34.9237 49.3895 78.0917 143.9940
«21 34.9237 49.3895 78.0917 143.9940
«22 55.8018 78.9157 124.7766 230.0767
a/b = 1.5
œn 10.1054 14.2912 22.5964 41.6657
«12 19.4217 27.4664 43.4282 80.0777
«21 31.0511 43.9129 69.4324 128.0271
«22 40.3420 57.0522 90.2074 166.3343
a/b = 2
«11 8.7459 12.3685 19.5563 36.0601
«12 13.9886 19.7828 31.2794 57.6764
«21 29.6953 41.9954 66.4006 122.4367
«22 34.9237 49.3895 78.0917 143.9940
Table 4
Comparison of frequencies of different modes of various nanoplates for different material length scale to thickness ratios (a/h = 30, a/b = 2)
Mode l/h
0 0.5 1 2
Mindlin plate
«11 8.7280 17.9862 32.5528 62.6001
«12 13.9429 28.7266 051.9052 99.1252
«21 29.4914 60.7224 109.1821 204.2795
«22 34.6425 71.3140 128.0217 237.9174
Kirchhoff plate
«11 8.7459 12.3685 19.5563 36.0601
«12 13.9886 19.7828 31.2794 57.6764
«21 29.6953 41.9954 66.4006 122.4367
«22 34.9237 49.3895 78.0917 143.9940
Third order shear deformation plate
«11 8.7284 12.3536 19.5411 36.0389
«12 13.9441 19.7447 31.2407 57.6223
«21 29.4967 41.8251 66.2277 122.1954
«22 34.6497 49.1546 77.8533 143.6613
Table 5
Comparison of frequencies of different modes of various nanoplates for different length to thickness ratios (l/h = 1, a/b = 1.5)
Mode a/h
20 30 40
Mindlin plate
«11 83.8680 37.5829 21.2022
«12 159.1488 071.8354 40.6324
«21 250.5691 114.0783 64.7336
«22 321.6951 147.4292 83.8680
Kirchhoff plate
«11 50.8001 22.5964 12.7141
«12 97.5592 43.4282 24.4419
«21 155.8295 69.4324 39.0903
«22 202.3037 90.2074 50.8001
Third order shear deformation plate
«11 50.6984 22.5761 12.7077
«12 97.1889 43.3538 24.4182
«21 154.8989 69.2435 39.0300
«22 200.7540 89.8902 50.6984
Conclusion
In this study the buckling and vibration of a Kirchhoffs nanoplate were investigated using the modified couple stress theory. As observed in the tables and figures, the Kirchhoffs nanoplate buckling under a bi-axial in x & y direction, decreases with an increase in length to thickness ratio of the nanoplate. Furthermore when the size effect parameter is neglected (classical theory), the value of critical force becomes constant and reaches its lowest value, but with an increase in the size parameter the critical force value increases. It is found that the value of critical force is the lowest for the first mode and increases gradually for the next modes. It was also observed that the critical buckling force is the lowest for the third order shear deformation plate and is the highest for the Mindlin's nanoplate.
Analysis of frequencies of different modes (ro^ -0>i2 -®21 ~®22) for Kirchhoffs nanoplate, showed that this values decrease due to an increase in material length scale to thickness and/or the plate's aspect ratios. Also, for the classical theory (neglecting the effect of size parameter) the frequency reaches its lowest value, but with an increase in the size effect, the frequency values increase. It was also found that the value of frequency is the lowest for the first mode and increases gradually for the next modes.
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15. Akgoz B., Civalek O. Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. International Journal of Engineering Science. 2011;49(1):1268-1280.
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17. Sahmani S., Ansari R. On the free vibration response of functionally graded higherorder shear deformable microplates based on the strain gradient elasticity theory. Composite Structure. 2013;95:430-442.
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Информация об авторах / Information about the authors
Majid E. Shahraki Маджит Эскандари Шахраки
Ph.D. Student, Faculty Аспирант, Инженерный факультет,
of Engineering, Ferdowsi University Мешхедский университет имени
of Mashhad, Iran Фирдоуси
E-mail: mjdeskandari@gmail.com
Jafar E. Jam
Professor, Faculty of Materials and Manufacturing Technologies, Malek Ashtar University of Technology, Tehran, Iran
Джафар Эскандари Джам профессор, факультет технологии производства материалов, Технологический университет Малека Аштара, Тегеран, Иран
Авторы заявляют об отсутствии конфликта интересов / The authors declare no conflicts of interests.
Поступила в редакцию / Received 06.12.2022
Поступила после рецензирования и доработки / Revised 03.07.2023 Принята к публикации / Accepted 19.09.2023
