Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 5989010, 9 pages https://doi.org/10.1155/2018/5989010
Research Article Closed-Form Solution of a Peripherally Fixed Circular Membrane under Uniformly Distributed Transverse Loads and Deflection Restrictions Teng-fei Wang 1
,1 Xiao-ting He ,1,2 and Yang-hui Li
1
School of Civil Engineering, Chongqing University, Chongqing 400045, China Key Laboratory of New Technology for Construction of Cities in Mountain Area, Chongqing University, Ministry of Education, Chongqing 400045, China
2
Correspondence should be addressed to Xiao-ting He;
[email protected] Received 4 January 2018; Accepted 26 March 2018; Published 8 May 2018 Academic Editor: Fabrizio Greco Copyright © 2018 Teng-fei Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The problem of axisymmetric deformation of a peripherally fixed and uniformly loaded circular membrane under deflection restrictions (by a frictionless horizontal rigid plate) was analytically solved, where the assumption of constant membrane stress adopted in the existing work was given up, and a closed-form solution of this problem was presented for the first time. The numerical analysis shows that the closed-form solution presented here has higher calculation accuracy than the existing approximate solution.
1. Introduction Elastic membrane structures and components are widely used in many fields [1–7]. The large deflection phenomena of membrane problem usually give rise to nonlinear differential equations [8–11]. These nonlinear equations generally present serious analytical difficulties when applied to boundaryvalue problems. Due to these somewhat intractable nonlinear equations, the analytical solutions of membrane problems are available in a few cases, but in practice they are often found to be necessary. Hencky [12] originally dealt with the problem of axisymmetric deformation of the circular membrane fixed at the outer edge under the action of a uniformly distributed transverse loads and presented the power series solution of the problem, as shown in Figure 1, where 𝑞 is the uniformly distributed transverse loads. A calculation error in [12] was corrected by Chien [13] and Alekseev [14], respectively. This problem and its solution are usually called well-known Hencky problem and well-known Hencky solution for short, which are often referred to or cited in a number of related studies [15–22]. However, if we use a frictionless horizontal rigid plate to restrict the deflection of the membrane in the well-known Hencky problem, as shown in Figure 2, then
such a problem will probably become somewhat complicated, where 𝑟 is the radial coordinate, 𝑤 is the transversal displacement, 𝑏 is the radius of the membrane contacting with the frictionless rigid plate, and 𝑔 is the gap between the frictionless rigid plate and the initially flat membrane. So, Xu and Liechti had to use the following four assumptions to deal with this problem [23]: (1) the membrane has negligible flexural rigidity and only membrane stresses are considered; (2) the slope angle 𝜃 of membrane is so small that the condition sin 𝜃 ≈ tan 𝜃 could approximately hold, that is, the socalled small-rotation-angle assumption; (3) a constant radial stress is assumed, that is, the radial stress in the membrane under loads has nothing to do with the radial coordinate; (4) the contact between the membrane and the rigid plate is frictionless. Obviously, assumption (3) above seems to be too harsh, and, in fact, it can be given up, as will be seen later. In the next section, the problem shown in Figure 2 was analytically solved, where the assumption of constant membrane stress adopted in the existing work was given up, and the closed-form solution of this problem was presented. In Section 3, based on numerical calculations, the reliability of the presented closed-form solution is verified and the beneficial effect of giving up the assumption of constant
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q
Figure 1: Sketch of the well-known Hencky problem.
q
r
r w(r) g
Frictionless rigid plate b
a
Figure 2: Sketch of confined deformation of the uniformly loaded circular membrane. r q
휎r ℎ
r w(r)
휃
휃
휎r ℎ
b r a
Figure 3: The equilibrium diagram of the central portion (𝑏 ≤ 𝑟 ≤ 𝑎) of the circular membrane.
membrane stress was discussed. Section 4 is the concluding remarks.
2. Membrane Equation and Its Solution Suppose that, an initially flat, linearly elastic, rotationally symmetric, taut circular membrane with Young’s modulus of elasticity 𝐸, Poisson’s ratio V, thickness ℎ, and radius 𝑎 is fixed at the outer edge and uniformly distributed transverse loads 𝑞 are quasi-statically applied onto the membrane surface (as shown in Figure 1, the well-known Hencky problem), and the applied loads 𝑞 especially have reached a value large enough so that the deflected membrane has a contact with a frictionless rigid plate being parallel to the initially flat membrane (as shown in Figure 2). Such a problem can be viewed as consisting of the two local problems in the central portion of 0 < 𝑟 ≤ 𝑏 and in the annular portion of 𝑏 ≤ 𝑟 ≤ 𝑎, which are connected by the continuity conditions at 𝑟 = 𝑏, where the problem in 0 < 𝑟 ≤ 𝑏 may be simplified as a plane tension or compression problem of membrane, while the problem in 𝑏 ≤ 𝑟 ≤ 𝑎 is still a problem of membrane deflection. The following assumptions are here made in order to reach a closed-form solution: (1) during deformation the variation in thickness of membrane could be ignored, as is seen usually in membrane problems; (2) the so-called small-rotation-angle assumption reported in the existing literatures [18–20] is still adopted here, that is, the slope angle 𝜃 of membrane is so small that the condition sin 𝜃 ≈ tan 𝜃 could approximately hold; (3) the contact between the deflected membrane and the rigid plate is frictionless. Here, we give up the assumption
of the constant radial stress reported in [23]. The continuous condition at 𝑟 = 𝑏 and the boundary conditions at 𝑟 = 𝑎 will be applied during the process of solution. As for the case prior to the contact between the deflected membrane and the frictionless rigid plate, that is, the well-known Hencky problem, it has been dealt with and its solution may be found in [12, 17]. Let us consider firstly the problem in the annular portion (𝑏 ≤ 𝑟 ≤ 𝑎) and take a piece of the central portion of the annular membrane whose radius is 𝑟; with a view of studying the static problem of equilibrium of this membrane under the joint action of the uniformly distributed loads 𝑞, the membrane force 𝜎𝑟 ℎ acted on the boundary and the reaction force from the rigid plate, just as it is shown in Figure 3, where 𝜎𝑟 is the radial stress and 𝜃 is the slope angle of membrane. Right here there are three vertical forces, that is, the total force 𝜋𝑟2 𝑞 (in which 𝑏 ≤ 𝑟 ≤ 𝑎) of the uniformly distributed loads 𝑞, the total reaction force 𝜋𝑏2 𝑞 from the rigid plate, and the total vertical force 2𝜋𝑟𝜎𝑟 ℎ sin 𝜃 which is produced by the membrane force 𝜎𝑟 ℎ. The out-plane equilibrium equation is 2𝜋𝑟𝜎𝑟 ℎ sin 𝜃 = (𝜋𝑟2 − 𝜋𝑏2 ) 𝑞,
(1)
where sin 𝜃 ≅ tan 𝜃 = −
𝑑𝑤 . 𝑑𝑟
(2)
Substituting (2) into (1), one has 2𝑟𝜎𝑟 ℎ
𝑑𝑤 = − (𝑟2 − 𝑏2 ) 𝑞. 𝑑𝑟
(3)
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3
The in-plane equilibrium equation is 𝑑 (𝑟𝜎𝑟 ℎ) − 𝜎𝑡 ℎ = 0, 𝑑𝑟
Substituting (11a) and (11b) into (4), (4)
where 𝜎𝑡 is the circumferential stress. The relations of strain and displacement of the large deflection problem are 𝑒𝑟 =
𝑑𝑢 1 𝑑𝑤 2 + ( ) , 𝑑𝑟 2 𝑑𝑟
(5a)
𝑒𝑡 =
𝑢 , 𝑟
(5b)
where 𝑒𝑟 is the radial strain, 𝑒𝑡 is the circumferential strain, and 𝑢 is the radial displacement. The relations of stress and strain are 𝐸ℎ 𝜎𝑟 ℎ = (𝑒 + ]𝑒𝑡 ) , 1 − ]2 𝑟
(6a)
𝐸ℎ 𝜎𝑡 ℎ = (𝑒 + ]𝑒𝑟 ) . 1 − ]2 𝑡
(6b)
Substituting (5a) and (5b) into (6a) and (6b), 𝜎𝑟 ℎ = 𝜎𝑡 ℎ =
𝐸ℎ 𝑑𝑢 1 𝑑𝑤 2 𝑢 [ + ( ) + ] ], 2 1 − ] 𝑑𝑟 2 𝑑𝑟 𝑟
(7a)
𝐸ℎ 𝑢 𝑑𝑢 ] 𝑑𝑤 2 [ + ] + ( ) ]. 1 − ]2 𝑟 𝑑𝑟 2 𝑑𝑟
(7b)
𝑟
𝑑 1 𝑑 2 𝐸ℎ 𝑑𝑤 2 [ (𝑟 𝜎𝑟 ℎ)] + ( ) = 0. 𝑑𝑟 𝑟 𝑑𝑟 2 𝑑𝑟
𝑑𝑢 , 𝑑𝑟 𝑢 𝑒𝑡 = . 𝑟
𝑢 = 𝑢 (𝑏)
(13b)
at 𝑟 = 𝑏,
𝑢 (𝑟) 𝑢 (𝑏) = . 𝑟 𝑏
(14)
Substituting (14) into (10a), (10b), (11a), and (11b), it may finally be found that 𝑒𝑟 = 𝑒𝑡 =
𝑢 (𝑏) , 𝑏
𝜎𝑟 = 𝜎𝑡 =
𝐸 𝑢 (𝑏) , 1−] 𝑏
in 0 ≤ 𝑟 ≤ 𝑏, in 0 ≤ 𝑟 ≤ 𝑏.
(15a) (15b)
Let us introduce the following nondimensional variables: 𝑞𝑎4 , 𝐸ℎ4 𝑤 𝑊= , ℎ 𝑆𝑟 =
(8)
𝛼=
𝐸 𝑢 𝑑𝑢 ( + ] ). 1 − ]2 𝑟 𝑑𝑟
(11b)
𝑏 , 𝑎
𝑄 (𝑥2 − 𝛼2 ) = −2𝑥𝑆𝑟 𝑥2
𝑑𝑊 , 𝑑𝑥
𝑑2 𝑆𝑟 𝑑𝑆𝑟 1 𝑑𝑊 2 + 3𝑥 + ( ) = 0, 𝑑𝑥2 𝑑𝑥 2 𝑑𝑥 𝑆𝑡 = 𝑆𝑟 + 𝑥
𝑑𝑆𝑟 , 𝑑𝑥
𝑑𝑆 𝑢 ℎ2 = [(1 − ]) 𝑆𝑟 + 𝑥 𝑟 ] . 𝑟 𝑎2 𝑑𝑥
(10b)
(11a)
(16)
and transform (3), (9), (4), and (8) into
(10a)
𝐸 𝑑𝑢 𝑢 ( 𝜎𝑟 = + ] ), 1 − ]2 𝑑𝑟 𝑟
𝜎𝑟 𝑎2 , 𝐸ℎ2
𝜎 𝑎2 𝑆𝑡 = 𝑡 2 , 𝐸ℎ 𝑟 𝑥= , 𝑎
(9)
Substituting (10a) and (10b) into (6a) and (6b),
𝜎𝑡 =
(13a)
𝑄=
The detailed derivation from (4) to (9) may be obtained from any general theory of plates and shells. It is not necessary to discuss this problem here. Obviously, in the central portion of 0 ≤ 𝑟 ≤ 𝑏, the membrane is always in a plane state of radial tensile or compression; in other words, it is in fact only a plane problem with a salient characteristic of 𝑑𝑤/𝑑𝑟 = 0. After substituting 𝑑𝑤/𝑑𝑟 = 0 into (5a) and (5b), it may be found that 𝑒𝑟 =
𝑢 = 0 at 𝑟 = 0,
the solution of (12) may be written as
If we substitute the 𝑢 of (8) into (7a), the compatibility equation may be written as 𝑟
(12)
So, considering the conditions
By means of (4), (7a), and (7b), one has 1 𝑑 1 𝑢 = (𝜎 ℎ − ]𝜎𝑟 ℎ) = [ (𝑟𝜎𝑟 ℎ) − ]𝜎𝑟 ℎ] . 𝑟 𝐸ℎ 𝑡 𝐸ℎ 𝑑𝑟
𝑑2 𝑢 𝑑𝑢 𝑢 + − = 0. 𝑑𝑟2 𝑑𝑟 𝑟
(17) (18) (19) (20)
The boundary conditions, under which (17) and (18) may be solved, are 𝑢 = 0, 𝑟
at 𝑥 = 1,
(21a)
𝑊 = 0,
at 𝑥 = 1,
(21b)
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Mathematical Problems in Engineering
(𝑆𝑟 )𝐵 = (𝑆𝑟 )𝐴 =
2
𝑢 (𝑏) 𝑎 , (1 − ]) ℎ2 𝑏
at 𝑥 = 𝛼,
(22a)
0.45
(22b)
0.4
(22c)
where the subscripts 𝐴 and 𝐵 denote the values of various variables on two sides of the interconnecting circle (𝑟 = 𝑏). The side of region (𝐴) is under the plane state of radial tensile or compression, while the side of region (𝐵) is under the deflection state of membrane. Eliminating 𝑑𝑊/𝑑𝑥 from (17) and (18), we obtain a differential equation which contains only 𝑆𝑟 : 𝑑2 𝑆 𝑥2 2𝑟 𝑑𝑥
𝑆𝑟 = 𝑄
𝑖
∑𝑐𝑖 (𝑥 − 𝛽) .
(24)
𝑖=0
∞
𝑖
𝑊 = 𝑄1/3 ∑𝑑𝑖 (𝑥 − 𝛽) ,
(25)
𝑖=0
and substituting (24) and (25) into (17), the coefficients 𝑑𝑖 (𝑖 = 1, 2, 3, . . .) can also be expressed into the polynomial of 𝑐0 and 𝑐1 , which are shown in Appendix B, while 𝑑0 is left as another undetermined constant, except 𝑐0 and 𝑐1 . With the given ] and 𝛼 = 𝑏/𝑎 all the undetermined constants can be determined by applying the continuous condition at 𝑟 = 𝑎, as follows. From (20) and (24), (21a) and (22a) give ∞
𝑖
𝑖−1
(1 − ]) ∑𝑐𝑖 (1 − 𝛽) + ∑𝑖𝑐𝑖 (1 − 𝛽) 𝑖=0
𝑄2/3
= 0,
(26)
𝑖=1
∞ ∞ ℎ2 𝑖 𝑖−1 [(1 − ]) 𝑐 (𝛼 − 𝛽) + 𝛼 𝑖𝑐𝑖 (𝛼 − 𝛽) ] ∑ ∑ 𝑖 𝑎2 𝑖=0 𝑖=1
(27)
𝑢 (𝑏) = . 𝑏 From (25), (21b) and (22b) give ∞
𝑖
𝑄1/3 ∑𝑑𝑖 (1 − 𝛽) = 0,
(28)
𝑖=0 ∞
𝑖
𝑄1/3 ∑𝑑𝑖 (𝛼 − 𝛽) = 𝑖=0
= 0.1 0.25
0.2 0.0001
0.1
0.2
𝑔 . ℎ
0.3 훼
0.4
0.5
0.6
From (24), (22c) gives ∞
∞
After substituting (24) into (23), the coefficients 𝑐𝑖 (𝑖 = 2, 3, 4, . . .) can be expressed into the polynomial of the undetermined constants 𝑐0 and 𝑐1 , which are shown in Appendix A. Further, expanding 𝑊 into the power series of 𝑥 − 𝛽,
∞
0.3
(23)
Letting 𝛽 = (1 + 𝛼)/2 and expanding 𝑆𝑟 into the power series of 𝑥 − 𝛽, one has 2/3
0.35
Figure 4: Variation of 𝑐0 with 𝛼.
2
2 2 𝑑𝑆 𝑄2 (𝑥 − 𝛼 ) + 3𝑥 𝑟 + = 0. 𝑑𝑥 8 (𝑥𝑆𝑟 )2
= 0.5
c0
𝑢 𝑢 (𝑏) 𝑢 ( ) =( ) = , at 𝑥 = 𝛼, 𝑟 𝐵 𝑟 𝐴 𝑏 𝑔 𝑊 = , at 𝑥 = 𝛼, ℎ
(29)
𝑖
𝑄2/3 ∑𝑐𝑖 (𝛼 − 𝛽) = 𝑖=0
𝑢 (𝑏) 𝑎2 . (1 − ]) ℎ2 𝑏
(30)
Eliminating the 𝑢(𝑏) from (27) and (30), one has ∞
𝑖−1
∑𝑖𝑐𝑖 (𝛼 − 𝛽)
= 0.
(31)
𝑖=1
Hence, for the given problem where 𝑎, 𝑏, ℎ, 𝑔, ], and 𝐸 are known in advance, the undetermined constants 𝑐0 and 𝑐1 can be determined by (26) and (31), then the coefficients 𝑐𝑖 (𝑖 = 2, 3, 4, . . .) and 𝑑𝑖 (𝑖 = 1, 2, 3, . . .) can also be determined, and consequently the undetermined constant 𝑑0 can be determined by (28). After this, the nondimensional variable 𝑄 can be determined by (29), that is, the applied loads q, corresponding to this 𝑏, can be determined. Thus, the problem dealt with can be solved.
3. Results and Discussions The undetermined constants 𝑐0 , 𝑐1 , and 𝑑0 depend, in fact, on only ] and 𝛼 = 𝑏/𝑎, as seen above. For the convenience of application, we here present the variation of the undetermined constants 𝑐0 , 𝑐1 , and 𝑑0 with 𝛼, as shown in Figures 4–6, where ] takes 0.1, 0.2, 0.3, 0.4, and 0.5, respectively. From the process of solution above we may see that, after giving up the assumption of the constant radial stress reported in [23], the solution to the problem dealt with becomes really complicated. Generally, the structural response is derived from an action of structure, but here the action, the applied loads 𝑞, has to be derived from the response, the contact radius 𝑏 of the membrane with the frictionless rigid plate. For verifying the reliability of the closed-form solution presented here, we made a comparison between the solution presented here and the well-known Hencky solution [12, 17], based on the numerical example of a polyethylene
Mathematical Problems in Engineering
5
−0.02 w (mm)
0
−0.04
0.6 0.9 1.2 10
= 0.5
−0.06
0.3
Frictionless rigid plate 8
6
4
2
0
2
4
6
8
10
c1
r (mm)
Figure 7: Variation of 𝑤 with 𝑟 when 𝑞 takes 2.69 KPa.
−0.08 = 0.1 −0.1
−0.12 0.0001
0.1
0.2
0.3 훼
0.4
0.5
0.6
Figure 5: Variation of 𝑐1 with 𝛼.
휎r (KPa)
26000 28000 30000 32000 34000 10
0.55
8
6
4
2
0 2 r (mm)
4
6
8
10
Figure 8: Variation of 𝜎𝑟 with 𝑟 when 𝑞 takes 2.998 KPa.
0.5 = 0.1 0.45
5
d0
0.4 0.35
4
= 0.5 b (mm)
0.3 0.25 0.2 0.0001
0.1
0.2
0.3 훼
0.4
0.5
3 2
0.6
Figure 6: Variation of 𝑑0 with 𝛼.
terephthalate thin-film (considered in [23]) with 𝑎 = 10 mm, ℎ = 0.003 mm, 𝑔 = 0.8 mm, 𝐸 = 4.65 GPa, and V = 0.34. When the contact radius 𝑏 is equal to 0.0001 mm, the action, the applied loads 𝑞, is about 2.69 KPa. Figure 7 shows the variation of 𝑤 with 𝑟, where the solid line represents the result obtained by the solution presented here, and the dashed line represents the result obtained by the well-known Hencky solution. From Figure 7 it may be seen that the two profiles 𝑤(𝑟) are very close to each other; this shows that the closed-form solution presented here is, to some extent, reliable. Further loading, when the contact radius 𝑏 is equal to 1 mm, the applied loads 𝑞 are about 2.998 KPa. Figure 8 shows the variation of the radial stress 𝜎𝑟 with the radial coordinate 𝑟, where the solid line represents the result obtained by the solution presented here, and the dashed dotted line represents the result obtained by the solution presented in [23] (the case without residual stress 𝜎0 ). This survey found a wide variation in the radial stresses 𝜎𝑟 (𝑟), which indicates that the assumption of the
1 2.12 KPa 0
0
1
2
2.69 KPa 3
4
5 6 q (KPa)
7
8
9
10
Figure 9: Variation of 𝑏 with 𝑞.
constant radial stress adopted in [23] should be given up. Figure 9 shows the variation of the contact radius 𝑏 with the applied loads 𝑞, where the solid line represents the result obtained by the solution presented here, and the dashed dotted line represents the result obtained by the solution presented in [23] (the case without residual stress 𝜎0 ). As we can see from Figure 9, the dashed dotted line shows that the deflected membrane will start touching the frictionless rigid plate at 𝑞 ≈ 2.12 KPa, while it starts touching the solid line at 𝑞 ≈ 2.69 KPa. This difference is caused mainly by the assumption of the constant radial stress adopted in [23].
4. Concluding Remarks In this study, the problem of axisymmetric deformation of a peripherally fixed circular membrane under uniformly
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Mathematical Problems in Engineering
distributed transverse loads was analytically dealt with under the condition of deflection restrictions, where the assumption of constant membrane stress adopted in [23] was given up, and the closed-form solution of this problem was presented for the first time. As far as well-known Hencky problem is concerned, the problem dealt with here can be understood as a new circular membrane problem under the combined loads, that is, the uniformly distributed transverse loads and the reaction force produced by the frictionless horizontal rigid plate. The work presented here makes a significant and new contribution to thin-film mechanics. It ought to have significant implication in the mechanical characterization of film/substrate surface and interface and the interpretation of film/substrate delamination experiment; it could be incorporated especially into the study work reported in [23]. Further studies are expected to focus on giving up the socalled small-rotation-angle assumption adopted usually in membrane problems, in order to reach a closed-form solution with higher calculation accuracy and wider application scope.
Appendix 2
Letting 𝑒 = 𝛽 − 𝛼 , then the coefficients 𝑐𝑖 (𝑖 = 2, 3, 4, . . .) may be written as 𝑐2 = − 𝑐3 =
− 1920𝛽5 𝑒3 𝑐02 𝑐1 + 816𝛽4 𝑒4 𝑐0 𝑐12 − 3456𝛽4 𝑒3 𝑐03 + 3000𝛽3 𝑒4 𝑐02 𝑐1 + 2856𝛽2 𝑒4 𝑐03 + 11𝑒6 ) 𝑐7 =
1 (507𝑒6 𝑐0 + 1315008𝑒2 𝛽6 𝑐05 𝑐12 1290240𝛽13 𝑐09
+ 42744𝑒4 𝛽2 𝑐04 − 66432𝑒3 𝛽4 𝑐04 + 30528𝑒2 𝛽6 𝑐04 − 3840𝑒𝛽8 𝑐04 + 785664𝑒2 𝛽4 𝑐07 − 949248𝑒𝛽6 𝑐07 + 324864𝛽8 𝑐07 + 292𝑒6 𝛽𝑐1 + 1310976𝑒2 𝛽5 𝑐06 𝑐1 − 77184𝑒3 𝛽5 𝑐03 𝑐1 + 18624𝑒2 𝛽7 𝑐03 𝑐1 + 61440𝛽11 𝑐04 𝑐13 − 1672704𝑒𝛽7 𝑐06 𝑐1 + 511488𝛽9 𝑐06 𝑐1 − 372𝑒5 𝛽2 𝑐0 + 40080𝑒4 𝛽4 𝑐02 𝑐12
− 1375488𝑒𝛽8 𝑐05 𝑐12 + 304128𝛽10 𝑐05 𝑐12 + 7632𝑒4 𝛽5 𝑐0 𝑐13 + 722688𝑒2 𝛽7 𝑐04 𝑐13 − 528384𝑒𝛽9 𝑐04 𝑐13 + 70584𝑒4 𝛽3 𝑐03 𝑐1
1 1 (24𝛽3 𝑐02 𝑐1 + 𝑒2 ) 16 𝛽4 𝑐02
1 1 (96𝛽3 𝑐03 𝑐1 − 4𝛽2 𝑒𝑐0 + 2𝛽𝑒2 𝑐1 + 7𝑒2 𝑐0 ) 48 𝛽5 𝑐03
𝑐4 = −
+ 113472𝛽5 𝑒2 𝑐05 𝑐1 + 88128𝛽4 𝑒2 𝑐06 + 960𝛽6 𝑒2 𝑐03
− 22656𝑒3 𝛽6 𝑐02 𝑐12 + 5160960𝛽7 𝑐09 𝑐1
A. The Coefficients 𝑐𝑖 (𝑖 = 2, 3,4,. . .) 2
− 127488𝛽7 𝑒𝑐05 𝑐1 + 81216𝛽6 𝑒2 𝑐04 𝑐12 − 99456𝛽6 𝑒𝑐06
1 1 (1920𝛽5 𝑐05 𝑐1 + 32𝛽6 𝑐03 − 64𝛽5 𝑒𝑐02 𝑐1 768 𝛽8 𝑐05
+ 203520𝑒2 𝛽8 𝑐03 𝑐14 − 76800𝑒𝛽10 𝑐03 𝑐14 + 23040𝑒2 𝛽9 𝑐02 𝑐15 ) 𝑐8 = −
1 (73𝑒8 + 14045184𝛽12 𝑐07 𝑐12 20643840𝛽16 𝑐011
+ 24𝛽4 𝑒2 𝑐0 𝑐12 − 160𝛽4 𝑒𝑐03 + 112𝛽3 𝑒2 𝑐02 𝑐1
− 39648𝑒5 𝛽4 𝑐03 + 12096𝑒4 𝛽6 𝑐03 + 1268112𝑒4 𝛽4 𝑐06
+ 188𝛽2 𝑒2 𝑐03 + 𝑒4 )
− 2297856𝑒3 𝛽6 𝑐06 + 1352448𝑒2 𝛽8 𝑐06
𝑐5 =
1 1 (11520𝛽5 𝑐06 𝑐1 + 192𝛽7 𝑐03 𝑐1 3840 𝛽9 𝑐06 6
− 288𝛽
𝑒𝑐02 𝑐12 5
− 1152𝛽
5 2
+ 96𝛽 𝑒
𝑒𝑐03 𝑐1
+
𝑐0 𝑐13
+
576𝛽4 𝑒2 𝑐02 𝑐12
− 282624𝑒𝛽10 𝑐06 + 15432192𝑒2 𝛽6 𝑐09 − 44393472𝑒𝛽9 𝑐08 𝑐1 − 19574784𝑒𝛽8 𝑐09
384𝛽6 𝑐04 4
− 1392𝛽
𝑒𝑐04
+ 1272𝛽3 𝑒2 𝑐03 𝑐1 + 1368𝛽2 𝑒2 𝑐04 − 16𝛽2 𝑒3 𝑐0 4
4
+ 11𝛽𝑒 𝑐1 + 22𝑒 𝑐0 ) 1 1 (9216𝛽10 𝑐04 𝑐12 − 12288𝛽9 𝑒𝑐03 𝑐13 𝑐6 = − 12 184320 𝛽 𝑐08
+ 7045632𝛽10 𝑐09 + 36912𝑒6 𝛽3 𝑐02 𝑐1 + 32200704𝑒2 𝛽9 𝑐06 𝑐13 + 2920128𝑒4 𝛽5 𝑐05 𝑐1 − 4179456𝑒3 𝛽7 𝑐05 𝑐1 + 1718784𝑒2 𝛽9 𝑐05 𝑐1 − 187392𝑒𝛽11 𝑐05 𝑐1 + 31984128𝑒2 𝛽7 𝑐08 𝑐1 + 15360𝛽12 𝑐06 + 15151104𝛽11 𝑐08 𝑐1
+ 3840𝛽8 𝑒2 𝑐02 𝑐14 + 645120𝛽7 𝑐08 𝑐1 + 32256𝛽9 𝑐05 𝑐1
+ 92897280𝛽9 𝑐011 𝑐1 + 10980𝑒6 𝛽4 𝑐0 𝑐12
− 66816𝛽8 𝑒𝑐04 𝑐12 + 28416𝛽7 𝑒2 𝑐03 𝑐13 + 31488𝛽8 𝑐06
− 2552832𝑒3 𝛽8 𝑐04 𝑐12 − 516096𝑒3 𝛽9 𝑐03 𝑐13
Mathematical Problems in Engineering
7
+ 2591424𝑒4 𝛽6 𝑐04 𝑐12 + 541440𝑒2 𝛽10 𝑐04 𝑐12
− 1216𝛽7 𝑒𝑐03 𝑐1 + 2104𝛽6 𝑒2 𝑐02 𝑐12 − 888𝛽5 𝑒3 𝑐0 𝑐13
+ 42056064𝑒2 𝛽8 𝑐07 𝑐12 + 30828𝑒6 𝛽2 𝑐03
− 1920𝛽4 𝑒𝑐07 − 1232𝛽6 𝑒𝑐04 + 5056𝛽5 𝑒2 𝑐03 𝑐1
− 24288𝑒5 𝛽5 𝑐02 𝑐1 − 50817024𝑒𝛽10 𝑐07 𝑐12
− 3588𝛽4 𝑒3 𝑐02 𝑐12 + 2596𝛽4 𝑒2 𝑐04 − 4416𝛽3 𝑒3 𝑐03 𝑐1
+ 1024128𝑒4 𝛽7 𝑐03 𝑐13 + 5898240𝛽13 𝑐06 𝑐13
− 1554𝛽2 𝑒3 𝑐04 + 48𝛽2 𝑒4 𝑐0 − 43𝛽𝑒5 𝑐1 − 56𝑒5 𝑐0 )
+ 150912𝑒4 𝛽8 𝑐02 𝑐14 − 30523392𝑒𝛽11 𝑐06 𝑐13
𝑑7 =
+ 14146560𝑒2 𝛽10 𝑐05 𝑐14 − 9216000𝑒𝛽12 𝑐05 𝑐14 +
921600𝛽14 𝑐05 𝑐14
−
1105920𝑒𝛽13 𝑐04 𝑐15
+
− 92160𝛽12 𝑒𝑐03 𝑐16 + 1198080𝛽12 𝑐05 𝑐14
3317760𝑒2 𝛽11 𝑐04 𝑐15 +
− 783360𝛽11 𝑒𝑐04 𝑐15 + 2856960𝛽11 𝑐06 𝑐13
322560𝑒2 𝛽12 𝑐03 𝑐16 ) . (A.1)
2
Letting 𝑒 = 𝛽 − 𝛼 , then the coefficients 𝑑𝑖 (𝑖 = 1, 2, 3, . . .) may be written as 𝑑1 = −
1 𝑒 2 𝑐0 𝛽
𝑑2 = −
1 1 (2𝛽2 𝑐0 − 𝛽𝑒𝑐1 − 𝑒𝑐0 ) 4 𝛽2 𝑐02
− 325440𝛽7 𝑒3 𝑐03 𝑐13 − 92160𝛽6 𝑒𝑐09 − 122496𝛽8 𝑒𝑐06 + 681984𝛽7 𝑒2 𝑐05 𝑐1 − 626208𝛽6 𝑒3 𝑐04 𝑐12 + 217440𝛽6 𝑒2 𝑐06 − 484608𝛽5 𝑒3 𝑐05 𝑐1
1 1 (48𝛽6 𝑐02 𝑐12 − 24𝛽5 𝑒𝑐0 𝑐13 + 96𝛽5 𝑐03 𝑐1 192 𝛽6 𝑐05
− 96𝛽
𝑒𝑐02 𝑐12
+
24𝛽4 𝑐04
2 2
3
− 108𝛽
3
𝑒𝑐03 𝑐1
2
− 24𝛽
𝑒𝑐04
3
+ 5𝛽 𝑒 𝑐0 − 4𝛽𝑒 𝑐1 − 5𝑒 𝑐0 ) 1 1 (384𝛽9 𝑐03 𝑐13 − 192𝛽8 𝑒𝑐02 𝑐14 𝑑5 = 1920 𝛽9 𝑐07 +
1344𝛽8 𝑐04 𝑐12
7
− 1056𝛽
𝑒𝑐03 𝑐13
+
1248𝛽7 𝑐05 𝑐1
− 1968𝛽6 𝑒𝑐04 𝑐12 + 192𝛽6 𝑐06 − 1344𝛽5 𝑒𝑐05 𝑐1 − 192𝛽4 𝑒𝑐06 − 40𝛽6 𝑒𝑐03 + 112𝛽5 𝑒2 𝑐02 𝑐1 4 3
− 58𝛽 𝑒
− 115968𝛽10 𝑒𝑐04 𝑐12 + 158976𝛽9 𝑒2 𝑐03 𝑐13
+ 22272𝛽10 𝑐06 − 259584𝛽9 𝑒𝑐05 𝑐1 + 600384𝛽8 𝑒2 𝑐04 𝑐12
− 40𝛽3 𝑒𝑐02 𝑐1 − 16𝛽2 𝑒𝑐03 − 𝑒3 )
4
− 3571200𝛽8 𝑒𝑐07 𝑐12 + 24576𝛽11 𝑐05 𝑐1
− 59328𝛽8 𝑒3 𝑐02 𝑐14 + 92160𝛽8 𝑐09 − 1244160𝛽7 𝑒𝑐08 𝑐1
1 1 𝑑3 = (32𝛽5 𝑐02 𝑐1 − 16𝛽4 𝑒𝑐0 𝑐12 + 16𝛽4 𝑐03 96 𝛽5 𝑐04
𝑑4 = −
− 2626560𝛽10 𝑒𝑐05 𝑐14 + 2972160𝛽10 𝑐07 𝑐12 − 4343040𝛽9 𝑒𝑐06 𝑐13 + 1198080𝛽9 𝑐08 𝑐1
B. The Coefficients 𝑑𝑖 (𝑖 = 1, 2,3,. . .) 2
1 1 (184320𝛽13 𝑐04 𝑐15 1290240 𝛽13 𝑐010
𝑐0 𝑐12
+
124𝛽4 𝑒2 𝑐03
−
152𝛽3 𝑒3 𝑐02 𝑐1
− 87𝛽2 𝑒3 𝑐03 − 𝑒5 ) 1 1 (3840𝛽10 𝑐03 𝑐14 − 1920𝛽9 𝑒𝑐02 𝑐15 𝑑6 = − 10 23040 𝛽 𝑐08
− 118656𝛽4 𝑒3 𝑐06 − 4288𝛽6 𝑒3 𝑐03 + 10336𝛽5 𝑒4 𝑐02 𝑐1 − 5416𝛽4 𝑒5 𝑐0 𝑐12 + 12608𝛽4 𝑒4 𝑐03 − 14440𝛽3 𝑒5 𝑐02 𝑐1 − 8896𝛽2 𝑒5 𝑐03 − 43𝑒7 ) 𝑑8 = −
1 (−322560𝑒𝛽6 𝑐010 5160960𝛽14 𝑐011
+ 5329152𝑒2 𝛽7 𝑐06 𝑐1 − 79216𝑒5 𝛽2 𝑐04 + 143840𝑒4 𝛽4 𝑐04 + 947𝑒6 𝛽2 𝑐0 − 1182𝑒7 𝑐0 + 14400𝑒2 𝛽8 𝑐04 − 611712𝑒3 𝛽4 𝑐07 + 1191888𝑒2 𝛽6 𝑐07 − 741120𝑒𝛽8 𝑐07 + 155904𝛽10 𝑐07 − 81664𝑒3 𝛽6 𝑐04 + 322560𝛽8 𝑐010 − 203272𝑒5 𝛽3 𝑐03 𝑐1 − 3225600𝑒𝛽12 𝑐04 𝑐16 + 248224𝑒4 𝛽5 𝑐03 𝑐1 − 70848𝑒3 𝛽7 𝑐03 𝑐1 − 3349056𝑒3 𝛽5 𝑐06 𝑐1
+ 19200𝛽9 𝑐04 𝑐13 − 13440𝛽8 𝑒𝑐03 𝑐14 + 31680𝛽8 𝑐05 𝑐12
− 322560𝑒𝛽13 𝑐03 𝑐17 + 645120𝛽14 𝑐04 𝑐16
− 35040𝛽7 𝑒𝑐04 𝑐13 + 18240𝛽7 𝑐06 𝑐1 − 40800𝛽6 𝑒𝑐05 𝑐12
− 2517504𝑒𝛽9 𝑐06 𝑐1 + 365568𝛽11 𝑐06 𝑐1
+ 1920𝛽6 𝑐07 − 19200𝛽5 𝑒𝑐06 𝑐1 + 160𝛽8 𝑐04
− 5644800𝑒𝛽7 𝑐09 𝑐1 + 5483520𝛽9 𝑐09 𝑐1
8
Mathematical Problems in Engineering − 159400𝑒5 𝛽4 𝑐02 𝑐12 + 98232𝑒4 𝛽6 𝑐02 𝑐12 − 6096192𝑒3 𝛽6 𝑐05 𝑐12 + 7421376𝑒2 𝛽8 𝑐05 𝑐12
[7]
− 28385280𝑒𝛽10 𝑐06 𝑐14 − 892𝑒7 𝛽𝑐1 + 177408𝛽12 𝑐05 𝑐12 − 20885760𝑒𝛽8 𝑐08 𝑐12 + 18144000𝛽10 𝑐08 𝑐12
[8]
− 39204𝑒5 𝛽5 𝑐0 𝑐13 − 4976064𝑒3 𝛽7 𝑐04 𝑐13 +
4110336𝑒2 𝛽9 𝑐04 𝑐13
−
[9]
665856𝑒𝛽11 𝑐04 𝑐13
− 33586560𝑒𝛽9 𝑐07 𝑐13 + 24514560𝛽11 𝑐07 𝑐13
[10]
− 1888128𝑒3 𝛽8 𝑐03 𝑐14 + 792000𝑒2 𝛽10 𝑐03 𝑐14 [11]
− 2371584𝑒𝛽10 𝑐05 𝑐12 + 16128000𝛽12 𝑐06 𝑐14 − 271872𝑒3 𝛽9 𝑐02 𝑐15 − 13224960𝑒𝛽11 𝑐05 𝑐15
[12]
+ 5160960𝛽13 𝑐05 𝑐15 ) . (B.1)
[13]
Conflicts of Interest The authors declare that they have no conflicts of interest.
Acknowledgments
[14]
[15]
This work was funded by the National Natural Science Foundation of China (Grants nos. 11772072 and 11572061). [16]
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