A Closed-Form Solution of Prestressed Annular Membrane Internally-Connected with Rigid Circular Plate and Transversely-Loaded by Central Shaft

In this paper, we analytically dealt with the usually so-called prestressed annular membrane problem, that is, the problem of axisymmetric deformation of the annular membrane with an initial in-plane tensile stress, in which the prestressed annular membrane is peripherally fixed, internally connected with a rigid circular plate, and loaded by a shaft at the center of this rigid circular plate. The prestress effect, that is, the influence of the initial stress in the undeformed membrane on the axisymmetric deformation of the membrane, was taken into account in this study by establishing the boundary condition with initial stress, while in the existing work by establishing the physical equation with initial stress. By creating an integral expression of elementary function, the governing equation of a second-order differential equation was reduced to a first-order differential equation with an undetermined integral constant. According to the three preconditions that the undetermined integral constant is less than, equal to, or greater than zero, the resulting first-order differential equation was further divided into three cases to solve, such that each case can be solved by creating a new integral expression of elementary function. Finally, a characteristic equation for determining the three preconditions was deduced in order to make the three preconditions correspond to the situation in practice. The solution presented here could be called the extended annular membrane solution since it can be regressed into the classic annular membrane solution when the initial stress is equal to zero.


Introduction
Elastic membrane structures and structural components have been widely used in many advanced fields due to the properties of lightweight, high flexibility and high toughness [1][2][3][4][5][6]. The problems of membrane structure are generally shown as large deflection problems, which makes it inevitable to produce nonlinear differential equations in solving. Generally, these nonlinear differential equations will bring serious analytical difficulties even in simple boundary-value problems [7][8][9][10][11][12][13][14][15][16][17]. Thus, the closed-form solutions of these membrane problems are usually difficult to be obtained. However, the closed-form solutions are often found to be necessary when designing membrane structures and structural components.
In the existing literature, there are some analytical solutions for circular and annular membrane problems. Hencky [18] originally studied the problem of axisymmetric deformation of the circular membrane fixed at the outer edge under the uniformly-distributed loads, and presented the power series solution of the problem. A computational error in Hencky [18] was corrected by Chien [19]  The paper is organized as follows. In Section 2, the membrane equations are firstly established, then the boundary conditions with initial stress are deduced, and finally, the resulting somewhat intractable nonlinear second-order differential equation is solved by creating the integral expressions of elementary function. In Section 3, some important issues are discussed, a numerical example is conducted, and it is proved that the solution presented here can be regressed into the classic annular membrane solution when the initial stress is equal to zero. Section 4 is the concluding remarks. Table 1 is the list of symbols used in this paper. Table 1. List of the symbols used in this paper.

Symbol Description
E, ν, θ Young's modulus, Poisson's ratio and slope angle of the annular membrane h, a, b Thickness, outer radius and inner radius of annular membrane r, x Radial coordinate with dimensions and without dimensions σ r , σ t , σ 0 Radial stress, circumferential stress and initial stress S r , S t Radial stress and circumferential stress without dimensions w, u Transversal displacement and radial displacement e r , e t Radial strain and circumferential strain P, γ Transverse load and proportional coefficient W, P 0 Transversal displacement and transverse load without dimensions B, k, R Integration constants ϕ 1 , ϕ α ϕ at x = 1 and at x = α

Establishment of Membrane Equations
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 extended a plane radial displacement u 0 , then, the extended circular membrane is fixed at the perimeter of radius a and the central part of the extended circular membrane is clamped by two rigid circular plates of radius b. An annular membrane structure with initial stress, the so-called prestressed annular membrane structure, is thus established. We study the problem of axisymmetric deformation of this annular membrane with initial stress under the action of transverse load P at the center (loaded by a shaft at the center of the circular membrane), as shown in Figure 2, where r is the radial coordinate, w is the transverse displacement and o is the origin of the coordinates. A piece of the annular membrane, whose radius is b ≤ r ≤ a, is taken with a view of studying the static problem of equilibrium of this membrane under the joint action of the load P and the membrane force σ r h acted on the boundary, just as shown in Figure 3, where σ r is the radial stress and θ is the slope angle of the deflected membrane. Right here, there are two vertical forces, i.e., the force P and the total vertical force 2πrhσ r sin θ produced by the membrane force σ r h.  The out-plane equilibrium condition is 2πrhσ r sin θ = P. (1) Considering the physical phenomenon that the slope angle θ is usually less than 15 o , the following approximate expression therefore holds Substituting Equation (2) into Equation (1), the out-plane equilibrium equation may be written as In the plane of the membrane, there are the actions of the radial membrane force σ r h and the circumferential membrane force σ t h, where σ t is the circumferential stress, the in-plane equilibrium equation is If the radial strain, circumferential strain, radial displacement and transversal displacement are denoted by e r , e t , u(r) and w(r), respectively, then there are the relations of the strain and displacement of the large deflection problem The relations of the stress and strain are Substituting Equation (6) into Equation (5), it may be obtained that From Equation (7), we can finally obtain By means of Equation (8) and Equation (4), one has If we substitute the u of Equation (9) into the first expression of Equation (8), then The detailed derivation from Equation (4) to Equation (10) may be obtained from any general theory of plates and shells. It is not necessary to discuss this problem here. Equations (3), (4) and (10) are three equations for the solutions of σ r , σ t and dw/dr. The boundary conditions, under which Equations (3), (4) and (10) may be solved, must take into account the initial stress in the initially flat circular membrane, and it may be determined based on the following analysis of the plane radial stretching problem.

Establishment of Boundary Conditions Considering Initial Stress
For the axisymmetric problem of plane radial stretching, i.e., the case where the initially flat circular membrane is extended a plane radial displacement u 0 from r = a, it is obvious that dw(r)/dr = 0. So, from Equation (5) it may be obtained that Substituting Equation (11) into Equation (6), it is found that From Equations (4) and (12), one has The boundary conditions, under which Equation (13) may be solved, are So, under the conditions of Equation (14a,b), the solution of Equation (13) may be written as Substituting Equation (15) into Equations (11) and (12), it may be obtained that where σ 0 is initial plane stress. Equation (16) indicates that for the axisymmetric plane stretching problem the radial strain e r is always equal to the circumferential strain e t at every point of the circular plane membrane, also the radial stress σ r is always equal to the circumferential stress σ t at every point of the circular plane membrane. Let us introduce the proportional coefficient γ, such that Hence, from Equations (16) and (17), the initial plane strain e 0 may be written as So, the boundary conditions, under which Equations (3), (4) and (10) may be solved, may finally be written as and

Nondimensionalization
Let us introduce the following dimensionless variables and transform Equations (10), (3) and (4) into and The boundary conditions Equation (19) can be transformed into and The boundary conditions (24a,b) may also be expressed in S r . So long as we eliminate e t from Equations (5) and (6), we may obtain u/r = (σ t − vσ r )/E. After non-dimensionalization, we may transform Equation (24a,b) into S t − vS r = e 0 , in which S t may be expressed in S r via Equation (16).
Further, eliminating dW/dx from Equations (21) and (22), we obtain an equation which contains only S r d 2 Let us substitute Z for xS r , i.e., let Substituting Equation (26) into Equation (25), we obtain a nonlinear equation From Equation (22), one has Multiplying the two sides of Equation (27) with dZ/dx, After integrating, 1 2 where B is an undetermined integration constant. We take the positive value in the square-root value, then
After integrating where k is another undetermined integration constant. Substituting Equation (20) into Equation (9), we may obtain Then, substituting Equation (26) into Equation (34) and considering Equation (32), one has Substituting Equation (33) into Equation (35), one has From Equations (27) and (33), one has Integration of Equation (37) gives where R is another undetermined integration constant. From Equations (26) and (33), one has When x = α 2 (i.e., r = b), Equation (24a) gives, from Equation (36), When x = 1 (i.e., r = a), Equation (24b) gives, from Equation (36), From Equation (38), Equation (24c) gives From Equations (40) and (41), it may be obtained that and 3 Equations (43) and (44) satisfy the form of y 3 + β 1 y + β 2 = 0, and each equation should have a real root and a pair of complex conjugate roots due to (β 2 /2) 2 + (β 1 /3) 3 > 0. We consider only the real root, such that From Equations (45) and (46), it is found that Further eliminating k from Equations (47) and (48), the important condition of B = 0 can be obtained So, for the concrete problem where the values of α, ν and γ are known in advance, if ∆ = 0 (i.e., Equation (49) holds), then this is the case of B = 0, and the value of the undetermined integration constant k can be determined by Equation (47) or Equation (48) with the known α, ν and γ (whose values must satisfy Equation (49), thus satisfying Equations (47) and (48)). The value of R may be determined by Equation (42) with the known k, and the deflection and stresses within the annular membrane can thus be determined by Equations (38), (39) and (23). Moreover, from Equations (38), (42), (45) and (46), the maximum deflection at the inner edge (x = α 2 ) may be written as Let us introduce the new variable ϕ, such that From Equation (30), we may see that the variable transformation is valid since Z ≤ 1/B while B > 0. Substituting Equation (51) into Equation (31), one has Integration of Equation (52) gives where k is another undetermined integration constant. Substituting Equation (20) into Equation (9) and eliminating xS r with the help of Equation (26), one has Making use of Equations (31), (51) and (53), Equation (54) may be simplified as From Equations (28), (51) and (52), one has Integration of Equation (56) gives where R is another undetermined integration constant. From Equations (26), (51) and (53), one has If we call ϕ at x = 1 as ϕ 1 and call ϕ at x = α as ϕ α , then from Equation (53) it may be obtained that and From Equations (55) and (59a), Equation (24b) gives From Equations (61) and (62), one has From Equation (59a,b), one has From Equations (59a) and (64), one has From Equations (63) and (64), one has From Equations (61), (62) and (64), one has Hence, for the concrete problem where the values of α, ν and γ are known in advance, using Equation (67) we may calculate the numerical value of ϕ α with a given ϕ 1 and the known γ and α. With this obtained value of ϕ α , we may further calculate the numerical values of B, k and v via Equations (64)-(66), respectively. If the numerical value of ν, obtained in this calculation, is just equal to its known value, then the corresponding numerical values of B and k are just the solution of the problem, otherwise, try another given value of ϕ 1 and continue the numerical calculation until the obtained numerical value of ν is just equal to its known value. As soon as B, R and k are determined, the displacement and stresses within the annular membrane can thus be calculated. Equation (53) is the condition for the determination of x from ϕ with the known values of k and B. All the calculations of numerical values can easily be finished with the help of a Microsoft Excel spreadsheet. Moreover, From Equations (57), (60) and (64), the maximum deflection at the inner edge (that is at x = α 2 and ϕ = ϕ α ) may be written as Integration of Equation (71) gives where k is the undetermined integration constant. Substituting Equation (20) into Equation (9) and eliminating xS r with the help of Equation (26), one has Making use of Equations (69), (70) and (72), Equation (73) may be simplified as From Equations (28), (70) and (71), one has Integration of Equation (75) gives where R is the undetermined integration constant. From Equations (26), (70) and (72), one has If we call ϕ at x = 1 as ϕ 1 and ϕ at x = α as ϕ α , then from Equation (72) it may be obtained that and From Equations (74) and (78a), Equation (24b) gives From Equations (80) and (81), one has (1 + ν) From Equation (78a,b), one has (1 − α 2 )(α 2 cot 2 ϕ 1 − cot 2 ϕ α ) − 1.
From Equations (80)-(82), one has 2Φ 3 −2 cot 2 ϕ 1 cos ϕ 1 −Φ 2 γ cos ϕ 1 2 cot 2 ϕ 1 cos ϕ 1 −Φ 2 γ cos ϕ 1 = 2α 2 Φ 3 −2 cot 2 ϕ α cos ϕ α −α 2 Φ 2 γ cos ϕ α 2 cot 2 ϕ α cos ϕ α −α 2 Φ 2 γ cos ϕ α Hence, for the concrete problem where the values of γ, ν and α are known in advance, using Equation (86) we may calculate the numerical value of ϕ α with a given ϕ 1 and the known γ and α. With this obtained value of ϕ α , we may further calculate the numerical values of R, B, k and v via Equations (79), (83)-(85), respectively. If the numerical value of ν, obtained in this calculation, is just equal to its known value, then the corresponding numerical values of R, B and k, are just the solution of the problem, otherwise, try another given value of ϕ 1 and continue the numerical calculation until the obtained numerical value of ν is just equal to its known value. As soon as R, B and k are determined, the displacement and stress within the annular membrane can be calculated. Equation (72) Thus, the problem of axisymmetric deformation of the so-called prestressed annular membrane can be solved.

Results and Discussions
Since Equation (31) was solved under the preconditions of B = 0, B > 0 and B < 0, respectively, then for solving the concrete problem where the values of γ, ν and α are known in advance, the conditions of B = 0, B > 0 and B < 0 should be firstly specified, otherwise, we still don't know how to use the solutions presented above. Based on a large number of numerical calculations we finally find that ∆ = 0, ∆ > 0 and ∆ < 0 corresponds to B = 0, B > 0 and B < 0, respectively, where ∆ was presented in Equation (49).

Comparison with Existing Work
An obvious difference between the annular membrane problem without initial stress and the one with initial stress (i.e., the difference between the classical problem and the problem dealt with here) is that, in the classical problem (without initial stress) the conditions of B = 0, B > 0 and B < 0 depends only on the Poisson's ratio ν (i.e., ν = 1/3, ν < 1/3 and ν > 1/3 correspond to B = 0, B > 0 and B < 0, respectively, see Figure 7 in reference [22]), while in the problem dealt with here (with initial stress) it depends on not only ν and α but also σ 0 and P (σ 0 and P are introduced by the proportional coefficient γ, see Equations (17) and (49) in this paper). When γ = 0 (corresponding to σ 0 = 0), however, Equation (49) can be regressed into Equation (34) in reference [22]. Moreover, in the classical problem the important integral constant B is determined by α and ν (see Equations (52) and (70) in reference [22]), but in the problem dealt with here it is determined by α, ν and γ (see Equations (62) and (81) in this paper). When γ = 0 (corresponding to σ 0 = 0), however, Equations (62) and (81) can be regressed into Equations (52) and (70) in reference [22], respectively.
From the derivation above it may easily seen that only the boundary conditions (see Equation (24)) were modified in comparison with the boundary conditions in reference [22]. All the expressions obtained here for displacements, strains and stresses have the same form as the expressions obtained in reference [22]. However, the initial stress σ 0 plays an important role in the determination of numerical values of the undetermined integral constants. If σ 0 = 0, however, Equation (24) can be regressed into Equation (14) in reference [22], and consequently all the expressions obtained here can be regressed into the corresponding expressions in reference [22]. This means that the solution obtained here can be regressed into the classic annular membrane solution when the initial stress is equal to zero. Therefore, the solution presented here could be called extended annular membrane solution.

Numerical Example
The following example shows the difference between the deflection curves of the same annular polymer thin-film without and with initial stress (under the same transverse load). The outer radius of the annular polymer thin-film is a = 10 mm, the inner radius is b = 1 mm, the thickness is h = 60 µm, the elastic modulus is E = 1100 MPa, the Poisson's ratio is ν = 0.4, the transverse load is P = 1 N, and the yield stress of the polymer thin-film is found to be σ y = 20 MPa. Suppose that the initial stress is σ 0 = 0 MPa and σ 0 = 5 MPa, respectively. Here α = b/a = 0.1. For the case of σ 0 = 5 MPa, from Equation (17) it may be obtained that γ = 2.3465141593 and hence Equation (49) gives ∆ < 0. Consequently, the case of σ 0 = 5 MPa corresponds to B < 0, and also the case of σ 0 = 0 MPa corresponds to B < 0 due to ν = 0.4 (see Figure 7 in reference [22]). So, the problems considered here should be approached in B < 0, i.e., the expressions obtained in the case of B < 0 should be adopted. All the numerical values of B, k, R, ϕ α and ϕ 1 have been calculated and are listed in Table 2. The maximum stress of the thin-film with σ 0 = 5 MPa, which is at r = b = 1 mm, is calculated to be about σ m = 15.86 MPa under P = 1 N. So, the thin-film is in the range of elastic deformation due to σ m < σ y = 20 MPa. A graphical representation of deflection results is shown in Figure 4, where the solid line corresponds to σ 0 = 5 MPa and the dashed line to σ 0 = 0 MPa (the classic annular membrane problem). From Figure 4 we can see that the initial stress has a large influence on the mechanical behavior of the annular membrane, and we may imagine, such an influence will increase as the initial stress increases.

Concluding Remarks
In this paper, the problem of axisymmetric deformation of a prestressed annular membrane internally-connected with a rigid circular plate and transversely-loaded by a central shaft was analytically dealt with. The prestress effect, that is, the influence of the initial stress in the undeformed membrane on the axisymmetric deformation of the membrane, was taken into account by establishing the boundary condition with initial stress, rather than by establishing the physical equation with initial stress, as done in the existing work. The numerical example shows that the initial stress has a large influence on the mechanical behavior of the membrane. The closed-form solution presented here is given in the form of elementary function, which is relatively rare in solving nonlinear differential equations. So, in this sense, the work presented here has positive significance to the mathematical modeling of mechanical problems, especially to studies such as the thin-film/substrate or film/film delamination and shape finding of building film structures.