# Closed-Form Solution of a Peripherally Fixed Circular Membrane under Uniformly Distributed Transverse Loads and Deflection Restrictions

Authors: Teng-fei Wang; Xiao-ting He; Yang-hui Li
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

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.


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## Body

## 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 boundary-value 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 so-called 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.Figure 1
Sketch of the well-known Hencky problem.Figure 2
Sketch of confined deformation of the uniformly loaded circular membrane.In the next section, the problem shown in Figure2 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 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 elasticityE, 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 σrh 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 πr2q (in which b≤r≤a) of the uniformly distributed loads q, the total reaction force πb2q from the rigid plate, and the total vertical force 2πrσrhsinθ which is produced by the membrane force σrh.Figure 3
The equilibrium diagram of the central portion (b≤r≤a) of the circular membrane.The out-plane equilibrium equation is(1)2πrσrhsinθ=πr2-πb2q,where(2)sin⁡θ≅tan⁡θ=-dwdr.Substituting (2) into (1), one has(3)2rσrhdwdr=-r2-b2q.The in-plane equilibrium equation is(4)ddrrσrh-σth=0,where σt is the circumferential stress. The relations of strain and displacement of the large deflection problem are(5a)er=dudr+12dwdr2,(5b)et=ur,where er is the radial strain, et is the circumferential strain, and u is the radial displacement. The relations of stress and strain are(6a)σrh=Eh1-ν2er+νet,(6b)σth=Eh1-ν2et+νer.Substituting (5a) and (5b) into (6a) and (6b),(7a)σrh=Eh1-ν2dudr+12dwdr2+νur,(7b)σth=Eh1-ν2ur+νdudr+ν2dwdr2.By means of (4), (7a), and (7b), one has(8)ur=1Ehσth-νσrh=1Ehddrrσrh-νσrh.If we substitute the u of (8) into (7a), the compatibility equation may be written as(9)rddr1rddrr2σrh+Eh2dwdr2=0.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 of0≤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(10a)er=dudr,(10b)et=ur.Substituting (10a) and (10b) into (6a) and (6b),(11a)σr=E1-ν2dudr+νur,(11b)σt=E1-ν2ur+νdudr.Substituting (11a) and (11b) into (4),(12)rd2udr2+dudr-ur=0.So, considering the conditions(13a)u=0atr=0,(13b)u=ubatr=b,the solution of (12) may be written as(14)urr=ubb.Substituting (14) into (10a), (10b), (11a), and (11b), it may finally be found that(15a)er=et=ubb,in0≤r≤b,(15b)σr=σt=E1-νubb,in0≤r≤b.Let us introduce the following nondimensional variables:(16)Q=qa4Eh4,W=wh,Sr=σra2Eh2,St=σta2Eh2,x=ra,α=ba,and transform (3), (9), (4), and (8) into(17)Qx2-α2=-2xSrdWdx,(18)x2d2Srdx2+3xdSrdx+12dWdx2=0,(19)St=Sr+xdSrdx,(20)ur=h2a21-νSr+xdSrdx.The boundary conditions, under which (17) and (18) may be solved, are(21a)ur=0,atx=1,(21b)W=0,atx=1,(22a)urB=urA=ubb,atx=α,(22b)W=gh,atx=α,(22c)SrB=SrA=a21-νh2ubb,atx=α,where the subscriptsA 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:(23)x2d2Srdx2+3xdSrdx+Q28x2-α22xSr2=0.Letting β=1+α/2 and expanding Sr into the power series of x-β, one has(24)Sr=Q2/3∑i=0∞cix-βi.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,(25)W=Q1/3∑i=0∞dix-βi,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 ν 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(26)1-ν∑i=0∞ci1-βi+∑i=1∞ici1-βi-1=0,(27)Q2/3h2a21-ν∑i=0∞ciα-βi+α∑i=1∞iciα-βi-1=ubb.From (25), (21b) and (22b) give(28)Q1/3∑i=0∞di1-βi=0,(29)Q1/3∑i=0∞diα-βi=gh.From (24), (22c) gives(30)Q2/3∑i=0∞ciα-βi=a21-νh2ubb.Eliminating the ub from (27) and (30), one has(31)∑i=1∞iciα-βi-1=0.Hence, for the given problem where a, b, h, g, ν, 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 loadsq, corresponding to this b, can be determined. Thus, the problem dealt with can be solved.


## 3. Results and Discussions

The undetermined constantsc0, c1, and d0 depend, in fact, on only ν 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 ν takes 0.1, 0.2, 0.3, 0.4, and 0.5, respectively.Figure 4
Variation ofc0 with α.Figure 5
Variation ofc1 with α.Figure 6
Variation ofd0 with α.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 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 wr are very close to each other; this shows that the closed-form solution presented here is, to some extent, reliable.Figure 7
Variation ofw with r when q takes 2.69 KPa.Further loading, when the contact radiusb 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 σrr, which indicates that the assumption of the constant radial stress adopted in [23] should be given up.Figure 8
Variation ofσr with r when q takes 2.998 KPa.Figure9 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].Figure 9
Variation ofb with q.


## 4. Concluding Remarks

In this study, the problem of axisymmetric deformation of a peripherally fixed circular membrane under uniformly 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 so-called 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.



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