Elastic Contact Between a Transversely, Uniformly Loaded Circular Membrane and a Spring-Reset Rigid Flat Circular Plate: An Improved Closed-Form Solution

Abstract

The closed-form solution of the problem regarding elastic contact between a transversely, uniformly loaded circular membrane and a spring-reset rigid flat circular plate has potential application value in sensor developments or bending-free shell designs, but it still needs to be further improved. In this paper, on the basis of existing studies, the plate/membrane elastic contact problem is reformulated by improving the system of differential equations governing the elastic behavior of a large deflection of a circular membrane. Specifically, the radial geometric equation used in the existing studies is improved by giving up the assumption of a small rotation angle for the membrane, and an improved closed-form solution to the plate/membrane elastic contact problem is presented. The convergence and validity of the improved closed-form solution are analyzed, and the difference between the closed-form solutions before and after improvement is graphically shown. In addition, the effect of changing some important geometric and physical parameters on the improved closed-form solution is investigated.

1. Introduction

A membrane, in mechanics, refers to a fully stretched plate or a plate with vanishing bending stiffness, regardless of the thickness of the plate; that is, it could be either a thick plate or a thin plate [1,2,3,4,5]. They are usually used in various fields to form a complete structure or a structural component that bears lateral loads [6,7,8,9,10] and are often accompanied by a large deflection if their thickness is very thin [11,12,13,14,15]. The large deflection tends to give rise to geometric nonlinearity, which often makes the membrane problem present serious analytical difficulties [16,17,18,19,20]. However, analytical solutions of membrane problems are often necessary for many engineering or technical applications, especially those with good precision [21,22,23,24,25].

The circular membrane problem, i.e., the problem of axisymmetric deformation of an initially flat, peripherally fixed, transversely uniformly loaded circular membrane, was analytically dealt with originally by the famous German scientist Hencky [26]. A computational error in the analytical solution presented by Hencky in [26] was corrected by Chien [27] and Alekseev [28], respectively. The well-known Hencky solution is often cited by related studies [29,30,31,32,33]. However, it was derived based on the assumption of a small rotation angle of the membrane; therefore, it is suitable only for membrane applications with relatively small elastic deflections [34,35,36,37,38]. In the last decade, the well-known Hencky solution was first extended from the special case without initial stress to the general case with initial stress [39], and then, to accommodate the membrane applications with relatively large elastic deflections, it was improved three times by giving up the assumption of a small rotation angle of a membrane [40,41,42].

If the initially flat, peripherally fixed circular membrane keeps an initial parallel gap from a rigid flat circular plate, and under the action of uniformly distributed transverse loads, deflects toward the rigid flat plate, then when the maximum membrane deflection is equal to the parallel gap, the deflected circular membrane will just come into contact with the rigid flat circular plate. This is the so-called contact problem between the rigid flat circular plate and the deflected circular membrane, or the plate/membrane contact problem for short. In more detail, if the rigid flat circular plate is fixed, that is, it does not move when pushed by the deflected circular membrane, then the plate/membrane contact problem is called the plate/membrane rigid contact problem, and if the rigid flat circular plate is connected to one end of a spring whose other end is fixed, that is, the rigid flat circular plate is reset by the spring when pushed by the deflected circular membrane, then the plate/membrane contact problem is called the plate/membrane elastic contact problem. The analytical solution of the plate/membrane rigid contact problem can be used to develop the capacitive pressure sensors that use the non-parallel plate variable capacitors [43,44,45,46,47], while the analytical solution of the plate/membrane elastic contact problem can be used to develop the capacitive pressure sensors that use the parallel plate variable capacitors [48,49]. In comparison with the non-parallel plate variable capacitors, the parallel plate variable capacitors have at least one clear advantage, that is, the edge effect of the capacitors can be more easily reduced. So, the improvement to the analytical solution of the plate/membrane elastic contact problem has potential application value and significance.

In mathematics, the elastic behavior of membranes is usually governed by equilibrium equations, geometric equations, and physical equations. The plate/membrane elastic contact problem was first proposed in [48], but it is formulated using the equilibrium and geometric equations that are derived based on the assumption of a small rotation angle of a membrane. More unfortunately, however, there is an error in the boundary and continuous conditions in [48], i.e., Equation (31) in [48] is wrong. Therefore, due to such an error, the analytical solution of the plate/membrane elastic contact problem, presented in [48], is actually a wrong solution. This error, i.e., Equation (31) in [48], was corrected in [49]; see Equation (A27) in [49] for details. Moreover, in [49], a more precise out-of-plane equilibrium equation was established by giving up the assumption of a small rotation angle of a membrane; the correct analytical solution of the plate/membrane elastic contact problem was, for the first time, derived, and finally, based on the derived analytical solution, the numerical design and calibration of a parallel plate variable capacitor-based circular capacitive wind pressure sensor were presented.

In this paper, the plate/membrane elastic contact problem is reformulated and analytically solved, and an improved analytical solution of the problem is presented. In comparison with the recent work presented in [49], the innovation of this study mainly lies in the use of the new membrane governing equations, thereby improving the analytical solution of the elastic contact problem. Specifically, the radial geometric equation used in this study is independent of the assumption of a small rotation angle of a membrane (see the following section for details). In Section 3, some important issues are discussed, such as the convergence and validity of the analytical solution derived in Section 2, and the effect of changing some important geometrical and physical parameters on the analytical solution. Concluding remarks are shown in Section 4.

2. Analytical Solution to the Elastic Contact Problem

The so-called plate/membrane elastic contact problem is detailed as follows. An initially flat circular membrane with radius a, thickness h, Poisson’s ratio v, and Young’s modulus of elasticity E is peripherally fixed, that is, the radial, circumferential, and transverse displacements at the outer edge of the initially flat circular membrane are all constrained rigidly. A rigid flat circular plate with radius a keeps an initial parallel gap g from the initially flat circular membrane and is connected to one end of a spring with original length L and stiffness coefficient k, and the other end of the spring is fixed, as shown in Figure 1a. A uniformly distributed transverse load q is applied onto the circular membrane, resulting in the membrane deflection (transverse displacement) towards the rigid flat circular plate, as shown in Figure 1b, where the initial compressed length ∆l of the spring is considered for the convenience of practical application (for example, ∆l is caused by the self-weight of the rigid flat circular plate). In Figure 1b, the dash-dotted line represents the geometric middle plane of the initially flat circular membrane, the dashed lines represent the initial position of the rigid flat circular plate (i.e., the position before the rigid flat circular plate is moved by ∆l), r, φ, w, and o denote the radial, circumferential, transverse coordinates, and coordinate origin of the introduced cylindrical coordinate system (r, φ, w) (where the polar plane (r, φ) is located in the plane in which the geometric middle plane is located, the coordinate origin o is at the centroid of the geometric middle plane, and the deflection of the circular membrane is also denoted by w, but φ is not shown due to the axisymmetric deformation), and w_m denotes the maximum deflection of the circular membrane under the uniformly distributed transverse load q. When the maximum deflection w_m of the circular membrane under the uniformly distributed transverse load q is equal to the sum of the initial parallel gap g and the initial compressed length ∆l, as shown in Figure 1c, then the circular membrane will just come into contact with the rigid flat circular plate. As the load q intensifies, the deflected circular membrane will push the rigid flat circular plate to move, resulting in a further compression of the spring, especially forming a contact radius b between the deflected circular membrane and the rigid flat circular plate, as shown in Figure 1d, where the second dashed lines represent the position after the rigid flat circular plate is moved by ∆l. It should be stated in particular that only the case of frictionless plate/membrane contact is considered here; that is, the friction force on the contact interface between the deflected circular membrane and the rigid flat circular plate is neglected. Obviously, if q = 0 and ∆l = 0, the rigid flat circular plate can be reset to its initial position by the spring, so it can be called a spring-reset plate. The situation shown in Figure 1d is the so-called plate/membrane elastic contact problem, where its limit state of b → 0 is the situation shown in Figure 1c, and its limit state of k → ∞ is the so-called plate/membrane rigid contact problem. It should be noted here that the direction of the uniformly distributed transverse load q is assumed to be always perpendicular to the polar plane (r, φ) in which the geometric middle plane of the initially flat circular membrane is located, and it is always independent of the deformation of the circular membrane, as shown in Figure 1b–d.

The plate/membrane interaction force between the rigid flat circular plate moved by (w_m − g) and the circular membrane under a uniformly distributed transverse load q can be analyzed, as shown in Figure 2, where q′ represents the plate/membrane interaction force per unit area. Obviously, since the circular membrane in r ≤ b is parallel to the polar plane (r, φ), the rotation angle of the membrane (it is assumed to be denoted by θ) at r = b is equal to zero. Therefore, the equilibrium condition of the upward and downward forces in the region of r ≤ b is πb² q′ = πb² q, that is, q′ = q. In addition, the equilibrium condition of the upward and downward forces acting on the rigid flat circular plate moved by (w_m − g) is πb² q′ = k(w_m − g − ∆l).

Now, let us take a free body with radius r (b ≤ r ≤ a) from the central portion of the circular membrane that is in contact with the rigid flat circular plate moved by (w_m − g), as shown in Figure 3, where σ_r denotes the radial stress, σ_rh is the membrane force acting on the boundary of the radius r, and θ denotes the meridional rotation angle of the membrane at radius r.

In Figure 3, the static equilibrium of the free body subjected to the uniformly distributed transverse loads q is governed in the vertical direction perpendicular to the polar plane (r, φ) by

$$2\pi r{\sigma }_{r}h\sin \theta -\pi r^{2}q+\pi b^2{q}^{\prime }=0,$$

(1)

where b ≤ r ≤ a. After considering the condition of q′ = q, Equation (1) can be reduced to

$$2\pi r{\sigma }_{r}h\sin \theta -\pi (r^2-b^2)q=0,$$

(2)

where

$$\sin \theta =1/\sqrt{1+1/{\tan }^2\theta }=1/\sqrt{1+1/{(-\mathrm{d}w/\mathrm{d}r)}^2}.$$

(3)

After substituting Equation (3) into Equation (2), the out-of-plane equilibrium equation can be written as

$$2r{\sigma }_{r}h-(r^2-b^2)q\sqrt{1+1/{(-\mathrm{d}w/\mathrm{d}r)}^2}=0$$

(4)

If the circumferential stress is denoted by σ_t, the in-plane equilibrium equation can be written as [48,49]

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}r}}(r{\sigma }_r)-{\sigma }_t=0.$$

(5)

The so-called geometric equations refer to the mathematical equations that describe the relationships between strains and displacements. The radial geometric equation used in [48,49] is derived based on the assumption of a small rotation angle of a membrane. In fact, it can also be derived without using the assumption of a small rotation angle of a membrane, which is detailed as follows. Suppose that the radial and circumferential strains are denoted, respectively, by e_r and e_t, and the radial and transverse displacements are denoted, respectively, by u and w. A radial micro straight-line element $\overline{A^{\prime }{B}^{\prime }}$ with length ∆r is assumed to be located initially in the polar plane (r, φ), and after deformation, it becomes a meridional micro curve element $\stackrel{⏜}{AB}$, as shown in Figure 4, where the radial coordinate of point A′ is r, the radial displacement from point A′ to the point A along the radial direction is equal to u (i.e., the radial displacement of point A′ is u(r)), and the transverse displacement from point A′ to point A along the transverse direction is equal to w (i.e., the transverse displacement of point A′ is w(r)). Therefore, the radial and transverse coordinates of point A are r + u and w, respectively. Let us expand the radial displacement u(r + ∆r) and transverse displacement w(r + ∆r) of point B′ into the Taylor series and ignore the higher-order terms therein

$$u(r+\Delta r)=u(r)+{\displaystyle \frac{\mathrm{d}u(r)}{\mathrm{d}r}}\Delta r+{\displaystyle \frac{1}{2!}}{\displaystyle \frac{{\mathrm{d}}^{2}u(r)}{\mathrm{d}{r}^2}}{(\Delta r)}^2+\cdots \cong u(r)+{\displaystyle \frac{\mathrm{d}u(r)}{\mathrm{d}r}}\Delta r$$

(6)

and

$$w(r+\Delta r)=w(r)+{\displaystyle \frac{\mathrm{d}w(r)}{\mathrm{d}r}}\Delta r+{\displaystyle \frac{1}{2!}}{\displaystyle \frac{{\mathrm{d}}^{2}w(r)}{\mathrm{d}{r}^2}}{(\Delta r)}^2+\cdots \cong w(r)+{\displaystyle \frac{\mathrm{d}w(r)}{\mathrm{d}r}}\Delta r.$$

(7)

If the radial displacement u(r) and transverse displacement w(r) of point A′ are abbreviated as u and w, then the radial and transverse coordinates of point B′ can be written as u + ∆r du/dr and w + ∆r dw/dr, respectively, as shown in Figure 4.

Figure 4. Sketch of the radial geometric relationship between the micro straight-line element $\overline{A^{\prime }{B}^{\prime }}$ and the micro curve element $\stackrel{⏜}{AB}$.

Figure 4. Sketch of the radial geometric relationship between the micro straight-line element $\overline{A^{\prime }{B}^{\prime }}$ and the micro curve element $\stackrel{⏜}{AB}$.

It can be found from Figure 4 that the length of the straight line $\overline{A^{\prime }{B}^{\prime }}$ is $L_{\overline{A^{\prime }{B}^{\prime }}}=\Delta r$, while the length of the curve $\stackrel{⏜}{AB}$ may be given by

$$L_{\stackrel{⏜}{AB}}\cong \sqrt{{(\Delta r+{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}\Delta r)}^2+{(-{\displaystyle \frac{\mathrm{d}w}{\mathrm{d}r}}\Delta r)}^2}.$$

(8)

Therefore, according to the definition of linear strain, the radial line strain for the radial deformation from the micro straight-line element $\overline{A^{\prime }{B}^{\prime }}$ to the micro curve element $\stackrel{⏜}{AB}$ may be written as

$$e_r=\underset{\Delta r\to 0}{\lim }{\displaystyle \frac{L_{\stackrel{⏜}{AB}}-L_{\overline{A^{\prime }{B}^{\prime }}}}{L_{\overline{A^{\prime }{B}^{\prime }}}}}=\underset{\Delta r\to 0}{\lim }{\displaystyle \frac{\sqrt{{(\Delta r+\frac{\mathrm{d}u}{\mathrm{d}r}\Delta r)}^2+{(-\frac{\mathrm{d}w}{\mathrm{d}r}\Delta r)}^2}-\Delta r}{\Delta r}}.$$

(9)

After eliminating ∆r from Equation (9), the radial geometric equation, the relationship between the radial strain and the radial and transverse displacements, may be written as

$$e_r=\sqrt{{(1+{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}})}^2+{(-{\displaystyle \frac{\mathrm{d}w}{\mathrm{d}r}})}^2}-1.$$

(10)

It can be seen from the above derivation of Equation (10) that the new radial geometric equation, Equation (10), is independent of the assumption of a small rotation angle of a membrane.

Since the circumferential geometric equation used in [48,49] is independent of the assumption of a small rotation angle of a membrane, it is still used here, given by

$$e_t={\displaystyle \frac{u}{r}}.$$

(11)

The relationships between the stresses and strains of the membrane material used are assumed to follow Hooke’s law

$${\sigma }_r={\displaystyle \frac{E}{1-{\nu }^2}}(e_r+\nu e_t)$$

(12)

and

$${\sigma }_t={\displaystyle \frac{E}{1-{\nu }^2}}(e_t+\nu e_r),$$

(13)

where E is the Young’s modulus of elasticity of the used membrane material.

After eliminating e_r and e_t from Equations (10)–(13), it is found that

$${\sigma }_r={\displaystyle \frac{E}{1-{\nu }^2}}[\sqrt{{(1+{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}})}^2+{(-{\displaystyle \frac{\mathrm{d}w}{\mathrm{d}r}})}^2}-1+\nu {\displaystyle \frac{u}{r}}]$$

(14)

and

$${\sigma }_t={\displaystyle \frac{E}{1-{\nu }^2}}[{\displaystyle \frac{u}{r}}+\nu \sqrt{{(1+{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}})}^2+{(-{\displaystyle \frac{\mathrm{d}w}{\mathrm{d}r}})}^2}-\nu ].$$

(15)

By means of Equations (5), (14) and (15), one has

$${\displaystyle \frac{u}{r}}={\displaystyle \frac{1}{E}}({\sigma }_t-\nu {\sigma }_r)={\displaystyle \frac{1}{E}}[{\displaystyle \frac{\mathrm{d}}{\mathrm{d}r}}(r{\sigma }_r)-\nu {\sigma }_r].$$

(16)

Substituting the u in Equation (16) into Equation (14) yields

$${[{\displaystyle \frac{1}{{\rm E}}}{\sigma }_r-{\displaystyle \frac{\nu }{{\rm E}}}{\displaystyle \frac{d}{dr}}(r{\sigma }_r)+1]}^2-{\left\{{\displaystyle \frac{1}{{\rm E}}}{\displaystyle \frac{d}{dr}}[r{\displaystyle \frac{d}{dr}}(r{\sigma }_r)]-{\displaystyle \frac{\nu }{{\rm E}}}{\displaystyle \frac{d(r{\sigma }_r)}{dr}}+1\right\}}^2-{(-{\displaystyle \frac{dw}{dr}})}^2=0.$$

(17)

Therefore, the radial stress σ_r and transverse displacement w in the plate/membrane non-contact region of b ≤ r ≤ a can be determined by simultaneously solving Equations (4) and (17).

On the other hand, since dw/dr = 0 in the plate/membrane contact region of 0 ≤ r ≤ b, Equations (10) and (11) can be reduced to

$$e_r={\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}$$

(18)

and

$$e_t={\displaystyle \frac{u}{r}}.$$

(19)

Similarly, using dw/dr = 0 in Equations (14) and (15) can be reduced to

$${\sigma }_r={\displaystyle \frac{E}{1-{\nu }^2}}({\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}+\nu {\displaystyle \frac{u}{r}}).$$

(20)

and

$${\sigma }_t={\displaystyle \frac{E}{1-{\nu }^2}}({\displaystyle \frac{u}{r}}+\nu {\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}).$$

(21)

Substituting Equations (20) and (21) into Equation (5) yields

$$r{\displaystyle \frac{{\mathrm{d}}^{2}u}{\mathrm{d}{r}^2}}+{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}-{\displaystyle \frac{u}{r}}=0.$$

(22)

The general solution of the Euler equation for Equation (22) may be written as

$$u=C_{1}r+C_2{\displaystyle \frac{1}{r}},$$

(23)

where C₁ and C₂ are two undetermined constants. Obviously, C₂ ≡ 0, since u = 0 at r = 0. If the radial displacement u at r = b is denoted by u(b), then C₁ = u(b)/b due to C₂ ≡ 0; see Equation (23). This results in

$$u(r)={\displaystyle \frac{u(b)}{b}}r,$$

(24)

where both b and u(b) need to be further determined. When substituting Equation (24) into Equations (18)–(21), then

$$e_r=e_t={\displaystyle \frac{u(b)}{b}}$$

(25)

and

$${\sigma }_r={\sigma }_t={\displaystyle \frac{E}{1-\nu }}{\displaystyle \frac{u(b)}{b}}.$$

(26)

Equations (25) and (26) indicate that in the plate/membrane contact region of 0 ≤ r ≤ b, both strains and stresses are uniformly distributed, and the radial strain (or stress) is always equal to the circumferential strain (or stress).

Obviously, on the two sides of the inter-connecting circle of r = b, the stresses, strains, and displacements in the circular membrane are continuous at the inter-connecting circle of r = b. Therefore, the boundary conditions and continuous conditions, under which the frictionless elastic contact problem under consideration can be solved, are

$$w=0 \mathrm{at} r=a,$$

(27)

$$e_t={\displaystyle \frac{u}{r}}={\displaystyle \frac{1}{E}}[{\displaystyle \frac{\mathrm{d}}{\mathrm{d}r}}(r{\sigma }_r)-\nu {\sigma }_r]=0 \mathrm{at} r=a,$$

(28)

$${(e_t)}_A={(e_t)}_B={\displaystyle \frac{u(b)}{b}} \mathrm{at} r=b$$

(29)

and

$${({\sigma }_r)}_A={({\sigma }_r)}_B={\displaystyle \frac{E}{1-\nu }}{\displaystyle \frac{u(b)}{b}}\mathrm{at} r=b,$$

(30)

where the subscript A refers to the plate/membrane non-contact region of b ≤ r ≤ a, and the subscript B refers to the plate/membrane contact region of 0 ≤ r ≤ b. In addition, the equilibrium conditions in Figure 2 or Figure 3 must also be used; that is, q′ = q and πb² q′ = k(w_m − g − ∆l), resulting in

$$\pi b^{2}q=k(w_m-g-\Delta l),$$

(31)

where w_m = w(b).

The following nondimensionalization is introduced

$$Q={\displaystyle \frac{qa}{Eh}}, W={\displaystyle \frac{w}{a}}, S_r={\displaystyle \frac{{\sigma }_r}{E}}, S_t={\displaystyle \frac{{\sigma }_t}{E}}, x={\displaystyle \frac{r}{a}}, \alpha ={\displaystyle \frac{b}{a}}, K={\displaystyle \frac{k}{\pi Eh}}, \overline{L}={\displaystyle \frac{\Delta l}{a}}, G={\displaystyle \frac{g}{a}},$$

(32)

where b ≤ r ≤ a, i.e., α ≤ x ≤ 1. Further, Equations (4), (17) and (27)–(31) can be transformed, from Equation (32), into

$$4x^2{S}_r^2{(-{\displaystyle \frac{\mathrm{d}W}{\mathrm{d}x}})}^2-{(x^2-{\alpha }^2)}^2{Q}^2[1+{(-{\displaystyle \frac{\mathrm{d}W}{\mathrm{d}x}})}^2]=0,$$

(33)

$${[S_r-\nu {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}(x{S}_{\mathrm{r}})+1]}^2-{\left\{{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}[x{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}(x{S}_{\mathrm{r}})]-\nu {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}(x{S}_{\mathrm{r}})+1\right\}}^2-{(-{\displaystyle \frac{\mathrm{d}W}{\mathrm{d}x}})}^2=0,$$

(34)

$$W=0 \mathrm{at} x=1,$$

(35)

$$(1-\nu )S_r+x{\displaystyle \frac{\mathrm{d}{S}_r}{\mathrm{d}x}}=0 \mathrm{at} x=1,$$

(36)

$$(1-\nu )S_r+x{\displaystyle \frac{\mathrm{d}{S}_r}{\mathrm{d}x}}={\displaystyle \frac{u(b)}{b}} \mathrm{at} x=\alpha ,$$

(37)

$$S_r={\displaystyle \frac{1}{1-\nu }}{\displaystyle \frac{u(b)}{b}} \mathrm{at} x=\alpha$$

(38)

and

$${\alpha }^{2}Q=K(W_m-G-\overline{L}).$$

(39)

Under conditions (35)–(39), the dimensionless radial stress S_r and deflection W can be determined by solving Equations (33) and (34) simultaneously. The power series method is a feasible and commonly used approach for solving differential equations analytically [50]. Obviously, S_r and W vary continuously within the interval α ≤ x ≤ 1. This implies that within the interval α ≤ x ≤ 1, S_r and W should be able to be analytically expressed in terms of the independent variable x. Therefore, S_r and W can be first expanded into the power series and substituted into Equations (33) and (34). If, under conditions (35)–(39), all the coefficients of the power series can be determined, then the analytical expressions of S_r and W can be determined; otherwise, the power series method is not applicable here. Now, let us expand S_r and W into the power series of the x − β (where β = (1 + α)/2, and 2β − 1 ≤ x ≤ 1 due to α ≤ x ≤ 1), i.e.,

$$S_r={\displaystyle \sum _{i=0}^{\infty }{c}_i{(x-\beta )}^i}$$

(40)

and

$$W={\displaystyle \sum _{i=0}^{\infty }{d}_i{(x-\beta )}^i}.$$

(41)

For convenience, Equations (40) and (41) can be reduced to the following by introducing X = x − β (where β − 1 ≤ X ≤ 1 − β due to 2β − 1 ≤ x ≤ 1)

$$S_r={\displaystyle \sum _{i=0}^{\infty }{c}_i{X}^i}$$

(42)

and

$$W={\displaystyle \sum _{i=0}^{\infty }{d}_i{X}^i}.$$

(43)

Moreover, Equations (33)–(39) should be transformed into

$$4{(X+\beta )}^2{S}_r^2{(-{\displaystyle \frac{\mathrm{d}W}{\mathrm{d}X}})}^2 -{[{(X+\beta )}^2-{(2\beta -1)}^2]}^2{Q}^2[1+{(-{\displaystyle \frac{\mathrm{d}W}{\mathrm{d}X}})}^2]=0,$$

(44)

$$\begin{array}{c}{\left\{S_r-\nu {\displaystyle \frac{\mathrm{d}}{\mathrm{d}X}}[(X+\beta )S_{\mathrm{r}}]+1\right\}}^2-{(-{\displaystyle \frac{\mathrm{d}W}{\mathrm{d}X}})}^2\\ -{\left\{{\displaystyle \frac{\mathrm{d}}{\mathrm{d}X}}[(X+\beta )S_{\mathrm{r}}]+(X+\beta ){\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{X}^2}}[(X+\beta )S_{\mathrm{r}}]-\nu {\displaystyle \frac{\mathrm{d}}{\mathrm{d}X}}[(X+\beta )S_{\mathrm{r}}]+1\right\}}^2=0,\end{array}$$

(45)

$$W=0 \mathrm{at} X=1-\beta ,$$

(46)

$$(1-\nu )S_r+(X+\beta ){\displaystyle \frac{\mathrm{d}{S}_r}{\mathrm{d}X}}=0 \mathrm{at} X=1-\beta ,$$

(47)

$$(1-\nu )S_r+(X+\beta ){\displaystyle \frac{\mathrm{d}{S}_r}{\mathrm{d}X}}={\displaystyle \frac{u(b)}{b}} \mathrm{at} X=\beta -1,$$

(48)

$$S_r={\displaystyle \frac{1}{1-\nu }}{\displaystyle \frac{u(b)}{b}}\mathrm{at} X=\beta -1$$

(49)

and

$${\left(2\beta -1\right)}^{2}Q=K[W(\beta -1)-G-\overline{L}].$$

(50)

Eliminating dW/dx from Equations (45) and (47) gives

$$\begin{array}{c}\left\{4{(X+\beta )}^2{S}_r^2-{[{(X+\beta )}^2-{(2\beta -1)}^2]}^2{Q}^2\right\}{\left\{S_r-\nu {\displaystyle \frac{\mathrm{d}}{\mathrm{d}X}}[(X+\beta )S_r]+1\right\}}^2\\ -\left\{4{(X+\beta )}^2{S}_r^2-{[{(X+\beta )}^2-{(2\beta -1)}^2]}^2{Q}^2\right\}\\ \times {\left\{{\displaystyle \frac{\mathrm{d}}{\mathrm{d}X}}[(X+\beta )S_r]+(X+\beta ){\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{X}^2}}[(X+\beta )S_r]-\nu {\displaystyle \frac{\mathrm{d}}{\mathrm{d}X}}[(X+\beta )S_r]+1\right\}}^2\\ -{\left[{(X+\beta )}^2-{(2\beta -1)}^2\right]}^2{Q}^2=0\end{array}$$

(51)

The power series coefficients c_i (i = 2, 3, 4, …) can be expressed in polynomials with regard to the undetermined constants c₀, c₁, and β, which can be easily (but very tediously) achieved: first substitute Equation (42) into Equation (51), combine like terms, and then let all the coefficients of the like terms be zero simultaneously (resulting in a system of equations consisting of an infinite number of algebraic equations); finally, solve this system of algebraic equations (resulting in the recursive formulas for the power series coefficients c_i (i = 2, 3, 4, …)). Due to the complexity of the frictionless elastic contact problem under consideration here, a general form of recursive formulas cannot be given, which is also what is usually encountered when solving complex problems with the power series methods for differential equations. So, how many recursive formulas need to be derived depends on how many recursive formulas are needed, and thus, only the recursive formulas for the power series coefficients c_i (i = 2, 3, 4, …, 8) are shown in Appendix A. Moreover, the power series coefficients d_i (i = 1, 2, 3, …) can be expressed in polynomials with regard to c₀, c₁, and β by substituting Equations (42) and (43) into Equation (44), as shown in Appendix B. The remaining coefficient d₀ depends on the undetermined constants c₀, c₁, and β; therefore, it is a dependent, undetermined constant.

The boundary conditions (Equations (46) and (47)), continuous conditions (Equations (48) and (49)), and equilibrium condition (Equation (50)) are used to determine the undetermined constants c₀, c₁, β, and d₀, which are detailed as follows. From Equation (43), Equations (46) and (50) give

$${\displaystyle \sum _{i=0}^{\infty }{d}_i}{(1-\beta )}^i=0$$

(52)

and

$${\displaystyle \sum _{i=0}^{\infty }{d}_i}{(\beta -1)}^i={\displaystyle \frac{{\left(2\beta -1\right)}^{2}Q}{K}}+\overline{L}+G.$$

(53)

Equation (53) minus Equation (52) yields

$${\displaystyle \sum _{i=1}^{\infty }{d}_i}[{(\beta -1)}^i-{(1-\beta )}^i]={\displaystyle \frac{{\left(2\beta -1\right)}^{2}Q}{K}}+\overline{L}+G.$$

(54)

Equations (42) and (47)–(49) give

$$(1-\nu ){\displaystyle \sum _{i=0}^{\infty }{c}_i{(1-\beta )}^i}+{\displaystyle \sum _{i=1}^{\infty }i{c}_i{(1-\beta )}^{i-1}}=0,$$

(55)

$$(1-\nu ){\displaystyle \sum _{i=0}^{\infty }{c}_i{(\beta -1)}^i}+\left(2\beta -1\right){\displaystyle \sum _{i=1}^{\infty }i{c}_i{(\beta -1)}^{i-1}}={\displaystyle \frac{u(b)}{b}}$$

(56)

and

$${\displaystyle \sum _{i=0}^{\infty }{c}_i{(\beta -1)}^i}={\displaystyle \frac{1}{1-\nu }}{\displaystyle \frac{u(b)}{b}}.$$

(57)

Eliminating u(b)/b from Equations (56) and (57) yields

$${\displaystyle \sum _{i=1}^{\infty }i{c}_i{(\beta -1)}^{i-1}}=0.$$

(58)

Therefore, under the condition that the values of a, h, E, v, and q are known in advance, the undetermined constants c₀, c₁, and β (β = (1 + α)/2 = (a + b)/2a) can be determined by simultaneously solving Equations (54), (55) and (58). Then, the undetermined constant d₀ can be determined by Equation (52) with the known c₀, c₁, and β values. Since β = (1 + α)/2 = (a + b)/2a and β and a are known, the radial displacement u(b) can be determined using Equation (57) with the known b.

After the undetermined constants c₀, c₁, α, and d₀ are determined, since the power series coefficients c_i (i = 2, 3, 4, …,) and d_i (i = 1, 2, 3, …,) are then all known, the expressions of S_r and W can be determined. From Equations (32), (40) and (41), and considering β = (1 + α)/2 = (a + b)/2a, the dimensional analytical expressions σ_r(r) and w(r) can finally be written as

$${\sigma }_r=E{\displaystyle \sum _{i=0}^{\infty }{c}_i{(\frac{r}{a}-\frac{a+b}{2a})}^i}$$

(59)

and

$$w=a{\displaystyle \sum _{i=0}^{\infty }{d}_i{(\frac{r}{a}-\frac{a+b}{2a})}^i}.$$

(60)

Then, with the known σ_r(r) and w(r), the other analytical expressions σ_t(r), u(r), e_r(r), and e_t(r) can be easily obtained from Equations (5), (16), (10) and (11). The elastic contact problem dealt with here is, thus, solved analytically.

In addition, it can be seen from Equation (26) and Figure 1d that the maximum stress and deflection, σ_m and w_m, can be determined by substituting r = b into Equations (59) and (60):

$${\sigma }_m=E{\displaystyle \sum _{i=0}^{\infty }{c}_i{(\frac{b-a}{2a})}^i}$$

(61)

and

$$w_m=a{\displaystyle \sum _{i=0}^{\infty }{d}_i{(\frac{b-a}{2a})}^i}.$$

(62)

3. Results and Discussion

By comparing the work presented in [48] with that presented in [49] and this paper, it can be seen that Equation (31) in [48] is wrong. Therefore, the analytical solution presented in [48] is actually a wrong solution, while the analytical solution presented in [49] is actually the only correct existing solution so far. Further, comparing the work presented in [49] with the work presented in this paper, it can be seen that the radial geometric equation used in [49] is improved in this paper. Therefore, relative to the analytical solution presented in [49], the analytical solution derived in Section 2 in this paper can be called an improved analytical solution. In this section, some important issues will be discussed, such as the convergence and validity of the improved analytical solution derived in Section 2, the difference between the analytical solutions before and after improvement, and the effect of changing some important structural parameters on the improved analytical solution derived in Section 2.

3.1. The Convergence of the Improved Analytical Solution Derived in Section 2

It can be seen from Equations (59) and (60) that the analytical solutions σ_r(r) and w(r) are in the form of a power series. It is well known that the convergence of a power series solution must be theoretically or numerically analyzed before its application. However, as can be seen in Appendix A and Appendix B, the expressions of the power series coefficients c_i (i = 2, 3, 4, …,) and d_i (i = 1, 2, 3, …,) are so complex that it is difficult to analyze the convergence of σ_r(r) or w(r) theoretically, so only a numerical analysis method can be used.

The geometric and physical parameters of the plate/membrane elastic contact problem under consideration are assumed to be as follows. The radius, thickness, Poisson’s ratio, and Young’s modulus of elasticity of the circular membrane are a = 70 mm, h = 0.3 mm, v = 0.45, and E = 3.01 MPa, respectively. The stiffness coefficient and initial compressed length of the spring are k = 0.5 N/mm and ∆l = 5 mm, respectively. The initial parallel gap between the initially flat circular membrane and the rigid flat circular plate is g = 5 mm. The uniformly distributed transverse loads q applied to the circular membrane are assumed to take 0.002 MPa, 0.008 MPa, and 0.18 MPa, respectively.

Therefore, it can be found from Equation (32) that K = 0.052875, L = 0.071429, G = 0.071429, and Q = 0.155039 when q = 0.002 MPa, Q = 0.620155 when q = 0.008 MPa, and Q = 1.395349 when q = 0.18 MPa. In order to calculate the precise numerical values of the undetermined constants c₀, c₁, β, and d₀, Equations (54), (55) and (58) are transformed into

$${\displaystyle \sum _{i=1}^n{d}_i}[{(\beta -1)}^i-{(1-\beta )}^i]={\displaystyle \frac{{\left(2\beta -1\right)}^{2}Q}{K}}+\overline{L}+G,$$

(63)

$$(1-\nu ){\displaystyle \sum _{i=0}^n{c}_i{(1-\beta )}^i}+{\displaystyle \sum _{i=1}^{n}i{c}_i{(1-\beta )}^{i-1}}=0$$

(64)

and

$${\displaystyle \sum _{i=1}^{n}i{c}_i{(\beta -1)}^{i-1}}=0.$$

(65)

Moreover, Equation (52) is transformed into

$$d_0={\displaystyle \sum _{i=1}^n{d}_i}{(1-\beta )}^i.$$

(66)

The numerical calculations of the undetermined constants c₀, c₁, β, and d₀ will start with the parameter n taking 2. First, the values of the undetermined constants c₀, c₁, and β are determined from Equations (63)–(65), and then, using the known values of c₀, c₁, and β, the value of the undetermined constant d₀ is determined from Equation (66). The results of the numerical calculations are listed in Table S1 in the Supplementary Materials for q = 0.002 MPa, in Table S2 in the Supplementary Materials for q = 0.008 MPa, and in Table S3 in the Supplementary Materials for q = 0.018 MPa. The variations of the undetermined constants β, c₀, c₁, and d₀ with the parameter n are shown in Figure 5, Figure 6, Figure 7 and Figure 8, respectively, where Figure 5a, Figure 6a, Figure 7a and Figure 8a are the cases for q = 0.002 MPa, Figure 5b, Figure 6b, Figure 7b and Figure 8b are the cases for q = 0.008 MPa, and Figure 5c, Figure 6c, Figure 7c and Figure 8c are the cases for q = 0.018 MPa.

It can be seen from Figure 5, Figure 6, Figure 7 and Figure 8 that the convergence of the undetermined constants β, c₀, c₁, or d₀ is affected by the magnitude of the uniformly distributed transverse loads q applied onto the circular membrane. Taking Figure 5 as an example, when q = 0.002 MPa, the undetermined constant β reaches saturation at approximately n = 15; when q = 0.008 MPa, it reaches saturation at approximately n = 25; and when q = 0.018 MPa, it reaches saturation at approximately n = 35. The trend that the n value increases with an increase in the load q is also clearly shown in Figure 6, Figure 7 and Figure 8. As can be seen from Figure 5, Figure 6, Figure 7 and Figure 8, in order to calculate the precise numerical values of the undetermined constants β, c₀, c₁, and d₀, when q = 0.002 MPa, the parameter n in Equations (63)–(66) should take at least 15; when q = 0.008 MPa, it should take at least 25; and when q = 0.018 MPa, it should take at least 35.

Now, let us investigate the convergence of the analytical solutions σ_r(r) and w(r) on the interval b ≤ r ≤ a. It is obvious that as long as σ_r(r) and w(r) are convergent when r = b and r = a, then σ_r(r) and w(r) are convergent on the interval b ≤ r ≤ a. From Equations (59) and (60), it can be found that when r = b and r = a, σ_r(b) and w(b) can be written as

$${\sigma }_r(b)=E{\displaystyle \sum _{i=0}^{\infty }{c}_i{(\frac{b}{a}-\frac{a+b}{2a})}^i}=E{\displaystyle \sum _{i=0}^{\infty }{c}_i{(\beta -1)}^i}$$

(67)

and

$$w(b)=a{\displaystyle \sum _{i=0}^{\infty }{d}_i{(\frac{b}{a}-\frac{a+b}{2a})}^i}=a{\displaystyle \sum _{i=0}^{\infty }{d}_i{(\beta -1)}^i}$$

(68)

and σ_r(a) and w(a) can be written as

$${\sigma }_r(a)=E{\displaystyle \sum _{i=0}^{\infty }{c}_i{(\frac{a}{a}-\frac{a+b}{2a})}^i}=E{\displaystyle \sum _{i=0}^{\infty }{c}_i{(1-\beta )}^i}$$

(69)

and

$$w(a)=a{\displaystyle \sum _{i=0}^{\infty }{d}_i{(\frac{a}{a}-\frac{a+b}{2a})}^i}=a{\displaystyle \sum _{i=0}^{\infty }{d}_i{(1-\beta )}^i}.$$

(70)

Therefore, it can be seen from Equations (67)–(70) that as long as c_i (β − 1)ⁱ and d_i(β − 1)ⁱ converge as the parameter i gradually increases from zero, then σ_r(b) and w(b) are convergent; additionally, as long as c_i (1 − β)ⁱ and d_i(1 − β)ⁱ converge as the parameter i gradually increases from zero, then σ_r(a) and w(a) are convergent.

For q = 0.002 MPa, the numerical calculation values of c_i (β − 1)ⁱ, d_i(β − 1)ⁱ, c_i (1 − β)ⁱ, and d_i(1 − β)ⁱ are listed in Table S4 in the Supplementary Materials, where the numerical values of the undetermined constants β, c₀, c₁, and d₀ used are β = 0.673057, c₀ = 0.113957, c₁ = −0.013409, and d₀ = 0.187881; see Table S1 in the Supplementary Materials. For q = 0.008 MPa, the numerical calculation values of c_i (β − 1)ⁱ, d_i(β − 1)ⁱ, c_i (1 − β)ⁱ, and d_i(1 − β)ⁱ are listed in Table S5 in the Supplementary Materials, where the numerical values of the undetermined constants β, c₀, c₁, and d₀ used are β = 0.636168, c₀ = 0.364218, c₁ = −0.026723, and d₀ = 0.318717; see Table S2 in the Supplementary Materials. For q = 0.018 MPa, the numerical calculation values of c_i (β − 1)ⁱ, d_i(β − 1)ⁱ, c_i (1 − β)ⁱ, and d_i(1 − β)ⁱ are listed in Table S6 in the Supplementary Materials, where the numerical values of the undetermined constants β, c₀, c₁, and d₀ used are β = 0.607791, c₀ = 0.780118, c₁ = −0.032533, and d₀ = 0.407823; see Table S3 in the Supplementary Materials. When the parameter i gradually increases from zero, the variations of c_i (β − 1)ⁱ, d_i(β − 1)ⁱ, c_i (1 − β)ⁱ, and d_i(1 − β)ⁱ with i are shown in Figure 9 for q = 0.002 MPa, in Figure 10 for q = 0.008 MPa, and in Figure 11 for q = 0.018 MPa.

It can be seen from Figure 9 that when q = 0.002 MPa, as the parameter i gradually increases from zero, c_i (β − 1)ⁱ, d_i(β − 1)ⁱ, c_i (1 − β)ⁱ, and d_i(1 − β)ⁱ are all convergent, i.e., σ_r(b) and w(b) and σ_r(a) and w(a) are all convergent. This suggests that when q = 0.002 MPa, the analytical solutions σ_r(r) and w(r) are all convergent on the interval b ≤ r ≤ a (i.e., 24.22798 mm ≤ r ≤ 70 mm due to β = 0.673057; see Table S1 in the Supplementary Materials). From Figure 10, it can be seen that when q = 0.008 MPa, as the parameter i gradually increases from zero, c_i (β − 1)ⁱ, d_i(β − 1)ⁱ, c_i (1 − β)ⁱ, and d_i(1 − β)ⁱ are all convergent, i.e., σ_r(b) and w(b) and σ_r(a) and w(a) are all convergent. This suggests that when q = 0.008 MPa, the analytical solutions σ_r(r) and w(r) are all convergent on the interval b ≤ r ≤ a (i.e., 19.06352 mm ≤ r ≤ 70 mm due to β = 0.636168; see Table S2 in the Supplementary Materials). Moreover, it can be seen from Figure 11 that when q = 0.018 MPa, as the parameter i gradually increases from zero, c_i (β − 1)ⁱ, d_i(β − 1)ⁱ, c_i (1 − β)ⁱ, and d_i(1 − β)ⁱ are all convergent, i.e., σ_r(b) and w(b) and σ_r(a) and w(a) are all convergent. This suggests that when q = 0.018 MPa, the analytical solutions σ_r(r) and w(r) are all convergent on the interval b ≤ r ≤ a (i.e., 15.09074 mm ≤ r ≤ 70 mm due to β = 0.607791; see Table S3 in the Supplementary Materials).

Therefore, it can be concluded from Figure 9, Figure 10 and Figure 11 that for the specific problem where the radius, thickness, Poisson’s ratio, and Young’s modulus of elasticity of the circular membrane are, respectively, a = 70 mm, h = 0.3 mm, v = 0.45, and E = 3.01 MPa, the stiffness coefficient and initial compressed length of the spring are, respectively, k = 0.5 N/mm and ∆l = 5 mm, and the initial parallel gap between the initially flat circular membrane and the rigid flat circular plate is g = 5 mm; the condition that the analytical solutions σ_r(r) and w(r) are both convergent is that the uniformly distributed transverse loads q applied to the circular membrane are less than or equal to 0.018 MPa.

3.2. The Validity of the Improved Analytical Solution Derived in Section 2

The validity of the improved analytical solution derived in Section 2 can be proven in the following way. It is obvious that if the uniformly distributed transverse loads q remain constant, then, when the spring stiffness coefficient k gradually approaches zero, the deflection shape of the circular membrane in the plate/membrane elastic contact problem will gradually approach the deflection shape when the same circular membrane is subjected only to the uniformly distributed transverse loads q (i.e., at this time, the rigid flat circular plate in Figure 1 is assumed to be removed). At the same time, if the uniformly distributed transverse loads q remain constant, then, when the spring stiffness coefficient k gradually approaches infinity, the deflection shape of the circular membrane in the plate/membrane elastic contact problem will gradually approach the deflection shape when the same circular membrane is in the plate/membrane rigid contact state under the uniformly distributed transverse loads q (i.e., at this time, the rigid flat circular plate in Figure 1 is assumed not to produce displacement under the external forces).

The calculation result based on the above-mentioned process is shown in Figure 12, where the four solid lines are the deflection curves of the circular membrane in the plate/membrane elastic contact state, the dashed line is the deflection curve of the same circular membrane in the plate/membrane rigid contact state, and the dotted line is the deflection curves of the same circular membrane subjected only to the action of q = 0.008 MPa. In Figure 12, the four solid lines are calculated by the improved analytical solution derived in Section 2 (where the relevant parameters used are, respectively, a = 70 mm, h = 0.3 mm, v = 0.45, E = 3.01 MPa, g = 5 mm, ∆l = 5 mm, and q = 0.008 MPa, and the spring stiffness coefficient k is equal to 0.05 N/mm, 1 N/mm, 5 N/mm, and 10,000 N/mm, respectively), where the dashed line is calculated by the well-established analytical solution of the plate/membrane rigid contact problem presented in [21] (the relevant parameters used are a = 70 mm, h = 0.3 mm, v = 0.45, E = 3.01 MPa, g = 5 mm, and q = 0.008 MPa, respectively), and the dotted line is calculated by the well-established analytical solution of the circular membrane problem presented in [41] (the relevant parameters used are a = 70 mm, h = 0.3 mm, v = 0.45, E = 3.01 MPa, and q = 0.008 MPa, respectively).

It can be seen by comparing the works presented in [21,41] and this paper, that the equations governing the elastic behavior of the circular membrane (i.e., the out-of-plane and in-plane equilibrium equations, the geometric equations, and the physical equations) used in [21,41] and in this paper are the same, while the conditions of solving the governing equations used are different. However, if the spring stiffness coefficient k takes infinity, then the solving conditions used in this paper will be equivalent to those used in [21], and if the spring stiffness coefficient k takes zero, then the solving conditions used in this paper will be equivalent to those used in [41]. This implies that under the same conditions, when k gradually approaches infinity or zero (obviously, k cannot be equal to infinity or zero directly), the improved analytical solution derived in Section 2 should be equivalent to the analytical solution presented in [21] or [41]. It can be seen from Figure 12 that the improved analytical solution derived in Section 2 can perfectly approach the two analytical solutions presented in [21,41], which, to some extent, indicates that it should be valid.

Similarly, the validity of the improved analytical solution derived in Section 2 can also be proven in the following way. It is obvious that as the uniformly distributed transverse loads q gradually decrease to zero, the deflection shape of the circular membrane in plate/membrane elastic contact state will gradually change into the deflection shape of the same circular membrane in the plate/membrane non-contact state. In other words, as the uniformly distributed transverse loads q continuously decrease to zero, the change in the deflection curve shape of the circular membrane from the plate/membrane elastic contact state to the plate/membrane non-contact state should be continuous, as shown in Figure 13; in this figure, the four dotted lines (in top-down order) are the deflection curves of the circular membrane in the plate/membrane non-contact state when the uniformly distributed transverse loads q are equal to 20 Pa, 50 Pa, 100 Pa, and 163 Pa, respectively, the four solid lines (in top-down order) are the deflection curves of the circular membrane in the plate/membrane elastic contact state when the uniformly distributed transverse loads q are equal to180 Pa, 250 Pa, 450 Pa, and 650 Pa, respectively, and the dashed line is the result of the finite element calculation of the deflection curve of the circular membrane in the plate/membrane elastic contact state when the uniformly distributed transverse load q is equal to 650 Pa. In Figure 13, the four dotted lines (in top-down order) are calculated using the well-established analytical solution of the circular membrane problem presented in [41] (where the relevant parameters used are a = 70 mm, h = 0.3 mm, v = 0.45, and E = 3.01 MPa, with q = 20 Pa, 50 Pa, 100 Pa, and 163 Pa, respectively), the four solid lines (in top-down order) are calculated using the improved analytical solution derived in Section 2 (where the relevant parameters used are a = 70 mm, h = 0.3 mm, v = 0.45, E = 3.01 MPa, g = 5 mm, ∆l = 5 mm, and k = 0.5 N/mm, with q = 180 Pa, 250 Pa, 450 Pa, and 650 Pa, respectively), and the dashed line is calculated using ABAQUS 2021/Explicit (where the relevant parameters used are a = 70 mm, h = 0.3 mm, v = 0.45, E = 3.01 MPa, g = 5 mm, ∆l = 5 mm, and k = 0.5 N/mm, with q = 650 Pa, respectively; the Membrane Element M3D4R is employed, and the circular membrane is uniformly cut into 30 parts along the radial direction and 100 parts along the circumferential direction).

As can be seen from Figure 13, the deflection shape change from the plate/membrane elastic contact state to the plate/membrane non-contact state is perfectly continuous, and the result of the finite element analysis agrees very well with the analytical result, which, to some extent, indicates that the improved analytical solution derived in Section 2 should be valid.

3.3. A Comparison Between the Analytical Solutions Before and After Improvement

On the basis of the existing studies [48,49], the analytical solution of the plate/membrane elastic contact problem (see Figure 1) is improved in this study. Since the analytical solution presented in [48] is a wrong solution (see Section 1 for details), the analytical solution presented in [49] is the only correct analytical solution for this plate/membrane elastic contact problem that is available in the literature. Therefore, the improved analytical solution derived in Section 2 only needs to be compared with the previous analytical solution derived in [49].

Figure 14 shows the deflection shapes of the circular membrane in the plate/membrane elastic contact state when the uniformly distributed transverse loads q, respectively, take 0.002 MPa, 0.008 MPa, and 0.018 MPa; for this, the three solid lines were calculated using the improved analytical solution derived in Section 2, and the three dot-dashed lines were calculated using the previous analytical solution for the plate/membrane elastic contact problem derived in [49]; the relevant parameters used were a = 70 mm, h = 0.3 mm, v = 0.45, E = 3.01 MPa, g = 5 mm, ∆l = 5 mm, and k = 0.5 N/mm, respectively. It can be seen from Figure 14 that when the uniformly distributed transverse load q is equal to 0.008 MPa, the difference in the deflection shapes becomes obvious, especially when q = 0.018 MPa.

Figure 15 shows the variations in the maximum deflection w_m with uniformly distributed transverse loads q, where the solid line was calculated using the improved analytical solution derived in Section 2, the dot-dashed line was calculated using the analytical solution of the plate/membrane elastic contact problem derived in [49], and the relevant parameters used were, respectively, a = 70 mm, h = 0.3 mm, v = 0.45, E = 3.01 MPa, g = 5 mm, ∆l = 5 mm, and k = 0.5 N/mm, with the uniformly distributed transverse loads q ranging from 164 Pa to 18,000 Pa; the numerical calculation results are listed in Tables S7 and S8 in the Supplementary Materials. As can be seen from Equation (4) in [49], the total capacitance C of the sensor proposed in [48,49] is in a one-to-one correspondence with the maximum deflection w_m. Therefore, the calculation accuracy of the maximum deflection w_m is crucial for the accurate determination of the total capacitance C of the sensor. However, it can be seen from Figure 15 that the difference between the values of the maximum deflection w_m under the same uniformly distributed transverse loads q increases with the increase in the uniformly distributed transverse loads q. This suggests that the improved analytical solution derived in Section 2 is superior to the analytical solution before the improvement derived in [49]. Therefore, the improved analytical solution derived in Section 2 should be used preferentially when designing the parallel plate variable capacitor-based circular capacitive wind pressure sensor proposed in [48,49].

The difference shown in Figure 14 and Figure 15 is mainly caused by the difference between the membrane governing equations used in Section 2 and those used in [48,49], as shown in

Table 1. The membrane governing equations used in Section 2 and in [48,49].

	Out-of-Plane Equilibrium Equation	In-Plane Equilibrium Equation	Geometric Equations	Physical Equations
In this paper	$2r{\sigma }_{r}h-(r^2-b^2)q\sqrt{1+1/{(-\mathrm{d}w/\mathrm{d}r)}^2}=0$	${\displaystyle \frac{\mathrm{d}}{\mathrm{d}r}}(r{\sigma }_r)-{\sigma }_t=0$	$\left\{\begin{array}{l}{e}_r=\sqrt{{(1+{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}})}^2+{(-{\displaystyle \frac{\mathrm{d}w}{\mathrm{d}r}})}^2}-1\\ e_t={\displaystyle \frac{u}{r}}\end{array}\right.$	$\left\{\begin{array}{l}{\sigma }_r={\displaystyle \frac{E(e_r+\nu e_t)}{1-{\nu }^2}}\\ {\sigma }_t={\displaystyle \frac{E(e_t+\nu e_r)}{1-{\nu }^2}}\end{array}\right.$
In [49]	$2r{\sigma }_{r}h-(r^2-b^2)q\sqrt{1+1/{(-\mathrm{d}w/\mathrm{d}r)}^2}=0$	${\displaystyle \frac{\mathrm{d}}{\mathrm{d}r}}(r{\sigma }_r)-{\sigma }_t=0$	$\left\{\begin{array}{l}{e}_r={\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}+{\displaystyle \frac{1}{2}}{(-{\displaystyle \frac{\mathrm{d}w}{\mathrm{d}r}})}^2\\ e_t={\displaystyle \frac{u}{r}}\end{array}\right.$	$\left\{\begin{array}{l}{\sigma }_r={\displaystyle \frac{E(e_r+\nu e_t)}{1-{\nu }^2}}\\ {\sigma }_t={\displaystyle \frac{E(e_t+\nu e_r)}{1-{\nu }^2}}\end{array}\right.$
In [48]	$2r{\sigma }_{r}h{\displaystyle \frac{\mathrm{d}w}{\mathrm{d}r}}+(r^2-b^2)q=0$	${\displaystyle \frac{\mathrm{d}}{\mathrm{d}r}}(r{\sigma }_r)-{\sigma }_t=0$	$\left\{\begin{array}{l}{e}_r={\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}+{\displaystyle \frac{1}{2}}{(-{\displaystyle \frac{\mathrm{d}w}{\mathrm{d}r}})}^2\\ e_t={\displaystyle \frac{u}{r}}\end{array}\right.$	$\left\{\begin{array}{l}{\sigma }_r={\displaystyle \frac{E(e_r+\nu e_t)}{1-{\nu }^2}}\\ {\sigma }_t={\displaystyle \frac{E(e_t+\nu e_r)}{1-{\nu }^2}}\end{array}\right.$

. As can be seen from Table 1, in comparison with the membrane equation used in [48], only the out-of-plane equilibrium equation is improved in [49], while in this paper, both the out-of-plane equilibrium equation and the radial geometric equation are improved.

3.4. The Effect of Changing the Structural Parameters on the Improved Analytical Solution

In this section, the effect of changing some of the geometric and physical parameters on the improved analytical solution of the plate/membrane elastic contact problem derived in Section 2 is investigated. The geometric and physical parameters, which will be changed, mainly include the radius a, thickness h, Poisson’s ratio v, and Young’s modulus of elasticity E of the circular membrane, as well as the stiffness coefficient k of the spring. By changing these structural parameters one by one, the effect of such changes on the relationship between the maximum deflection w_m and the uniformly distributed transverse loads q (hereinafter abbreviated as the q–w_m relationship) can be investigated.

3.4.1. The Effect of Changing the Radius a on the q–w_m Relationship

In this section, the values of the relevant structural parameters are, respectively, a = 70 mm, 80 mm and 90 mm, h = 0.3 mm, v = 0.45, E = 3.01 MPa, g = 5 mm, ∆l = 5 mm, and k = 0.5 N/mm, and the uniformly distributed transverse load q takes different values. The numerical calculation results are listed in Table S8 in the Supplementary Materials, where a = 70 mm, in Table S9 in the Supplementary Materials, where a = 80 mm, and in Table S10 in the Supplementary Materials, where a = 90 mm. The effect of changing the radius a on the q–w_m relationship is shown in Figure 16. It can be seen from Figure 16 that under the same uniformly distributed transverse loads q, the increase in the radius a results in an increase in the maximum deflection w_m.

3.4.2. The Effect of Changing the Thickness h on the q–w_m Relationship

In this section, the values of the relevant structural parameters are, respectively, a = 70 mm, h = 0.1 mm, 0.3 mm, and 0.5 mm, v = 0.45, E = 3.01 MPa, g = 5 mm, ∆l = 5 mm, and k = 0.5 N/mm, and the uniformly distributed transverse load q takes different values. The numerical calculation results are listed in Table S8 in the Supplementary Materials, where h = 0.3 mm, in Table S11 in the Supplementary Materials, where h = 0.1 mm, and in Table S12 in the Supplementary Materials, where h =0.5 mm. The effect of changing the thickness h on the q–w_m relationship is shown in Figure 17. It can be seen from Figure 17 that under the same uniformly distributed transverse load q, the increase in the thickness h results in a decrease in the maximum deflection w_m.

3.4.3. The Effect of Changing the Young’s Modulus of Elasticity E on the q–w_m Relationship

In this section, the values of the relevant structural parameters are, respectively, a = 70 mm, h = 0.3 mm, v = 0.45, E = 3.01 MPa, 5 MPa and 7.84 MPa, g = 5 mm, ∆l = 5 mm, and k = 0.5 N/mm, and the uniformly distributed transverse load q takes different values. The numerical calculation results are listed in Table S8 in the Supplementary Materials, where E = 3.01 MPa, in Table S13 in the Supplementary Materials, where E = 5 MPa, and in Table S14 in the Supplementary Materials, where E = 7.84 MPa. The effect of changing the Young’s modulus of elasticity E on the q–w_m relationship is shown in Figure 18. It can be seen from Figure 18 that under the same uniformly distributed transverse load q, the increase in the Young’s modulus of elasticity E results in a decrease in the maximum deflection w_m.

3.4.4. The Effect of Changing the Poisson’s Ratio v on the q–w_m Relationship

In this section, the values of the relevant structural parameters are, respectively, a = 70 mm, h = 0.3 mm, v = 0.15, 0.3 and 0.45, E = 3.01 MPa, g = 5 mm, ∆l = 5 mm, and k = 0.5 N/mm, and the uniformly distributed transverse load q takes different values. The numerical calculation results are listed in Table S8 in the Supplementary Materials, where v = 0.45, in Table S15 in the Supplementary Materials, where v = 0.15, and in Table S16 in the Supplementary Materials, where v = 0.3. The effect of changing the Poisson’s ratio v on the q–w_m relationship is shown in Figure 19. It can be seen from Figure 19 that under the same uniformly distributed transverse load q, the increase in the Poisson’s ratio v results in a decrease in the maximum deflection w_m.

3.4.5. The Effect of Changing the Stiffness Coefficient k on the q–w_m Relationship

In this section, the values of the relevant structural parameters are, respectively, a = 70 mm, h = 0.3 mm, v = 0.45, E = 3.01 MPa, g = 5 mm, ∆l = 5 mm, k = 0.1 N/mm, 0.5 N/mm, and 1 N/mm, and the uniformly distributed transverse load q takes different values. The numerical calculation results are listed in Table S8 in the Supplementary Materials, where k = 0.5 N/mm, in Table S17 in the Supplementary Materials, where k = 0.1 N/mm, and in Table S18 in the Supplementary Materials, where k = 1 N/mm. The effect of changing the stiffness coefficient k on the q–w_m relationship is shown in Figure 20. It can be seen from Figure 20 that under the same uniformly distributed transverse load q, the increase in the stiffness coefficient k results in a decrease in the maximum deflection w_m.

4. Concluding Remarks

In this paper, a further theoretical study on the problem of elastic contact between a transversely uniformly loaded circular membrane and a spring-reset rigid flat circular plate is presented, where the plate/membrane elastic contact problem is reformulated and then analytically solved, and an improved analytical solution of the plate/membrane elastic contact problem is also presented. From this study, the following conclusions can be drawn.

The improved analytical solution in the form of a power series, derived in Section 2, is convergent, but its convergence rate depends on the magnitude of the uniformly distributed transverse loads applied to the circular membrane; that is, when the applied load is relatively small, it converges very quickly, but as the applied load intensifies, it converges very slowly; see Section 3.1 for details. From Section 3.2, it can be concluded that the improved analytical solution derived in Section 2 is valid. The improved analytical solution derived in Section 2 is superior to the previous analytical solution before improvement, and the difference between the analytical solutions after and before improvement increases with the increase in the uniformly distributed transverse loads applied to the circular membrane; see Section 3.3 for details. The maximum deflection in the plate/membrane elastic contact state of a circular membrane can be increased by increasing the radius of the circular membrane, but it can be decreased by increasing the thickness, Poisson’s ratio, and Young’s modulus of elasticity of the circular membrane or by increasing the stiffness coefficient of the spring; see Section 3.4 for details.

The analytical solution of the plate/membrane elastic contact problem is potentially useful for pressure sensor developments or bending-free shell designs. However, the work presented here mainly focuses on theoretical study, mainly devoted to providing the basic or general theory needed for potential applications. Therefore, further research is still needed, especially experimental studies combined with specific applications.

Finally, special attention should be paid to the limitation of this study: the analytical solution of the plate/membrane elastic contact problem presented in this paper applies only to those cases where the stress–strain relationship of the membrane material used follows the generalized Hooke’s law (i.e., the material linearity assumption), and the friction force on the contact interface between the deflected circular membrane and the rigid flat circular plate is neglected (i.e., frictionless plate/membrane elastic contact).