1. Introduction
Rotationally symmetric membranes, as structures or structural components, are of much interest in a variety of applications such as bulge tests [
1,
2,
3], blister tests [
4,
5,
6] or constrained blister tests [
7,
8,
9,
10], and capacitive pressure sensors [
11,
12,
13,
14]. Attention in the literature has focused generally on circular membranes [
15,
16,
17,
18,
19], relatively less on annular membranes [
20,
21,
22,
23]. The external loads applied to circular membranes cover lateral loading and normal loading, but only a few cases of normal loading are available in the literature [
24,
25]. Axisymmetric deformations with large deflections are allowed and formulated usually in terms of out-of-plane and in-plane equilibrium equations, geometric equations, and physical equations. During the establishment of these equations, some approximations or assumptions have to be considered, due to the strong nonlinearity of large deflection phenomena.
The first solution of circular membrane problems was presented originally by H. Hencky, who dealt analytically with the problem of axisymmetric deformation of a peripherally fixed circular membrane under uniformly distributed transverse loads and gave an analytical solution in the form of power series in 1915 [
26]. Chien and Alekseev corrected a computational error in [
26], a wrong value of a power series coefficient, in 1948 [
27] and 1953 [
28], respectively. This solution is often cited by related studies [
5,
15,
16,
17,
18,
19,
29,
30,
31], and usually called the well-known Hencky’s solution. During the derivation of the well-known Hencky’s solution, in addition to the assumption of microdeformation of materials that is adopted in the establishment of physical equations, the assumption that the deflected circular membrane only produces a relatively small rotation angle under uniformly distributed transverse loads is adopted in the establishment of out-of-plane and in-plane equilibrium equations and geometric equations. Therefore, later scholars have done a lot of improvement on well-known Hencky’s solution [
4,
18,
32]. The other two cases of transverse loading for peripherally fixed circular membranes are the concentrated loading at the central point of the circular membranes and the local uniform loading in the central portion of the circular membranes [
15]. As for the normal loading of peripherally fixed circular membranes, the most typical example is gas pressure loading on membrane surfaces [
24,
25]. No matter how large the deflections are, the direction of action of gas pressure is always perpendicular to the deflected circular membranes under gas pressure, while the direction of transverse loading is always perpendicular to the undeflected circular membranes before loading.
The first person to deal with annular membrane problems is S. A. Alekseev, an academician of the former Soviet Union Academy of Sciences, who algebraically solved the large deflection problem of an annular membrane whose outer edge is fixed and inner edge is connected centrally with a stiff, weightless, concentric, circular thin plate, where the external loads is applied at the central point of the stiff circular thin plate [
20]. The algebraic solution, which was presented by Alekseev [
20], is valid only when the Poisson’s ratio of membranes is less than 1/3. Sun et al. [
21] algebraically solved the problem considered originally by Alekseev [
20] again and presented the algebraic solution suitable for Poisson’s ratios less than, or equal to, or greater than 1/3. The Alekseev-type annular membrane structure, i.e., the one with an inner edge connected to a weightless, stiff, concentric, circular thin plate, was used for developing a membrane elastic deflection and parallel plate capacitor-based capacitive pressure sensor by Lian et al. [
23], where the annular membrane and the circular thin plate are synchronously subjected to the uniformly distributed transverse loads, resulting in the parallel movement of the circular thin plate. Therefore, due to such a parallel movement the circular thin plate can be used to form a parallel plate capacitor and works as a movable electrode plate of the capacitor, thus achieving the capacitive pressure sensor mechanism of detecting pressure by measuring capacitance. The closed-form solution, which was given by Lian et al. [
22], is not an algebraic solution, but in the form of power series. This closed-form solution is also the first power series solution for annular membrane problems. The annular membrane problems, if solved by using the power series method, are often more difficult to converge than circular membrane problems. This is because the stress or deflection variables can be expanded into a power series at the center of circular membranes (at the center of the domain of the independent variable), but in annular membrane problems, the stress or deflection variable has to be expanded into a power series at a point on the annular membranes (not at the center of the domain of the independent variable). Therefore, annular membrane problems often converge more slowly than circular membrane problems. This limitation means that for an annular membrane problem, the convergence of its power series solution can only be tested after the convergence of its undetermined constants has been confirmed. So, annular membrane problems, if solved by using power series method, are often much more complex than circular membrane problems.
In this paper, we will deal with a plate/membrane contact problem, in which an initially flat, peripherally fixed circular membrane, which elastically deflects under uniformly distributed transverse loads, comes into contact with a concentric circular rigid flat plate that keeps a certain parallel gap from the initially flat circular membrane. Before touching the rigid plate, the circular membrane freely deflects, which is a standard circular membrane problem, the well-known Hencky problem. After coming into contact with the rigid plate, the circular membrane undergoes two classes of axisymmetric deformation: one is out-of-plane axisymmetric deflection in the plate/membrane non-contact area (an annular area); the other is in-plane axisymmetric stretching with or without plate/membrane friction in the plate/membrane contact area (a circular area). The case with plate/membrane friction is less studied because of its complexity, while the frictionless case is relatively more studied, such as the membrane/substrate delamination [
5,
6], adhesion [
7,
9,
10], and capacitive pressure sensors [
11,
12,
13,
14], etc. It should be pointed out that there must be friction in the movement of a deflected circular membrane in contact with a rigid flat plate and the larger the contact area, the greater the friction. Therefore, the frictionless case is just an ideal case, and in practice is an approximation to relatively small friction. The plate/membrane friction can often be reduced, for example, by coating the rigid flat plate with a smooth substance, and this is where the studies on frictionless contact problems are of value. Earlier studies on frictionless contact problems often tended to use some strict approximations or assumptions, such as assumptions of equi-biaxial constant stress state and small rotation angle of membrane [
33]. Wang et al. gave up the assumption of equi-biaxial constant stress state and presented a closed-form solution of this plate/membrane frictionless contact problem for the first time [
34]. Lian et al. gave up the assumption of equi-biaxial constant stress state and analytically solved this plate/membrane frictionless contact problem again, where the assumption of small rotation angle of membrane was also given up during the establishment of the out-of-plane equilibrium equation [
35]. In this paper, this plate/membrane frictionless contact problem is further solved analytically and a new and more refined closed-form solution is presented. The further and specific applications of this new and more refined closed-form solution are mainly the development of conductive membrane-based capacitive pressure sensors [
36,
37,
38,
39].
The assumptions used in the existing studies have been further given up in this study. Firstly, the assumption that the stress state is equi-biaxial and constant is completely given up during the derivation of the closed-form solution. Secondly, the assumption that the rotation angle of membrane is small enough is given up in the establishment of both the out-of-plane equilibrium equation and the geometric equations. In the following section, the problem under consideration is reformulated and is analytically solved in terms of the plate/membrane non-contact region and the plate/membrane contact region. In
Section 3, some important issues are addressed, such as whether the power series solutions for stress and deflection converge, whether the new and more refined closed-form solution presented can return to a classical circular membrane solution when the plate/membrane contact area approaches zero, and how accurate the present closed-form solution is in comparison with the previous closed-form solution. Concluding remarks are given in
Section 4.
2. Membrane Equations and Its Solution
As shown in
Figure 1, an initially flat, peripherally fixed circular membrane with Young’s modulus
E, Poisson’s ratio
ν, thickness
h, and radius
a, which elastically deflects under uniformly distributed transverse loads
q, comes into contact with a concentric circular frictionless rigid flat plate that keeps a certain parallel gap
g from the initially flat circular membrane, resulting in a circular contact area with radius
b. In
Figure 1, the dash-dotted line represents the geometric middle plane of the initially flat circular membrane,
o is the origin of the introduced cylindrical coordinate system (
r,
φ,
w) and is located in the centroid of the geometric middle plane,
r is the radial coordinate,
φ is the angle coordinate (but is not given in
Figure 1 due to the characteristic of axisymmetric deformation),
w denotes the axial coordinate and also denotes the deflection (transversal displacement) of the circular membrane under loads
q, the polar coordinate plane (
r,
φ) passes through the geometric middle plane.
The size of the uniformly-distributed transverse loads
q that makes the circular membrane go from free deflection to confined deflection depends on the size of the gap
g, and for a given gap
g, can be determined by using a circular membrane solution, that is, before the circular membrane comes into contact with the rigid plate, its maximum deflection (i.e., the deflection at
r = 0) under the loads
q should be exactly equal to the given gap
g. The value of the loads
q, thus determined, corresponds to the lower limit of the pressure measurement range of touch mode capacitive pressure sensors [
13,
14]. After coming into contact with the rigid plate, the deflected circular membrane may be divided into two portions: one is an annular area without plate/membrane contact and the other is a circular area with plate/membrane contact. In the plate/membrane non-contact area, the circular membrane undergoes the out-of-plane axisymmetric deflection, while in the plate/membrane frictionless contact area, it undergoes the in-plane axisymmetric stretching, a plane problem in the plane in which the rigid plate is located. On the connecting ring between the annular region and the circular region, the stresses, strains, and displacements should be continuous. Such a continuity may be used as definite solution conditions. For convenience, let us deal first with the annular portion of the deflected circular membrane, then with its circular portion. Throughout the derivation, it is assumed that the circular membrane always has a constant thickness
h during its deflection, i.e., the change of membrane thickness is ignored.
In the plate/membrane non-contact annular area, a free body of radius
b ≤
r ≤
a is taken from the annular portion of the deflected circular membrane to study its static problem of equilibrium, as shown in
Figure 2, where
σr is the radial stress (the mean stress over the cross section of the deflected membrane) and
θ is the rotation angle of the deflected membrane, the meridional rotation angle.
In the vertical direction perpendicular to the initially flat circular membrane and the rigid plate (see
Figure 2), there are three vertical forces: the total force
πr2q of the uniformly distributed transverse loads
q within radius
r, the total reaction force
πb2q from the rigid plate, and the total vertical force 2
πrσrhsin
θ produced by the membrane force
σrh, where
b ≤
r ≤
a. Thus, the out-of-plane equilibrium condition is
If the deflection (transversal displacement) of the deflected circular membrane under the uniformly distributed transverse loads
q is denoted by
w, then
Substituting Equation (2) into Equation (1) one has
The in-plane equilibrium condition is the same as before
where
σt denotes circumferential stress. As for the relationships of the strain and displacement (the geometric equations), we here replace the previous geometric equation in [
35] with a more accurate one in [
18]
and
where
er,
et, and
u denote the radial and circumferential strain, and radial displacement, respectively. During the derivation of Equation (5), the assumption of small rotation angle of membrane was given up, see reference [
18] for details. Moreover, the circular membrane is still assumed to be linearly elastic, so the relationships between stress and strain satisfy the generalized Hooke’s law
and
Substituting Equations (5) and (6) into Equations (7) and (8) yields
and
By means of Equations (4), (9) and (10), one has
After substituting the
u in Equation (11) into Equation (9), then the consistency equation may be written as
Therefore, the radial stress σr and deflection w(r) within b ≤ r ≤ a can be determined by simultaneously solving Equations (3) and (12). Their definite solution conditions are the boundary conditions at r = a and the continuity conditions at r = b. However, the continuity conditions at r = b can only be determined after the plane problem in the plate/membrane circular area is solved.
In the plate/membrane contact area with radius 0 ≤
r ≤
b, the circular portion of the deflected circular membrane always undergoes the in-plane axisymmetric stretching in the plane in which the rigid plate is located. Obviously, for a plane problem the first derivative of the deflection
w(
r) is always zero, i.e., d
w/d
r = 0. Therefore, Equations (5) and (6) give
and
Substituting Equations (13) and (14) into Equations (7) and (8) yields
and
Substituting Equations (15) and (16) into Equation (4), one has
Equation (17) satisfies the form of the Euler equation, therefore its general solution of Equation (17) may be written as
where
C1 and
C2 are two unknown constants. It is not difficult to understand that in order for the radial displacement u to be finite at
r = 0,
C2 has to be equal to zero. If
u = u(
b) at
r = b, then
C1 =
u(
b)/
b, thus
Therefore, substituting Equation (19) into Equations (13)–(16) yields
and
Equations (20) and (21) show that the strain and stress are always uniformly distributed in the plate/membrane contact area within 0 ≤ r ≤ b.
Now, let us proceed to the determination of the definite solution conditions in the plate/membrane non-contact annular area within
b ≤
r ≤
a. The boundary conditions at
r =
a are
and
The continuity conditions at
r =
b are
and
where ( )
A and ( )
B represent the values of various variables on two sides of the interconnecting circle with radius
b, and the subscript
A refers to the side of plate/membrane non-contact while the subscript
B refers to the side of plate/membrane contact.
Let us introduce the following dimensionless variables
and transform Equations (3), (4), (12) and (22)–(26) into
and
It can be obtained by eliminating (d
w/d
r)
2 from Equations (28) and (30) that
Since both stress and deflection are finite in the region of
x ≤ 1 (i.e.,
r ≤
a),
Sr and
W can be expanded as a power series of
x − (1 +
α)/2. If letting
β = (1 +
α)/2, then
and
By substituting Equation (37) into Equation (36), the coefficients
ci (
i = 2, 3, 4, …) can be expressed as the polynomials of
c0,
c1, and
α, which are listed in
Appendix A. Then, substituting Equations (37) and (38) into Equation (28) or Equation (30), the coefficients
di (
i = 1, 2, 3, …) can also be expressed as the polynomials of
c0,
c1, and
α, which are listed in
Appendix B.
The remaining three coefficients
c0,
c1, and
d0 are usually known as undetermined constants, while the dimensionless variable
α (
α =
b/
a) also plays the role of an undetermined coefficient here. The
c0,
c1,
α, and
d0 can be determined by using the boundary conditions and continuity conditions as follows. From Equation (38), Equations (31) and (33) give
and
Equation (40) minus Equation (39) is
From Equations (29) and (37), Equations (32), (34) and (35) yield
and
Eliminating the
u(
b)/
b from Equations (43) and (44), one has
For the given problem where
a,
h,
E,
υ,
g, and
q are known in advance, the undetermined constants
c0,
c1, and
α can be determined by the simultaneous solutions of Equations (41), (42) and (45). Furthermore, with the known
c0,
c1, and
α, the undetermined constant
d0 can be determined by Equation (39) or Equation (40). The problem dealt with here is thus solved. In addition, from Equations (27), (37) and (38), the expressions for dimensional stress and deflection may finally be written as
and