Analysis of wrinkled membrane structures based on a wrinkle-wave model

Thin membranes that stretch under tension are widely used as lightweight structures in the aerospace industry.¹ However, thin membranes have negligible bending stiffness, which results in wrinkles when they are subjected to a compressive load.² Wrinkles significantly influence the stability and dynamic characteristics of these structures; thus, wrinkling has become a popular research topic.

The first study on wrinkling was performed by Wagner in 1929 and this was complemented by Reissner in 1938.³ Their research led to the development of the tension field theory, parameters were introduced to modify the constitutive relation on the membrane. To measure partially wrinkled membranes, Stein and Hedgepeth introduced the concept of a viable Poisson’s ratio,⁴ which was extended by Mikulas⁵ in a numerical and experimental study. On the basis of these works, Ding and Yang proposed a new membrane model with viable Young’s modulus and Poisson’s ratio,⁶ which was later studied by Rossi et al.⁷ and Jarasjarungkiat et al.⁸ However, although tension field theory can describe certain characteristics of wrinkling, it cannot predict wrinkle amplitude, number, or wavelength.⁹ 

To overcome the difficulty of predicting the wrinkling behavior, a stability theory was developed.¹⁰ One method involves analyzing the post-buckling response of a thin-shell with the help of finite element analysis (FEA) software.¹⁰ However, the finite element method cannot express taut, slack, and wrinkled states in a systematic and continuous fashion, which can lead to divergent results.⁶ Diaby et al. used another method, bifurcation analysis, to analyze the buckling and wrinkling phenomena of pre-stressed membranes.¹¹ The bifurcation theorem was introduced to track the bifurcation path, which requires considerable computational effort. Therefore, a simpler and more accurate analytical model is desired.

In this paper, we propose the concept of a wrinkle wave. For an infinite triangular structure under corner tension stress, radial wrinkle waves will occur when the geometric strain is greater than the strain capacity of the material.¹² After analyzing the characteristics of the wrinkle waves, we considered the influence from the other corners and introduced a tensile force ratio to determine the radius of the wrinkling radiation region. In addition, an easily solvable wrinkle-wave model was built for a square-membrane structure subjected to radial forces, which can also be used for other polygonal membrane structures. More recently, experiments of a square structure were performed using a Kapton membrane;¹³ this method was shown to be accurate and could overcome the difficult of convergence or non-convergence during calculations.

A triangular structure under tension, T, is shown in Figure 1. It is assumed that the membrane is isotropic with Young’s modulus E and Poisson’s ratio v, and that the constitutive material has linear elasticity. Similar to mechanical waves, a source and medium are necessary for wrinkle waves to occur. The tension stress, T, defined as wave source, is constant in time and generates static wrinkle waves. Also shown in Figure 1 are the radius of the propagation area (R_w), wrinkle number $(η)$⁠, wrinkle half-wavelength $(λ)$⁠, and amplitude (A). Wrinkle waves transfer through the medium and there are no wrinkles outside the membrane structure.

In contrast to mechanical waves, the wrinkle number η is a constant, while the corner tension T keeps constant. Wong and Pellegrino¹⁴ obtained an expression of the wrinkle number as

Accordingly, the angle of one wrinkle can be expressed as $α¯=2α/η$⁠, as shown in Figure 2. The wrinkling half-wavelength is uniform on an arbitrary arc, $⌢AB$⁠, of radius r; however, on a straight line $A′B′¯$⁠, it is smallest close to the central line and increases to the sides, Therefore, we define the half-wavelength vector as:

where $λi=r(tan(iα¯)−tan((i−1)α¯),(i=1,2,⋯,ceil(η/2))$⁠, and $ceil(η/2)$ is the nearest integer to $η/2$⁠.

A square membrane with side length L is a typical solar sail structure, as shown in Figure 3. This structure is pre-stressed by two pairs of equal and opposite corner forces. According to the stress field assumption obtained by Wong and Pellegrino,¹⁴ which consists of identical corner regions subject to purely radial stress and a central region under uniform biaxial stress. In purely radial stress regions, wrinkles will form when the geometric hoop strain, $εθg$⁠, is larger than the material strain, $εθm$⁠; thus, the wrinkles propagation radius can be obtained as:¹⁴ 

A general case is provided in Figure 4. We assume that $T1≤T2$ and define a tensile force ratio, $k=T2/T1≥1$⁠. As k increases, the radius of wrinkle propagation, $Rw1$ $(Rw1≤L/2ev)$⁠, caused by T₁ becomes smaller; by contrast, the propagation radius $Rw2$$(L/2ev≤Rw2≤2L)$⁠, caused by T₂ becomes larger, Both radii can be expressed as functions of the ratio k:

When $k≥4$⁠, $Rw2→2L$⁠, wrinkles overlap in the central region (Figure 5). The width of the overlapping region b is dependent on k. We introduced an exponential function, $exp(−tk(|yp|−λ1/2))$⁠, where y_p is the y coordinate of the point P on line $CD¯$⁠. The wrinkling amplitude on line $CD¯$ can be defined as $Aw=Aexp(−tk(|xp|−λ1/2))sin(βλπ)$⁠, where β is the angle from the central line, $λ1$ is the first half-wavelength close to the central line, and the maximum amplitude is located at $±λ1/2$ (Figure 6).

For vertex 1, it is assumed that the wrinkle waveform caused by T₁ is w₁ and that the maximum amplitude is A₁. Similarly, the wrinkle waveform caused by T₂ at vertex 2 is w₂ and the maximum amplitude is A₂. $λ1$ is used to express the first half-wavelength close to the central line, different values are obtained according to Eq. (2). Accordingly, the wrinkle waveform functions can be written as follows:

A wrinkling hoop strain was introduced, which is defined as follows:² 

The maximum amplitude, A₁(A₂), is obtained at $|xp|=λ1/2(|yp|=λ1/2),r=Rw1/2(r=Rw2/2),$ and the term $∫0λ(∂wi∂r)2dr→0$⁠.² Substituting Eq. (5) into Eq. (6) gives

To satisfy the condition $εθg=εθm+εθwrin$⁠, it has $σrln(Rwiexp(v)r)=−vσrE−π2Ai24kexp(v)λ12$⁠,¹⁴ According to the basic definition of an Airy stress field,¹⁵ the radial stress, $σr$⁠, can be expressed as $σr=2Tπrtcosβ$⁠. Then the wrinkle amplitude, $Ai$⁠, can be obtained:

The final wrinkle deformations are gained after superposition of the wrinkle waveform functions:

where

An n-sided regular polygonal membrane structure is shown in Figure 7, where n is even and the structure is subjected to $n/2$ pairs of equal and opposite corner forces. Assuming the minimum force is T₁, the other forces are defined as $T2$ ,$T3$ , $⋯$ ,$Tn2$ in a clockwise direction with a side length L. The ratios of the tensile forces can then be defined as:

The influence of all forces leads to the deflection of the central line of each corner, it is assumed that the deflection angle corresponding to T₁ is $δ1=0$⁠. Taking T₂ as an example, based on the inscribed angle theorem, $φ1=φ2=⋯=φ=πn$ and considering the influence of the other forces, a general expression of deflection angle, $δi$⁠, can be determined as:

Based on Eq. (4), the propagation radii can be expressed as:

where $𝐿′$ is the length of the deflected central line, as shown in Figure 7.

The wrinkle deformations generated by T_i at point Q along the y-axis in Figure 8 can be calculated after rotation around the origin (θ; that is, the angle between T₁ and T_i). Point Q(x, y) is then converted to $Q′(x′,y′)$ using the following formulae:

The maximum wrinkle amplitude, A_i, can be obtained according to Eq. (8). A wrinkle-wave model of a regular polygonal membrane structure can then be obtained as follows:

The parameters of the membrane structure are listed in Table I.

Comparison among the FEA, experimental, and analytical results are made below. In the experiments, vertexes were locally thickened using a short Kapton tab to avoid local tearing.¹⁰ The tensions were determined using tension transducers and the out-of-plane displacement was measured with photogrammetry. For a load ratio of k = 1, the membrane was loaded uniformly and incrementally. The wrinkle shape, as shown in Figure 9, was T₁ = T₂ = 10N, and the cross-wise displacements at section D–D in Figure 9, which are 70 mm from the vertex, are shown in Figure 10.

Slack states are observed experimentally near the edges. In the central region, the amplitude determined analytically, experimentally and using the FEA are consistent. However, the wrinkling half-wavelengths proposed in this paper are in better agreement with the experimental results than those obtained using the FEA simulation. This is because in the FEA model, at least eight elements per wrinkling half-wavelength are needed to capture the rapidly varying displacement.¹⁶ Otherwise, the initial imperfections added to the membrane must be large enough to ensure that wrinkling occurs. Initial imperfections and the low number of elements lead to inexact or divergent wrinkling half-wavelengths.

For the case of k = 4, T₁ = 4N, and T₂ = 16N, the wrinkle shapes are shown in Figure 11 and the out-of-plane deformation in the central cross section is shown in Figure 12. The maximum amplitude and deformation root mean square (Drms) values are compared in Table II. The analytical values are also closer to the experimental values than the FEA results, which further validates the analytical model.

For regular hexagonal membrane structures, loading cases of T₁ = T₂ = T₃ = 5N, T₁ = 2N,T₂ = T₃ = 12N, and T₁ = T₂ = 2N, T₃ = 12N were analyzed. The corresponding wrinkle shapes are shown in Figures 13 and 14 and the maximum amplitudes are listed in Table III. The agreement between analytical and FEA results proves that this model is applicable to various loading cases and other polygonal membrane structures.

We proposed the concept of wrinkle waves and introduced an Airy stress function to analyze the characteristics of the wrinkle waves. The tensile force ratios were defined to describe the influence from the other corners, and the out-of-plane displacement around each vertex was calculated after rotation.We constructed an analytical wrinkle-wave model for a square and other polygonal membrane structures. The proposed model is easily solvable and does not lead to divergent results. In addition, the analytical model can be applied to various loading cases, and the wrinkling amplitude and half-wavelength correlate more closely with the experimental results than with the results of the FEA simulation.

This work was supported by the National Natural Science Foundation of China under grants 51575419 and 51490661.