Wrinkling of Cylindrical Membranes with Varying Thickness

The numerical treatment of wrinkling in membranes can be handled by two approaches, see Ref. 1. In the first approach, out-of-plane geometric nonlinearities are treated as constitutive nonlinearities through a modification of the strain energy density, which is called the relaxed strain energy approach. In contrast, the second method involves direct modifications of the deformation gradient instead of the constitutive relation. The second approach is more general and applicable to anisotropic membranes, and this method is implemented in COMSOL Multiphysics.

In this example, wrinkling is studied in a cylindrical membrane of nonuniform thickness that is being stretched axially and filled with water internally. The membrane material is described with an incompressible Mooney–Rivlin model. Depending on the level of axial stretch and internal pressure, certain portions of the membrane undergo wrinkling. The results of the two approaches of handling wrinkling are compared with each other as well as with the results from the example presented in Ref. 1. In the case of the relaxed strain energy approach, the total strain energy for a Mooney–Rivlin material, which is a combination of the full and the relaxed strain energy, is taken from Ref. 1.

Model Definition

The model example is taken from Ref. 1. A cylindrical membrane of radius 10 mm and initial height of 80 mm is first stretched axially and subsequently filled with water. The membrane is modeled with an incompressible two-parameter Mooney–Rivlin hyperelastic material. The material properties are given in Table 1.

Table 1: Material properties.
Property	Variable	Value
Mooney–Rivlin parameter C1	C1	0.2111 MPa
Mooney–Rivlin parameter C2	C2	2.111 kPa

The membrane thickness varies along the height as th = thm(2(1 − M)(Z/Hi) + M), where Hi is initial the height, thm is the mean thickness of 0.1 mm, and M is a parameter controlling the variation in thickness, here taken to be 0.5.

Results and Discussions

Figure 1 and Figure 2 show the wrinkled regions in the stretched and inflated cylinder by both approaches at different levels of water column height. The results match with each other approximately; the slight variation in the wrinkled region with the relaxed strain-energy approach comes from the transition zone approximation of the step function.

After stretching, the central portion of the membrane is wrinkled. As the water level increases, the extent of the wrinkling region reduces and finally disappears at around zw= 70 mm. The results are in agreement with those published in Ref. 1.

The first and second principal stresses in the inflated cylinder are shown in Figure 3 and Figure 4, respectively. The stresses from the modified deformation gradient approach matches with the relaxed strain energy approach (see the results from the model). Note that the second principal stress is nonnegative in the whole membrane.

Figure 5 and Figure 6 show the variations in the principal stresses along the height of cylinder after the prestretch. The results from both approaches agree with each other; moreover, the second principal stress is nonnegative in the wrinkled region. Similarly, Figure 7 shows the variation in the third principal strain after prestretch, and both approaches give the same results.

Figure 1: Wrinkled region in the inflated cylinder at different water heights computed with the modified deformation gradient approach.

Figure 2: Wrinkled region in the inflated cylinder at different water heights computed with the relaxed strain energy approach.

Figure 3: First principal stress in the inflated cylinder computed with the modified deformation gradient approach.

Figure 4: Second principal stress in the inflated cylinder with the modified deformation gradient approach.

Figure 5: First principal stress in the cylinder after prestretch.

Figure 6: Second principal stress in the cylinder after prestretch.

Figure 7: Third principal strain in the cylinder after prestretch.

Notes About the COMSOL Implementation

A wrinkling model based on the modified deformation gradient is incorporated within the membrane theory using the feature, which solves a set of nonlinear equations with the Newton–Raphson method.

The strain energy density based on the relaxed strain energy approach is taken from Ref. 1 and implemented through the option under the node. A weak contribution is added to enforce material incompressibility. Furthermore, wrinkling is identified to occur in the limit when the second principal stress turns negative (uniaxial stress conditions). Because the material is isotropic and incompressible, the wrinkling condition can thus be formulated in terms of the in-plane principal stretches as

(1)

Reference

1. A. Patil, “Inflation and instabilities of hyperelastic membranes,” PhD Thesis, KTH, 2016.

Nonlinear_Structural_Materials_Module/Hyperelasticity/membrane_varying_thickness

Modeling Instructions

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Global Definitions

Model Parameters

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Browse to the model’s Application Libraries folder and double-click the file membrane_varying_thickness_parameters.txt.

Definitions

Step 1 (step1)

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Variables 1

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Membrane (mbrn)

Thickness and Offset 1

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To model wrinkling using the built-in modified deformation gradient approach, add a Mooney–Rivlin hyperelastic material with a subfeature.

Hyperelastic Material (Modified Deformation Gradient Approach)

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Wrinkling 1

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Hyperelastic Material (Relaxed Strain Energy Approach)

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Weak Contribution 1

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Mesh 1

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Prestretch

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Inflation

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In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
z_w (Height of water column)	range(0,5,80)	mm

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Add Study

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Study (Relaxed Strain Energy Approach)

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Prestretch

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Inflation

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In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
z_w (Height of water column)	range(0,5,80)	mm

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Add two predefined plots, one 2D and one 3D representation. Edit the first to show the wrinkled region in the membrane.

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Add Predefined Plot

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Results

Wrinkled Region

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Line 1

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Wrinkled Region

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Table Annotation 1

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In the table, enter the following settings:


x-coordinate	y-coordinate	Annotation
-0.01	0	\[\textrm{z}_\textrm{w}=\textrm{0[mm]}\]
0.055	0	\[\textrm{z}_\textrm{w}=\textrm{20[mm]}\]
0.12	0	\[\textrm{z}_\textrm{w}=\textrm{40[mm]}\]
0.18	0	\[\textrm{z}_\textrm{w}=\textrm{80[mm]}\]