Uniaxial Stretching of a Rectangular Membrane

The numerical treatment of thin structures with a membrane model is much simpler than with a shell model due to the assumption of zero bending stiffness. However, for some load cases this assumption is disadvantageous. For instance, when a membrane is subjected to compressive stresses, it may trigger wrinkling when these reach a critical level defined by the bending stiffness.This undesirable limitation can be overcame with the incorporation of a wrinkling model that removes these instabilities.

In this example, a rectangular membrane is stretched uniaxially, which results in wrinkling in the central region. The wrinkling region and stress distribution depends on the geometry, loading conditions, and material properties. The analytical results are used to verify the numerical solution.

Model Definition

A rectangular sheet 1 m in length, 0.5 m in width, and 1 mm thick is stretched in the longitudinal direction.

One of the short edges is fixed, while a prescribed displacement of 1 mm is applied on the opposite edge. The long edges are unconstrained.

The sheet is made of a linear elastic material, and both isotropic and orthotropic scenarios are analyzed. The material properties are given in Table 1.

Material Properties

Table 1: Material properties.
Property	Variable	Isotropic Case	Orthotropic Case
Young’s modulus, 1 direction	E1	100 kPa	100 kPa
Young’s modulus, 2 direction	E2	100 kPa	20 kPa
Young’s modulus, 3 direction	E3	100 kPa	20 kPa
Poisson’s ratio, 12 direction	ν12	0.3	0.3
Poisson’s ratio, 23 direction	ν23	0.3	0
Poisson’s ratio, 13 direction	ν13	0.3	0
Shear modulus, 12 direction	G12	E1/(2(1+ν12))	38.5 kPa
Shear modulus, 23 direction	G23	G12	G12
Shear modulus, 13 direction	G13	G12	G12

Results and Discussions

The numerical and analytical results are compared side by side in the same figure. For this particular problem, wrinkling occurs in the 2nd local direction as uniaxial stretching is applied in the 1st local direction.

The analytical expression for the wrinkling measure for isotropic and orthotropic materials is derived from the equality

which gives the expression for the wrinkling measure as

Figure 1 and Figure 2 show the wrinkled regions for the isotropic and orthotropic cases. The results match the analytical values for both scenarios. For the isotropic case, the whole central region is wrinkled, while for the orthotropic case, only the areas along the long edges are wrinkled.

Figure 3 and Figure 4 show the measure of wrinkling in the membrane. Again, in both isotropic and orthotropic cases the results match the analytical expressions. The region of maximum wrinkling is the region where maximum compressive stress could have developed without a wrinkling model.

The first principal stress and the tensile direction for the isotropic and orthotropic cases are shown in Figure 5 and Figure 6. Figure 7 and Figure 8 show the second principal stress along the wrinkling direction. As expected, the direction of wrinkling is perpendicular to the loading direction in uniaxial stretching. The lowest value of the second principal stress is nearly zero. Both figures show no compressive stresses.

Figure 1: Wrinkled region, isotropic material.

Figure 2: Wrinkled region, orthotropic material.

Figure 3: Measure of wrinkling, isotropic material.

Figure 4: Measure of wrinkling, orthotropic material.

Figure 5: First principal stress, isotropic material.

Figure 6: First principal stress, orthotropic material.

Figure 7: Second principal stress, isotropic material.

Figure 8: Second principal stress, orthotropic material.

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.

Since the unstressed membrane does not have stiffness in the normal direction, a very small spring support is added in order to stabilize the model.

Structural_Mechanics_Module/Buckling_and_Wrinkling/membrane_uniaxial_stretching

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Geometric Parameters

In the window, under click .

In the window for , type Geometric Parameters in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Value	Description
th	1[mm]	0.001 m	Thickness of rectangular sheet
L	1[m]	1 m	Length of rectangular sheet
W	0.5[m]	0.5 m	Width of rectangular sheet

Isotropic Material Properties

In the toolbar, click

and choose .

In the window for , type Isotropic Material Properties in the text field.

Locate the section. Click

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

Orthotropic Material Properties

In the toolbar, click

and choose .

In the window for , type Orthotropic Material Properties in the text field.

Locate the section. Click

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

Definitions

Variables 1

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
Beta_ana	max(-(mbrn.D12mbrn.eel11+mbrn.D22mbrn.eel22+mbrn.D23*mbrn.eel33)/mbrn.D22,0)		Wrinkling measure, analytical
iswrinkled_ana	Beta_ana>0		Is wrinkled, analytical

Geometry 1

Work Plane 1 (wp1)

In the toolbar, click

Work Plane 1 (wp1)>Plane Geometry

In the window, click .

Work Plane 1 (wp1)>Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type L.

In the text field, type W.

Click

Membrane (mbrn)

Linear Elastic Material 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Wrinkling 1

In the toolbar, click

and choose .

Thickness and Offset 1

In the window, under click .

In the window for , locate the section.

In the d text field, type th.

Fixed Constraint 1

In the toolbar, click

and choose .

Select Edge 1 only.

Prescribed Displacement 1

In the toolbar, click

and choose .

Select Edge 4 only.

In the window for , locate the section.

Select the check box.

In the u 0 x text field, type 1[mm].

Select the check box.

Add a spring support in the thickness direction in order to achieve numerical stability for this problem.

Spring Foundation 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the kA table, enter the following settings:


0	0	0
0	0	0
0	0	1e-3[N/m^3]

Add a node in order to run same study with different materials.

Materials

Material Switch 1 (sw1)

In the window, under right-click and choose .

Isotropic Material

In the window, right-click and choose .

Orthotropic elastic properties are asked by the material due to the settings in the physics features. The isotropic material can however be defined by adding the isotropic properties manually. The orthotropic properties will be calculated automatically by synchronization rules.

In the window for , type Isotropic Material in the text field.

Locate the section. In the tree, select .

Click

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	E_iso	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	nu_iso	1	Young’s modulus and Poisson’s ratio
Density	rho	0	kg/m³	Basic

may show a warning. You can ignore it since the orthotropic material properties are well defined by the synchronization rules.

Orthotropic Material

In the window, right-click and choose .

In the window for , type Orthotropic Material in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	{Evector1, Evector2, Evector3}	{E1_orth, E2_orth, E3_orth}	Pa	Orthotropic
Poisson’s ratio	{nuvector1, nuvector2, nuvector3}	{nu12_orth, nu23_orth, nu13_orth}	1	Orthotropic
Shear modulus	{Gvector1, Gvector2, Gvector3}	{G12_orth, G23_orth, G13_orth}	N/m²	Orthotropic
Density	rho	0	kg/m³	Basic