Sme.hyperelastic Seal

Solved with COMSOL Multiphysics 4.0a. © COPYRIGHT 2010 COMSOL AB.

Hype r e l a s t i c S e a l

Introduction

In this model you study the force-deflection relation of a car door seal made from a soft rubber material. The model uses a hyperelastic material model together with formulations that can account for the large deformations and contact conditions.

T H E O R E T I C A L B A C K G R O U N D

You find the theory for hyperelastic materials in the section “Hyperelastic Materials” on page 103 of the Structural Mechanics Module User’s Guide.

Model Definition

The seal is compressed between a stationary plane surface and an indenting cylinder. It is of special interest to investigate the effect of air confined within the seal. Figure 1 below shows the undeformed geometry of the seal and the contacting surfaces.

Figure 1: Model geometry.

The model describes a cross section of the seal with an assumption of plane strain. The contacting surfaces are rigid when compared to the seal.

When computing the pressure from the air compressed inside the seal, the current cross-section area is required. One useful method for computing an area is by using Gauss’ theorem, and converting to a contour integral.

R = 4.5

R = 12

1.5

R = 2.58

H Y P E R E L A S T I C S E A L | 1


2 | H Y P E R E L A S

(1)

You need to compute the integral in the deformed geometry, which is the default.

M A T E R I A L P R O P E R T I E S

• The rubber is hyperelastic and is modeled as a Mooney-Rivlin material with C10 = 0.37 MPa and C01 = 0.11 MPa. The material is almost incompressible, so the bulk modulus is set to 104 MPa. A mixed formulation is automatically used for this material model.

• The compression of the confined air is assumed to be adiabatic, giving the pressure-density relation

(2)

Here the cross-section area is denoted by A, with the undeformed value A0 = 123.63 mm2. The constant γ has the value 1.4 and p0 = 0.1 MPa is the standard air pressure. The load acting on the interior of the seal is then

(3)

C O N S T R A I N T S A N D L O A D S

• The lower straight part of the seal is glued to the car body, so all displacements are constrained there.

• One contact pair between the cylinder and the seal.

• One contact pair between the stationary plate and the seal.

• The rigid cylinder is lowered using the parameter of the parametric solver as the negative y displacement. It starts with a gap of 0 mm and is lowered 4 mm.

A 1 ad∫ ∇ x0

⋅⎝ ⎠⎜ ⎟⎛ ⎞

ad∫ xnxˆ dl∫°= = =

pp0------

ρρ0------⎝ ⎠⎛ ⎞ γ A0

A-------⎝ ⎠⎛ ⎞

γ= =

∆p p p0– p0A0A-------⎝ ⎠⎛ ⎞

γ1–⎝ ⎠

⎛ ⎞= =

T I C S E A L


Results and Discussion

The deformed shape at the lowest cylinder position—corresponding to an indentation of 4 mm—without internal pressure is shown in Figure 2. The deformation scale is 1:1, that is, a true shape.

Figure 2: Seal deformation at 4 mm indentation with internal pressure neglected.




In Figure 3 you can compare the deformed shapes at 2 mm indentation. In the right figure, where the pressure is included, the seal profile appears “inflated.”

Figure 3: Seal deformation at 2 mm indentation without internal pressure (above) and with internal pressure (below).

Figure 4 contains a plot of the force (per unit length) versus compression (displacement of the rigid cylinder) with and without the internal pressure taken into

T I C S E A L


account.

Figure 4: Force per unit length versus compression with (dashed) and without (solid) internal pressure.

Notice that the forces needed to compress the seal can be up to one order of magnitude larger when the effect of the air is taken into account.

In reality, a car door seal contains small holes through which the air can escape as long as the compression is not too fast. Thus the computed values are the limits corresponding to very slow and very fast compression, respectively.

Model Library path: Structural_Mechanics_Module/Nonlinear_Material_Models/hyperelastic_seal

Modeling Instructions

M O D E L W I Z A R D

1 Go to the Model Wizard window.




2 Click the 2D button.

3 Click Next.

4 In the Add Physics tree, select Structural Mechanics>Solid Mechanics (solid).

5 Click Next.

6 In the Studies tree, select Preset Studies>Stationary.

7 Click Finish.

G E O M E T R Y 1

1 In the Model Builder window, click Model 1>Geometry 1.

2 Go to the Settings window for Geometry.

3 Locate the Geometry Settings section. Find the Units subsection. From the Length

unit list, select mm.

Circle 11 Right-click Model 1>Geometry 1 and choose Circle.

2 Go to the Settings window for Circle.

3 Locate the Size section. In the Radius edit field, type 6.

4 Locate the Position section. In the y edit field, type 6.

Circle 21 In the Model Builder window, right-click Geometry 1 and select Circle.



4 Locate the Position section. In the x edit field, type 8.

5 In the y edit field, type 4.






6 In the Model Builder window, right-click Circle 3 and select Build All.

7 Click the Zoom Extents button on the Graphics toolbar.

T I C S E A L


Rectangle 11 Right-click Geometry 1 and choose Rectangle.

2 Go to the Settings window for Rectangle.

3 Locate the Size section. In the Width edit field, type 8.

4 In the Height edit field, type 12.

Rectangle 21 In the Model Builder window, right-click Geometry 1 and select Rectangle.






7 In the Model Builder window, right-click Rectangle 2 and select Build All.

Union 11 Right-click Geometry 1 and choose Boolean Operations>Union.

2 Select the objects c2, c1, c3, r1, and r2 only.

3 In the Model Builder window, right-click Union 1 and select Build All.

4 Go to the Settings window for Union.

5 Locate the Union section. Clear the Keep interior boundaries check box.

6 Click the Build All button.



3 Locate the Size section. In the Radius edit field, type 4.5.

4 Locate the Position section. In the y edit field, type 6.


















5 Locate the Position section. In the y edit field, type 1.5.



3 Locate the Size section. In the Width edit field, type 3.5.





Difference 11 Right-click Geometry 1 and choose Boolean Operations>Difference.

2 Go to the Settings window for Difference.

3 Locate the Difference section. Under Objects to add, click Activate Selection.

4 Select the object uni1 only.

5 Under Objects to subtract, click Activate Selection.

6 Select the objects c4, c5, c6, r3, and r4 only.

7 In the Model Builder window, right-click Difference 1 and select Build All.

Rectangle 51 Right-click Geometry 1 and choose Rectangle.


T I C S E A L



4 In the Height edit field, type 2.5.

5 Locate the Position section. In the x edit field, type -7.

6 In the y edit field, type -2.5.










5 Locate the Position section. In the x edit field, type -7.



Intersection 11 Right-click Geometry 1 and choose Boolean Operations>Intersection.

2 Select the objects r6 and c7 only.

3 In the Model Builder window, right-click Intersection 1 and select Build All.


Form Union1 In the Model Builder window, click Form Union.

2 Go to the Settings window for Finalize.

3 Locate the Finalize section. From the Finalization method list, select Form an assembly.

4 Click the Build All button.

G L O B A L D E F I N I T I O N S

1 In the Model Builder window, right-click Global Definitions and select Parameters.

Add a parameter that you can use to gradually increase the vertical displacement.



10 | H Y P E R E L A

Parameters1 Go to the Settings window for Parameters.

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

D E F I N I T I O N S

Selection 11 In the Model Builder window, right-click Model 1>Definitions and select Selection.

2 Go to the Settings window for Selection.

3 Locate the Geometric Scope section. From the Geometric entity level list, select Boundary.

4 Select Boundaries 9, 10, 12, 16, 17, 19, and 20 only.

5 In the Model Builder window, right-click Selection 1 and select Rename.

6 Go to the Rename Selection dialog box and type Inner Boundary in the New name edit field.

7 Click OK.

8 Right-click Definitions and choose Contact Pair.

9 Go to the Settings window for Contact Pair.

10 Locate the Pair Name section. In the Pair name edit field, type upper.

11 Select Boundaries 6 and 7 only.

12 In the upper-right corner of the Destination Boundaries section, click Activate

Selection.

13 Select Boundaries 11, 15, and 21 only.

14 In the Model Builder window, right-click Definitions and select Contact Pair.

15 Go to the Settings window for Contact Pair.

16 Locate the Pair Name section. In the Pair name edit field, type lower.

17 Select Boundary 3 only.

18 In the upper-right corner of the Destination Boundaries section, click Activate

Selection.

19 Select Boundaries 14 and 18 only.

NAME EXPRESSION DESCRIPTION

para 0 Vertical displacement parameter

S T I C S E A L


S O L I D M E C H A N I C S

1 In the Model Builder window, click Model 1>Solid Mechanics.

2 Go to the Settings window for Solid Mechanics.

3 Locate the Thickness section. In the d edit field, type 50[mm].

For the plane strain approximation, this setting only affects total forces.

Hyperelastic Material Model 11 Right-click Model 1>Solid Mechanics and choose Hyperelastic Material Model.

2 Go to the Settings window for Hyperelastic Material Model.

3 Locate the Hyperelastic Material Model section. From the Material model list, select Mooney-Rivlin.

4 Select Domain 3 only.

5 In the κ edit field, type 1e4[MPa].

Linear Elastic Material Model 11 In the Model Builder window, click Linear Elastic Material Model 1.

2 Go to the Settings window for Linear Elastic Material Model.

3 Locate the Geometric Nonlinearity section. Select the Include geometric nonlinearity check box.

Fixed Constraint 11 In the Model Builder window, right-click Solid Mechanics and select More>Fixed

Constraint.


Prescribed Displacement 11 In the Model Builder window, right-click Solid Mechanics and select More>Prescribed

Displacement.


3 Go to the Settings window for Prescribed Displacement.

4 Locate the Prescribed Displacement section. Select the Prescribed in X direction check box.

5 Select the Prescribed in Y direction check box.

6 In the V0 edit field, type -para*1[mm].

Fixed Constraint 21 In the Model Builder window, right-click Solid Mechanics and select Fixed Constraint.




2 Select Boundary 8 only.

Contact 11 In the Model Builder window, right-click Solid Mechanics and select Pairs>Contact.

2 Go to the Settings window for Contact.

3 Locate the Pair Selection section. In the Pairs list, select Contact Pair 1.

As no modulus of elasticity is available for this material, you must change the default for the penalty factor. 100 MPa is a reasonable magnitude for the stiffness of rubber.

4 Locate the Normal Contact section. In the pn edit field, type min(1e-3*5^segiter, 1)*100[MPa]/solid.hmin_dst.

Friction 11 Right-click Contact 1 and choose Friction.

2 Go to the Settings window for Friction.

3 Locate the Friction section. In the pt edit field, type min(1e-3*5^segiter, 1)*100[MPa]/solid.hmin_dst.

4 In the µstat edit field, type 0.2.

Contact 21 In the Model Builder window, right-click Model 1>Solid Mechanics and choose

Pairs>Contact.

2 Go to the Settings window for Contact.

3 Locate the Pair Selection section. In the Pairs list, select Contact Pair 2.

4 Locate the Normal Contact section. In the pn edit field, type min(1e-3*5^segiter,1)*100[MPa]/solid.hmin_dst.

M A T E R I A L S

Material 11 In the Model Builder window, right-click Model 1>Materials and select Material.


3 Go to the Settings window for Material.

S T I C S E A L


4 Locate the Material Contents section. In the Material contents table, enter the following settings:

Because all displacements on the rigid parts are prescribed, the choice of material is irrelevant. For the sake of clarity, you can select steel. It is also possible to use the rubber material for the entire model.

5 In the Model Builder window, right-click Materials and select Open Material Browser.

6 Go to the Material Browser window.

7 Locate the Materials section. In the Materials tree, select Built-In>Structural steel.

8 Right-click and choose Add Material to Model from the menu.

Structural steel1 In the Model Builder window, click Structural steel.

2 Select Domains 1 and 2 only.

M E S H 1

Free Triangular 1In the Model Builder window, right-click Model 1>Mesh 1 and select Free Triangular.

Size 11 In the Model Builder window, right-click Free Triangular 1 and select Size.

2 Go to the Settings window for Size.

3 Locate the Geometric Scope section. From the Geometric entity level list, select Domain.


5 Locate the Element Size section. Click the Custom button.

6 Locate the Element Size Parameters section. Select the Maximum element size check box.

7 In the associated edit field, type 0.5.

8 In the Model Builder window, right-click Size 1 and select Build All.

PROPERTY NAME VALUE

Model parameters C10 0.37[MPa]

Model parameters C01 0.11[MPa]

Density rho 1100[kg/m^3]




S T U D Y 1

Step 1: Stationary1 In the Model Builder window, expand the Study 1 node, then click Step 1: Stationary.

2 Go to the Settings window for Stationary.

3 Locate the Study Settings section. Select the Continuation check box.

4 Under Continuation parameter, click Add.

5 Go to the Add dialog box.

6 In the Continuation parameter list, select Vertical displacement parameter (para).

7 Click the OK button.

8 Go to the Settings window for Stationary.

9 Locate the Study Settings section. In the Parameter values edit field, type range(0,0.5,4).

Because the initial value of the contact pressure is zero, you need to use manual scaling of the variables. It is reasonable to expect displacements of the order 1 mm and contact pressures of the order 0.1 MPa.

10 In the Model Builder window, right-click Study 1 and choose Show Default Solver.

Solver 11 In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1

node, then click Dependent Variables 1.

2 Go to the Settings window for Dependent Variables.

3 Locate the Scaling section. From the Method list, select Manual.

4 In the Scale edit field, type 0.1E6.

5 In the Model Builder window, expand the Dependent Variables 1 node, then click mod1_u.

6 Go to the Settings window for Field.


8 In the Scale edit field, type 0.001.

9 In the Model Builder window, click Dependent Variables 1>mod1_solid_Tt_p1.

10 Go to the Settings window for Field.


S T I C S E A L


12 In the Scale edit field, type 0.02E6.

In contact problems, it is essential that the displacements are solved with a high degree of accuracy.

13 In the Model Builder window, expand the Stationary Solver 1>Segregated 1 node, then click Segregated Step 1.

14 Go to the Settings window for Segregated Step.

15 Click to expand the Damping and Termination section.

16 In the Tolerance factor edit field, type 1e-5.

17 In the Model Builder window, right-click Study 1 and choose Compute.

R E S U L T S

2D Plot Group 11 In the Model Builder window, expand the 2D Plot Group 1 node.

2 Right-click Surface 1 and choose Deformation.

3 Go to the Settings window for Deformation.

4 Locate the Scale section. Select the Scale factor check box.

5 In the associated edit field, type 1.

6 In the Model Builder window, right-click Surface 1>Deformation 1 and select Plot.


Modify the model so that the internal pressure can be taken into account.

D E F I N I T I O N S

Integration 11 In the Model Builder window, right-click Model 1>Definitions and select Model

Couplings>Integration.

2 Go to the Settings window for Integration.

3 Locate the Operator Name section. In the Operator name edit field, type AreaInt.

4 Locate the Source Selection section. From the Geometric entity level list, select Boundary.

5 From the Selection list, select Inner Boundary.

Variables 1a1 In the Model Builder window, right-click Definitions and select Variables.

2 Go to the Settings window for Variables.


francisco.felis

Highlight



3 Locate the Geometric Scope section. From the Geometric entity level list, select Boundary.

4 From the Selection list, select Inner Boundary.

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

S O L I D M E C H A N I C S

1 In the Model Builder window, click Model 1>Solid Mechanics.

2 Go to the Settings window for Solid Mechanics.

3 Locate the Thickness section. In the d edit field, type 50[mm].

Boundary Load 11 In the Model Builder window, right-click Model 1>Solid Mechanics and select Boundary

Load.

2 Go to the Settings window for Boundary Load.

3 Locate the Boundaries section. From the Selection list, select Inner Boundary.

4 Locate the Force section. From the Load type list, select Follower pressure.

5 In the p edit field, type int_p.

S T U D Y 1

Before solving again, copy the current solution so that you can compare the results with and without internal pressure.

Solver 1In the Model Builder window, right-click Solver Configurations>Solver 1 and select Solution>Copy.

R E S U L T S

Data Sets1 In the Model Builder window, expand the Data Sets node.

2 Right-click Solution 1 and choose Rename.

3 Go to the Rename Solution dialog box and type With Pressure in the New name edit field.

4 Click OK.

NAME EXPRESSION

EnclosedArea AreaInt(-x*solid.nx)

int_p 0.1[MPa]*((123.63[mm^2]/EnclosedArea)^1.4-1)

S T I C S E A L



6 Go to the Rename Solution dialog box and type Without Pressure in the New name edit field.

7 Click OK.

S T U D Y 1

Solver 1In the Model Builder window, right-click Solver Configurations>Solver 1 and select Compute.

R E S U L T S

2D Plot Group 11 In the Model Builder window, click 2D Plot Group 1.

2 Go to the Settings window for 2D Plot Group.

3 Locate the Data section. From the Parameter value list, select 2.

4 Click the Plot button.

2D Plot Group 21 In the Model Builder window, right-click Results and select 2D Plot Group.

2 Go to the Settings window for 2D Plot Group.

3 Locate the Data section. From the Data set list, select Without Pressure.

4 From the Parameter value list, select 2.

5 In the Model Builder window, right-click 2D Plot Group 2 and select Surface.

6 Right-click Surface 1 and choose Deformation.

7 Go to the Settings window for Deformation.

8 Locate the Scale section. Select the Scale factor check box.

9 In the associated edit field, type 1.

10 In the Model Builder window, right-click Surface 1>Deformation 1 and select Plot.

The force used for the compression can be computed as the sum of all vertical reaction forces on the indentor. Prepare data sets for this result.

Data Sets1 In the Model Builder window, right-click Data Sets and select Solution.





3 Go to the Rename Solution dialog box and type Indentor with Pressure in the New name edit field.

4 Click OK.

5 Right-click Solution 3 and choose Add Selection.




9 In the Model Builder window, right-click Data Sets and select Solution.


11 Go to the Rename Solution dialog box and type Indentor Without Pressure in the New name edit field.

12 Click OK.

13 Go to the Settings window for Solution.

14 Locate the Solution section. From the Solution list, select Copy 2.

15 Right-click Solution 4 and choose Add Selection.




19 In the Model Builder window, right-click Data Sets and select Evaluation>Integral.

20 Right-click Integral 1 and choose Rename.

21 Go to the Rename Integral dialog box and type Sum with Pressure in the New name edit field.

22 Click OK.

23 Go to the Settings window for Integral.

24 Locate the Settings section. From the Method list, select Summation.

25 Locate the Data section. From the Data set list, select Indentor with Pressure.

26 In the Model Builder window, right-click Data Sets and select Evaluation>Integral.

27 Right-click Integral 2 and choose Rename.

28 Go to the Rename Integral dialog box and type Sum Without Pressure in the New

name edit field.

S T I C S E A L


29 Click OK.

30 Go to the Settings window for Integral.

31 Locate the Data section. From the Data set list, select Indentor Without Pressure.

32 Locate the Settings section. From the Method list, select Summation.

1D Plot Group 31 In the Model Builder window, right-click Results and select 1D Plot Group.

2 Right-click 1D Plot Group 3 and choose Point Graph.

3 Go to the Settings window for Point Graph.

4 Locate the Data section. From the Data set list, select Sum Without Pressure.

5 Locate the Expression section. In the Expression edit field, type -solid.RFy.

This is the negative of the reaction force's y-component.

6 Click to expand the Coloring and Style section.

7 Find the Line style subsection. In the Width edit field, type 2.

8 Click to expand the Legends section.

9 Select the Show legends check box.

10 From the Legends list, select Manual.

11 In the table, enter the following settings:

12 In the Model Builder window, right-click Point Graph 1 and select Plot.

13 Right-click 1D Plot Group 3 and choose Point Graph.

14 Go to the Settings window for Point Graph.

15 Locate the Data section. From the Data set list, select Sum with Pressure.

16 Locate the Expression section. In the Expression edit field, type -solid.RFy.

17 Locate the Coloring and Style section. Find the Line style subsection. In the Width edit field, type 2.

18 From the Line list, select Dashed.

19 Locate the Legends section. Select the Show legends check box.

20 From the Legends list, select Manual.

LEGENDS

Without pressure




21 In the table, enter the following settings:

22 In the Model Builder window, click 1D Plot Group 3.

23 Locate the Plot Settings section. Select the Title check box.

24 In the associated edit field, type Compressive force as a function of compression.

25 In the x-axis label edit field, type Indentation (mm).

26 In the y-axis label edit field, type Force (N/mm).

27 Click the Plot button.

LEGENDS

With pressure

S T I C S E A L

Documents

Sme.hyperelastic Seal