nonlinear structural materials model library manual

VERSION 4.3b

Nonlinear Structural Materials ModuleModel Library Manual

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Part number: CM022902

N o n l i n e a r S t r u c t u r a l M a t e r i a l s M o d e l L i b r a r y 19982013 COMSOL

Protected by U.S. Patents 7,519,518; 7,596,474; and 7,623,991. Patents pending.

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Version: May 2013 COMSOL 4.3b

Solved with COMSOL Multiphysics 4.3b

2 0 1 3 C O M S O L

I n f l a t i o n o f a S ph e r i c a l Rubb e r Ba l l o o n

Introduction

This model aims to investigate the inflation of a rubber balloon using different hy

Are

T(R

M

Tfo

Fito

InN 1 | I N F L A T I O N O F A S P H E R I C A L R U B B E R B A L L O O N

perelastic material models, and to compare the results to analytical expressions.

controlled inflation could be of importance in clinical applications, cardiovascular search, and medical device industry (Ref. 2), among others.

he example is taken from the book Nonlinear Solid Mechanics, by G. Holzapfel ef. 1).

odel Definition

his model compares the hoop stress and inflation pressure as a function of the stretch r a spherical rubber balloon.

gure 1: Model geometry. The initial inner radius is set to 10 cm, and the initial thickness 1 mm.

this example, the following four hyperelastic material models are compared: eo-Hookean, Money-Rivlin, Ogden, and Varga.

Thickness

20

Symmetry

Inner Radius


2 | I N F L A T I O N

Due to the spherical symmetry, an arbitrary sector in the azimuthal direction can be used. Here, a 20 degrees sector is modeled in a 2D axial symmetry plane.

F

R

Tinth

Faneqar

Dstin O F A S P H E R I C A L R U B B E R B A L L O O N 2 0 1 3 C O M S O L

igure 2: 2D axisymmetric geometry and mesh.

esults and Discussion

he results are compared to the analytical expression for a thin-walled vessel. The flation pressure is a function of the hoop stress , current inner radius r and current ickness h

or spherical balloons, the hoop stress is equal to the largest principal stresses 1 d 2. Two of the principal stretches lay on the plane tangential to the sphere and are ual, 12r/R, which is typical for equibiaxial deformation. Here, r and R e the current and initial inner radii, respectively.

ue to the nearly incompressibility assumption, the third principal stretch (this is the retch in the radial direction) 31/2h/H, where h and H are the current and itial balloon thicknesses, respectively.

pi 2hr---=


2 0 1 3 C O M S O L

The analytical expression for the hoop stress for the Ogden material model becomes (Ref. 1)

where p and p are Ogden parameters, and is the largest principal stretch.Sicara

Tpo

Tvaagrem

p p 2p p 1=

N

= 3 | I N F L A T I O N O F A S P H E R I C A L R U B B E R B A L L O O N

nce rR and hH/2, the analytical expression for the inflation pressure is lculated as a function of Ogden parameters, stretch, initial thickness and initial inner dius

he results are in excellent agreement with experimental results and the figures rtrayed in Ref. 1.

he experiments show a rapid rise in the internal pressure until reaching a maximum lue, followed by a pressure decrease until reaching a minimum, and then increasing ain. This phenomenon is similar to snap-through buckling, and can only be covered by the Ogden material model. The Neo-Hookean and the Varga material odels can only reproduce balloon inflations at small strain levels.

pi 2hr--- 2HR----- p

p 3 2p 3 p 1=

N

= =



Fd O F A S P H E R I C A L R U B B E R B A L L O O N 2 0 1 3 C O M S O L

igure 3: Computed inflation pressure as a function of circumferential stretch for ifferent material models, compared to the analytical expression for Ogden material.


2 0 1 3 C O M S O L

Fim

B

N

Dstprfo

Tis

Inre 5 | I N F L A T I O N O F A S P H E R I C A L R U B B E R B A L L O O N

gure 4: Computed hoop stress as a function of circumferential stretch for different aterial models, compared to the analytical expression for Ogden material.

oth plots are in a excellent agreement with the results described in Ref. 1, page 241.

otes About the COMSOL Implementation

ifferent hyperelastic material models are constructed by specifying different elastic rain energy expressions. The Nonlinear Structural Materials module provides several edefined material models together with an option to enter user defined expressions r the strain energy density.

he predefined nearly incompressible version of the Neo-Hookean material uses the ochoric invariant and the initial bulk modulus

this example, 422.5 kPa and 105The Lam parameter can be seen as presenting the shear modulus at small strains.

I1 Cel

Ws12--- I1 3 12--- Jel 1

2+=



The predefined nearly incompressible Mooney-Rivlin material has an elastic strain energy density written in terms of the two isochoric invariant of the elastic right Cauchy-Green deformation tensors and , and the elastic volume ratio Jel

The material parameters C10 and C01 are related to the shear modulus C10C01 InC

Tis

w

T

Wsup

Inex

Geqin

TA

p

1

2

3

I1 Cel I2 Cel

Ws C10 I1 3 C01 I2 3 12--- Jel 1 2

+ += O F A S P H E R I C A L R U B B E R B A L L O O N 2 0 1 3 C O M S O L

this example, they are set as C10and C01, so that the relation 10C01 is fulfilled.he predefined nearly incompressible Ogden material is implemented with the ochoric elastic stretches and the initial bulk modulus

ith N, and the Ogden parameters as written in Table 1

he Varga material model is implemented with a user defined strain energy density

hen the relation between the applied load and the displacement is not unique, a itable modeling technique is to use an algebraic equation that controls the applied

ressure, so that the model reaches the desired displacement increments.

this example, a Global Equation uses the radial displacement at point 3 to add an tra degree of freedom for the inflation pressure.

lobal equations are a way of adding an additional equation to a model. A global uation can be used to describe a load, constraint, material property, or anything else the model that has a uniquely definable solution. In this example, the model is

BLE 1: OGDEN PARAMETERS

p p (kPa)1.3 630

5.0 1.2

-2.0 -10

Wspp------ el1p el2p el3p 3+ +

p 1=

N

12--- Jel 1 2+=

Ws 2 el1 el2 el3 3+ + 12--- Jel 1 2

+=


2 0 1 3 C O M S O L

augmented by a global equation which solves for the inflation pressure to achieve a desired applied stretch.

References

1. G.A. Holzapfel, Nonlinear Solid Mechanics: A Continuum Approach for Engineering, Wiley, 2000.

2. H. Azarnoush et al. Real-time control of angioplasty balloon inflation based on fear

MHy

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Pa1 7 | I N F L A T I O N O F A S P H E R I C A L R U B B E R B A L L O O N

edback from intravascular optical coherence tomography: preliminary study on an tery phantom. IEEE Trans Biomed Eng. 59, pp. 697-705, 2012.

odel Library path: Nonlinear_Structural_Materials_Module/perelasticity/balloon_inflation

odeling Instructions

O D E L W I Z A R D

Go to the Model Wizard window.

Click the 2D axisymmetric button.

Click Next.

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

Click Next.

Find the Studies subsection. In the tree, select Preset Studies>Stationary.

Click Finish.

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

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

Begin by defining model parameters.

rametersIn the Parameters settings window, locate the Parameters section.



2 In the table, enter the following settings:

Va1

2

3

G

D

C1

2

3

4

5

6

7

D1

Name Expression Description

Ri 10[m] Inner radius

H 0.1[m] Thickness

mu 4.225e5[Pa] Shear modulus

kappa 1e5*mu Bulk modulus

stretch 1 Applied stretch

N

u

L

L O F A S P H E R I C A L R U B B E R B A L L O O N 2 0 1 3 C O M S O L

Setting the bulk modulus to 105 times the shear modulus is based on the assumption that the material is incompressible.

riables 1Right-click Global Definitions and choose Variables.

In the Variables settings window, locate the Variables section.

In the table, enter the following settings:

Use the stretch and geometry parameters to calculate the applied displacement.

E O M E T R Y 1

ue to symmetry, it suffices to model a 20-degree sector of the balloon.

ircle 1In the Model Builder window, under Model 1 right-click Geometry 1 and choose Circle.

In the Circle settings window, locate the Size and Shape section.

In the Radius edit field, type Ri+H.

In the Sector angle edit field, type 20.

Click to expand the Layers section. In the table, enter the following settings:

Click the Build All button.

Click the Zoom Extents button on the Graphics toolbar.

elete Entities 1In the Model Builder window, right-click Geometry 1 and choose Delete Entities.

ame Expression Description

_appl (stretch-1)*Ri Applied displacement

ayer name Thickness (m)

ayer 1 H


2 0 1 3 C O M S O L

2 In the Delete Entities settings window, locate the Entities or Objects to Delete section.

3 From the Geometric entity level list, choose Domain.

4 On the object c1, select Domain 1 only.

5 Click the Build All button.

6 Click the Zoom Extents button on the Graphics toolbar.

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

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dd the four hyperelastic material models.

yperelastic Material 1In the Model Builder window, under Model 1 right-click Solid Mechanics and choose Hyperelastic Material.

Select Domain 1 only.

In the Hyperelastic Material settings window, locate the Hyperelastic Material section.

Select the Nearly incompressible material check box.

In the edit field, type kappa.From the list, choose User defined. In the associated edit field, type mu.Right-click Model 1>Solid Mechanics>Hyperelastic Material 1 and choose Rename.

Go to the Rename Hyperelastic Material dialog box and type Neo-Hookean in the New name edit field.

Click OK.

yperelastic Material 2Right-click Solid Mechanics and choose Hyperelastic Material.



From the Material model list, choose Mooney-Rivlin, two parameters.

From the C10 list, choose User defined. In the associated edit field, type 0.4375*mu.

From the C01 list, choose User defined. In the associated edit field, type 0.0625*mu.

In the edit field, type kappa.Right-click Model 1>Solid Mechanics>Hyperelastic Material 2 and choose Rename.

Go to the Rename Hyperelastic Material dialog box and type Mooney-Rivlin in the New name edit field.

Click OK.


10 | I N F L A T I O

Hyperelastic Material 31 Right-click Solid Mechanics and choose Hyperelastic Material.

2 Select Domain 1 only.

3 In the Hyperelastic Material settings window, locate the Hyperelastic Material section.

4 From the Material model list, choose Ogden.

5 Select the Nearly incompressible material check box.

6 Click Add twice.

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-N O F A S P H E R I C A L R U B B E R B A L L O O N 2 0 1 3 C O M S O L

In the Ogden parameters table, enter the following settings:

In the edit field, type kappa.Right-click Model 1>Solid Mechanics>Hyperelastic Material 3 and choose Rename.

Go to the Rename Hyperelastic Material dialog box and type Ogden in the New name edit field.

Click OK.

yperelastic Material 4Right-click Solid Mechanics and choose Hyperelastic Material.



From the Material model list, choose User defined.

Select the Nearly incompressible material check box.

In the Wsiso edit field, type 2*mu*(solid.stchelp1+solid.stchelp2+solid.stchelp3-3).

In the Wsvol edit field, type 0.5*kappa*(solid.Jel-1)^2.

Right-click Model 1>Solid Mechanics>Hyperelastic Material 4 and choose Rename.

Go to the Rename Hyperelastic Material dialog box and type Varga in the New name edit field.

Click OK.

pply symmetry conditions.

hear modulus (Pa) Alpha parameter

.3e5 1.3

.012e5 5

0.1e5 -2


2 0 1 3 C O M S O L

Symmetry 11 Right-click Solid Mechanics and choose Symmetry.

2 Select Boundaries 1 and 2 only.

Control the inflation of the balloon by the pressure.

Boundary Load 11 Right-click Solid Mechanics and choose Boundary Load.

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In1

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Va1

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S

1

N

u 11 | I N F L A T I O N O F A S P H E R I C A L R U B B E R B A L L O O N

Select Boundary 3 only.

In the Boundary Load settings window, locate the Force section.

From the Load type list, choose Pressure.

In the p edit field, type p_f.

You will define the pressure p_f using a Global Equation feature shortly. First, define an integration coupling operator to evaluate the displacement at Point 3.

E F I N I T I O N S

tegration 1In the Model Builder window, under Model 1 right-click Definitions and choose Model Couplings>Integration.

In the Integration settings window, locate the Source Selection section.

From the Geometric entity level list, choose Point.

Select Point 3 only.

riables 2In the Model Builder window, right-click Definitions and choose Variables.



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

In the Model Builder windows toolbar, click the Show button and select Advanced Physics Options in the menu to allow to add a global equation and other advanced modeling features.


b intop1(u) Radial displacement, inner boundary



Global Equations 11 In the Model Builder window, under Model 1 right-click Solid Mechanics and choose

Global>Global Equations.

2 In the Global Equations settings window, locate the Global Equations section.


M

MIn

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Name f(u,ut,utt,t) Initial value (u_0) Initial value (u_t0)

p_f ub-u_appl 0 0N O F A S P H E R I C A L R U B B E R B A L L O O N 2 0 1 3 C O M S O L

E S H 1

apped 1 the Model Builder window, under Model 1 right-click Mesh 1 and choose Mapped.

istribution 1In the Model Builder window, under Model 1>Mesh 1 right-click Mapped 1 and choose Distribution.


In the Distribution settings window, locate the Distribution section.

In the Number of elements edit field, type 3.

istribution 2Right-click Mapped 1 and choose Distribution.




In the Model Builder window, right-click Mesh 1 and choose Build All.


he first study solves the problem with a Neo-Hookean material model.

T U D Y 1

tep 1: StationaryIn the Model Builder window, under Study 1 click Step 1: Stationary.

In the Stationary settings window, locate the Physics and Variables Selection section.

Select the Modify physics tree and variables for study step check box.


2 0 1 3 C O M S O L

4 In the Physics and variables selection tree, select Model 1>Solid Mechanics>Mooney-Rivlin, Model 1>Solid Mechanics>Ogden, and Model 1>Solid Mechanics>Varga.

5 Click Disable.

Define a parametric analysis where the applied stretch varies from 1 to 10.

6 Click to expand the Study Extensions section. Select the Continuation check box.

7 Click Add.

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s 13 | I N F L A T I O N O F A S P H E R I C A L R U B B E R B A L L O O N


odify the default solver to improve convergence.

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

Use manual scaling to help the nonlinear solver at the first steps. A constant predictor is also suitable for nonlinear materials.

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

In the Dependent Variables settings window, locate the Scaling section.

From the Method list, choose Manual.

In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1>Stationary Solver 1 node, then click Direct.

In the Direct settings window, locate the General section.

From the Solver list, choose PARDISO.

In the Model Builder window, under Study 1>Solver Configurations>Solver 1>Stationary Solver 1 click Parametric 1.

In the Parametric settings window, locate the General section.

From the Predictor list, choose Constant.

In the Model Builder window, under Study 1>Solver Configurations>Solver 1>Stationary Solver 1 click Fully Coupled 1.

In the Fully Coupled settings window, click to expand the Method and Termination section.

ontinuation parameter Parameter value list

tretch range(1, 0.1, 2) range(2.2, 0.2, 10)



13 From the Nonlinear method list, choose Constant (Newton).

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

Add a second study to solve for the Mooney-Rivlin material model, then repeat the steps described above.

R O O T

In the Model Builder window, right-click the root node and choose Add Study.

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sN O F A S P H E R I C A L R U B B E R B A L L O O N 2 0 1 3 C O M S O L

O D E L W I Z A R D



Click Finish.

T U D Y 2

In the Model Builder window, click Study 2.

In the Study settings window, locate the Study Settings section.

Clear the Generate default plots check box.

T U D Y 2




In the Physics and variables selection tree, select Model 1>Solid Mechanics>Neo-Hookean, Model 1>Solid Mechanics>Ogden, and Model 1>Solid Mechanics>Varga.

Click Disable.

Locate the Study Extensions section. Select the Continuation check box.

Click Add.

For the Mooney-Rivlin material, use a parametric analysis that changes from 1 to 5 in 50 steps.



tretch range(1, 0.1, 5)


2 0 1 3 C O M S O L

Solver 21 In the Model Builder window, right-click Study 2 and choose Show Default Solver.

2 In the Model Builder window, expand the Solver 2 node, then click Dependent Variables 1.

3 In the Dependent Variables settings window, locate the Scaling section.

4 From the Method list, choose Manual.

5 In the Model Builder window, expand the Study 2>Solver Configurations>Solver

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2>Stationary Solver 1 node, then click Direct.







In the Fully Coupled settings window, locate the Method and Termination section.

From the Nonlinear method list, choose Constant (Newton).

In the Model Builder window, right-click Study 2 and choose Compute.

O O T

ontinue with a third study for the Ogden material model.

In the Model Builder window, right-click the root node and choose Add Study.

O D E L W I Z A R D



Click Finish.

T U D Y 3






Step 1: Stationary1 In the Model Builder window, under Study 3 click Step 1: Stationary.

2 In the Stationary settings window, locate the Physics and Variables Selection section.

3 Select the Modify physics tree and variables for study step check box.

4 In the Physics and variables selection tree, select Model 1>Solid Mechanics>Neo-Hookean, Model 1>Solid Mechanics>Mooney-Rivlin, and Model 1>Solid Mechanics>Varga.

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sN O F A S P H E R I C A L R U B B E R B A L L O O N 2 0 1 3 C O M S O L

Click Disable.


Click Add.


olver 3In the Model Builder window, right-click Study 3 and choose Show Default Solver.


In the Dependent Variables settings window, locate the Scaling section.


In the Model Builder window, expand the Study 3>Solver Configurations>Solver 3>Stationary Solver 1 node, then click Direct.












2 0 1 3 C O M S O L


R O O T

Finally, add a study for a Varga material.

1 In the Model Builder window, right-click the root node and choose Add Study.

M O D E L W I Z A R D

1 Go to the Model Wizard window.

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Click Finish.

T U D Y 4




ep 1: StationaryIn the Model Builder window, under Study 4 click Step 1: Stationary.



In the Physics and variables selection tree, select Model 1>Solid Mechanics>Neo-Hookean, Model 1>Solid Mechanics>Mooney-Rivlin, and Model 1>Solid Mechanics>Ogden.

Click Disable.


Click Add.


lver 4In the Model Builder window, right-click Study 4 and choose Show Default Solver.






3 In the Dependent Variables settings window, locate the Scaling section.

4 From the Method list, choose Manual.

5 In the Model Builder window, expand the Study 4>Solver Configurations>Solver 4>Stationary Solver 1 node, then click Direct.

6 In the Direct settings window, locate the General section.

7 From the Solver list, choose PARDISO.

8 In the Model Builder window, under Study 4>Solver Configurations>Solver

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4N O F A S P H E R I C A L R U B B E R B A L L O O N 2 0 1 3 C O M S O L

4>Stationary Solver 1 click Parametric 1.






In the Model Builder window, right-click Study 4 and choose Compute.

he first default plot shows the von Mises stress on the modeled 2D cross section for e Neo Hookean material at maximum inflation. When you adjust the scaling, the

lot should become similar to the one below.

E S U L T S

tress (solid)In the Model Builder window, expand the Results>Stress (solid)>Surface 1 node, then click Deformation.

In the Deformation settings window, locate the Scale section.

In the Scale factor edit field, type 0.05.

Click the Plot button.


2 0 1 3 C O M S O L

5 Click the Zoom Extents button on the Graphics toolbar.

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ress, 3D (solid)The second default plot shows the von Mises stress in a 3D revolved plot.

o reproduce Figure 3 proceed as follows.

D Plot Group 3Right-click Results and choose 1D Plot Group.

In the 1D Plot Group settings window, locate the Plot Settings section.

Select the x-axis label check box.

In the associated edit field, type Applied stretch.

Select the y-axis label check box.

In the associated edit field, type Inflation pressure (100 Pa).

In the 1D Plot Group settings window, click to expand the Title section.

From the Title type list, choose Manual.

In the Title text area, type Inflation pressure vs. prescribed stretch..

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

In the Point Graph settings window, locate the Data section.



12 From the Data set list, choose Solution 1.

13 Select Point 3 only.

14 Click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Solid Mechanics>State variable p_f (p_f).

15 Locate the y-Axis Data section. In the Expression edit field, type p_f/100.

16 Click Replace Expression in the upper-right corner of the x-Axis Data section. From the menu, choose Definitions>Applied stretch (stretch).

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ON O F A S P H E R I C A L R U B B E R B A L L O O N 2 0 1 3 C O M S O L

Click to expand the Legends section. Select the Show legends check box.

From the Legends list, choose Manual.



Right-click Results>1D Plot Group 3>Point Graph 1 and choose Duplicate.


From the Data set list, choose Solution 2.

In the Point Graph settings window, locate the Legends section.



Right-click Point Graph 1 and choose Duplicate.


From the Data set list, choose Solution 3.

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


Right-click Point Graph 1 and choose Duplicate.


egends

eo-Hookean

egends

ooney-Rivlin

egends

gden


2 0 1 3 C O M S O L

8 From the Data set list, choose Solution 4.

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

10 Click the Plot button.

1 Right-click Point Graph 1 and choose Duplicate.

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Legends

Varga

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In the Point Graph settings window, locate the y-Axis Data section.

In the Expression edit field, type 2*(H/Ri)*((6.3e5[Pa]*(stretch^(1.3-3)-stretch^(-2*1.3-3)))+(0.012e5[Pa

]*(stretch^(5-3)-stretch^(-2*5-3)))-(0.1e5[Pa]*(stretch^(-2-3)-st

retch^(2*2-3))))/100.

Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose None.

From the Color list, choose Black.

Click to expand the Coloring and Style section. Find the Line markers subsection. From the Marker list, choose Asterisk.

In the Number edit field, type 40.

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


o reproduce Figure 4 proceed as follows.

D Plot Group 4In the Model Builder window, right-click 1D Plot Group 3 and choose Duplicate.

In the 1D Plot Group settings window, locate the Title section.

In the Title text area, type First principal stress versus prescribed stretch..

Locate the Plot Settings section. In the y-axis label edit field, type First principal stress (MPa).

Click to expand the Axis section. Select the Manual axis limits check box.

In the y maximum edit field, type 40.

egends

nalytical



7 In the Model Builder window, expand the 1D Plot Group 4 node, then click Point Graph 1.

8 In the Point Graph settings window, locate the y-Axis Data section.

9 In the Expression edit field, type solid.sp1.

10 From the Unit list, choose MPa.

11 In the Model Builder window, under Results>1D Plot Group 4 click Point Graph 2.

12 In the Point Graph settings window, locate the y-Axis Data section.

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27N O F A S P H E R I C A L R U B B E R B A L L O O N 2 0 1 3 C O M S O L

In the Expression edit field, type solid.sp1.

From the Unit list, choose MPa.

In the Model Builder window, under Results>1D Plot Group 4 click Point Graph 3.










In the Expression edit field, type ((6.3e5[Pa]*(stretch^(1.3)-stretch^(-2*1.3)))+(0.012e5[Pa]*(stret

ch^(5)-stretch^(-2*5)))-(0.1e5[Pa]*(stretch^(-2)-stretch^(2*2)))).




2 0 1 3 C O M S O L

P l a s t i c De f o rma t i o n Du r i n g t h e Expan s i o n o f a B i omed i c a l S t e n t

Introduction

Percutaneous transluminal angioplasty with stenting is a widely spread method for the trbybath

Sttonoseal

T

w

T

he

Opa

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hele 1 | P L A S T I C D E F O R M A T I O N D U R I N G T H E E X P A N S I O N O F A B I O M E D I C A L S T E N T

eatment of atherosclerosis. During the procedure, a stent is deployed into the artery using a balloon as an expander. Once the balloon-stent package is in place, the lloon is inflated to expand the stent. The balloon is then deflated and removed, but e stent remains expanded to act as a scaffold, keeping the blood vessel open.

ent design is of significance for this procedure, since serious damage can be inflicted the artery during the expansion procedure. One of the most common defect is the n-uniform deformation of the stent, where the ends expand more than the middle

ction, phenomenon which is also called dogboning. Foreshortening of the stent can so damage the artery, and it could make the positioning difficult.

he dogboning is defined according to

here rdistal and rcentral are the radii at the end and middle of the stent, respectively.

he foreshortening is defined as

re, L0 is the original length of the stent and Lload is the deformed length of the stent.

ther common parameters in stent design are the longitudinal and radial recoil. These rameters give information on the stent behavior when removing the inflated balloon.

he longitudinal recoil is defined as

re, Lunload is the length of the stent once the balloon is removed, and Lload is the ngth of the stent when the balloon is fully inflated.

dogboningrdistal rcentral

rdistal---------------------------------------=

foreshorteningL0 Lload

L0--------------------------=

LrecoilLload Lunload

Lload--------------------------------------=


2 | P L A S T I C D

The radial recoil can be defined as follow

here, Runload is the radius of the stent once the balloon is removed, and Rload is the radius of the stent when the balloon is fully inflated.

To check the viability of a stent design, you can study the deformation process under thbto

M

Tangd

F

RrecoilRload Runload

Rload---------------------------------------=E F O R M A T I O N D U R I N G T H E E X P A N S I O N O F A B I O M E D I C A L S T E N T 2 0 1 3 C O M S O L

e influence of the radial pressure that expands the stent. With this model you can oth monitor the dogboning and foreshortening effects, and draw conclusions on how change the geometry design parameters for optimum performance.

odel Definition

he model studies the Palmaz-Schatz stent model. Due to the stents circumferential d longitudinal symmetry, it is possible to model only one twenty-forth of the

eometry. Figure 1shows the geometry used in the study, represented with the meshed omain.

igure 1: reduced geometry used in the study (meshed) and full stent geometry.


2 0 1 3 C O M S O L

The main focus of the study consists in the stress evaluation in the stent. The angioplasty balloon is assumed to stretch with a maximum expansion radius of 2 mm.

M A T E R I A L

The stent is made of stainless steel. The material parameters are given in the following table.

L

Aba

R

Tth

Fth

MATERIAL PROPERTY VALUE

Y

P

I

I 3 | P L A S T I C D E F O R M A T I O N D U R I N G T H E E X P A N S I O N O F A B I O M E D I C A L S T E N T

O A D S

pply a radial outward pressure on the inner surface of the stent to represent the lloon expansion.


he stent is expanded from an original diameter of 0.74 mm to a diameter of 2 mm in e middle section.

igure 2 shows the stress distribution at maximum balloon inflation. Figure 3 shows e residual stress after the balloon deflation.

oungs modulus 193[GPa]

oissons ratio 0.27

nitial yield stress 207[MPa]

sotropic tangent modulus 692[MPa]


4 | P L A S T I C D

F

FE F O R M A T I O N D U R I N G T H E E X P A N S I O N O F A B I O M E D I C A L S T E N T 2 0 1 3 C O M S O L

igure 2: Maximum stress in the stent during the balloon inflation.

igure 3: Remanent stress in the stent after balloon deflation.


2 0 1 3 C O M S O L

Figure 4 shows the effective plastic strains at the maximum balloon inflation.

Fi

Inw

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gure 4: Effective plastic strain after stent deformation.

Figure 5, you can see the evolution of the dogboning and foreshortening effects ith respect to the pressure during the balloon inflation.

he longitudinal recoil is about 0.9%, the distal radial recoil is about 0.4%, and the ntral radial recoil is about 0.7%.


6 | P L A S T I C D

Fa

N

Tthb

Fimcodth

MbE F O R M A T I O N D U R I N G T H E E X P A N S I O N O F A B I O M E D I C A L S T E N T 2 0 1 3 C O M S O L

igure 5: Stent dogboning (blue) and foreshortening (green) versus pressure inside the ngioplasty balloon.


he maximum radius of the angioplasty balloon is represented with a step function: e pressure is applied as long as the stents inner radius is lower than the maximum

alloon radius. Above this limit the pressure is set to zero.

or a highly nonlinear problem like this, the choice of the continuation parameter can prove the convergence during the computation of the solution. A displacement ntrol parameter is usually better than a load parameter. In this model, the average

isplacement of the stents inner radius is prescribed, and a global equation computes e equivalent pressure to be applied.

odel Library path: Nonlinear_Structural_Materials_Module/Plasticity/iomedical_stent


2 0 1 3 C O M S O L

Modeling Instructions



2 Click Next.

3 In the Add physics tree, select Structural Mechanics>Solid Mechanics (solid).

4 Click Next.

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Click Finish.

E O M E T R Y 1

port 1In the Model Builder window, under Model 1 right-click Geometry 1 and choose Import.

In the Import settings window, locate the Import section.

From the Geometry import list, choose COMSOL Multiphysics file.

Click the Browse button.

Browse to the models Model Library folder and double-click the file biomedical_stent.mphbin.

Click the Import button.



asticity 1In the Model Builder window, under Model 1>Solid Mechanics right-click Linear Elastic Material 1 and choose Plasticity.

In the Plasticity settings window, locate the Plasticity Model section.

From the Plasticity model list, choose Large plastic strains.

A T E R I A L S

aterial 1In the Model Builder window, under Model 1 right-click Materials and choose Material.

In the Material settings window, locate the Material Contents section.


8 | P L A S T I C D


D

S1

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Va1

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A1

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P1

Property Name Value

Young's modulus E 193[GPa]

Poisson's ratio nu 0.27

Density rho 7050

Initial yield stress sigmags 207[MPa]

Is

N

rE F O R M A T I O N D U R I N G T H E E X P A N S I O N O F A B I O M E D I C A L S T E N T 2 0 1 3 C O M S O L

E F I N I T I O N S

tep 1In the Model Builder window, under Model 1 right-click Definitions and choose Functions>Step.

In the Step settings window, locate the Parameters section.

In the Location edit field, type 2e-3.

In the From edit field, type 1.

In the To edit field, type 0.

Click to expand the Smoothing section. In the Size of transition zone edit field, type 1e-5.

riables 1In the Model Builder window, right-click Definitions and choose Variables.



verage 1Right-click Definitions and choose Model Couplings>Average.

In the Average settings window, locate the Source Selection section.

From the Geometric entity level list, choose Edge.

Select Edge 28 only.

iecewise 1Right-click Definitions and choose Functions>Piecewise.

otropic tangent modulus Et 692[MPa]


sqrt(y^2+z^2) Radial distance from x-axis


2 0 1 3 C O M S O L

2 In the Piecewise settings window, locate the Function Name section.

3 In the Function name edit field, type r0.

4 Locate the Definition section. In the Argument edit field, type t.

5 Find the Intervals subsection. In the table, enter the following settings:

6

S

Sy1

2

B1

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G1

2

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Start End Function

0 1 (2e-3-7.1e-4)*t+7.1e-4

1

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mmetry 1In the Model Builder window, under Model 1 right-click Solid Mechanics and choose Symmetry.

Select Boundaries 5, 12, 18, 24, 30, and 31 only.

oundary Load 1In the Model Builder window, right-click Solid Mechanics and choose Boundary Load.


In the Boundary Load settings window, locate the Force section.

From the Load type list, choose Pressure.

In the p edit field, type p*step1(r).

In the Model Builder windows toolbar, click the Show button and select Advanced Physics Options in the menu.

lobal Equations 1Right-click Solid Mechanics and choose Global>Global Equations.

In the Global Equations settings window, locate the Global Equations section.


2 (2e-3-7.1e-4)*(1-t)+2e-3

ame f(u,ut,utt,t) Description

aveop1(r)-r0(t) Pressure


10 | P L A S T I C D

M E S H 1

Free Triangular 11 In the Model Builder window, under Model 1 right-click Mesh 1 and choose More

Operations>Free Triangular.

2 Select Boundary 3 only.

Size 11 Right-click Model 1>Mesh 1>Free Triangular 1 and choose Size.

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In the Size settings window, locate the Element Size section.

Click the Custom button.

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

In the associated edit field, type 6e-5.

Select the Minimum element size check box.

In the associated edit field, type 5e-6.

Select the Maximum element growth rate check box.

In the associated edit field, type 1.4.

Select the Resolution of curvature check box.

In the associated edit field, type 0.2.

wept 1 the Model Builder window, right-click Mesh 1 and choose Swept.

istribution 1In the Model Builder window, under Model 1>Mesh 1 right-click Swept 1 and choose Distribution.




E F I N I T I O N S

reate variables for the results processing.

tegration 1In the Model Builder window, under Model 1 right-click Definitions and choose Model Couplings>Integration.


2 0 1 3 C O M S O L

2 In the Integration settings window, locate the Source Selection section.

3 From the Geometric entity level list, choose Point.

4 Select Point 57 only.

5 Locate the Operator Name section. In the Operator name edit field, type central.

Integration 21 In the Model Builder window, right-click Definitions and choose Model

Couplings>Integration.

2

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Va1

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Pa1

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In the Integration settings window, locate the Source Selection section.

From the Geometric entity level list, choose Point.

Select Point 3 only.

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

riables 1In the Model Builder window, under Model 1>Definitions click Variables 1.




rametersIn the Model Builder window, right-click Global Definitions and choose Parameters.

In the Parameters settings window, locate the Parameters section.



ogboning (distal(r)-central(r))/distal(r)

Dogboning

ength 2*abs(distal(x)-central(x))

Length of the deformed stent

0 2*abs(distal(X)-central(X))

Length of the undeformed stent

oreshortenng

(length-L0)/length Foreshortening


0 Time



S T U D Y 1

Step 1: Stationary1 In the Model Builder window, under Study 1 click Step 1: Stationary.

2 In the Stationary settings window, click to expand the Results While Solving section.

3 Select the Plot check box.

4 Click to expand the Study Extensions section. Select the Continuation check box.

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mE F O R M A T I O N D U R I N G T H E E X P A N S I O N O F A B I O M E D I C A L S T E N T 2 0 1 3 C O M S O L

Click Add.


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

In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1>Dependent Variables 1 node, then click Pressure (mod1.ODE1).

In the State settings window, locate the Scaling section.


In the Scale edit field, type 1e6.

In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1>Stationary Solver 1 node, then click Parametric 1.



Add a stop condition to prevent the computed pressure from becoming negative.

Right-click Study 1>Solver Configurations>Solver 1>Stationary Solver 1>Parametric 1 and choose Stop Condition.

In the Stop Condition settings window, locate the Stop Expressions section.

Click Add.


Specify that the solution is to be stored just before the stop condition is reached.

arameter value list

ange(0,1e-2,1.5)

top expression

od1.p


2 0 1 3 C O M S O L

13 Locate the Output at Stop section. From the Add solution list, choose Step before stop.


R E S U L T S

Data SetsUse Mirror 3D and Sector 3D data sets to display the solution on the entire geometry.

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In the Model Builder window, expand the Results>Data Sets node.

Right-click Data Sets and choose More Data Sets>Mirror 3D.


In the Mirror 3D settings window, locate the Data section.

From the Data set list, choose Mirror 3D 1.

Locate the Plane Data section. From the Plane list, choose zx-planes.

Right-click Data Sets and choose More Data Sets>Sector 3D.

In the Sector 3D settings window, locate the Data section.


Locate the Axis Data section. In row Point 2, set x to 1.

In row Point 2, set z to 0.

Locate the Symmetry section. In the Number of sectors edit field, type 6.

ress (solid)In the Model Builder window, under Results click Stress (solid).

In the 3D Plot Group settings window, locate the Data section.

From the Data set list, choose Sector 3D 1.


Click the Go to Default 3D View button on the Graphics toolbar.

In the Model Builder window, click Stress (solid).


From the Time list, choose 1.


ress (solid) 1Right-click Stress (solid) and choose Duplicate.



2 In the Model Builder window, expand the Stress (solid) 1 node, then click Surface 1.

3 In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Solid Mechanics>Strain (Gauss points)>Effective plastic strain, Gauss-point evaluation (solid.epeGp).

4 In the Model Builder window, right-click Stress (solid) 1 and choose Rename.

5 Go to the Rename 3D Plot Group dialog box and type Effective plastic strain in the New name edit field.

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6E F O R M A T I O N D U R I N G T H E E X P A N S I O N O F A B I O M E D I C A L S T E N T 2 0 1 3 C O M S O L

Click OK.

Right-click Stress (solid) 1 and choose Plot.



From the Time selection list, choose From list.

In the Times list, select all the solution steps between 0 and 1.

Right-click Results>1D Plot Group 3 and choose Global.

In the Global settings window, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Definitions>Dogboning (dogboning).

Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Definitions>Foreshortening (foreshortening).

Click Replace Expression in the upper-right corner of the x-Axis Data section. From the menu, choose Solid Mechanics>Pressure (p).


valuate the length recoil, the distal radial recoil, and the central radial recoil.

erived ValuesIn the Model Builder window, under Results right-click Derived Values and choose Global Evaluation.

In the Global Evaluation settings window, locate the Data section.

From the Time selection list, choose From list.

In the Times list, select 1.

Locate the Expression section. In the Expression edit field, type (length-with(103,length))/length.

Select the Description check box.


2 0 1 3 C O M S O L

7 In the associated edit field, type Longitudinal recoil.

8 Click the Evaluate button.

9 Right-click Results>Derived Values>Global Evaluation 1 and choose Rename.

10 Go to the Rename Global Evaluation dialog box and type Longitudinal recoil evaluation in the New name edit field.

11 Click OK.

12 Right-click Results>Derived Values>Global Evaluation 1 and choose Duplicate.

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Right-click Results>Derived Values>Longitudinal recoil evaluation 1 and choose Rename.

Go to the Rename Global Evaluation dialog box and type Distal radial recoil evaluation in the New name edit field.

Click OK.

In the Global Evaluation settings window, locate the Expression section.

In the Expression edit field, type (distal(r)-with(103,distal(r)))/distal(r).

In the Description edit field, type Distal radial recoil.

Click the Evaluate button.

Right-click Results>Derived Values>Longitudinal recoil evaluation 1 and choose Duplicate.

Right-click Results>Derived Values>Distal radial recoil evaluation 1 and choose Rename.

Go to the Rename Global Evaluation dialog box and type Central radial recoil evaluation in the New name edit field.

Click OK.

In the Global Evaluation settings window, locate the Expression section.

In the Expression edit field, type (central(r)-with(103,central(r)))/central(r).

In the Description edit field, type Central radial recoil.


he steps below illustrate how to display the geometry as in Figure 1.

D Plot Group 4In the Model Builder window, right-click Results and choose 3D Plot Group.



2 Right-click 3D Plot Group 4 and choose Surface.

3 In the Surface settings window, locate the Data section.

4 From the Data set list, choose Sector 3D 1.

5 Locate the Coloring and Style section. From the Coloring list, choose Uniform.

6 From the Color list, choose Gray.

7 In the Model Builder window, right-click 3D Plot Group 4 and choose Rename.

8 Go to the Rename 3D Plot Group dialog box and type Full geometry and mesh in

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6E F O R M A T I O N D U R I N G T H E E X P A N S I O N O F A B I O M E D I C A L S T E N T 2 0 1 3 C O M S O L

the New name edit field.

Click OK.

ll geometry and meshRight-click 3D Plot Group 4 and choose Surface.

In the Surface settings window, locate the Coloring and Style section.

From the Coloring list, choose Uniform.

From the Color list, choose Black.

Select the Wireframe check box.



2 0 1 3 C O M S O L

B r a c k e t Geome t r y

This is a template MPH-file containing the bracket geometry. For a description of this model, including detailed step-by-step instructions showing how to build it, see the section Modeling Contact Problems in the book Introduction to the Structural Mechanics Module.

Mbr 1 | B R A C K E T G E O M E T R Y

odel Library path: Nonlinear_Structural_Materials_Module/Plasticity/acket_basic


2 | B R A C K E T G E O M E T R Y 2 0 1 3 C O M S O L


2 0 1 3 C O M S O L

B r a c k e tPla s t i c i t y Ana l y s i s

Introduction

The elastoplastic material model represents the material behavior when the stresses are higher than the yield stress of the material. Above this value, inelastic strains develop and extra parameters are required to represent the behavior.

Intoth

Itwre

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Dge 1 | B R A C K E T P L A S T I C I T Y A N A L Y S I S

this example you will learn how to perform a stress analysis of a bracket subjected external load. The applied load is increased so that the stresses locally become higher an the yield stress.

is recommended you review the Introduction to the Structural Mechanics Module, hich includes background information and discusses the bracket_basic.mph model levant to this example.

odel Definition

his model is an extension to the model example described in the section The undamentals: A Static Linear Analysis in the Introduction to the Structural echanics Module.

ue to symmetry in the model settings, you can consider only one half of the full ometry, see Figure 1.


2 | B R A C K E T

F

T

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FP L A S T I C I T Y A N A L Y S I S 2 0 1 3 C O M S O L

igure 1: Geometry of the bracket

he bracket displacements are fixed in the bolt regions.

he maximum pressure applied at the bracket hole is 90 MPa.


igure 2 below shows the von Mises stress for the final value of the applied load.


2 0 1 3 C O M S O L

Fi

Yoa

Fw 3 | B R A C K E T P L A S T I C I T Y A N A L Y S I S

gure 2: Von Mises stress distribution

u can see that the maximum stress is above the yield stress level (260 MPa) and that high stress region is located at the base of the bracket and around the hole.

igure 3 below shows the plastic region in the bracket. These regions correspond to here the plastic strain is above 0.


4 | B R A C K E T

F

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igure 3: Plastic region in the bracket


lastoplastic problems are path dependent, which means that the current plastic strain aluation depends on a previous result. To reach good accuracy in the solution you en need to include a continuation in the solver settings in order to ramp up the plied load to the structure.

odel Library path: Nonlinear_Structural_Materials_Module/Plasticity/racket_plasticity


From the View menu, choose Model Library.

Go to the Model Library window.

In the Model Library tree, select Nonlinear Structural Materials Module>Plasticity>bracket basic.


2 0 1 3 C O M S O L

4 Click Open.

M O D E L 1

In the Model Builder window, right-click Model 1 and choose Add Physics.



2 In the Add physics tree, select Structural Mechanics>Solid Mechanics (solid).

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Click Add Selected.

Click Next.


Click Finish.

E O M E T R Y 1

port 1In the Model Builder window, under Model 1>Geometry 1 click Import 1.

In the Import settings window, locate the Import section.

From the Geometry import list, choose COMSOL Multiphysics file.

Click the Browse button.

Browse to the models Model Library folder and double-click the file bracket_symmetry.mphbin.

Click the Import button.



In the Model Builder window, expand the Global Definitions node.

In the Parameters settings window, locate the Parameters section.


near Elastic Material 1he plastic material definition is available as a subnode of the linear elastic material odel node.


heta0 135[deg] Direction of load


6 | B R A C K E T

1 In the Model Builder window, expand the Model 1>Solid Mechanics node.

2 Right-click Linear Elastic Material 1 and choose Plasticity.

Plasticity 1In the Plasticity Settings page you can define plasticity properties such as yield function, initial yield stress, and hardening model. For this example the default settings (von Mises stress yield function and Isotropic hardening model) are fine.

M

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A T E R I A L S

he elastoplastic material model requires additional material properties, in this case itial yield stress and the isotropic tangent modulus.

tructural steelIn the Model Builder window, under Model 1>Materials click Structural steel.




ou can now add a displacement constraint, a boundary load and symmetry condition the model.

xed Constraint 1In the Model Builder window, under Model 1 right-click Solid Mechanics and choose More>Fixed Constraint.

In the Fixed Constraint settings window, locate the Domain Selection section.

From the Selection list, choose Box 1.

ymmetry 1In the Model Builder window, right-click Solid Mechanics and choose Symmetry.

Select Boundaries 43 and 44 only.

oundary Load 1Right-click Solid Mechanics and choose Boundary Load.

In the Boundary Load settings window, locate the Boundary Selection section.

From the Selection list, choose Box 2.

roperty Name Value

ield stress level sigmags 260[MPa]

otropic tangent modulus Et 200[MPa]


2 0 1 3 C O M S O L

4 Locate the Coordinate System Selection section. From the Coordinate system list, choose Rotated System 2.

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

M

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0 x1

loadIntensity x2

0 x3 7 | B R A C K E T P L A S T I C I T Y A N A L Y S I S

E S H 1

se a refined mesh in the geometry region where you can expect the material to enter e plastic region.

ee Tetrahedral 1 the Model Builder window, under Model 1 right-click Mesh 1 and choose Free trahedral.

ze 1In the Model Builder window, under Model 1>Mesh 1 right-click Free Tetrahedral 1 and choose Size.

In the Size settings window, locate the Geometric Entity Selection section.

From the Geometric entity level list, choose Boundary.


In the Size settings window, locate the Element Size section.

From the Predefined list, choose Finer.


T U D Y 1

ep 1: StationaryIn the Model Builder window, expand the Study 1 node, then click Step 1: Stationary.

In the Stationary settings window, click to expand the Results While Solving section.

Using the Plot while solving option you have the possibility to visualize how the solution develops while the solver is running.

Select the Plot check box.

Click to expand the Study Extensions section. Select the Continuation check box.

Click Add.


8 | B R A C K E T



R E S U L T S

The default plot shows the von Mises stress for the final value of the applied load.

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Continuation parameter Parameter value list

P0 range(2e7,5e6,9e7)P L A S T I C I T Y A N A L Y S I S 2 0 1 3 C O M S O L

D Plot Group 2ollow the step below to generate Figure 3.

In the Model Builder window, right-click Results and choose 3D Plot Group.

In the Model Builder window, under Results right-click 3D Plot Group 2 and choose Surface.

In the Surface settings window, locate the Expression section.

In the Expression edit field, type solid.epe>0.



2 0 1 3 C O M S O L

Comp r e s s i o n o f an E l a s t o p l a s t i c P i p e

Introduction

In offshore applications, it is sometimes necessary to quickly seal a pipe as part of the prevention of a blowout. This examples shows a simulation, in which a circular pipe is squeezed between two flat stiff indenters.

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he model serves as an example of an analysis with very large plastic strains and ntact.

odel Definition

he pipe has an external radius, R0, of 200 mm and a wall thickness of 25 mm. The aterial is a stainless steel with the following properties

he hardening curve is available as a text file containing pairs of data (plastic strain, ress) which can be imported as a function. The function is shown in Figure 1below. he stress is measured as true (Cauchy) stress, and the strain is measured as true ogarithmic) strain. The data can thus be used directly as input to COMSOL when rge strain plasticity is used. Note that if the strain is above the ultimate strain, then e curve is implicitly flat, so that deformation will continue under constant stress.

BLE 1: MATERIAL DATA

ROPERTY VALUE

oungs modulus 195 GPa

oissons ratio 0.3

ield stress 250 MPa

ltimate tensile stress 616 MPa

ltimate strain 0.52


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igure 1: The hardening curve.

he pipe is compressed between two flat indenters that can be considered as rigid. The riginal position of each indenter is 1 mm from the outer pipe wall. During the


2 0 1 3 C O M S O L

compression of the pipe, the distance between the indenters is decreased by 300 mm, and then they are retracted to their original positions.

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gure 2: The geometry; pipe and indenters.

ue to the symmetries, only one quarter of the geometry needs to be modeled. The oblem is considered as 2D with the plane strain assumption.


his model exhibits extremely large plastic strains. The deformation and stress state at e maximum compression are shown in Figure 3. The maximum stress displayed is ightly above the ultimate tensile stress (616 MPa). This is caused by the extrapolation



of the results from the integration points inside the elements, where the constitutive law is exactly fulfilled.

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igure 3: Effective stress at maximum compression.

he distribution of the effective plastic strains is shown in Figure 4 and Figure 5. As n be seen, the peak value (0.95) is far above the ultimate strain (0.52). All values ove the ultimate strain are however on the inside of the pipe, where the strain state

mainly in compression. At the outer edge of the pipe, the plastic strain approximately aches the ultimate strain. There is thus a certain risk that cracks could start forming. he values of ultimate stress and strain are related to the specimen used for the testing sually a cylindrical bar) and cannot directly be transferred to general multiaxial nditions.


2 0 1 3 C O M S O L

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gure 4: Effective plastic strain at maximum compression.

gure 5: Effective plastic strain at maximum compression, detail. The black contour dicates the ultimate strain (0.52).



The final state after the retraction of the indenters is shown in Figure 6. There is some reversed yielding during the unloading process. The springback is very small.

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igure 6: Deformed shape and residual stresses at the end of the process.

he load used to compress the pipe is computed from the reaction force in the denter, and it is shown in Figure 7.


2 0 1 3 C O M S O L

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gure 7: The applied force as function of the loading parameter.

odel Library path: Nonlinear_Structural_Materials_Module/Plasticity/mpressed_elastoplastic_pipe


O D E L W I Z A R D


Click the 2D button.

Click Next.

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

Click Next.


Click Finish.



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

Parameters1 In the Model Builder window, right-click Global Definitions and choose Parameters.

2 In the Parameters settings window, locate the Parameters section.


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Name Expression Description

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dd the hardening curve for the elastoplastic material.

terpolation 1Right-click Global Definitions and choose Functions>Interpolation.

In the Interpolation settings window, locate the Definition section.

Click Load from File.

Browse to the models Model Library folder and double-click the file compressed_elastoplastic_pipe_stress_strain.txt.

In the Function name edit field, type hardFcn.

Locate the Interpolation and Extrapolation section. From the Interpolation list, choose Piecewise cubic.

Locate the Units section. In the Arguments edit field, type 1.

In the Function edit field, type MPa.


dd a function for the displacement if the indenting part.

terpolation 2Right-click Global Definitions and choose Functions>Interpolation.

In the Interpolation settings window, locate the Definition section.

In the Function name edit field, type compr.

o 200[mm] Outer radius

hic 25[mm] Pipe wall thickness

i Ro-thic Inner radius

ara 0 Solution parameter


2 0 1 3 C O M S O L


5 Locate the Units section. In the Arguments edit field, type 1.

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In the Function edit field, type m.


E O M E T R Y 1

ircle 1In the Model Builder window, under Model 1 right-click Geometry 1 and choose Circle.


In the Radius edit field, type Ro.


ircle 2In the Model Builder window, right-click Geometry 1 and choose Circle.


In the Radius edit field, type Ri.


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

Select the object c1 only.

In the Difference settings window, locate the Difference section.

Under Objects to subtract, click Activate Selection.

Select the object c2 only.


ectangle 1Right-click Geometry 1 and choose Rectangle.

In the Rectangle settings window, locate the Size section.

In the Width edit field, type 0.05.


10 | C O M P R E S S

4 In the Height edit field, type 1.3*Ro.

5 Locate the Position section. In the x edit field, type Ro+0.001.

Form Union1 In the Model Builder window, under Model 1>Geometry 1 click Form Union.

2 In the Finalize settings window, locate the Finalize section.

3 From the Finalization method list, choose Form an assembly.

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E F I N I T I O N S

In the Model Builder window, under Model 1 right-click Definitions and choose Pairs>Contact Pair.


In the Pair settings window, click Activate Selection in the upper-right corner of the Destination Boundaries section. Select Boundary 4 only.


lasticity 1In the Model Builder window, under Model 1>Solid Mechanics right-click Linear Elastic Material 1 and choose Plasticity.


In the Plasticity settings window, locate the Plasticity Model section.

From the Plasticity model list, choose Large plastic strains.

From the Isotropic hardening list, choose Use hardening function data.

In the h(pe) edit field, type hardFcn(solid.epe).

A T E R I A L S

aterial 1In the Model Builder window, under Model 1 right-click Materials and choose Material.



2 0 1 3 C O M S O L


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Property Name Value

Young's modulus E 195[GPa]

Poisson's ratio nu 0.3

Density rho 8000

Initial yield stress sigmags 250[MPa] 11 | C O M P R E S S I O N O F A N E L A S T O P L A S T I C P I P E


ontact 1In the Model Builder window, under Model 1 right-click Solid Mechanics and choose Pairs>Contact.

In the Contact settings window, locate the Pair Selection section.

In the Pairs list, select Contact Pair 1.

In the Contact settings window, locate the Penalty Factor section.

From the list, choose Speed.

mmetry 1In the Model Builder window, right-click Solid Mechanics and choose Symmetry.


escribed Displacement 1Right-click Solid Mechanics and choose More>Prescribed Displacement.


In the Prescribed Displacement settings window, locate the Prescribed Displacement section.

Select the Prescribed in x direction check box.

Select the Prescribed in y direction check box.

In the u0x edit field, type -compr(para).

E S H 1

In the Model Builder window, under Model 1 click Mesh 1.

In the Mesh settings window, locate the Mesh Settings section.

From the Sequence type list, choose User-controlled mesh.



Free Triangular 11 In the Model Builder window, under Model 1>Mesh 1 right-click Free Triangular 1 and

choose Delete.

2 Click Yes to confirm.

Mapped 1Right-click Mesh 1 and choose Mapped.

Distribution 11

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In the Model Builder window, under Model 1>Mesh 1 right-click Mapped 1 and choose Distribution.




One element is enough on the indenter, since the whole domain is under displacement control.



From the Distribution properties list, choose Predefined distribution type.



In the Element ratio edit field, type 2.

Select the Symmetric distribution check box.






T U D Y 1



2 0 1 3 C O M S O L

2 In the Stationary settings window, click to expand the Study Extensions section.

3 Select the Continuation check box.

4 Click Add.


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Continuation parameter Parameter value list

para range(0,0.02,1) range(1.005, 0.005, 1.05) 1.1 2 13 | C O M P R E S S I O N O F A N E L A S T O P L A S T I C P I P E

Click to expand the Results While Solving section. Select the Plot check box.

Click the Compute icon on the toolbar.

irror the solution twice to get a full view of the pipe.

E S U L T S

ata SetsIn the Model Builder window, under Results right-click Data Sets and choose More Data Sets>Mirror 2D.


In the Mirror 2D settings window, locate the Data section.


Locate the Axis Data section. In row Point 2, set x to 1.

In row Point 2, set y to 0.

ress (solid)In the Model Builder window, under Results click Stress (solid).





Plot the stresses at the maximum compression. This gives Figure 3.

From the Parameter value (para) list, choose 1.


Create an animation of the compression process.

Click the Player button on the main toolbar.



ExportCreate a graph of the applied force as function of the compression.

Derived Values1 In the Model Builder window, under Results right-click Derived Values and choose

Integration>Surface Integration.

2 Select Domain 2 only.

3 In the Surface Integration settings window, locate the Expression section.

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In the Expression edit field, type -solid.RFx*2/1E6.


In the Table window, click Table Graph.

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

In the 1D Plot Group settings window, locate the Plot Settings section.

Select the y-axis label check box.

In the associated edit field, type Force/unit length [MN/m].


In the Model Builder window, right-click 1D Plot Group 2 and choose Rename.

Go to the Rename 1D Plot Group dialog box and type Compression force in the New name edit field.

Click OK.


In the Model Builder window, under Results right-click 2D Plot Group 3 and choose Surface.

In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Solid Mechanics>Strain (Gauss points)>Effective plastic strain, Gauss-point evaluation (solid.epeGp).

Set the plot scale limit to the ultimate strain.

Click to expand the Range section. Select the Manual color range check box.

In the Maximum edit field, type 0.52.

Right-click Results>2D Plot Group 3>Surface 1 and choose Deformation.

In the Deformation settings window, locate the Scale section.


2 0 1 3 C O M S O L

8 Select the Scale factor check box.

9 In the associated edit field, type 1.

Add a contour showing where the ultimate strain is exceeded.

1 In the Model Builder window, right-click 2D Plot Group 3 and choose Contour.

2 In the Contour settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Solid Mechanics>Strain (Gauss points)>Effective plastic strain, Gauss-point evaluation (solid.epeGp).

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Locate the Levels section. From the Entry method list, choose Levels.

In the Levels edit field, type 0.52.

Locate the Coloring and Style section. From the Coloring list, choose Uniform.

From the

nonlinear structural materials model library manual

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