nonlinear structural materials model library manual

5/20/2018 Nonlinear Structural Materials Model Library Manual

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VERSION 4.3b

Nonlinear Structural Materials Module

Model Library Manual


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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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2 0 1 3 C O M S O L 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

I n f l a t i o n o f a Sph e r i c a l Rubb e r

Ba l l o on

Introduction

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

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

A controlled inflation could be of importance in clinical applications, cardiovascular

research, and medical device industry (Ref. 2), among others.

The example is taken from the book Nonlinear Solid Mechanics, by G. Holzapfel

(Ref. 1).

Model Definition

This model compares the hoop stress and inflation pressure as a function of the stretch

for a spherical rubber balloon.

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

In this example, the following four hyperelastic material models are compared:

Neo-Hookean, Money-Rivlin, Ogden, and Varga.

Thickness

20

Symmetry

InnerRadius


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2 | 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 2 0 1 3 C O M S O L

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.

Figure 2: 2D axisymmetric geometry and mesh.

Results and Discussion

The results are compared to the analytical expression for a thin-walled vessel. The

inflation pressure is a function of the hoop stress , current inner radius rand current

thickness h

For spherical balloons, the hoop stress is equal to the largest principal stresses 1and 2. Two of the principal stretches lay on the plane tangential to the sphere and are

equal, 1 2 r/R, which is typical for equibiaxial deformation. Here, randR

are the current and initial inner radii, respectively.

Due to the nearly incompressibility assumption, the third principal stretch (this is the

stretch in the radial direction) 3 1/2 h/H, where handHare the current and

initial balloon thicknesses, respectively.

pi 2h

r---=


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The analytical expression for the hoop stress for the Ogden material model becomes

(Ref. 1)

where pand pare Ogden parameters, and is the largest principal stretch.

Since rRand hH/2, the analytical expression for the inflation pressure is

calculated as a function of Ogden parameters, stretch, initial thickness and initial inner

radius

The results are in excellent agreement with experimental results and the figures

portrayed in Ref. 1.

The experiments show a rapid rise in the internal pressure until reaching a maximum

value, followed by a pressure decrease until reaching a minimum, and then increasing

again. This phenomenon is similar to snap-through buckling, and can only be

recovered by the Ogden material model. The Neo-Hookean and the Varga material

models can only reproduce balloon inflations at small strain levels.

p p

2p

p 1=

N

=

pi 2h

r--- 2

H

R----- p

p 3 2p 3

p 1=

N

= =


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


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

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

Notes About the COMSOL Implementation

Different hyperelastic material models are constructed by specifying different elastic

strain energy expressions. The Nonlinear Structural Materials module provides several

predefined material models together with an option to enter user defined expressions

for the strain energy density.

The predefined nearly incompressible version of the Neo-Hookean material uses the

isochoric invariant and the initial bulk modulus

In this example, 422.5 kPa and 105 The Lam parameter can be seen as

representing the shear modulus at small strains.

I1 Cel

Ws1

2--- I1 3

1

2--- Jel 1

2+=


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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 ratioJel

The material parameters C10and C01are related to the shear modulus C10C01

In this example, they are set as C10 and C01 , so that the relation

C10 C01is fulfilled.The predefined nearly incompressible Ogden material is implemented with the

isochoric elastic stretches and the initial bulk modulus

withN , and the Ogden parameters as written in Table 1

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

When the relation between the applied load and the displacement is not unique, a

suitable modeling technique is to use an algebraic equation that controls the applied

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

In this example, a Global Equation uses the radial displacement at point 3 to add an

extra degree of freedom for the inflation pressure.

Global equations are a way of adding an additional equation to a model. A global

equation can be used to describe a load, constraint, material property, or anything else

in the model that has a uniquely definable solution. In this example, the model is

TABLE 1: OG DEN PARAMETE RS

p p p(kPa)

1 1.3 630

2 5.0 1.2

3 -2.0 -10

I1 Cel I2 Cel

Ws C10 I1 3 C01 I2 3 1

2--- Jel 1

2+ +=

Wspp------ el1

pel2

pel3

p3+ +

p 1=

N

1

2--- Jel 1

2+=

Ws 2 el1 el2 el3 3+ + 1

2--- Jel 1

2+=


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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

feedback from intravascular optical coherence tomography: preliminary study on an

artery phantom. IEEE Trans Biomed Eng. 59, pp. 697-705, 2012.

Model Library path: Nonlinear_Structural_Materials_Module/

Hyperelasticity/balloon_inflation

Modeling Instructions

M O D E L W I Z A R D

1 Go to the Model Wizardwindow.

2 Click the 2D axisymmetricbutton.

3 Click Next.

4 In the Add physicstree, select Structural Mechanics>Solid Mechanics (solid).5 Click Next.

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

7 Click Finish.

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

1 In the Model Builderwindow, right-click Global Definitionsand choose Parameters.

Begin by defining model parameters.

Parameters

1 In the Parameterssettings window, locate the Parameterssection.


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2 In the table, enter the following settings:

Setting the bulk modulus to 105times the shear modulus is based on the

assumption that the material is incompressible.

Variables 1

1 Right-click Global Definitionsand choose Variables.

2 In the Variablessettings window, locate the Variablessection.


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

G E O M E T R Y 1

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

Circle 1

1 In the Model Builderwindow, under Model 1right-click Geometry 1and choose Circle.

2 In the Circlesettings window, locate the Size and Shapesection.

3 In the Radiusedit field, type Ri+H.

4 In the Sector angleedit field, type 20.

5 Click to expand the Layerssection. In the table, enter the following settings:

6 Click the Build Allbutton.

7 Click the Zoom Extentsbutton on the Graphics toolbar.

Delete Entities 1

1 In the Model Builderwindow, right-click Geometry 1and choose Delete Entities.

Name Expression DescriptionRi 10[m] Inner radius

H 0.1[m] Thickness

mu 4.225e5[Pa] Shear modulus

kappa 1e5*mu Bulk modulus

stretch 1 Applied stretch

Name Expression Descriptionu_appl (stretch-1)*Ri Applied displacement

Layer name Thickness (m)

Layer 1 H


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2 In the Delete Entitiessettings window, locate the Entities or Objects to Deletesection.

3 From the Geometric entity levellist, choose Domain.

4 On the object c1, select Domain 1 only.



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

Add the four hyperelastic material models.

Hyperelastic Material 11 In the Model Builderwindow, under Model 1right-click Solid Mechanicsand choose

Hyperelastic Material.

2 Select Domain 1 only.

3 In the Hyperelastic Materialsettings window, locate the Hyperelastic Materialsection.

4 Select the Nearly incompressible materialcheck box.

5 In the edit field, type kappa.

6 From the list, choose User defined. In the associated edit field, type mu.

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

8 Go to the Rename Hyperelastic Materialdialog box and type Neo-Hookeanin the New

nameedit field.

9 Click OK.

Hyperelastic Material 21 Right-click Solid Mechanicsand choose Hyperelastic Material.



4 From the Material modellist, choose Mooney-Rivlin, two parameters.

5 From the C10list, choose User defined. In the associated edit field, type 0.4375*mu.

6From the

C01list, choose

User defined. In the associated edit field, type

0.0625*mu.



9 Go to the Rename Hyperelastic Materialdialog box and type Mooney-Rivlinin the

New nameedit field.

10 Click OK.


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Hyperelastic Material 3

1 Right-click Solid Mechanicsand choose Hyperelastic Material.



4 From the Material modellist, choose Ogden.


6 Click Addtwice.

7 In the Ogden parameterstable, enter the following settings:



10 Go to the Rename Hyperelastic Materialdialog box and type Ogdenin the New name

edit field.

11 Click OK.

Hyperelastic Material 4

1 Right-click Solid Mechanicsand choose Hyperelastic Material.



4 From the Material modellist, choose User defined.


6 In the Wsisoedit field, type

2*mu*(solid.stchelp1+solid.stchelp2+solid.stchelp3-3) .

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


9 Go to the Rename Hyperelastic Materialdialog box and type Vargain the New name

edit field.

10 Click OK.

Apply symmetry conditions.

Shear modulus (Pa) Alpha parameter

6.3e5 1.3

0.012e5 5

-0.1e5 -2


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Symmetry 1

1 Right-click Solid Mechanicsand choose Symmetry.

2 Select Boundaries 1 and 2 only.

Control the inflation of the balloon by the pressure.

Boundary Load 1

1 Right-click Solid Mechanicsand choose Boundary Load.

2 Select Boundary 3 only.

3 In the Boundary Loadsettings window, locate the Forcesection.

4 From the Load typelist, choose Pressure.

5 In thepedit field, type p_f.

You will define the pressure p_fusing a Global Equationfeature shortly. First, define

an integration coupling operator to evaluate the displacement at Point 3.

D E F I N I T I O N S

Integration 11 In the Model Builderwindow, under Model 1right-click Definitionsand choose Model

Couplings>Integration.

2 In the Integrationsettings window, locate the Source Selectionsection.

3 From the Geometric entity levellist, choose Point.

4 Select Point 3 only.

Variables 21 In the Model Builderwindow, right-click Definitionsand choose Variables.




1 In the Model Builderwindows toolbar, click the Showbutton and select Advanced

Physics Optionsin the menu to allow to add a global equation and other advanced

modeling features.

Name Expression Description

ub intop1(u) Radial displacement, inner

boundary


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Global Equations 1

1 In the Model Builderwindow, under Model 1right-click Solid Mechanicsand choose

Global>Global Equations.2 In the Global Equationssettings window, locate the Global Equationssection.


M E S H 1

Mapped 1

In the Model Builderwindow, under Model 1right-click Mesh 1and choose Mapped.

Distribution 1

1 In the Model Builderwindow, under Model 1>Mesh 1right-click Mapped 1and choose

Distribution.

2 Select Boundary 2 only.3 In the Distributionsettings window, locate the Distributionsection.

4 In the Number of elementsedit field, type 3.

Distribution 2

1 Right-click Mapped 1and choose Distribution.


3 In the Distributionsettings window, locate the Distributionsection.


5 In the Model Builderwindow, right-click Mesh 1and choose Build All.


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

S T U D Y 1

Step 1: Stationary

1 In the Model Builderwindow, under Study 1click Step 1: Stationary.

2 In the Stationarysettings window, locate the Physics and Variables Selectionsection.

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

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

p_f ub-u_appl 0 0


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4 In the Physics and variables selectiontree, 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 Extensionssection. Select the Continuationcheck box.

7 Click Add.


Modify the default solver to improve convergence.

Solver 1

1 In the Model Builderwindow, right-click Study 1and 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.

2 In the Model Builderwindow, expand the Solver 1node, then click Dependent

Variables 1.

3 In the Dependent Variablessettings window, locate the Scalingsection.

4 From the Methodlist, choose Manual.

5 In the Model Builderwindow, expand the Study 1>Solver Configurations>Solver1>Stationary Solver 1node, then click Direct.

6 In the Directsettings window, locate the Generalsection.

7 From the Solverlist, choose PARDISO.

8 In the Model Builderwindow, under Study 1>Solver Configurations>Solver

1>Stationary Solver 1click Parametric 1.

9 In the Parametricsettings window, locate the Generalsection.

10 From the Predictorlist, choose Constant.


1>Stationary Solver 1click Fully Coupled 1.

12 In the Fully Coupledsettings window, click to expand the Method and Termination

section.

Continuation parameter Parameter value list

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


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13 From the Nonlinear methodlist, choose Constant (Newton).

14 In the Model Builderwindow, right-click Study 1and 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 Builderwindow, right-click the root node and choose Add Study.




3 Click Finish.

S T U D Y 2

1 In the Model Builderwindow, click Study 2.

2 In the Studysettings window, locate the Study Settingssection.

3 Clear the Generate default plotscheck box.

S T U D Y 2

Step 1: Stationary





Mechanics>Neo-Hookean, Model 1>Solid Mechanics>Ogden, and Model 1>Solid

Mechanics>Varga.

5 Click Disable.

6 Locate the Study Extensionssection. Select the Continuationcheck box.

7 Click Add.For the Mooney-Rivlin material, use a parametric analysis that changes from 1 to 5

in 50 steps.



stretch range(1, 0.1, 5)


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Solver 2


2 In the Model Builderwindow, expand the Solver 2node, then click DependentVariables 1.



5 In the Model Builderwindow, expand the Study 2>Solver Configurations>Solver

2>Stationary Solver 1node, then click Direct.

6 In the Directsettings window, locate the Generalsection.7 From the Solverlist, choose PARDISO.







12 In the Fully Coupledsettings window, locate the Method and Terminationsection.



R O O T

Continue with a third study for the Ogden material model.

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




3 Click Finish.

S T U D Y 3





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Step 1: Stationary





Mechanics>Neo-Hookean, Model 1>Solid Mechanics>Mooney-Rivlin, and Model 1>Solid

Mechanics>Varga.

5 Click Disable.

6 Locate the Study Extensionssection. Select the Continuationcheck box.7 Click Add.


Solver 31 In the Model Builderwindow, right-click Study 3and choose Show Default Solver.


Variables 1.



5 In the Model Builderwindow, expand the Study 3>Solver Configurations>Solver3>Stationary Solver 1node, then click Direct.














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R O O T

Finally, add a study for a Varga material.

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




3 Click Finish.

S T U D Y 4




Step 1: Stationary1 In the Model Builderwindow, under Study 4click Step 1: Stationary.




Mechanics>Neo-Hookean, Model 1>Solid Mechanics>Mooney-Rivlin, and Model 1>Solid

Mechanics>Ogden.

5 Click Disable.

6 Locate the Study Extensionssection. Select the Continuationcheck box.

7 Click Add.


Solver 4



Variables 1.




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4>Stationary Solver 1node, then click Direct.





9 In the Parametricsettings window, locate the Generalsection.10 From the Predictorlist, choose Constant.





14 In the Model Builderwindow, right-click Study 4and choose Compute.The first default plot shows the von Mises stress on the modeled 2D cross section for

the Neo Hookean material at maximum inflation. When you adjust the scaling, the

plot should become similar to the one below.

R E S U L T S

Stress (solid)

1 In the Model Builderwindow, expand the Results>Stress (solid)>Surface 1node, then

click Deformation.

2 In the Deformationsettings window, locate the Scalesection.

3 In the Scale factoredit field, type 0.05.

4 Click the Plotbutton.


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Stress, 3D (solid)

The second default plot shows the von Mises stress in a 3D revolved plot.

To reproduce Figure 3proceed as follows.

1D Plot Group 3

1 Right-click Resultsand choose 1D Plot Group.

2 In the 1D Plot Groupsettings window, locate the Plot Settingssection.

3 Select the x-axis labelcheck box.

4 In the associated edit field, typeApplied stretch.

5 Select the y-axis labelcheck box.

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

7 In the 1D Plot Groupsettings window, click to expand the Titlesection.

8 From the Title typelist, choose Manual.

9 In the Titletext area, type Inflation pressure vs. prescribed stretch..

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

11 In the Point Graphsettings window, locate the Datasection.


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12 From the Data setlist, choose Solution 1.

13 Select Point 3 only.

14 Click Replace Expressionin the upper-right corner of the y-Axis Datasection. Fromthe menu, choose Solid Mechanics>State variable p_f (p_f).

15 Locate the y-Axis Datasection. In the Expressionedit field, type p_f/100.

16 Click Replace Expressionin the upper-right corner of the x-Axis Datasection. From

the menu, choose Definitions>Applied stretch (stretch).

17 Click to expand the Legendssection. Select the Show legendscheck box.

18 From the Legendslist, choose Manual.19 In the table, enter the following settings:


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



4 In the Point Graphsettings window, locate the Legendssection.



1 Right-click Point Graph 1and choose Duplicate.



4 Locate the Legendssection. In the table, enter the following settings:




Legends

Neo-Hookean

Legends

Mooney-Rivlin

Legends

Ogden

S l d i h COMSOL M l i h i 4 3b


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2 In the Point Graphsettings window, locate the y-Axis Datasection.

3 In the Expressionedit 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.

4 Click to expand the Coloring and Stylesection. Find the Line stylesubsection. From

the Linelist, choose None.

5 From the Colorlist, choose Black.

6 Click to expand the Coloring and Stylesection. Find the Line markerssubsection.

From the Markerlist, choose Asterisk.

7 In the Numberedit field, type 40.



To reproduce Figure 4proceed as follows.

1D Plot Group 4

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

2 In the 1D Plot Groupsettings window, locate the Titlesection.

3 In the Titletext area, type First principal stress versus prescribedstretch..

4 Locate the Plot Settingssection. In the y-axis labeledit field, type First principal

stress (MPa).

5 Click to expand the Axissection. Select the Manual axis limitscheck box.

6 In the y maximumedit field, type 40.

Legends

Varga

Legends

Analytical

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7 In the Model Builderwindow, expand the 1D Plot Group 4node, then click Point

Graph 1.

8 In the Point Graphsettings window, locate the y-Axis Datasection.9 In the Expressionedit field, type solid.sp1.

10 From the Unitlist, choose MPa.

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


13 In the Expressionedit field, type solid.sp1.












25 In the Expressionedit field, type

((6.3e5[Pa]*(stretch^(1.3)-stretch^(-2*1.3)))+(0.012e5[Pa]*(stretch^(5)-stretch^(-2*5)))-(0.1e5[Pa]*(stretch^(-2)-stretch^(2*2)))) .



Solved with COMSOL Multiphysics 4 3b


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2 0 1 3 C O M S O L 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

P l a s t i c D e f o rma t i o n Du r i n g t h e

E xpan 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

treatment of atherosclerosis. During the procedure, a stent is deployed into the artery

by using a balloon as an expander. Once the balloon-stent package is in place, the

balloon is inflated to expand the stent. The balloon is then deflated and removed, but

the stent remains expanded to act as a scaffold, keeping the blood vessel open.

Stent design is of significance for this procedure, since serious damage can be inflicted

to the artery during the expansion procedure. One of the most common defect is the

non-uniform deformation of the stent, where the ends expand more than the middle

section, phenomenon which is also called dogboning. Foreshortening of the stent can

also damage the artery, and it could make the positioning difficult.

The dogboning is defined according to

where rdistaland rcentralare the radii at the end and middle of the stent, respectively.

The foreshortening is defined as

here,L0is the original length of the stent andLloadis the deformed length of the stent.

Other common parameters in stent design are the longitudinal and radial recoil. These

parameters give information on the stent behavior when removing the inflated balloon.

The longitudinal recoil is defined as

here,Lunloadis the length of the stent once the balloon is removed, andLloadis the

length of the stent when the balloon is fully inflated.

dogboningrdistal rcentral

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

foreshorteningL0 Lload

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

LrecoilLload Lunload

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



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p y

2 | P LA ST I C D EF OR M AT I O N D UR I N G T HE E XP AN S I O N O F A B I O ME D I C A L S T E NT 2 01 3 C O MS OL

The radial recoil can be defined as follow

here,Runloadis the radius of the stent once the balloon is removed, andRloadis 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

the influence of the radial pressure that expands the stent. With this model you can

both monitor the dogboning and foreshortening effects, and draw conclusions on how

to change the geometry design parameters for optimum performance.

Model Definition

The model studies the Palmaz-Schatz stent model. Due to the stents circumferential

and longitudinal symmetry, it is possible to model only one twenty-forth of the

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

domain.

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

Rrecoil

Rload Runload

Rload---------------------------------------

=



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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 2mm.

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 O A D S

Apply a radial outward pressure on the inner surface of the stent to represent the

balloon expansion.


The stent is expanded from an original diameter of 0.74mm to a diameter of 2mm in

the middle section.

Figure 2shows the stress distribution at maximum balloon inflation. Figure 3shows

the residual stress after the balloon deflation.

MATERIAL PROPERTY VALUE

Youngs modulus 193[GPa]

Poissons ratio 0.27

Initial yield stress 207[MPa]

Isotropic tangent modulus 692[MPa]



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

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



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

Figure 4: Effective plastic strain after stent deformation.

In Figure 5, you can see the evolution of the dogboning and foreshortening effects

with respect to the pressure during the balloon inflation.

The longitudinal recoil is about 0.9%, the distal radial recoil is about 0.4%, and the

central radial recoil is about 0.7%.



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Figure 5: Stent dogboning (blue) and foreshortening (green) versus pressure inside theangioplasty balloon.


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

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

For a highly nonlinear problem like this, the choice of the continuation parameter can

improve the convergence during the computation of the solution. A displacement

control parameter is usually better than a load parameter. In this model, the average

displacement of the stents inner radius is prescribed, and a global equation computes

the equivalent pressure to be applied.

Model Library path: Nonlinear_Structural_Materials_Module/Plasticity/

biomedical_stent



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M O D E L W I Z A R D1 Go to the Model Wizardwindow.

2 Click Next.

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

4 Click Next.


6 Click Finish.

G E O M E T R Y 1

Import 1

1 In the Model Builderwindow, under Model 1right-click Geometry 1and choose

Import.

2 In the Importsettings window, locate the Importsection.

3 From the Geometry importlist, choose COMSOL Multiphysics file.

4 Click the Browsebutton.

5 Browse to the models Model Library folder and double-click the file

biomedical_stent.mphbin.

6 Click the Importbutton.



Plasticity 1

1 In the Model Builderwindow, under Model 1>Solid Mechanicsright-click Linear Elastic

Material 1and choose Plasticity.

2 In the Plasticitysettings window, locate the Plasticity Modelsection.

3 From the Plasticity modellist, choose Large plastic strains.

M A T E R I A L S

Mater ial 1

1 In the Model Builderwindow, under Model 1right-click Materialsand choose Material.

2 In the Materialsettings window, locate the Material Contentssection.



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Step 1

1 In the Model Builderwindow, under Model 1right-click Definitionsand choose

Functions>Step.

2 In the Stepsettings window, locate the Parameterssection.

3 In the Locationedit field, type 2e-3.

4 In the Fromedit field, type 1.5 In the Toedit field, type 0.

6 Click to expand the Smoothingsection. In the Size of transition zoneedit field, type

1e-5.

Variables 1

1 In the Model Builderwindow, right-click Definitionsand choose Variables.



Average 1

1 Right-click Definitionsand choose Model Couplings>Average.

2 In the Averagesettings window, locate the Source Selectionsection.

3 From the Geometric entity levellist, choose Edge.

4 Select Edge 28 only.

Piecewise 1

1 Right-click Definitionsand choose Functions>Piecewise.

Property Name Value

Young's modulus E 193[GPa]

Poisson's ratio nu 0.27

Density rho 7050

Initial yield stress sigmags 207[MPa]

Isotropic tangent modulus Et 692[MPa]


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



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2 In the Piecewisesettings window, locate the Function Namesection.

3 In the Function nameedit field, type r0.

4 Locate the Definitionsection. In the Argumentedit field, type t.

5 Find the Intervalssubsection. In the table, enter the following settings:



Symmetry 1


Symmetry.

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

Boundary Load 1

1 In the Model Builderwindow, right-click Solid Mechanicsand choose Boundary Load.


3 In the Boundary Loadsettings window, locate the Forcesection.

4 From the Load typelist, choose Pressure.

5 In thepedit field, type p*step1(r).

6 In the Model Builderwindows toolbar, click the Showbutton and select AdvancedPhysics Optionsin the menu.

Global Equations 1

1 Right-click Solid Mechanicsand choose Global>Global Equations.

2 In the Global Equationssettings window, locate the Global Equationssection.


Start End Function

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

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

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

p aveop1(r)-r0(t) Pressure



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10 | P LA S T I C D E F O R M A T I ON D U R I NG T H E E X PA NS I O N O F A B I OM E D I C AL S T E N T 2 01 3 C OM S O L

M E S H 1

Free Triangular 1

1 In the Model Builderwindow, under Model 1right-click Mesh 1and choose MoreOperations>Free Triangular.


Size 1

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

2 In the Sizesettings window, locate the Element Sizesection.

3 Click the Custombutton.

4 Locate the Element Size Parameterssection. Select the Maximum element sizecheck

box.

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

6 Select the Minimum element sizecheck box.

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

8 Select the Maximum element growth ratecheck box.9 In the associated edit field, type 1.4.

10 Select the Resolution of curvaturecheck box.

11 In the associated edit field, type 0.2.

Swept 1

In the Model Builderwindow, right-click Mesh 1and choose Swept.

Distribution 1

1 In the Model Builderwindow, under Model 1>Mesh 1right-click Swept 1and choose

Distribution.





Create variables for the results processing.

Integration 1

1 In the Model Builderwindow, under Model 1right-click Definitionsand choose Model




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3 From the Geometric entity levellist, choose Point.

4 Select Point 57 only.5 Locate the Operator Namesection. In the Operator nameedit field, type central.

Integration 2

1 In the Model Builderwindow, right-click Definitionsand choose Model



3 From the Geometric entity levellist, choose Point.4 Select Point 3 only.

5 Locate the Operator Namesection. In the Operator nameedit field, type distal.

Variables 1

1 In the Model Builderwindow, under Model 1>Definitionsclick Variables 1.




Parameters





dogboning (distal(r)-central(r))/

distal(r)

Dogboning

length 2*abs(distal(x)-central

(x))

Length of the deformed

stent

L0 2*abs(distal(X)-central

(X))

Length of the undeformed

stent

foreshorten

ing

(length-L0)/length Foreshortening


t 0 Time



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S T U D Y 1

Step 1: Stationary


2 In the Stationarysettings window, click to expand the Results While Solvingsection.

3 Select the Plotcheck box.


5 Click Add.


Solver 1



1>Dependent Variables 1node, then click Pressure (mod1.ODE1).

3 In the Statesettings window, locate the Scalingsection.


5 In the Scaleedit field, type 1e6.


1>Stationary Solver 1node, then click Parametric 1.



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

9 Right-click Study 1>Solver Configurations>Solver 1>Stationary Solver 1>Parametric 1

and choose Stop Condition.

10 In the Stop Conditionsettings window, locate the Stop Expressionssection.

11 Click Add.


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

Parameter value list

range(0,1e-2,1.5)

Stop expression

mod1.p


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13 Locate the Output at Stopsection. From the Add solutionlist, choose Step before

stop.


R E S U L T S

Data Sets

Use Mirror 3Dand Sector 3Ddata sets to display the solution on the entire geometry.

1 In the Model Builderwindow, expand the Results>Data Setsnode.

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


4 In the Mirror 3Dsettings window, locate the Datasection.

5 From the Data setlist, choose Mirror 3D 1.

6 Locate the Plane Datasection. From the Planelist, choose zx-planes.

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

8 In the Sector 3Dsettings window, locate the Datasection.


10 Locate the Axis Datasection. In row Point 2, set xto 1.

11 In row Point 2, set zto 0.

12 Locate the Symmetrysection. In the Number of sectorsedit field, type 6.

Stress (solid)

1 In the Model Builderwindow, under Resultsclick Stress (solid).2 In the 3D Plot Groupsettings window, locate the Datasection.

3 From the Data setlist, choose Sector 3D 1.


5 Click the Go to Default 3D Viewbutton on the Graphics toolbar.

6 In the Model Builderwindow, click Stress (solid).

7 In the 3D Plot Groupsettings window, locate the Datasection.8 From the Timelist, choose 1.


Stress (solid) 1

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



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

3 In the Surfacesettings window, click Replace Expressionin the upper-right corner of

the Expressionsection. From the menu, choose Solid Mechanics>Strain (Gausspoints)>Effective plastic strain, Gauss-point evaluation (solid.epeGp).

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

5 Go to the Rename 3D Plot Groupdialog box and type Effective plastic strain

in the New nameedit field.

6 Click OK.

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

1D Plot Group 3

1 Right-click Resultsand choose 1D Plot Group.

2 In the 1D Plot Groupsettings window, locate the Datasection.

3 From the Time selectionlist, choose From list.

4 In the Timeslist, select all the solution steps between 0 and 1.

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

6 In the Globalsettings window, click Replace Expressionin the upper-right corner of

the y-Axis Datasection. From the menu, choose Definitions>Dogboning (dogboning).

7 Click Add Expressionin the upper-right corner of the y-Axis Datasection. From the

menu, choose Definitions>Foreshortening (foreshortening).

8 Click Replace Expressionin the upper-right corner of the x-Axis Datasection. From

the menu, choose Solid Mechanics>Pressure (p).


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

Derived Values

1 In the Model Builderwindow, under Resultsright-click Derived Valuesand choose

Global Evaluation.

2 In the Global Evaluationsettings window, locate the Datasection.

3 From the Time selectionlist, choose From list.

4 In the Timeslist, select 1.

5 Locate the Expressionsection. In the Expressionedit field, type

(length-with(103,length))/length .

6 Select the Descriptioncheck box.



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7 In the associated edit field, type Longitudinal recoil.

8 Click the Evaluatebutton.

9 Right-click Results>Derived Values>Global Evaluation 1and choose Rename.10 Go to the Rename Global Evaluationdialog box and type Longitudinal recoil

evaluationin the New nameedit field.

11 Click OK.

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

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

Rename.

14 Go to the Rename Global Evaluationdialog box and type Distal radial recoil

evaluationin the New nameedit field.

15 Click OK.

16 In the Global Evaluationsettings window, locate the Expressionsection.

17 In the Expressionedit field, type (distal(r)-with(103,distal(r)))/

distal(r).

18 In the Descriptionedit field, type Distal radial recoil.


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

Duplicate.

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

Rename.

22 Go to the Rename Global Evaluationdialog box and type Central radial recoilevaluationin the New nameedit field.

23 Click OK.

24 In the Global Evaluationsettings window, locate the Expressionsection.

25 In the Expressionedit field, type (central(r)-with(103,central(r)))/

central(r).

26 In the Descriptionedit field, type Central radial recoil.


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

3D Plot Group 4

1 In the Model Builderwindow, right-click Resultsand choose 3D Plot Group.



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

3 In the Surfacesettings window, locate the Datasection.

4 From the Data setlist, choose Sector 3D 1.5 Locate the Coloring and Stylesection. From the Coloringlist, choose Uniform.

6 From the Colorlist, choose Gray.

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

8 Go to the Rename 3D Plot Groupdialog box and type Full geometry and meshin

the New nameedit field.

9 Click OK.

Full geometry and mesh

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

2 In the Surfacesettings window, locate the Coloring and Stylesection.

3 From the Coloringlist, choose Uniform.

4 From the Colorlist, choose Black.

5 Select the Wireframecheck box.6 Click the Plotbutton.



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

B r a c k e t G eome 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.


bracket_basic



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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


B k P l i i A l i


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2 0 1 3 C O M S O L 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

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.

In this example you will learn how to perform a stress analysis of a bracket subjected

to external load. The applied load is increased so that the stresses locally become higher

than the yield stress.

It is recommended you review the Introduction to the Structural Mechanics Module,

which includes background information and discusses the bracket_basic.mphmodel

relevant to this example.

Model Definition

This model is an extension to the model example described in the section The

Fundamentals: A Static Linear Analysis in the Introduction to the Structural

Mechanics Module.

Due to symmetry in the model settings, you can consider only one half of the full

geometry, see Figure 1.



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2 | 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 2 0 1 3 C O M S O L

Figure 1: Geometry of the bracket

The bracket displacements are fixed in the bolt regions.

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


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



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Figure 2: Von Mises stress distribution

You can see that the maximum stress is above the yield stress level (260 MPa) and that

a high stress region is located at the base of the bracket and around the hole.

Figure 3below shows the plastic region in the bracket. These regions correspond to

where the plastic strain is above 0.



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Figure 3: Plastic region in the bracket


Elastoplastic problems are path dependent, which means that the current plastic strain

evaluation depends on a previous result. To reach good accuracy in the solution you

then need to include a continuation in the solver settings in order to ramp up the

applied load to the structure.


bracket_plasticity


1 From the Viewmenu, choose Model Library.

2 Go to the Model Librarywindow.

3 In the Model Librarytree, select Nonlinear Structural Materials

Module>Plasticity>bracket basic.


4 Click Open.


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p

M O D E L 1

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




3 Click Add Selected.

4 Click Next.


6 Click Finish.

G E O M E T R Y 1

Import 1

1 In the Model Builderwindow, under Model 1>Geometry 1click Import 1.

2 In the Importsettings window, locate the Importsection.

3 From the Geometry importlist, choose COMSOL Multiphysics file.

4 Click the Browsebutton.


bracket_symmetry.mphbin.

6 Click the Importbutton.



1 In the Model Builderwindow, expand the Global Definitionsnode.



Linear Elastic Material 1

The plastic material definition is available as a subnode of the linear elastic material

model node.


theta0 135[deg] Direction of load


1 In the Model Builderwindow, expand the Model 1>Solid Mechanicsnode.


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2 Right-click Linear Elastic Material 1and 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 A T E R I A L S

The elastoplastic material model requires additional material properties, in this case

initial yield stress and the isotropic tangent modulus.

Structural steel

1 In the Model Builderwindow, under Model 1>Materialsclick Structural steel.

2 In the Materialsettings window, locate the Material Contentssection.



You can now add a displacement constraint, a boundary load and symmetry condition

to the model.

Fixed Constraint 11 In the Model Builderwindow, under Model 1right-click Solid Mechanicsand choose

More>Fixed Constraint.

2 In the Fixed Constraintsettings window, locate the Domain Selectionsection.

3 From the Selectionlist, choose Box 1.

Symmetry 1

1 In the Model Builderwindow, right-click Solid Mechanicsand choose Symmetry.


Boundary Load 1

1 Right-click Solid Mechanicsand choose Boundary Load.

2 In the Boundary Loadsettings window, locate the Boundary Selectionsection.

3 From the Selectionlist, choose Box 2.

Property Name Value

Yield stress level sigmags 260[MPa]

Isotropic tangent modulus Et 200[MPa]


4 Locate the Coordinate System Selectionsection. From the Coordinate systemlist,


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choose Rotated System 2.

5 Locate the Forcesection. In the FAtable, enter the following settings:

M E S H 1

Use a refined mesh in the geometry region where you can expect the material to enter

the plastic region.

Free Tetrahedral 1

In the Model Builderwindow, under Model 1right-click Mesh 1and choose Free

Tetrahedral.

Size 1

1 In the Model Builderwindow, under Model 1>Mesh 1right-click Free Tetrahedral 1

and choose Size.

2 In the Sizesettings window, locate the Geometric Entity Selectionsection.

3 From the Geometric entity levellist, choose Boundary.


5 In the Sizesettings window, locate the Element Sizesection.

6 From the Predefinedlist, choose Finer.


S T U D Y 1

Step 1: Stationary

1 In the Model Builderwindow, expand the Study 1node, then click Step 1: Stationary.

2 In the Stationarysettings window, click to expand the Results While Solvingsection.

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

3 Select the Plotcheck box.


5 Click Add.

0 x1

loadIntensity x2

0 x3




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R E S U L T S

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

3D Plot Group 2

Follow the step below to generate Figure 3.

1 In the Model Builderwindow, right-click Resultsand choose 3D Plot Group.

2 In the Model Builderwindow, under Resultsright-click 3D Plot Group 2and choose

Surface.

3 In the Surfacesettings window, locate the Expressionsection.

4 In the Expressionedit field, type solid.epe>0.



P0 range(2e7,5e6,9e7)


Comp 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


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2 0 1 3 C O M S O L 1 | 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

Comp 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

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.

The model serves as an example of an analysis with very large plastic strains and

contact.

Model Definition

The pipe has an external radius,R0, of 200mm and a wall thickness of 25mm. The

material is a stainless steel with the following properties

The hardening curve is available as a text file containing pairs of data (plastic strain,

stress) which can be imported as a function. The function is shown in Figure 1below.

The stress is measured as true (Cauchy) stress, and the strain is measured as true

(logarithmic) strain. The data can thus be used directly as input to COMSOL when

large strain plasticity is used. Note that if the strain is above the ultimate strain, then

the curve is implicitly flat, so that deformation will continue under constant stress.

TABLE 1: MATERIAL DATA

PROPERTY VALUE

Youngs modulus 195 GPa

Poissons ratio 0.3

Yield stress 250 MPa

Ultimate tensile stress 616 MPa

Ultimate strain 0.52



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2 | 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 2 0 1 3 C O M S O L

Figure 1: The hardening curve.

The pipe is compressed between two flat indenters that can be considered as rigid. The

original position of each indenter is 1mm from the outer pipe wall. During the


compression of the pipe, the distance between the indenters is decreased by 300mm,

and then they are retracted to their original positions


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and then they are retracted to their original positions.

Figure 2: The geometry; pipe and indenters.

Due to the symmetries, only one quarter of the geometry needs to be modeled. The

problem is considered as 2D with the plane strain assumption.


This model exhibits extremely large plastic strains. The deformation and stress state at

the maximum compression are shown in Figure 3. The maximum stress displayed isslightly above the ultimate tensile stress (616MPa). 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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law is exactly fulfilled.

Figure 3: Effective stress at maximum compression.

The distribution of the effective plastic strains is shown in Figure 4and Figure 5. As

can be seen, the peak value (0.95) is far above the ultimate strain (0.52). All values

above the ultimate strain are however on the inside of the pipe, where the strain state

is mainly in compression. At the outer edge of the pipe, the plastic strain approximately

reaches the ultimate strain. There is thus a certain risk that cracks could start forming.

The values of ultimate stress and strain are related to the specimen used for the testing(usually a cylindrical bar) and cannot directly be transferred to general multiaxial

conditions.



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Figure 4: Effective plastic strain at maximum compression.

Figure 5: Effective plastic strain at maximum compression, detail. The black contourindicates 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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Figure 6: Deformed shape and residual stresses at the end of the process.

The load used to compress the pipe is computed from the reaction force in the

indenter, and it is shown in Figure 7.



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


compressed_elastoplastic_pipe




2 Click the 2Dbutton.

3 Click Next.


5 Click Next.


7 Click Finish.



Parameters


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Parameters




Add the hardening curve for the elastoplastic material.

Interpolation 1

1 Right-click Global Definitionsand choose Functions>Interpolation.

2 In the Interpolationsettings window, locate the Definitionsection.

3 Click Load from File.


compressed_elastoplastic_pipe_stress_strain.txt .

5 In the Function nameedit field, type hardFcn.

6 Locate the Interpolation and Extrapolationsection. From the Interpolationlist,

choose Piecewise cubic.

7 Locate the Unitssection. In the Argumentsedit field, type 1.8 In the Functionedit field, type MPa.


Add a function for the displacement if the indenting part.

Interpolation 2

1 Right-click Global Definitionsand choose Functions>Interpolation.

2 In the Interpolationsettings window, locate the Definitionsection.

3 In the Function nameedit field, type compr.


Ro 200[mm] Outer radius

thic 25[mm] Pipe wall thickness

Ri Ro-thic Inner radiuspara 0 Solution parameter



t f(t)


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5 Locate the Unitssection. In the Argumentsedit field, type 1.

6 In the Functionedit field, type m.


G E O M E T R Y 1

Circle 1

1 In the Model Builderwindow, under Model 1right-click Geometry 1and choose Circle.


3 In the Radiusedit field, type Ro.


Circle 2

1 In the Model Builderwindow, right-click Geometry 1and choose Circle.


3 In the Radiusedit field, type Ri.


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

2 Select the object c1only.

3 In the Differencesettings window, locate the Differencesection.

4 Under Objects to subtract, click Activate Selection.

5 Select the object c2only.


Rectangle 1

1 Right-click Geometry 1and choose Rectangle.

2 In the Rectanglesettings window, locate the Sizesection.

3 In the Widthedit field, type 0.05.

t f(t)

0 0

1 0.15

2 0


4 In the Heightedit field, type 1.3*Ro.

5 Locate the Positionsection. In the xedit field, type Ro+0.001.


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

2 In the Finalizesettings window, locate the Finalizesection.

3 From the Finalization methodlist, choose Form an assembly.




1 In the Model Builderwindow, under Model 1right-click Definitionsand choose

Pairs>Contact Pair.


3 In the Pairsettings window, click Activate Selectionin the upper-right corner of the

Destination Boundariessection. Select Boundary 4 only.


Plasticity 1

1 In the Model Builderwindow, under Model 1>Solid Mechanicsright-click Linear Elastic

Material 1and choose Plasticity.


3 In the Plasticitysettings window, locate the Plasticity Modelsection.

4 From the Plasticity modellist, choose Large plastic strains.

5 From the Isotropic hardeninglist, choose Use hardening function data.

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

M A T E R I A L S

Mater ial 1

1 In the Model Builderwindow, under Model 1right-click Materialsand choose Material.2 In the Materialsettings window, locate the Material Contentssection.



Property Name Value


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Contact 1


Pairs>Contact.

2 In the Contactsettings window, locate the Pair Selectionsection.

3 In the Pairslist, select Contact Pair 1.

4 In the Contactsettings window, locate the Penalty Factorsection.

5 From the list, choose Speed.

Symmetry 1

1 In the Model Builderwindow, right-click Solid Mechanicsand choose Symmetry.


Prescribed Displacement 1

1 Right-click Solid Mechanicsand choose More>Prescribed Displacement.


3 In the Prescribed Displacementsettings window, locate the Prescribed Displacement

section.

4 Select the Prescribed in x directioncheck box.

5 Select the Prescribed in y directioncheck box.

6 In the u0xedit field, type -compr(para).

M E S H 1

1 In the Model Builderwindow, under Model 1click Mesh 1.

2 In the Meshsettings window, locate the Mesh Settingssection.

3 From the Sequence typelist, choose User-controlled mesh.

p y

Young's modulus E 195[GPa]Poisson's ratio nu 0.3

Density rho 8000

Initial yield stress sigmags 250[MPa]


Free Triangular 1

1 In the Model Builderwindow, under Model 1>Mesh 1right-click Free Triangular 1and

choose Delete.


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choose Delete.

2 Click Yesto confirm.

Mapped 1

Right-click Mesh 1and choose Mapped.

Distribution 1

1 In the Model Builderwindow, under Model 1>Mesh 1right-click Mapped 1and choose

Distribution.




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

displacement control.

Distribution 2



3 From the Distribution propertieslist, choose Predefined distribution type.



6 In the Element ratioedit field, type 2.

7 Select the Symmetric distributioncheck box.

Distribution 3






S T U D Y 1

Step 1: Stationary



2 In the Stationarysettings window, click to expand the Study Extensionssection.

3 Select the Continuationcheck box.

4 Cli k Add


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4 Click Add.


6 Click to expand the Results While Solvingsection. Select the Plotcheck box.

7 Click the Computeicon on the toolbar.

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

R E S U L T S

Data Sets

1 In the Model Builderwindow, under Resultsright-click Data Setsand choose More

Data Sets>Mirror 2D.


3 In the Mirror 2Dsettings window, locate the Datasection.


5 Locate the Axis Datasection. In row Point 2, set xto 1.

6 In row Point 2, set yto 0.

Stress (solid)1 In the Model Builderwindow, under Resultsclick Stress (solid).

2 In the 2D Plot Groupsettings window, locate the Datasection.




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

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


Create an animation of the compression process.

8 Click the Playerbutton on the main toolbar.


para range(0,0.02,1) range(1.005,0.005, 1.05) 1.1 2


Export

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

D i d V l


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Derived Values

1 In t

nonlinear structural materials model library manual

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