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Model Library Manual
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C o n t a c t I n f o r m a t i o n
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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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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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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.
3 In the table, enter the following settings:
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.
5 Click the Build Allbutton.
6 Click the Zoom Extentsbutton on the Graphics toolbar.
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.
2 Select Domain 1 only.
3 In the Hyperelastic Materialsettings window, locate the Hyperelastic Materialsection.
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.
7 In the edit field, type kappa.
8 Right-click Model 1>Solid Mechanics>Hyperelastic Material 2and choose Rename.
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.
2 Select Domain 1 only.
3 In the Hyperelastic Materialsettings window, locate the Hyperelastic Materialsection.
4 From the Material modellist, choose Ogden.
5 Select the Nearly incompressible materialcheck box.
6 Click Addtwice.
7 In the Ogden parameterstable, enter the following settings:
8 In the edit field, type kappa.
9 Right-click Model 1>Solid Mechanics>Hyperelastic Material 3and choose Rename.
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.
2 Select Domain 1 only.
3 In the Hyperelastic Materialsettings window, locate the Hyperelastic Materialsection.
4 From the Material modellist, choose User defined.
5 Select the Nearly incompressible materialcheck box.
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.
8 Right-click Model 1>Solid Mechanics>Hyperelastic Material 4and choose Rename.
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.
2 In the Variablessettings window, locate the Variablessection.
3 In the table, enter the following settings:
S O L I D M E C H A N I C S
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.
3 In the table, enter the following settings:
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.
2 Select Boundary 3 only.
3 In the Distributionsettings window, locate the Distributionsection.
4 In the Number of elementsedit field, type 50.
5 In the Model Builderwindow, right-click Mesh 1and choose Build All.
6 Click the Zoom Extentsbutton on the Graphics toolbar.
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.
8 In the table, enter the following settings:
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.
11 In the Model Builderwindow, under Study 1>Solver Configurations>Solver
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.
M O D E L W I Z A R D
1 Go to the Model Wizardwindow.
2 Find the Studiessubsection. In the tree, select Preset Studies>Stationary.
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
1 In the Model Builderwindow, under Study 2click 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.
4 In the Physics and variables selectiontree, select Model 1>Solid
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.
8 In the table, enter the following settings:
Continuation parameter Parameter value list
stretch range(1, 0.1, 5)
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Solver 2
1 In the Model Builderwindow, right-click Study 2and choose Show Default Solver.
2 In the Model Builderwindow, expand the Solver 2node, then click DependentVariables 1.
3 In the Dependent Variablessettings window, locate the Scalingsection.
4 From the Methodlist, choose Manual.
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.
8 In the Model Builderwindow, under Study 2>Solver Configurations>Solver
2>Stationary Solver 1click Parametric 1.
9 In the Parametricsettings window, locate the Generalsection.
10 From the Predictorlist, choose Constant.
11 In the Model Builderwindow, under Study 2>Solver Configurations>Solver
2>Stationary Solver 1click Fully Coupled 1.
12 In the Fully Coupledsettings window, locate the Method and Terminationsection.
13 From the Nonlinear methodlist, choose Constant (Newton).
14 In the Model Builderwindow, right-click Study 2and choose Compute.
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.
M O D E L W I Z A R D
1 Go to the Model Wizardwindow.
2 Find the Studiessubsection. In the tree, select Preset Studies>Stationary.
3 Click Finish.
S T U D Y 3
1 In the Model Builderwindow, click Study 3.
2 In the Studysettings window, locate the Study Settingssection.
3 Clear the Generate default plotscheck box.
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Step 1: Stationary
1 In the Model Builderwindow, under Study 3click 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.
4 In the Physics and variables selectiontree, select Model 1>Solid
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.
8 In the table, enter the following settings:
Solver 31 In the Model Builderwindow, right-click Study 3and choose Show Default Solver.
2 In the Model Builderwindow, expand the Solver 3node, 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 3>Solver Configurations>Solver3>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 3>Solver Configurations>Solver
3>Stationary Solver 1click Parametric 1.
9 In the Parametricsettings window, locate the Generalsection.
10 From the Predictorlist, choose Constant.
11 In the Model Builderwindow, under Study 3>Solver Configurations>Solver
3>Stationary Solver 1click Fully Coupled 1.
12 In the Fully Coupledsettings window, locate the Method and Terminationsection.
13 From the Nonlinear methodlist, choose Constant (Newton).
Continuation parameter Parameter value list
stretch range(1, 0.1, 2) range(2.2, 0.2,10)
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14 In the Model Builderwindow, right-click Study 3and choose Compute.
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.
M O D E L W I Z A R D
1 Go to the Model Wizardwindow.
2 Find the Studiessubsection. In the tree, select Preset Studies>Stationary.
3 Click Finish.
S T U D Y 4
1 In the Model Builderwindow, click Study 4.
2 In the Studysettings window, locate the Study Settingssection.
3 Clear the Generate default plotscheck box.
Step 1: Stationary1 In the Model Builderwindow, under Study 4click 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.
4 In the Physics and variables selectiontree, select Model 1>Solid
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.
8 In the table, enter the following settings:
Solver 4
1 In the Model Builderwindow, right-click Study 4and choose Show Default Solver.
2 In the Model Builderwindow, expand the Solver 4node, then click Dependent
Variables 1.
Continuation parameter Parameter value list
stretch range(1, 0.1, 2) range(2.2, 0.2,10)
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3 In the Dependent Variablessettings window, locate the Scalingsection.
4 From the Methodlist, choose Manual.
5 In the Model Builderwindow, expand the Study 4>Solver Configurations>Solver
4>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 4>Solver Configurations>Solver
4>Stationary Solver 1click Parametric 1.
9 In the Parametricsettings window, locate the Generalsection.10 From the Predictorlist, choose Constant.
11 In the Model Builderwindow, under Study 4>Solver Configurations>Solver
4>Stationary Solver 1click Fully Coupled 1.
12 In the Fully Coupledsettings window, locate the Method and Terminationsection.
13 From the Nonlinear methodlist, choose Constant (Newton).
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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5 Click the Zoom Extentsbutton on the Graphics toolbar.
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:
20 Click the Plotbutton.
1 Right-click Results>1D Plot Group 3>Point Graph 1and choose Duplicate.
2 In the Point Graphsettings window, locate the Datasection.
3 From the Data setlist, choose Solution 2.
4 In the Point Graphsettings window, locate the Legendssection.
5 In the table, enter the following settings:
6 Click the Plotbutton.
1 Right-click Point Graph 1and choose Duplicate.
2 In the Point Graphsettings window, locate the Datasection.
3 From the Data setlist, choose Solution 3.
4 Locate the Legendssection. In the table, enter the following settings:
5 Click the Plotbutton.
6 Right-click Point Graph 1and choose Duplicate.
7 In the Point Graphsettings window, locate the Datasection.
Legends
Neo-Hookean
Legends
Mooney-Rivlin
Legends
Ogden
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8 From the Data setlist, choose Solution 4.
9 Locate the Legendssection. In the table, enter the following settings:
10 Click the Plotbutton.
1 Right-click Point Graph 1and choose Duplicate.
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.
8 Locate the Legendssection. In the table, enter the following settings:
9 Click the Plotbutton.
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.
12 In the Point Graphsettings window, locate the y-Axis Datasection.
13 In the Expressionedit field, type solid.sp1.
14 From the Unitlist, choose MPa.
15 In the Model Builderwindow, under Results>1D Plot Group 4click Point Graph 3.
16 In the Point Graphsettings window, locate the y-Axis Datasection.
17 In the Expressionedit field, type solid.sp1.
18 From the Unitlist, choose MPa.
19 In the Model Builderwindow, under Results>1D Plot Group 4click Point Graph 4.
20 In the Point Graphsettings window, locate the y-Axis Datasection.
21 In the Expressionedit field, type solid.sp1.
22 From the Unitlist, choose MPa.
23 In the Model Builderwindow, under Results>1D Plot Group 4click Point Graph 5.
24 In the Point Graphsettings window, locate the y-Axis Datasection.
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)))) .
26 From the Unitlist, choose MPa.
27 Click the Plotbutton.
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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
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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.
Results and Discussion
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.
Notes About the COMSOL Implementation
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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Modeling Instructions
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.
5 Find the Studiessubsection. In the tree, select Preset Studies>Stationary.
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.
7 Click the Zoom Extentsbutton on the Graphics toolbar.
S O L I D M E C H A N I C S
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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3 In the table, enter the following settings:
D E F I N I T I O N S
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.
2 In the Variablessettings window, locate the Variablessection.
3 In the table, enter the following settings:
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]
Name Expression Description
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:
6 Click the Plotbutton.
S O L I D M E C H A N I C S
Symmetry 1
1 In the Model Builderwindow, under Model 1right-click Solid Mechanicsand choose
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.
2 Select Boundary 4 only.
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.
3 In the table, enter the following settings:
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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M E S H 1
Free Triangular 1
1 In the Model Builderwindow, under Model 1right-click Mesh 1and choose MoreOperations>Free Triangular.
2 Select Boundary 3 only.
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.
2 In the Distributionsettings window, locate the Distributionsection.
3 In the Number of elementsedit field, type 2.
4 Click the Build Allbutton.
D E F I N I T I O N S
Create variables for the results processing.
Integration 1
1 In the Model Builderwindow, under Model 1right-click Definitionsand choose Model
Couplings>Integration.
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2 In the Integrationsettings window, locate the Source Selectionsection.
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
Couplings>Integration.
2 In the Integrationsettings window, locate the Source Selectionsection.
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.
2 In the Variablessettings window, locate the Variablessection.
3 In the table, enter the following settings:
G L O B A L D E F I N I T I O N S
Parameters
1 In the Model Builderwindow, right-click Global Definitionsand choose Parameters.
2 In the Parameterssettings window, locate the Parameterssection.
3 In the table, enter the following settings:
Name Expression Description
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
Name Expression Description
t 0 Time
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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, click to expand the Results While Solvingsection.
3 Select the Plotcheck box.
4 Click to expand the Study Extensionssection. Select the Continuationcheck box.
5 Click Add.
6 In the table, enter the following settings:
Solver 1
1 In the Model Builderwindow, right-click Study 1and choose Show Default Solver.
2 In the Model Builderwindow, expand the Study 1>Solver Configurations>Solver
1>Dependent Variables 1node, then click Pressure (mod1.ODE1).
3 In the Statesettings window, locate the Scalingsection.
4 From the Methodlist, choose Manual.
5 In the Scaleedit field, type 1e6.
6 In the Model Builderwindow, expand the Study 1>Solver Configurations>Solver
1>Stationary Solver 1node, then click Parametric 1.
7 In the Parametricsettings window, locate the Generalsection.
8 From the Predictorlist, choose Constant.
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.
12 In the table, enter the following settings:
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.
14 In the Model Builderwindow, right-click Study 1and choose Compute.
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.
3 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.
9 From the Data setlist, choose Mirror 3D 2.
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.
4 Click the Plotbutton.
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.
9 Click the Plotbutton.
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).
9 Click the Plotbutton.
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.
19 Click the Evaluatebutton.
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.
27 Click the Evaluatebutton.
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.
Model Library path: Nonlinear_Structural_Materials_Module/Plasticity/
bracket_basic
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B k P l i i A l i
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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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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.
Results and Discussion
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
Notes About the COMSOL Implementation
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.
Model Library path: Nonlinear_Structural_Materials_Module/Plasticity/
bracket_plasticity
Modeling Instructions
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.
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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.
M O D E L W I Z A R D
1 Go to the Model Wizardwindow.
2 In the Add physicstree, select Structural Mechanics>Solid Mechanics (solid).
3 Click Add Selected.
4 Click Next.
5 Find the Studiessubsection. In the tree, select Preset Studies>Stationary.
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.
5 Browse to the models Model Library folder and double-click the file
bracket_symmetry.mphbin.
6 Click the Importbutton.
7 Click the Zoom Extentsbutton on the Graphics toolbar.
G L O B A L D E F I N I T I O N S
1 In the Model Builderwindow, expand the Global Definitionsnode.
2 In the Parameterssettings window, locate the Parameterssection.
3 In the table, enter the following settings:
Linear Elastic Material 1
The plastic material definition is available as a subnode of the linear elastic material
model node.
Name Expression Description
theta0 135[deg] Direction of load
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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.
3 In the table, enter the following settings:
S O L I D M E C H A N I C S
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.
2 Select Boundaries 43 and 44 only.
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]
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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.
4 Select Boundaries 11 and 13 only.
5 In the Sizesettings window, locate the Element Sizesection.
6 From the Predefinedlist, choose Finer.
7 Click the Build Allbutton.
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.
4 Click to expand the Study Extensionssection. Select the Continuationcheck box.
5 Click Add.
0 x1
loadIntensity x2
0 x3
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6 In the table, enter the following settings:
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7 In the Model Builderwindow, right-click Study 1and choose Compute.
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.
5 Click the Plotbutton.
Continuation parameter Parameter value list
P0 range(2e7,5e6,9e7)
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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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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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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
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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.
Results and Discussion
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
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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).
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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.
Model Library path: Nonlinear_Structural_Materials_Module/Plasticity/
compressed_elastoplastic_pipe
Modeling Instructions
M O D E L W I Z A R D
1 Go to the Model Wizardwindow.
2 Click the 2Dbutton.
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.
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G L O B A L D E F I N I T I O N S
Parameters
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Parameters
1 In the Model Builderwindow, right-click Global Definitionsand choose Parameters.
2 In the Parameterssettings window, locate the Parameterssection.
3 In the table, enter the following settings:
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.
4 Browse to the models Model Library folder and double-click the 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.
9 Click the Plotbutton.
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.
Name Expression Description
Ro 200[mm] Outer radius
thic 25[mm] Pipe wall thickness
Ri Ro-thic Inner radiuspara 0 Solution parameter
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4 In the table, enter the following settings:
t f(t)
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5 Locate the Unitssection. In the Argumentsedit field, type 1.
6 In the Functionedit field, type m.
7 Click the Plotbutton.
G E O M E T R Y 1
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 Ro.
4 In the Sector angleedit field, type 90.
Circle 2
1 In the Model Builderwindow, right-click Geometry 1and choose Circle.
2 In the Circlesettings window, locate the Size and Shapesection.
3 In the Radiusedit field, type Ri.
4 In the Sector angleedit field, type 90.
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.
6 Click the Build Allbutton.
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
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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.
4 Click the Build Allbutton.
5 Click the Zoom Extentsbutton on the Graphics toolbar.
D E F I N I T I O N S
1 In the Model Builderwindow, under Model 1right-click Definitionsand choose
Pairs>Contact Pair.
2 Select Boundary 5 only.
3 In the Pairsettings window, click Activate Selectionin the upper-right corner of the
Destination Boundariessection. Select Boundary 4 only.
S O L I D M E C H A N I C S
Plasticity 1
1 In the Model Builderwindow, under Model 1>Solid Mechanicsright-click Linear Elastic
Material 1and choose Plasticity.
2 Select Domain 1 only.
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.
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3 In the table, enter the following settings:
Property Name Value
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S O L I D M E C H A N I C S
Contact 1
1 In the Model Builderwindow, under Model 1right-click Solid Mechanicsand choose
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.
2 Select Boundaries 1 and 2 only.
Prescribed Displacement 1
1 Right-click Solid Mechanicsand choose More>Prescribed Displacement.
2 Select Domain 2 only.
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]
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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.
2 In the Distributionsettings window, locate the Distributionsection.
3 In the Number of elementsedit field, type 1.
4 Select Boundaries 5 and 6 only.
One element is enough on the indenter, since the whole domain is under
displacement control.
Distribution 2
1 Right-click Mapped 1and choose Distribution.
2 In the Distributionsettings window, locate the Distributionsection.
3 From the Distribution propertieslist, choose Predefined distribution type.
4 Select Boundaries 1 and 2 only.
5 In the Number of elementsedit field, type 8.
6 In the Element ratioedit field, type 2.
7 Select the Symmetric distributioncheck box.
Distribution 3
1 Right-click Mapped 1and choose Distribution.
2 Select Boundary 3 only.
3 In the Distributionsettings window, locate the Distributionsection.
4 In the Number of elementsedit field, type 80.
5 Click the Build Allbutton.
S T U D Y 1
Step 1: Stationary
1 In the Model Builderwindow, under Study 1click Step 1: Stationary.
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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.
5 In the table, enter the following settings:
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.
2 Right-click Data Setsand choose More Data Sets>Mirror 2D.
3 In the Mirror 2Dsettings window, locate the Datasection.
4 From the Data setlist, choose Mirror 2D 1.
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.
3 From the Data setlist, choose Mirror 2D 2.
4 Click the Plotbutton.
5 Click the Zoom Extentsbutton on the Graphics toolbar.
Plot the stresses at the maximum compression. This gives Figure 3.
6 From the Parameter value (para)list, choose 1.
7 Click the Plotbutton.
Create an animation of the compression process.
8 Click the Playerbutton on the main toolbar.
Continuation parameter Parameter value list
para range(0,0.02,1) range(1.005,0.005, 1.05) 1.1 2
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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
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