Workshop 1

Crack in a Three-point Bend Specimen

© Dassault Systèmes, 2009 Modeling Fracture and Failure with Abaqus

Workshop 7

Modeling Crack Propagation in a Pressure Vessel

Introduction

In this workshop, we will model crack propagation in a steel pressure vessel using XFEM.

The procedure is similar to that used earlier, but the ease of modeling as compared to

conventional methods will become more evident here in three dimensions. In the

postprocessing section of this workshop, we will get acquainted with tools and features

available in the Visualization module that allow one to effectively probe the cracked

geometry in a three-dimensional solid.

Figure W7–1 The pressure vessel

The structure being modeled here is a 10m thick cylindrical pressure vessel with an inner

diameter of 40m at the base with a hemispherical cap. The entire structure is ~94m high

and is modeled using reduced-integration solid continuum elements (C3D8R). The

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meshed model is shown in Figure W7–1. The pressure vessel is constrained at the bottom

against movement in all directions, and a uniform pressure of 210 MPa is applied on all

the interior surfaces. We will assume the material to be linear elastic; failure initiates

when the maximum principal stress reaches a critical value (the MAXPS damage

initiation criterion is used). We will use an energy-based damage evolution criterion that

accounts for mode mixing.

An initial crack is located in one of the nozzles near the bottom of the pressure vessel, as

shown in Figure W7–2. As done previously, the initial crack is defined using a part

constructed in the shape of the crack and instanced in the assembly at the desired location.

The crack geometry, i.e., the crack surface and the crack front are defined by means of

two level set functions φ and ψ which Abaqus/CAE calculates using the geometric

feature — in this case the part instance — used to define the crack. Note that this part

need not be meshed or assigned material properties; it is a dummy part present only for

the purpose of defining the initial crack.

Figure W7–2 Initial crack in the nozzle shown in (a) the unmeshed part (b) the meshed part

Preliminaries

1. Enter the working directory for this workshop: ../fracture/vessel.

2. Run the script named ws_press_vessel_xfem.py.

The model created by this script contains the part geometry, model assembly, mesh and

the sets and surfaces necessary for defining the crack, boundary conditions and loads. We

will make the following additions to configure the model.

Material and section properties

Here we will define a linear elastic material named steel with a Young’s modulus of 210

GPa and Poisson’s ratio of 0.3, and specify damage initiation, evolution and stabilization.

We will then create a solid section referencing this material and assign it to the part.

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1. In the Model Tree, double-click Materials; in the material editor that appears,

enter steel as the name.

2. Select Mechanical → Elasticity → Elastic. Enter 210.0E9 and 0.3 as the

Young’s modulus and the Poisson’s ratio, respectively.

3. Select Mechanical → Damage for Traction Separation Laws → Maxps

Damage. As shown in Figure W7–3, change the tolerance to 0.1 and enter 8.44E7

as the maximum principal stress.

Figure W7–3 The material editor

4. Select Suboptions → Damage Evolution. In the suboption editor that appears,

select Energy as the type and Power Law as the mixed mode behavior. Toggle

on Power and enter 1 in the data field. Enter 4220 in the three data fields

corresponding to fracture energy. The editor should resemble Figure W7–4. Click

OK.

5. Select Suboptions → Damage Stabilization Cohesive. In the suboption editor

that appears, enter 1.0E-4 as the viscosity coefficient and click OK.

6. Click OK in the material editor.

7. In the Model Tree, double-click Sections and create a homogeneous solid section

named Solid with steel as the material.

8. Assign the section Solid to the predefined set named vessel. This set encompasses

the entire model.

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Figure W7–4 Specifying damage evolution using the suboption editor

Step, time incrementation, and analysis controls

We will now create a general static step. The default choices for time incrementation are

usually not sufficient for crack propagation analyses that employ XFEM. We will

reduce the sizes of the minimum time increment as well as the initial increment. In

general, the discontinuous nature of crack propagation causes convergence difficulties,

which can be alleviated by specifying certain analysis controls. These analysis controls

may not always be necessary; but more often than not, they prove useful in bringing an

analysis to completion.

Three-dimensional XFEM analyses are usually time intensive and may require a large

number of increments. Here we will run the analysis just long enough to produce some

crack propagation for illustration purposes.

1. In the Model Tree, double-click Steps. In the Create Step dialog box that appears,

select Static, General as the procedure type and click Continue.

2. In the step editor that appears, toggle on Nlgeom and set the time period to 1.

3. Switch to the Incrementation tabbed page of the editor. Enter 0.05 as the initial

and the maximum time increment sizes. Reduce the minimum increment size to

1.0e-12. Enter 10 as the maximum number of increments and click OK.

4. From the main menu bar in the Step module, select Other → General Solution

Controls → Edit → Step-1. Abaqus/CAE displays a warning message. Review

it and click Continue.

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5. In the General Solutions Controls Editor that appears, go to the Time

Incrementation tabbed page and toggle on Specify. Then, toggle on

Discontinuous Analysis.

Note: This increases I0 and IR to 8 and 10, respectively. While solving the

equations in any given increment, the automatic time integration algorithm will

check the behavior of residuals from iteration to iteration to gauge the likelihood

of convergence and decide whether or not to abandon iterations and begin again

with a smaller time increment. A check is made for quadratic convergence after I0

iterations and if quadratic convergence is not achieved, then a check is made to

maintain logarithmic convergence after IR iterations. In discontinuous analyses

convergence is generally slow and we are simply postponing these checks to

account for this by increasing I0 and IR.

6. Click the first More tab on the left to display the default values of time

incrementation parameters. Increase the value of IA, the maximum number of

attempts before abandoning an increment, from the default value of 5 to 20.

This data field is highlighted in Figure W7–5. Click OK.

Figure W7–5 The general solution controls editor

Output requests

The output variables required to visualize and probe an XFEM crack are not included in

the default output. Edit the default field output request to include the output variables

PHILSM, PSILSM and STATUSXFEM. The first two are found under the category

Failure/Fracture, and the latter is found under State/Field/User/Time, as shown in

Figure W7–6.

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Figure W7–6 Output requests

XFEM crack definition

Create a frictionless interaction property for the crack surfaces and define a propagating

XFEM crack in the Interaction module using the part instance crack-1.crack as the initial

crack location.

1. In the Model Tree, double-click Interaction Properties. In the Create

Interaction Property dialog box that appears, enter noFric as the name and

Contact as the type. Click Continue.

2. In the interaction editor that appears, select Mechanical → Tangential Behavior.

Accept the default friction formulation Frictionless.

3. Select Mechanical → Normal Behavior. Accept the default selection for the

pressure-overclosure relationship and click OK.

4. From the main menu bar in the Interaction module, select Special → Crack →

Create. In the Create Crack dialog box that appears, choose XFEM as the type

as shown in Figure W7–7 and click Continue.

5. Choose Single instance as the crack domain in the prompt area and select the

instance of the pressure vessel in the viewport. If the Region Selection dialog box

appears, click Select in viewport in the prompt area to select the instance directly

from the viewport.

6. In the crack editor that appears, toggle on Allow crack growth.

7. Toggle on Crack location and click ; then click Sets in the prompt area. In

the Region Selection dialog box that appears, select crack-1.crack and click

Continue.

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8. Toggle on Specify contact property in the crack editor. If it is not already

selected, select the contact property noFric. The crack editor should appear as

shown in Figure W7–8. Click OK.

Figure W7–7 Creating an XFEM crack

Figure W7–8 The crack editor

Boundary conditions and loads

Create an encastre boundary condition and apply it to the bottom of the pressure vessel in

the initial step. Use the predefined set named pressure_vessel-1.bottom for this purpose.

1. In the Model Tree double-click BCs. In the Create Boundary Condition dialog

box that appears, enter fixed as the name. Select Initial as the step and

Symmetry/Antisymmetry/Encastre as the type, and click Continue.

2. Click Sets in the prompt area and select the set pressure_vessel-1.bottom in the

Region Selection dialog box that appears. Click Continue.

3. In the boundary condition editor, select ENCASTRE and click OK.

Apply a pressure of 210 MPa on the interior surface of the pressure vessel. Use the

predefined surface named pressure_vessel-1.interior.

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1. In the Model Tree double-click Loads. In the Create Load dialog box that

appears, enter Pressure as the name. Select Step-1 as the step and Pressure as the

type, and click Continue.

2. Select the predefined surface pressure_vessel-1.interior in the Region Selection

dialog box and click Continue.

3. In the load editor, enter 2.1E8 as the magnitude and click OK.

Job

1. In the Model Tree, double-click Jobs to create a job for this model. Name the job

vessel.

2. Save your model database.

3. Click mouse button 3 on the job name and select Submit from the menu that

appears. From the same menu, you may also select Monitor to monitor the

progress of the job and Results to automatically open the output database file for

this job (vessel) in the Visualization module.

Results

As we limited the maximum number of increments to 10, the job will exit with the error

message, Error in job vessel: Too many increments needed to complete the step. Ignore

the message and open vessel.odb in the Visualization module.

4. Plot the deformed shape and contour the stress distribution in the specimen.

Animate the response. Figure W7–9 shows the Mises stress at the end of the 10th

increment.

When enriched elements are used and PHILSM is requested as an output variable,

Abaqus/CAE automatically creates an isosurface named Crack_PHILSM where

the value of the signed distance function is zero corresponding to the surface of

the crack. This isosurface cut is turned on by default so that the crack is visible

upon opening the output database.

5. Contour and animate the variable STATUSXFEM to visualize crack propagation.

The last frame is shown in Figure W7–10. STATUSXFEM varies between 0 and

1, with 0 for elements where a crack has not initiated and 1 for elements that have

cracked completely. This allows us to pin-point the crack location at any given

time and to assess the extent of failure in a particular region.

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Figure W7–9 Mises stress distribution in the pressure vessel

Figure W7–10 STATUSXFEM showing progressive damage and failure

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6. Change the common plot options to display only the feature edges and contour the

output variable PHILSM. This allows us to view the crack in the pressure vessel

more clearly.

a. From the toolbar click to open the Common Plot Options dialog box.

b. Select Feature edges as shown in Figure W7–11 and click OK.

c. In the field output toolbar choose PHILSM. The resulting contour plot near

the cracked region is displayed in Figure W7–12.

7. Make the assembly translucent to visualize internal crack surfaces.

a. Click the Toggle Global Translucency icon to turn this feature on.

b. Click the Translucency value icon next to . Abaqus/CAE displays a

slider which can be used to set the translucency level. Adjust the slider until

the crack surfaces can be seen clearly. Rotate the model for better clarity if

necessary.

c. Animate PHILSM to view crack propagation on the exterior as well as in the

interior. The last frame is shown in Figure W7–13.

Figure W7–11 Changing common plot options

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W7–12 Contour plot of PHILSM near the nozzle

W7–13 Contour plot of PHILSM with global translucency turned on

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8. Using the View Cut Manager, it is possible to display the model on the cut,

which in the case of an XFEM crack will show only the crack surface without the

surrounding material.

a. From the main menu bar, select Tools → View Cut → Manager.

b. In the View Cut Manager that appears, toggle off for the cut named

Crack_PHILSM as shown in Figure W7–14. The resulting crack surface is

displayed in the viewport. Figure W7–15 shows the crack surface without the

surrounding material.

Figure W7–14 The view cut manager

Figure W7–15 The crack surface

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Note: A script that creates the complete model described in these instructions is available for your convenience. Run this script if you encounter difficulties following the instructions outlined here or if you wish to check your work. The script is named

ws_press_vessel_xfem_answer.py

and is available using the Abaqus fetch utility.