Surface Cracked Cylinder

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Surface Cracked Cylinder

This model reproduces a NAFEMS benchmark (Ref. 1) in which a pressurized cylinder has a horizontal semi-elliptic crack on its inner surface. The energy release rate is calculated along the crack front using the J-integral method. The stress intensity factors calculated from the J-integral are then compared to the reference data.

Model Definition

Since the model consists on a cylinder with a horizontal crack on its mid plane, only a quarter of the whole geometry is built. A tube around the elliptical crack front is built to generate a swept mesh along it. The swept mesh makes the J-integral curve along the crack front smoother.

A condition is applied on the thickness boundaries of the cylinder. On the crack boundaries, it is overridden by a feature. The crack is symmetric to define explicitly the crack front. Due to the symmetry, the stress intensity factors for modes II and III will be zero.

The pressure load can be applied with a feature on the inner face of the cylinder, and with the subfeature on the crack face .

The J-integral at a point of the crack front is composed of a closed contour integral and a surface integral. The closed contour integral is composed of a circular integral around the crack front and an integral on the crack face to take into account the force applied on it.

The stress intensity factor in mode I is then computed from the J-integral

where E* is the equivalent Young’s modulus. A 2D plane strain condition is assumed, so

Results and Discussion

Figure 1 shows the default stress plot, where a stress concentration around the crack front is visible. Figure 2 shows a closer view of this stress concentration

Figure 1: Von Mises stress in the cylinder and at the crack front.

Figure 2: Close view of the stress concentration at the crack front.

The J-integral calculated along the crack front represents the energy release rate per surface area when the crack is locally extended in the crack plane direction. The crack extension direction, the J-integral intensity and the surface contour used for integral calculation are plotted in Figure 3

Figure 3: 3D plot of J-integral: crack growth direction, intensity and contour used for the calculation.

Figure 4 plots the J-integral as function of the parametric angle along the elliptic crack. A jump in the curve at zero angle can be noticed. This glitch is often seen near free boundaries due to the singularities that contribute to the surface integral.

Figure 4: J-integral along the crack front.

From the J-integral the stress intensity factor in mode I can be calculated and compared to the benchmark in Ref. 1, see Figure 5.

Figure 5: Stress intensity factor along the crack front compared to the benchmark.

1. R. Judge and B. Mardsen, Three Dimensional Test Cases in Linear Elastic Fracture Mechanics, part 4: Surface Cracked Cylinder, NAFEMS, 1993.

Structural_Mechanics_Module/Fracture_Mechanics/surface_cracked_cylinder

Modeling Instructions

From the menu, choose .

In the window, click

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In the window, click

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In the tree, select .

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In the tree, select .

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In the toolbar, click

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Browse to the model’s Application Libraries folder and double-click the file surface_cracked_cylinder.mph.

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In the toolbar, click

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Global Definitions

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In the window, under click .

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In the window for , locate the section.

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In the table, enter the following settings:


Name	Expression	Value	Description
p	1[MPa]	1E6 Pa	Pressure load

Material 1 (mat1)

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In the toolbar, click

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In the window for , locate the section.

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In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	207[GPa]	Pa	Basic
Poisson’s ratio	nu	0.3	1	Basic
Density	rho	8000	kg/m³	Basic

Solid Mechanics (solid)

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In the window, under right-click and choose .

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Select Boundaries 1, 2, 5, 6, 8, 9, 11, 12, 15, 22, and 26 only.

The condition is applied on the whole thickness of the cylinder. It will be overridden by the feature on the crack surface.

Prescribed Displacement 1

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In the toolbar, click

and choose .

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Select Point 11 only.

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In the window for , locate the section.

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Select the check box.

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In the toolbar, click

and choose .

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Select Boundaries 6 and 9 only.

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In the window for , locate the section.

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From the list, choose .

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Click to expand the section. Click

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Select Edge 13 only.

In the toolbar, click

and choose .

Boundary Load 1

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In the toolbar, click

and choose .

Apply pressure load on the inner face of the cylinder. The pressure on the crack will be applied under the feature.

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Select Boundaries 4, 7, 17, 24, 27, and 31 only.

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In the window for , locate the section.

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From the list, choose .

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In the p text field, type p.

In the window, click .

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In the toolbar, click

and choose .

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In the window for , locate the section.

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In the p text field, type p.

Free Triangular 1

In the toolbar, click

and choose .

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In the window, click .

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In the window for , locate the section.

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Click the button.

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Locate the section. In the text field, type th/2.

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In the text field, type a/200.

Free Triangular 1

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In the window, click .

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Select Boundary 8 only.

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Right-click and choose .

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In the window for , locate the section.

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From the list, choose .

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Select Point 5 only.

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Locate the section. Click the button.

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Locate the section. Select the check box.

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In the associated text field, type a/100.

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Select the check box.

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In the associated text field, type a/200.

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Select the check box.

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In the associated text field, type 1.2.

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In the toolbar, click

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In the window for , locate the section.

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From the list, choose .

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Select Domain 3 only.

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Right-click and choose .

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In the window for , locate the section.

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From the list, choose .

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In the text field, type 40.

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In the text field, type 6.

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Select the check box.

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Free Tetrahedral 1

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In the toolbar, click

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In the window for , locate the section.

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From the list, choose .

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Select Domain 2 only.

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In the toolbar, click

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In the window for , locate the section.

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From the list, choose .

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Select Domains 1 and 4 only.

In the toolbar, click

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Right-click and choose .

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In the window for , locate the section.

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From the list, choose .

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In the text field, type 12.

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In the text field, type 4.

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In the toolbar, click

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In the window, expand the node, then click .

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In the window for , locate the section.

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From the list, choose .

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Click to expand the section. Select the check box.

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In the text field, type 0.

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In the text field, type 20.

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In the window, click .

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In the window for , locate the section.

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Clear the check box.

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In the toolbar, click

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The first default plot shows von Mises stress. On a closer view the maximum stress is visible along the crack front, see Figure 2.

There is also a default arrow plot that shows the crack direction and the intensity of J-integral, Figure 3. The contour used to calculate the J-integral is displayed as well.

To plot the J-integral (Figure 4) along crack front, you may introduce the parametric angle along the ellipse.

1

In the toolbar, click

and choose .

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In the window for , locate the section.

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From the list, choose .

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Select Domains 2 and 3 only.

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


Name	Expression	Unit	Description
angle	atan2((Z-R1)/a,X/c)	rad	Parametric angle

In the toolbar, click

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In the window, expand the node, then click .

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In the window for , locate the section.

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From the list, choose .

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In the text field, type angle.

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From the list, choose .

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In the toolbar, click

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Duplicate the last plot group to plot and compare the stress intensity factor to the reference value, Figure 5

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In the window, right-click and choose .

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In the window for , type Crack 1, KI in the text field.

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In the window, expand the node, then click .

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In the window for , type KI in the text field.

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Click in the upper-right corner of the section. From the menu, choose .

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Locate the section. In the text field, type solid.crack1.jint1.KI/1e6.

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In the window, click .

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In the window for , locate the section.

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Select the check box.

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Select the check box.

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In the associated text field, type Stress intensity factor.

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Locate the section. Select the check box.

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In the text field, type 0.

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In the toolbar, click

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Import the stress intensity factors from benchmark to compare with the computed results.

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In the toolbar, click

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In the window for , locate the section.

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Browse to the model’s Application Libraries folder and double-click the file surface_cracked_cylinder_results.txt.

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Right-click and choose .

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In the window for , locate the section.

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From the list, choose .

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Find the subsection. From the list, choose .

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Find the subsection. From the list, choose .

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From the list, choose .

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In the toolbar, click

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1

In the window, click .

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In the window for , locate the section.

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Select the check box.

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From the list, choose .

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In the window, click .

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In the window for , click to expand the section.

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Select the check box.

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From the list, choose .

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In the table, enter the following settings:


Legends
KI

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In the window, click .

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In the window for , click to expand the section.

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Select the check box.

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From the list, choose .

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In the table, enter the following settings:


Legends
KI, reference