final report_as submitted

Final Project
CRACK ANALYSIS AND STUDY OF EFFECTS OF
CRACK LENGTH AND LOADING ON STRESS
INTENSITY FACTOR’S.
MAE 5060
Instructor– Dr. Bo Yang
July 01, 2008
Hassan Alessa
Abstract
Study a three-point bending specimen model and the effects of the crack length and
load location on the stress intensity factor. Use ANSYS to calculate the first and second
mode stress intensity factors (KI and KII) and compare with theoretical values.

MAE 5060
July 1, 2008 Final Project- Crack Analysis
1
CRACK ANALYSIS
PROBLEM DEFINITION:
The problem is to use ANSYS to construct a three-point bending specimen model to
carry out an analysis of the first and second mode stress intensity factors (SIF‟s)(KI, KII)
with „a‟ and „s‟ as the varying parameters.
Given:
Modulus of Elasticity (E) = 100
Poisson‟s ratio () = 0.3
Note: All units are in SI units.
Figure 1: Specimen
0.05m
P
0

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DISCUSSION
There are three basic modes of crack loading, that is, three ways the forces can act on
a material, and correspondingly three modes of fracture they can cause.
Mode I:
This mode is referred to as the “crack opening” mode. In this mode, forces are applied
perpendicular to the crack, as a result, pulling the crack open. This results in the stress
intensity factor, KI.
Figure 2: Three modes of crack loading.

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3
Mode II:
This mode is referred to as “in plane shear.” Forces are applied parallel to the crack;
one pushing the top half and the other pulling the bottom half, while keeping it in the
vertical plane. This results in a shear crack and the stress intensity factor, KII.
Mode III:
This mode is referred to as “out of plane shearing.” The forces are applied
perpendicular to the plane. One is pulling the top half left and the other is pulling the
bottom half right. This results in a shear crack and the stress intensity factor, KIII.
For the case in which s = 0, the problem partially reduces to that of a three-point
bending (Mode I) problem. When s > 0, then we have mode II crack opening.

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ANALYTICAL CALCULATIONS (For KI only)
𝐾𝐼 =
𝑃
𝐵 𝑊
f 𝑎
𝑊
8.19)(
1
5.0
02.0
05.0
:
7.293.315.2199.1
1212
3
)(
2
2
3






















































W
af
P
S
a
mW
Where
W
a
W
a
W
a
W
a
W
a
W
a
W
a
W
S
W
af
The SIF (KI) is calculated to be 1772.6. KI for all investigated crack lengths:
Crack length
KI
(Theoretical)
0.01 1050.8
0.02 1772.6
0.03 3373.4
0.04 9662.7
Figure 3: Single Edge Notched Beam
(SENB)
Table 1: Analytical KI results
Where, B is the specimen thickness = 0.05

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NUMERICAL SOLUTION USING ANSYS
ASSUMPTIONS:
Crack length
KI
(ANSYS)
KI
(Theoretical)
% Error
0.01 1090 1050.8 3.73
0.02 1845.1 1772.6 4.09
0.03 3465.4 3373.4 2.72
0.04 9972.8 9662.7 3.20
Analysis Type Structural
Material Linear elastic isotropic
Model Type 2-D full model with boundary conditions defined in figure 1
Elements (Global) Quad 8 node 183 (Triangular Mesh)
(plane stress with thickness of 0.05).
Elements (Crack tip)  Radius of 1st
row: a/8 (a/32 for 2nd
solution)
 Number of elements around crack tip: 12
 Radius ratio: 1
 Mid-side node position: Skewed ¼ point
Table 2: Crack Length vs. KI

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PRE-PROCESSING
The most important region in a fracture model is the region around the edge of the crack
or the crack tip. In linear elastic problems, it has been shown that the stresses near the
crack tip are inversely proportional to r , where r is the distance from the crack tip. The
stresses and strains that each mode of loading produces are singular at the crack tip,
varying with respect to 1/ r . To pick up the singularity in the strain, ANSYS uses
singular elements along a defined crack path in which the crack faces are coincident.
The singular elements are quadratic with mid-side nodes placed at the quarter points.
Two rectangular areas are constructed which share line 3 (L3). The crack faces are
shown as line 11 (L11, node 2 – 3) and line 4 (L4, node 2 – 1).
Figure 4: Model Lines and Key-points
Coincident Crack Faces (L11) & (L4)
Common Line for 2 Areas (L3)
2 Areas (L3)

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For 2-D fracture models, the recommended element type is the PLANE2, 6-node
triangular solid element. The PLANE2 element has a quadratic displacement behavior,
and is well suited to model irregular meshes and can be used for plane stress models.
The element is defined by six nodes having two degrees of freedom at each node. But
since it is not available in ANSYS ver. 11 we use Plane 183 and mesh using Triangular
mesh.
In order to maintain a mesh about the crack tip such that elements are not distorted, a
refined spider mesh is developed. The first row of elements around the crack tip should
have a radius of approximately a/8 or smaller for reasonable results:
Figure 5: ANSYS window for Conc. Keypoints

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Figure 6: Spider Mesh
Coincident Crack Faces (L11) & (L4) Crack Tip Key point 2 Common Line for 2 Areas (L3)

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The models constraints are applied to Key-points 10 and 4. Key-point 10 is constrained
in all degrees of freedom (all DOF‟s) and Key-point 4 is free to move in the Y-direction
but constrained in the X (Ux=0).For the initial model, the applied force is at Key-point 11
Figure 7: Loads and Boundary Conditions
Ux = Uy = 0
Ux = 0
P = 1

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10
SOLUTION
Once the model is ready, we do the static analysis to estimate the fracture parameters-
the SIF‟s- KI, KII.
Its important to define a local crack-tip (or crack-front for 3-D) coordinate system, with X
parallel to the crack face (perpendicular to the crack front in 3-D models) and Y
perpendicular to the crack face, as shown in the following figure.
First, a defined path along the crack face is created. The first node on the path should
be the crack-tip node. For a half-crack model, two additional nodes are required, both
along the crack face. For a full-crack model, where both crack faces are included, four
additional nodes are required: two along one crack face and two along the other. The
next step is to calculate KI and KII for the full crack model assuming plane stress
condition. ANSYS calculates the SIF‟s from the nodal displacements on opposite sides
of the crack plane. Here we use the same mesh size that we had obtained to show
convergence. We now calculate KI, KII at various positions of point load and initial crack
size and study the effect of the two parameters - the location of application of point load
and the crack length on the two SIF‟s.
Figure 8: 2-D Full Model Crack Coordinate
System

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Crack Length ‘a’ 0.01 0.02 0.03 0.04
Point Load Location ‘s’ KI
0.00 1090.00 1845.10 3465.40 9972.80
0.05 900.52 1520.40 2828.60 8056.80
0.10 674.18 1139.40 2132.70 6042.70
0.15 449.41 759.60 1414.20 4028.50
0.20 224.71 379.80 707.09 2213.40
0.25 0.00 0.00 0.00 0.00
0
2000
4000
6000
8000
10000
12000
0.00 0.05 0.10 0.15 0.20 0.25 0.30
KI
Point Load Location from Crack Tip 's'
a = 0.01
a = 0.02
a = 0.03
a = 0.04
Figure 9: Variation of KI
Table 4: Variation of KI with ‘a’ and‘s’

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12
Crack Length ‘a’ 0.01 0.02 0.03 0.04
Point Load Location ‘s’ KII
0.00 0.87 0.10 0.29 0.53
0.05 20.52 48.90 77.28 116.93
0.10 14.64 35.51 59.73 87.54
0.15 9.76 23.65 38.10 58.36
0.20 4.88 11.83 19.05 29.25
0.25 0.00 0.00 0.00 0.00
0
20
40
60
80
100
120
140
0 0.05 0.1 0.15 0.2 0.25 0.3
KII
Point Load Location from Crack Tip 's'
a = 0.01
a = 0.02
a = 0.03
a = 0.04
Figure 10: Variation of KII
Table 5: Variation of KII with ‘a’ and ‘s’

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CONCLUSION
 As shown in the graph the values of KI and KII are independent of each other. KI
relates to the bending moment whereas KII relates to the maximum shear in the
crack planes.
 KI and KII are inversely proportional to the crack length.
 To prolong the life of a structure we would have to keep the crack length smaller
and away from the crack tip, preferably the load be applied at the supports as
shown in the graph.
 The stress intensity factors (SIF‟s) are severely affected by continuously moving
loads and the structure might fail pre-maturely.
REFERENCES:
[1] ANSYS Help files.
[2] Anderson, T., L, Fracture Mechanics Fundamentals and Applications, CRC,
2005.
[3] Barsom, J., M, and Rolfe, S., T, Fracture and Fatigue Control in Structures,
Prentice Hall, NJ, 1987

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