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Pardo-FinalReport.docx
Crack Growth under Tensile Residual Stresses due to
Compressive Cyclic Loading
by
Rafael E. Pardo
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the Degree of
MASTER OF MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
August, 2014
(For Graduation August 2014)
© Copyright 2014
by
Rafael E. Pardo
All Rights Reserved
ii
CONTENTS
LIST OF SYMBOLS ......................................................................................................... v
LIST OF TABLES ............................................................................................................ vi
LIST OF FIGURES ......................................................................................................... vii
ACKNOWLEDGMENT .................................................................................................. ix
KEYWORDS ..................................................................................................................... x
ABSTRACT ..................................................................................................................... xi
1. Introduction.................................................................................................................. 1
2. Theory/Literature ......................................................................................................... 3
2.1
Material Strength ................................................................................................ 3
2.2
Plastic Deformation ............................................................................................ 4
2.3
Stress-Strain Curve Model used in ANSYS....................................................... 8
2.4
Fatigue Failure ................................................................................................... 8
2.5
Fracture Mechanics .......................................................................................... 11
2.5.1
Linear Elastic Fracture Mechanics (LEFM) ........................................ 11
2.5.2
Fracture Toughness .............................................................................. 12
2.5.3
Stress Intensity Factor .......................................................................... 13
2.5.4
Behavior at Crack Tips......................................................................... 13
2.5.5
Inspections ........................................................................................... 14
3. Methodology .............................................................................................................. 15
3.1
Geometry .......................................................................................................... 15
3.2
Material Properties ........................................................................................... 16
3.2.1
3.3
Definition of Material Properties in ANSYS ....................................... 17
Loading Conditions .......................................................................................... 19
4. Results and Discussion .............................................................................................. 21
4.1
Stress-Strain Results ........................................................................................ 21
4.1.1
First Half of Load Cycle: Compression ............................................... 23
iii
4.1.2
Second Half of Load Cycle: Return Sample Back To Its Original
Length .................................................................................................. 24
4.2
Crack Growth Results (FRANC3D) ................................................................ 26
4.3
Crack Growth Results (CTOD) ........................................................................ 30
5. Conclusion ................................................................................................................. 34
6. References.................................................................................................................. 36
7. Appendices ................................................................................................................ 37
7.1
Appendix 1: INCO 718 Material Properties .................................................... 37
7.2
Appendix 2: ANSYS Macro ............................................................................ 40
7.3
Appendix 3: Calculate Nominal Stress Using ANSYS Linearization ............. 44
iv
LIST OF SYMBOLS
E
= Modulus of Elasticity (psi)
ε
= Strain (-)
σo
= Yield Point in Tension (psi)
σoc
= Yield Point in Compression (psi)
σuts
= Ultimate Tensile Strength (psi)
σyts
= Yield Tensile Strength (psi)
Kc
= Critical Value Fracture Toughness (Ksi√in)
K
= Stress Intensity Factor (Ksi√in)
ΔK
= Stress Intensity Factor Range (Ksi√in)
R
= Ratio of Minimum to Maximum (-)
Kth
= Fatigue Crack Growth Threshold Stress Intensity Factor (Ksi√in)
da/dN = Crack Growth Rate (inch/cycles)
a
= Crack Length (inch)
LS
= Load Step (-)
Ep
= Plastic Modulus
̅
= Effective Stress (psi)
̅
= Effective Plastic Strain (-)
v
LIST OF TABLES
Table 1: INCO 718 Mechanical Material Properties ....................................................... 16
Table 2: Stress and Strain Results at Notch Center Node ............................................... 21
Table 3: Results for Cycle 1 (Load Step 1 & 2) .............................................................. 32
Table 4: Results for Cycle 2 (Load Step 3 & 4) .............................................................. 33
vi
LIST OF FIGURES
Figure 1: Crack Growing by Two Different Stress Fields ................................................. 1
Figure 2: Engineering Stress-Strain Curve; based on figure from [1] ............................... 3
Figure 3: Stress-Strain Diagram Showing Elastic and Plastic Deformation ..................... 5
Figure 4: Unloading Stress-Strain Curve where Bauschinger Effect Causes Yielding at A
Prior to the Compressive Yield Strength σoc ..................................................................... 7
Figure 5: Differing Unloading Behavior for Kinematic and Isotropic Hardening ............ 7
Figure 6: Elastic, Linear Hardening Material Behavior .................................................... 8
Figure 7: Fatigue Failure Mechanism ................................................................................ 9
Figure 8: da/dN vs ΔK ..................................................................................................... 10
Figure 9: Fracture Modes [6] ........................................................................................... 12
Figure 10: Redistribution of Stresses at Crack Tip .......................................................... 14
Figure 11: ANSYS Square Bar Finite Element Model (FEM) ........................................ 15
Figure 12: Notch Dimensions .......................................................................................... 16
Figure 13: ANSYS v15.0 Bilinear Material Property Definition Windows .................... 17
Figure 14: ANSYS v15.0 Bilinear Material Property Definition Windows .................... 17
Figure 15: ANSYS Bilinear Kinematic Model Intersecting Slope at ε=0.005 ................ 18
Figure 16: Bilinear Kinematic Model Intersecting Slopes at ε=0.002 and ε=0.005 ........ 19
Figure 17: Body Diagram of Boundary Conditions Applied in ANSYS v15.0 ............. 20
Figure 18: ANSYS FEM Showing Location of Applied Constraint (Left) and
Displacement (Right) ....................................................................................................... 20
Figure 19: Cyclic Stress-Strain Curve from ANSYS Results Table 1 ............................ 21
Figure 20: ANSYS Results, Stress vs. Load Step ........................................................... 22
Figure 21: Displacement, X-direction; Load Step 1 ........................................................ 23
Figure 22: Stress, X-direction; Load Step1 ..................................................................... 23
Figure 23: Total Mechanical Strain, X-direction; Load Step 1 ....................................... 24
Figure 24: Displacement, X-direction; Load Step 2 ........................................................ 24
Figure 25: Stress, X-direction; Load Step 2 .................................................................... 25
Figure 26: Total Mechanical Strain, X-direction; Load Step 2 ....................................... 26
Figure 27: ANSYS Finite Element Model with Edge Crack in the Notch ...................... 27
Figure 28: Zoomed View of ANSYS FEM with Edge Crack ......................................... 27
vii
Figure 29: Stress with Edge Crack, X-direction; Load Step 1 ........................................ 28
Figure 30: Stress with Edge Crack, X-direction; Load Step 2 ........................................ 28
Figure 31: ANSYS Residual Stress Distribution at Edge Crack Faces; Load Step 2...... 29
Figure 32: FRANC3D Negative Stress Intensity Factor Results..................................... 30
Figure 33: Crack Arrest (Crack Will No Longer Continue to Grow) .............................. 33
viii
ACKNOWLEDGMENT
I want to dedicate this effort to my mother, for all the things she has done for me and for
supporting me during the most difficult years of my life. She always believed in me.
Thanks Hilda.
ix
KEYWORDS
Crack Growth
Cyclic Loading
Finite Element Model (FEM)
ANSYS
FRANC3D
Edge Crack
Liner Elastic Fracture Mechanics (LEFM)
Stress Intensity Factor (K)
J-Integral
Close Tip Opening Displacement (CTOD)
Plasticity
x
ABSTRACT
The purpose of this project is to demonstrate that crack propagation occurs under special
cases of compressive loads. In most books, crack propagation is only studied under
tensile loads and a common mistake in the industry is to overlook crack propagation
issues when parts or components are stressed by compressive loads.
This project demonstrates that is not safe to ignore crack propagation if a part is
subjected to high compressive loads because these loads may plastically deform the part
leaving high tensile residual stresses. If these high compressive loads are repeated in a
cyclic manner; and the part has a preexisting crack. The residual tensile stresses left in
the part after each cycle may propagate the crack some small length until the crack
arrests itself.
To prove this concept a finite element model of a prismatic bar with a notch was
prepared in ANSYS ver.15 and plastic properties for an INCO718 material were defined
for the part. A cyclic compressive displacement was applied to generate the required
amount of stress intensity and maintain a controlled strain environment.
Using the crack simulation program FRANC3D to demonstrate that the residual
tensile stresses generated by this compressive cyclic load could propagate a crack, a 0.02
inch deep edge crack was introduced inside the notch geometry. Unfortunately it was
not possible to propagate the crack in FRANC3D because some of residual stresses
computed by ANSYS along the crack faces, once the edge crack was inserted into the
FEM analysis, were compressive and produced negative stress intensity factors,
impeding crack propagation.
Instead the more robust Closure Tip Opening Displacement (CTOD) fracture
mechanic method was used to demonstrate edge crack propagation. This method
successfully calculated fatigue crack growth due to tensile residual stress.
xi
1. Introduction
Cracks found in a part under the influence of a compressive stress field are often ignored
due to the common knowledge that a crack only grows when its surfaces are being torn
apart, as is the case of Fracture Mode 1 (opening). Also, because of the fact that when
the crack increases up to a certain size, under compressive load, it will arrest itself (it
will no longer continue to grow). Under these assumptions cracks inside compressive
stress fields are often overlooked. The purpose of this project is to demonstrate that
tensile residual stresses generated by very high compressive strain cycles, in which a
part deforms plastically, are large enough to propagate a crack up to some point.
In many industries, like the automobile, jet engine, power plants, aerospace,
submarine and ships, ignoring cracks under a compressive cyclic load could be
catastrophic. In these industries parts are designed to stand multiple loading conditions at
the same time, for example: temperature, speed, blade-out, pressures, vibrations,
maneuvers, etc. It might happen that a crack under compression might grow long enough
that will eventually be intersected by a tensile stress field from one of the other loading
conditions and continue to propagate in a different direction, Figure 1 shows this
example. This situation has been seen in the field and sometimes is the root cause of part
failure, especially in thin heat shields. In this project a part failure will be considered
when a part is completely broken or the crack has increased to some pre-established
limit that it no longer serves its original purpose.
Crack initially grows due
to Load 1 , second half
due to Load 2.
Load 2
Load 1
Load 1
Crack into
the part
Load 2
Figure 1: Crack Growing by Two Different Stress Fields
1
Other examples in which compressive stress fields are of concern are at welded
locations. Residual stresses left in the part after welding processes can be, in many cases,
large enough to initiate crack growth. But this case, as well as many others are outside
the scope of this study.
During the reference gathering phase to initiate the project a lot of information
was found on the subject of ratcheting, which is defined as the accumulation of cyclic
deformation. Most of the studies found only treated the topic of ratcheting under purely
tensile loads. And just a few studies were found to relate the topic of crack propagation
to compressive loads.
N.A. Fleck, in his study [4], used 2mm BS4360 50B steel plates with center
crack as specimens and despite the fact that the load was fully compressive, cracks grew
in regions of completely tensile residual stresses, but his experimental results were not in
good agreement with his predicted results.
In another study [5], Xiaoping Huang used high strength steel HST-A 6mm
specimens with center crack and with double edge cracks under uniaxial compressive
fatigue loading. He showed that cracks can be propagated under fluctuating compressive
loads to a certain length and then arrested completely. In this study the test data was in
better agreement with the predicted results. But his team put a lot of effort in
experimental studies to come up with a unique material crack growth rate method
(constants) that matched their experimental values.
Both authors had difficulty in calculating the stress intensity factor from their
models due to the complexity of the distribution of the residual stresses. The same
problem was encountered in this study, as will be explained later.
2
2. Theory/Literature
Topics of material strength, elastic-plastic theory, fatigue and fracture mechanics will be
discussed in this section in order to explain to the reader the crack growth studied in this
project.
2.1 Material Strength
The standard tensile test is the most common engineering test used to obtain material
properties for the design of metal parts. One of these properties is the material strength,
and it is obtained by making a stress-strain curve as the one shown below from
experimental tension tests data.
u
f
y
el
pl
Stress, σ
O a
εy
εu
εf
Strain, ε
Figure 2: Engineering Stress-Strain Curve; based on figure from [1]
Stress-Strain curves exhibit different behavior for different materials but for all metals
the behavior is fairly similar, Figure 2. Point ‘pl’ is the proportional limit. This the point
at which the curve first begins to deviate from a straight line. No permanent set is
observed in the specimen if the load is removed. In this linear range, the material
behavior is governed by Hooke’s law and the uniaxial stress-strain relation is:
3
 = 
Eq. 1
Where E, the Young Modulus is the slope of the linear part of the stress-strain curve and
is measure of the stiffness of the material.
Point ‘el’ is called the elastic limit and if the specimen is loaded beyond this point,
the deformation is said to be plastic and the material will take on permanent set when the
load is removed. Point ‘y’ or yield point, is the point at which the strain begins to
increase very rapidly without a corresponding increase in stress. This point is not
obvious and is defined by the offset method shown in Figure 2. Point ‘a’ corresponds to
a definite or stated amount of permanent set, usually 0.2 percent of the original gauge
length although 0.01, 0.1 and 0.5 percent are sometimes used. Point ‘u’ corresponds to
the ultimate tensile strength Su and is the maximum stress reached on the stress-strain
diagram. Finally, point ‘f’ refers to complete fracture of the test specimen.
This project assumed that the ‘el’, ‘pl’ and ‘yield’ points are located in the same
place as the ‘yield’ point, since it is difficult to distinguish a difference between these
three points. An extension to this assumption is that the magnitude of the material yield
strength used for the compression cycle was taken from Appendix 1, which clearly states
came from a tension test (+150,000psi).
2.2 Plastic Deformation
Cold working is the process of plastic straining below the re-crystallization temperature
in the plastic regions of the stress-strain diagram. Consider Figure 3, here the material
has been stressed beyond its yield point (σo) to some point into the plastic region, and
then the load is removed. At this location the material has a permanent plastic
deformation εp. If the load is reapplied, the material will elastically deform by the
amount εe. Thus the total strain consists of two components εp and εe and is given by the
equation:
 =  + 
Eq. 2
The material now has a higher yield point, is less ductile and is said to be strainhardened. Stress-strain behavior beyond yielding is called plasticity. The result of plastic
4
deformation (yielding) is that atoms in the material change neighbors due to the motion
of dislocations and return to a stable configuration with new neighbors after the
dislocation has passed. This is a completely different process than elastic deformation,
which is merely the stretching of chemical bonds. Elastic deformation occurs as an
essentially independent process along with plastic deformation. When a stress that
causes yielding is removed, the elastic strain is recovered just as if there had been no
yielding, but the plastic strain is permanent.
σ
Yield, σ0
ET
E
0
εp
εe
ε
Figure 3: Stress-Strain Diagram Showing Elastic and Plastic Deformation
During plastic deformation, stresses and strains are no longer proportional, so
relationships more general than Hooke’s law are needed to provide an adequate
description of the stress-strain behavior. In order to calculate the correct strains when
undergoing plastic deformation both strain portions, elastic and plastic, need to be added
up as in Equation 2.
5
Elastic strain components portion:
1
 =  [ − ( +  )]
1
 =  [ − ( +  )]
Eq. 3
1
 =  [ − ( +  )]
Plastic strain components portion, specifically for linear plasticity:
1
 =  [ − .5( +  )]

1
 =  [ − .5( +  )]

Eq. 4
1
 =  [ − .5( +  )]

Where Ep is known as the plastic modulus and is defined by:
̅

 = ̅

Eq. 5
̅ = Effective Stress
̅ = Effective Plastic Strain
Since plastic deformation causes residual stresses to remain after unloading, residual
stresses can either decrease or increase the subsequent resistance of a component to
fatigue or environmental cracking, depending on whether the residual stress is tensile or
compressive. If the direction of straining is reversed, after yielding has occurred, the
stress-strain path that is followed differs from the initial monotonic one, Figure 4. This
6
early yielding behavior is called the Bauschinger effect, after the German engineer who
first studied it in the 1880s.
σot
σ
ε
A
σoc
Figure 4: Unloading Stress-Strain Curve where Bauschinger Effect Causes Yielding at A Prior to
the Compressive Yield Strength σoc
Kinematic Hardening predicts that yielding in the reverse direction occurs when the
stress change from the unloading point is twice the monotonic yield strength. Isotropic
hardening predicts yielding later where σ’ is the highest stress reached prior to
unloading. Refer to Figure 5. In this project we will implement the kinematic model,
since it more closely approaches real metals behavior.
σ’
σo
E
2σo
o
2σ’
kinematic
isotropic
Figure 5: Differing Unloading Behavior for Kinematic and Isotropic Hardening
7
2.3 Stress-Strain Curve Model used in ANSYS
The stress-strain curve model used in this project was the elastic, linear-hardening
behavior. It is a useful approximation of the rise after yielding. The slope before yielding
being the Elastic Modulus, E, and the one after yielding being the Tangent Modulus, ET.
If ET = 0 we get the special case that corresponds to an elastic, perfectly plastic behavior.
σ
σo
ET
E
O
ε
Figure 6: Elastic, Linear Hardening Material Behavior
2.4 Fatigue Failure
Often machine members are found to fail under the action of repeated or fluctuating
loads, even when the stresses generated by these loads are well below the yield strength
of the material. This type of failure is known as fatigue. Some of the early scientists who
firs studied this subject were Albert in 1828, which first tested mine hoist chains. And
then later, in the 1850’s Wohler, who tested railroad axes.
Fatigue failure is caused by repeated loading and it involves three phases. The first
phase of fatigue cracking is designated as stage I. It consists of crystal slip that extends
though several contiguous grains, inclusions, and surface imperfections. The second
phase, stage II, the one under analysis in this project is crack growth. And the last phase
or stage III, is when the crack is sufficiently long that K = KIC, the crack becomes
unstable and catastrophic fracture occurs, Figure 7.
8
Crack Length (a)
Fracture
Crack Growth
da
dN
Crack Nucleation
Elapsed Cycles N
Figure 7: Fatigue Failure Mechanism
There are three approaches for fatigue life methods used today in design and analysis of
components, the stress-life method, the strain-life method and linear-elastic facture
method. Each method attempts to predict the life in number of cycles to failure, N. If the
number of repetitions is large say millions of cycles, the term high-cycle fatigue is used.
But if the number of cycles is small, say tens, hundreds or thousands, the term low-cycle
fatigue is used. Low cycle fatigue is generally accompanied by significant amounts of
plastic deformation; whereareas high cycle fatigue is associated with relatively small
deformations that are primarily elastic.
The stress-life method, based on stress levels only, is the least accurate of the
three. However it is the most traditional method since it is the easiest to implement, there
are lots of available data and it represents high cycle applications fairly accurately. The
strain life method involves more detailed analysis of the plastic deformation at localized
regions where the stresses and strains are considered for life estimates. This method is
good for low cycle fatigue applications. Third is the fracture mechanics method, it
assumes a crack is already present and predicts crack growth with respect to stress
intensity.
This project focused on the third method, which assumes a crack is already
present.
9
The fatigue crack growth approach ignores the crack initiation process and assumes a
component is cracked before cycling begins. Crack growth caused by cyclic loading is
called fatigue crack growth. Fatigue crack growth is used in conjunction with damage
tolerance design which is the ability of a structure to resist damage for a specified period
of time; it protects parts from service damage. Engineering analysis of fatigue crack
growth is done using the stress intensity concept, K, of fracture mechanics which is
explained in the next section.
The ratio da/dN, or fatigue crack growth rate, is the slope at a point on the a
versus N curve, Figure 7. It is convention in Fracture Mechanics analysis to determine
the stress range Δσ and the stress ratio R to which the part is exposed, and use these
values to calculate the Stress Intensity Factor Range, ΔK.
Crack growth behavior is best described by the relationship between cyclic crack
growth rate da/dN and the Stress Intensity Factor Range ΔK. Figure 8 below shows this
relationship. First curves representing this relationship were done by Paul Paris in the
da/dN, Crack Growth Rate (in/cycle)
1960’s, who was influential in the first application of fracture mechanics to fatigue.
Region I
Crack
Initiation
Region II
Crack
Propagation
da
dN
Region III
Crack
Unstable
Kc
(ΔK)th
ΔK, Stress Intensity Range (ksi-in1/2 )
Figure 8: da/dN vs ΔK
10
At low growth rates, the curve generally becomes steep and appears to approach a
vertical asymptote denoted ΔKth, which is called fatigue crack growth threshold. This
quantity is interpreted as a lower limiting value of ΔK, below which crack growth does
not occur. At high growth rates, the curve may again become steep, due to rapid unstable
crack growth just prior to final failure, in this case the curve approaches and asymptote
corresponding to ΔK = Kc, Kc being the fracture toughness of the material. Intermediate
values of ΔK, are shown as a diagonal line on a log-log plot similar to the one shown
above, and if the calculated stress intensity factor range is inside this region, crack
propagation will occur.
2.5 Fracture Mechanics
Fracture mechanics is the mathematical analysis of solids with notches, cracks or
defects. Fracture occurs when small cracks in parts start growing until they completely
break or the crack length increases to an already established size that it has to be taken
out of operation and replaced. Prevention of fatigue fracture is a vital aspect of design
for turbines, automobiles, airplanes, helicopters and all kind of structures subjected to
repeated loading or vibration
We have discussed three basic categories of mechanical response to an applied
load: elasticity, plasticity, and fracture. Elasticity is defined by a fully recoverable
response, a component loaded and unloaded without any permanent change to its shape.
Plasticity and fracture both involve permanent shape changes under load. Plasticity is
shape change without cracking, whereas fracture involves the creation or propagation of
a crack that separates a portion of the component from the remainder. The occurrence of
flaws in a structural component is an unavoidable circumstance of material processing,
fabrication, or service. Flaws may appear as cracks, voids, metallurgical inclusions, weld
defects, design discontinuities or some combination of these. Cracks are especially likely
to be found in any service hardware after some usage has occurred.
2.5.1
Linear Elastic Fracture Mechanics (LEFM)
If the material is assumed to behave in a linear elastic manner according to Hooke’s law
Eq. 1, the method use to analyze crack propagation is LEFM. This approach uses the
11
flaw size and shape, component geometry, loading conditions and the material property
fracture toughness to determine the rate of growth.
There are three different fracture modes of loading in fracture as shown on
Figure 9. Mode 1: opening mode, Mode II: sliding and Mode III: tearing. Mode I
loading is the condition in which the crack plane is normal to the direction of largest
tensile loading. This is the most commonly encountered mode and, therefore, is the one
consider here.
Figure 9: Fracture Modes [6]
2.5.2
Fracture Toughness
The fracture toughness, Kc, is a measure of the resistance of the material to brittle
fracture. Is also an indication of the amount of stress required to propagate a preexisting
crack. As with other mechanical properties, fracture toughness is dependent upon alloy
type, processing variables, product form, geometry, temperature, loading rate, and other
environmental factors. K1c is a measure of a given material to resist fracture in the
presence of a crack under mode I loading condition.
12
2.5.3
Stress Intensity Factor
The parameter stress intensity factor (K) is a measure of the severity of a crack as
affected by crack size, stress and geometry. A given material can resist a crack without
brittle fracture occurring as long as the stress intensity factor (K) is below the critical
value of fracture toughness (Kc).
 = √
Eq. 6
a = crack length
σ = nominal stress
β = crack length and component geometry factor
The relationship between stress K, and fracture toughness Kc, is similar to the
relationship between stress and tensile strength. The stress intensity K, represents the
level of stress at the tip of the crack and the fracture toughness Kc, is the highest value of
stress intensity that a material can withstand without fracture. As the stress intensity
factor reaches Kc value, an unstable fracture occurs. In this project (K) will be calculated
using mode I loading condition, no difference will be assumed between K and KI.
2.5.4
Behavior at Crack Tips
An infinite stress cannot exist in real materials. In ductile materials like metals large
plastic deformations occur in the vicinity of the crack tip. This yielded region is called
the plastic zone. Intense deformation at the crack tip results in the sharp tip being
blunted to a small, but non-zero radius. Hence the stress is not infinite and the crack
opens near its tip by a finite amount δ, called crack-tip opening displacement (CTOD).
The very high stress that would theoretically exist is spread over a larger region and is
said to be redistributed, Figure 10.
13
σx
theoretical elastic stress
Yielded, redistributed stress
σ0
2r0σ
y
δ
Plastic zone
Figure 10: Redistribution of Stresses at Crack Tip
2.5.5
Inspections
When speaking about fracture mechanics it is imperative to speak about inspections of
service parts. It is necessary to find cracks before they can grow to a dangerous size.
Methods of inspections include visual examination, X-ray photography, reflection of
ultrasonic waves, and application of electric currents. Repairs of these cracks sometimes
are possible using processes such as machining to leave a smooth surface but most of the
times replacement of the whole part is required.
14
3. Methodology
Four steps were followed to conduct the analysis.
1. Create the model
2. Define and apply the correct material properties in ANSYS v15.0
3. Generate stresses
4. Introduce a crack and check propagation.
The subsections below explain in detail each one of these steps.
3.1 Geometry
The geometry shown in Figure 11 was created and meshed in ANSYS v15. It is a
prismatic bar with square cross section with dimensions 1 in x 1 in x 12 in. The bar cross
section thickness of 1 inch was chosen having in mind the crack plane strain condition
and to obtain a uniform plastic zone through the entire thickness.
Figure 11: ANSYS Square Bar Finite Element Model (FEM)
A notch of depth 0.2 inch and radius 0.05 inch as shown in Figure 12 was introduced in
the model to increase stresses in the region of interest without yielding at other points of
the bar.
15
r =0.05 inch
0.2 inch
Figure 12: Notch Dimensions
3.2 Material Properties
The material chosen for the analysis was INCO718, being the most important reason the
availability of the crack growth rate data that drives this study. All the material
properties where found in [3], which is a metallic materials design data Handbook
acceptable to Government procuring and certification agencies. This 2,288 page
document is available for buying on the World Wide Web (projects.battelle.org). This
document supersedes the more known MIL-HDBK-5. A copy of the material properties
for INCO 718 can be found in the Appendix 1 of this document. Table 1, below,
summarized the mechanical material properties used for the analysis in this project.
Table 1: INCO 718 Mechanical Material Properties
Specification
AMS 5663
Form
Forging
Elastic Modulus, E
29,400,000 psi
Poisson Ratio, ν
0.29
Density, ρ
0.297 lbm / in3
Yield Tensile Strength (YTS)
150,000 psi
Ultimate Tensile Strength (UTS)
185,000 psi
% of Elongation, e
12
16
3.2.1
Definition of Material Properties in ANSYS
The stress-strain relationship selected to model the plasticity of the INCO718 was the
elastic, linear-hardening behavior which is explained in a previous section. Figures 13
and 14 below show how the bilinear kinematic hardening behavior was defined in
ANSYS v15.0.
Figure 13: ANSYS v15.0 Bilinear Material Property Definition Windows
Figure 14: ANSYS v15.0 Bilinear Material Property Definition Windows
The material ultimate tensile strength σuts, yield tensile strength σyts and % of elongation
to fracture were used to calculate the Tangent Modulus (ET), Equation 7.
17

−


 = %−0.002
Eq. 7
Figure 15 shows the stress-strain curve defined in ANSYS. The results provided by
ANSYS were no longer linear and followed this bilinear stress-strain curve. One thing
that was noted when properties were applied in ANSYS was that the elastic modulus E
and the tangent modulus ET didn’t intersect at the yield point offset ε = 0.002 as you
would normally expect, but instead they intersect at ε = 0.005, Figure 15. As the theory
section mentioned, offsets of 0.01, 0.1 and 0.5 percent are sometimes used in the
industry. To compensate for this difference Equation 9 was modified to calculate the
plastic slope ET with a Δε of (%Elong - 0.005).

−


 = %−0.005
Figure 15: ANSYS Bilinear Kinematic Model Intersecting Slope at ε=0.005
18
Eq. 8
Figure 16 below demonstrate the offset method used to determine the yield point at 0.5%
strain. As mentioned previously not all materials have obvious yield points, especially
brittle ones.
σ
Yield, σ 0
ET
E
0
εp
εe
ε
0.2% 0.5%
Figure 16: Bilinear Kinematic Model Intersecting Slopes at ε=0.002 and ε=0.005
3.3 Loading Conditions
The square bar was fixed in all directions at its left end (see Figure 17). At the other end
along the upper edge of the bar a compressive displacement of 0.045 inch was applied.
The part was compressed until stresses were above the yield strength of INCO718, YS=
150,000 psi. Then the bar was pulled back to its original position, applying a 0.00 inch
displacement on a second load step. Then the cycle was repeated. Ten loads steps were
created using this loading sequence, which means 5 complete cycles were applied to the
part. The idea behind this cyclic load was to create a controlled strain loading condition.
19
ux = -0.045 inch
Face Fixed –
All DOF
Y
Z
X
Figure 17: Body Diagram of Boundary Conditions Applied in ANSYS v15.0
Figure 18: ANSYS FEM Showing Location of Applied Constraint (Left) and Displacement (Right)
The sample was meshed using quadratic elements (mid nodes) to get more accuracy for
the stress and strain results. The large amount of displacement applied was needed in
order to plastically deform the part. Due to bilinear kinematic hardening material
property used, ANSYS performs a nonlinear solution to solve the problem.
20
4. Results and Discussion
4.1 Stress-Strain Results
Table 2 and Figure 19 below show the results obtained from ANSYS v15 for each one of
the load steps applied to the FEM. The center node in the middle of the notch was
selected to get stress and strain results. As explained previously, one cycle is composed
of two load steps. Load step 1 is the compressive portion of the cycle and load step 2 is
the return of the sample to its original length.
Table 2: Stress and Strain Results at Notch Center Node
Stress vs Strain History
200,000
150,000
100,000
Stress (psi)
50,000
-0.0200
-0.0180
-0.0160
-0.0140
-0.0120
-0.0100
-0.0080
-0.0060
-0.0040
-0.0020
0
0.0000
-50,000
-100,000
-150,000
-200,000
-250,000
-300,000
Strain (in/in)
LS1
LS2
LS3
LS4
LS5
LS6
LS7
LS8
LS8
LS9
Figure 19: Cyclic Stress-Strain Curve from ANSYS Results Table 1
21
LS10
The even load steps show residual tensile stresses left in the notch after the part is
brought back to its original length. On the first half of the cycle or first load step the part
is compressed past the yield point in compression up to -247,374 psi. When the
compressive displacement is removed the part is pulled back to its original length. The
tensile residual stress left in the part is +145,813 psi. Refer to Table 2 and Figure 19
above for details.
Figure 20 shows the material hardening behavior after each load cycle. It can be
seen that stress increase after each load step using the same amount of displacement,
0.045 inch.
Figure 20: ANSYS Results, Stress vs. Load Step
22
4.1.1
First Half of Load Cycle: Compression
Figure 21, below, shows the displacement of the sample when compressed in the xdirection, Load Step 1. The same direction as the load was applied.
Figure 21: Displacement, X-direction; Load Step 1
Figure 22, below, shows compressive stress results for Load Step 1.
Figure 22: Stress, X-direction; Load Step1
23
Figure 23, below, shows compressive strain results for Load Step 1.
Figure 23: Total Mechanical Strain, X-direction; Load Step 1
4.1.2
Second Half of Load Cycle: Return Sample Back To Its Original Length
Figure 24, below, shows the computed displacements in the x-direction for the second
half of the cycle, Load Step 2.
Figure 24: Displacement, X-direction; Load Step 2
24
Figure 25, below, shows stress results for Load Step 2. Note the red tensile residual
stresses right at the rim of the notch, the stress magnitude in this region is around
149,000 psi, close to the yield strength of the material. The green area right below the
red is still dominated by tensile residual stress but of less intensity (around 63,000 psi).
And just a few thousands of an inch below the green area, in the blue area, a sudden drop
to compressive stresses is seen. In this region any introduced crack in the notch is
expected to arrest itself if it propagates.
Figure 25: Stress, X-direction; Load Step 2
Figure 26, below, shows plastic strain results for Load Step 2, after the part is pulled
back to its original length. It can be seen that the part deformed plastically and
compressive strains remained after unloading the compressive displacements.
25
Figure 26: Total Mechanical Strain, X-direction; Load Step 2
Only the result plots for the first two load steps were shown, since we are dealing with a
cyclic load problem and the plots for the following load steps are very similar, except
that the magnitudes of the strains and stresses are different. Refer to Table 2 for result
values of the other load steps.
4.2 Crack Growth Results (FRANC3D)
The first method used to investigate if the residual stresses seen in ANSYS were large
enough to propagate a crack was using the open source computer code, FRANC3D. It
was used to insert a crack in the original ANSYS finite element model and calculate the
stress intensity factor. To do this, the original ANSYS mesh, used to generate the above
results of Section 4.1, was input into FRANC3D and an edge crack was inserted inside
the notch geometry, across the thickness of the sample. Figure 27, below, shows the new
mesh in the notch region once the 0.02 inch deep edge crack was inserted.
26
This new FEM with an edge crack in the notch was run in ANSYS using the same first
cycle as the FEM without the edge crack, Figure 18.
Edge Crack
Figure 27: ANSYS Finite Element Model with Edge Crack in the Notch
Figure 28: Zoomed View of ANSYS FEM with Edge Crack
The results for the first two Load Steps, or first cycle, are shown in Figures 29 and 30,
below. Larger stress magnitudes in compression and tension, than the ones computed for
Figures 22 and 25 respectively, were expected, due to the high plastic deformation at the
crack tip. But, during the compression portion of the cycle, Load Step 1, the crack faces
27
penetrated and crossed each other, leaving plastic deformation in the crack faces.
Contact elements, in the crack faces were needed to avoid this penetration. Due to time
constraints however, this will be left for future studies.
Edge Crack
Figure 29: Stress with Edge Crack, X-direction; Load Step 1
Edge Crack
Figure 30: Stress with Edge Crack, X-direction; Load Step 2
28
Compressive Stress at
Crack Face = -40,050 psi
Max Stress along Crack Tip Edge
= +329,811 psi
Crack Tip
Figure 31: ANSYS Residual Stress Distribution at Edge Crack Faces; Load Step 2
Figure 31, above, show the remained tensile residual stress close to the crack once the
sample was unloaded. It can be seen that the area around the crack is green, which
means that tensile residual stress dominates the region. But in the crack faces, as can be
seen in the figure, a small compression stress field was generated due to the missing
contact elements. The stress at the crack tip is a very high tensile stress (+329,811 psi)
but on the track faces of the crack there is a small amount of compressive stress (-40,050
psi). As mentioned before, it was concluded that the crack faces crossed each other,
deformed plastically and did not follow the rest of the material when the sample was
returned to its original length.
Figure 32, below, shows the negative stress intensity factor calculated by
FRANC3D when the displacements form the stress field around the crack where
exported from ANSYS into FRANC3D for stress intensity computations. Since some of
the displacements around the crack were negative, no crack propagation was seen,
because they produced negative stress intensity factors.
29
Figure 32: FRANC3D Negative Stress Intensity Factor Results
In reality, this crack should grow under this type of loading and it has been demonstrated
in some previous studies, as mentioned in the Introduction. In order to the evaluate crack
growth problem, an alternative approached was used for this study.
4.3 Crack Growth Results (CTOD)
Besides K there are two other fracture mechanics parameters that can be used to
calculate crack propagation driving forces: the path independent integral, J, and the
crack tip opening displacement (CTOD) δ. These two parameters are used primarily in
the elastic-plastic regime, whereareas the stress intensity parameter, K, is used primarily
in the linear elastic regime.
The path independent J-integral proposed by Rice and used by FRANC3D is a
method for characterizing the stress-strain field at the tip of a crack by an integral path
taken at some distance from the crack to be analyzed elastically, and then substituted for
the inelastic region close to the crack tip. This also explains why the stress intensity
factor was negative in FRANC3D, since this method replaces the stress region at the
crack tip with a nearby stress field. It is not clear exactly what near means for
FRANC3D, but the another conclusion is that it is replacing the stresses at the crack tip
with the compressive stresses enclosing the crack, Figure 31.
In 1961, Wells proposed that the fracture behavior in the vicinity of a sharp crack
could be characterized by the opening of the notch faces. This method is known as crack
30
tip opening displacement or CTOD. Because CTOD measurements can be made when
there is considerable plastic flow ahead of a crack, as would be expected for elasticplastic or fully plastic behavior, this technique can be used to establish crack sized in a
quantitative manner similar to that of LEFM.
The CTOD relationship for an edge crack is [7]:
 = 1.122
2 

Eq. 10
If the stress intensity factor, K, for an edge crack is used:
 = 1.12 √
2
 = 

Eq. 11
Eq. 12
Since elastic-plastic methods, such as the J-integral and CTOD are used to calculate
crack propagation driving forces, some engineers prefer to measure fracture toughness in
terms of Jc and δc for convenience. There are relations available to convert those
measurements to values of Kc. Since the material crack growth rate data available in this
study was already in terms of Kc in Appendix 1, we simplified the methodology and
calculated δ first and then converted to K as shown in Equation 13. Finally we compared
the calculated K against the material Kc to determine if there was indeed crack growth.
Several relations between the various fracture parameters were found in [7],
among them the one shown below, between K and δ. It can be obtained by rewriting
Equation 12:
 = √ 
Eq. 13
To evaluate the crack propagation problem, ANSYS stress results without the insertion
of the edge crack (Figure 25) were used because the theoretical equations [7] to calculate
CTOD are based in nominal stresses through the thickness and do not require stresses at
31
the crack tip. In order to check for crack growth, it was necessary to calculate the stress
intensity factor range ΔK.
∆ =  − 
Eq. 14
To simplify the stress range factor calculations for this study and because the crack
growth rate data available in Appendix 1 is for stress R ratios close to zero (R=0.05),
Kmin was considered to be zero. This turns out to be conservative because Kmin is
negative, if the stresses from the first half of the cycle are used, it would produce an even
larger ΔK.
Using the above equations, assuming an edge crack of 0.006 inch and solving for
ΔK, which in this case is Kmax, since we made the assumption of Kmin = 0, the results are
shown in the tables below. These equations use nominal stresses; in this case we
assumed the nominal stresses to be mean stresses across the thickness of the bar, opening
the crack, up to the compressive region. Refer to Appendix 3 for details on how these
nominal/mean stresses were calculated.
Table 3: Results for Cycle 1 (Load Step 1 & 2)
σ (psi)
59,682
a (inch)
0.006
E (psi)
29,400
σys (psi)
150,000
δI (inch)
0.019
E (psi)
29,400
σys (psi)
150,000
δI (inch) KI (ksi*in^1/2)
0.019
9,177
Comparing the calculated K with INCO718 ‘da/dN vs. ΔK’ data from table in Appendix
1, it was seen that K was below the ℎ = 10.59  √ value for this material and
therefore no crack propagation was expected.
32
Table 4: Results for Cycle 2 (Load Step 3 & 4)
σ (psi)
69,615
a (inch)
0.006
E (psi)
29,400
σys (psi)
150,000
δI (inch)
0.026
E (psi)
29,400
σys (psi)
150,000
δI (inch) KI (ksi*in^1/2)
0.026
10,705
On the second cycle, since the tensile residual stresses increased due to the hardening of
the material, the calculated K value was over the ΔKth of the material and therefore the
crack was expected to grow.
It is only logical to assume that the crack is going to grow up to a certain size
until the crack reaches the compressive stress field and no more crack propagation takes
place. Refer to Figure 33, below.
Compressive stress field
(No crack propagation will
occur in this region)
Figure 33: Crack Arrest (Crack Will No Longer Continue to Grow)
33
5. Conclusion
A sample of a prismatic bar with square cross section (12x1x1 inch) and a center notch
was modeled using ANSYS. INCO718 elastic and plastic material properties were
defined in ANSYS for the sample, in order to obtain the kinematic hardening behavior of
the material. A compressive strain controlled cyclic load was applied in ANSYS as
well. This strain controlled cycle consisted of two load steps: in the first portion of the
cycle the sample was compressed way passed its material yield strength of negative
150,000 psi in the notch region and the second portion of the cycle consisted of returning
the sample back to its original length. The idea behind this cyclic loading was to
generate enough tensile residual stresses in the notch area to make and edge crack
propagate.
Two approaches were used to verify crack propagation. One of them consisted of
using the open source code from Cornell University FRANC3D and the other approach
consisted of using the theoretical equations of CTOD [7]. The FRANC3D approach did
not work well, due to ANSYS computations of compressive residual stresses once a
crack was inserted in the mesh and the load cycle applied. The CTOD approach,
however, on the other hand successfully calculated fatigue crack growth.
This study proved that residual tensile stress values can be large enough to
produce crack propagation if a high cyclic compressive load is applied. Also we have
made designers aware of the importance of having this design requirement in mind when
making parts that will be subjected to this type of loading.
Other findings of importance in this experiment are:

It can be seen that the material cyclically hardens, meaning that ANSYS
accurately implements cyclic hardening behavior if the right properties are input.

The residual stress caused by a compression cycle loading can be estimated
accurately from finite element analysis.

Stress intensity factor is difficult to calculate from residual stress distribution.
Also, it has been shown that there is plasticity limitation on the LEFM approach. If the
plastic zone is too large, it eliminates the K-field and the LEFM approach is no longer
34
valid. If the plastic zone surrounding the crack tip is quite small compared with
thickness, and is very well isolated relative to the boundaries of the sample, then a state
of plane strain is established.
Modest amounts of yielding can be handled using
mathematical LEFM adjustments. However, for fully plastic deformation, more general
methods such as the J-integral or the Crack Tip Opening Displacements are needed.
More importantly, it has been demonstrated that the CTOD concept provides an
engineering approach to fracture beyond linear elasticity.
35
6. References
[1] Joseph E. Shigley, Charles R. Mishcke and Richard G. Budynas, Mechanical
Engineering Design, McGraw-Hill, 2004.
[2] Richard W. Hertzberg, Richard P. Vinci and Jason L. Hertzberg, Deformation and
Fracture Mechanics of Engineering Materials, John Wiley & Sons, 2013.
[3] MMPDS 07 - Metallic Materials Properties Development and Standardization,
Battelle Memorial Institute (www.projects.battelle.org), 2012.
[4] NA Fleck, CS Chin, and RA Smith, Fatigue Crack growth under compressive
loading. Engineering Fracture Mechanics.1985;21(1):173-175
[5] Xiaoping Huang, Anqing Wang, Weicheng Cui, and Rugang Bian, The Fatigue
Crack Growth under Compressive to Compressive Fluctuating Loading, ASME
29th International Conference on Ocean, Offshore and Arctic Engineering, 2010.
[6] Norman E. Dowling, Mechanical Behavior of Materials, Pearson Education Inc,
2013.
[7] John M. Barsom and Stanley T. Rolfe, Fracture and Fatigue Control in Structures,
ASTM, April 2006
36
7. Appendices
7.1 Appendix 1: INCO 718 Material Properties
37
38
39
7.2 Appendix 2: ANSYS Macro
/TITLE,squarebeam
/PREP7
BLOCK,0,12,0,1,0,1,
/USER, 1
/FOC,
1,
5.87758934376
,
0.751659985393
/VIEW, 1, 0.311246201990
,
0.901406086448
/ANG,
1,
2.02795323311
/FOC,
1,
5.53535082672
,
1.94773449141
/VIEW, 1, 0.521944551532E-01,
0.987599023613
/ANG,
1, -1.31454634782
lplo
wpoff,6
wpro,,,90.000000
VSBW,
1
wpoff,1
lplo
wpoff,-1
wpoff,0,0,1
VSBW,
3
wpoff,0,0,-2
VSBW,
2
lplo
WPCSYS,-1,0
FLST,3,1,3,ORDE,1
FITEM,3,12
KGEN,2,P51X, , , ,-.2, , ,0
FLST,3,1,3,ORDE,1
FITEM,3,12
KGEN,2,P51X, , ,-.05, , , ,0
FLST,3,1,3,ORDE,1
FITEM,3,12
KGEN,2,P51X, , ,0.05, , , ,0
FLST,3,3,3,ORDE,3
FITEM,3,12
FITEM,3,22
FITEM,3,-23
KGEN,2,P51X, , , ,-.15, , ,0
KWPAVE,
24
CSWPLA,11,1,1,1,
L,
21,
25
L,
21,
26
csys,0
L,
25,
22
L,
26,
23
lplo
0.371111992053
,
0.300986493156
,
0.274311128118
,
0.148067239486
,
40
FLST,2,4,4,ORDE,4
FITEM,2,13
FITEM,2,-14
FITEM,2,16
FITEM,2,-17
ADRAG,P51X, , , , , ,
20
FLST,3,2,5,ORDE,2
FITEM,3,7
FITEM,3,-8
VSBA,
5,P51X
FLST,3,2,5,ORDE,2
FITEM,3,9
FITEM,3,-10
VSBA,
4,P51X
FLST,2,2,6,ORDE,2
FITEM,2,2
FITEM,2,5
VDELE,P51X, , ,1
WPSTYLE,,,,,,,,0
lplo
!*
ET,1,SOLID186
!*
!*
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,DENS,1,,7.6224E-005
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,1,,2.94E+007
MPDATA,PRXY,1,,.29
TB,BKIN,1,1,2,1
TBTEMP,0
TBDATA,,1.5e5,3.2309e5,,,,
lplo
/ZOOM,1,RECT,0.275771,0.307707 ,0.603760789149 ,0.0936498150432
FLST,5,2,4,ORDE,2
FITEM,5,13
FITEM,5,-14
CM,_Y,LINE
LSEL, , , ,P51X
CM,_Y1,LINE
CMSEL,,_Y
!*
LESIZE,_Y1, , ,6, , , , ,0
!*
ESIZE,.1,0,
FLST,5,4,6,ORDE,4
FITEM,5,1
FITEM,5,3
FITEM,5,6
FITEM,5,-7
41
CM,_Y,VOLU
VSEL, , , ,P51X
CM,_Y1,VOLU
CHKMSH,'VOLU'
CMSEL,S,_Y
!*
VSWEEP,_Y1
!*
CMDELE,_Y
CMDELE,_Y1
CMDELE,_Y2
!*
/UI,MESH,OFF
/AUTO,1
/REP,FAST
!*
eplo
/ZOOM,1,SCRN,-0.867016,0.017694,-0.808323,-0.120407
FLST,2,341,1,ORDE,6
FITEM,2,6307
FITEM,2,-6317
FITEM,2,10124
FITEM,2,-10413
FITEM,2,15850
FITEM,2,-15889
!*
/GO
D,P51X, ,0, , , ,ALL, , , , ,
/AUTO,1
/REP,FAST
/ZOOM,1,SCRN,1.532491,0.059125,1.556658,0.000432
FLST,2,117,1,ORDE,12
FITEM,2,4717
FITEM,2,-4725
FITEM,2,4776
FITEM,2,4827
FITEM,2,9854
FITEM,2,-9937
FITEM,2,15664
FITEM,2,-15669
FITEM,2,15724
FITEM,2,-15729
FITEM,2,15840
FITEM,2,-15849
!*
/GO
D,P51X, ,-.045, , , ,UX, , , , ,
/AUTO,1
/REP,FAST
nplo
eplo
/COLOR,U,BMAG
42
/COLOR,ROT,ORAN
/COLOR,TEMP,ORAN
/COLOR,VOLT,YGRE
/COLOR,F,RED
/COLOR,M,CBLU
/COLOR,HEAT,GCYA
/COLOR,MAST,BMAG
/COLOR,CP,GREE
/COLOR,CE,MRED
/COLOR,NFOR,MRED
/COLOR,NMOM,GREE
/COLOR,RFOR,MAGE
/COLOR,RMOM,BMAG
/COLOR,PATH,WHIT
/REPLOT
!*
LSWRITE,1,
LSWRITE,3,
LSWRITE,5,
LSWRITE,7,
LSWRITE,9,
/VIEW,1,,,1
/ANG,1
/REP,FAST
/ZOOM,1,SCRN,1.522133,0.069482,1.529038,0.007337
FLST,2,117,1,ORDE,12
FITEM,2,4717
FITEM,2,-4725
FITEM,2,4776
FITEM,2,4827
FITEM,2,9854
FITEM,2,-9937
FITEM,2,15664
FITEM,2,-15669
FITEM,2,15724
FITEM,2,-15729
FITEM,2,15840
FITEM,2,-15849
!*
/GO
D,P51X, ,0, , , ,UX, , , , ,
LSWRITE,2,
LSWRITE,4,
LSWRITE,6,
LSWRITE,8,
LSWRITE,10,
SAVE
/SOL
LSSOLVE,1,10
43
7.3 Appendix 3: Calculate Nominal Stress Using ANSYS Linearization
A path was created at the center of the notch from the max tensile stress node to the min
compressive stress node. Stresses in the x-direction were map onto the path.
Stress Path
An ANSYS post process stress linearization was done to determine membrane stress,
bending stress and total stress (membrane + bending) along the defined node path.
Mean Stress acting on
defined stress path
Only membrane portion of the stress was assumed to act as the nominal stress since is
the one opening the crack.
44
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