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Pardo-FinalReport rev11.pdf
Crack Growth under Tensile Residual Stresses produced by
High Compressive Strain Controlled Cycle
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 ENGINNERING
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 ................................................................................................... 9
2.4.1
2.5
Fatigue Crack Growth Approach ......................................................... 11
Fracture Mechanics .......................................................................................... 12
2.5.1
Linear Elastic Fracture Mechanics (LEFM) ........................................ 12
2.5.2
Fracture Toughness .............................................................................. 13
2.5.3
Stress Intensity Factor .......................................................................... 13
2.5.4
Behavior at Crack Tips......................................................................... 14
2.5.5
Inspections ........................................................................................... 15
3. Methodology .............................................................................................................. 16
3.1
Geometry .......................................................................................................... 16
3.2
Material Properties ........................................................................................... 17
3.2.1
3.3
Definition of Material Properties in ANSYS ....................................... 17
Loading Conditions .......................................................................................... 20
4. Results and Discussion .............................................................................................. 21
4.1
Stress-Strain Results Discussion ...................................................................... 21
iii
4.1.1
First Half of Load Cycle: Compression ............................................... 23
4.1.2
Second Half of Load Cycle: Return sample back to its original length
.............................................................................................................. 24
4.2
Crack Growth Results with mesh edge crack (FRANC3D)............................. 26
4.3
Crack Growth Results without mesh edge crack (CTOD) ............................... 29
5. Conclusion ................................................................................................................. 32
6. References.................................................................................................................. 33
7. Appendices ................................................................................................................ 34
7.1
Appendix 1: Material Properties ...................................................................... 34
7.2
Appendix2: ANSYS Macro ............................................................................. 37
7.3
Appendix 3: Calculate nominal stress using ANSYS linearization. ................ 41
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 (√)
K
= Stress Intensity Factor ( √)
ΔK
= Stress Intensity factor range ( √)
R
= Ration of minimum to maximum (-)
Kth
= Threshold stress intensity factor ( √)
da/dN = Crack growth rate (-)
a
= Crack Length (inch)
LS
= Load Step (-)
v
LIST OF TABLES
Table 1: Stress and Strain Results at node 25135 ............................................................ 21
Table 2: Results for Cycle 1 (Load Step 1 & 2) .............................................................. 30
Table 3: Results for Cycle 2 (Load Step 3 & 4) .............................................................. 30
vi
LIST OF FIGURES
Figure 1: Crack growing by two different stress fields ..................................................... 1
Figure 2: Engineering Stress-Strain Curve ........................................................................ 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................................................................................... 8
Figure 7: Fatigue Failure Mechanism ................................................................................ 9
Figure 8: da/dN vs ∆K ..................................................................................................... 11
Figure 9: Fracture Modes................................................................................................. 13
Figure 10: Redistribution of Stresses at Crack Tip .......................................................... 14
Figure 11: ANSYS Square Bar Finite Element Model (FEM) ........................................ 16
Figure 12: Notch Dimensions .......................................................................................... 17
Figure 13: ANSYS v15.0 Bilinear Material Property Definition Windows.................... 17
Figure 14: ANSYS v15.0 Bilinear Material Property Definition Windows.................... 18
Figure 15: ANSYS Bilinear kinematic model intersecting slope at ε=0.005 .................. 19
Figure 16: Bilinear kinematic model intersecting slopes at ε=0.002 and ε=0.005 .......... 19
Figure 17: Sketch of Boundary Condition applied in ANSYS ....................................... 20
Figure 18: ANSYS FEM showing location of applied constraints and displacements ... 20
Figure 19: Cyclic Stress-Strain Curve from ANSYS Results Table 1 ............................ 21
Figure 20: ANSYS Results, Stress vs. Load Step ........................................................... 22
Figure 21: LS 1 Displacement X-direction ...................................................................... 23
Figure 22: LS 1 Stress x-direction ................................................................................... 23
Figure 23: LS 1 Total Mechanical Strain x-direction ...................................................... 24
Figure 24: LS 2 Displacement x-direction....................................................................... 24
Figure 25: LS 2 Stress x-direction ................................................................................... 25
Figure 26: LS 2 Total Mechanical Strain x-direction ...................................................... 25
Figure 27: FRANC3D Edge Crack mesh ........................................................................ 26
Figure 28: ANSYS residual stress distribution at crack faces ......................................... 26
Figure 29: FRANC3D Negative Stress Intensity Factor Results..................................... 27
vii
Figure 30: Crack arrest .................................................................................................... 31
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
Edge Crack
Liner Elastic Fracture Mechanics (LEFM)
Stress Intensity Factor (K)
x
ABSTRACT
The purpose of this project is to demonstrate that crack propagation occur 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. In many cases high compressive loads plastically
deformed 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 after each cycle will propagate the crack up to some small length or until the
crack arrest itself.
To proof this concept a finite element model of a square bar with a notch was
prepared in ANSYS ver.15, plastic properties for an INCO718 material were defined for
the part and a compressive displacement was applied to generate the require amount of
stress intensity and maintain a controlled strain environment.
An edge crack was then induced into the geometry using the crack simulation
program FRANC3D to demonstrate that the stresses generated by this cycle could
indeed propagate the crack but unfortunately the crack propagation correlation use by
FRANC3D does not support the stress fields generated by ANSYS and produced
negative stress intensity factors.
An explanation for the behavior seen in FRANC3D was proposed and alternate
theoretical method for crack propagation CTOD (Closure Tip Opening Displacement)
was used to proof the concept.
xi
1. Introduction
There is a general assumption in the field that cracks found in a part under the influence
of a compressive stress field, are often ignore due to the common knowledge that a crack
only grows when its surfaces are being tear 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). 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. Under these assumptions cracks inside compressive
stress fields are often overlooked.
But 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 show this
example. This situation has been seen in the field and sometimes is root cause of parts
failure, especially in thin heat shields. In this project a part failure will be considered
when a part is completely broken or broke to such degree 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 is 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 is outside the scope
of this project and will be material for future research.
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 “Fatigue Crack Growth under Compressive Loading” [4]
used 2mm BS4360 50B steel plates with center crack as specimens and despite the fact
that the load was fully compressive, cracks grow in regions of completely tensile
residual stresses but his experimental results were not in good agreement with his
predicted results.
In another study “The fatigue Crack Growth under Compressive to Compressive
Fluctuating Loading” [5] Xiaoping Huang used high strength steel HST-A 6mm
specimens with center crack and with double edge crack 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 on
experimental studies to come up with a unique material crack growth rate method
(constants) that match 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. Same problem
was encountered here 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 under studied
in this project.
2.1 Material Strength
The standard tensile test is the most common engineering test used to obtain material
properties that are used in 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
Stress-Strain curves exhibit different behavior for different materials but for 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. The ‘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 ‘el’, ‘pl’ and ‘yield’ point are located in the same place
the ‘yield’ point since it is difficult to distinguish a difference between these three points.
An extension to this assumption in this project is that the material yield strength is the
same in tension as in compression; which is a very common assumption for engineering
metals.
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 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 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, σ 0
ET
E
0
εe
εp
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.
= + = + = + Eq. 2
In order to calculate the correct strains when undergoing plastic deformation both strain
portions, elastic and plastic need to be add up.
5
Elastic strain components portion:
= − + = − + Eq. 3
= − + Plastic strain components portion:
= − .5 + = − .5 + Eq. 4
= − .5 + 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
early yielding behavior is called the Bauschinger effect, after the German engineer who
first studied it in the 1880s.
6
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 gives the special case that corresponds to an elastic, perfectly plastic behavior.
σ
σo
ET
E
O
ε
Figure 6: Elastic, Linear Hardening
8
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 that 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 is crack growth and the one under study in this project. And the last phase
or stage III, is when the crack is sufficiently long that KI = KIC, the crack becomes
unstable and catastrophic fracture occurs. Figure 7.
Crack Length (a)
Fracture
Crack Growth
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 stain-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
9
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 and
there is lots of available data and represent 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.
10
2.4.1
Fatigue Crack Growth Approach
Incomplete
Crack growth approach ignores initiation and assumes component is cracked before
cycling begins. Crack growth is used in conjunction with damage tolerance design.
Damage tolerance is the ability of a structure to resist damage for a specified period of
time; it protects parts from service damage.
Crack growth cause by cyclic loading, is called fatigue crack growth.
Engineering analysis of crack growth is done with the stress intensity concept, K, of
fracture mechanics.
The rate of fatigue growth is controlled by K. Hence the
dependence of K on a and F causes cracks to accelerate as crack grow. Predictions of life
to grow a crack to failure
Log da/dN
Region I
Region II
Crack
Initiation
Crack
Propagation
Region III
Crack
Unstable
Kc
(ΔK)th
Log ΔK
Figure 8: da/dN vs ∆K
11
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 a part or the crack increases to a limit size that it has to be taken out 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
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.
12
Figure 9: Fracture Modes
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.
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 the stress intensity factor (K) is below the critical value of
fracture toughness (KIc).
= √ !
a = crack length
σ = nominal stress
β = crack length and component geometry factor
13
Eq. X
The relationship between stress KI, and fracture toughness KIc, is similar to the
relationship between stress and tensile strength. The stress intensity KI, represents the
level of stress at the tip of the crack and the fracture toughness KIc, is the highest value
of stress intensity that a material can withstand without fracture. As the stress intensity
factor reaches KIc value, unstable fracture occurs.
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 is
open near its tip by a finite amount δ, called crack-tip opening displacement (CTOD). A
crack experiences an intense deformation and develops a finite separation near its tip.
The very high stress that would theoretically exist is spread over a larger region and
redistributed.
Incomplete
Figure
Figure 10: Redistribution of Stresses at Crack Tip
14
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.
15
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 explained in detail each one of the steps.
3.1 Geometry
The geometry shown in Figure 11 was created and meshed in ANSYS v15. It is a square
bar 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 obtained 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 (Figure 12) was introduced in the model
to increase plastic stresses in the region of interest without yielding at the opposite
portion side of the bar.
16
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 the MMPDS-07 [3] ‘Metallic Materials Properties
Development and Standardization’, the latest metallic materials design data acceptable
to Government procuring and certification agencies. This 2,288 pages document is open
to public and can be found for download on the World Wide Web. This document
supersedes the more known MIL-HDBK-5. A copy of the materials used can be found in
the Appendix 1 of this document.
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 on 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
17
Figure 14: ANSYS v15.0 Bilinear Material Property Definition Windows
The material ultimate tensile strength σuts, yield strength σyts and % of elongation to
fracture were used to calculate the Tangent Modulus (ET), Equation X.
#
'#
()
$%&
" = %+,-.'/.//0
Eq. X
Figure 15 shows the stress-strain curve used by ANSYS. The results provided by
ANSYS were no longer linear and followed the defined 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
industry. To compensate for this difference Equation X was modified to calculate the
plastic slope ET with a Δε of (%Elong - 0.005).
#
'#
()
$%&
" = %+,-.'/.//1
Eq. X
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.
18
Figure 15: ANSYS Bilinear kinematic model intersecting slope at ε=0.005
σ
Yield, σ0
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
19
3.3 Loading Conditions
The square bar was fixed in all direction in of its end (left side on Figure 17). At the
other end a compressive displacement of 0.045 inch was applied. The part was compress
until stresses were above the yield strength of INCO718, YS= 150,000 psi. Then the bar
was pull back to its original position applying a 0.00 inch displacement on 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 create to create a controlled strain loading condition.
ux = -0.045 inch
Face Fixed –
All DOF
Y
Z
X
Figure 17: Sketch of Boundary Condition applied in ANSYS
Figure 18: ANSYS FEM showing location of applied constraints and displacements
The sample was meshed using quadratic elements (mid nodes) to get less error estimates
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 Discussion
The table 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 1: Stress and Strain Results at node 25135
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
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
compress pass the yield point in compression up to -247,374 psi. When the compressive
displacement is removed the part is pull back to its original length. The tensile residual
stress left in the part is +145,813 psi. Refer to Table 1 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 in the x-direction. Same direction
as the load was applied.
Figure 21: LS 1 Displacement X-direction
Figure 22 shows compressive stress results for load step 1.
Figure 22: LS 1 Stress x-direction
23
Figure 23 shows compressive strain results for load step 1.
Figure 23: LS 1 Total Mechanical Strain x-direction
4.1.2
Second Half of Load Cycle: Return sample back to its original length
Figure 24 below shows the displacements in the x-direction for the second half of the
cycle.
Figure 24: LS 2 Displacement x-direction
24
Figure 25 shows tensile residual stress results for load step 2.
Figure 25: LS 2 Stress x-direction
Figure 26 show strain results for load step 2, after part is brought back to its original
length. It can be seen that the part deformed plastically and compressive strains are left
after unloading.
Figure 26: LS 2 Total Mechanical Strain x-direction
25
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 1 for values of the
other load steps.
4.2 Crack Growth Results with mesh edge crack (FRANC3D)
Open source computer code FRANC3D was used to proof crack propagation without
success. The plastic deformation left in the part after the compressive displacements
were retrieved did not allow the program to calculate a positive stress intensity factor,
then resulting in no crack propagation.
The mesh below was created by FRANC3D when a 0.2 inch depth edge crack was added
to the notch geometry.
Figure 27: FRANC3D Edge Crack mesh
Figure 28: ANSYS residual stress distribution at crack faces
26
Figure 28 above shows tensile residual stress around the crack. It can be seen that the
area around the crack is green which means that tensile residual stress dominate the
region. But right next to the crack as can be seen on both pictures a small compression
region exists enclosing the crack. The stress at the crack tip is a very high tensile stress
but on the track faces of the crack there is a small amount of compressive stress. One
reason for this behavior might be that contact elements where required at the faces of the
crack to prevent penetration of the crack faces when the load is applied.
Figure 29 below, shows the negative stress intensity factor calculated by
FRANC3D when displacement where exported from ANSYS into the open source code.
Since crack faces were closing and crossing as seen in the picture above due to tensile
compressive stress right next to the crack. Displacements around the crack tip were
negative. These negative displacements were exported into FRANC3D and no crack
propagation was seen because they produce negative stress intensity factor.
Figure 29: FRANC3D Negative Stress Intensity Factor Results
In reality this crack should grow under this type of loading and it has been demonstrated
in previous studies. In order to evaluate crack growth an alternative approached was
used.
Besides KI there are two other fracture mechanics parameters that can be used to
calculate crack propagation driving forces. The path independent integral JI and the
crack tip opening displacement (CTOD) δI. These two parameters are used primarily in
27
the elastic-plastic regime, whereareas the stress intensity parameter KI, is used primarily
in the linear elastic regime.
The path independent J-integral proposed by Rice and used by FRAN3D is a
method for characterizing the stress-strain filed at the tip of a crack by an integral path
taken at some distance from the crack to be analyzed elastically, and then substitute for
the inelastic region close to the crack tip. This might be another reason why the stress
intensity factor was not calculated correctly by 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 conclusion is that it is replacing stresses at the crack
tip with compressive stresses enclosing the crack, Figure 28.
“This technique estimate the fracture characteristics of materials exhibiting elasticplastic behavior and is a means of extending fracture-mechanics concepts from linear
elastic behavior to elastic-plastic behavior” J.M. Barson, Fracture and Fatigue Control
in Structures. Since the J integral method was developed for plastic regions under tensile
stresses and not for tensile residual stresses. It was concluded that it was not the best
method to be use in the study of crack propagation for residual stresses.
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
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 expressed in textbook as:
5#6 7
2 = 1.120
#
89
Eq. 10
If the stress intensity factor for an edge crack is used,
= 1.12√ !
28
Eq. 11
;6
2 = <
#
89
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 is already in terms KIc, Appendix 1, we will simplify the methodology and
calculate δI first and then convert to KI as shown in Equation 12. Finally we compared
the calculated K1 against material KIc to determine if there was indeed crack growth.
Several relations between the various fracture parameters were found in J.M.
Barson, Fracture and Fatigue Control in Structures [4] among them is the one shown
below, between K and δ. It can be obtained by rewritten Equation 12:
= => 2
Eq. 13
4.3 Crack Growth Results without mesh edge crack (CTOD)
To calculate crack propagation the ANSYS stresses without the insertion of the edge
crack were used. Since that set of results do not create compressive residual stresses
around the crack tip when the part is unloaded and pull back to its original length or
second half of the strain cycle, where the tensile residual stresses are assumed to
propagate the crack.
In order to check for crack growth, is necessary to calculate the stress intensity
factor range ΔK.
∆ = @7 − @ATo simplify the calculations 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
for this study. 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 ∆K.
29
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, mean stresses across
the thickness of the bar were used. Refer to Appendix 3 for details on how these nominal
stresses were calculated.
Table 2: 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 calculate KI with INCO718 ‘da/dN vs. ΔK’ table from Appendix 1 it can be
seen that K1 is still below the BCℎ = 10.59G √ value for this material and no
crack propagation is expected.
Table 3: 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, KI value is over the ΔKth of the material and the crack is expected to grow.
30
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 will take
place. Refer to Figure 30.
Figure 30: Crack arrest
31
5. Conclusion
Incomplete.
In this experiment it can clearly be seen that the material cyclically hardens, since that’s
the behavior it was assumed in ANSYS. Fatigue crack propagated from a notch even
when the external loading is completely compression. The residual stress caused by a
compression cycle loading can be estimated accurately from FEA. Stress intensity factor
is difficult to calculate from residual stress distribution.
32
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, 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.
33
7. Appendices
7.1 Appendix 1: Material Properties
34
35
36
7.2 Appendix2: 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
,
37
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
38
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
39
/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
40
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.
An ANSYS post process stress linearization was done to determine membrane stress,
bending stress and total stress (membrane + bending) along the defined node path.
Only membrane portion of the stress was assumed to act as the nominal stress since is
the one opening the crack.
41
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