The Effect of Stress Ratio on Fatigue Cracks Growth Rate
in Aluminum Alloy
EMAD TOMA KARASH
Northern Technical University/ Technical Institute Mosul, IRAQ
Abstract: - The financial and societal consequences of the sudden failure of various mechanical structures and components
can occasionally be severe. As a result, there has been extensive research done on fatigue failure in both the scientific and
industrial fields. The impact of stress ratio adjustments on the rates of fatigue fracture propagation for an aluminum alloy has
been investigated in the current study. This aluminum alloy was extracted from a Russian MI25 helicopter's primary blade.
The stress intensity factor was calculated and employed by creating an initial crack in the test specimens after a specimen of
MI25 blade was cut into testing specimens with standard dimensions.
A special program was written specifically for this purpose to draw a graph of the stress intensity factor range (k) with
value of the propagation of fatigue cracks (da / dN) at each stress ratio (R); Propagation of fatigue cracks has been studied.
Finally, the relationship between the two different stress ratios has been discovered. Between the two, a comparison has been
established. The findings indicated that when the stress ratio rises, so too does the rate at which fatigue cracks form.
Conversely, with a negative stress ratio, the negative increase leads to a reduction in the rate at which fatigue cracks grow.
The data have been examined, and during the second stage of crack propagation, where (R = 0.0), pertinent equations that are
compatible with the Paris equation have been discovered.
Keywords: - Applied Mechanics, Stress Ration, Fatigue, Fatigue Cracks, Alloys
Received: December 26, 2021. Revised: November 8, 2022. Accepted: December 5, 2022. Published: December 31, 2022.
1. Introduction
One of the principal reasons of failure, particularly in
rotating axles, springs, connecting rods, aircraft wings,
etc., is vehicle fatigue cracking. This phenomenon is
known as fatigue and it occurs because the engineering
material is exposed to reciprocating or periodic stresses
that are much lower than the material's tensile resistance.
The majority of failures of engineering parts occur at
stress levels well below the maximum design stress limit,
at points where it has a high stress concentration, or when
its operational life is increased. When the critical stress
density coefficient grows and failure occurs, it includes
the growth of tiny cracks in the form of microscopic
cracks. Failure fatigue happens when a material's elastic
limits are exceeded and the stress changes are below the
material's tensile strength. The increase in slit length per
cycle (da/dN), which is proportional to the amplitude of
the stress intensity component (k) during the cycle, is
used to define the fatigue growth rate. Given the
significance of this subject, extensive study has been done
on it recently and is still being done, but despite this, we
continue to hear of tragic incidents, the primary cause of
which is weariness [1], [2]. The life of the
investigated feather is estimated with a restricted time of
1,000 hours in the case of flight or seven years in the
case of non-flight in order to work for a limited time. A
tire life for a certain number of cycles (N), where the
compounds composed of aluminum, copper, etc. are
depicted in this type of diagram for nonferrous metal
models. They are made for a specific lifespan, but
ferrous metal models (Ferro) show the limits of
fatigue under lower stress because iron can withstand
an infinite number of cycles before failing. There is
enough proof that many engineering constructions
have pre-existing flaws that develop during
manufacturing or piping, thus conjecture starts with the
initiation and growth of fatigue from these flaws in the
metals used to make automobile and aircraft hulls [3],
[4],[5]. Due to its significance in defining the lifetime
necessary for these devices, the growth of the crack in a
location where the value of (∆k) is low is extremely
necessary, making this region the most controlling for
determining the amount of fatigue life that concerns
aircraft [6],[7]. Numerous studies have focused on
the development of fatigue cracking in aluminum alloys
used to make aviation vehicles at various frequencies
and
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temperatures [8], [9], [10]. The effect of the stress strength
coefficient ( ) of an aluminum ingot that is
identical to the type (AA- AA- 6063T832) of aluminum
ingot used in the production of (MI25) helicopter blades on
the development of fatigue fractures. [11],[12],[13]
investigated the rate of FCG on intermetallic TiAl at low
and high temperatures. Three different Rs were used to set
the increased temperature at 750 °C. It was discovered that
lowering fatigue thresholds and raising FCG rates are both
caused by increasing the stress ratio (R ratio).
Additionally, a study on the impact of the tiny FCG strain
ratio in Ti-6Al-4V was carried out by [14],[15],[16],[17].
As a result, when the strain ratio is plotted with the slit
size function, the effect of R is observed on the tiny slit
growth rate, but it has essentially little effect when
plotted with the K function. [18] investigated the
behavior of crack growth by testing typical CT
specimens made of 2024-T4 aluminum alloy under
loading in the first mode with various R ratios and
crack tips, and discovered that the stress intensity factor
was responsible for the increase in FCG rates with
rising R ratios. Research in this area is ongoing and
not just experiments. There have been several
successful simulation research series. In a recent analysis
using overloading and simulated plane spectrum loading
(mini-Falstaff+), [19] FCG was examined. [20] They
studied the beginning and progression of stress cracks
in 16MnR steel. The study of the mechanical
characteristics of aluminum alloys has been the subject of
extensive research [21],[22],[23],[24],[25],[26],[27].
Other models that have been created thus far include
[28], [29], [30], [31], [32]. However, the complete
models created do not account for the influence of all
parameters simultaneously and are not relevant to all
materials [28], [29], [30], [31], [32].
In order to understand the fatigue crack growth at various
stress ratios and compare them, we will analyze the
fatigue crack growth of the aluminum alloy (AA- AA-
6063T832) used to make helicopter wings in this article.
2. Materials and working methods
All studies used metal obtained from a helicopter type
(MI25) blade, and Figure (1) shows parts in the main
tested blade (Main Blade). Figure (3) shows the model's
dimensions, which were (5 * 20 * 150) mm, and shows
that the model was cut into the necessary number of
pieces. On the width of the model, a scratch (slit) was
formed with a depth of (0.5) mm and a thickness of (0.2)
mm. With the help of these scratches, a sufficient stress
intensity was created in the area of highest curvature to
mimic real-world conditions. According to its chemical
make-up and mechanical characteristics, the metal is an
aluminum alloy type (AA- 6063T832). The alloy was
examined, and Table provides information on the
chemical makeup of the alloy (1). A reputable source was
used to determine the alloy's mechanical characteristics
[5]. Table provides information on this alloy's mechanical
characteristics (2). Moreover, Figure (2) shows models of
the main blades of destroyed helicopters, showing the
locations and shape of the crack,its growth and collapse of
the main blade. Table (1) shows the chemical composition
of the aluminum alloy used in the test. Lastly, Table (2)
shows the mechanical properties of (6063-T832)
aluminum alloy and that of the tested aluminum alloy.
Figure (1) shows sections of the main blade of the
helicopter on which the test was conducted
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Figure (2) shows models of the main blades of destroyed
helicopters, showing the locations and shape of the crack,
its growth and collapse of the main blade
Figure (3) shows the dimensions of the model used in the
test
Table (1) shows the chemical composition of the aluminum alloy used in the test
Table (2) shows the mechanical properties of (6063-T832) aluminum alloy and that of the tested aluminum alloy
Aluminum Alloy
Density, Kg/m3
Tensile Yield
Strength, MPa
Ultimate Tensile
Strength MPa
% EL
Modulus of
Elasticity, GPa
Shear modulus,
Gpa
Hardness (HBW),
typical value
Passion’s ratio,
Nominal value
AA- 6063T832 [33]
2780
≥ 160
≥ 215
≥ 10
70
28
70
0.33
Actual value
AA- 6063T832
2750
150
200
9
69
28
68
0.31
Elements
Materials
Zn
%
Si
%
Fe
%
Cu
%
Mn
%
Mg
%
Cr
%
Ti
%
Al
%
Nominal value
AA- 6063T832[33]
0.21
-
0.25
0.30
-
0.6
0. 1
-
0.35
≤ 0.1
≤ 0.1
0.35
-
0.6
≤
0.05
0.13
-
0.25
Rem.
Actual value
AA- 6063T832
0.22
0.31
0.32
0.2
0.09
0.53
0.09
0.15
98.09
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The reverse bending machine was used in the test. A
mobile optical microscope with a numbered lens (ocular
lens) was used in this research, through which the
formation and growth of the fissure was observed.
The stress intensity coefficient (ΔK) used to define the
stress level was:
)b/a(Faπ)σσ(=KΔminmax
(1)
432 )b/a(0.14)b/a(08.13-)b/a(33.7)b/a(40.1122.1)b/a(F
The calibration of the above equation was given by
scientists (Bueckher1960), (Gross1965), (emery,1969),
(Benther,1972) and accuracy (0.2%) when the value of
(a/b) is equal to or less than (0.6), meaning that (a/b0.6
[3] . All tests were carried out in the laboratory
atmosphere and at room temperatures around (25°C) and
the tests were conducted at a frequency of (5 Hz). As for
the slit length measurements, they were calculated using a
mobile light microscope with a magnification power
(25X-50X) and using a numbered eyepiece installed on
the light microscope used in the test along the fissure
growth area, where at each stress ratio of the ratios used,
which was (
) The model is polished and its surface
smoothed well before use, and then the polished surface is
divided by clear signs using a vernier caliper on the model
with a distance of (0.5) mm between one scratch and
another scratch by recording the number of cycles
Necessary for the access of the slot to each visa in order
to find the relationship between the length of the slot and
the number of courses [9], [10].
3. Results and discussion
3-1. First-mode fissure growth
The optical method was used to discover and monitor the
growth of the fissure, as a mobile optical microscope was
used. The crack growth was monitored on the side where
the crack begins to grow, and in calculating the value of
the range of stress coefficient for the first phase, equation
No. (1) given by Tada (1973) was used. Before
conducting the test, the model was polished in the area
where the crack is expected to develop, with a good
polishing and smoothing of the surface, in order to
facilitate the reading process. The surface in which the
crack is expected to appear is indicated by means of a
vernier with a distance of (0.5 mm) between one visa and
another visa. The number of cycles is recorded when the
length of the incision reaches to a specified distance by
means of a counter on the device and is recorded each
time the number of revolutions (Nc) with the length of the
slit (a) and until the complete collapse of the model.
Figure (4) shows a comparison between the slit length
with the number of cycles for the different stress ratios (
) when the stress intensity coefficient
range value was (∆K=5.25 MPa). From the figure, we
notice that the rate of collapse of the model increases with
the increase in the stress ratio, that is, the number of
cycles required for the collapse of the model decreases
with the increase in the stress ratio at the same stress
intensity coefficient (∆K).
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Figure (4) Comparison of the fatigue crack growth curves
at different stress ratios
The above results were analyzed by entering a special
program. In order to obtain values for the fatigue cracking
growth rate ( ) with the range values of the stress
strength modulus (K) and calibration equation (1) was
used in the program to obtain the values of the strain
intensity modulus range (K), while the (Finite
difference) method was used to obtain the values of
Fatigue crack growth rate ( ). For the first point,
the (Forward difference) method was used, while for the
other points, except for the last point, the (Central
difference) method was used, and the last point was
obtained by the (Backward difference) method.
Through this program, values for the range of stress
intensity coefficient (K) were obtained with values for
the fatigue crack growth rate (). Figure (5)
illustrates a comparison between the test results. We note
from the figure that the positive increase in the stress ratio
(R) leads to an increase in the fatigue crack growth rate,
while at the stress ratio (R = -1) the value of the fatigue
crack growth rate has decreased. Therefore, it can be said
that the increase Negative stress ratio leads to a decrease
in fatigue crack growth rate.
Figure (5) A comparison between the fatigue crack
growth rate data for different stress ratios for an aluminum
alloy type (AA- 6063T832)
The effect of the stress ratio on the growth of the fatigue
crack can be seen from Figure (6). When the values of the
stress ratios are positive, the positive increase in the stress
ratios (R) leads to an increase in the growth rate of the
fatigue cracking as well. The rate of fatigue crack growth
at ( ) stress increases from (0.0 - 2.2 * 10-4 mm/
cycle). At the strain ratio ( ), the fatigue crack
growth rate increases from (0.0 - 2.4 * 10-4 mm/cycle),
while at the strain ratio ( ) it increases from (0.0 -
4.3 * 10-4 mm/cycle)) As for the stress ratio (R = 0.5), the
fatigue crack growth rate increases from (0.0 - 1.3 * 10-3
mm/cycle), and all models were tested at the same stress
intensity factor (∆k = 5.25 Mpa). The positive increase in
the stress ratio from ( ) the fatigue crack
growth equations also increase from (0.0 - 2.2 * 10-4 mm/
cycle) at the stress ratio (R = 0.0) to (0.0 - 1.3 * 10-3).
mm/cycle) at a stress ratio ( ), but at a stress ratio
( ), the crack growth velocity increased from (0.0 -
1.8 * 10-4 mm/cycle), and therefore it can be said that the
negative increase in the stress ratio Lead to a decrease in
the growth rate of fatigue cracking.
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Figure (6) Comparison of fatigue crack growth
rate with crack length for different stress ratios
3-2. Effect of the stress ratio on the crack
growth rate in the linear region
Early on, the researchers (Lindley & Richards) believed
that the crack growth rate in the linear region is relatively
sensitive to the effect of the stress ratio (R). In this study,
the (Paris) equation was deduced in the linear part of the
fatigue crack growth rate for many models at different
stress rates [10].
All tests were for prototype slit models. Figure (7) shows
the real origin of these equations, which were extracted
using a special program called (Curve Export). Figure (8)
shows the practical test results and their comparison with
(Paris) equation in the second formation stage, the linear
stage. Figure (9) shows the results of the tests The
different process of stress ratios ( )
and its comparison with (Paris) equation and through the
above figures it is clear that an increase in the growth
rates of fatigue cracking with the increase of the stress
ratio and many previous works confirm an increase in the
growth rates of fatigue cracking Increasing the positive
stress ratio as the work of researchers (Moddox),
(Pearson), (James), (Frost), (March & Pook) and
(Nishioka)
and others. The increase in crack growth rates is due to an
increase in (Kmax) as it approaches the value of (Kic) for
the metal, as well as an increase in the stress rate as well
as a decrease in the crack closing effect when the stress
ratio increases.
Figure (7) shows how to derive the fatigue crack growth
equation for the first phase when (R = 0.0)
Figure (8) comparing the fatigue crack growth rate with
(Paris) equation at stress ratio (R = 0.0).
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Figure (9) is the growth rate of fatigue cracking in
models with an initial crack at different stress ratios in the
phase-first tests
The Paris equation for the linear region of fatigue crack
growth for the first phase was given as follows:
m
KC
dN
da
(2)
Figure (6) shows a drawing of this equation for a model
with an initial slit. In this section, the curve of the fatigue
crack growth rate ( ) against the full range of the
stress intensity modulus (K) will be taken, and it was
considered that the lowest rate of fatigue cracking growth
rate at the beginning of the fatigue cracking (Threshold) is
(1*10-10 m/cycle).
The practical values were taken from the value of (Kth)
and the values of the fatigue crack growth rate in the first
and second stages to the last point in the linear region of
the fatigue crack growth and at a stress ratio (R = 0.0),
which is in similar behavior at different stress ratios
because the effect of stress ratios (R) In the linear phase
of slightly fatigue crack growth. The following equation
has been considered:
DKC
dN
da m
(3)
This equation was taken to represent the fatigue crack
growth behavior for the first and second stages, and
(Curve Expert) program was used to derive the values of
the constants. The following equation was obtained:
902.29 10*714.1K10*144.0
dN
da
(4)
This equation describes the growth of fatigue cracking in
the first and second stages, i.e., the phase of onset and
progression of fatigue and the linear phase of fatigue
crack growth, which is consistent with (Paris) equation.
Figure (10) shows a comparison between the fatigue
crack growth equation (4) for the first phase with the
practical results and the (Paris) equation in the linear
region and the (Paris) equation in the linear region are
The figure shows an acceptable match between the
practical and inferred values.
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Figure (10) Comparison of the equation for the crack
growth rate of the whole for the first phase with the
practical results and an equation (Paris)
4. Conclusions
1. To describe the crack growth in the first and
second phases of the first phase testing, a new
equation for the fatigue crack growth was
developed.
2. The results of the fatigue crack growth tests
show that the number of cycles required to
collapse the model at positive values of the stress
ratios (R) reduces as the positive stress ratio (R)
increases at the same values of the used stress
intensity coefficient (Kth). The more negative
the stress ratios (R) are, the more cycles must
pass until the model collapses.
3. According to the findings of the analysis of
fatigue crack growth, the fatigue crack growth
rate (Kth) reduces at stress ratios (R = -1),
whereas it increases with a positive increase in
the stress ratios (R)..
Acknowledgment
This work was supported through Northern Technical
University, Iraq. by the Research Program of the
Engineering Science, (No. 001233- 2020).
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6060/.
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.28
Emad Toma Karash
E-ISSN: 2224-3429
244
Volume 17, 2022
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the Creation of a Scientific Article
(Ghostwriting Policy)
The author(s) contributed in the present
research, at all stages from the formulation
of the problem to the final findings
and solution.
Sources of Funding for Research Presented
in a Scientific Article or Scientific Article
Itself
This work was supported through Northern
Technical University, Iraq. by the Research
Program of the Engineering Science, (No.
001233- 2020).
Conflict of Interest
The author(s) declare no potential conflicts of
interest concerning the research, authorship, or
publication of this article.
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International, CC BY 4.0)
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