16

Chapter 2

SURVEY OF LITERATURE In order to assess the trend and level of research work done till date, in the area of

titled work, an exhaustive literature has been reviewed. A gist of some of the most relevant

research work is presented in this chapter under various classified headings.

2.1 Fatigue

Fatigue refers to the structural damage, which results from repeated or otherwise

varying stresses that are well below the static yield strength of the material. Fatigue

involves the initiation and growth of a crack, or growth from a pre-existing defect, which

progresses until a critical size is reached. As per ASTM E-1823 [57], the term fatigue is

defined as, “The process of progressive localized permanent structural change occurring in

a material subjected to conditions that produce fluctuating stresses and strains at some point

or points and that may culminate in cracks or complete fracture after a sufficient number of

fluctuations.”

2.1.1 Historical perspective

The work on fatigue goes back to 1837, when Wilhelm Albert established a

correlation between applied loads and durability in the context of chains used in mines [58].

Around 1839, Jean-Victor Poncelet, who designed cast iron axles for mill wheels, described

metals being “tired” in his lectures at the military school at Metz [58]. These lectures were

never published, but have been quoted in the works of many eminent authors [59]. Poncelet

also made the first official reference to the term “fatigue” [1] in his French book titled,

“Introduction a la Mécanique Industrielle,” (Introduction to Industrial Mechanics). The

phenomenon of fatigue was widely observed in 17th century during the failure of railway

structures that claimed many lives, as reported by Gray [60] and Smith [61]. Rankine [62]

17

investigated the fatigue failure of railway axles and suggested that the axles should be

forged with a hub of enlarged diameter and large radii, so as to reduce the cutting of grain-

flows during machining, which would improve the fatigue life of axles.

The first organized research on fatigue, concerning failure of railway axles, was

carried out by German engineer August Wöhler in 1860s. Besides developing the S-N

curve, Wöhler also fabricated apparatus for repeated loading of axles [63]. An extensively

cited piece of his work published in German, is titled, “Versuche zur Ermittlung der auf die

Eisenbahnwagen Achsen einwirkenden Kräfte und die Widerstandsfähigkeit der Wagen-

Achsen,” (Tests to determine the forces acting on railway car axles and the resistance of

car-axles). Wöhler was also the first to arrive at the modern terms of “fatigue life” and

“scatter” in the context of design for fatigue life [1].

Subsequent to the initial work cited above, scientists and engineers in the

industrialized countries started working on morphological and material aspects of fatigue.

Batson and Bradley [64] undertook an organized research on fatigue of laminated springs,

which were subjected to different heat treatments as well as various surface finising

operations. Thum and coworkers did initial research on the role of stress concentration on

fatigue by investigating fatigue behaviour of shafts having rapidly varying cross-sections

[65-67]. Almen [68] did pioneerg work on the effect of surface modification on endurance

limit. Based on the data obtained by Wöhler, Basquin [69] represented the finite life region

of the fatigue curve in the form of stress vs number of load cycles on a double-log scale.

Later on, laboratory fatigue tests were performed by many researchers, notably Haigh

[70], Gough and Pollard [71] and Gough [72]. A well-known damage accumulation

hypothesis, commonly known as Palmgren-Miner rule, was developed by Palmgren [73]

and Miner [74]. Investigations by Tavernelli and Coffin [75], Coffin [76] and Manson [77]

led to the development of well-known Coffin-Mansion law of damage accumulation in low-

cycle fatigue, which was later extended by Manson himself [78], as well as subsequent

researchers, notably Brechet et al. [79]. The low cycle fatigue analysis has been

18

successfully applied to structures subjected to small number of fatigue cycles during their

lifetime, viz. electrical equipment subjected to thermal stresses. Around the same time,

Neumann [80-82] developed the slip-plane model of crack initiation, while Schijve and

Brock [83] investigated the factors affecting crack propagation in aircraft structures, which

were subjected to variable amplitude or gust-spectrum loading. Subsequently, Schijve

developed detailed model of fatigue crack growth and published a series of lectures on the

topic [84-87]. The developments during twentieth century are comprehensively covered in

a review article by Schijve [88].

Subsequent researchers further extended the concepts developed by pioneers, as

mentioned above. The prominent names among them include Brechet et al. [79] for

extension of Coffin-Manson law, Toth and Krasowsky [89] for damage process analysis on

the basis of Paris law and Coffin-Manson equation, Todinov [90] for work on Palmgren-

Miner rule, and Shodja and Kamalzare [91] for their work on Griffith-Inglis model of crack

growth.

The organized study of fatigue, beginning with the historical work of Wöhler on

railway axles, has come a long way to the safe life fatigue resistant design of modern

aircrafts. However, the problem of fatigue still continues to cause catastrophic failures and

claim lives wherever material properties or service loads deviate from the design values.

2.1.2 Fatigue of metallic materials

Microscopic investigations have revealed that the nucleation of fatigue cracks occurs

at a very early stage of fatigue life. Cheng and Laird [92] demonstrated that the microcracks

commence in the form of slip bands within a grain. The cyclic slip occurs as a result of

cyclic shear stress. On a micro-scale, this shear stress is not distributed uniformly, but

varies from grain to grain, depending on its size and shape, crystallographic orientation and

associated elastic anisotropy of the material. The slip occurring on surface grains leads to

formation of slip steps, as revealed in the study of Villechaise et al. [93] on 316L austenitic

stainless steel. In the presence of oxygen, the freshly exposed surface of the material in slip

19

steps gets oxidized, which prevents slip reversal. The slip reversal in this case occurs in

some adjacent slip plane, thereby leading to formation of extrusions or intrusions on the

surface of material, as observed by Gross [94] and Terentev [95] and depicted

schematically in Figure 2.1.

The fatigue life is generally divided into three stages: Stage-I or crack initiation stage,

Stage-II or crack growth stage and Stage-III, which involves rapid crack growth till final

fracture [Refer Figure 2.2]. Stage-I usually comprises maximum portion of fatigue life of

the component.

Figure 2.1: Formation of intrusion and extrusion marks on the material surface.

During crack initiation, fatigue is essentially a surface phenomenon. In this period,

the growth of crack is slow and erratic owing to its interaction with the grain boundaries.

However, a more regular growth of crack occurs afterwards. The periods of crack initiation

and growth are distinct from each other in many ways, viz. while the surface conditions do

affect the initiation period, they do not have much influence on the crack growth period.

Stress concentration factor is an important parameter during the crack initiation phase,

while the stress intensity factor is important for predictions of crack growth [96].

20

Figure 2.2: Phases of fatigue life

The difference between microcracks observed during the initiation period and

macrocracks developed during the propagation period is that while the former grow along

the plane of maximum shear, the growth of macrocracks occurs along the plane of

maximum tensile stress. However, although the crack propagates along the plane of

maximum tensile stress, the crack tip exhibits shear as well as tensile opening

displacements [97]. The cyclic opening of the crack tip leads to the formation of fatigue

striations [Refer Figure 2.3]. According to Davidson and Lankford [98], the cyclic crack

growth may equal the striation spacing at higher crack growth rates, while it may be much

less for slower growth. Grinberg [99] observed that regardless of the material, the minimum

striation spacing observed remains around 0.1µm.

During Stage-III of fatigue crack growth, the striation-forming mode is gradually

replaced by the static fracture modes, viz. rupture or cleavage. Since stages I and II

comprise service life of a component subjected to fatigue, most of the research work has

been dedicated to the study of these modes. For a steadily propagating stage II fatigue

crack, the Paris relation [100,101] of crack growth rate can be applied:

mKCdNda )(∆=

Equation 2.1

where a is the crack length, N the cycle number, ∆K is the stress intensity factor, C and m

are constants. A value of da/dN =1 represents formation of striation in each loading cycle.

Berkovitz [102] extended this hypothesis in the opposite sense to develop a technique for

estimating the loads from the observations of fractured surfaces. The technique was further

developed by Ruckert et al. [103].

21

Figure 2.3: Mechanism of fatigue crack propagation

Elber [104,105] introduced the concept of crack closure, according to which, when

the fatigue crack is loaded in tension, there will be Poisson contraction in the transverse

(through-thickness) direction. During unloading of the crack tip, this sucked-in material will

lead to establishment of contact between the two crack surfaces before complete unloading,

thereby resulting in compressive stress in the vicinity of crack tip, over the region of

premature contact. These compressive stresses have to be first overcome for the crack to

propagate further in the next loading cycle. The effective range of the tensile cycle is thus

the range between the opening load and the maximum load of the cycle. Different

researchers have further developed the concept of crack closure. Notable among them are

the experimental and numerical studies carried out by Blom and Holm [106] and the finite

element study by Chermahini et al. [107].

22

2.2 Design of fatigue test experiments

Different investigators have proposed different designs of fatigue tests in order to

meet the intended engineering and research objectives. While the engineering objectives

involve determination of fatigue properties of materials or structural elements, the research

objectives are usually concerned with determination of influence of one or more variable

factors, viz. composition, processing parameters, environment, load spectra etc., on the

resulting fatigue properties.

2.2.1 Fatigue tests

The design of fatigue tests depends on the following parameters:

• Type of test: Stress based or strain based.

• Type of loading: Tensile, torsional or bending.

• Type of fatigue test piece: Actual components or standard specimens.

• Nature of load cycle: Constant amplitude (CA), variable amplitude (VA) or

service simulation spectrum.

• Environmental conditions: Temperature, humidity, presence of corrosives etc.

The choice of various parameters for a given application is based on the underlying

objective behind the fatigue tests. The stress-based tests are useful for situations involving

high cycle fatigue, where loading is confined to elastic stresses and associated strains.

However, plastic strains may exist in the vicinity of stress raisers, such as notches or holes,

even when bulk of the component is subjected to elastic loading. As proposed by Lee et al.

[108], such situations are better analyzed through strain-based tests, which take into account

local strains as the governing fatigue parameters.

In their study on constant and variable amplitude fatigue loading of standard

aluminum alloy specimens, Schijve et al. [109] state that the CA stress tests are generally

used for obtaining material-specific information pertaining to fatigue performance, while

23

VA loading is helpful in providing information on crack closure. However, in case of

critical assemblies, viz. aircraft structures, testing of actual components subjected to actual

or simulated service load spectra is indispensable for ascertaining their fatigue performance.

The nominal stress (S) for a given fatigue loading (tensile or shear) is given by:

APS = (For axial load P)

JTrS = (For torsional load T)

IMcS = (For bending moment M)

Equation 2.2

where A is the area of cross section, J is the polar moment of inertia, r is the distance from

the centre of cross section to the point of interest, I is the moment of area about neutral axis,

and c is the distance of the fibre from the neutral axis.

Since the loading in case of rotating bending fatigue tests on round specimens closely

resembles actual components such as axles, these tests have been extensively applied since

the early phase of fatigue research [110].

2.2.2 Test specimens and loading conditions

Each one of the basic types of fatigue tests requires a test specimen having specific

geometry, as shown in Figure 2.4. The test specimens can either be notched or unnotched.

Notched specimens are supposed to be representative of the real life structures and contain

stress raisers in the form of holes or notches with well-defined geometry. On the other

hand, as opined by Schijve [96], the fatigue data for unnotched specimens is considered to

be representative of the material’s fatigue properties, but does not provide information

regarding notch sensitivity of the material.

24

As the fatigue behaviour of unnotched specimens depends largely on the surface

conditions, the tests on these specimens in various states can be used for making a

comparative study of the effect of different surface treatments, viz. nitriding, carburizing

etc., on the resulting fatigue performance.

Figure 2.4: Various types of fatigue test specimens

As stated earlier in section 2.2.1, the rotating bending fatigue tests are most

commonly employed for assessing fatigue properties of the materials. These tests can be

further classified on the basis of the type of load applied, viz. single point, two-point or

four-point. The first one employs a cylindrical, toroidal or tapered cantilever specimen,

while the remaining two tests employ cylindrical or toroidal beam type specimens of the

form prescribed in IS:5075 [111], as shown in Figure 2.5.

In case of materials exhibiting high hysteresis losses during straining, excessive

heating may result in considerable rise in temperature of the test piece, thereby leading to

inconsistent fatigue life data. Accordingly, toroidal shaped specimens are used for fatigue

testing, so that relatively smaller volume of the specimen material is subjected to maximum

stresses. ISO1143 standard [112] recommends that wherever possible, the shape which

subjects the largest volume of material to maximum stress should be preferred, while a

liberal allowance is kept on the portion of specimen meant for holding purpose.

25

Figure 2.5: (a), (b) and (c): Cylindrical fatigue test specimens under single, 2-point and 4-point bending respectively; (d), (e) and (f): Toroidal fatigue test specimens in single, 2-point and 4-point bending respectively.

In case of single point loading, as shown in Figure 2.5 (a) and (d), the cantilever beam

specimen is also subjected to shear stress, while in case of two or four point loading, the

neck portion of the specimen does not have any shear stress and is subjected to pure and

uniform bending moment. When four point bending is applied to cylindrical specimens,

Figure 2.5 (c), it subjects the largest volume of material to maximum stress, which is

desirable according to ISO1143 standard [112].

2.3 Collection, analysis and presentation of fatigue data

The stress cycle during fatigue tests is characterized by stress amplitude (Sa), mean

stress component (Sm) and wave shape of the stress cycle. The stress ratio (R=Smin/Smax) is

1− for completely reversed rotating bending fatigue test.

The wave-shapes resembling actual loading conditions will provide more realistic

estimates of the fatigue performance of components during a real-life fatigue loading, but

the laboratory fatigue tests, aimed at determining material’s fatigue properties, are generally

performed under simple load cycles, viz. the wave-shape of stress cycle for a rotating

bending test is a sine wave.

26

In order to generate realistic data for fatigue analysis, the fatigue tests are carried out

on several specimens at different levels of maximum alternating stresses. The fatigue data is

plotted on a semi-log or log-log scale in the form of S-N curve (i.e. stress vs. number of

cycles to failure curve), also known as Wöhler curve. The portion of curve with negative

slope constitutes the finite life region and represents fatigue strength of the material for a

given number of stress cycles, while the horizontal portion represents infinite life region.

The stress level corresponding to horizontal portion (i.e. infinite life) is known as fatigue

limit of the material. The changeover point, or the knee, signifies the phenomenon where

crack nucleation is essentially arrested by some microstructural features, as suggested by

McGreevy et al. [113] and Murakami and Nagata [114]. A typical log-log S-N curve is

mathematically expressed by the following relation proposed by Basquin [69]:

bffa NSS )2('= Equation 2.3

where Sa is the stress amplitude, Nf is the number of cycles to failure, b is the fatigue

strength exponent and fS ' is the fatigue strength coefficient.

2.3.1 Analysis and determination of finite life

Owing to inherent microstructural inhomogeneity in the material properties,

differences in surface finish and test conditions, the fatigue data exhibits scatter, as

mentioned by Troschenko [115] and Rodopoulos and Chilveros [116]. The variance of log

life generally increases with decreasing stress levels, particularly for unnotched specimens.

This makes it necessary to take into account the statistical nature of the fatigue data, as

suggested originally by Weibull [117,118].

The objective of this section is to provide statistical background for constructing a

median S-N curve and associated lower and upper bound curves, that characterize the

minimum and the maximum fatigue lives (i.e. stress cycles) at a given level of fatigue

strength (i.e. stress amplitude), so that the majority of fatigue data falls within these lower

and upper bound curves. In this regard, various techniques have been prescribed in the

27

literature [119-122]. Out of these, the techniques provided in JSME [119] and ASTM [121]

standards are widely used by researchers for obtaining S-N curves and making fatigue life

predictions. These techniques are briefly described in the following sub-sections.

2.3.1.1 JSME S 002 standard

The median S-N test method, as described in JSME standard, involves testing of 2

fatigue specimens at 4 stress levels for finite life region, while 6 specimens are used for

determining the fatigue limit through staircase method. The recommended test sequence is

shown in Figure 2.6, where the numbers next to the data points represent the order for

conducting the fatigue tests. The fatigue limit is determined by taking the average of the

stress levels employed during the staircase test. The details of staircase method are

described in section 2.3.2.4.

Figure 2.6: S-N testing according to JSME standard

2.3.1.2 ASTM E739 standard

The guidelines for generation of statistical S-N curve are provided in ASTM E 739

standard [121]. In order to assess the variability and statistical distribution of fatigue life,

28

the standard recommends replication of fatigue tests, i.e. testing more than one specimen at

each stress level. The percentage replication (PR) depends on the size of test batch (k) and

number of stress levels (L) used during fatigue testing, and can be determined from the

following relationship:

PR = 100(1-L/k) Equation 2.4

The guidelines recommend employing 6-12 specimens for preliminary, research and

development tests and 12-24 specimens for design allowable and reliability tests. The

recommendations regarding percentage replication for various tests are as follows:

• 17 – 33% for preliminary and exploratory work

• 33 – 50% for research and development tests

• 50 – 75% for design allowable data tests

• 75 – 88% for reliability data tests

Once the requisite fatigue data has been collected, statistical analysis prescribed in

ASTM E739 standard can be applied. The analysis assumes that the fatigue life at a given

amplitude of stress follows log-normal distribution and the variance of log-life is constant

within the test range. The resulting regression model of fatigue is expressed as:

Y = A + BX + ε,

where, Y = log Nf , X is the maximum alternating stress and ε is a random variable

representing error.

The linear regression model of fatigue is given by:

XBAY ˆˆ += Equation 2.5

where the over-bar () symbol signifies average values and the caret (^) symbol represents

estimated values, obtained by minimizing the sum of squares of deviations of the observed

values of Y from its predicted values, i.e.,

29

2

11

22 )ˆˆ()ˆ( i

k

ii

k

iii XBAYYY −−=−=∆ ∑∑

==

Equating the partial derivatives of ∆2 with respect to the estimators A and B to zero,

we get:

0)1)(ˆˆ(2ˆ1

2

=−−−=∂∆∂ ∑

=

k

iii XBAY

A and

0))(ˆˆ(2ˆ1

2

=−−−=∂∆∂ ∑

=

k

iiii XXBAY

B

Therefore, the least squares method gives the estimators A and B as:

∑

∑

=

=

−

−−= k

ii

k

iii

XX

YYXXB

1

2

1

)(

))((ˆ

XBYA ˆˆ −= Equation 2.6

The variance, which is assumed to be constant within the range of Xi, can be

estimated as follows:

∑=

−−

=k

ii YY

k 1

22 )ˆ()2(

1σ

Equation 2.7

Taking logarithm of both sides of the Wöhler relation (Equation 2.3), we get:

)log(1)'log(1)2log( aff Sb

Sb

N +−=

Equation 2.8

Comparing this equation with the regression Equation 2.5, we get:

X=log(Sa), Y=log(2Nf), )'log(1ˆfS

bA −= and

bB 1ˆ =

Equation 2.9

In other words, the fatigue strength exponent (b) is inverse of the linear regression

constant B , while the fatigue strength coefficient )ˆ(' Abf eS −= . The coefficient of variation

of fS ' , defined as the ratio of standard deviation to mean, is calculated as:

30

1)ˆ('

22

−= σbS eC

f

The double-sided confidence band for the entire median S-N curve can be computed

from the following equation:

2/1

1

2

2

)(

)(1ˆ2ˆˆ

−

−+±+

∑=

k

ii

ip

XX

XXk

FXBA σ

Equation 2.10

where Fp is taken from statistical tables abstracted in ASTM E 739 [121]. The plots of these

confidence bands take hyperbolic form, as shown in Figure 2.7. The confidence band will

become narrowest when ∑=

−k

ii XX

1

2)( is maximum. This is possible if the fatigue tests are

conducted at the farthest possible points, i.e., at the extreme stress levels of S-N curve,

provided the linearity of S-N curve is already ascertained.

Figure 2.7: S-N data along with ASTM regression fit and confidence bands.

2.3.2 Analysis and determination of infinite life

Just as observed for the finite life regime, the endurance limit too exhibits scatter. The

objective of this section is to provide statistical analysis for estimating the endurance limit

at a specific high cycle fatigue life. Depending upon the availability of time, number of test

31

specimens and the accuracy required, any one of the methods, as described in the following

sub-sections can be employed for estimating the endurance limit.

2.3.2.1 Estimation of endurance limit from material properties

This method provides a technique, developed by various researchers, notably

Bannantine et al. [123], Dowling [124] and Stephens et al. [125], for estimating the

endurance limit on the basis of limited information, viz. ultimate strength, hardness and

available S-N data for various materials. The strength value thus obtained is modified to

account for other factors, such as loading, surface finish, size and desired reliability on S-N

curve.

2.3.2.2 Step test

As discussed in the preceding sections, due to statistical nature of fatigue, it usually

becomes necessary to test many specimens at stress levels in the vicinity of fatigue limit.

However, it is possible to make an approximate determination of fatigue limit by

employing very small number of specimens, if one adopts the step-test method. The method

is also useful for estimating the fatigue limit in situations where only one specimen of its

kind is available, as suggested by Bellows et al. [126]. In this method, a specimen is tested

for a large number of cycles (~107) at a stress amplitude Sa0, slightly lower than the

expected value of endurance limit. If the specimen does not undergo fatigue failure after

completion of stipulated number of cycles, Sa0 may be considered to be slightly below the

endurance limit. The test is then repeated for the stipulated number of cycles, by

successively increasing the stress amplitude by small amount ∆Sa each time, till failure

occurs. The endurance limit (Sf) is then considered to be between (Sa - ∆Sa) and Sa; where

Sa is the final stress amplitude at which fatigue failure occurred. The method is graphically

represented in Figure 2.8.

32

Figure 2.8: Step test for approximating the fatigue limit using a single specimen.

2.3.2.3 Probit method

The Probit method is described in ASTM Special Technical Publication 91-A [127].

In this method, a number of specimens are tested at various stress levels in the vicinity of

endurance limit. For the specimens, which do not fail, the test is stopped at some high value

of N (say, 107 cycles), which is to be associated with the endurance limit. In this way, the

percentage of failed specimens is recorded at various stress amplitudes and thus, provides

the estimated probability of failure at that particular stress amplitude. These probabilities

are then plotted on a probability graph, as shown in Figure 2.9, so as to obtain the

distribution function of the endurance limit associated with N = 107 cycles. The mean value

of endurance limit is taken against a 50% probability of failure, while the standard deviation

is calculated from the slope of the probability graph.

33

Figure 2.9: Probit method for determining the endurance limit.

2.3.2.4 Staircase method

The staircase method provides an estimate of the endurance limit by taking into

account its statistical nature. Also known as the up-and-down method, it has been adopted

and prescribed by many standards, viz. JSME [119] and British Standard [128], for

assessing the statistical properties of endurance limit.

In this test method, first of all, the mean endurance limit is estimated. Following this,

a specimen is tested at stress amplitude Sa slightly (~5%) higher than the expected

endurance limit. If the specimen fails before completion of stipulated number of cycles, say

2×106, then the next specimen is tested at lower stress amplitude. However, in the event of

survival of a specimen, the test is suspended after completion of stipulated number of

cycles and the next specimen is tested at higher amplitude of maximum alternating stress.

Thus, the stress amplitude of each successive test is based on the outcome of its previous

test. The graphical representation of the method is given in Figure 2.10.

34

Figure 2.10: Staircase method for determining endurance limit.

The statistical parameters of the test results can be obtained either by Dixon-Mood

[129] or Zhang-Kececioglu [130] techniques. The Dixon-Mood method is based on the

maximum likelihood estimation and assumes that the normal distribution curve best fit the

data pertaining to endurance limit. The Zhang-Kececioglu [130] analysis is based on either

the maximum likelihood estimation or the suspended-items analysis method. The method

can also be applied to staircase tests with variable stress-steps.

The Dixon-Mood method on the other hand, requires uniform stress-steps in the

staircase. This method is widely used as it is easy to apply and provides good estimate of

the endurance limit [131], with slight bias towards conservative side. It assumes that the

data pertaining to endurance limit has normal distribution. This method provides formulae

for estimating the mean (µs) and standard deviation (σs) of the endurance limit (Se). The

mean and standard deviation are estimated by using the data of less frequent event out of

the two possible events, i.e. survivals or failures. The individual stress amplitudes (Si),

which are spaced uniformly at an interval of d, are numbered as i, where i=0 is used to

35

denote the lowest stress amplitude, (So). The stress increment (d) should be in the range of

SS d σσ 2

2≤≤ . The estimate of mean endurance limit is given by:

++=

∑ 21

,iDM

DMOS n

AdSµ , if survival is the less frequent event

Equation 2.11

−+=

∑ 21

,iDM

DMOS n

AdSµ , if failure is the less frequent event

Equation 2.12

Here, ( )( )∑= iDMDM niA , , while iDMn , is the count of less frequent event

corresponding to the ith stress amplitude.

The standard deviation is estimated from either of the following two expressions:

( )

+

−=

∑∑ 029.062.1 2

,

2,

iDM

DMiDMDMS

n

AnBdσ

Equation 2.13

if ( ) 3.02,

2, ≥−

∑∑

iDM

DMiDMDM

n

AnB

or

dS 53.0=σ Equation 2.14

if ( ) 3.02,

2, <−

∑∑

iDM

DMiDMDM

n

AnB

where ( )( )∑= iDMDM niB ,2

Once the values of mean and standard deviation are known, the one-sided confidence

levels (lower bounds) for the endurance limit can be determined by the expression:

SsCRe KS σµ −=,, Equation 2.15

where K is the factor for one-side tolerance limit for normal distribution, which can be

obtained from the works of Lieberman [132] or Link [133]. The lower bound endurance

36

limit ( CReS ,, ) signifies that with a confidence level of C%, R% of the tested fatigue limit

would be expected to exceed the stress amplitude CReS ,, .

2.4 Effect of surface modification on fatigue

A number of researchers have investigated the influence of various types of surface

modification processes on the fatigue behaviour of substrate material. Bruzzone et al. [134],

in their paper concerning state-of-the-art in surface engineering, have discussed in detail the

properties of functional surfaces, their applications and the technologies available to

engineer these surfaces. They have observed an expansion in the importance as well as

application of engineered surfaces to new domains. These developments are attributed to

the greater knowledge of the phenomena occurring on the engineered surfaces as well as the

availability of affordable technologies for mass production of engineered surfaces.

As described earlier in section 1.2, the surface modification processes can be broadly

classified into two categories. The first category involves modification of the material

properties upto a certain depth, as in case of heat treatment processes such as case

carburizing or nitriding etc. The second category involves application of various types of

coatings on the substrate surface. Following sub-sections present the literature pertaining to

the effect of both types of surface modification processes on the resulting properties,

particularly fatigue.

2.4.1 Effect of heat treatment

A variety of heat treatment processes have been developed for various grades of

steels. Since the response of steels to heat treatment is governed largely by the percentage

of carbon present in it, therefore the selection of process, viz. physical or chemical heat

treatment, depends primarily on content of carbon in steel. For most of the low carbon alloy

steels that find widespread application in machine tool and automotive industries, the heat

37

treatment processes like nitriding, carburizing or a combination thereof are widely

employed for achieving the desired properties of hardness and strength.

Although nitriding is possible on many grades of steels, but very high hardness is

obtained in certain special grades of steels that contain nitride forming elements, viz.

aluminium, chromium, molybdenum or vanadium, which help in maintaining a high

concentration of nitrides near the surface and prevent their diffusion further deep into the

core. Thus, while nitriding of case achieves higher hardness in comparison to case

carburization, the depth of nitrided case is less. Nitriding is also known to cause substantial

improvement in fatigue performance, as reported by many researchers, viz. Czerwiec and

coworkers [135,136], Suh et al. [137] and Ashrafizadeh [138]. However, owing to the

associated shape distortions, the use of nitriding is generally restricted to tribological

applications [139].

Case carburization is the most common surface modification technique and finds

widespread application as it leads to tremendous improvement in mechanical properties of

hardness, strength and endurance [140,141].

While high hardness of martensitic case is important from wear resistance standpoint,

the enhancement in fatigue properties is attributed mainly to the presence of residual

compressive stresses within the carburized case. In his historical work on case carburized

steels, Ebert [142] correlated these residual stresses, induced during carburization process,

to the time lag in the occurrence of martensitic transformations in the layers at various

depths, which have different carbon concentration. The work by Krauss [143] shows that

the factors responsible for this favourable shift in properties root from the martensitic

transformations taking place during the course of heat treatment process. Bag et al. [144], in

their study on dual phase steels, found that the maximum fracture toughness is obtained for

steels having martensitic volume fraction in the range of 0.7 to 0.8.

Yang et al. [145] in their study involving case carburized 20CrMnTi specimens, have

reported the shift in residual stress field resulting from case carburization process. The

38

study revealed that in the non-carburized quenched specimen, the residual stress is tensile at

the surface and compressive in the center, however, in carburized specimen, it is observed

to be compressive in the surface zone. The authors also proposed a simulation model, which

accounts for percentage of carbon and calculates the residual stress resulting from

martensitic transformation.

Since the magnitude of residual stresses in steel components is dependent on the

substrate microstructure, viz. percentage of retained austenite, the same can be modified

through some suitable processing, viz. cold treatment, as employed by Surberg et al. [146]

and Stratton and Graf [147], so as to reduce the amount of retained austenite. The resulting

change in the stress field has been worked out by Bensely and coworkers in different

studies [148,149]. They investigated the distribution of residual stress in case carburized

EN353 steel subjected to cryogenic treatment prior to tempering and observed an increase

in the compressive residual stress for the specimens subjected to cryogenic treatment.

In his work on bending fatigue strength of carburizing SS2506 steel, Preston [150]

reported that the fatigue performance is largely dependent on the presence of residual

compressive stresses in the surface layer. It was reported that as the case depth is increased,

the magnitude of compressive stresses in the outermost layers gets reduced, thereby

increasing the chances of surface crack initiation. Genel and Demirkol [151] further

extended the work to investigate the effect of case depth on fatigue behaviour of case

carburized SAE8620 steel specimens. They reported an optimal relative case depth, which

results in best performance under fatigue loading.

In their study on fatigue of carburized steels, Farfan et al. [152] reported a general

trend of improvement in endurance limit with increasing depth of case. In another similar

work, Asi et al. [153] divided their standard fatigue test specimens into three batches and

subjected them to a different type of heat treatment by varying the parameters such as

carburizing temperature, carburizing time and holding time at 850°C. The study reveals that

the higher depth of case leads to greater depth of surface oxidation, which in-turn enhances

39

the non-martensitic transformations, thereby adversely affecting the distribution of residual

stresses and leading to a lower fatigue life. In another study [154], they reported

degradation in fatigue performance with increasing carburization temperature, owing to

unfavorable distribution of residual stresses within the carburized case. Woods et al. [155]

suggest that a favourable modification in stress field can be utilized for enhancing the

performance of machine elements, such as gears, which are subjected to bending fatigue

during their normal operation.

For certain grades of alloy steels, which contain higher percentages of oxide forming

elements, such as manganese and chromium, internal oxidation can occur, which is often

accompanied by the formation of high temperature transformation products (HTTP) viz.

pearlite, upper and lower bainite up to a certain depth below the surface [156]. For steels

containing silicon, manganese and chromium as major alloying elements, oxides of

chromium are generally formed in the outermost layers in the form of precipitates within

the grains. This is followed by oxides of manganese, which form at some depth below the

surface and may appear either in the form of precipitates or as grain boundary oxides. The

oxides of silicon penetrate to maximum depths, generally in the form of grain boundary

oxides [157,158]. The oxidation potentials of major alloying elements, as presented by

Kozlovskii et al. [159], are shown in Figure 2.11.

As a consequence of internal oxidation, non-martensitic microstructures are likely to

form within the alloy-depleted regions in the vicinity of oxides, thereby resulting in loss of

hardness, even though the microhardness tests performed on the outer surface may not

reveal such phenomenon. Further, the high temperature transformation products are usually

the first to undergo transformation and their coefficient of thermal expansion is also less

than that of the surrounding material. As a result, these products are likely to come under

compression when the component cools down to room temperature. Owing to differential

contraction during cooling, the residual compressive stresses in the carburized layer get

compromised. This reduction in residual compressive stress in turn can lead to deterioration

of fatigue performance, as reported by Arkhipov et al. [160] in their work involving fatigue

40

testing of case carburized gears. In his comparative study on fatigue performance of

20MnCr5 and 15CrNi5 steels, Brugger [161] observed that the fatigue limits of these steels

were substantially different at 680 and 780 MPa respectively in the absence of HTTPs, but

dropped in the range of 520 to 540 MPa for both the steels when similar amounts of HTTPs

were present in the surface layer. Naito et al. [162] in their work on SCM415 steel, also

reported a negative influence of internal oxidation and HTTP on the resulting fatigue

performance. Besides, they also observed the phenomenon of double-knee formation in the

fatigue curve in the presence of HTTP.

Figure 2.11: Oxidation potentials of alloying elements and iron in steel, heated in endothermic gaseous environment.

2.4.2 Effect of surface coating

Coating materials of various types and deposited by different means onto various

substrates have been extensively used in the cutting-tool, aerospace and automotive

applications for enhancing specific properties of a given component, such as resistance to

corrosion, heat and wear. Corrosion resistance has traditionally been achieved by means of

41

nickel or chrome plating. However, owing to the environmental concerns associated with

this process, application of physical vapour deposition is on the rise.

The resistance to heat is usually achieved through the application of thermal barrier

coatings (TBCs) [163]. These coatings are usually applied by means of EB-PVD process,

as explained in section 1.4.4.4. The state-of-the-art in thermal barrier coatings is described

in the review by Schulz et al. [164].

Over the course of last couple of decades, the application of hard coatings for

enhancing tribological properties has become very popular in the cutting tool industry. In

this context, TiN is one of the most well-established coating, which gained widespread

popularity for its excellent tribological properties [165].

Besides TiN, coatings based on amorphous carbon [166] or diamond-like carbon

(DLC) [167] have also been developed and successfully applied. A number of researchers

have evaluated the properties as well as performance of these coatings. Gupta and Bhushan

[168] and Bouzakis et al. [169] evaluated mechanical properties, Wellman et al. [170]

investigated hardness and Sundararajan and Bhushan [171], Precht et al. [172] and Shum et

al. [173] evaluated tribological behaviour of these coatings.

Since fatigue is an important performance parameter and is closely linked to the

prevalent state of material at the surface of a component, a number of researchers have

studied the effect of coatings on the resulting fatigue properties. The influence of some

conventionally applied as well as recently developed coatings on the substrate properties,

with particular reference to fatigue, is being discussed in the following sub-sections.

2.4.2.1 Electroplating

Chrome plating is the most commonly eletrodeposited coating, known for its

corrosion and wear resistance, along with low coefficient of friction. However,

environmental and mechanical problems concerning chrome plating are resulting in

research aiming to identify alternative baths [16,17], remedial treatments [174,175] as well

42

as economically viable alternatives [176]. An important characteristic of chrome plating is

the presence of high tensile residual stresses, which are detrimental for fatigue performance

of the chromium plated components [177-179].

2.4.2.2 Thermal spray coatings

With a view to offset the detrimental effects of chromium plating on the environment

as well as component’s fatigue properties, Nascimento and coworkers [180,181] have

proposed the application of thermal spray coatings as a viable alternative [182]. Thermal

spray coatings are deposited by processes such as detonation gun, High Velocity Oxy-Fuel

(HVOF) arc etc. Coatings applied by means of HVOF technique possess high mass density,

good adhesion, good mechanical strength and improved fatigue properties of the coated

components.

An early study correlating the residual stress with fatigue life of thermal sprayed steel

and aluminium substrates was conducted by McGrann et al. [183]. In their extensively cited

study, the authors have established a positive correlation between the observed fatigue life

and the magnitude of residual compressive stresses present within the coating. The authors

have observed a ten-fold improvement in fatigue life as a result of WC-Co thermal spray

coating as compared to conventional or accelerated chrome plating. The authors have

attributed this enhancement to the presence of residual compressive stresses induced during

thermal spray process.

In another study involving WC-Co thermal spray coating, Ahmed and Hadfield [184]

conducted rolling contact fatigue tests and identified four modes of failure, viz. abrasive,

delamination, bulk deformation and spalling. The delamination failure is stated to be the

most catastrophic mode of failure. The study also confirmed the presence of protective

residual compressive stresses within the coating.

43

2.4.2.3 Chemical vapour deposition

The CVD process overcomes many of the shortcomings of thermal spray process, as a

result of which, the CVD coated components exhibit enhanced tribological performance

[185,186]. However, CVD process can also cause surface embrittlement, thereby leading to

detrimental effect on mechanical properties [187].

The effect of CVD coatings on fatigue performance is also detrimental. During

rotating bending tests of PE-CVD coated specimens, Baragetti and Tordini [188] have

reported a small decrease in fatigue limit. Schlund et al. [189] have also observed a decline

in fatigue properties of CVD coated specimens, which could be overcome by applying a

combination of PVD and CVD coatings [190].

2.4.2.4 Physical vapour deposition

The physical vapour deposition (PVD) process offers numerous advantages over the

older processes such as electroplating or thermal spray coating. It has led to the

development of a number of commercially and technologically successful tribological

coatings, such as TiN, TiCN, TiAlN, TiZrN, DLC, TiC etc., which have found widespread

application in the cutting tool industry [191].

The influence of PVD coatings on fatigue behaviour of the substrate is also governed

by factors similar to those discussed for thermal spray coatings. In their investigation on the

high temperature fatigue performance of TiAlN coatings applied to 1Cr-1Mo-0.25V steels,

Suh et al. [192] conclude that the fatigue performance is associated mainly to the presence

of residual compressive stresses in the coating. Ferreira et al. [193], in their study involving

fatigue behaviour of 42CrMo4 steel coated with W, WN, WTi and WTiN deposits,

observed a general trend showing improvement in fatigue performance at lower levels of

maximum alternating stresses. The gain in fatigue performance is attributed to the presence

of residual compressive stresses within the coating. At higher stresses, cracks get initiated at

places where flaws exist within the coating and then find their way into the bulk substrate.

Similar observations have been reported by Su and co-workers [10,11] for TiN PVD

44

coatings, where fatigue properties were found to be adversely affected below 5×105 cycles,

but observed to improve beyond this point.

Puchi-Cabrera et al. [194] conducted rotating bending fatigue tests on the substrate

coated with a PVD film of TiCN and observed 140 – 180% gain in fatigue life, which is

attributed to compressive residual stresses and good adhesion of the deposit to the substrate.

Similar findings have been reported by Berríos-Ortiz et al. [195,196] and Puchi-Cabrera et

al. [197] in their works on fatigue behaviour of AISI 316L stainless steel coated with: (i)

different TiNx [195,197] and (ii) ZrNy [196] understoichiometric films deposited by

unbalanced close field magnetron sputtering (UCFMS) process and (iii) TiN films

deposited by plasma assisted PVD (PAPVD) tehnique. The studies reported an

improvement in fatigue life of the coated specimens in comparison to their uncoated

counterparts.

Baragetti et al. [198] investigated the influence of CrN PVD coating on fatigue

behaviour of 2205 duplex stainless steel substrate, subjected to four-point bending. The

study also investigated the effect of substrate preparation processes (rolling, polishing and

shot peening) on surface residual stress as well as fatigue properties. The investigation

revealed an enhancement in the fatigue life of the coated specimens, which is attributed to

the presence of compressive surface residual stress field induced by PVD coatings. In

another work [199] regarding the influence of CrN PVD coatings on fatigue behaviour of

steel substrate, a 15% improvement in endurance limit was reported by the same authors.

They were also successful in developing FEM model for the problem. However, in their

study on Plasma Enhanced Chemical Vapour Deposition (PECVD) coatings, Baragetti and

Tordini [188] reported a slight reduction in endurance limit.

Multilayer coatings of various compositions have also been developed by various

researchers. The chief advantage of multilayer coatings over single layer coatings lies in

that they can be better customized to accomplish the desired properties. In their work

involving multilayer metallic coatings on copper substrates, Stoudt et al. [200] observed

45

improvement in fatigue properties due to retardation of fatigue crack initiation. They

hypothesized that the fatigue properties can be enhanced by depositing an ideal surface

layer, which possesses the desired properties of hardness, toughness, cyclic work

hardenability, residual compressive stresses and good adherence. However, higher cost

associated with multilayer coatings is a major prohibitive factor in their application.

A recent addition to the class of PVD coatings is the WC/C coating [50], described

earlier in section 1.4.4.4. The microstructural and mechanical properties of these coatings

have been reported by Carvalho and DeHosson [201] and Park et al. [202]. It is reported

that the amorphous phase dominates in the layers of carbon as well as WC. By varying the

concentration of WC particles and correlating the resulting electrical conductivity with

mechanical properties, Park et al. [202] showed that the physical contact between the

carbide particles is an important structural factor that governs the physical properties of the

nanocomposite films. The hardness as well as residual compressive stresses within the film

were found to increase only at higher concentrations of WC phase, which was attributed to

physical contact between the WC particles present within the matrix of amorphous carbon.

The applicability of WC/C coating on machine elements appears to be justified for a

number of supporting reasons. First, the range of processing temperatures for this coating

renders it suitable for heat-sensitive machine elements, viz. heat treated steels. Secondly,

the coating possesses compressive stresses as well as good tribological properties that are

often required in tandem for the machine elements such as gears, which are subjected to

wear as well as fatigue during their service life. The favourable effect of WC/C coating on

the performance of gear-pairs working under loss of lubrication condition has been

demonstrated by Murakawa et al. [203]. The tribological performance of WC/C coatings

depends on the hardness of WC particles and the lubricating properties of amorphous

carbon matrix. The effect of this coating on the fatigue properties has been reported in a

study by Baragetti et al. [204] concerning 2011-T6 aluminium alloy coated with PVD

WC/C, PA-CVD DLC and PE-CVD SiOx. The authors conducted rotating bending fatigue

tests and reported an improvement in fatigue limit of WC/C coated specimens. The

46

observed improvement was attributed to the presence of residual compressive stresses in the

coating, apart from its excellent adhesion to the substrate surface.

In a recent study concerning application of WC/C coating on gears made of case

carburized steel, Fujii et al. [205] made comparative tests on three types of pinions, labeled

as: NT (case carburized, uncoated), WT (case carburized and WC/C coated) and ST (case

carburized and WC/C coated, with an interlayer of CrN). The authors reported better

fatigue performance for WT pinion than the NT pinion, under a maximum hertzian

pressure, pmax = 1700 MPa. The WT pinion exhibited poor performance in comparison to

NT pinion at pmax > 1900 MPa, which was attributed to peeling of the coated layer. The

peeling was found to occur at relatively lower loads for the ST pinion. The authors

concluded that the occurrence of peeling in the coated layer is the dominant factor affecting

surface durability of WC/C coated gears.

2.5 Residual stress and its estimation

It is revealed from the literature that hardness and adhesion of the coated deposit

alone play a dominant role in determining tribological properties of the coated components,

such as cutting tools and gears, but the influence of coating on fatigue properties of the

coated substrate is a more complex function of various factors, viz. (a) properties of

coating, (b) surface and bulk properties of the substrate, (c) influence of coating process on

the substrate’s properties, (d) distribution of residual stresses in the coating and substrate

and (e) adhesion of coating to the substrate. The last two parameters are actually inter-

related in the sense that the adhesion of deposit is affected by the presence of residual

stresses in the coating/substrate system.

In their article concerning interactions between residual stress and crack growth,

Fitzpatrick and Edwards [206] point that though residual stress plays a major role in

determining the fatigue behaviour of a component, but the growing crack can in-turn affect

the residual stress field at the crack tip. They observed that in order to know the influence

of stress field on crack propagation, the exact effect at the crack tip should be estimated

47

through some suitable model. However, the approach has limited validity due to the

changes imminent in the residual stress field once the crack begins to grow.

The inter-relation between residual stress and integrity of PVD coatings has been

described in two different studies by Teixeira [207,208]. The author has analyzed the stress-

induced cracking in thin PVD coatings, originating from the presence of residual stresses

and has also presented a numerical model to estimate the distribution of residual stress

within a layered metal-ceramic composite coating.

Thus, it is evident that for a given composition and properties of material in the

surface layer, magnitude of residual stresses plays an important role in determining the

mechanical properties, especially fatigue of a heat treated or coated material. In view of

this, it is extremely important to measure residual stresses within the component for

predicting its fatigue behaviour.

As stress is an extrinsic property of the material, it cannot be measured directly, but is

rather estimated indirectly by measuring some other physical quantity, viz. velocity of

sound in the material, which in turn is affected by the presence of stress. The residual

stresses can be divided into two categories – micro-stresses and macro-stresses. Micro-

stresses are those stresses, whose range is on the order of the microstructure, viz. crystallites

or grains. Macro-stresses, on the other hand, cast their influence over ranges, which extend

far beyond the dimensions of grains.

2.5.1 Methods for residual stress measurement

A number of destructive and non-destructive methods have been developed for

estimation of residual stresses [209]. While mechanical test methods, such as hole drilling

[210] or curvature measurements on thin strips [211] can estimate the magnitude of macro-

stresses, the diffraction techniques, viz. X-ray and neutron diffraction, can provide an

estimate of both the micro as well as macro-stresses. Besides, diffraction methods being

non-destructive in nature, can be applied to any number of components, whether freshly

48

manufactured, in service, or failed. In case of large components that cannot be

accommodated in X-ray diffraction equipment, or where stresses are to be determined in

layers that are well below the surface where X-rays cannot penetrate, it becomes necessary

to either cut a section of material from an appropriately selected location, or remove surface

layers – either mechanically or chemically. In such situations, the method obviously

becomes destructive. However, owing to the importance of non-destructive testing, portable

X-ray diffraction equipment for residual stress measurement has also been developed and

successfully employed in industry to address the problems associated with accommodation

of large components inside the equipment [212]. Besides XRD, some other non-destructive

methods for residual stress measurement have also been developed, viz. ultrasonic

techniques [213-215], micro-magnetic measurements [216,217]) etc.

2.5.2 X-ray diffraction residual stress measurement

Among the various non-destructive methods of residual stress estimation, X-ray

diffraction is a well-developed and widely applied technique [218-223]. X-ray diffraction

method for estimation of residual stress involves the measurement of strain within the

crystal lattice, which is used to estimate the residual stress causing the observed strain. The

presence of micro-strains, such as those resulting from plastic deformation processes like

cold-working etc., affects the lattice spacing within individual grains in a random manner,

thereby leading to broadening of diffraction lines.

On the other hand, presence of macro-strains systematically alters the spacing

between a particular set of lattice planes as a function of their orientation with respect to the

direction of macro-stress [Refer Figure 2.12]. This in-turn causes a shift in the

corresponding diffraction peak. The strain along the normal to reflecting planes varies with

the angle ψ between the normals to lattice planes and the surface normal. As a result, there

are two slightly different diffraction angles, 2θ1 and 2θ2 corresponding to the lattice planes

having normals N1 and N2 on the lower and higher sides of the detector respectively. The

resulting Debye ring from the specimen no longer remains perfectly circular, but becomes

49

slightly distorted, so that the radii S1 and S2 are no longer equal. The measurement of radii

S1 and S2 at various values of tilt angle (ψ) provides an estimate of the strain and hence

corresponding stress.

Figure 2.12: Diffraction from a set of hkl planes for residual stress measurement.

The estimation of strains and corresponding stresses requires application of classical

continuum mechanics. The technique is based on the Reuss or Voigt linear elastic distortion

models for the crystal lattice [224,225]. The Reuss model assumes stress to be

homogeneous throughout the polycrystal, as a consequence of which, the strain distribution

becomes heterogeneous. Voigt model, on the other hand, is based on homogeneous strain

distribution and associated heterogeneous stress distribution.

2.5.2.1 Relationship between stress and strain

If a cylindrical bar of initial diameter D, is loaded axially to induce uniaxial stress σy

acting along the y-direction, the corresponding strain, εy, in the y-direction is given by:

50

Ey

y

σε =

Equation 2.16

where E is the Young’s modulus. If the Poisson’s ratio of material of the bar is ν, the

corresponding strains in the x and z directions, for the material assumed to be isotropic,

would be given by:

yzx νεεε −== Equation 2.17

The measurement of εy by X-ray diffraction is not practically feasible since it would

require recording the diffraction pattern of planes which are normal to the axis of the bar.

On the other hand, measurements of strain are made normal to the cylindrical surface, i.e. in

the x or z-direction, (or any combination thereof if the direction is not important and the

material is isotropic) and are given by:

0

0

dddn

zx−

== εε

Equation 2.18

where dn is the spacing of hkl planes, whose normal is perpendicular to the surface of the

bar, and d0 is the value of spacing for undeformed lattice.

The longitudinal strain εy and hence the corresponding stress σy can thus be deduced

from the treatment given in Appendix A.

For such a measurement, only a set of selected grains, whose hkl planes are parallel

(within the range of angle of divergence of the incident beam) to the specimen surface, will

contribute to the diffraction cone. For these hkl planes, which are oriented parallel to the

specimen surface, the inter-planar spacing is reduced as a result of Poission’s ratio.

However, for the grains whose hkl planes are oriented normal to the specimen axis, this

spacing between the planes would be extended. In other words, the spacing dhkl is a function

of crystal orientation with respect to the direction of macro-strain [Refer Figure 2.12].

One basic problem with steels is that there does not exist any standard value of lattice

constant d0 against which the lattice strain can be measured. One alternative for determining

51

the value of d0 is to make measurements on a small, stress-free portion cut from the

specimen, but it will make the method destructive. Another method is to use calibration

against known values of stress (hence strain) on identical specimens, but this may not

always be practical. A practical workaround for this problem is that uniaxial internal strain

can be estimated by comparing the diffraction patterns obtained from the same location on

specimen at different tilt angles (ψ) [Refer Figure 2.12], typically ψ =0° and at some higher

value, viz. ψ =45°. The relationship between stress and strain on a free surface is developed

in Appendix A.

In practice, it is usually desired to determine stress σΦ [Refer Figure 2.13] in a given

direction of interest (on the specimen surface), where the Greek alphabet Φ represents the

azimuth angle between σΦ and the direction of principal stress σ11. This can be achieved by

recording two diffraction patterns: one for hkl planes whose normal is parallel to the surface

normal and the other for hkl planes whose normal is at an angle ψ with the surface normal.

The corresponding measured strains are termed as ε33 and εψ. The X-ray diffraction

technique thus involves working with three coordinate systems: crystallite coordinate

system (Ki), equipment or laboratory coordinate system (Li) and specimen coordinate

system (Si).

The diffraction peaks are obtained according to the orientation of lattice planes hkl

within the volume of material contributing towards diffraction pattern. The coordinates Ki

for these planes of the crystallite should satisfy Bragg’s relation with the equipment or

laboratory coordinates Li in order to obtain a diffraction peak. Further, since the Bragg’s

relationship between these coordinate systems is only a function of hkl and wavelength of

X-rays, so it is completely independent of the specimen coordinate system, Si.

Thus, stress and strain referred to in the preceding discussion refer to the crystallite or

laboratory frame of reference. But in actual practice, it is required to determine their values

in relation to the specimen’s frame of reference, viz. with respect to direction of rolling etc.

52

This can be achieved through the application of suitable transformation matrices [Refer

Appendix B], which lead to the following result:

( ) { }

{ } { } ψφσφσνσσσν

σνψφσσφσφσνε φψ

2sincoscos1

1sin2sinsincos1'

2313332211

332

12332

222

1133

++

+++−

+++−+

+=

EE

EE

Equation 2.19

where the prime superscript denotes strain in the specimen coordinate system.

Figure 2.13: Relationship between equipment coordinate system, principal stresses and stress to be measured, σΦ.

2.5.2.2 Determination of biaxial residual stresses from diffraction peaks

Following the treatment given in Appendix C, Equation 2.19 provides the relationship

between biaxial residual stress at the free surface, in terms of lattice-spacings corresponding

to various ψ-tilts. The Equation C.5 derived in Appendix C is reproduced below:

( ) φφψ σνψ E

dd

+=

∂

∂ 1sin

12

0

Equation 2.20

53

If the values of d0, E and ν are known, σΦ can be obtained by measuring dΦψ at two

values of ψ. However, as stated earlier, determination of d0 is a problem in itself, especially

for materials containing alloying elements, such as alloy steel. So, for practical purposes, d0

is taken as equal to dΦ0 which is nothing but d corresponding to ψ = 0.

The factor E/(1+ν) in the above equation is known as effective elastic parameter or

Eeff and it relates the macroscopic stress to the strain measured along a particular

crystallographic direction in a polycrystalline material. Where required, the value of this

parameter can be determined with the help of a calibrating specimen according to the

procedure outlined in ASTM E 1426 [226].

If multiple values of dΦψ are recorded and plotted against sin2ψ, a more accurate value

of σΦ can be obtained. If an isotropic material is subjected to plane stress condition, the (dΦψ

vs sin2ψ) graph would be in the form of a straight line, as shown in Figure 2.14a. A positive

slope of the graph indicates tensile stresses, while the negative slope implies compressive

nature of residual stresses.

It should, however, be noted that since X-rays can penetrate to a considerable depth

below the free surface, triaxial stress states might also be encountered in certain cases,

under the influence of which, different lattice spacings would be observed for positive and

negative values of ψ, as explained by Skrzypek and Baczmanksi [227]. Such a case is

known as ψ-splitting, as shown in Figure 2.14b. The plot shown in Figure 2.14c indicates

the presence of crystallographic texture in the material, as described by Cullity [220] and

Prevey [221].

Since the Reuss and Voigt models are based on the assumption of isotropic character

of material, the presence of texturing, which leads to anisotropy, would adversely affect the

accuracy with which residual stresses can be determined [225]. Presence of strong texturing

may also lead to low intensities of diffracted beams and thus, further increase the

uncertainties in estimated values of residual stress. Such problems can, however, be

overcome through some suitable treatments. For example, Kapoor et al. [228] have

54

developed an approach for determining the stress tensor in textured material, while Smith et

al. [229] have presented an analysis for obtaining complete multi-axial residual stress

distribution from limited measurements.

Figure 2.14: Various forms of d vs sin2ψ plots; (a): Under uniaxial or biaxial stresses in

isotropic material, (b):Under triaxial stresses and (c): For textured material.

When residual stress measurements are to be made on thin films, specialized

techniques have to be employed, viz. grazing-incidence diffraction technique [227,230],

micro-area diffraction [231] or Debye ring analysis [232].

2.5.2.3 Diffraction peak location

According to Lonsdale [233], the reliability of residual stress estimation is closely

related to the precision and accuracy with which peak-positions of the X-ray diffraction

pattern are determined; along with the applicability of the linear model that relates peak

shift to the specimen orientation. A number of factors influence the accuracy of this

measurement, as discussed briefly in the following text.

The first consideration is related to the source of X-rays, as the transition metal targets

(viz. copper) used in the X-ray tubes produce white radiation, along with the characteristic

radiation of Kα1, Kα2 and Kβ. While the wavelength of Kβ is sufficiently different to

facilitate its filtering-out (by using β-filters), the separation between Kα1 and Kα2 is too

small and hence, the Kα doublet contributes towards formation of diffraction peaks, which

in case of broad peaks would remain unresolved. Further, peak broadening is a common

55

phenomenon among the most hardened and fine-grained steels, as reported by Pineault et

al. [234]. Thus, the estimation of residual stresses in heat treated steels would involve

locating the peak-positions for unresolved pairs of diffraction peaks.

The modern day, position-sensitive detectors make is possible to record diffracted

intensity at various locations spanning the diffraction peak. While sharp diffraction peaks

can be located rather easily using low precision data, broad peaks, on the other hand,

mandate the collection of large number of X-rays at each location, thereby increasing the

time required for stress measurement. Subsequent to recording, the data pertaining to

intensities at various locations is corrected for polarization, absorption and sloping variation

in background radiation by assuming a linear variation beneath the peak. Once the raw data

has been refined, the next step involves determination of peak position. An old method,

proposed by Koistinen [235] and recommended in SAE literature, involves calculating the

vertex of parabola defined by three points confined to the top 15% of the peak.

With the refinement in resolution of position detectors to the tune of 0.01°, it is now

possible to record a reasonably large number of points within close proximity of the peak,

to which a parabola can be fitted by least squares regression. This method provides better

location of the peak position. The fitted parabola provides the best estimate of peak position

in case of symmetrical peaks. However, its accuracy gets compromised in certain situations,

viz. when beam is defocused as a result of ψ – tilting, or where the doublet is not fully

combined or in case of asymmetrical peaks [236]. Prevey [237] recommends that for such

situations, fitting Pearson VII functions separately to the Kα1 and Kα2 peaks provides the

best estimate of peak location. Such corrections can be conveniently applied with the help

of computer programs or spreadsheet software, such as Microsoft® Excel [238].

For the intensity of data recorded at high angular resolution, the peak position can

also be calculated as the centroid of the area above the background. This area-integration

method is independent of the peak shape, but is extremely sensitive to the precision with

which the tails of the diffraction peak can be determined.

56

2.6 Fracture toughness and its estimation

Fracture toughness signifies the ability of a material to resist fracture when a crack is

present. It describes the intensification of applied stress at the tip of a crack of known size

and shape at the onset of rapid crack propagation. The fractured surfaces of carburized

steels failed under bending fatigue are known to consist of well-defined stages of (i) crack

initiation, (ii) stable crack propagation and (iii) unstable crack propagation. In the high

carbon case of carburized steels, crack initiation usually occurs through intergranular

cracking at prior austenitic grain boundaries [239]. This occurs as soon as the applied stress

at the surface exceeds the sum of surface compressive residual stresses and the cohesive

strength at the prior austenitic grain boundaries. The crack thus initiated usually progresses

as stable transgranular crack for a few hundred micrometers, till it reaches a critical size.

The size and shape of stable transgranular crack propagation region is a function of fracture

toughness (KIC) of the material. Beyond a critical size, the crack begins to progress in

unstable intergranular mode, which continues upto low carbon portion of the case,

whereupon cleavage-like ductile fracture becomes the dominant mode.

Though the sites of initiation and stable crack propagation are often too small and

difficult to find, but they do provide information regarding fracture toughness of the

material. The fracture toughness can be estimated from the relationship put forth by Hyde et

al. [240]:

QaK a

ICπσ2.1

=

Equation 2.21

65.12

2

2

464.11and,

212.0where,

+=

−=

ca

Qys

a

φ

σσφ

Here, a and c are depth and width of the stable crack propagation region, σys is the

yield strength of material and σa is the amplitude of stress at which the given specimen

failed under fatigue.

57

2.7 Problem formulation

The literature consulted so far reveals that the application of numerous thin hard

coatings to cutting tools is a reliable solution for improving efficiency and productivity in

high demand applications. However, except for cutting tools, as far as machine elements are

concerned, the application of coatings is limited to very few components, viz. components

of fuel injection system [241]. Though, many well-established surface engineering methods

have found widespread application in the automotive sector, owing to performance

improvement [242], but the full potential of thin hard coatings in improving performance is

far from being realized.

The literature concerning application of coatings on automotive components is rather

scant, but nevertheless provides an insight regarding their potential towards performance

enhancement [243,244]. The influence of coatings on many potential automotive

components such as piston rings, bearings and gears has still not been explored. The

foremost advantages of thin coatings are their excellent friction and wear behaviour, which

are of great importance for coated parts exposed to intense as well as varying normal and

tangential loads.

Besides friction and wear, the critical parameter to be taken into account during the

selection of an appropriate coating-substrate combination for a specific application is the

effect of coating on fatigue strength and endurance limit of the component. The research

available on fatigue behaviour of coatings is rather scarce in terms of their application to

machine elements (viz. gears, cams, shafts etc.), which are subjected to various standard

processes, such as heat-treatment, prior to their application involving fatigue loading.

Therefore, it is important to investigate the effect of coatings on fatigue behaviour of

materials, which are used for manufacturing such machine elements.

Further, the research in this area is also motivated by environmental concerns

associated with the manufacturing sector. These concerns are addressed in terms of

enhanced service life of the components as well as through replacement of environment-

58

damaging materials and processes such as chrome and cadmium plating with the eco-

friendly process such as PVD. Besides, the PVD process induces residual compressive

stresses within the surface layer, which can lead to better fatigue performance as compared

to electroplated components, which possess tensile residual stresses within the electroplated

layer. In their study concerning EB-PVD coatings, Reinhold et al. [245] point out that PVD

is the most promising coating process and its use is increasing gradually.

The case-carburizing alloy steels are extensively used in automobile industry for

manufacturing machine elements viz. cams, gears, shafts etc. [246-248]. In addition to

increasing surface hardness, the case-carburizing process also imparts residual compressive

stresses on the surface [249-251], which contribute towards enhanced fatigue performance

of the component [252]. However, these residual stresses as well as surface hardness are

compromised to some extent during the subsequent tempering process [246]. A surface

coating deposited through the PVD process, in addition to enhancing tribological

properties, also imparts compressive residual stresses [253] and therefore may be employed

to further enhance the fatigue life of heat-treated components. An important consideration is

that the temperature during the coating process should not be high, as it might impair the

substrate properties gained through the heat treatment process. Among the various PVD

coatings, the recently developed carbon based tungsten carbide doped coating (WC/C) is

accomplished at a relatively low substrate temperature [254], which renders it suitable for

heat-treated components. In addition to its high hardness and low coefficient of friction,

ranging from 0.1 to 0.2 [255,256], the small thickness (1-4µm) achievable in this coating

makes it suitable for application over finished components without casting any sizeable

effect on their dimensions [257].

In the light of above-stated facts, the present study has been taken up to develop an

understanding of the effect of WC/C PVD coating on fatigue performance of substrates

made of various low-alloy steels. The study attempts to investigate the influence of WC/C

coating on the fatigue performance of case carburized low-alloy steels in finite life as well

as infinite life regimes. It aims at quantifying the influence of coating on the fatigue

59

strength corresponding to a given fatigue life for the case carburized low alloy steels. An

attempt is made to understand the mechanisms responsible for the observed fatigue

behaviour by correlating the fatigue properties with measurable parameters, viz. hardness,

residual compressive stress and fracture toughness. Efforts are also made to investigate the

crack propagation mechanisms in case carburized (uncoated) as well as case carburized –

PVD coated specimens made of low alloy steels.