A study of very high cycle fatigue in high strength steels

Mohamed Sadek

A study of very high cycle fatigue in high strength steels

Until 1970´s, fatigue properties of materials were usually studied and evaluated up to the HCF regime (N<106 cycles) beyond which failure was not expected. However, in the late decades of the 20th century proof of fatigue failure in the VHCF regime (N>106 cycles) was presented by different material scientists. This led to an aroused interest in developing a new testing technique where a very high number of loading cycles is achieved within reasonably short time. The ultrasonic fatigue testing system was developed for this purpose, testing materials in the VHCF regime, where specimens are loaded at 20 kHz reaching 1010 cycles in less than a week. In the ultrasonic fatigue testing system, an electric sinusoidal signal is generated and then converted to a mechanical sinusoidal vibration. The mechanical vibration is led through a magnifying horn to the specimen. The system vibrates at resonance frequency, hence all individual parts of the system are designed and dimensioned according to the system resonance frequency.

RESEARCH REPORT | Karlstad University Studies | 2020:22

Faculty of Health, Science and Technology

Materials Engineering

RESEARCH REPORT | Karlstad University Studies | 2020

ISSN 1403-8099

ISBN 978-91-7867-135-9 (pdf)

ISBN 978-91-7867-130-4 (print)

Research report | Karlstad University Studies | 2020:22

A study of very high cycle fatigue in high strength steels

Mohamed Sadek

Print: Universitetstryckeriet, Karlstad 2020

Distribution:Karlstad University Faculty of Health, Science and TechnologyDepartment of Engineering and PhysicsSE-651 88 Karlstad, Sweden+46 54 700 10 00

© The author

ISSN 1403-8099

Karlstad University Studies | 2020:22

RESEARCH REPORT

Mohamed Sadek

A study of very high cycle fatigue in high strength steels

WWW.KAU.SE

ISBN 978-91-7867-135-9 (pdf)

ISBN 978-91-7867-130-4 (print)

2

Abstract

Until 1970´s, fatigue properties of materials were usually studied and

evaluated up to the HCF regime (N<106 cycles) beyond which failure

was not expected. However, in the late decades of the 20th century

proof of fatigue failure in the VHCF regime (N>106 cycles) was pre-

sented by different material scientists. This led to an aroused interest

in developing a new testing technique where a very high number of

loading cycles is achieved within reasonably short time. The ultrason-

ic fatigue testing system was developed for this purpose, testing mate-

rials in the VHCF regime, where specimens are loaded at 20 kHz

reaching 1010 cycles in less than a week. In the ultrasonic fatigue test-

ing system, an electric sinusoidal signal is generated and then con-

verted to a mechanical sinusoidal vibration. The mechanical vibration

is led through a magnifying horn to the specimen. The system vibrates

at resonance frequency, hence all individual parts of the system are

designed and dimensioned according to the system resonance fre-

quency.

Fatigue testing commonly implies testing for fatigue strength and

testing fatigue crack growth. The staircase test method is used to de-

termine a materials fatigue strength at a certain fatigue life. To meas-

ure the crack growth rate the crack is continuously monitored and

measured and the stress intensity at the crack tip is computed for the

growing crack. FEM has been used when the specimen geometry is

complicated or when dynamic simulations are required.

Results from fatigue tests in the VHCF regime have showed that inte-

rior non-metallic inclusions are the most common initiation sites.

This led to further investigations on the inclusion type, shape and size

and the effect they have on the fatigue strength. An inclusion initiated

fatigue crack normally creates a fish-eye with a Fine Granular Area

(FGA) around the inclusion. It has been confirmed that 90-99% of the

total fatigue life is due to the formation of the FGA area.

This study contains an assemblage of theories and models for, and

results of, different VHCF tests.

3

Table of content

ABSTRACT ................................................................................. 2

1. VERY HIGH CYCLE FATIGUE ............................................. 4

2. ULTRASONIC FATIGUE TESTING EQUIPMENT ................. 6

2.1 ASSOCIATED EQUIPMENT ................................................................. 9

3. DIFFERENT TESTING METHODS ....................................... 9

3.1 TESTING FOR FATIGUE STRENGTH................................................... 10

3.1.1 TC/TT-testing of hourglass shaped specimen ........................ 10

3.1.2 Three-point bending................................................................ 14

3.1.3 Torsion .................................................................................... 17

3.1.4 Estimation of fatigue life .......................................................... 18

3.1.5 Estimation of fatigue strength – Staircase method ................. 20

3.1.6 SN-curve ................................................................................. 21

3.2 TESTING FOR FATIGUE CRACK GROWTH........................................... 23

3.2.1 Crack growth testing at 20 kHz ............................................... 24

4. MODELLING AND SIMULATION .......................................26

4.1 CALCULATION OF THE STRESS AMPLITUDE AT 20 KHZ ....................... 26

4.2 COMPUTATION OF THE STRESS INTENSITY FACTOR FOR 20 KHZ CRACK

GROWTH TESTING ..................................................................................... 28

4.2.1 Resonance frequency correction ............................................ 29

4.2.2 Crack tip opening displacement - CTOD ................................ 29

4.2.3 J contour integral .................................................................... 32

5. FATIGUE PROPERTIES OF HIGH STRENGTH STEELS ...... 37

5.1 VHCF TESTING OF HIGH STRENGTH STEELS .................................... 38

5.2 FATIGUE STRENGTH ...................................................................... 40

5.3 CRACK INITIATION ......................................................................... 41

5.4 CRACK GROWTH ........................................................................... 43

5.5 FREQUENCY EFFECT ..................................................................... 47

5.6 DAMPING ...................................................................................... 48

5.6.1 Damping measurement .......................................................... 50

CONCLUSION ........................................................................... 52

REFERENCES ........................................................................... 53

4

1. Very high cycle fatigue

In the late 19th century, the German railway engineer August Wöhler

introduced his famous SN-curve containing a relationship between

load and expected fatigue life, it was limited to a fatigue life up to 106-

107 cycles, see Figure 1. The fatigue strength at 107 cycles was typically

determined by the staircase method, hence the infinite fatigue limit

(probability of fracture close to zero) was assumed to µ̂y-3σ̂y where σ̂y

is the standard deviation (often 10 MPa). This assumption is not the

best way to determine the fatigue limit since new research clearly

shows that the difference between µ̂y at 107 and µ̂y at 109 is much high-

er than 3σ̂y=30 MPa [1].

Figure 1. August Wöhler´s first introduced SN-curve [2, 3].

In the late 20th century, researchers managed to show evidence of fa-

tigue failure of metals after 108 cycles. Consequently, an interest grew

among metal researchers to extend the primary SN-curve (Figure 1)

and construct the Very High Cycle Fatigue (VHCF)-regime of the

curve. Bathias and co-workers [4] and Stanzl-Tschegg and co-workers

[5] were leading researchers in developing a system for testing metals

for fatigue at very long fatigue lifes within manageable testing times.

The desire of high loading frequency led to the idea of using the ultra-

sonic booster (first constructed by Mason in year 1950) to load small,

specially designed metal specimens. Such machine contains, most

importantly, a generator that generates a 20 kHz sinusoidal electric

Predominant surfaceinitiations

Infinite fatigue life

5

signal and a piezoelectric transducer that transforms the electric sig-

nal into oscillating mechanical vibration at the same frequency. Later,

with the development and introduction of computer science into the

ultrasonic fatigue testing, a control unit was developed enabling sev-

eral important functions such as: varying the strain amplitude, load-

ing ratio and loading cases (bending and torsion). With a 20 kHz

loading frequency it is reasonable to test for VHCF where 1010 cycles

are reached within 7-days of running time. Today there are a handful

of well-established laboratories in France, Austria, Germany, Japan,

China, USA and Sweden specialized in VHCF [3-5].

The first VHCF-dedicated conference took place in Paris in the year

1998 (organized by Bathias and co-chaired by Stanzl-Tschegg) and

was the first one, VHCF1 (at the time called Euromech 382: Fatigue

Life in Gigacycle regime), in a successful conference series reaching

its eight conference, VHCF8, in Sapporo 2020. During the first con-

ference in 1998, several researchers, including Bathias, have showed

evidence of unexpected fatigue failure occurring at fatigue lifes be-

yond 107 cycles under load amplitudes lower the conventional fatigue

limit. Researchers have showed a change in failure mechanism in high

strength steels related to longer fatigue lifes. A transition from surface

initiations for short lifes to subsurface initiations for long lifes has

been observed. Surface initiations are usually caused by surface de-

fects or internal concentrations of plastic deformation forming persis-

tent slip bands (PSBs) that are transferred to the surface. The subsur-

face initiated cracks often starts at material defects such as non-

metallic inclusions, pores or triple points [2, 6, 7].

A new extended SN-curve, shown in Figure 2, proposed by some re-

searchers includes the VHCF-regime [1-3, 8-10]. While the previously

defined SN-curve (Figure 1) assumed a fatigue limit at 106 cycles, the

new SN-curve (Figure 2) eventually reaches a lifetime where internal

crack initiations mechanisms starts to occur at around 108 cycles. This

is a typical SN-curve for high strength metals containing internal de-

fects such as non-metallic inclusions or even pores.

6

Figure 2. Typical SN-curve for high strength metals with internal defects [1-3, 8-

10].

2. Ultrasonic fatigue testing equipment

Figure 3 shows schematic model of a typical modern ultrasonic fa-

tigue testing machine and the parts included; a generator, a piezoelec-

tric transducer, a magnifying horn, a booster and a control unit. The

horn magnifies the displacement amplitude by a factor specific for its

dimensions calculated with FEM. The booster is necessary when

mean load is required or when high amplitudes are required since it

too contains a magnification factor. The control unit enables the op-

erator to control the voltage input and hence the displacement ampli-

tude with a ±1% accuracy. A recording device provides data such as;

displacement amplitude, system frequency (to a ±1 Hz accuracy),

number of cycles and with additional equipment the temperature of

the specimen. A capacitive displacement sensor is used for calibration

of the displacement at the bottom of the horn or at the bottom of the

specimen and a relationship between the input voltage and the dis-

placement is determined.

The ultrasonic fatigue testing system vibrates at the uniaxial reso-

nance frequency of the whole load train including oscillator, horn,

booster and specimen. This is only possible when all parts included

are appropriately designed and dimensioned.

7

Figure 3. Ultrasonic fatigue testing equipment with stress and displacement dis-

tribution along the loading axis [4, 11].

All parts need to have a resonance frequency of the longitudinal vi-

brating mode matching the resonance frequency of the system, i.e. 20

kHz.

Different shapes of axially loaded specimens, axisymmetric and rec-

tangular, varying and constant cross-sections, are the most common

ones. Bending and torsion specimens are also used although special

fixtures will be required for conducting such tests [4, 5]. In conven-

tional fatigue testing the loading frequency is set by the testing ma-

chine and forced onto the specimen, i.e. the specimen is under forced

vibration. However, in ultrasonic fatigue testing the loading frequency

is one of the natural frequencies of the specimen, i.e. free vibration.

The free vibration is analysed by the elastic wave theory, where a lon-

gitudinal wave travels through the whole load train (oscillator, horn

8

and specimen). The ultrasonic fatigue testing system, including the

specimen, is considered as a one-dimensional system where the wave

velocity is expressed in Equation 1 and the resulting displacement and

strain distribution shown in Figure 3 and for the specimen only in

Figure 4.

𝑐 = √𝐸𝑑/𝜌 [1]

From the solution of the differential equation of a general isotropic

elastic body and the simplification to a one-dimensional system the

following expressions of the displacement and the strain (Equations 2

and 3) are extracted.

𝑈(𝑥) = 𝑈0 cos(𝑘𝑥) [2]

휀(𝑥) = −𝑘 𝑈0 sin(𝑘𝑥) [3]

where U0 is the applied amplitude at the end of the specimen, k = π/l

and l is the length of the specimen expressed by Equation 4.

𝑙 =1

2𝑓0√

𝐸𝑑

𝜌 [4]

where f0 is the resonance frequency of the system. For x=l/2, i.e. at

the middle of the specimen we have;

𝑈 = 0 [5]

휀 = −𝑘 𝑈0 [6]

Thus, the stress at the middle of the specimen is given by;

𝜎 = −𝐸𝑑 𝑘 𝑈0 [7]

9

Figure 4. Displacement and strain distribution along an elastic bar [11].

2.1 Associated equipment

For fully reversed tension-compression (R=-1) loading the specimen

is fixed at one end only where the top end is attached to the horn and

the bottom end is free to vibrate. Varying load ratio (R≠-1) are

achieved by adding a twin horn to the bottom of the specimen and

inserting the whole ultrasonic system into a tensile testing rig for su-

perposing a static load.

Additional equipment commonly used includes; strain gages, dis-

placement measuring system, cooling system and crack growth de-

tecting and measuring device. The displacement measuring system

usually consists of a capacitive displacement gage (placed under the

horn measuring the maximum displacement) calibrated with laser

measurement technique. Depending on the cooling rate required, un-

dercooled pressurized air or cooled water is used cooling the speci-

men. For the crack measurement, a CCD camera, strain gage or a DC

potential measurement device is most often used. Crack growth over-

all is easily detected by simply noticing the frequency drop since the

resonance frequency of the system is very sensitive to any specimen

stiffness changes [3-5].

3. Different testing methods

Testing material fatigue properties involves measurements and inves-

tigations of different parameters. Material fatigue properties mainly

10

includes information on fatigue strength, fatigue crack propagation

rates and fatigue crack initiation mechanisms. Fatigue strength is an

estimation of fatigue strength at given fatigue life, commonly deter-

mined by staircase testing, see sections 3.1.4 and 3.1.5. Testing for fa-

tigue crack growth rates and evaluate at 20 kHz loading frequency

provides crack growth data in the low ΔK regime of the Paris law

curve. The standardized methodology of such testing, and an adapta-

tion to the ultrasonic fatigue testing system is explained in section 3.2.

Furthermore, this chapter gives a detailed description of some differ-

ent types of ultrasonic fatigue testing methods and result evaluation

models.

3.1 Testing for fatigue strength

The fatigue life behaviour of a specific material is best presented by a

SN-curve. Microstructural inhomogeneity and surface defects are two

reasons behind the significant scatter in the fatigue life results. There-

fore a series of test specimens must be tested for each material and a

statistical SN-curve needs to be generated [12]. Randomness in selec-

tion of specimens and preparing test batches is important. For a relia-

ble test, a batch of minimum 28 specimens is recommended [13].

3.1.1 TC/TT-testing of hourglass shaped specimen

Hourglass shaped specimens, cylindrical or plane form, smooth or

notched, are mounted to the horn in the ultrasonic fatigue testing de-

vice. For R=-1 loading ratio, Figure 5, the specimens top end is

mounted to the horn while the bottom end is free.

11

Figure 5. Ultrasonic fatigue testing setup for hourglass shaped specimen in R=-1

loading ratio [4, 11].

The interaction between the ultrasonic load train and the mounting

rig is at the zero displacement point within the horn, see Figures 3

and 5 [11].

To vary the loading ratio, an identical second horn is mounted to the

bottom end of the specimen, see Figure 6. To superpose a static load

(i.e. mean load) the whole load train is installed in a tensile testing rig.

By adding a booster to the system, compressive forces are possible [4,

5]. The booster is not necessary when tensile mean load (R>-1) is de-

sirable. Two identical horns are enough when mounted as the bottom

horn in Figure 6. For a compressive mean load (R<-1) however, two

identical boosters are required, one above the top horn and one under

the bottom horn. The boosters are fully constrained in the mounting

rig like the one in Figure 6 [11].

12

Figure 6. Ultrasonic fatigue testing setup for hourglass shaped specimen in vary-

ing loading ratio R≠-1 [4, 11].

Theoretically, using sheet specimens (Figure 7b) is not different from

cylindrical specimens. However, practically speaking, there are some

difficulties that needs to be overcome. The finishing of the specimens

must be done with delicacy for the specimen to vibrate at the right

frequency in the right mode. Another challenge is the connection to

the horns. A specially designed connecting feature is required, and

high precision is again required to avoid any change in the vibration

frequency and vibration mode. Furthermore, there are some con-

straints regarding the thickness of the specimens. It has been proofed

13

that a thickness to length (or largest dimension) ratio as low as 6-8%

is manageable. A smaller thickness will cause perturbation in the vi-

bration frequency and will require special measure regarding the at-

tachment to the horns [11, 14, 15].

Analytical stress-strain calculations for cylindrical bar specimen

Fatigue specimens used in ultrasonic testing machines usually have a

varying cross-sectional area. The hourglass shape of the specimen

(Figure 7a-b) enhances the desired stress and displacement distribu-

tion shown in Figure 3. This distribution is desirable for mainly two

reasons; the position of the maximum stress is narrowed down to the

middle of the specimen (at x=l/2) and, the displacement at that posi-

tion (x=l/2) is zero enabling for instance the recording of a growing

crack. The length of the constant cross-section part of the specimens

(the resonance length) are usually determined by numerical analysis

(FEM). However, if the mid-part of the specimen has an exponential

profile, the resonance length can be calculated analytically. Such ana-

lytical solution also provides the strain and stress distribution [11].

For the axially loaded circular cross-section specimen, see Figure 7a,

the analytical solution is presented in Equations 8-16.

Figure 7. Axially loaded a) cylindrical and b) plane hourglass shaped specimens

with 20kHz resonance frequency [11, 14, 15].

a) b)

W

14

𝑘 =𝜔

𝑐 [8]

𝛼 =1

𝐿2 𝑎𝑟𝑐𝑐𝑜𝑠ℎ(

𝑅2

𝑅1) [9]

𝛽 = √𝛼2 − 𝑘2 [10]

𝜑(𝐿1, 𝐿2) =cos(𝑘𝐿1) 𝑐𝑜𝑠ℎ(𝛼𝐿2)

𝑠𝑖𝑛ℎ(𝛽𝐿2) [11]

The resonance length of the specimen (L1) becomes

𝐿1 =1

𝑘 arctan [

1

𝑘(𝛽 coth(𝛽𝐿2) − 𝛼 tanh(𝛼𝐿2))] [12]

The strain and stress functions for the end-part of the specimen (L2 <

x < L) are expressed in Equations 13 and 14.

휀(𝑥) = 𝑘 𝑈0 𝑠𝑖𝑛(𝑘(𝐿 − 𝑥)) [13]

𝜎(𝑥) = 𝐸𝑑 𝑘 𝑈0 𝑠𝑖𝑛(𝑘(𝐿 − 𝑥)) [14]

and for the mid-part of the specimen (0 < x < L2), see Equations 15

and 16.

휀(𝑥) = 𝑈0 𝜑(𝐿1, 𝐿2) 𝛽 𝑐𝑜𝑠ℎ(𝛽𝑥) 𝑐𝑜𝑠ℎ(𝛼𝑥)−𝛼 𝑠𝑖𝑛ℎ(𝛽𝑥)𝑠𝑖𝑛ℎ (𝛼𝑥)

𝑐𝑜𝑠ℎ2(𝛼𝑥) [15]

𝜎(𝑥) = 𝐸𝑑 𝑈0 𝜑(𝐿1, 𝐿2) 𝛽 𝑐𝑜𝑠ℎ(𝛽𝑥) 𝑐𝑜𝑠ℎ(𝛼𝑥)−𝛼 𝑠𝑖𝑛ℎ(𝛽𝑥)𝑠𝑖𝑛ℎ (𝛼𝑥)

𝑐𝑜𝑠ℎ2(𝛼𝑥) [16]

3.1.2 Three-point bending

The most common loading type used in the ultrasonic fatigue testing

system is uniaxial. However, many components in today’s mechanical

systems are subjected to flexural cyclic loading hence the investiga-

tion of the fatigue properties of such components are important. This

has led to the development of special rigs for other loading types in

ultrasonic fatigue testing system, i.e. bending and torsion, in various

laboratories. In [16, 17] a three-point bending specimen with a trans-

15

verse resonant frequency at 20kHz is used to measure crack propaga-

tion rates in different metals. The results have been compared to

measurements done at lower frequency (60Hz) using different

equipment and the results have showed a good agreement.

In [11, 18] a three-point bending ultrasonic fatigue testing system, il-

lustrated in Figure 8. The converter, booster and horn are under uni-

axial resonant vibration while the rectangular specimen is under

transverse resonant vibration. The booster is connected to the frame

of a tensile testing machine to superpose a static load for various load

ratios.

To transfer the load from the horn to the specimen a special tip is de-

signed to be bolted into the end of the horn and pushes down onto the

specimen, as illustrated in Figure 9. Where L is the specimen length

and L1 is the resonance length of the specimen, i.e. the distance be-

tween the supporters. These two dimensions (L and L1) are calculated

by either FEM or analytically with the use of the free flexural wave

function for a beam with uniform cross-section, Equation 17.

𝐸𝐼𝜕4𝑢(𝑥,𝑡)

𝜕𝑥4+ 𝜌ℎ𝑏

𝜕2𝑢(𝑥,𝑡)

𝜕𝑡2= 0 [17]

where I=bh3/12, ρ is the density of the material, b and h are the width

and height of the cross-section and u(x,t) is the displacement of point

x (along the specimen length) at time t. By using the correct boundary

conditions, this differential equation is solved, and the length and

resonance length of the specimen are extracted and given in Equa-

tions 18 and 19.

2𝐿 = 0,507 (𝐸ℎ2

𝜌𝑓2)

1/4

[18]

2𝐿1 = 0,28 (𝐸ℎ2

𝜌𝑓2)

1/4

[19]

16

Figure 8. Three-point bending fatigue testing setup [4, 11, 18]

Figure 9. Rectangular three-point bending specimen with the load-transferring

tip [18].

17

3.1.3 Torsion

A new technique in torsion loading using ultrasonic fatigue system

was introduced by [19] in the late 1980´s and has been reused and

developed by other researchers [15, 20-22]. Special design of the me-

chanical parts of the equipment is required to achieve torsion reso-

nance vibration in the specimen, Figure 10. The amplitude control is

basically the same as for the axial loading tests. Since the resonance

vibration in torsion is depending on the shear modulus, which is

smaller than the elastic modulus, the vibrating parts in the system

needs to be redesigned for 20 kHz resonance twisting oscillation [11].

Figure 10. Ultrasonic fatigue testing setup for torsion; a) direct system and b)

indirect system [4, 11, 15].

The torsion specimen (Figure 11) has the same design as the axially

loaded specimen (Figure 7a) but with different dimensions. For the

specimen to have a twisting resonance frequency at 20 kHz it is de-

signed shorter and thicker than the uniaxial specimen is with the

same resonance frequency.

Torsional horn

Specimen

Twist oscillator

Axial horn

Torsional horn

Specimen

Connecting pinAxial oscillator

a) b)

18

Figure 11. Hourglass shaped torsion specimen with 20 kHz resonance frequency

[11, 15].

Two different setups exists today, direct and indirect systems, see

Figure 10. In the direct system, the converter/oscillator generates

twist oscillations transferred to the specimen through a torsion horn,

see Figure 10a [20, 21]. In the indirect system an axially oscillating

converter/oscillator is connected to a torsion horn through an axially

vibrating horn, Figure 10b. The axially oscillating horn magnifies the

displacement oscillation from the converter, and the perpendicularly

attached torsion horn transform the axial vibration to torsional vibra-

tion. The specimen is attached to the torsional horn and vibrated at its

and the systems resonance frequency. Just as for the axially loaded

specimen the maximum displacement is at the ends of the specimen

while the maximum strain is at the middle of the specimen [15, 22].

3.1.4 Estimation of fatigue life

It is often observed that the fatigue life at a given stress level follow a

normal distribution, expressed in Equation 20. A statistical estima-

tion of the distribution parameters is done by calculation of the sam-

ple mean µ̂ and the sample standard deviation �̂�, Equations 21 and 22

[13].

𝑃(𝑥) =1

𝜎√2𝜋∫ 𝑒(−

1

2(

𝑥−µ

𝜎)2)𝑥

−∞𝑑𝑥 [20]

19

µ̂ =∑ 𝑥𝑖

𝑛𝑖=1

𝑛 [21]

�̂� = √∑ (𝑥𝑖−µ̂)2𝑛

𝑖=1

𝑛−1 [22]

where n is the number of data points, xi is the Log(Ni) of each data

point.

An estimation of lower limit of fatigue life at a given probability of

failure P and confidence level (1-) is done according to Equation 23.

�̂�(𝑃,1−𝛼) = µ̂𝑥 − 𝑘(𝑃,1−𝛼,𝜐)�̂�𝑥 [23]

where the coefficient k(P,1-,) is a tabulated value of a one-sided toler-

ance limit for a normal distribution and is the number of degrees of

freedom (n-1). To evaluate the results and the assumption of a normal

distribution, the fatigue lives can be plotted on log-normal probability

coordinate. The data points should line up in a straight line, any devi-

ation means invalid data or distribution model. The failure mecha-

nism of the specimen should be analysed to see if the deviated data

point is valid or if it should be removed, or simply try with another

statistical distribution model e.g. Weibull.

The Weibull distribution function, Equation 24, is a flexible function

suitable for a variety of practical situations including life length dis-

tribution.

𝐹𝑋(𝑥) = 1 − 𝑒−(𝑥

𝑎)𝑐

[24]

where a is a scale factor and c is the form parameter describing differ-

ent forms of probability density. The parameter c cannot be negative

and for the special case c=1, a constant failure tendency and an expo-

nential distribution is obtained [23].

The two parameters a and c are estimated either graphically or analyt-

ically. The graphic method starts by rearranging and taking the loga-

rithm of the Weibull distribution function:

ln[1 − 𝐹𝑋(𝑥)] = −(𝑥

𝑎)𝑐 [25]

20

ln[− ln(1 − 𝐹𝑋(𝑥))] = 𝑐 ln(𝑥) − 𝑐 ln(𝑎) [26]

and consequently:

ln [−ln (1 −𝑖−1/2

𝑛)] ≈ 𝑐 ln(𝑥(𝑖)) − 𝑐 ln(𝑎) [27]

where

𝑢𝑖 = ln [−ln (1 −𝑖−1/2

𝑛)] [28]

𝑧𝑖 = ln [𝑥(𝑖)] [29]

so

𝑢𝑖 ≈ 𝑐 𝑧𝑖 − 𝑐 ln(𝑎) [30]

ui is calculated for every zi and plotted in a diagram. The data points

should be distributed on a straight line from which the parameters a

and c are extracted.

3.1.5 Estimation of fatigue strength – Staircase method

When estimating the fatigue strength at a given fatigue life, the stair-

case method [24, 25] is appropriate. This method starts by a rough

estimation of the mean stress and the standard deviation. At least 15

specimens are required for a reliable test where the first specimen is

tested at a stress level close to the estimated mean stress. If the spec-

imen fails within the set fatigue life limit, the next specimen is tested

at lower stress level. If the specimen reaches the fatigue life limit be-

fore failure, i.e. run-out, then the next specimen is tested at a higher

stress level. The stress increments should be close to the standard de-

viation or simply 5% of the mean stress.

After counting the failures and non-failures of the specimens tested,

the group of the least occurred event is selected for the analysis. The

stress levels used in the test are denoted in an ascending order by S0 <

S1 < S2 < …. < Sl, where l is the number of stress levels. The number of

21

events at each stress levels is denoted fi. The statistical distribution

parameters are estimated by Equations 31 and 32.

µ̂𝑦 = 𝑆0 + 𝑑 (𝐴

𝐶±

1

2) [31]

where +1

2 is used when event analysed is failure and -

1

2 when the event

analysed is non-failure.

�̂�𝑦 = 1,62 𝑑 (𝐵𝐶−𝐴2

𝐶2+ 0,029) [32]

where d is the stress step and A, B and C as in Equations 33-35:

𝐴 = ∑ 𝑖 𝑓𝑖𝑙𝑖=1 [33]

𝐵 = ∑ 𝑖2𝑙𝑖=1 𝑓𝑖 [34]

𝐶 = ∑ 𝑓𝑖𝑙𝑖=1 [35]

In [24] it is stated that Equation 32 is valid only if 𝐵𝐶−𝐴2

𝐶2 > 0,3. This is

generally the case when 𝑑

�̂�𝑦 is chosen within the range of 0,5-2.

3.1.6 SN-curve

Linear fatigue response and curvilinear fatigue response are two

mathematical relationships used to describe most of all produced SN-

data. However, occasionally there are some cases where these two

mathematical relationships are not sufficient. The linear and curvilin-

ear fatigue responses are presented in Equations 36 and 37, respec-

tively [13].

𝐿𝑜𝑔10(𝑁) = 𝑏0 + 𝑏1 𝐿𝑜𝑔10(𝑆) [36]

𝐿𝑜𝑔10(𝑁) = 𝑏0 + 𝑏1 𝐿𝑜𝑔10(𝑆) + 𝑏2 𝐿𝑜𝑔102 (𝑆) [37]

where b0, b1 and b2 are regression coefficients and S is either stress or

strain.

22

Linear fatigue response

Define Xi and Yi as follows:

𝑋𝑖 = 𝐿𝑜𝑔10(𝑆𝑖) [38]

𝑌𝑖 = 𝐿𝑜𝑔10(𝑁𝑖) [39]

Calculate the regression parameters b1 and b0 with the use of Equa-

tions 40 and 41.

𝑏1 =∑ 𝑋𝑖𝑌𝑖

𝑛𝑖=1 −

∑ 𝑋𝑖𝑛𝑖=1 ∑ 𝑌𝑖

𝑛𝑖=1

𝑛

∑ 𝑋𝑖2𝑛

𝑖=1 −(∑ 𝑋𝑖

𝑛𝑖=1 )2

𝑛

[40]

𝑏0 =∑ 𝑌𝑖

𝑛𝑖=1 −𝑏1 ∑ 𝑋𝑖

𝑛𝑖=1

𝑛 [41]

The predicted value of the dependent variable, �̂�𝑖, is defined in Equa-

tion 42.

�̂�𝑖 = 𝑏0 + 𝑏1𝑋𝑖 [42]

The standard deviation is defined in Equation 43.

�̂� = √∑ (𝑌𝑖−�̂�𝑖)2𝑛

𝑖=1

𝑛−𝑝 [43]

where n is, as mentioned above, the number of data points and p is

the number of parameters (2 parameters for the linear case).

The evaluation of the quality of the fit is performed by the computa-

tion of the correlation coefficient, R2, by the expression in Equation

44.

𝑅2 =(∑ 𝑋𝑖𝑌𝑖

𝑛𝑖=1 −

∑ 𝑋𝑖𝑛𝑖=1 ∑ 𝑌𝑖

𝑛𝑖=1

𝑛)2

(∑ 𝑋𝑖2𝑛

𝑖=1 −(∑ 𝑋𝑖

𝑛𝑖=1 )2

𝑛)(∑ 𝑌𝑖

2𝑛𝑖=1 −

(∑ 𝑌𝑖𝑛𝑖=1 )2

𝑛)

[44]

The quality of the fit is considered acceptable for R2 > 0,9.

23

3.2 Testing for fatigue crack growth

In [26] an ASTM standard method for crack growth testing is de-

scribed. The method assumes the theory of linear elasticity and does

not consider the local crack-tip effects such as crack closure and re-

sidual stresses. Crack growth rates are revealed from near-threshold

to Kmax, expressed by the crack growth rate (da/dN) versus the crack

tip stress-intensity factor range (K). The stress-intensity factor rang-

es from Kmin to Kmax for positive loading ratios (R = Kmin/Kmax =

Pmin/Pmax > 0) and from zero to Kmax for negative loading ratios.

The stress-intensity threshold is the value where the crack growth rate

approaches zero, or an arbitrary value of 10-10 m/cycle. This arbitrary

threshold value is estimated from the linear regression of the log

da/dN vs log K curve where a minimum of five data points with

equal spacing between 10-9 to 10-10 m/cycle. It is required that the K

is the dependent variable since the range is specified by the da/dN.

In the low crack growth rate regime (da/dN < 10-8 m/cycle) a variabil-

ity of the rate can be as high as a factor of five or more due to high

sensitivity to small changes in K and other factors such as micro-

structural differences, residual stresses, changing crack tip geometry,

force precision and environmental control.

The tests start with the pre-cracking of the specimen. It is important

that the force distribution during pre-cracking is symmetrical with

respect to the notch (this is important throughout the test and not on-

ly during pre-cracking). The pre-crack should not be less than 10% of

the width, 10% of the thickness or 1mm, whichever is greater. Prefer-

ably, the pre-cracking is done at the lowest stress-intensity possible,

pre-cracking growth rates less than 10-8 m/cycle are recommended.

After pre-cracking, the test proceeds with incremental crack growth

under K-increasing or K-decreasing procedure. For the K-increasing

procedure the force amplitude is kept constant while the stress-

intensity is increased with the growth of the crack. This procedure is

not suitable for crack growth rates below 10-8 m/cycle since prior

loading history may influence near-threshold growth rates. However,

the K-decreasing procedure is recommended for growth rates below

10-8 m/cycle.

The K-decreasing test procedure is conducted by incremental de-

crease of the force amplitude. It is recommended that the shedding of

24

the force amplitude is conducted by steps of maximum 10% of previ-

ous Pmax and an in-step crack growth of at least 0.5mm allowing the

establishment of at least five da/dN vs K data point per decade of

crack growth rate with approximately equal spacing.

For a constant normalized K-gradient the K is expressed by Equa-

tion 45.

∆𝐾 = ∆𝐾0𝑒[𝐶(𝑎−𝑎0)] [45]

where K0 is the initial K of the test and a0 is the corresponding

crack length.

The crack length is measured on both sides of the specimen, and the

difference should not be greater than 0,5R1. The crack growth line

should not deviate more than ±20 degrees. The crack length meas-

urement should be conducted with a resolution of 0,1mm or 0,2% of

the width, whichever is greater. Recommendations states that the test

should not be interrupted during crack length measurement. Howev-

er, if the stoppage is necessary for the measurement, any crack growth

due to i.e. creep or growth under static load is to be avoided.

3.2.1 Crack growth testing at 20 kHz

The procedure described above is the appropriate procedure even at

very high loading frequencies (20 kHz). Crack growth rate measure-

ments at 20 kHz loading frequency have been conducted by some ma-

terial engineering laboratories. In [16] fatigue crack propagation at

rates as low as 10-12 m/cycle has been measured using a three-point

bending specimen in the ultrasonic fatigue testing system. Sheet and

bar tensile specimens (see bar specimen in Figure 12) were designed

by [11, 27-29] for crack growth rate measurements at 20 kHz resulting

in successful measurements of fatigue crack growth rates and stress

intensity threshold values of different metals.

An analytical method of computing the resonance length of a rectan-

gular cross-section specimen, see Figures 12 and 7b, and the stress-

strain distribution is presented below.

25

𝛼1 =1

2𝐿2𝑙𝑛(

𝑅2

𝑅1) [46]

𝛽1 = √𝛼12 − 𝑘2 [47]

𝜑1(𝐿1, 𝐿2) =𝑐𝑜𝑠(𝑘𝐿1) 𝑒𝑥𝑝(𝛼1𝐿2)

𝑠𝑖𝑛ℎ(𝛽1𝐿2) [48]

(values according to Figure 12)

Figure 12. Crack propagation measurement specimen designed for 20 kHz reso-

nance frequency [11, 15].

the resonance length of the specimen (L1) becomes

𝐿1 =1

𝑘 arctan [

1

𝑘(𝛽1 coth(𝛽1𝐿2) − 𝛼1)] [49]

and the strain and stress functions for the mid-part of the specimen

(0 < x < L2) are expressed in Equations 50 and 51.

휀(𝑥) = 𝑈0 𝜑1(𝐿1, 𝐿2) [𝛽1𝑐𝑜𝑠ℎ(𝛽1𝑥) − 𝛼1 𝑠𝑖𝑛ℎ(𝛽1𝑥)] 𝑒𝑥𝑝(−𝛼1𝑥)

[50]

𝜎(𝑥) = 𝐸𝑑 𝑈0 𝜑1(𝐿1, 𝐿2) [𝛽1𝑐𝑜𝑠ℎ(𝛽1𝑥) − 𝛼1 𝑠𝑖𝑛ℎ(𝛽1𝑥)] 𝑒𝑥𝑝(−𝛼1𝑥)

[51]

26

FEM analysis is another, often much faster, method of computing the

stress-strain conditions in such specimens. The stress intensity factor

at the crack tip is calculated numerically by either the use of the J-

integral or the CTOD-method after measuring the crack tip opening

close to the tip [29, 30].

4. Modelling and simulation

To test for fatigue, or any other material property for that matter, it is

necessary to investigate the solid mechanics of the specific material

and specific specimen shape. This is done either analytically or using

FEM. The latter is appropriate when the specimen geometry and/or

testing characteristics are complicated.

For the fatigue strength testing, it is necessary to determine the rela-

tionship between the displacement amplitude at the top of the speci-

men and the stress amplitude at the middle of the specimen. This can

be done analytically according to the method described in sections

3.1.1 and 3.2 or by a FEM simulation as described in section 4.1.

When testing for fatigue crack growth rates, e.g. Paris law parameters,

the relationship between the displacement amplitude at the top of the

specimen and the stress intensity factor at the crack tip at the middle

of the specimen is now required. Two appropriate methods are de-

scribed in sections 4.2.2 and 4.2.3 where both methods require FEM

assistance, at least when specimen geometry is non-uniform.

4.1 Calculation of the stress amplitude at 20 kHz

The ultrasonic fatigue testing machine is displacement controlled. The

stress amplitude is altered by altering the amplitude of the displace-

ment of the bottom of the horn. For that, a multiplication factor (m-

factor) between the displacement of the horn bottom and the maxi-

mum stress in the specimen is required. The m-factor can be calculat-

ed analytically [11] or with FEM analysis where a model of the speci-

men is loaded with a sinusoidal displacement at its resonance fre-

quency [14]. The m-factor is obviously different for different types of

specimens. Figure 13 shows the stress distribution in the hourglass

27

specimens under a sinusoidal displacement placed on the top surface.

The maximum stress is of course located at the mid part of the speci-

men where the cross-section is minimum.

Figure 13. Stress and displacement distribution of hourglass shaped smooth and

notched specimens [15].

Figure 14 shows the displacement distribution in the rectangular 20

kHz bending specimen under a 20 µm sinusoidal displacement placed

at the middle of the top surface. The maximum stress is located at the

middle of the bottom surface.

Figure 14. Stress and Displacement distribution of rectangular three-point bend-

ing specimen [15, 18].

The m-factor is computed for the torsion specimen with the same

principle as for the uniaxial and bending specimens. Figure 15 shows

a FEM model of the torsion specimen with the relating displacement,

angle of twist and shear stress distribution. As clearly seen, the dis-

placement and the angle of twist is zero at the middle of the specimen

with maximum values at the ends. Expectedly, the maximum value of

the shear stress is at the middle of the specimen and minimum at the

ends [15, 22].

Stress Displacement Stress Displacement

Smooth specimen Notched specimen

distribution distribution distribution distribution

Displacement distributionStress distribution

28

Figure 15. Displacement, angle of twist and shear stress distribution in the hour-

glass shaped torsion specimen [15].

4.2 Computation of the stress intensity factor for 20 kHz crack

growth testing

The conventional linear elastic fracture mechanics (LEFM) is appro-

priate when the nonlinear/plastic deformation is confined to a small

region around the crack tip. For most material, especially under very

high load frequency (as in VHCF testing), this is not the case. For high

toughness materials, crack tip blunting caused by local plastic defor-

mation invalidates the LEFM. There are other models better fitting to

the fracture mechanics of elastic-plastic materials, such as the crack

tip opening displacement method (CTOD) and the J-integral method

[31]. With the CTOD method, the crack tip opening close to the crack

tip is measured. The plastic behaviour of a material affects the crack

tip opening and are thus considered in the measured CTOD. The J-

integral method is a path-dependent integral for plastic materials.

However, the path-dependence can be measured and in the case

where path-independence (negligible path-dependence) occur, the J-

integral method is valid for plastic materials.

29

4.2.1 Resonance frequency correction

The prediction of the eigenfrequencies in VHCF-testing is important

due to the frequency sensitivity of the ultrasound fatigue testing sys-

tem. During fatigue crack growth testing, a crack is initiated and

grows under controlled conditions. The existence of a crack and the

growth of it directly affects the stiffness and hence the resonance fre-

quency of the specimen. Under longitudinal vibration, a cracked spec-

imen behaves as a longer specimen and the eigenfrequency conse-

quently decreases. When modelling for the computation of the stress

intensity factor, it is desired to have a valid model for the frequency

change due to the crack growth, i.e. prediction of the eigenfrequencies

of a fatigue crack growth specimen containing cracks of different

lengths [32]. Gudmundson [33] has described a first order perturba-

tion method predicting change in eigenfrequencies of structures con-

taining small cracks, notches or other cut-outs of material. A method

correcting for the perturbation (geometrical change of a structure) has

been derived and used for three different examples.

The linear modal analysis model of a cracked beam in FEM does not

allow any boundary conditions, hence the crack surface inter-

penetration is inevitable and results in a large difference between the

computed modal frequencies of a cracked specimen and the experi-

mentally measured ones. Another method used for resonance fre-

quency correction of cracked beams has been introduces by [34]. A

simple formula (Equation 52) has been derived for calculation of the

so-called “effective natural frequency, ω0”.

𝜔0 =2𝜔1𝜔2

𝜔1+𝜔2 [52]

where ω1 and ω2 are the eigenfrequencies of an un-cracked and crack

specimen respectively.

4.2.2 Crack tip opening displacement - CTOD

In LEFM´s solutions of stresses and displacements (in Mode I)

around a crack tip Equation 53 is found. This equation expresses the

asymptotic solution and the higher order terms for the displacement

30

(uy). The stress intensity factor is proportional to crack tip opening

and is calculated by Equation 53 with measured uy [31, 35].

𝑢𝑦(𝑟, 𝜃) =(1+𝜈)

4𝜋𝐸𝐾𝐼√2𝜋𝑟 [(2𝜅 + 1)𝑠𝑖𝑛

𝜃

2− 𝑠𝑖𝑛

3𝜃

2] + 𝑂(𝑟) + 𝑂 (𝑟

3

2) …

[53]

where G is the shear modulus, E is the elastic modulus, ν is Poisson´s

ratio, κ = (3 – 4ν)for plane strain and 3D, r and 𝜃 are according to

Figure 16a. With 𝜃 = 𝜋 and 𝐺 = (𝐸

2(1+𝜈)) Equation 54 is derived where

the crack lip displacement is directly related to the stress intensity

factor.

𝑢𝑦(𝑟) =𝜅+1

2𝐺𝐾𝐼√

𝑟

2𝜋 [54]

In this way, the stress intensity factor KI is computed at several points

at the crack lip with equal spacing close to the crack tip. The values

are plotted and linearly extrapolated back to the crack tip, see Figure

16b. The shape of the curve is not necessarily as presented. It rather

depends on the neglected higher order terms whom in turn depends

on specimen geometry, 3D effects and other factors. Anyhow, a linear

extrapolation is often used to determine KI [36].

Figure 16. a) x, y- and r, 𝜃-coordinate systems at the crack tip [31] and b) linear

extrapolation of KI [36].

Distance from crack tip, r

KI

KI, CT

y

x

r

q

Crack tip

a) b)

31

In an attempt to measure the fracture toughness KIC in different struc-

tural steels, Wells [37] has observed the crack tip blunting and noticed

that the material fracture toughness is proportional to the CTOD. Us-

ing Equation 54 with ry as r relates the fracture toughness to the

CTOD. The plastic zone at the crack tip is presumed to make the crack

to behave as if it were slightly longer, see “the Irwin plastic zone cor-

rection, ry” in Figure 17b [31, 38].

𝐶𝑇𝑂𝐷 = 𝛿 = 2𝑢𝑦 =𝜅+1

𝐺𝐾𝐼√

𝑟𝑦

2𝜋 [55]

𝑎𝑒𝑓𝑓 = 𝑎 + 𝑟𝑦 [56]

Figure 17. a) Crack tip blunting and b) The Irwin plastic zone correction [31].

In a linear elastic material, the normal stress (σyy) in Mode I is accord-

ing to Equation 57.

𝜎𝑦𝑦 =𝐾𝐼

√2𝜋𝑟cos (

𝜃

2) [1 + 𝑠𝑖𝑛 (

𝜃

2) 𝑠𝑖𝑛 (

3𝜃

2)] [57]

On the crack plane (𝜃 = 0) equation simplifies to

𝜎𝑦𝑦 =𝐾𝐼

√2𝜋𝑟 [58]

Sharp crack

Blunted crack

ry

uy

Plastic zone

a) b)

rp

32

Substituting σyy with the yield strength (σys) gives the Irwin plastic

zone correction, see Equations 59 and 60.

𝑟𝑝

2= 𝑟𝑦 =

1

2𝜋(

𝐾𝐼

𝜎𝑦𝑠)

2

(Plain stress) [59]

𝑟𝑝

2= 𝑟𝑦 =

1

6𝜋(

𝐾𝐼

𝜎𝑦𝑠)

2

(Plain strain and 3D) [60]

Substitution of Equations 55 and 60 (for 3D) yields the relationship

between the CTOD and the stress intensity factor in Equation 61.

𝐶𝑇𝑂𝐷 = 𝛿 =4

√3𝜋

𝐾𝐼2

𝜎𝑦𝑠 𝐸´ [61]

where E´= (𝐸

1−𝜈2) for plane strain and 3D.

4.2.3 J contour integral

Elastic-plastic materials are approximated as nonlinear elastic mate-

rials extending the validity of LEFM to cover the fracture behaviour of

elastic-plastic materials. Figure 18 shows the uniaxial stress-strain

behaviour of elastic-plastic and nonlinear elastic materials.

The two material types stress-strain behaviour are identical during

the loading process and differs only during the unloading part of the

graph. The stress-strain curve for nonlinear elastic materials is re-

versible while elastic-plastic materials exhibit irreversible plasticity.

However, during tests where no unloading occurs (i.e. cyclic loading)

this difference is correctly neglected and the approximation of elastic-

plastic to nonlinear elastic material becomes valid. Furthermore, Rice

[39] has showed that the energy release rate could be expressed as a

path-independent integral, the J-integral.

33

Figure 18. Stress-strain behaviour of elastic plastic and nonlinear elastic materi-

als [31].

Similarly, to how the energy release rate is related to the stress inten-

sity factor in linear elastic materials, the J-integral relates the energy

release rate to the stress intensity factor in nonlinear elastic materials,

Equation 62.

𝐽 = −𝑑Π

𝑑𝐴 [62]

where 𝛱 is the potential energy defined as the difference between the

strain energy stored in the body (U) and the work done by external

forces (F), Equation 63. A is the crack lip area.

Π = 𝑈 − 𝐹 [63]

The work done by external forces is F=0 for displacement-controlled

testing (i.e. fixed displacement) and F=P for load-controlled testing.

Where P is the external force and is the displacement. The load-

displacement curve in Figure 19 gives a schematic representation of

the strain energy [31].

Strain

Str

ess

Nonlinear elastic

Elastic-plastic�material

material

34

Figure 19. Nonlinear energy release rate [31].

The potential energy for load-controlled testing is according to Equa-

tion 64 and for displacement-controlled testing according to Equation

65.

Π = 𝑈 − 𝑃Δ [64]

Π = 𝑈 [65]

U* is the complementary strain energy given by Equation 66.

𝑈∗ = ∫ Δ 𝑑𝑃𝑃

0 [66]

From Figure 19, the complementary strain energy (U*) is expressed as

in Equation 67 (∆a → 0).

Π = 𝑈 − 𝑃Δ = −𝑈∗ [67]

The J-integral is expressed in Equation 68 and 69 for load and dis-

placement controlled testing, respectively.

𝐽 = (𝑑𝑈∗

𝑑𝑎)𝑃 [68]

𝐽 = −(𝑑𝑈

𝑑𝑎)Δ [69]

35

Finally, by invoking the definitions of U and U*, the J-integral can be

expressed as in Equation 70 and 71.

𝐽 = ∫ (𝜕∆

𝜕𝑎)𝑃 𝑑𝑃

𝑃

0 [70]

𝐽 = − ∫ (𝜕𝑃

𝜕𝑎)∆ 𝑑∆

∆

0 [71]

Computation of the J-contour integral

The J-integral is expressed in Equation 72.

𝐽 = ∫ (𝑤 𝑑𝑦 − 𝑇𝑖 𝜕𝑢𝑖

𝜕𝑥 𝑑𝑠)

Γ [72]

where Γ is an arbitrary path around the crack tip (Figure 20), w is the

strain energy given by Equation 73.

𝑤 = ∫ 𝜎𝑖𝑗𝑑휀𝑖𝑗𝜀𝑖𝑗

0 [73]

Ti defines the stresses acting on the path boundaries

𝑇𝑖 = 𝜎𝑖𝑗𝑛𝑗 [74]

and

𝑇𝑖𝜕𝑢𝑖

𝜕𝑥= 𝑇𝑥

𝜕𝑢𝑥

𝜕𝑥+ 𝑇𝑦

𝜕𝑢𝑦

𝜕𝑦 [75]

Figure 20. J contour integral.

36

Path independence

The path independence of the J-integral indicated that the path be-

tween A and B could equivalently be chosen along (Γ1) or along (Γ- +

Γ2 + Γ+), see Figure 20, also shown in Equation 76.

∫ (𝑤 𝑑𝑦 − 𝑇𝑖 𝜕𝑢𝑖

𝜕𝑥 𝑑𝑠)

Γ1

= ∫ (𝑤 𝑑𝑦 − 𝑇𝑖 𝜕𝑢𝑖

𝜕𝑥 𝑑𝑠)

Γ−

+ ∫ (𝑤 𝑑𝑦 − 𝑇𝑖 𝜕𝑢𝑖

𝜕𝑥 𝑑𝑠)

Γ2

+ ∫ (𝑤 𝑑𝑦 − 𝑇𝑖 𝜕𝑢𝑖

𝜕𝑥 𝑑𝑠)

Γ+

[76]

We further know that for the boundaries Γ+ and Γ- we have (Ti = 0)

and (dy = 0) giving

∫ (𝑤 𝑑𝑦 − 𝑇𝑖 𝜕𝑢𝑖

𝜕𝑥 𝑑𝑠)

Γ+ = ∫ (𝑤 𝑑𝑦 − 𝑇𝑖 𝜕𝑢𝑖

𝜕𝑥 𝑑𝑠)

Γ− = 0 [77]

Inserting Equation 77 in 76 yields the following proof of the path in-

dependency of the J-integral.

∫ (𝑤 𝑑𝑦 − 𝑇𝑖 𝜕𝑢𝑖

𝜕𝑥 𝑑𝑠)

Γ1 = ∫ (𝑤 𝑑𝑦 − 𝑇𝑖 𝜕𝑢𝑖

𝜕𝑥 𝑑𝑠)

Γ2 [78]

Stress intensity factor

For homogeneous, isotropic materials, the stress intensity factor is

extracted from the computed energy release rate, or the J-integral us-

ing Equation 79.

𝐽 =1

𝐸´(𝐾𝐼

2 + 𝐾𝐼𝐼2) +

1

2𝐺𝐾𝐼𝐼𝐼

2 [79]

During uniaxial loading, i.e. KII and KIII are much smaller than KI,

Equation 79 simplifies to Equation 80.

𝐽 =𝐾𝐼

2

𝐸´ [80]

37

In validating the J-integral in the estimation of the stress intensity in

nonlinear elastic materials, Hutchinson, Rice and Rosengren all used

the following power-law relationship between plastic strain and stress

[39-41]:

𝜀

𝜀0=

𝜎

𝜎0+ 𝛼(

𝜎

𝜎0)𝑛 [81]

where 0 is usually the yield strength, is a dimensionless constant

and n is the strain hardening exponent [31].

5. Fatigue properties of high strength steels

A fatigue crack´s total life is divided into three major stages, Figure

21. The first stage is of course the crack initiation where the crack is

starting at a surface or at a sub-surface stress concentration point.

The second stage is the crack growth. After initiation phase, the crack

grows at significantly higher rates. Finally, the crack has grown to a

final length where the remaining, un-cracked area is simply too small,

to carry the load and the last stage, the brittle final crack occurs. For

internal fatigue crack initiations, growth rates during the crack

growth phase are significantly higher than during the initiation phase.

The crack growth portion of the total fatigue life has been reported to

be less than 1% of the total fatigue life in high strength steels [42].

Hence, the average crack growth rate over the entire lifetime will not

be correct, in fact often smaller than the lattice spacing da

dN< 10−11

m/cycle.

Figure 21. Crack initiation and crack growth portions of the total fatigue life [15].

38

As mentioned in chapter 1, the new SN-curve (Figure 2) contains two

regions where fatigue strength is decreasing. In the first region, i.e.

N<106 cycles, the crack initiation mechanism is being proposed as

surface initiations [2, 3, 10]. In the third and last region, i.e. N>108

cycles, the stresses are lower and insufficient for the crack to be initi-

ated at the surface. However, at the very high cycle regime a new initi-

ation mechanism starts to occur. Even at low stress levels, after 108

loading cycles, sub-surface non-metallic inclusions in the material are

now possible initiation sites and replaces the surface initiations as the

dominant initiation site [1-3, 8-10].

5.1 VHCF testing of high strength steels

The study of fatigue behaviour in the VHCF-regime is mandatory in

the modern world. Automotive industries need to gain information

regarding this to be able to produce safe and light components. Many

have been dedicating their research to increase the knowledge on the

VHCF-behaviour of high strength automotive steels. Bayraktar et al.

[43] studied the VHCF-behaviour of different automotive metallic al-

loys; a ferritic-pearlitic (D38MSV5) and a martensitic (42CrMo4)

steel grades with tensile strengths at 878 and 1535 MPa respectively.

The results have showed that for both grades, fatigue failure occurred

beyond 107 cycles. For the ferritic-pearlitic steel, a smooth decrease

with low scatter in the fatigue strength has been observed between 106

and 109 cycles with a total difference of ca 100 MPa. For the marten-

sitic steel, fatigue failure has been occurring between 107 and 1010 cy-

cles initiated from interior inclusions, where the fatigue behaviour

depends on the inclusion size, position, type and number. Optical

Dark Areas (ODA´s) and fish-eye regions have been observed in all

specimens that failed at interior non-metallic inclusions. It has been

concluded that Paris´s model of predicting the growth life (section

5.6) in the fish-eye works well. The effect of inclusions on fatigue

crack initiations in the gigacycle regime has been studied by [44]

where the results has showed that all fatigue crack initiations of high

strength low alloy steels in the gigacycle regime has originated by sub-

surface crack initiations. The fatigue crack initiation-life (Ni) was at

least 95% of total fatigue lifetime. A model predicting the fatigue crack

initiation life has been presented and is summarized in Equation 82.

39

𝑁𝑖 =9𝐺𝑊𝑠

(𝜎𝑎−𝜎𝐹𝐿𝑅 )2𝑎0

[82]

where G is the shear modulus, Ws =∆Kth

2

2E is the fracture energy, ais

the applied stress amplitude, σFLR is the fatigue limit at R stress ratio

and a0 is the crack initiation size considered as the inclusion size. By

implementing the Paris´s fatigue crack growth law, an expression

(Equation 83) for the prediction of the total fatigue life (Nf) has been

derived:

𝑁𝑓 =9𝐺𝑊𝑠

(𝜎𝑎−𝜎𝐹𝐿𝑅 )2𝑎0

+𝑎0

(1−𝑛/2)

𝐶𝜎𝑎𝑛𝛽1

𝑛𝜋𝑛2(

𝑛

2−1)

[83]

where C and n are material parameters and 1 is a geometry constant.

The prediction model has reasonably agreed with the experimental

results and implied that the fatigue crack initiation portion (Ni/Nf) is

over 99%.

Other researchers [45-48] have tested high carbon-chromium,

quenched and tempered martensitic bearing steel at different fre-

quencies to cover the 105-1011 region of the SN-curve. They have all

concluded that fatigue failure occurred above 109 cycles, surface dom-

inant initiation sites up to 107 cycles and subsurface dominant initia-

tion sites in the very high cycle regime (107-1011 cycles). The subsur-

face initiations have originated at non-metallic inclusions mainly

Al2O3, CaO, SiO2 and MgO. Fine granular areas around the inclusions

have always been visible during the fractography analysis.

In [49], an investigation of the initiation mechanisms of a martensitic

steel grade quenched and tempered in two different ways, 37HRC and

57HRC, has been made. The results have showed that for the low

strength grade (37HRC), the initiations have been exclusively located

at the surface and caused by the formation of slip bands. However, for

the high strength grade (57HRC), the fatigue crack have tended to ini-

tiate at non-metallic inclusions with formation of FGA around the in-

clusions.

In [50], the effect of a carburized layer in a martensitic Cr-Ni steel on

the fatigue behaviour and the initiation mechanisms has been investi-

40

gated. The same material has been tested under 100Hz uniaxial load-

ing at three different loading ratios, R=-1, 0 and 0.3. From the results,

the authors have concluded that during R=-1 testing the initiations

have occurred only at the surface, hence a continuously decreasing

SN-curve. While during the R=0 and 0.3 tests, a mixture of surface

and subsurface initiations has been observed. In fact, the probability

of subsurface initiations has increased with the increasing mean load.

The reason behind this effect has been due to the contained compres-

sive residual stresses in the carburized layer restraining crack initia-

tion. However, during increased mean load, i.e. increased applied

maximum stress, it has been easier to overcome the compressive re-

sidual stresses at the surface and the likelihood of surface initiations

increase. The surface initiations from the R=0 and 0.3 tests have been

related to short fatigue lives, and the subsurface initiations (all with

FGA formation) have been related to long fatigue lives. For the R=-1

test however, FGA formation has occurred at the long-life region

(N>106) while at the short life (N<5*105) no FGA formation has oc-

curred. Generally, it has been concluded that the fatigue strength of

the material has decreased with the increasing loading ratio.

5.2 Fatigue strength

Murakami introduced the √areaODA parameter model where the size

of the ODA around an inclusion is correlated with the stress intensity

threshold Kth and the fatigue limit FL, Equations 84 and 85 [8, 51,

52].

∆𝐾𝑡ℎ = 𝐶 (𝐻𝑉 + 120) (√𝑎𝑟𝑒𝑎𝑂𝐷𝐴)1/3

[84]

𝜎𝐹𝐿 = 𝐴(𝐻𝑉+120)

(√𝑎𝑟𝑒𝑎𝑂𝐷𝐴)1/6 (

1−𝑅

2)𝛼 [85]

where A is 1,43 for surface crack and 1,56 for subsurface crack, C is

0,0033 for surface crack and 0,00254 for subsurface crack, R is the

stress ratio and =0,226+HV*10-4.

The morphology of the ODA contains of granular-like looking, zigzag

shaped surface, hence the alternative designation fine granular area

41

(FGA). This differs from the flat transgranular morphology outside

the ODA. The crack growth inside the ODA is clearly not cycle-by-

cycle growth. However, Murakami introduced the hypothetical cycle-

by-cycle rate equation (Equation 86) based on the master curve of

ODA presented in [52].

𝑑√𝑎𝑟𝑒𝑎𝑖𝑛𝑐

𝑑𝑁𝑖𝑛𝑠𝑖𝑑𝑒𝑂𝐷𝐴=

1

𝐴𝐵(

√𝑎𝑟𝑒𝑎𝑂𝐷𝐴

√𝑎𝑟𝑒𝑎𝑖𝑛𝑐− 1)1−𝐵 √𝑎𝑟𝑒𝑎𝑖𝑛𝑐

2

√𝑎𝑟𝑒𝑎𝑂𝐷𝐴 [86]

where √areainc represents the size of the inclusion and the constants A

and B are extracted from the master curve of ODA.

Paris introduced a simple model (Equation 87) for estimation of crack

growth life (Np) which is considered as the number of cycles between

a0 (initial crack size/inclusion size) and af (final crack size) [11].

𝑁𝑝 =𝜋 𝐸2

2(𝜎𝐹𝐿)2 [87]

5.3 Crack initiation

For high strength metals, fatigue cracks can initiate either from a sur-

face or from a subsurface point in the bulk material. Earlier in the

VHCF community, researchers [3, 10, 43] believed that a transition of

crack initiation sites from surface to interior occurred at 106-107 cy-

cles. In [8] it was even stated that the SN-curve for high strength steel

could be plotted with two straight lines, one for the short life tests and

surface initiated fatigue cracks at high stress levels, and one for the

long life tests at low stress levels with subsurface crack initiations.

However, since the growing of the number of VHCF laboratories and

the amount of experiments conducted, this rule has been somewhat

disproved. In [1, 10] it has been explained that different initiation

types occurs at different stages in the SN-curve. While surface initia-

tion are the most common in the LCF regime, subsurface crack initia-

tion becomes more dominant at the HCF and VHCF regime.

Surface initiations are typically originated at surface defects such as;

scratches, corrosion, inclusions located at the surface, mechanical

stress concentration sites (i.e. notches) or simply when bending loads

are present, Figure 22. In [53] it has been stated that for low strength

42

steels, cracks initiates at irreversible persistent slip bands (PSB’s).

These cracks obtain good correlation between the fatigue limit and the

ultimate tensile strength and hardness. Equation 88 presents a linear

relationship between the fatigue limit and the hardness of a material

with HV < 400 regardless of microstructure, with FL is in MPa and

HV in kgf/mm2 [51, 52].

𝜎𝐹𝐿 = 1,6 𝐻𝑉 [88]

For high strength steels (HV > 400) however, fatigue failure will most

probably originate at a defect, e.g. a non-metallic inclusion and the

above displayed relation between the fatigue limit and the hardness is

no longer valid. Anyhow, the fatigue limit is strongly influenced by the

defect size.

Figure 22. Fatigue crack surface initiation [15].

Subsurface crack initiations, commonly at longer life lengths, occurs

at stress concentrations points found in the bulk material. For metals,

these points can typically be a non-metallic inclusion (Figure 23), a

triple point connecting three nearby grains, or even a pore if porous

material.

43

Figure 23. Fatigue crack sub-surface initiation [15].

Subsurface initiations in the VHCF regime are mostly located at non-

metallic inclusions. Murakami et. al. [51, 52] defined the area around

the inclusion as the Optical dark area (ODA), see Figure 23. This area

defines the first stage of a fatigue failure, proposed as cyclic loading

assisted by hydrogen trapped by the inclusion causing the very slow

crack growth. The border of the ODA indicates the start of second

stage where cycle-by-cycle fatigue crack growth occurs [52]. The cause

for the formation of the FGA is yet to be fully revealed. Different re-

searchers have presented different models explaining the formation of

the FGA. In [54] researchers have referred to continuous grain re-

finement around the inclusion caused by stress concentrations and

cyclic loading as the mechanism behind the formation of the FGA.

The refinement of the grains decreases the stress intensity threshold

value around the initiation site resulting in a crack initiation followed

by fine crack propagation inside the FGA. In [55] researchers argued

that repeated crushing of the fracture surfaces due to crack closure

causing grain refinement by Numerous Cyclic Pressing (NCP). In [56]

a duplex stainless steel and a martensitic stainless steel have been

tested in the VHCF regime. Fracture surface investigation has re-

vealed a layer of nano-sized grains (5-20 nm) inside the FGA satisfy-

ing the idea of grain refinement.

5.4 Crack growth

The crack growth phase (Nfish-eye) is the sum of, growth of internal flaw

(Nint), small crack growth (Na0−ai) and long crack growth (Nai−af

), i.e.

small crack growth occurs between a0 and ai while long crack growth

44

occurs between ai and af. The total fatigue crack growth life is ex-

pressed in Equation 89 [42, 57].

𝑁𝑓𝑖𝑠ℎ−𝑒𝑦𝑒 = 𝑁𝑖𝑛𝑡 + 𝑁𝑎0−𝑎𝑖+ 𝑁𝑎𝑖−𝑎𝑓

[89]

Bathias, Paris and co-workers have investigated the relative portions

of growth of internal flaw, small crack growth and long crack growth

phase by integrating the expression in Equation 90, where b is the

Burger’s vector.

𝑑𝑎

𝑑𝑁= 𝑏 (

∆𝐾𝑒𝑓𝑓

𝐸√𝑏)

3

[90]

The relative portions of the different phases of the total fatigue life are

estimated from Equations 91-93.

𝑁𝑖𝑛𝑡 =𝜋 𝐸2

2 (∆𝜎)2

1

2(𝛼

2−1)

[(𝑎0

𝑎𝑖𝑛𝑡)

(𝛼

2−1)

− 1] [91]

𝑁𝑎0−𝑎𝑖=

𝜋 𝐸2

2 (∆𝜎)2 [1 − √

𝑎0

𝑎𝑖 ] [92]

𝑁𝑎𝑖−𝑎𝑓=

𝜋 𝐸2

2 (∆𝜎)2 [𝑥3√

𝑎0

𝑎𝑖− 𝑥3

√𝑎0

𝑎𝑓] [93]

Figure 24. Relative fatigue crack propagation for small and long cracks [42, 57].

45

The used parameters are according to Figure 24. Ni is computed from

Equation 94 (Ni = Nexperimental - Nfish-eye), where Nexperimental is the total

fatigue life determined experimentally.

𝑁𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 = 𝑁𝑓 = 𝑁𝑖 + 𝑁𝑓𝑖𝑠ℎ−𝑒𝑦𝑒 [94]

In the work of [42, 57], fracture surfaces of different types of steel

were evaluated where fish-eye features are found and measured. The

results of this work confirm that the ratio Nfish-eye/Nexperimental is less

than 0.01.

Crack growth rate testing have been conducted on high strength steels

by different laboratories [15, 28]. Within the previously mentioned

FREQTIGUE project [15], crack growth rate measurements on auto-

motive steels have been performed at three different load frequencies

(35 Hz, 130 Hz and 20 kHz). The 20 kHz tests have been carried out

on six high strength steels, three bar grades (38MnSiV5, 50CrV4 and

16MnCr5) and three sheet grades (M800HY, CP1000 and DP1180), at

R=-1 and R=0.1 load ratios using ultrasonic fatigue equipment. The

results have been analysed according to the Paris law.

𝑑𝑎

𝑑𝑁= 𝐶 (∆𝐾)𝑚 [95]

where C and m are material parameters.

The 20 kHz testing yields data points only at low crack growth rates

and is therefore insufficient for the estimation of C and m. However,

considering all data point from all three frequencies, and defining a

common Paris law curve, the results have showed better estimation of

the parameters. Figure 25 shows two examples of such plot.

46

Figure 25. da/dN vs ΔK plot for 38MnSiV5 steel grade at R=-1 and R=0,1 load

ratio respectively [15].

As seen in Figure 25, the data points from the 20 kHz testing covers

the lower ΔK regime. This was the case for nearly all six materials.

In [28], crack growth rate measurements has been conducted using

martensitic chromium steels (AISI410). Measurements have been car-

ried out at three different loading ratios (R=0.05, R=0.5 and R=0.8)

and in two different environments (ambient air and vacuum). The re-

sults of these measurements have been plotted in a da/dN vs ΔK dia-

gram, see Figure 26.

As seen in Figure 26, decreasing load ratio (R=0.8, R=0.5 and

R=0.05) yields increasing stress intensity threshold values (2.77, 3.39

and 4.23 MPa√m for the vacuum tests and 2.18, 2.64 and 3.98

MPa√m for the ambient air tests, respectively), see Figure 26. It is al-

so noticed that the curves for the different load ratios in ambient air

tests converges at higher ΔK values. Crack propagation took place at

rates as low as 5*10-13 m/cycle in vacuum and 6*10-12 – 2*10-11 m/cycle

in ambient.

47

Figure 26. da/dN vs ΔK plot for a) vacuum tests and b) ambient air tests at dif-

ferent loading ratios [28].

5.5 Frequency effect

The high load frequency during ultrasonic fatigue testing brings a

concern about the frequency effect on the experimental results.

Bathias and co-workers [11, 45] have stated that experience and pre-

vious results shows a very small frequency effect, at least for low dis-

placement amplitudes. Other researchers [5, 6] have stated that fre-

quency effect on the fatigue strength increases with ductility, strain-

rate sensitivity and other time-dependent processes like corrosion

and high temperature, and is more pronounced in BCC materials.

Hoffelner and Gudmundson [17] have performed a crack growth anal-

ysis of two austenitic materials, Hastelloy X and nickelbased alloy IN

800, at two different loading frequencies 60Hz and 20kHz. The re-

sults showed slim to none frequency effect regarding da/dN vs K and

stress intensity threshold values.

Zhao et al. [46] have tested a bearing steel with different tempering

temperatures, i.e. different strengths, at different load frequencies.

The results showed that the frequency effect becomes higher when the

material strength is decreased. This was explained by the fact that the

lower the frequency is the more time the material spends at the max-

imum stress level, where most of the damage occurs, resulting in a

lower fatigue strength. However, for the high strength specimens the

frequency effect was very small.

48

Guennec et al. [58] have studied the frequency effect on the ductile

ferritic-pearlitic S15C steel using five different loading frequencies

(0,2Hz, 2Hz, 20Hz, 140Hz and 20kHz). The resulted SN-curves

showed a decreasing fatigue strength with decreasing load frequency

(more pronounced at lower strength levels for the 0,2-140Hz tests)

where the largest gap was between 140Hz and 20kHz. However, an

increase in the yield strength due to higher strain-rates was observed

in the 0,2-140 Hz frequency range. Taking this into account, the SN-

curves were normalized by plotting a/ys against life time and the

differences between the SN-curves of 0,2-140 Hz diminished and the

gap between 140Hz and 20kHz decreased but was still pronounced.

In the extensive research project (FREQTIGUE) [15], the major ob-

jective was to investigate the influence of the load frequency on fa-

tigue strength and the effect of the 20 kHz load frequency of the ultra-

sonic fatigue testing system. Six different automotive steels were test-

ed at different frequencies (1-20 000 Hz) at seven different facilities.

The results of the project showed evidence of a load frequency influ-

ence on the fatigue strength. It was concluded that the fatigue

strengths for several steel grades was slightly lower at 20 kHz than at

lower load frequencies. This effect is thought to be due to softening of

the material caused by localized internal friction at this high frequen-

cy.

5.6 Damping

When computing the stress or the stress intensity factor, one needs to

consider the energy dissipation (internal friction Q-1) due to anelastic

relaxation during dynamic computations. In crystalline materials,

there are different mechanisms producing anelastic relaxation . Point

defect relaxation, dislocation relaxation and interface relaxation are

three mechanisms caused by interaction between the motion of a de-

fect and the applied stress [59, 60]. Point defect relaxation implies

diffusive redistribution of point defects under an applied external

stress. The redistribution occurs due to mainly two phenomenon,

Gorsky (1935) relaxation where an interstitial atom (IA) moves from

compressed to dilated regions (observed only for hydrogen) and the

more common phenomenon where point defects are reoriented due to

anisotropy of the applied stress and the defect-induced distortion.

49

Snoek (1941) relaxation describes the anelastic relaxation due to in-

teraction between IA and host metal atoms that leads to deformation

and relaxation of the crystal lattice and, hence, energy losses (i.e. car-

bon atoms in α-Fe).

Dislocations are line defects in crystals with complex distortion

and/or stress fields [60]. Dislocation relaxation occurs in plastically

deformed metals and alloys with high density of dislocations and is

caused by movement of the dislocations themselves due to nucleation

and propagation of thermal or geometrical kinks (Bordoni and Nib-

lett-Wilks peaks), jointly by dislocations and proper lattice point de-

fects (Hasiguti peaks), jointly by dissolved foreign IA and dislocations

at elevated temperatures (Snoek-Köster relaxation and dislocation

enhanced Snoek effect), or dislocation climbing at elevated tempera-

tures. Interface relaxation occurs in polycrystalline materials due to

grain boundary relaxation where sliding of grain boundaries is caused

by shear stresses along the boundary of two adjacent crystals. A spe-

cial type of grain boundary relaxation is the twin boundary relaxation

where the sliding boundary is the separation of two twin crystals that

are mirror image of each other. The last main type of anelastic relaxa-

tion in crystalline materials is the thermoelastic relaxation where irre-

versible heat flow relaxes the temperature gradient arisen due to spa-

tially inhomogeneous stress variations leading to entropy production

and dissipation of mechanical energy.

The internal friction 𝑄−1 is related to the ratio between the absorbed

energy ∆𝑊 and the maximum elastic stored energy 𝑊 during one cy-

cle, Equation 96.

𝑄−1 =∆𝑊

2𝜋𝑊 [96]

The absorbed energy ∆𝑊 is represented by the area under the stress-

strain hysteresis loop and the maximum elastic stored energy is calcu-

lated as 𝑊 = 𝜎𝑎2/2𝐸 where 𝜎𝑎 is the stress amplitude and 𝐸 is the

elastic modulus.

In FEM computation however, the internal friction is approximated

as viscous damping and is implemented into the material model. The

FEM software ABAQUS/CAE uses Rayleigh damping as the energy

dissipation model. The Rayleigh damping model contains two damp-

50

ing factors; R for mass proportional damping and R for stiffness

proportional damping. For a given angular frequency (ω) the Rayleigh

damping is expressed as in Equation 97 and illustrated in Figure 27

[59, 61].

𝜉 =𝛼𝑅

2𝜔+

𝛽𝑅𝜔

2 [97]

This equation implies that the effect on the damping is dominated by

the density at lower frequencies and by the stiffness at higher fre-

quencies.

Figure 27. Rayleigh damping versus frequency [59].

5.6.1 Damping measurement

Several different methods and apparatuses are designed to measure

the internal friction. The impulse excitation technique, IET, where the

vibration is induced to a specimen (with known dimensions) by strik-

ing it with a small steel ball, allowing it to “ring”. The vibration is then

recorded by a piezo crystal or by a microphone and the decaying sig-

nal is analysed. The IET-apparatus used during this research is de-

scribed by [62, 63]. For flexural mode, the specimen is placed on two

wires acting as supporters, Figure 28. The supporters are positioned

in the nodes of the specific vibration mode. In the next step, the spec-

imen is struck and allowed to vibrate at its natural frequencies. A mi-

crophone connected to a LabVIEWTM program using Fourier analysis

Frequency, w

Da

mp

ing

ra

te,x

Stiffness-proportional�damping

x=Rw + R

w

x=Rw, R = 0

Mass-proportionaldamping

x=R, R = 0w

51

and iterative optimization to compute the resonance frequency, f0,

and the exponential decay parameter, k, of the signal. The relation-

ship in Equation 98 is then used to calculate the mechanical loss coef-

ficient (Q-1) of the material.

Figure 28. Schematic view of the damping measurement setup with the IET-

apparatus [62].

𝑄−1 =𝑘

𝜋 𝑓0 [98]

The dynamic elastic modulus, Ed, is obtained from Equation 99.

𝐸𝑑 = 0,9465(𝑚 𝑓0

2

𝑏)(

𝐿3

𝑡3)𝑇1 [99]

where m is the mass and L, b and t are the dimensions of the speci-

men. T1 is a correction factor depending on the (t/L) ratio.

𝑇1 = 1 + 6,858 (𝑡

𝐿)2 [100]

For low damping rates (x<<1) the Rayleigh damping factor x is half

the loss coefficient parameter Q-1 measured with IET-apparatus.

𝜉 =𝑄−1

2 [101]

The IET equipment can provide damping and elastic modulus of spec-

imens with resonance frequencies approximately 100-25 000 Hz. Alt-

hough experience have showed that specimens with resonance fre-

52

quency below 250 Hz are difficult to use. Such low frequencies can

instead be measured with the dynamic mechanical analyser, DMA.

The specimen is under forced vibration at pre-chosen frequencies

while the DMA measures the damping ratio as the ratio between the

loss modulus (G´´) and the storage modulus (G´). This ratio is plot-

ted as the phase distortion tan() and is equal to the internal friction

(Q-1) parameter [64].

tan(𝛿) =𝐺´´

𝐺´ [102]

tan(𝛿) = 𝑄−1 [103]

Conclusion

The ultrasonic fatigue testing system enables fatigue research in the

gigacycle regime by significantly shortening the testing time. Leading

scientists in material fatigue recognized the opportunity with using

piezoelectric fatigue testing system at very high frequencies and took

interest in evaluating such testing method. Development and evalua-

tion of the system continued, and today, the ultrasonic fatigue testing

system is accepted worldwide as a trustworthy testing method.

Precisely designed specimens, and in some cases specimen holders,

are today developed for the three main loading modes i.e. axial, bend-

ing and torsion. FEM modelling and simulation has been effectively

used in designing the specimens. Computation of specimens natural

frequency, stress-strain conditions and stress intensity factor values

are some of the tasks FEM usefully provides.

Several models, both conventional and new methods specially engi-

neered for VHCF are today used to evaluate different aspects of the

results of the VHCF testing.

53

References

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Nomenclature Acronym Description

LCF Low cycle fatigue

HCF High cycle fatigue

VHCF Very high cycle fatigue

FGA Fina granular area

ODA Optical dark area

E Elastic modulus

Ed Dynamic elastic modulus

G Shear modulus

Poisson´s ratio

ρ Density

c Wave velocity

U Displacement

U0 Displacement amplitude

ε Strain

σ Stress

f Frequency

f0 Resonance frequency

ω Angular frequency

ω0 Resonance angular frequency

ω0 Effective natural frequency

ω1 Resonance angular frequency of an un-cracked beam

ω2 Resonance angular frequency of a cracked beam

L1 Specimen resonance length

k Exponential decay parameter

µ̂x Estimated sample mean stress

�̂�x Estimated sample standard deviation

µ̂y Estimated mean stress

�̂�y Estimated standard deviation

KI Stress intensity factor in loading mode I

KII Stress intensity factor in loading mode II

KIII Stress intensity factor in loading mode III

Kmin Lower limit of the stress-intensity factor

Kmax Upper limit of the stress-intensity factor

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K Stress-intensity factor range

Kth Stress intensity threshold

Pmin Lower limit of the force amplitude

Pmax Upper limit of the force amplitude

Stress range

min Applied minimum stress

max Applied maximum stress

a Applied stress amplitude

R Loading ratio

C Normalized K-gradient

da/dN Crack growth rate

LEFM Linear elastic fracture mechanics

CTOD Crack tip opening displacement

uy Crack tip opening displacement

Crack tip opening displacement

ry Irwin plastic zone correction

ys Yield strength

FL Fatigue limit

FLR Fatigue limit at given stress ratio R

J J-contour integral

Π Potential energy

U Strain energy

U* Complementary strain energy

F Work done by external forces

P External forces

Δ Displacement

√𝑎𝑟𝑒𝑎𝑖𝑛𝑐 Inclusion area parameter

√𝑎𝑟𝑒𝑎𝑂𝐷𝐴 ODA area parameter (including the inclusion)

NinsideODA Number of cycles inside the ODA

Ni Fatigue crack initiation life

Nint Growth of internal flaw

Nfich-eye Number of cycles from the start of crack growth to Nf

Nf Total fatigue life

Nexperimental Experimental fatigue life

Ws Fracture energy

aint Internal flaw size

a0 Crack initiation size/defect size

59

ai Transition point from small crack to long crack

af Final crack size

aeff Effective crack size, including the Irwin plastic zone cor-

rection

ξ Rayleigh damping ratio

R Mass proportional Rayleigh damping factor

R Stiffness proportional Rayleigh damping factor

Q-1 Loss coefficient

A study of very high cycle fatigue in high strength steels

Mohamed Sadek

A study of very high cycle fatigue in high strength steels

Until 1970´s, fatigue properties of materials were usually studied and evaluated up to the HCF regime (N<106 cycles) beyond which failure was not expected. However, in the late decades of the 20th century proof of fatigue failure in the VHCF regime (N>106 cycles) was presented by different material scientists. This led to an aroused interest in developing a new testing technique where a very high number of loading cycles is achieved within reasonably short time. The ultrasonic fatigue testing system was developed for this purpose, testing materials in the VHCF regime, where specimens are loaded at 20 kHz reaching 1010 cycles in less than a week. In the ultrasonic fatigue testing system, an electric sinusoidal signal is generated and then converted to a mechanical sinusoidal vibration. The mechanical vibration is led through a magnifying horn to the specimen. The system vibrates at resonance frequency, hence all individual parts of the system are designed and dimensioned according to the system resonance frequency.

RESEARCH REPORT | Karlstad University Studies | 2020

Faculty of Health, Science and Technology

Materials Engineering

RESEARCH REPORT | Karlstad University Studies | 2020:22

ISSN 1403-8099

ISBN 978-91-7867-135-9 (pdf)

ISBN 978-91-7867-130-4 (print)